Ecometric Reviews EDITOR DALE

Ecometric Reviews EDITOR DALE J. POIRIER Department of Econo mics University of Toronto 150 St. George Street Toronto, Canado M5S lAI ASS@IATE EDITORS . ALBERTO HOLLY U"luersit€de Lausanne Ecole des Hautes Eudes Commerciales 1015 l^ousanner;f::tr CHENG HSIAO of Toronto Instirute for Policy Analysis University 150 St. George Street Toronto, Ontario, MsS IAI ROGER KOENKER Department of Economics University of lllinois Champaign, Illinois U.S.A. 6 I 80 I TAKAMITSU SAWA Kyoto Universiry Institute of Economic Research fukyo-Ku, Kyoto 606, fapan Canada PETER SCHMIDT Michigan Stat e University Department of Eonomics Marshall Hall East Lansing, Mich$an 48824 U.S,A. ECONOMETRrc REVIEWS Volunc l,Nubcr I, l9ts CONTENTS !. reaarrr of -Bgnlnctricr and-1':P'Tl:.11*'^;;-:' von zur Muehlen K. Conwsy, {i v. i.$loern/, R. $i;;-, o. thc Forndrtionr of Econometrics - Arc Thete $ _ 5t Doraltd iii or rbc Foundrttonr of Economctricr - Aro Thcre on thc Ponndrtlonr of Econonctricr - Arc Thetc 'lmpfopcr" Prlorr, rnd Finltc 69 75 Addittvity on thc Poundrtl,on of Econometricr - El Are Therc 93 jtt, t. Smlhy :fcply lo Conrncntr on thc Foundrtirons of Econometricr IttfrtAny? l'"'.IrI A. Y. B, Are l0l Swomy, R. K. Conway, ond P. von zur Muehlen j.: ||tb - . rod Llmltrtlonr of Prncl t2t Drtr . . i-ttbo r Gomot Dttr . . t7s Limitrtiorrc of Prncl Dttr . . t79 of Prnel Drtr . . 183 on Eenefitr rnd Limitrtions of Penel i" O. lrtuoton and N. M. Kiefer I I' ;i0mncrt ol lcrefitr ud \gtrt*rn ! Srrot on Dcocfitr rnd Limitrtionr 7 tr. Sobn 'il.pty to Gomrncnts on Bencfitc rnd Limitrtiong of Prncl Detr . i.C, Iltbo I IARCEL DEKKER,INC. New York end Bqcc! ,, chrrgc 187 ECoNOMETRTC REVTEWS, 4(I), !-6L (1985) TITE FOUNDATIONS OF ECONCI'IETRICS--ARX THERE ANY? P.A.V.B. Roger Swany Special Studles K. Conway Bureau of Economic Analysis Department of Conmerce Washington, DC 20030 Federal Reserve Board Washington, DC 20551 P. von zur Muehlen Special StudLes Federal Reserve Board I'lashington, DC 20551 of eeonornetrLes; the h,to-ttalued Logie; the problen of induetion; inetnunentalien; Loeal eoheneneel ila.7y-uaLued l,ogie; eoidential interpretation. Keg llords arld Phr.ases: foundations ASSTRACT of a 1ogically conslstent economic theory strictly adheres to Aristotlers axioms of logic are factually true if its sufficient conditlons are all factually true. Alternatively, if a conclusion of such a theory is false, then at least one of its assumptlons is false. Unfortunately, the factual truth of sufficient conditlons carnot be established because the probleu of induction is impossible to solve. It is alpo true that the falsity of a conclueion cannot be established in the presence of uncertainty. Whlle the philosophy of lnstrumentalisn applied to The concluslons which Copyright @ 1985 by Marccl Dekkcr, Inc. 07 47 4938 | 8s I 0401 -000 I $3.50/0 SWAMY, CONWAY, AND VON ZUR MUEHLEN sufficient and logically consistent explanations may provide useful solutlons to immediate practical problems' the principles of simpliclty, parsinony and profligacy--al1 of them requiring conditional deductive arguments--are useless as crlEerla for rnodel choice. Furthermore, Arlstotlers axioms can give rise to difficulties. For example, as G6delrs lnconpleteness Eheorem shows, if our supposed axioos for a Eheory do not permit any contradictions, then the theory ls incomplete in the sense that Ehere are proposiEions that are undecLdable relatlve to those axioms' If we adhere gq Popperts view that sclentiflc actlvity is inpossible wlthout the adoption of Aristotlerg axion of noncontradiction, then we nay achleve 1oca1 coherence (i.e., consistency within a glven model) by adopting the axiom and conblning a prior distribuEion satisfying certain conditions with the lnformation on a sufficient and logically consistent model via the Bayes theorem. Intultlonlsts are correct ln denying the unlversal validity of lndirect proofs and thus precluding the use of Aristotlers axLon of the excluded-rniddle and we have denonstrated that roany-valued loglc applies whenever this axlorn is not used. Probabllity theory ls a version of many-val"ued loglc and one can provide strong or weak (but inconcl_uslve) evidence for one statistical hypothesis against another using thls theory. The actual predlctive perforrnance is a good crlterion for Judging dlfferent approaches to lnference' CONTENTS 1. Introductlon z. The Merit of Two-Valued or Deductive Logic - Sone First Principles 2.7 Nature of loglc 2.2 Aristotlets axloms of loglc 2,3 Loglcally valid arguments 2.4 Modus ponens or nodus tollens 2.5 Exanples of the vlolations of Aristotlers axioms FOUNDATIONS OF ECONOMETRICS 3. Sufflcient and Logically Conslstent Models 3.1 The necessity of logical consLstency and an exanple 3.2 Alternatlve theories 4. Inductlvlsm and Approxlmations 4.L The problen of induction 4.2 The retreat Eo approximations 4.3 Attempts to solve the probl-ern of lnduction 5. Popperrs Falsificatlonlsm and Related Problems 5.1 Asynnetry 5.2 Loglcal anbigulty regarding refutatlons 5.3 The problem caused by uncertalnty 6. ObJective Knowledge, Instrumentalism, Simplicity and Parsimony or ProfLigacy 6.1 Bolandrs (1982, p. 179) view 6.2 Instrumentallsm 6.3 Slnpllclty 6.4 Principle of parslmony vs. pri.nclple of profllgacy 7. Restrlctive Nature of Arlstotlers Axiorns and methods of Relaxlng Then 7.t G6delts chali.enge to the axiom of noncontradlctlon 7.2 Local vs. globa1 coherence 7.3 Locally coherent inference procedures 8. Intultionistsr crittclsm of the axion of 7.4 Brouwerrs and 7.5 Many vaLued 7.6 excluded-mlddle Birnbaumrs (1977) the excluded-nlddle or ftzzy logic-relaxlng the axiom of the evidential lnterpretation vs. the llkellhood prlnclple or Bayes procedures Conclusions Appendlx: InterpreLations of Probability 1. Importance of probabLllty lnterpretations 2. Alternative meanings of the word "probabt1lty" 3. Dl-fferences and sinl-larltles of results based on dlfferent vlews 4. CondLtlong for the relevance of frequency lnterpretation 5. Conditlons for the relevance of subJectlve lnterpretatlons 6. On frequentlst vs. posterl-or probablllty References SWAMY, CONWAY, AND VON ZUR MUEHLEN 1. Introduction Eltle asks such a questlon is that few econometrlc textbooks clarlfy or even conslder the foundatl-ons of econometrics.l l{ence, our purpose in thls paper 1s to artlculate these foundatlons. By 'foundations" we mean the econometrtclansr vlew of the relationship between r'he economic theorles or asaumptions upon which they base thelr models and the statistical methods Ehey use to reach concluslons about the nature of the real world' I{hat is usually discussed under the rubric of economic methodology is nore concerned with the nonstochastlc cases than wlth lhe stochastlc cases lBlaug (1980) and Boland (1982)]' In contrast' our vLew ls that a proper study of econometric foundatlons should be concerned strictly nlth the foundatlons a8 manlfested ln the theorLes. nature of stochastic ry!3!9 In this paper' we argue that Ehe accepted views of the approprtate rnethodologles for enpirlcal lnvestigation of neoclasslcal econonlc theorl.es are lnadequate to clarlfy the foundatione of The reason why our cannot understand econometric nethodology \tithout first understandlng economic theory; thus, in Sections 2 chrough 4 we dlscuss the foundatl'ons of economics, which are anchored on two dlfferent but not necessarlly mutually economeErlcs. We presume one exclusive nethodological topics. The flrst toplc is the Aristotellan logtcal system conelsting of only true or false staEements and the other ls the "problem of induction,'both of whlch are Ehe subJect of critical examination by philosophers of science. Some examples of vlolatlons of Arlstotlers axLous of logic will also be glven ln the6e sections to show thaL econometric models wLth all givens incorporated ln thelr "unlverse of discourse" do not fa11 under the category of Arlstotellan systems. A consLstent set of distrlbutional and behavloral assumptl-ons should be added to nake such nodels logical'Ly valld. The loglcal conslstency and valldicy of a nodel is a necessary conditlon for -T-.ffi .-c"ptron is Zellner's ln econonlcs. (1971 , Ch. 1) renarks on inference FOUNDATIONS OF ECONOMETRICS iis truth. Related questions about Popperts falsificatlonism and Friednanrs views based on Lnstrumentallsm wii-1 be examined in Sections 5 and 6, respectively. It will be shown that these philosophles applted to Arlstotelian systems have the sort of logical Justificatlon which phllosophers generally insist on, but ihey do not have a slmllar Justlfication when applied to econometric models. Section 6 also contains a discusslon about the non- logical principles of slnpllcity, parsl-mony and profligacy slnce they are frequently advocated as approprlate criteria for economet- ric models. The restrictive nature of Arlstotlets axioms will be discussed in Section 7. Specifically, it will be shown in this section that not a1l- of Aristotlers axioms can be satl.sfied ln all econometric applications and failure to satLsfy all of Aristotlets axLoms renders the argument in favor of true conclusions inadequate. Fortunately, in the present century there have been several investlgatlons into an alternative logical system known as "nanyvalued 1ogic." The axioms of nany-valued logic, though weaker than those of Aristotle, can be shown to form a valid foundation for econometrics if a system of rules of inference can be established which will insure that what is provable is exactly what is valid; however, it should be noEed that there is no unique way of extending the notion of validity to the many-valued case. Fina11y, in Sectlon 8 we offer our sunnary and conclusions. 2. The Merit of Tvo-Valued or Deductive LoglcSome First Principles 2.1 Nature of loglc Loglc deals Itith the correctness of arguments. In logic, an argument is a group of indlvidual statements standing ln relatLon to each other. One of theo ls the conclusl-on and the relnalning statements are assunptLons or premises. The latter present the evidence for the former. Obvlously, the truth of any statenents SWAMY, CONWAY, AND VON ZUR MUEHLEN not guarantee that a conclusLon ls true' Such statements must also have gome bearlng upon that concluslon' It ls this is connection beEween premises and conclusions wlth which loglc or conconcerned and not wlth the truth or falsity of premises clusions by themselves lsee Salmon (1973, pP' 3-4)]' Thus' 1ogic, as a dlscipline, is preoccupied ltlth establlehing the valiclity of arguments. Truth or falsity is a property of lndividual statements and not of arguments' 2 does 2.2 Aristotlets axl-ons of logic Aristotle poslted three rules that are only ry939a cooditions for the adnissibility of statements into logical arguments' "the axion These "rulegr" "axioms" or "canons" of logic are: (1) ofidentity.':differentstatementscannotusedifferentdefinitionsofthesamewordsl(2)'.theaxlomoftheexc]-uded.niddle'.: statements that are always neither true nor false nor both are prohibited;and(3)'.theaxiomofnoncontradiction..:stateDents cannot be allowed to be both true and false' Ite mean "factual truth"' In addition' there is -z'J_i-Gall anothei type of truth known as "logicai' truth" (or truth lnfollows truth whose sEatements are truths Logical clrcurnstances). frorn the deflnltlons of the words that occur ln them' Such statements are called analytlc Etatements or tautolologies [see A r.rccawley (198r, p. 7o)ffi.-130)1. 1,o"..?91i!9!v statenent is one whose "1ogica1 falsity" follows fron the Treanings are stateoenEs i?EET.as rhat occur in lt. By contrast, there whose truth or falsity is not determined solely by rhe meanings ofthewordstheycontaln.Theseareknownas..syntheticstate. ments." Synthetic statenents are not logical truths or falsehoods; they are iactual statenentsr,each of whlch is either factually ana1ytic statement is necessarily crue' trrrl o, f;G;ffi;-an it wl1l be true regardless of what cl-rcumstances hold' Its truth does not exclude any inaglnable poesibillties' Loglcal truth is always much stronger than factual truth because no empLrical obeeivation could ever refute the former, but it night refute the latter. We sha11 show Ln thl-s sectlon that no synthettc conclueLons can be valtdly deduced frorn analytic premises alone lsee Salnon (1973, p. 132)1. Further, we wLll u8e the convention of letiing .'true" or "faise" be "factually true" or "factually fa1se" throughout the PaPer. FOUNDATIONS OF ECONOMETRICS 2.3 Iogicblly valid arguments In Arl-stotellan 1ogic, an argument which does not violate any of Arlstotlers axioms is loglcal whenever lt is a sufficient argument in favor of its conclusions in the following sense: If an argument is logicai-, then whenever 4 of its assumptlons are true, all of its conclusions will be true as we11. Having established the logical sufflciency of a formal argument, it can be used as part of a larger enplrical argument that agrees wlth any particular conclusion of the formal argument. In this sense, logical validity is a necessary (but not sufficient) condltion for the truth of the concluslon of an enpirl-cal argument. 2.4 or modus tollens The mere sufficiency of the assumptions of a logicaL argument shows that whenever all the assumptLons are true then every conclusion which logica11y follows from their conJunctLon must be true or, equivalenti.y, whenever any concluslon turns out to be false then the conJunction of all of Lts assumptions cannot be true. Using a loglcal argument in favor of the truth of any of Lts conclusions by arguing fron the truth of its assunptione Ls said to be using the argument in modus ponens. Alternatlvely' usLng a logical argument against the truth of an assumptlon by arguing from the falslty of a concluslon is cal1ed nodus to11ens. Put differently, any use of modus ponens is ca11ed "affirming is called "denying the antecedent" and any use of glglglfg the consequent." It ls a logicaL fallacy to "affirn the coneequent" (reverse modus ponene) or to "deny the antecedent" (rylg tollene). A further discussion of these ldeas is given ln SaLmon Modus ponens (1973), Boland (1979), and Blaug (1980). A loglcal argument Ln modus ponens or modus tollens cannot be developed if Aristotlers axloms of logic are lgnored. For example, if we actually arrive at an erroneous conclusion, at least one of our assumptions ls guaranteed to be false, provlded any of our statements which Ls not true l-s false, i.e., all of SWAI'IY, CONWAY, AND VON ZUR MUEHLEN satisfy Arl-stotlers axioms of the excluded-nlddle d c trrrdiction. In this sense, Aristotellan logle can be mltbi e tw*alued logic in which statex0ents are assigned only c orf the tsso values: "true" or "fa1se". rDur s6.GsrrEEs 2-t Era+les of the violations of Aristotlers axiorns Unfortunately, Aristotlers logical system is of linited use in econorretrics because econonetric practices in general cannot strictly adhere to Arlstotlers axloms. The following lllustrations of the violations of these axioras form a representative sample of size four drawn from the econorneEric literature.3 Exanple 1. Approxiuate demand systems. Theil (1975) has shown that logarithndc differentials can be used to investigate the general features of conplete sets of demand functions. Goldberger (1967) has pointed out a robustness property of this approach by arguing that a "differential formrlation quite possibly provides an adequale approxination to utility-maxirnizing behavior over a range of conceivably true utllity functLons; this without being exactly appropriate for any particular one" (italtcs added). In the same vein, Barnett (1983) has derived a demand sysEem from a new Laurent series approximation to Ehe reclprocal of an indirect utllity function on the contention that the Laurent series expansion provLdes a better-behaved remaioder term than the Taylor series used by Thei1. C1-ear1y, both theories vlolate ArLstotlers axiom of the excluded=niddle: the approxirnately true demand systems of Theil and Barnett are neither absolutely true nor absolutely fa1se. Exanple 2. Causality tests. Typically, econometric modelbutldlng posits a set of autonomous conjectures as to basic behavioral relationships, including an indicatlon of the relevant varlables and a partitioning of those variables lnto endogenous -T--Th-t purpose of presenting these four examples is not to accuse econometrLcians of violating Arlstotlets axioms but to point out that many-valued logic (to be discussed later in Section 7) is needed Eo Justlfy econometrl-c practice. FOUNDATIONS OF ECONOMETRICS and exogenous sets. This structural modellng approach has been recently chall.enged by an alcernative nethodology utillzing vector autoregressive (VAR) nodeling Ln conjunction with "exogenelty" tesEs based on Grangerrs definitions of causality Isee Sins (1980)1. I{owever, the implied statistlcal test results for "exogeneity" can hold only wlth probabtlity less than one even in sufficiently large sanples Isee Sargent (1979) and Conway, Swamy, Yanagida, and von zur Muehlen (1984)1. Thls alternative nethodology by naking the exogeneity assunption dependent on the outcoxle of an inconslstent statistlcal test permtts assumptions that are neither absolutely false nor absolutely true and therefore, vlolates the axlom of the excluded-niddle Isee Boland (1982, p. l24)1. Example 3. Ratlonal expectations. In 1961, Muth (1961) posited an equilibrium nodel in which agents forrned expectations rationally. AB stated by hin, "expectations of firrns (or, more generally, the subJective probabllity distribution of outcomes) tend Eo be distributed, for Ehe same information set, about the predictlon of the theory (or the robJectLver probability di6tri-- bution of outcomes)" [Muth (1961, p. 316)]. Lucas (1976, p. 27), among others, adopted thLs concept by posltlng that expectations of individuals are "ratLonal" if a subjective distributLon on whlch decisions are based is equal to a true (objective) probability distribution of Ehe event(s) under conslderation. Such a Juxtaposltlon is lnapproprlate if, as SItaEy, Barth, and Tinsley (1982) have shown, "subJective probability" is based on the approach of subJectlvisEic BayesLans and "obJective probability" ls based on long-term frequency considerations, because these probabillty concepts are not conpatible \rithln the same definition of "probabi1lty.'4 Consequently, rational- expectatlons arguments that are based on these two competing definltlone vlolate Arlstotletg axiom of identity. ffiore conprehensive discussion of subJective probability and frequentist probabiltty ts gl-ven in the Appendix t.o the paper. SWAMY, CONWAY, AND VON ZUR MUEHLEN 4. Specificatlon of rational expectations models. can be seen from Sargent (1976) and Wal1ls (1980) that ratlonal expectations models include Ehe following sets of equatLons: (i) A set of structural equations specifled as of some time, t, where sone rlght-hand side variables represent anticipatlons formed by lndividuals about unobservable values of speciflc variables in period t or Lacer. (1i) A sec of corresponding reduced forxn equations. (lil) A set of equations statlng that Ehe unobserved antlcipatlons or expectations of indlvlduals are equal to the condltional neans of the daLa-generatlng processes described by the structural equatlons Ln set (i) given the data aval1ab1e when Example It the expectatLons were formed. (1v) A set of time series models representing the processes that generated the exogenous varlables. The argument presented in Swany, Barth, and Tlnsley (1982' pp. 139-141) and Conway, Swamy, Yanaglda, and von zur Muehlen (1984) shows that the conJunction of the sets (f)-(iv) of equations can violate Aristotlers axlom of noncontradl-ction because the identlfylng restrictions inposed on the structural equations may conlradict the necessary and sufficient conditlons under which the corresponding reduced form equatloos and the specified tLme serles models exist. For addltlonal examples of econometric pracEice possibly involvlng concradictlons, see Swany (1980), Swamy and Mehta (1983a), and Swamy, von zur Huehlen, Tinsley, and Farr (I983). 3. Sufflclent 3.1 and Logically CoosLstent l{odels of loglcal consietency and an eraTle It follows from the scheoe of Aristotellan loglc presented earlLer ln Sectlon 2 that the theoretlcal knorledge on shich a uodelts speciflcation l-s based uuet b€ loglcally consistent if lt The necesslty FOUNDATIONS OF ECONOMETRICS 11 is to provlde a "true" explanation of anything. Although the logical consistency of a given expl-anatlon does not necessarily i.nply lts truth, it is a necessary prior condition to any explanation of the reil world. Boland (L982, p. 24) expresses the same view by argulng eloquently that "the only objective and nonarbitrary test to be applled to theories or models ls that of 1ogica1, consistency and validity. Even if we caonot prove a theory or model is true, at the very mlni-mum to be true Lt must be 1oglca11y consistent." If the assunptions underlying a nodel contradict each other and/or violate Aristotlers axlom of identity, as ln Exarnples 4 and 3, then our knowledge leading to these assunptions is not 1ogica1Ly consistent and cannot provide the true distribuElons of economic variables. For this reason, we should specify a nodel without violatlng Arlstotlers axioms of identity and noncontradlctLon as far as possible. Exanple 5. A suffl-clent and logically consistent mode1. We now consider an exarnple of a sufficient and 1ogical1y consistent mode1 from orthodox statistics that conforng wlth Aristotlers axiomg. In what fo11ows, we utllize an g1g!9.5g!9,9. calculus of probablltty baeed on Kolmogorovre axlons [Rao (1973, 80-87)1.5 Suppose that x1 ls a k-dimensional. random vector having the decomposltlon --m=ta;Aard interpretations of probabtllty violate the axion of the excluded-middle lsee Appendix to the paper]. Consequently, to avold this vlolation, we have chosen here to work wlth an unlnterpreted calculus of probabillty dedtrced fron Kolmogorovrs axions of probability. Howeverr we should point out that not every statl-stician would llke to work with Kolnogorovts axlons wl-thout a modificatlon. For example, L. J. Savage and de Finetti (L974) replaced Kolmogorovre countable addttlvity conditlon by a finite additivity conditi-on. Jeffreys (1961) and Cox (1951) regarded the theory of probabillty as a generalization of ordinary Boolean loglc. Jeffreys was also the first to propose an axlomatlc foundation for "inproper" distrlbutions (1.e., distributlone whoee deneities lntegraEe to lnflnlty inetead of to unity), whereLn certalnty is represented by lnflnlty lnstead of by unity; for a nasterly exposltion of Jeffreyst axioms of probablllty, see Zellner (1971, pp. 42-53; 1983, pp. 74-8f). FOOTNOTE 5 CONTINUED ON NEXT PAGE SWAMY, CONWAY, AND VON ZUR MUEHLEN t2 t=x;+vr (t=L,z,... , T) (r) xl represents anticipatlons of lndividuals formed about the values of xg in perlod t-1 (as in Wallist model), and vg l-s the diecrepancy of individualsr antl-clpatlons from lhe actual observable xg. Further, 1et where xt* = efg (2) is a k x r matrix of less Ehan fu11 rank, every colunn of whlch containe at l-east two nonzero elements and no row of whlch ls nu1-1, and f1 is a r x 1 random vector conslsting of the psychologlcal factors that determlne lndividual anticlpatlone' Suppose also that the elernents of the vector (f!, vl)t are nutuallylndependent.LetYgbeascalarrandomvarlabledeflned where A by Yt=nrfg*€6gr (3) lr is a 1 x r fixed vector and €6g is a random varlable independent of fr. WithouE Loss of generallty, 1et E fg = 0, Evg = 0, and EGgg = 6. Then the following theoren due to Kagan' Linnlk and Rao (KLR) (1973, p. 320) gives the conditl-one under which the conditional rrnean E(Yglxg) extsts and is linear' Theoren (KLR): Let k ) 2 and let the regression of € 9g on vs be llnear: E(€pglvg) = 6tvs. Then E(Ygl*t) = x{8 if and only if the followlng conditlons are satisfl-ed: where FbbTIi6'IE-5Tn-mvunl Ilowever, the probabillty calculus based on Boolean algebra or a finite addttlvtty conditlon is less general than that based on Kolmogorovrs axloms. For example, the theory of conditional alistributlons for flnitely addltlve probabillties 1s relatively new and stlll incornplete [see l{eath and Sudderth (1978)]' we sha11 say more about finlte addltivlty Later in thls paper' To avold the vlolation of the axlom of identity, we use only Kolr,oogorovrs axioms here. FOUNDATIONS OF ECONOMETRICS 13 (f) If the Jth element of I - A'B ls not zero, Ehen the Jth elenent of f1 is normal. (fi) If the JEh element of I - 6 is not zero, then the Jth element of v1 ls noroal-. (iii) If the Jth elernents of fg and vg ar€ normal, then the vector B and the variances of the elernents of fg and vt are reLated according to an identity glven in KLR (1973, p. 327,10.5.7). This theoren has been extended by KLR to the cases where A has fu11 colurnn rank, r=1, or k=1. The last case is especlalLy interesting because when k=1, the randorn variables tx, vg r and €6a need not be normal Ln order for the regression of Yg on xg to be lLnear. Further, this regressl.on is not linear for all A and X. Thuer,unllke Sargentrs or lJaLl-lsr rational expectations nodelthe expectations nodel n(v.lx.) = xiB (4) in (3) and (l) respectively, follows 1oglca1l-y fron the conJunctl-on of the conditions of KtRrs Theorem. ModeL (4) is true tf the condl-tlons of KLRrs Theoren are true; l-n this sense, lt is sufficient and 1oglca1Ly conslstent. Notice that the specification of nodel (4) does not violate any of Aristotlets axions.6 KLR's Theoren ls an example of an argunent ln modus ponens. In fact, all the characterization theorens - - Mathernatfcfans, working with detentrinistic rnodels, give sufflcient, necessary, or necessary and sufficient conditions for their results. They do not violate any of Aristotlers axions. In the same rtay, Kolmogorov deduces the entire probabiltty calculus from a conelstent set of axions without vlolating Arlstotlers axioms. Econometricians can, in principle, follow the sane derivatl-on. They can take a consistent set of behavioral assumptions nade by economists and conbLne then wlth Kofuoogorovrs axlonrs of probability tn a consistent manner to obtain a full set of consistent behavioral and stochastic assumptions. From these assumptions they can deduce a 1ogical1y consistent, sufficLent model, as ln Example 5. In this case, stochasticism does not necessarily vLolate the axlom of the excluded-nidd1e. where Yg and xg are aa defined SWA},IY, CONWAY, AND VON ZUR MUEHLEN L4 proved by KLR yield several different arguments in modus Ponens, thereby renderlng the srochastic rnodels in their theorems sufficient and loglca11y consistent. 3.2 Alternative theories At any Point in time there nay not be only one theory of the economy. There can be several conpeting theorles favoring sone glven propositions or speciflc predictlons. Kuhnrs view oo how gcience works Ls very well known among nethodologists Isee Boland (1982, p. l6f) and Blaug (1980, pp. 27-33)1. According to hls vLew, scLentific progress is not sfunp1y the cunulative reeult of normal sclentific actlvity. The scientlfic revolutions and the controversLes that arl,se when one paradigm Ls deemed to be a blt stale and a new and more promlsing paradigm takes ite place' clearly do not flt lnto an lncrementallst view of the world' In the case of a revolution in scientific paradigm ln this sense' we slmply have one more theory to add to the existlng (economic) theories. If each of these Eheorles is sufficient and 1ogical1y conslstent, then its concluslons are true whenever all of lts assumptions are true. If two theorles contradict each other, then both of thern cannot be true. 4. Inductivlsn and Approxinations of induction far has not yet addressed the neans by which one knows the truth of the assumptions of a sufficlent and logically coneistent explanatlon. Unfortunately, Arlstotellan loglc ls of 1lttle help ln deEerroining the truth of an assumptlon' It can only help by "passlng" along known truthe from assumptLons Thls llnitatlon of tradltlonal to conclusions as 1. glq-ry. logic leads to a consideration of the so-ca11ed probleo of inductlon: the problern of findlng a form of loglcal argument l-n 4.1 The problem The discussion so FOUNDATIONS OF ECONOMETRICS which (a) the conclusion ls a general statement and (b) the assumptions lnclude gla slngular statements of partLcuLars [Boland (7979, p. 506)]. Inductl-visn is the methodologlcal doctrlne that asserts that any Justlfication of oners knowledge must be 1ogica1ly based glg!1 on experiential evidence con6isting of partlcular or sl.ngular obgervational statements IBoland (1982, p. 14)1.7 Given inductivi-sm, any straightforward solution to the problern of induction requlres an l-nducEive 1ogic. That ls to say, there must be a forn of logic whlch permits correct (inducti.ve) arguments consisting of ooly singular statements (..g., a finlte number of observatlons) to yield true conclusions that are general statements (..g., "the conditions of KLR!s Theorem are absolutely true"). Unfortunately, there is no solution to the problern of induction: no matter how maly "facts" one has, one cannot ever prove the absol-ute truth of the conditLons of KLR's Theoren. Popper (1972, pp. 23'29) considered induction to be a "mythr" and, indeed, said so in no uncertaLn terms. The personal (or subjective) probability theory also supports the conclusion that there ls no solutl-on to the problem of lnductlon. In the vlew of a personall-stic Bayesian like Savage (f98f)' our opinions today are the (rational) consequence of what they were yesterday and of what we have seen since yesterday. Similarly' yesterdayrs opinions can be traced to the day before' but the theory of personal probability does not pretend to say what system of opinions ne ought to have been born wlth. Thus, there are no oblective grounds for any specific belief beyond recent experience and there are none for believing a unlversal proposltion other than one that is tautological", gLven what has been observed. Knowledge of a unlversal ls acceptance with a hlgh personal probabtlity of a unlversal with flnlte domaLn or of many such, vaguely speclfied. But Justification of opinione ln the -7--ET;E'-how philosophers define the problern of inductlon. Unless othernlse quallfied, this le the sense in whlch we use the terrn "problem of lnductlon" throughout the paper. I6 SWAMY, CONWAY, AND VON ZUR MUEHLEN that one ltho rejects them is guilty of an error to a loglcal fallacy is not aval lab1e. sense retreat to approxinations Since there is no solution to the problem of induction' tenptlng to argue that if the condltions of the real world 4.2 ls conparable The it sufflciently well the conditions of KLRrs Theorern (whatever this means), model (4) w111 be approxlmately true. This tenptatlon 6hou1d be reslsted Lf we are lnterested in applylng Aristotellan two-valued logic because there ls no valid approxinate modus poneng.8 Modus ponens is not valid for arguments consistlng of statements Ehat are only "approxirnately correct." The reagon is that auch staEements vl-olate Ariatotlers axiom of the excluded-mtddle and hence are not adnisslble into Aristotellan type loglcal arguments [see Boland (1981, p. 85)]. The valldity of cerEain results under certain departures fron certaLn assumptLons proved by Rao and Mitra (L97L' pp. 155-167)' Box and Tiao (1964) and FLsher (1961) does not establlsh the validity of an approxinate modus ponens, but, rather, stnply demonstrates the fact that a result can be true even lf tts sufflclent condltlon is fa1se. approxLmate 4.3 Attenpts to solve the Problem of induction Jeffreys (1961, p. 43) has constructed an axiomatlc system for a Bayesian probabllity theory to formallze Lnductive loglc ln such a way that lt lncludes deductlve logic as a special llnltlng case [see Zellner (L97I, p. 4; f983' p. 74)1. An alternative statemetrt of Jeffreys'argument given by Good (1962, p.488) ls that an inducttve inference can approach certalnty or, aymbolica11y, that P(pnlpl,..., pn-l)+I, where Plr P2r... are the results of experinents, all of which were "successful." Good (1962' pp. 488-489) offers the criticisn that "it seems somewhat 8 An approprlate nethodology for handLing lnexact concepts such as approxLmatlons is provided by the nany-valued logic franework discussed in Section 7. FOUNDATIONS OF ECONOMETRICS inconsistent to deny [as Jeffreys does] the validity of a limttlng definition for physical probabilities, and then to use [again as Jeffreys doesl a lirnittng argument for the justificatlon of lnduction." Good then goes on Eo argue that ln practice (after a very 1ong, but of course, finlte sequence of successes) repeated veriflcations of the consequencee of a hypothesls would not make it practically certain that the next consequence of lt would be verifled if the new consequence were kooltn to be of an entlrely different character frorn Ehe previous ones. To ilLustrace his inductive 1ogLc, Jeffreys (1961, p. i28) considers the hypothesis lhat all animals with feathers have beaks and clairns that the probabllity that this hypothesls ls true will approach I as Ehe number of successes, wlthout fal-lures, increases. Comrnenting on Jeffreysr l-l1ustratlon, Good (1962, p. 489) polnts out that "[t]he argument and the conclusion are both undermlned by the fact that the dlstribution of essentl-al1y distlnct species of feathered anirnals rnight be very skewIed]..., or even norse, so tha! there would always renaln species that had not been sarnpled until about half of the entire population of feathered aninals had been examined." Accordingly, Good (L962, p. 489) says, "a general 1aw which does not itself refer to probablllties will not usually tend to become certain, however often lt ls verified. If the 1aw could be qualified by saying that lt l-s to be lnterpreted as applying only to experiments and observations of a sinllar nature to those already rnade, then it night well tend to certainty, but lt l-s not easy to make this klnd of qualification precise." Berk (1970) has attempted to nake this quallflcation preclse by describtng the conditions under which a sequence of poeterior distrlbutLons converges weakly Eo a degenerate dlstrlbutlon. Therefore, under Berkrs (1970) conditlons, a 1aw could be properly qualified so that it tends to certalnty. llowever, the problen of establishing the truth of Berkrs conditlons ls identlcal to the problem of induction. Thus, Goodrs deecrlptlon of the problem with Jeffreyst 77 SWAMY, CONWAY, AND VON ZUR MUEHLEN i.nductive infereace appears to be consistent r'lEh Popperrs or Boland's (1982) description of the problem with the philosophersr type lnduction. Boland (1979) discusses several unsuccessful attempts to solve the phii-osophersr problem of induction' Therefore, we conclude that the problem of inductlon as stated by Boland is not solvable and the attenpts to solve it are not successful. 5. 5.1 Popperrs Falsiflcationism and Related Problems Asynnetry is a fundamental asymmetry between establishing the truth of a statement and refuting it. No unlversal statement can be 1ogica1ly derived from or conclusively established by any finite number of singular statenents, but any unlversal statement can be 1ogica11y contradicted or refuted with the aid of deductive logic by only one singular statement. Popper exploits this fundamental asynmetry in formulating his demarcatlon cricerlon tlhich states: sclence is that body of synthetic proposltl-ons about the real world that can, at least in prlnclple, be falsified by enpirical observatlons because, in effect, Ehey rule out cerLain events from occurrlng [B1aug (1980, p. 12)]. This demarcatLon crlterlon then leads to the following program: Whenever lt is shown t.hat one of the predictlons of a theory ls fa1se, then, by modus tollens' lte can conclude that at least one of its assunpcions must be fa1se, provided all of Arl-slotlers axloms are satisfied. Predlctions have an overrlding importance for Popper in refuting explanatory theories. Accordingly' sclentl-sts seek to explain their observatlons, and they derlve logical predlctlons that are lnherent in thelr explanatlons ln order to refute thelr theories; all "true" theories are merely provlsionally true, having so far defied falsiflcatioo; or' reetating Popperts pol-nt' all the naterial tsruth we possess is packed into lhose theorles that have not yet been falsified There FOUNDATIONS OP ECONOMETRICS 19 lflaug (1980, p. 17)]. In econometrLcs, the question also arises: falsify economic theories? If a nodel violates Aristotlets axion of identity or noncontradictlon, then certainly, we can reJebt it outright on the grounds of loglcal lnvalidity, as we proposed in the preceding sections. If a model violates Aristotlers axiom of the excluded-middle. then rte cannot uae We can use many-valued logic, !o be modus tollens to refute it. dlscussed in Section 7 below, to analyze euch a nodel, but we cannot refute Lt. The difficult problem remalns: IIow do hre refute a sufficient and 1ogica1Ly consistenE model of the forn (4)? llow does one 5.2 Loglcal ambiguity regarding refutatlons Equat.lon (4) nay represent the behavior of some unlt of the economy. If lt does,lt will only partly do so because, at best, economic theory suggests only lthlch variables are deflnltely reLevant and whlch are possibly relevant. Theory night, in addition, outline aome broad features of the approprlate functional form. In general, economl"c theories do not u8ually aasert a specific functlonal form nor do they suggest a specific distrlbutlon for the pertlnent variables. Indeed, as a practical matter, the epread of available observatlons and thelr number wl11 reetrLct onets abllity to discrlml-nate anong general-Lzed. functional forns. Therefore, in order to bul1d a eufficlent and logically conslsten! model of the form (4), we mus! add extraneous assumptions of whlch the conditiono of KLRrs Theorem are examples. One problen wlth eufficlent and Logically coflslstent rnodels is their inherent logical anbigulty regardlng refutatlone. In other words, if nodel (4) were refuted, one would not know whether the source of the fallure rtas the set of extraneous assumptlone or the seE of original behavioral assumptions underlylng (4) [Boland (L982, p. 120)1. 5.3 The problen caused by uncertainty Any eufficient and loglcally cons{stent econornetric nodel of 20 SWAMY, CONI.JAY, AND VON ZUR MUEHLEN the foru (4) represents a sltuation of uncertainty' In the case ofcertainty'therewouldbeaone-to-onecorrespondencebetween actions and consequences. In the presence of uncertalnty, model (4) nerely generates a probabiltty dlstrlbutlon of coosequences resulting fron a change in the exogenous variables, l-'€', 3o exogenous change in xg in (4) generates a (Predictlve) dlstrlbutlon of Y1. Uncertainty, then' means that we [ust dlscriminate and choose among sufficient and logtcally conslstent models by comparingthepredictivedistributionsofavariableofinterest inplled by those nodels. But then the basis of conparlson and falslflcatlon becomes an issue. First, to dlstlnguish model (4) from another nodel Lt must be identifiable. That is, if the variabLe Y6 has different distributioas for dlfferent values of p,thennodel(4)isidentlfiableandl-nformativestatemenlsabout E can be nade uslng an observed value of Ya'9 Seuinal work on (1971' comparlng predletlve dlstrlbutions can be found ln Zellner pp. 306-317). ThLs work has been taken further by Gelsel (1975) and others. Second, as shown by DeGroot (1970, pp' 86-115)' the speciflcation of a cornplete orderlng of a class of predictive distrlbutlons requlres an approprl-ate utllity functlon' Apart fron the obvl-ous practlcal difftculcies aasoclated with the determinarLon of an approprlate uttltty functlon, such an orderlng rnay not concluslvely falelfy a nodel lGeisel (1975)]' that identiftablllty ln this sense ls lmportant T/f-Etlfeve Fron.the ft"t i.ti the non-Bayesian and Eayeslan vl-ewpolnts' quantif lcations neanlngful stanipoint, op.."iio""ff subJectivlst, "t by (coherent) lndividuals about of uncertainty are Lhose nade observables. fte."for", .tt op"t"tionali'y lnterpretabfe !11Uffiuatirrtydistribut1onoveranyunobservab1e(11kea is possltle lf there la a one-to-one correspondence ;;;;;";;.) betweentheunobservableandthecondltlonaldistrlbutionofsooe bets observables given the unobservable because in thie case the bets the lnto translated unlquely be can observables the about there ls about the ,rrrobse..r"ii".- rtt the unldlntlfiabiltty case' (subinterpretable no such correspondence and an oPeratlonally j""ii".l-ltotalrrrty distrlbutlon over the parameters ilAy not be possible. FOUNDATIONS OF ECONOMETRICS 2T falsify a model arises if a sequence of posterior distributions for the nodel converges weakly to a distribution which ls degenerate at a correct point in the paraneter space whenever the nodel is true. Sufficlent conditlons for this convergence are glven in Berk (f970), as sre have already polnted out. In an earlier paper, Berk (1966) has also shown that the posterior distrlbutl-on tends to be confined to a special set in the paraneLer space when the nodel is incorrecc. Verlficatlon of the truth of Berkrs conditions is unlikely siace it would seem to require a solution to the problem of lnduction. The situatl-on lrhere one can conclusively 6. Objective Knowledge, Instrunentalisn, Sinplicity or Profligacy and Parsinony 6.1 Bolandrs (1982, p. 179) view A role of theoretical knowledge is one of providing sufficient and 1ogica11y consistent explanations of the real wor1d. Boland (L982, p. I79) proposes that this role be separated from one havl-ng to do with truth (i.e., whether statements derived from theoretical knowledge are true or false). This separation is Justified because the truth of someonefs knowledge is not always necessary for successful action, as we show below ln this section. By separatlng the role of knowledge fron its truth status, Boland is not suggestlng that theories or knowledge cannot be true. Rather he is asserting LhaL a theory can be true even Lhough its truth status ls usually unknown to us. The truth status of anyoners theoretlcal knowledge is necessarj.ly conJectural, but by irs loglcal nature Lt must be capable of at least belng staEed in language or in "other repeatable forms" to the extent that it concerns the real world [Popper (L972, pp. 106ff) l. 22 SWAMY, CONWAY, AND VON ZUR MUEHLEN 6.2 Instrumentaliso There is a vLew that, as long as a 1ogically va1ld rnodel does its lntended Job, there ls no apparent need to argue ln lte favor, or on behaLf of any of its constituent parts. For poLicyorlented economLsts' the stated purpose ls the generat{on of "Erue" or comparatl.vely successful predictlons. In this case a nodelrs predlctlve success ls certainly a sufflcient argument ln lts favor. This view of the gfg of models is ca11ed "instrumentall-sm. " Instrumentalism holds that nodels do not have to be considered true statements about the nature of the worLd, but, instead, nay be considered convenlent ways of (logtcal1y) generating what have lurned out to be true (or successful) predictions or conclusions lBoland (1982, p. 114)]. A prlori truth of the assumptLons ls not required if it is already known that the concLuslons are "true" or acceptable by sone conventionalist crl-terlon. It is ln this manner that instrumentalism' such as the view presented by Friednan (1953), responds to the failure to solve the probLem of lnduction. For the followers of Lnstrunentall"t, glg po* wt11 now necessarlly be eeen as irrelevant because they begin thelr analysls wlth a search--not for the true assurnptlons--but for "true" or useful conclusions. t{odus tollens ls likewlse irrelevant because lte use can only begtn with false concluslons IBoland (1982, pp. 144-145)1. Priedman concludes Ehat testing of assumptions ls Lrrelevant for true conclusions since nodua tollens cannot be used in reverse whenever tests reject assumptions, and statistlcal testing vlolates Arlstotlets axlom of the excluded-niddle in any eca11 saqle case. This leads Friedman to discuss the possibllity that a false assumption nl-ght be applled as part of an explanatl-on of sore observed phenomenon. llere he introduces hls fanous version of the "as lf" theory of explanatLon: "[I]f we are trylng to explain the effect of the assumed behavlor of sone lndividua1s..., eo long as the effect Le in fact observed and lt rould be the effect lf they were Ln fact to behave as ite assume' FOUNDATIONS OF ECONOMETRICS se can use our behavioral assumption even when the assumption is fa1se." IBoland, (1979, p. 513)]. Thus, under lnstrumentalism, the use of 1ogically consistent but lncorrect assumptions ls Justtfiable. While FrLedrnanrs instrumentalisrn does not necessarily violate any loglcal princlples, Lt can, nevertheless, give rise to the sane difficulties as Popperra falslficationism when applied to stochastic nodels of the type (4). For example, in the presence of uncertalnty, the meanlng of the term "true or useful prediction" ls not c1ear. Also, past auccess ln prediction ni11 not guarantee future success. Therefore, DeGrootrs (1970, pp. 86-115) conplete orderlng of predictive dlstributions can change over tine IGelsel (1974)1. Furthermore, there exists no slngle econometric model which predlcts all varlables better than any other nodel for all tirne periods. Flnally, ae Boland (L982, p. l-96) points out, LnsErumentalisn is approprlate only for lnrnedlate practlcal problems and not for long-terrn phllosophical questLons and, as Kyburg (1983, p. 27) argues, a "strongly instrunentalist vl-ew of science is perfectly adequate to the design of experlmental apparatus as well as to the creation of engLneering wonders." But to understand the world, we need a nodel that repreaents it adequately. There is no guarantee that Lnstrumentallsm leads to such a model. "Repeated successes (or failed refutations) of instrumentalisn... [are] 1ogtca11y equivalenE to repeated aucceaaful predictLone or true conclusions. We sti11 cannot conclude 1oglcaL1y that the assumptLone, l-.e., the basee of lnstrumentalisn Itself, are true. They could very well be false, and in the future someone nay be able to find a refutatlon" [Boland (L979, p. 522)1. 6.3 Sinpl-lcity Expllclt acknowledgement of modeling fallures can be indefinltely postponed by such patchwork devlces as dumy variables, ratchet variables, Judgenental conetant-term adJustments, and SWAMY, CONWAY, AND VON ZUR MUEHLEN in applied econometrlcs. Followlng Lakatos [Blaug (1980, P. 36)], thls practice ls cal1ed "degenerating," since it involves the endless addition of 35|@ adjustrnents thaE nerely accommodate whatever new facts become avallable. Ilistoricallyr thls type of resPonse Eo fallure has given lrnpetus to a yearning for sl-nplicity in nodeling' Unfortunately, as Blaug (1980, 9. 25) polnts out' attenpEs to deflne precisei,y what ls meant by a simple theory have so far falled largely because the very notion (of the sinplielty of a theory) is itself highly conditloned by the hlstorical perspective of scientists. This view is shared by Good (I980b' P' 403) who' after some heroic but unguccessful attenpts by Nagel IGood (1980b' p. 402)l and hlmself [Good (l98ob, pp. 422'423)]' adnits that it ls difflcult to specify sharply whether one 1aw Ls more general than another. As ln physics and other highly developed sciences, where it has become largely accepted that sords unaccompanl-ed by operational meanlng are suspect and 1tke1y to prove meanlngless IBridgman (1928)], we reject unquallfled termlnoLogy llke "simpliclty" and "cornplexity. " Indeed, from the polnt of view of l-nstrumentalisn, there is no need to imPose such undefinable crlteria, since the only relevant criterion ought to be dogmatic priors whlch are commonly used efficaclousness: use whatever works, provided the meana are derived from a 1ogtca11y valld nodel and are not influenced patctrerork devlces designed to flt speclfic needs. by vs. principle of profllgacy Occasionally, elther the "prlnciple of parsLmony' or the "prlnclple of profligacy" is advocated as a crlterlon for model choice. An example of the forrner vlew is given by Box (1976, p. 792) who argues, "Just as the abiltty to devlse sinple but evocative nodels ls the signature of the great eclentlst 8o overelaboraEion and overParameterization is often the mark of nedlocrlty." An advocate of the prlnciple of profligacy 18 Slns (1980, p. 15). Where pure lnductlvisn requiree a final (absolute) 6.4 Prl-nciple of parsimony FOUNDATIONS OF ECONOMETRICS 25 inductive proof for any true model, these tlto so-called prlnci-p1-es requ1reon1y@deductl-veargumentstoshowthatthe chosen model ls alternatively "slmple but evocatlve" or "extravagantly paraneterlzed." Ilowever, thls solution poses a new problem. Since knowledge is Lmperfect, condltlonal deductive argumenta necessarlly contain assumptLons. Therefore, the choice of a rnodel is always open to question. In partlcular, one can, crlterla used to deflne "parsimony" or "profligacy." So thls posslbility then presents the new problen of an lnflnLte regress ln whlch the next step lnvolves the choice of a rnetacrlterLon to be used for defining what is evocaLlve stnpllcity or paranetTic extravagance. A flnal unintended consequence of an appllcatlon of either of these "prlnciples" is, lndeed, a circularlty whereby some operatlve crlteria are held to be appropriate because Lhey are sufficient to Justtfy our choice. The "prlnclple of parslmony" rnay nislead the lnvestlgator, sLnce the true theory might actually be different frorn the one he considers on the basls of his definition of the prlnclple of parslnony. On the other hand, a profusely parameterized model rnay not be falslflable or identiflable ln the econometric sense. Therefore, without further qualifications, the nonlogical princlples of parsimony and profligacy are vacuous, and, Lndeed, have the smack of nunerology rather than of sclence. as we1l, questlon the 7. Restrlctive Nature of Aristotlers Axioms and Methods of Relaxlng Then 7.L Giidelrs challenge to the axion of noncontradiction In thls section we begin to address the following questions: (1) can we satisfy Aristotlets axlome all the tine? and (2) what are the conaequences of relaxing some of hls axloms? The restriction imposed by the axiom of noncontradlctl.on can best be expressed ln terms of a conpleteness concept. [The following discossion ls borrowed from McCawley (1981, p. 76) and 26 swAMY, CoNWAY, AND VON ZUR MUEHLEN pp. 4-5, 81 and 85)1. A systen is "deductively complete" lf it inposes the most strlngent possible restrictlon on states of affalrs: that there is only one sEate of affairs in which the rulee of the system lead only Eo true conclusLons when applied to true assuoptions, as l"n modus Ponens. A system ls deductlvely complete if for every proposltlon p of the system' either p or not-p is a theorern of the systen. The question of "deductive cornpleteness" arl-ses ln connection with any system that includes axlons for a particular subJecE matter. one exanple concerns a formal system that ls intended as an axiomatizatlon of the arithnetic of posltlve lntegers. In a most celebrated resul,t Lyndon (1966, regarding deductive completeness, G6delrs incompleteneas theorem shows that any axiomatizatl-on of the arithneclc of positive integers is elther deductively incomplete or inconsistent; that ie, if our supposed axioms for arithnetic are consistent (i.e', they dontt al1ow us to deduce any contradlctions), then there are propositions of arithnetlc that are "undecidable" relative to those axloms: proposiElons A such that neither A nor not-A can be proven from those axioms. ciiDEL's TNcoMPLETENESS TEEoREM: If the theory T ls valid' it l-s not complete. Thus, there ls a conflict betlteen deductlve completenese and logical consistency. We have to glve up elther the goal of achieving deductlve completeness or the goal of achieving logical consistency. As we have already argued, lf at the very minlmum to be true a theory or model must be 1oglca11y conslstent' Ehen lre cannot give up logtcal consLstency. This means that we have to give up deductlve cornpleteness. Once we give up deductlve completeness, there are propositlons of econometrics that are undecidable rel.ative to any assumed econonetric axloms tha! do not permlt ue to deduce any contradictlons. The usefulness of GiJdelrs theoren resldes ln lte abllity to indicat.e a llnitatlon of an econometrlc approach, where one hypothesls is tested agalnst another by conbining the trto rival FOUNDATIONS OF ECONOMETRICS 27 in a general nodel such that each of the hypotheses can be derlved as speclal cases of the conbined nodel [see Pesaran (1982)1. In other words, tn applied research Ln econometrics there is a natural llmlt to the process of buildtng a general nodel by enbedding a number of otherwise nonnested models tf the general nodel is required to be logicall-y conslstent. hypotheses 7.2 Local vs. g1oba1 coherence G6delre lncompletenesa theorem noEwithstanding, the discussion in Sections 2-6 shows that there 1s much to be said for conducting the study of econometrics even within the weakest possible netatheory that wl11 serve. In support of thls klnd of proposltion, Barnard (I976, p. 121) states that "...a useful distlnction can be drawn between r1oca1 coherencer and rglobal coherencer. ...[T]he total process of naking eclentiflc inferences... involves, among other thlngs, the cholce of a rcde1, and rrnodel critlclsn. I The procees of nodel critlcLsn may se11 lead to a change of model-, and in that case, we rnay well cotrtradlct propositlons prevlously taken (tentatively) as true. Such a change of model could lead to rLncoherencerr... But so long as we contlnue to use the same mode1, our reasonl-ng should of course, satisfy the rcoherencyt requirement. Thus, we should be 1oca11y coherent (withtn the sane model) but good senae may rel1 require Ehat we should be prepared Eo be rincoherentr when e change of rnodel ls ca11ed for." Good (1983, p. f6), using the t€n "type 2 princlple of ratlonality" to refer to Barnardrs term 'local coherencer" states that the procesa of enlargLng bodies of bcliefs and of checking them for consl"stency can never be conpleted crn l-n prlnciple, by vlrtue of Gijdelrs theoren. Ilence, the type 2 principle of ratlonaltty is a logical necessi.ty. Denpster (1968, pp. 244-245) draws a distinction between cmaistency ln the weak sense and consl-stency in the strong oltrs€. L.J. Savage and other Bayesians have deecribed conslstent bcbavior ln the senee of consistency wlth plausibl,e sets of SWA},IY, CONWAY, AND VON ZUR MUEHLEN 28 is consistency ln the weak s€ns€r Accordi.ng to Dempster' conslstency ln the atrong sense referg to consistency rtith some unique true laws of thought and behavior, rthl-1e consistency ln the weak sense Ls a mLnor achlevenent and provides no grounds for chooeLng among approaches' Tverskyrs (1974, p. f58) view is sinllar. Ile doee not belleve that (locaL) coherence, or lnternaL conslstency, of a given set of probability Judgnents is the only criterion for their adequacy' In his view "Judgnenta muat be conpatlble wlth the entire web of bellefs held by the tndividual, and not only consiEtent among themselves. Conpattbility anong beliefg is the essence of rational that Judgnent...l0 One of the lnpllcatlons of Gijdel.rs theorem is Dempsterrs criEerlon of consistency ln the strong eense and Tverskyte crl-terlon of the compatibllity of a set of probablllty nay be Judgrnents with the Judgets total systern of bellefs lnpossible to satlsfY. axLonts which ln Dempsterts teros 7.3 Locally coherent Lnference procedures Local coherence can be achleved by applying de Finettire (Ig74) eubJectlve Bayesian method to a sufficlent and 1ogical1y consistent model deftned earlier in sectlon 3. A declgion-maker who employs such a model ls seen to be locally coherent lf antl only lf his probabillties are conputed in accordance nith some finirely additlve prior Isee l{eath and sudderth (1978)]. Unfortunately, there are technical difficulties lnvolved in erryloying flnitely addltlve distrlbutlons which are not countably addltive. For example, such prlors nay fall to yleld poeterlors; and even lf poeEerlors exlst for a finitely additlve prlor, there may be polnt about savagets axioms that Llndley (1974) brlngs out ls that they refer to a large wor1d, l-n Savagers language' Llndley warns that we should not take our worlde too sna11. When thinklng about one probabllity that ls needed, Lindley says that -e we should introduce othersr assess these as well, and then see if the Judgnents cohere. Els guess ls that typicaLly-they-w111 not' and ievlsion to make then so will be ca11ed for. One of the lnplicatl-ons of Giidelts theoreo is that coherence cannot be achleved ln sufficlently large worlds. FOUNDATIONS OF ECONOMETRICS no available al,gorLthn for coraputing thern [see Ileath and Sudderth (L978, p. 336)1. To avoid such difficultLes, Lane and Sudderth (1983) describe a fairl-y general inferential setting ln whlch all locally coherent Lnferences can be obtained as posteriors from proper, countably additive priors. Speciflcally, they prove that if the (sufficlent and 1ogical1y consistent) sanpling nodel is a contlnuous function of a parameter and if either the parameter apace or the observatl-on space is compact, then a coherent Lnference whlch ls a continuous function of the obeervation nu6t be the posterior of a proper, counEably additlve prior. These conditions are not always satlsfied. Whenever the conditions of lleath and Sudderthrs (1978, p. 335) Theorem 1 or Lane and Sudderthrs (1983, p. 118) Corollary 3.1 are true and the sanpling nodel is grounded in a euffLcient and logtcally consistent erplanation, Bayesian lnferences are 1,ocally coherent and do not violate Arl-stotlets axlorn of noncontradiction for a gLven nodel. An alternatl-ve nethod euggested by Jeffreys (1961) is based on a calculus of credlbilities (= unique ratlonal degrees of bellef or inteneLties of convictLon = logical probabilities). In order to apply thls calculus to statlstics an assumptlon concerning inltial probabllittes of scientific lans or nu11 hypotheses ls necessary in each appltcation [see Good (L962, p. 488)]. Once initial probabtlities have been speclfl-ed, ernpirical evidence lodifles such probabilittes via Bayee theoren, provided these lnitlal (or prtor) probabilities possess posterl-or probabilltles. Although it can be said that 1ogical1y inconslstent theories or hypotheses have zero prlor probabtllty of being true, one philosophical difftculty here concerns the unprovable exlstence of logical prlor probabilities for all (1ogica1ly conelstent) scientlfic lawe or null hypotheses, whlch Jeffreys and his followers stnply assutne. We say "unprovable" because the available evidence is confllctlng. For example, Tversky (1974) reports sone results of a research proJect showing that lndividual subJects Judge the probability of a hypothesls by the degree to which SWAMY, CONWAY, AND VON ZUR MUEHLEN tt represents the evidence' Irith llttle or no regard for lts prior probabtlicy, whereas Salmon (1973, p. f13) states that scientistsr 3udgments about the reasonablenesa or plauslbility of hypotheses constltute assessmenta of prior probabilltiee. DeGroot (I974, p. 1053) asserts that "the person's rlnltialr probabilltlee for X will no! be formed until he realLzes that X affects hls llfe, and Ehese probabillties will then be personal ones based on hls individual curuulative observatlonal experLence. There are no correct initlal probabilities"; for a Justification of thls assertation, see savage (1981, pp. 519-520). Another difflculty is that ne may noE be abl-e to recognize inltial probabilltiea even lf they exl-st. Agaln' as Salmon (L973, pp. 114-116) observes, the assessment of prior probabilitlee Ls often a difflcult and subEle matter. Jeffreysr own attempt to overcome thls dlfftculty by means of some rules or l-nvariance theory (whlch says that the prlor should be invariant under a rnodel transformation tha! does not change the parameter space and the structure of the rnodel) proved not entirely satisfactory even to hineelf and forced him to use ad hoc adJustments when deciding wheEher or not to use the theory lsee Good (1962, p. 488)1. More irnportantly' Good (1980a' pp. 24-25) interpreEs Jeffreyst use of rules for deEerninlng prLor probabllities as an application of "Type 2 rationallty"' It ls lnterestlng, then, that de Finettlre and Jeffreysr approaches make use of the notion of "locally coherent inferences." It should be noted that Jeffreyst lnvarlance principle wll-1 not ahtays lead to unique prlor probablltties. Although the invarlance prlnciple determlnes a uniquely defined prior for any 1l-near mode1, it is known that addltional princlples of inference may be needed to flnd unlquely defined prl-ors for other models whlch do not have the rlch algebralc atructure of linear models [eee Vtllegas (f977b)]. one princlple that may be used for this purpose is a compattbility prlnclple defined in Villegas (L977b, p. 652) or a certain group structure defined in Vlllegas (1981' p. 775, Proposition 2). It is aLeo true that nany of the prlors FOUNDATIONS OF ECONOMETRICS Jeffreys (1961) uaes are not entLrely based on the invarlance principle and the lnner structure of the nodel belng analyzed for they may also be based on external Judgments lVillegas (1977a' pp. 454)1. Posterior probabllltles nay have some desirable interpretatlon lf these external Judgmenta are not imposed lVillegas L977 a, l98l) I. The invarl-ant prlor dlstrlbutione that produce the invarlant procedures are, of course' typlcally lmproper. Renyi (1970) provides an axiomatic system of probabillty that accomodates inproper prlors. Howeverr not every irnproper prior can lead to coherent inferences. This, J.n our oplnion, is a loglcal obJection to the use of lnvariant or improper prlors. If tt is nathematically convenLent and reasonable to uae Lmproper, countably additLve prlors ln practlcal problems, then, in vlew of Eeath and Sudderthrs (1978, pp. 336-337) Corollary I and Theorem 2, l-t ls necesaary to tho\t that the chosen improper prlors have posterLors shich could also be obtalned from properr flnitely additive priors. If an inproper prLor does not lead to a posterior whLch could have been obtained from a proper, flnltely additive prl-or, then such a prlor results Ln lncoherence and should not be ueed. As alternatives to lnproper prLors, Box and Ttao (1973) and Zellner (1971) consider prlors whlch we call seeningly inproper. That i6, by supposlng that to a eufflclent approxlmatlon the prior follows the form of a partlcular lnproper prlor only over the range of appreciable llkelthood and that it Euitably fa1ls to zero outslde that range' Box and Tiao (1973' P. 21) and also Zellner (1971) ensure that the prlors actually used are proper. The posterior inferences for local-ly uniform, Proper prlors llke those for l-mproper prlors are not coherent unless they could also be obtalned from proper, finltely additlve priors, as lleath and Sudderthrs (1978) and Stonere (1976) regults lndlcate. See aleo Stonefs (L976, pp. L24'125) response to Barnard. Thus, ln the final analyels, the cholce of a prlor should be dependent not ooly on the nature of the argunents used to derlve it but also on the coherence of the posterLor lnference lt leads to. 31 32 SWAMY, CONWAY, AND VON ZUR MUEHLEN flnltely addltive probabillty for a glven sarnpllng model p lf and only lf p and q are conslstent. A rlgorous proof of this result is glven by Lane and Sudderth (1983' pp. 115-116) who have shown that, roughly speaking, p and q are conslstent if they are the condltional dlstrlbutiona corresponding to sone Joint distribution. An analogous result is Kol,rnogorovrs consLstency theorem for countably additive dlstributlons stated in Rao (1973' p. 108). It should be noted that coherence nay provlde a necessary condltlon for reasonable lnference but ls not eufficlent. Lane and Sudderth (1983, p. lf9) ilLustrate this polnt uslng an example where the lnferrer wlth a prior ls capable of drawing coherent lnferences, though he ls'free to be stupid in some clrcumstances." Thls ls analogous to the result that loglcal conslsEency ls a necessary condition for an argument to be true' as pointed out Ln Section 3. l. Hill- (f984) ls of rhe opi-nlon that lt is too restrlctive to always restrLct oneself to proper prlor dlstributlons and that lnproper or flnltely additlve prior dlstrlbutlons may provide satlsfactory approxlnations ln obtainlng our posterlor distrlbutlons. Ile also presents an Lnteresting discusslon of an example showlng that lf a Bayeslao uses the uniform prlor dlstrlbutlon for the parameter 0 ln this example, then hls pogterl-or probabllity for an interval, given any data, ls at Least .95, while given any 0, the frequentlst probabtlity for the interval is very tiny. This ls a real paradox and ls known as the phenonenon of non-conglonerability. The facE that the uniform lnproper prlor dlstribution can be glven a finitely addltive lnterpretation explalns why a non-congl,onerabllity occurs in the above example. This means that nith de FinetLits subJective theory which ln prlnclple does not rule out any flnltely addltive prlor distrlbution, we must Learn to 1lve wlth non-conglonerabil"ity orr ln particular, we must accept that we can have Pr(a physical theoryl data) = 0 for all poselble data, while Pr(a physlcal theory) ) 0. Furthermore, if we only consider measures, an lnference q is coherent FOUNDATIONS OF ECONOMETRICS A rigorous proof of this result appears tn IIil1 (1981) who ends his paper with the followlng conclusion: "For those of us who wlsh to retaLn the subJectlve Bayeslan model for learning and decLsLon-maklng there appear to be three main paths open. Flrst, we can restrict Ehe model io finltletlc appllcations and/or to bounded loss functl-ons and proper prior dlstrlbutlons ln Ehe lnfinite case; second, we can persist lrith conventional improper prlor distrlbutions ln the infinlte case, ignoring inadnisstbiHty (and even exlended inadnisslbility) problems; thlrd, we can develop the ftnitely addltive theory, learnlng to live wlth nonconglonerabllity. The flrst path ls qulte restrictive and nay be unreallstlc even as an approxirnatlon. The second path, at least ln sorne applicatlons, wl11 lead to unnecesgary losses. Are Lhere any real obJectlone to the thlrd path?" 7.4 Brouwerrs and IntuitionLstsr critlcisn of the axion of the excluded-niddle It ls clear from Section 2.2 that, for any proposltlon p which satisfies both the excluded-mtddle and the noncontradLctlon axions, the falslty of the falsity of p impliee the truth of p. That is, slnce an adnlssible statement canoot be both true and false and since lL can only be true or false, then lt is either true or false. If it is not false, then slnce it cannot be falee too, lt must be true. This ls the basls of all of the lndlrect proofs that are used in nathematlcs. ll Brouwer and the lntuitlonlsts would not accept this proof. To see why, conslder the followlng argument which ls borrowed liberally fron Lyndon (1966, pp. 34-35) and Wl1der (L952, pp. 243-244). Inrultiontsm ls a well-developed phllosophical posltion which proposes that nathenatical reasonlng, especlally concerning inflnite sete, be restrlcted to certain conetructLve and intuLtively tnmedlate --Ti- Ar-Cuments in which we prove p by supposlng not-p and showlng that a contradtctlon fol-lows are generally referred to as argunents by leductlo ad absurdun [see Mc0aw]_ey (1981, pp. 27-29)1. 33 SWAMY, CONWAY, AND VON ZUR 34 I,TUEHLEN principle. Ae a very rough exanple, consider the assertion that there exists a natural nurnber n which can be found and shown to have the property P. The fornal negation of the aesertton can be underetood constructlvely only as statlng that there ls a means of showing, for each n' that n does not have the property p. Ivere G a flnite eet of numbers, one could eay it was "intultively clear" that e{ther G contalned a number rtlth Ehe property p or no elenent of G had the property p (the nurnber of Ilowever, elemente in G is unlmPortant as long as it ls finite). if G were infinite, lt could be entlrely conceivable that nelther the assertl-on nor its negation Ls true in thls congtructl-ve sense. 0n thls basls, intultionlsts deny lndirect proofs and thus wl-sh to preclude the uee of excl'uded-mtddle eo as to nake lndirect proofs lmposslble. Heyting subsequently axlomatized the intultlonlstic proposttional or sentential Loglc. Ilis axlooatlzation differs frorn Arigtotellan sententiaL loglc only in the onisslon of the axion of the excluded-mtddle. Additional examples of such an axLomatlc system are those fron which elther frequentlst or Bayeslan statlstical rnodels are derlved Isee Lyndon (1966' pp. 33-34) l. 7.5 Many-valued or fuzzy loFic - relaxing the axion of the excludedaiddle Whlle the earlier discusslon ln Sections 2-6 descrlbed a loglcal syeten obeying Arletotlefs axlon (2) of the excludedmiddle-that for any statement s, "8 or not- a"--' it is lnstructive to consider the consequences of abandonlng that axiom so that statenents can be sonethlng other than true or false. There is a sizeable Llterature ln whlch rules of inference and princlples of truth value assignnents are formulated and lnvestlgated for the aet or aets of truth values under consideration. An excellent survey of thls llterature ls given by HcCawley (1981, Chapter 12). Once we devl-ate fron Arlstotlete axl-om (2)' the conventLonal FOUNDATIONS OF ECONOMETRICS 35 problenatlcal. Thusr "t.ruth valuer" whlle firmly established and thus hard to avoid, is mlsleading because -values" assigned to proposltions need not be TRUTII values .ELE, but can be values for other parameters or combinations of paratsters auch aa probabllitles. Thus, the probabllltles assigned 3o sets belonging to the Borel field of sete ln a sample space in .athematlcal statlstl-cs [Rao (1973, pp. 82-83)] do not always i41y only sure or impossibl-e events. As Lyndon (1964, p. 33) ootes, probablllty theory, in its most prinLtLve form, resembles rany-valued logic in that it attaches to each event a probability taken as a number tn [0, l]. Thus, probabllities can represent degrees of certalnty applytng to unknown factual inforrnatlon. lThere is an lnportant distinctLon between probabllitles as ilegrees of certainty about unestablished facts and probabilities as indices of eurprLse about facts which have become known. I Statisticians treat all nlsslng values of observable varlables lncludlng the unobserved future values of variabl-es as random variables. Therefore, the probablltty that a prediction lnterval rill cover the future value of a varl-able cannot be equal go 0 or I and thus deserves a different va1ue, lnternediate between 0 and I lsee Thel1 (1971, pp. 134-f35)]. Thle probabllity does not hcome 0 or I even after the future value ls real-ized. By contrast, frequentiste refuse to regard (after the experlment) the leyoan-Pearson confidence coefficl-ent as a "probabllity" of correctly covering the true parameter valuer lnstead interpretlng it8 oeaning before the experfunent Ls conducted Ln terns of the .ctua1 frequency of successes in many (not necessarily identlcal) crperLnents as Justifled by the law of Large numbere [see Kiefer (1977b, pp. L73-L74)1. Thus, before the experinent i8 conducted, e confl.dence lnterval ls random and it l-s reasonable for l-te confidence coefflcient to take a value between 0 and 1. Another craqle of "truth values" that do not repreaent truth or falslty ranld be a subJectlve Bayeslan assl-gnment of a real number (greater ihan or equal to 0 and less than or equal to 1) to a propositlon oooenclature becomea 36 SWAMY, CONWAY, AND VON ZUR MUEHLEN to expresa an indj.vidualts degree of belief that thls ProposlElon is true. The statenent that an event occurs wifh probabiUty p or that a varlabLe takes an lnterval of values wiLh probabllity p vlolates the axlom of the excluded-middle unless p ls always elther 0 or 1. Dempster (1968, p. 244) has offered an interpreEatlon apPlytng Jointly to the pair of lnterlocklng concepts' probability and betting. It ls that "probabillties are degrees of certainty applylng to unknown factual informaLlon, while associated betLing rules relaLe to decislon-making appropriate to those degrees of certainty. ....The related use of the lerm EL!g!, as in degrees of belief,... means only Ehat he who adopts any probabillty nusL There ls a sl-mllar have a commLtment to it or bellef in lt. notion of bellef trnpliclt ln factual information, for he who quotes a fact (as chough true) must feel conrniLted to it (though' of course, Lt rnay not be true). Thus, the controversial concept of probablllstl-c knowledge is no dlfferent as regards bellef from the noncontroverslal, concePL of determlnistic knowleilge...'12 If what we know today about the future Ls not a part of our deterministlc knowledge' then our statement about the future cannot properly be ca11ed either true or false (at least, not by our speaking ln the present) and so deserves a different truth value' intermedlate between truth and falsity. Thus' there is need for a many-valued logic concerned ltith the assignment of "values" Lo propositions without regard to whether those values are appropriately ca11ed truth values, although such "values" wlll nornally have sone relation to inference. For exanple, if "values" are degrees of confidence ln propositions, one would be interested in developing rules of lnference such that each rule of inference ytelds conclusions whose confidence leve1 is at least that of assumptions. --iZ- a; FG-e i. (L974, pp. 7O'72) has used "predlctlon" and "prevision" to deoote indivldualsf determlnlstlc and probablListic knowledge, respectively. FOUNDATIONS OF ECONOMETRICS no unique way of extending the notion of valldity the many-valued case, although one may say that a formula is There is to valid tf and only if lt receives the same value t (truth) under all adnlssible asslgnnent.s of truth values to the underl-ying assumptions. Alternatlvel-y, one can a11ow a set of truth values to be "deslgnated" and take a formula to be valid lf and only lf 1t recelves a designated truth value under all admissible assignments of truth values to the assunptlons. McCawley (1981, p. 368) discusses the important question of what it means to say that a systen of rules of lnference and a set of condltlons on truth The criterion of fit in the two-valued value assignrnent "fit.' case (truth and false) consldered, for exarnple, in Sectlon 2 was that, for any admisslble asslgnnent of Eruth values, the conclusions that can be lnferred fron assumptions that are true (l-n that asslgnment) are also true. This crlterlon, in conJunction sith the speclflc rules of inference of Section 2, imposed very strong constrainLs on how truth values could be assigned to complex propositlons. The constralnts were so strong largely because there were only two truth values. I{owever, if truth values can be any number from 0 to 1 (where 0 corresponds to 'falsityr" 1 to "truthr" and interrnediate val,ues to varlous greater and lesser degrees of certal.nty), a less stringent criterion of fit nay be admitted. A system with this range of truth vaLues is useful in coplng wlth Lnexact concepts enumerated in UcCawley (1981, pp. 360-394), such as APPROXII{ATION discussed in Sectlon 4.2. In de Flnettirs (1974) Bayesian theory, ltself a rurny-valued loglc, the condition of coherency serves the same functLon as the criteria of flt ln the verslons of fuzzy logic covered by McCawley (1981). To produce rules of inference it fuzzy logLc, the prlnciple of "semantLc conpleteness" has been lnvoked [McCawley (1981, p. 369)1. It requires that statements be provable if they are valid ln the nany-valued sense. Clear1y, any system of rules of l-nference designed to raake a ftzzy logic semanttcally complete 37 38 SWAMY, CONWAY, AND VON ZUR MUEHLEN nu6t neceasarily differ from che rules of Sectlon 2. I{ere it must be enphasized that 7rr frs.zzy 1ogic, concLusl-ons are restrLcted to the status of belng only g!_fgg.11 as certain as at least one of the assumptions. This llnltation ls the price we Pay for relaxing Aristotlete axLom of the excl.uded-midd1e. As we potnted out l-n footnote 5, the frequency or subJective l-nterpretation of probablllty violates the axlom of the excludedniddLe. Our Exanple 5 glves a nodel whi.ch, when conblned with the frequency interpretatl-on of probabllity, rejects the axlom of the excluded-middle but ls not lnpllclEly constructed on the basis of the acceptance of that axlom. Surely, this nodel gives conclusions which hold wlth probabillty less than I in srna11 samples, but doee not mean that we should reject probablllty theory. These concLusl-ons may hold wlth probability I ln sufflclently large samples. 7.6 Blrnbaunrs (1977) evldential interpretation vs. the llkellhood princlple or Bayes proceduree Inductlon based upon a statistical analysis of a model can easlly yleld false concluslons from true premlses. A11 we can do is try to conatruct our l-nductive arguments ln a way that lti11 rnl-ninize the chances of obtainlng false concluslons from true premises. Most statistical nethods applled to research data for this purpose have been given their systematic nathernatical Justlflcation ltithln the Neyrnan-Pearson theory. In thls theory the nu11 and alternatlve statlsttcal hypotheses whlch are speclfied by the values of a parameter and which nay be "reJected" or "accepted" on the basls of a testlng procedure can be assocl-ated nlth the reepective "decisions" appearing in the formal rnodel' of a decision problem. The slnplest nodels of declslon probleros may be descrlbed in terms of schenata of the followlng forn: FOUNDATIONS OF ECONOMETRICS 39 Slmple hypotheses: IIg Posslble declslons: d1 [=re Error probabillties: It1 ject 116l c = Pr[d1lltgl d2[= do not reJect ltgl B = Pr [d2lttl l tern "reject" expresses here an lnterpretatlon of the statlstlcal evidence, as glvlng appreclable, but linlted support to one statietical hypothesis agal-nst another. As Birnbaurn rightly points out, statLstical evidence is adequately The declsl-on-llke represented noc by d1 and d2 but by synboJ.s 1lke dl: (reJect 116 in favor of Il1, c, B) dit (t"J."t Ht in favor of H6, o, B). and p. 23) uses the term "evidential interpretatlon" of the decislon concept to refer to such applicatione of models of decision problerns and uses the term "confldence concept" of statistlcal evidence to refer to euch Lnterpretations of statistical evidence. A definttion of Birnbaumrs "confl-dence concept" is as follons. The confidence concept: A concept of statistical evidence ls not plausible unless it finds "strong evidence for H1 agalnst E6'with snal1 probabiilty when IIg is true, and with rnuch larger probabillty (l-B) when II1 is true [Birnbaum (Lg77, p. 24)].13 In evidentl-al lnterpretation, the result of a test is not llteral1y an action, but ls lnetead a relative evidential- evaluatlon of tno or nore hypotheses. For exanple, Birnbaum (1977, Birnbaurn (1,977, -i5-Teyoan and Pearson suggest that a test procedure be selected guch that B ls as sma11 as possible for a gLven value of c. The probabllity I tends to 0 as the sanple size tends to € if the test procedure is consLsteIrt. It is not possible to reduce both s and B to zero. For an intultlve explanatl-on of Birnbaunrs confidence concept, see Salmon (L973, pp. 105-f12). SWAMY, CONWAY, AND VON ZUR MUEHLEN (reject H6 ln favor of Il1' p. 25) interprets the decislon 'li: .01, .2) as very strong (but inconclusive) evidence for E1 against llgr and interprets the declslon df: (reJect II9 ln favor of IIl, .5, .5) as rtorthless evidence. The latter declsion Ls no more useful than ls the result of a toss of a fair coin, since the error probabllitles (.5, .5) also represent the experlment of tossing a fair coin, with one side labeled "reject Hg" and the other "reJect II1." Consequentl-y, the exact vaLues of c and F go a long way toward interpretlng statistlcal evidence' Althoughtyptcalapplicationsofstandardstallsticalnethods in econometric research report decislons of these types' the exact evaluation of o and B is exEremely difficult, lf not imposslble, in several settings. For this reason' the probabilttles c and E play thelr basic roles largely lnpllcitly and unsystematlcally in guiding applications and interpretatlons of econonetrlc evidence. The failure in currenE econometrlc practlce of not reportlng the values of boEh a and B Leaves much to be desired in achleving the fu11 discLoeure needed to nake a study convincl-ng to a crLtlcal reader and the palatabiltty needed to reach a casual reader. Blrnbaun (1977, pp. 26-27) Itarns that statistlcal evldence is one among several basee for validating scientiflc conclusions. The concluslons of a sclentific investigatlon ehould be based not only on statlstical evidence of eufficiert strength concernlng the appropriate statistical hypotheses, but also on (1) the adequacy of the employed (sufflcienE and logica11y consistent) nodel to represent the empirical situation ln ftnportant respects and (2) the coupatibillty with other knowledge and evidence of a conclusion that ls supported by statlstlcal evldence provided by the current lnvestigation. The nain difficulty is that the problen of providing avallable statlstical evidence of the form al or al on the hypothesle of interest when thls evidence le nlxed up wlth those on nuigance parameters is indeed complex lsee Basu (L977)1. The reason ls that a nuisance paraneter may eubstantially FOUNDATIONS OF ECONOMETRICS accuracy with which one can evaluate o and 0 for the hypothesis of interest. [For an incl"sive discusslon on this problern, see Kieferrs (L977a, pp. 824-825) reply to Denpsterrs affect the comments. ] dlfficulty which should also be stressed here ls that Lhe values of a and B may depend not only on the evidence but also on the path by which it was reached. Put differently, o and B may depend iodividually on Lhe pariicular long run [Cornfteld (1970, pp 4-6)1. In view of rhis dependence, Cornfleld (1970' p.6) argues Lhat ooe cannot always equate the concept of choosing between hypotheses on the basis of evidence and the concepi of reJectlng a hypothesl-s at a fixed a with minimum B but one shoul,l use the Savage and Lindley Bayeslan rule [Cornfield (1970' pp. l5-16)l whlch is necessary and sufftcient to minilnize a llnear compound a + lB, no matter what the long run is. The Another posit.ive quantity tr can then be interpreted as the weight one nould be willing to assign to B as compared to ct. The SavageLindley argument depends on the notion that L\to test procedures, one of which has a srnaller a and the other a snaller B may be equally acceptable. Ilowever, a criticlso of the Savage-Lindley argumenE stas given by Birnbaum (1977 pp. 37-39). CruciaL to this criticism ' is the distinction beEween the evidentiaL interpretations defined above and the following behavioral inEerpretations of the confidence concept. If the result of a test ls not a statement about a hypothesis but a decision to lake some particular course of action, then Birnbaum says that this result irnplles a behavloral inLerpretatlon of the confidence concePt. The Savage-Llndley argument concerns JudgrrnenLs of preference or else lndifference (equivalence) betrteen alLernatlve decislon functlons (tests) in problems of two simple hypotheses, with each declsion functlon represented by a polnt p = (ct, 8) in the unlt square, determined by lts error probabilitiee c and B. Birnbaum argues that in some situatlons he would strongly prefer to u8e a Eest characterized SWAMY, CONWAY, AND VON ZUR MUEHLEN by (.05, .05) rather than one characLerized by (.1,0), although lndiffereot between (.1, 0) and (0, .I), i.e., that (.05, .05) > (.1,0) - (0, .l). This paLtern of preferences ls incompatible with the assurtrpcion of the Savage-Lindley argument that the equivalence of sets in the unlt square deterrnined by error probabilities (c,B) ie invariant under probabilistic mixing. Thls assumptlon holds only for "decislons" under the behavioral interpretations, but not under the evidential lnterpretations which constitute the Neynen-pg3r'son statistical practice. Thus, Blrobaum does not belleve that a 50-50 rnixture of (say) (0, .1) and (.1, 0) error probabilities will have the same evidential meaning for al-1 practltioners as the (.05, .05) error probabillties, but this may not be relevanL to the consistency with which they would bet, lf forced Eo do so, based on such nunbers. They do not necessarily achleve a utility that is linear in (c,B). The polnt which Birnbaum ls making here is that while behavl-ora1 lnterpretatLons of "decislons" nay play a very valuable heuristic role ln the nathematical developments of the Neynan-Pearson theorles, statlstlcal nethods based on those theories must be lnterpreted etiLh care when considered for posslble use with evidential loterpretaEions. This ls particularly true when the utilltles of individuals are nonlinear. Such seens to have been Blrnbaumts later view. IIis earlier viev was that the process of interpretaElon of experinental evidence could be reduced to that of lnterpretlng the likelihood function. Birnbaunrs earller argument Lakes as the rnodel of an experlment E the triplet (n, S, p) where S = {y} is the (discrete) sanple space, CI = {0} is the parameLer space, and p = p(y,0) is the probability functlon of |, for each 0. Then he takes the palr (E,y), when y is observed, Lo coostitute a uodel of statlstical evldence. The statlgttcs h(Y), t(y) are defLned, respeetiveLy, to be (a) g!g!$g., or (b) sufflcienr if (a) p(y,O) = g(h) p(yltr,O), where g(h) does not depend on 0,0G0, or (b) p(y,O) = g(t,e) p(ylt), where p(ylr) does nor depend on 0, 0€fi. Then some of the axioms of statistical evldence consldered are he would be FOUNDATIONS OF ECONOMETRICS CoNDITIoNALITY PRINCIPLE (Cp): If h(y) is ancillary, rhen Ev(E,y) = Ev(85,y) where h = h(y), Ev(E,y) denores rhe evidence deternined by an outcone y of the experiment E and E5 = (e, 56, ph) with S5 = {y:h(y) = h} and p5 = p(ylh,o). LIKELIHOOD PRINCIPLE (Lp): If, for some c)0, p(y,O) = cp*(y*,0) for all 0 in 0, then Ev(E,y) = Ev(E*,y*). MATIIEMATICAL EQUMLENCE (ME): If p(y,O) = p(y*,e) f or all I 0, then Ev(E,y) = Ey(f,,y*). Wlth these formulations Birnbaum proved that Cp and ME Jotntly tnply LP [Barnard and Godambe (1982, p. 1035)]. In words, rhe Lp says that the ltkeLihood function for the data that happen to occur Ls alone an adequate descriptlon of an experlnent wLthout any statement of the probablllty that thls or another ltkelihood functlon would arise under various values of 0. Birnbaun also considers the extensions of the above argument (which ls malnly concerned lrlth the case where the posslble observations form a discrete set) to the continuous case. Ilere it is polnted out that the WEAK SUFFICIENCY PRINCIPLE [If, for some c)0, p(y,O) = cp(y*,O) for a1-1 0 ln 0, then Ev(E,y) = Ev(E, y*)l rnay be needed Eo replace ME in the derivation of LP fron CP. However, some conpllcations rnay arise from the llmited degree of arbitrarlness that exists ln the choice of a denslty functlon. The LP ls very controversLal. Crltics havg pointed out that in practice specific statistical nodels can never be who11y trusted so that a statistlc sufflclent on the hypothesls of a gl-ven rnodells not sufficient under the wider hypothesls that that rnodel may oot actually obtaln. A naJor crlticlsm of the Lp cane fron Birnbaun hinself. Specifically, he recognized that the Lp is inconpatible with hLs confLdence concept--under no 0 sha11 there be high probablltty of outcomes lnterpreted as "strong evidence aSainst 0"--deflned earlier. Any adequate concept of statistlcal evldence must meet at least certain rnlnlmal verslons of both the LP and the confidence concept, but it has become clear to Birnbaurn that no such concept of statistical evLdence can exlst. This does SWAMY, CONWAY, AND VON ZUR MUEHLEN not mean that adequate concepts could not be found in speciflc cases, chough Birnbaum thought that the domain to whlch such concepEs could be applied woul-d far from cover the rnaJor aPpll-catlons of statlstlcal- rnethodl for further dlscusslon, see Barnard and Godarnbe (1982) who give an exampLe where the ltkelthood function provides nlsleadtng evidence wlth probabtllty l. Thts type of likelthood functl-ons also arise ln the context of some econonetrLc nodels ar.a1-yzed by Maddala (1983, p. 300) and others. Furthermore, "the recording of the llkelihood function ls not an answer, 8lnce l-n a subsequent decilt begs the question of gqlong@ sion or stage of lnvestigation to be determined frorn the data"' pointed out by Kiefer (L977a, p. 823). The problem of generallzing the LP so that the generallzed version avoids some of the dtfficulties mentloned above is taken up by Berger and Wolpert (1984). Ilowever, Lane (1984) ln his dLscussion of Berger and Wo1-pertts (1984) work raises the important issue of the neaning of 0 ln the (0, S,p)-paradign. According to Lane, there are at leasE Ehree possible lnterpretations of the elenents of o: (a) e ts the distrtbutton p; (b) fl l-s an abstract 6et and 0 rnerely indexes the distributlon p; (c) 0 ls a possi-b1e value for some "rea1" physical Parameter' and p l-s to be regarded as the digtrlbutlon of the random quantity 9 should 0 be Ehe true value of that parameter. InterpretatLon (c) ralses the difficult phlloaophlcal questlon: when--and in what sense--do "rea1" physlcai, pararleters exlet. More l-mportantly, as Lane points out, depending on whether one adopts interpretatLon (a), (b), or (c), the (generalized) LP is devoid of lnterestlng consequences, wrong or severely and amblguously restrLcted in ics domain of appllcabtltty. In their reply to Laners comDents, Berger and Wolpert (1984) adrnlt that while the (generaltzed) LP will apply under l-nterpretatlon (c), tt also applles when 0 ie onJ-y defined by sone aspect of the experiment. The fundanental dlfflculty is that the information l-n (0, S, p) says nothing about how the rnodel represents reall-ty, and it ts hard to aee how FOUNDATIONS OF ECONOMETRICS a prlnclple of lnference can disregard the details of this representatlon. Slnce the (generallzed) LP lgnores alL euch details, Lane belleves that lt ls inadequate. IIe says that not enough lnformatLon ls encoded ln (O, S, p) upon whlch to base a general princlple of lnference. Lane further shows the irrelevance of the (0, S, p)-paradlgrn by argulng that the real aln of Lnference is usuaLly not to make statements about the "true" value of an unobservable parameter 0 on the basls of an observed quantlty y, hlt to generate a predictlon about the value of some future observables and the (generalized) LP does not address the question of how to generate such a predictlon directly. Another reason rhy Lane thlnks that the (0, S, p)-paradigm is lrrelevant ls that especially in applications arising in nonexperinental scLencee like econometrics, the nodel Ls sculptured either fron data already in hand or perhaps fron a realistic view of what data are potentlally available. In such cases, there is no rray to separate nhat (E, y) Eays about 0 from "prlor" information about e. When the "parameter" dependa upon the experLment for its existence and oeaning, the (generalized) LP does not apply and the (o, S, p)paradigrn is lrrelevant. For all these reasons we shoul-d not take the LP very serl-ously unless rle know lte are in a sltuatLon where interpretation (c) is approPrlate. The Bayeslan method applies under all lnterPretations of 0. Sl-nce the appeal of the confidence concept is undeniabl,e, we now lnvestigate whether a Bayesian nethod can be developed which is conslstent wlth that concep!. In hls conmentary on Birnbaurnfs (L977) paper, Llndley (1977) conatructs one example showing that the descrlptlon of evidence in the forn of al or al is unsatisfactory and another exanple showing that the particular preference pattern for teets [(.05' .05) > (.1' 0) - (0, .l)] nentioned by Birnbaum amounts to preferring evidence that uses the toss of a coin whose reeult ls irrelevanE to 116 or E1 to that whlch does not. Accordlng to Ltndley (L977), the meanlng Bl-rnbaun attaches to evLdence is not as satisfactory as that based on the LP and the SWAMY, CONWAY, AND VON ZUR MUEHLEN 46 of Bayesf theoren. follows from the Bayesian argument and an important property of the llkellhood functlon ls that lt does not depend on which long run rre conslder the result as ernbedded ln [see Cornfield (1970, pp. f3-15)]. However, since every posterior distribution, glven by Bayeer theorem, depends on a likelihood functLon, the former also gives rnleleadl-ng evidence whenever Che latter does, depending oo the cholce of prior. In fact, thle statement agrees rrith the conclusion of Lindleyrs (1977, p. 58) shorE derivation, the purpose of which is to ehow that the interpretation of a large value of the posterior odde ratlo in favor of II1 relatlve to 116 as an evidence agalnst IIg if the prlor odds ratio agreea wlth Birnbaunts g!!!g_ggg.9. l-n favor of H1 relative to It6 ls sufftclently snall. Consequently, the posterlor odds ratl-os based on arbltrary prior odds ratl-os may give nisleading evidence. As Kiefer (I977b, pp. L74-L75) notes, the basic problen ls that "statistlcs ls too couplex to be codlfled ln terme of a slmple prescriptlon that is a panacea for all Bettlngs, and... one mu6t look as careful-ly as posslble at a varlety of possible procedures..." In view of Kyburgrs (1983, pp. 91-93) conpelling argumenls lsee Appendlx to the paperl, it is difficult to show that a partlcular prlor distrtbutlon represents Eomeonets degrees of beltef; but fron the well establlshed results that under sultable regularity condltlone the Bayesian procedurea and adnisslble procedures coincide, lt appears reaeonable to conclude, aa Wolfowltz [Kiefer (L977b, pp. 17t-L73)] doee, that the use of a prlor distrtbutlon reflects, at the very least, a statistlclanrs ability to plck an adnissible or a preferred 1ocal1y coherent procedure fron thoee available or hie truth value assignnent. Coneequently, lt does not seem unreaeonable to us that one ghould make a tentatlve firet step at selecting a procedure ln accordance with Klefer'e (1977b, p. 171) euggestion: Flret, chooge a prior dlatrlbution that reflects eonethlng of what one would regard as a desirable rlsk functlon (operating characteristlc), deflned in use The LP FOUNDATIONS OF ECONOMETRICS of the relatlve importance of the risk values at various states of nature. Then compare the rlsk function of the resulting Bayes procedure with those of other candj.dates to make sure that lt does not have a subadnisstbllity defect. For an appllcation, terms see Swany and Tinsley (1980) and Swamy and Mehta (1983b) who have attempted to follorr this suggestion ln econonetrlc contexts. 8. of Conclusions The theory and practLce of econometrics draws on Dany sources knowledge lncludlng economic theory, statlstlcal theory, and data. Often consideratLons describing lnstitutlonal or psychological aspects of a problem must be entered. The intersection of all these sources forns a gray area which, as we have descrlbed in the preceding pages, defles the established rlgors of Aristotl-ers principles of logic. The foundations of econometrics economic are weaker than those so1e1y supporEing econonic Eheory under certainty. Thus, while Aristotelian principles may be applied to determine the loglcal valldlty of economlc theory under certainty, ihe truth of its conclusions so constructed cannot be determined slnply by inspectlon of empirical results for two reasons: first, the problem of induction (as defined by Boland) is lnpossible to solve and, second, the uncertainty theorles and approximations comonly adopted in econometrics violate Arlstotle's axiom of the excluded-middle. To our eventual dislllusionment we have found ihat econometricians cannot establlsh the truth or falsehood of economl-c theorles unless they are 1oglca11y inconslstent, in which case they are false. There are, oevertheless, real conirlbutlons to be rnade. Econometricians can develop sufficient and 1ogica11y consistent theorles and lmpose them on thelr enpirlca1 nodels. Care should be taken to ensure that the behavloral assuuptions of economic theory and the statlstlcal assumptlons nade do not contradict each other. Many-valued logic, whlch ls reaker than Aristotellan 1ogic, provldes a basls for the anaLysis 47 SWAMY, CONWAY, AND VON ZUR MUEHLEN models. Probability theory, in lts nost prlnltive form, resembles many-valued logic. lle have also noted the key lmportance of interpretatlons of probabllity in providing the modus operandL for fitting a sufficient and 1ogical1y conslsten! model to glven data. Rules of lnference that lead to "senantlcally compLete" many-valued logic in econometrlcs, thereby ensuring that nhat is provable is exactly what is va11d, ought to fit a set of condltions regardlng truth value asslgnments. Our maln recomrnendation for achleving such senantic completeness is to fo11ow Kieferrs advice and choose a prlor distrlbution (a truth value assignnent) that reflects aomething of what one would regard as a deslrable risk functlon and then compare the risk function of the resulting Bayes procedure with those of other candidates to make sure that it does not have a subadmlssibllity defect. Sueh a prescription would, in our view, produce 1oca11y (wlthin the same model) coherent inferences. I{owever, in ltght of G'ddelrs incompleteness theorem' g1oba1 coherence or consLstency among all possible cornpeting models is not possible. To facilitate cholce among competing nodels' we suggest the following maxim: When a given 1ogically consistent model produces better predictlons than any other logica11y consistent rnodel, a researcher may favor Lt over less successful cornpetitors. At the same tlme, the researcher should recognize thaE because of uncertalnty, the currently successful performance of a given nodel nay prove as evanescent as the morning dew. Thls w111 requlre constant vigil-: one must always be prepared to check the predictlons of a model against those of its competltors' new and o1d, provlded the model and iis competitors are logica11y consistent. of such Appendix Interpretations of Probabllity 1. and of probabllity lnterpretations Indeed, the meaning of a crlterion of fttting a sufflclent 1oglca11y consistent model to glven data and even the con- ftnportance FOUNDATIONS OF ECONOMETRICS sequences of a fitting criterion will depend fundanentally on the philosophical interpretation which is glven to the concept of probability. Furthermore, there are interpretaEions of "probability" under whlch lt makes sense to look for a Justificatlon of a fittlng crlterl-on, while there are other inferpretatlons under shich a fitting criterion may be regarded as a part of the definttion of "rationalityr" and sti11 other interpretations under rrhlch a criterlon l-s not always applicable. 2. Alternative rneanings of the word "probabl1lty" The word "probablllty" has been used in various senses by a nunber of writers [see Good (1983, Chapter 6)]. To quote Good (1962, p. 487), "Ic]he most inportant distinction ls that between physical, material, or lntrlnsic probabilities, chances, or propensities on the one hand, and non-physical or intuitive probabilities on the other... Intuitive probabilities can be subdivided into credibilities = loglcal probabllities = unlque rational degrees of be1lef or inEensltles of convlction on the one hand, and on the other hand subJective or personal degrees of bellef to nhich some canons of conslstency, honesty, and maturity have been applled, in which case they are cal1ed subjectlve or personal probabllities. " Probabilttles in Ehese senses can be covered as special cases of two fairl,y standard interpretaEions of probabllity: one is an empirical-frequency and the other ls a subJectivistlc interpretaiion. BoEh of these are falrly well-known, each of them havlng been clalmed by advocaEes to hold the key to the problem of decl-sLon-rnakLng under situations of uncertalnty. Let us consider ihen in turn. The Frequency Interpretation: Probability statenents are statistical hypotheses about the relative frequencles of occurrence of events, subJect to confirmation and disconfirmation. They are offen unknown. The arguments of the probablllty functlon (what we refer to when we speak of the "probability of") are certaln classes of events. st^,AMY, CoNWAY, AND VON ZUR MUEHLEN Interpretatlon: Probability statements express personal opinions or subJective degrees of belief, and are open to potentlal modiflcatlon so as to conform to Lhe rules of the probability ca1cu1us. They can in no event be unknown. The argumenis of the probablllty function are usually taken to be statements, but they nay also be taken to be events.14 The SubJectivistlc 3. Differences and similarities of results based on different vlews While there are surprislng cases of agreenent among the various points of view, there are also many cases of disagreenent.l5 For example, the subJectivistst approach to statlstl.cal lnference is essentially Bayesian; but it may, in view of a connection between the frequentistic criLerion of adnissibility and fhe subjectivistic coodition of coherency polnted out by Ei11 (1975, pp. 556-557), Lead to neEhods formally ldent.l-ca1 wlth methods put forward from a frequentist point of view.15 As pointed out by T-so-me personalists, taking the betElng background rnore serlously, had doubts about lncluding ln the dornain of che probability function statements whose truth or falsity cannot be settled ln a predlctable flnite length of time, see Kyburg (1983, pp. 23-24). 15 As we shal1 see below, a frequency interpretatlon of probablltty ls valld under the conditlons of a law of large numbers. But any assertion of the "appropriateness" of the conditlons of a law of large numbers is itself, lnescapably, an act of personal Judgnent. On the other hand, ln the subJectlvlst approach, the nixing meaaure p on [0,1] 1n terms of whlch the law of exchangeable dichotomous trials can be represented as a mi-xEure of lndependently and ldentlcally distributed Bernoulll processes, can be lnterpreLed as a bellef about long-run relative frequency. Thus, subJective Judgnents as well as the notion of long-run relatlve frequency enter Lnto both frequentist aod subJectivist approaches. 16 Rigorous statementa of this connectioo are given by Lehrnann (1983, p. 263) and Heath and Sudderth (1978). Speciflcally, Lehrnann has presented a simple proof to show that any unique Bayes estimator ls adnlssible, and lleath and Sudderth prove that lf a bounded loss functlon ls speclfled, then a decision rule is extended adnisslble (1.e., not uniformly domlnated) if and only 1f lt ts Bayee for some flnitely addltive prior. [We have already stated lleath and Sudderthrs (feZAl result which establlshes the close connectlon between the coherent lnferences and flnitely additlve prlors. l FOUNDATIONS OF ECONOMETRICS 51 Efron (1978), it is also true that while statistics, by R.A. Fisherts deflnitlon, is lnterested ln sumnary statements about large populations of objects, frequentlsts and Bayesians have produced fundamentally dlfferent answers to the basic queeti.on concernLng the cholce of sunmary statements most relevant to drawing inferences from data. 4. Conditions for the relevance of frequency interpretatLon The frequency lnterpretatlon is most prevalent. It takes probabllity statements to be ernplrical statements that describe lhe relative frequencles of occurrence of certain classes of objecEs or events encountered in the world and does noE require genuinely repetitive situations because the 1aw of large numbers In can also hold for nonidentical (and even dependent) trials. where of law of large numbers do any sltuaLlon the condltlons a aoL hold (such as the variable that Eakes only one value), the frequency interpretation of probabtlity nay be lncorrect. Even in those situations where the condltions of a 1aw of large numbers do ho1d, we may not knon exactly the lLmltlng frequencles ln infinlte reference classes; thus our probabllity statements entall assertlons about unknown rel-ative frequencies. Thls neans t.hat probabillcy statements can at best be approxlmatlons to long-run relative frequencies, and hence ArlsE.ote1lan logic does not apply under a frequeney interpretation. The many-valued loglc applies but requires the condltions of a law of large numbers and a sufficientl-y large sanple to make the frequentist procedures senantlcally conplete. [For a useful survey of the frequentistic approaches to infereoce, see Efron (1982). I The c and B in Elrnbaumrs evidential lnterpretatlon presented ln the preceding section denotes frequentist probabllity. 5. Condltions for the relevance of subjectlve interpretations Whlle the frequency interpretatlon posslbly conveys something obJective about events, the subJectlvlstic lnterpretation is baslc- 52 SWAMY, CONWAY, AND VON ZUR MUEHLEN ally psychologlcal. The representatlon of degrees of individualst beliefs by probability statements ls fundamental to the latter interpretation. Jeffreys and hls followers assume that unique rational degrees of bellef exlst common to all rational mlnds given the same factual lnforrnation. Good (1962) ca11s these unique rational degrees of beltef crediblltries' a term whlch we have already introduced. By eonErast, L.J. Savage, de Flnetti and thelr fo110wers believe that different lndlviduals can have different degrees of belief for the same event even when they are glven the sane factual inforrnatlon. But these indivldually varying subJective or personal degrees of belief are, in a sense abstractl-ons. They are not necessarlly the actual degrees of belLef of a llving individual but a rnodified version that satlsfies de Flnetti's conditl-on of coherency. To illustrate' suppose that an lndlvidual assesses dlrectly the probabllity Pi of each event A1 in a partltlon of the universe, and then discovers that [iP1 = P>1. If he were forced co bet simultaneousLy on each Ai occurring' paying P1 for the chance to rtln a unit if A1 occurs, and winning nothlng if A1 does not occur' then no matter which A1 actually does occur, he would pay P unlts and receive one unit, and thus be a sure 1oser. Using this example very effectively, Ilill (1975, p. 557) argues that a rnethod of evaluatlng probabilities which would make a llvlng human being a sure loser if he or she had to act upon those probablllties (even hypothetically) would seero suspicious and untrustworthy for any Purposes whatsoever, tncluding casual thought, and lt wou1d seem deslrable to remove such incoherencles sherever possible. The theorems of total probabillty and compound probability are only the lmrnediaLe coroLlarl-es of the condition of coherency; that is, the nunbers representing degrees of coherent bellef must satisfy a1-1 the axloms of the probabil-1ty calculus. While we are persuaded by llilLts (1975, p. 557) demonstration that incoherence is symptomatlc of a "basically unsound" attitude, there ls no evidence that indlvldualsr degreee of be1lef FOUNDATIONS OF ECONOMETRICS do in fact satisfy the probablllty calculus, and, to the contrary' conslderable evldence has been presented by Kyburg (t983' pp. 91-93) to the effect that they do not. That indlvldualsr bellefs foLlows from the -y oot conform to probablllty calculus also study eonducted by Tversky (L974) who descrlbes three heurlstics, or mental operatlons, that are enployed ln Judgment under uncer- heuristics are not dlscarded even though they occaslonaLly lead to errors. Tversky exhlbits the failure of both laynen (untutored in the laws of probablltty) and experts to iofer from life-1ong experl-ence fundamental statlstical rules such as the role of prior probablllty or Lhe effect of earnple size on sampling varlab111ty.17 This nay lead to a probl-en ln Bayeslan inference. If we suppose a ful-1 preference ranking aEong acts there are two posslbllitles. Elther the preference ranklng ls coherent, or lt ls not. If lt ls coherent, we are al-1 set. If tt is not' then somethtng must be changed; unfortunately' subJectlvlstlc theory wil-1 not te11 us erhat to change. As Kyburg (1983, pp. 81-85) polnts out' there may indeed be a rough lntuitLve connectlon between an lndividualrs degree of belief in a starement S and the least odds he or she is willing to offer ln a bet on S. But this connectlon is much too loose to generate by itself a set of numbers confornlng to the probabillty calculus. The Dutch Book argument cl-ted by Kyburgr glves excellent reasons for adopting a table of odds or publishlng a list of preferences which conform to the baslc axloms of probabl1lty, but coherence ls inposed at the coet of destroylng the lmedlate and intuitlve connectLon between odds and degrees of belief that the argument orlglnally depend on. Indeed, at thls point, we may fl-nd ourselves wonderlng lf there ls such a thlng as "degree of be11ef." Thts skepticism is also Justified by the fact that L.J. Savage uses two "structure" axloms which are exlstential ln character, ln conJunction wlth five "rationality' tainty. These reason, Lindley (7974, p. 181) thinks that a thor-I-F;Eis ough drilling in the prlnclples of maxlrnun expected utlllty ln the Laissez-faire schoolroons of today would not be anlss. 53 swAMY, CONI^JAY, AND VON ZUR MUEHLEN 54 to prove the exlstence of a unique probabillty distributlon on states of nature [see Suppes (1974)]. Thus, "sorne of Savagets axloms do not in any dlrect sense represent rationality that should be satlsfied by an ldeally ratlonal person but, rather they represent structural assumptlons about the environment that may or nay not be satlsfled ln glven appLlcations" [Suppes (l-974, p. ]'62)l' It certalnly follovs fron the rlgorous derivatlon presented in DeGroot (1970, pp. 70-82) that a subJectlvlst cannot asslgn a unique subJective probablllty to each event in an unamblguous nanner if he ls unable to lmaglne an ideal auxillary experlment in which a unlformly dlstributed random varlable X can be generated, and if he is unable to compare the reLattve likellhood of any event whl-ch was origLnally of lnterest to hin with that of some other event of the forn {XeI = an lnterval}. Consequently' if the distributions conatructed ln DeGrootrs roanner are construed as yieldlng subJective probabillties--i.e., degrees of bellef--this means that subjective probability asslgnments do not requl-re a unique degree-of-be1lef functlon.18 Thus, approxlmatlons are necessary to represent someoners oplnlons by a distribution, thereby ruling out any apPlicatlon of Aristotellan loglc to this axioms sl.tuation. frequentLst v8. posterior probabllity Once we have a sufflclent and 1ogica11y conslstent sanpllng model and a coherent prior, the derlvation of a posterlor distributlon, lf lt exlsts, can be stralghtforward [Ze11ner (1971)]. So we sklp this derivation and proceed to point out the dlstlnction between the frequentlst probabillty and posterlor probabillty' since we have made rnuch of this dlstlnctlon in Example 3 and in 6. On T'-Tt ;froffi-be noted that de Flnetti only recommends the use of finitely addltlve prlors, whereas I{111 (1976' p. 1000) Justifles the use of countably additlve prlor distrlbution on the basis of the usefulness of the approxlmations to whlch one ls led by means of the latter distrlbutlons. However, a countably addltlve distrlbution nay provlde an adequate approxinatlon to more than one degree-of-bellef function, thus destroylng a unlque connectlon between prior dlstrlbutlons and degrees of bellef. FOUNDATIONS OF ECONOMETRICS 55 the beginning of thls Appendix. As lndlcated by Kiefer (L977b, p. 774), frequentlst probabi.l-ity (e.g., that a confidence lnterval w111 by chance cover the true parameter value) refers to a chance experlnent yet to be conducted, whl1e posterior probability (re1ative to even a physical prlor 1aw and given some observed value of the varlable under conslderation, say Y = 3.17) does not. The fact that one may bet in a siurilar fashion in the two circumstances does not alter this dlstlnctlon because the nature of bets based on confidence coefficlents ls dlfferent fron the nature of bets based on posterlor probabllities. For a frequentist, it ls the posterlor probablllty given the yet-to-be-observed Y that is descrlbed in terms of an experlnent etll-1 to be conductedr gnd shose 1aw of large nurnbers behavior Justlfles the way he uses it when Y = 3.17. Of course, subJectlvtsts requlre no such distlnction. ACKNOWLEDGEI'{ENTS in this paper are those of the authors and reflect the views of the Board of Governors or the staff of the Federal Reserve System, nor do they reflect the vl-ews of the Departnent of Commerce. For thelr editorlal assistance and helpful comments, the authorg are grateful lndeed to James Barth, L.A. Boland, John Craven, Clark Edwards, Richard llaldacher, Charles llallahan, Richard Heifner, Dennl-s EenLgan, Anl1 Kashyap, Judlth Lathan, Hichael LeBlanc, DennLs Llndley, Thonas Lutton, Jitendar Mann, Darrel Parke, Dale Poirier, Paul Prentice, Lloyd Vlews expressed do not Teigen and Arnold Ze11ner. The authors are especially grateful also to Michael Weiss for hls rnetlculoue revlew of our paper and for lntroduclng us to ftzzy set theory arrd f'tzzy loglc. Thanks are due to Nadlne Loften, ERS, USDA and Sharon Sherbert, Speclal Studies, FRB, for thelr consclentious preparation of the manuacript. REFEPJNCES A., (1976). Comment on "Strong Inconslstency Fron Uniforn Priors" by Stone. J. Amer. Statist. Assoc., 7L, L2L-L22. Barnard, G. SWAUY, CONWAY, AND VON ZUR MUEHLEN 56 Barnard, G. A. & Godambe, V. P.' (1982). Memorial article, Allan Birnbaon L923-L976. Ann. Statl6t., 10, f033-1039. Barnett W. A., (1983). New lndlces of money supply and the flexldernand system. J. Bus. & Econ. 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At the outsel f wish to note, perhaps for obvious r€ssonsr that I accepl alnost everything SCM discuss concernj.ng Arlstotlefs axions for adnlsslbility of statenents into logical argunents. However, 1t ls not necessary to base one I s appreclatlon of logic on the suspicious nanalytlcal-synthetlcn distinctlon as SCM do in thelr footnote 2 (see instead, Qulne 1961). Furthernorer any conslderatlon of stat,enents about the future does not necessitate a reJection of the traxion of the excluded-middlen slnce an adnisslble statenent ls stlll either true or false even though we nay not yet know 1ts truth status (see Bo1and 1982, Chapter 6). l{ith these two exceptions notedr I will expLaln why, while I agree with SCMfs reconnendatlon that the naxlom of noncontradictiontt should not be abandonedr f dlsagree with thelr reconmendation that ve should reject Aristotlers axlon of the excluded-mlddle in favor of some form of nnany-valued logietr. The najor polnt to be nade in favor of the axion of noncontradiction ls that any argumenl that vlolates it can be used to prove anything (see Popper 1965). Startlng with two contradlcfory assumptions, nthe sun is shining nowr and nthe sun is not shining nowrr, Popper shows how easy it is to infer that iCaesar was a traitorn or if one wishesr to infer its denial rCaesar was not a traitorrt using the sane two (contradlctory) assurnptions. f doubt anyone wishlng to construct a convinclng logical argunent ln favor of the truth of a given statenent rrould 63 Copyrtht @ 1985 by Marcol Dekkcr, Inc. 07 47 -493818s 10401{063$3.s0/0 ever be satisfied nilh an argunent that 1s also capable of providing support for a denial of that given statenent' The only theorists that are ever satisfied with bulldlng argunents that vlolate the axlon of noncontradiction are those desiring lo bu11d Marxian nodels of soclal dynanlcs (where contradictlons are necessary elenents in a dialecttcal process). unless this ls oners purpose for employing econonetrlc techniquesr there is no need to reJect the axion of noncontradiction' While lt is dlfflcult to see why one should give up the axion of noncontradlction simply because G6del wants us to worry about the loglcal conpleteness of fornal systems' it ls even more difficult to see why the anti-fornalistts rejection of indirect proofs nust necessitate a rejection of the axion of the excluded-niddle . To avoid any nisunderstandlng herer let ne illustrate the role of the axions of noncontradictlon and the excluded-niddle in lndirect proofs. Consider the form of an lndirect argunent in favor of proposlllon P glven the truth of assunptlons A' and At' An lndirect argunent is (like any arguloent) a conpound statenent' It says that since all its parls are true, any conJunction formed by a valid rearrangement of its parts is true' In lhe case of an indlrect argunent ln favor of P the coniunction is forned by conjolning the denlal of P (not-P) wlth A.| and Ar' An indirect proof shows tha! such a conjunction contains a contradictlon. And since A.| and A, are accepted as truer by the axlon of noncontradictionr one cannot also accept the truth of not-P. That 1s, accepting the truth A, and A, reQuires the rejection of the falsity of P. ff P ls not falser what is it? Wel1r lf you also accept the axion of lhe excluded-nlddle, then P ls true or it is false. In thls case then, not-false is the sane as true since there ls no olher optlon. Thls ihen constltutes a proof of the truth of P (given A' and Ar) bf showing that not-P is false' Historicalllfr aoo€ nathemaiicians (e.g. r rintultionistsr) would always have us reJect indirecl proofs. Nevertheless, Euclid FOUNDATIONS OF ECONOMETRICS-COMMENT 65 supposedly provided an indlrect proof of his sixth proposition that nif two angles of a triangle are equal then the sides opposite are also equaln. I rnentlon this because there are many propositions that are conmonly accepted on the basis of indirect proofs. We should also recognize that every indirect proof lnvolves both the axiom of noncontradlction and axion of the excluded-niddfe. It 1s true thal if we were to give up eilher axionr we would have to avoid relylng on any lenna or theoren vbose proof ls indirect, but avoidlng lhe use of indi.rect proofs of the axion of the excluded-middle ! Whlle it night be possible to axionatize the intuillonistsl denial of the axiom of the excluded-niddle based on a denial of indirect proofs, the real issue with the intuitionists was the use of the concept of lnfinity ln proofs since the concepL of infinity always refers to an impossible quantity (see Boland 1985' Chapter 5). It ntght nake nore sense to reject the use of the concept of lnfinlty and retain the axlom of the excluded-niddle. By advocaiing sone forrn of nany-val'ued or nfuzzyn loglc' SCM yish us to replace Aristotlers axion of the excluded-niddle with another axion whlch woufd aIlow adnlsslble statenents to be sonething other than true or fa1se. Aristotfefs axiorn of the excluded-niddle would need to be replaced because it would seen to contradlct the econonetric model-bullderts use of Bayesr theorem. Could we not avold such a contradlction by ensuring that whenever does not require abandonnent tJe use Bayeslan estimatlon technlques neither we nor the economic theories that we have modelled have enployed a lenna or theoren rhich has to be pqoven indirectly? If instead we htere to fol1ow lhelr advicer we night also need sornething to replace the axj.on of nonconbradictlon. If statenents can be nnany-valuednr what conslitutes a contradiction? Presumably' a statement cannot have two different lruth values. Butr can we say that a staLenent nhj.ch is, by any neaaurement, .5 true is also .5 false? Is this a contradiclion? Just rhat does the axion of noncontradicllon mean rhen ne abandon the axlon of the excluded-nlddle? BOLAND by SCM ln Sectlon 6 of their paperr we could retreat fron the problen of ensuring that the concepts and assunptions of econouetrie theory satisfy Aristotlers axions of logic by instead 1i-miting applications of econonetrics to ninnediate practlcal problemsn. That is, we could slnply take an instrumentalist-type stance and thereby reject any need to ensure that our'econonetric assunptlons are adequately based on accepted logleal principles such as the axion of the excluded-midd1e. But scM reject such instrumentallsn, quite correctly I thinkr because they do not wish lo restrict econonetrics to inrnediate practical As reeognlzed problens. But is current econonetric theory capable of dealing with anything nore than lnnediate practlcal problens? Even if we limit our extention beyond innediate practical problems to hypothesis testingr it is not clear that nuch ean be acconplished. Every econonetric nodel of an econonic theory yields anbiguous results whenever one ls realty trying to test whether the theory ls lrue or false (see Boland'1977 and Cross 1982). At bestr the queslion of testing involves strategies and preferences concerning whether alpha-type or beta-type error is the least preferred. Neynan and dld not actually solve a theoretlcal problen. when rre recognize the question of strategies and error-type preferences' we are actuall-y offering an instrunentalist strategy to deal with practical problens (e.g., whether lo vaccinate a population agalnst a vlrus and thereby risk infecting that population)' No matter holr nuch we nay disllke lnstrunentali.smr il seens difficult to avold instrunentallsrn whenever econonetrics is invol"ved. What more do SCM want econonetrics to do? The founders of the Econonetric Society dld have greater thlngs in nind. For exanple, Frj.sch saw econornetrlcs as a nunificatlon of the theoretlcal-quantltative and the enplrical-quantitative approach to econonic problensn with a nconstrucLive and rlgorous thinklng slmilar to that whlch has cone to doninate in the natural sciencest (1933' p.1). Schunpeter saw the ains of econonetrics to Pearson FOUNDATIONS OF ECONOMETRICS-COMMENT 67 be nflrst and last sclentificn while stressing nthe nunerical aspectn to be able t,o nexpectr from constant endeavor to cope wlth the difflcultles of nunerical work, a wholesoroe disclpline ... [that] helps tn building up the econonlc theory of the futurer (1933, p. 12). It would seem that SCM have shown that the econonetrics of the founders wiII forever renain as 111-founded npipe-dreansn. And so long as econonic theory is built using ordlnary loglc as represented by Aristotlets axions, to be consistently applled lo current econonic lheory, econonetrlc lheory nust also be based on Aristotlers axions. For thls reasonr a logical foundation for econonetric theory that denles any of Aristotlers axions cannot be used to nodel ordinary econonic theory. Thus' lhe applicability of nfuzzyn econonetric theory would seen to be linited to only bullding econonetric nodels of lfuzzyn econonic theory. Lawrence A. Boland Simon Fraser Unlversily ADDITIONAL REFERENCES Boland, L. A., (1977). Testabllity in Boland, L. A., (1985). African J. of Econ., tl5, 93-105. Methodotogr Boston: Allen and Unwln. Econonic Sci.ence. So. for a New Microecononlcs. Cross, R., (1982). The Duhen{uine thesls, Lakatos and the appralsalrof theorles in nacroecononics. Econ. J., 92, 320-\0. Frlsch, R., (1933). Editorial. Econometrica, 1, 1-4. K., (1965). What is dlalectic? ConJectures Refutations. New York: Baslc Books, 31'2-35. Popper, and I{. V. 0., (1961). Tvo dognas of enpiricisn. From a Loglcal Point of Vlew. New lork: Harper Torchbooks, 20-46. Quine, Schunpeter, J., (1933). Econonetriea, 1, 5-12. The couron sense of econonetrics. ECONOMETRTC REVTEWS, 4(1), 69-74 (1985) COl.tl'tENT I shal,l refer to the pa.per under disctrssion ' and t'o ibs authorship ' by scu. It was nob clear Lo me rrhether it had ansuered lhe question in iLs bible. since the empllasis was on multivalued logic' Perhaps it is neither true nor false that bhey ansflered [he question. The comment in the conclu.sions, "The foundabions of econometrics are ireaker than those solely $rpporting economic theory under celtainLy" sholrs lhat scril believe bhat hhe foundaLions do exisb but are not solid. It is qrestionable rrhebher the foundahions of :ury science are enbirely solid' not even bhe fotrndabions of rnathemalics or logic. The ambiguiLies o{ language almosL preclude the solidity of foundations. This is rrhy il wa-s srch a good idea of Turing's to base lhe foundations of malhematics on a carefully specuied comFrter, br! even so he had to rtake lhe irdgement thal any Possible calculation corld be carried out on such a coDF:ter. is so much in the Paper thal in this discussion I can louch cr only a fraction of i[. I agree wilh rnany oi the comments' but in ttds discJssicn I shall concentrate more on points of disagleemenb. one criterion that scu have used, to eursarer lhe question in lheir title, is whether Aristotelean tro-valued logic is adequate for lesting econometric bheories. Buu lhe discussion is really more general since it deals wibh scientific inference as a whole, and not only with inference in econometrics. Inference usually requires a theory of There probabitiby lthich certainly goes beyond Aristotelean logic. Aristotle said, if the tlanslalion is Lo be Lrusted, "It, is Lherefore pfain that it is not necessaty that of an affinnalion and a denial one stpuld be true and lhe other false. For in lhe casb of bhab lthich 69 Copyrigbt @ 1985 by Marcol Dckker, Inc. o7 47 4938 l8s lo4o I {069$3.5010 IU GOOD exists potenllatly tnrt nob actually, the rule which applies to lhat which existsactrrallydoesnobholdgoo,d.,'(Thereferenceisgivenafterthe where next quobablon. ) But bhi.s is conLradicted in the table of conlents ib is staled (and I agree somewhaL more with this) that "propositions il is not whj.ch refer to fulure [ime musb be either true or false ' brrl 194I' deternined hrhich must be Lrue and which false"' (l{cKeon' ed" pp. 38 and /.8; oE Eubchins, €d., 1952' Volume 8' PP' 23 and 29') ebo Aristotle discussed probability Lo some exLent' Thus Arisbotle' unlike Iike other humans, went beyond Arislotelean logic brt PerhaPs' others, vrithout often contradicting it' I don't know whether Aristotle ever discussed the vagueness and a anbiguily of ordinary language. For example' lhe question rdhether yes or no man has a beard cannot alrrays be ansrrered with a clear The because beardednes€, like mosb Lhings, is a matber of degree' man'sfacemaybemerely'|'zzyaflelseveraldayswithotltshaving. This is a clear and simple exanPle showing Lhe need in princiPle for must non-ArisboLelean fuzzy logic. Tha! Uhere are degrees of meaning havebeenrecognizedcen|uriesago.Istillbelievel,ha|Isaidin diJfiorlt Good (f95o' page ln) that "there are senlences for which it is to decide rrhelher lhey are proPosilions. such 'par[ial ProPositions' ofbenoccurinthepioneeringworkonnegscienti'ictheories''. Aristotelean logic could be largely resLored to Lhe question of beardedrressbydefiningabeardbymeansofarrarbilrarylhresholdon theweighboffacialhair.Ittotldbenecessaryboshavelhemanto breigh Lhe hair, so Uhe Uest would be "destruc[ive" (to use a term from qualitycon|rol).AlsofurbherquestiorrswouldariseslrchasiJheredoes Lhe face end and the neck or head begin' If something atst concrete as a beard c'ur be ^tzzy ' the fuzziness of more abstract concePts i-s all the more Lo be exPecbed' An example of a very imPortanl fuzzy concept in the PhilosoPhy of science is that of slnpu,city or comPlexily, and bhe allied concept of parsimony of in assumPtions in hyPotheses or bhecries' Is simPucity to be defined LermsofnumbetofParameters'shot.fdthecomPlexiuyof[henumerical valuesoftheparametersbeincluded,issirnpticityjust'brevityandU }.OUNDAT]ONS OF ECONOMETRICS-COMMENT 7L so in hrhat language? But, in my oPinion' the fuzziness of simplicity ot IErsimony (which is Lhe same thing) and comPlexity does not by any leans make them "useless as crileria for model choice" as staled by SCH in lheir Abstrac!. on bhe other hand I hhink bhey are absolutely essenbial for [hat purPose. To take a familiar examPle' given dala (xt, yt), (x2, Yz), ..., (xn, Yn)' we can find a Polynomial fil y = f(x) of lhe (n - 1)th degree ttEb exaclly fits the data. I hope scr't nrld nob alhrays prefer lhis fib, for prediclive [rrrPoses ' over polynomials of lower degree (which are more parsimonious) ' what reason is Lhere, other than simplicity or parsimony , tor preferring a polynomial of lower degree? Another exannPle where parsimony (or sirpficiby) is of the essence is lhe Preference for lhe BuUiau/Nerrton rrilrerse square law of gravita[ion (Brrlliau' 1645, p. 23) over the epicycle bheory of uhe motion of lhe planebs. scM say in their conctusions' and elsewhere, that Probability tJEory i.s a version of nany-valued logii, and ib certainly can be so regarded. But ib also obeys the u.sral larrs of Arisbolelean logic. If a poposition is now only probable for us, lhen we don't know now whether t is Urue or false, though we might knoh, later, and it can be true or rrlce sysn if we never find or.rt which. similarly lhe statement Lhat *thing i-s approximately true can itself be absolutely true if anvthing caa be. scl,t say that 'The statement that an event occurs with Probabiliby p ... violaLes bhe axiom cit rer O or 1. " It of bhe excluded-middle unless p is seems to me !hai. no violabion of the law alt{ays of bhe cacluded middle follows from this argument, whebher the probability is ld€rpreted in the long-run frequency sense or in the srbjective (personal) sense. Ijf uhere is a violalion j.t is because Probabilities right nob take sharp values so bhat uhe sLatement Lhab a Probability is P !s somewhal unclear. It night mean bhaL lhe probabilily Ues between p j| €, for some €. It night be true or false thaL the long-run relative frequency of getting heads, when bossing a coin, Iies bebween O.45 and O.55 (provided bhat lhe long run is not too long). I{e don'U r€d double-headed or double-tailed coins in order lo use bhe lan of 72 the excluded middle. An interesting question is how one nighb measure degrees of mearring, or degrees of belonging to a set as in the theory of frtz.zy sets. Take the excrmple of the fuzzy beard. we night define it as belonging to degree x, to the class of beards if a fraction x of some specified pop$ation of people would caII i.t a beard if they were forced to say either Uhab it is or is not a beard. According to Uhis definition, degrees of belonging depend on nhat poprlation of itrdges is assumed. I do not know whether bhis proposal ttas been made in bhe extensive liberature on fuzzy sets. Also I do not know enough of the liberature to judge \dhether bhe theory of f,Lzzy sebs is useful . ft is not, by Lhe way, the same as a theory of partially-ordered probabilities. This I once chec*ed in conversation with zadeh who was probably the first person to study the theory of, fvz.zy sets intensively. rt deals with degrees of meaning, not degrees of probabiliby. on another topic, scM state that the problern of induction is impossible to solve. lf they had said it has no complete solution' I would have agreed. Part of the solution is bo drae a distinction betireen induction from previous experiences to a general law on the one hand ("universal induction") and only to bhe next case on lhe other ( "predictive inducbion" ) . The la[er form of induction is more secr.rre ! see also Good (1983, pp.2O5-2O7) where an abtempb is made to quantiJy the matter. complete solutions to Uhe problems of bhe philosophy of science are hard to come by. In view of Lhe comrnents in their conclusions, scl,l might partly agree qualitabivel.y wilh ny comments on induction. They refer bo ny criticisn of Jeffreys's argument for (predictive) induc[ion. l,ly argument did show that Jeffreys's argumenl did not enbirely solve the problem of induction. Bub I Uhink ScU overstate their case when they repeatedly say there is no solution bo bhe problem of induction. Parbial solutions can b€ valuable. I an sympathetic to the views of SCU on instrumentalisn, bub would like to add a cotrple of comments. (i) An instrumentalist theory can be true or false provided that ib is formulated as a neans for naking predictions. Iite don't have bo give concrete interpretations to FOUNDATIONS OF 73 ECONOMETRICS-COMMENT uE linear operators or wave functions Schroedinger fonntrlabions of of the Eeisenberg and quantum mechanics' yet bolh formulabions brue. The basic philosophical question is rrhether all bheories are insbrumentalist. (ii) when discttssing Friedman's instrurentalisn' sclt say bhat "in bhe presence of uncertainty' the neaning of 'Lrue or refuI predicbon' is not clear." I agree' yet some clariJication is pcible. For example, iJ we have bwo predictive theories !11 and E2r ttEre are various ways Uo decide which is in some sense the better tlEory. one apProach is by means of the concept of explicativity (c@d, I97?; Good & ucuichael' 1984), though ib does nob stfficiently u*e into account bhe question of comPlexity of bheories. The problens d philosophy are seldom completely solved, bub partial soluLions are esilable and can be useful . There are some commenbs in scH's can be cdrlusions that are consisbent with this somerrhal mundane sentiment. ADDIEONAI, REFERENCES brlliau, Ismail (Bullialdus) (f6/.5). -@ philolaica opus nowm. in quo motus planetarun nouam ac veram hypothesim demonstrantur. . . (Pariis, SunPtibus s. Piaget). cited by Jeffreys (1957, P. I32n). Goodr I. J. (1950). Probabilibv and the weishing of Evidence (tondon: charles Griffin; New York3 Hafners). eood, f . J. (f97?). "ExPlicativity: a mathemabical theory of explanation Ltith stabistical apPlications"' Proc. Rov. soc' (London) A 354, 3o3-33O. [RePrinted as (198Ob).] {bd, I. J. (f983). 'The of a Nerarchical model for nrlLinomials and contingency tables", in Fclenti4g.-&l9g ' Data Analysis. and Rohrsbness (G. E. P. Bo:<, Tom f,eonard & Chien-Rr lfu, eds.; Academic Press). bd, I. J. & UcHichael, Alan (1984). "A Pragmatic modification of explicabiviby for lhe acceptance of hyPotheses" ' Philosophv of science 51, L2o-L27. Elrins, R. ll . (edilor in chiet) (1952). -@ robustness 74 World.VolumeS.Aristotlel.chicago:EncycloPediaBritannica Inc. Jefireys, H. (1957). Fdentific hference, 2nd edn' cambridg€l University Press. llcKeon, R. (ed,) (1941). The Basic works of Aristotle' New York: Random llouse. I. J. va. Good of stabistics Polytechnic lIlst. DePt. and Stale Univ. Blac*str.rrg, Va. 2t O5L ECoNoMETRTC REVTEWS, 4(1), 75-80 (1985) col{uElrT oN "THE FOI'NDATIONS OF ECONOUETRICS _ ARE TIIERE AI[T?' \ION ZT'R MT'EIII,ET Swamy, Conway and Muehlen (sCM), have written a thought provoking paper addressing a number of jmportant philosophical j'ssues that surround the theory and practice of econometrics. l.Iany of the issues that they deal with are of general philosophical interest and have been the subject of an intensive on-going debate in the philosophy of science literature.l ttre problem of induction, the J-funitations of naive falsificationism, the issues concerning Aristotle's tlro-valued aygtem of Logj-c, and the alternative interpretations of probability discussed by SCIII are al]. matters that have pre-occupi-ed philosopherg and logicians for centuriea and remain largely un-resolved. As Chalmers puts. it, with characteristi.c clarity, "Ttrere is just no nethod that enabLeB sci.entific theories to be proven true or even probabLy true .... attempts to give a sinple and straightfonrard logical reconatruction of the "scientifi.e method" encounterg further difficulties wlren it is realized that there i.a no method that enables scientific theories to be conclugively disgroved either.' (p. xiv). seen from thiE general perspective, one of scM's nain ooncluaions that the truth or fa.l-eehood of economic theories can Copyrtbt @ 1985 by Marcel Dckkcr, Inc. 07 47 4938 I 8s 1040 I -0075 $3.50/O 76 l PESARAN not be estabfished by econonetric methods is hardly surprising, and follows directl"y from what we already 'know" from the philosophy of science u.terature. If even in (hard) sciences Iike physics, there is no way of conclusively Proving the truth or falsehood of theories then thi"s must be even rnore so in (soft) sciences uJ<e economics which are subject to additional ambiguities arising largely from the socio-psychological nature of hunan behaviour. 1[he methodological Problems surrounding the use of econotnetric methods for the Purpose of testing economic theories have long been recognized in the economic literature. the early debate between Keynes and Tinbergen over the role of econornetrics in testing of businegs cycle theories is one prominent example urhich clearly brings out the limitatj.ons of econometrics as a nethod of testing economic theories.2 Faced with the inevitable concLusion that econometricians can not conclusively eatab]-ish the truth or falsehood of economic theories, sclil advance the controversial view that the Aristotelian system of two-valued fogic is inapplicable to econometric rnodels, and that a system of many-valued logic is needed if one is to justify the econonetric Practice. According to scM the econor€tric practice violates Aristotle's axioms prfunarily because of the uncertainty invo).ved in the nodeUing In particular they argue that of economic behaviour. Aristotle's axiom of "the excluded-middle" is violated in the case of econometric nodets as these npde].s can only be specifj-ecl in a probabilistic manner. They regard any probabilistic stater€nt as a violation of the axiom of the excluded-middle. In their own words "t'he statement that an event occurs with Probability p or that a variable takes on interval of values rtith probability p violates the axiom of the e:(cludedmiddle unless p is alnays either o or l' (P.36). FOUNDATIONS OF ECONOMETRICS-COMMENT point of view. The fact that the truth or falsehood of a propositj.on i9 not known a priori, does not necessarily render that proposition indeterminabe (i.e. neither true nor false). I'he axiom of the excluded-middle states that every proposition ia ej.ther true or else false, although which of theae j,s the case may well be indeterminable as regards our knowledge of it. I'he problen with SCI-t's argument lies in the fact that they regard the indeterminacy of the truth-value of a proposition relative to our knowledge aEr evidence of itg indeterminasy in general. I|lrey state ftlis is €rn extreme "If what xte know today about the future is not a Part of our deterministic knowledge, then our statenrent about the future c.rn not properly be called either true or false (at least not by our speaklng 7n tlra present) and so deserves a different truth value, intermediate between truth and falsity" (p.36). In this quotation the authors justify their viey about the violation of the a>(iom of excludedqniddle on the grounds that relative to our knowledge "today", the truth or falsehood of a statenent about the future is not determinate. But this does not necessarily imply that the truth-status of the proposition about the future is intrinsically indeterminate. lrthether a sealed envelolE contains a red or a bihite colour tid(et may (relative to our knowledge) be indeterminate, but this does not Ean that the proposition "Irhe envelope contains a red coloured ticket" does not adnit a determi-nate truth-status. Ille econornetric exanples of the vj-olations of Aristotle's axions given by sCM are also baaed on a sileilar kind of misunderatanding. In exirmple I the authors regald the existence of 78 PESARAN two alternative forms of aPproxinations for the denand systems as evidence of the violation of Aristotle's axiom of excludedmiddle. Ihey do not, however, erq)Iain why this should be so' If the original ctenand system is taken to be the truth, then from a strictly logicat viewpoint both approximations are false. The adequacy of aPProximations is an empirical rather than a logical queation. t'lre fact that t e may not be able to conclusj.vely reject one aPProxination in favour of the other in particular apptj.cationa does not rnean that the truth-statuE of the approxinations relative to the original exact denand sltstem is indeterminate. In exanl[)le 3, the authors argue that the Rational Expectations tlPothesis (REH) violates Aristotle's axiom of the identity of meaning, because it involves a juxtaEosition of the concept of the subjective probability distriSution with the concept of the objective Probability distriSution. lfhey regard this as evidence that two different notions or meanings are used What the authors overlook, for one concept of probability. hoyrever, is the fact that under the REH the two concepts are by ctefinition indistinguishable from one another as they will be one and the same thing. only in situations where learning is j-ncoru,lete, is there .rny possibility for the subjective to diverge from the objective probability distrilution probability distribution. But in such a circumstance tlre REH, in the strong sense advanced ry iluth, will not be applicable in any c.rse. Ehe issue of whether the nEH is a valiA h1E>otheeis or not is again an erqPirical mater for wtrich tlrere seerns to be no conclusive angwer. In short I am highly scetr'tical of the authors' claifi that a system of nany-valued logic providee a fir.trler foundation for the analygis of econonetric models. sernantj,cg al)art, I altl also not at all sure what relevance all thege philosophical consider- FOUNDATIONS OF ECONOMETRICS-COMMENT 79 ations have for econonEtrj-c practice. The Problemg the aPPlied econolptrician faces in obtaining a model which is relevant to the particular question, is derived from a logically consistent econonic theory and which rePresents the data adequately' are not lilety to be resolved by a course in nany-valued logic' (:Ihese problens are discusEed in PeEaran and Smith (1985b))' rhite I fully endorae the authors' vien that rte muEt alwaya be prepared to check the Predictions of our nrodel against those of it8 Cotu)etitors, I am not convinced that reaort to many-valued logic is going to heIP us rtith this alifficult task. Any nove away from the logical rigour of the Aristotelian Principlea is lilely to make the outcorne of confronting competing theories more rather ttran less inconclusive. u. Hashem Pesaran Trinity college Cambridge FOCrII|C'TES lFor a highly readable introductory survey of the philosoPhy of science literature see Chalnerg (19?8), where relevant referenceg to original sourcesr can also be found. Many of the lEthodological issues raiged by SCU are also covered etctensively by Imre LatakoS' (19?O) ercellent account of the basis of the nethodological claah between PoPPer and Kuhn. 2por the relevant references and re-examj,natj.on of the KelmeETiribergen debate in the light of recent developrnents in econolptrics nee Pes€rran and Snith (1985a). PESARAN 80 ADDTTIONAI, REFERENCES A.F., (I9?8), What is This Thing Called Science? University Press. Chalmerg, OIEn f11e Lakatos, I., (1970), Falsification and the nethodologD, of scientific rese€rrch prograrmes. Criticism and the Growth of Knowledge (I. Iakatos and A. Itusgrave, Eds. ), cafibridge University Press. Pesaran, M.H. and Snith, R.P., (1985a), Kelmes on econometrics. Kelmes' Econoroica: Methodological, Igsues (T. IJawson and !!.t1. Pes€rran, Eds. ), Croom tlelm. Pesaran, U.H. and $nith, R.P., (1985b), Evaluation of nacroeconometric tnodels. Economic llodelling, 2, L25-L3+, ECONOMETRTC REVTEI{S, CoHERENCE, 4(1), 81-91 (1985) "II'fROPER" PRTORS, AND FrNrrE ADDTTTVTTY claims span important contributions from Aristotle to Zel-l"ner. I shal1 confine myself, however, to issues of coherence of (finiteLy additive) probability, "improper" priors, and representation of Thus, my comment is focused "ignorance" by such probability. 57,3 on of their essay. As Jeffreys showed nearly half a century ago, many farniliar (textbook) "orthodox" statistical procedures have a Bayesian nodel under an "improper" prior. (see 52,1 of his Ttreory of Probability (1961).) Exampler: Let x 'r, N(0rl), with 0 unknown. Ttren flJeffreys, (1961, p. 137)] ordinary confidence intervals (and also fiducial intervals) for Q are Bayes posterior probabiLities, provided the "prior" for 0 (used in calculation with Bayes' Theorem for densities) is the uniform, "improper" (Lebesgue) density. Jeffreys (1961, chapter 3) argued for the existence of "ignorancet' priors, Eo be used to represent an agentts uncertainty quantities prior to the reLevant abou! unknown statistical just the general background assurnptions of, observations and given e.g., B statistical mode1. His theory of Invariants (1946), incorporated in the 2nd edition (1948) of the lheory of Probability, offers a tractable nethod for cal.culating such "ignorance" priors Not the and gives reasons on behalf of the program of Invariants. least of these reasons is the important, pragmatic consideration mentioned above, to wit: Jeffreyst "ignorancett priors provide a analyses. Bayesian model for sorne basic "orthodox" statistical (1979, p, 421) I have argued that Jeffreys' theory Elsewhere The authors' many interesting 81 Copyright @ 1985 by Marccl Dckkcr, Inc' 07 47 4938 185 1O401{08 I S3.s0/O SEIDENFELD of Invariants is not Bayesian after aLL. The problem, in a nutshell,, is that the theory of Invariants makes the prior a function of the data through the likelihood. This leads to Bayesian contradictions when, for instance, the data are composite, Lack a common sufficient statistic, yet admit (unique) "ignorance" priors for separate components of the data. then different posterior probabilities arise from Ehe same data depending merely upon which component of the data is used to fix the (invariant) prior. Thus, I do not find Jeffreys' program acceptable for rnaking t'objectiver" i.e., observer invariant, a sense of statistical "ignorance.t' Better, I think, to use sets of probabilities to capture uncertainty than to rely on some one distinguished distribution to model a lack of knowledge. (See [tevi, (1980, chapter 9)] for a rich discussion of this approach, including its decisiontheoreEic consequences and variations due to Dempster, Good, Kyburg, and C.A.B. Smith.) Nonetheless, there are important questions worth asking about ttimpropert' priors without supposing they serve as canonical representations for states of (near) ignorance. (Recall their How pragmatic vaLue ir: linking Bayesian and orthodox statistics.) do "improper" densities translate into probability distributions? And do they stand the Bayesian test of coherence? Of course, the 'rimproper" uniform (Lebesgue) density from example, is no probability density: the sure-event carries infinite measure. But, as Sir Harold points out (1961, p. 119), There is "... vre must use o instead of I to denote certainty ... no difficulty in this because the number assigned to certainty is conventional.rr Ttrere is no difficulty so long as we keep our conventional shifts consistent with one another. Towards this end, and following Levi's (1980, 95.10-5.11) excel"lent presentation, let us partially define a probabiLity P(.) frorn a o-finite measure U(.) (arising from an "irnproper" density) as fol-lows: (1) P(E) = I if u(E) = @ and u(E) < -. As Levi (1980, p,129) shows, Lhis transl-ation from an "improper" COHERENCE prior to a probability gives a reconslruction of Jeffreys' formal calculations--where the "improper" density is used in Bayest theorem for densities. It also makes sense of other of Jeffreys' analyses, €.g., that the probability is 0 that a < 0< b, for all -- < a < b < o,, under the "improper" density from example, [J"ffr"ys, (1961, p. 122)). However, as has been noted by many: Heath and Sudderth (i978); Hill (1980); Levi (1980), to mention chree, the resulting probability l(') is merely finitely (not countably) additive. In the exampler, by translation according to (t), P(') is purely finitely additive (p.f.a.) as P(i < (r. i +l)=0fori=0,J1, Why should mathematical issues of additivit.y of measures be of interest in a discussion of foundations of statistical inference? After all, even Kolmogorov (1956, P. 15) thinks l-sdditivity is an expedient. So what if we alter the convention that probabiliry is councably additive in reaction to the other shift in convention: !o use o-finite but not finite measures to depict certainty? The question is more than a mathematical nicety given the serious allegations, €.8., of Dawid, Stone and Zidek (1973), that "improper" priors engender inconsiscency and Bayesian incoherence, Ttre so-called "marginalization paradoxes" and "strong inconsistencv" attributed to Jeffreys-styled analysis using "improper" priors are, in my opinion, traceable to two distinct features of "impropriety." FirsE, inadequate mathematical techniques have been employed to facror joint distributions for continuous random variables into a product of a "oarginal" and "conditional" density. Ihe inadequacy is evident through leparzrmeterization. (I have discussed this in (1982).) Second, finirely additive probabilities admit failures of what deFinetti (L972, p. 99) calls "conglornerability" of distributions. b" pefinition let n = {t'': i=I,'..) .1g14:'"."bi1ijl: a (denumerable) partition of the sure-event. If P(Elh.) < (>) k for each i, then P(E) < (>) t. 83 SEIDENFELD additive probability i'(') sa!isfies conglomerability in a partition n if and only if P(') is disintegrabLe in n.) the following is an elemenrary illustration of non-conglornerability of a merely finitel'y additive probability (reported in lderinetti (1972, p. 205)] and fDubins (Dubins (1975) shoved that a finitely (t975, p. e2)l). ExamoLe^: Let the field be the set of (a11) subsets of Let P-(a- ) = l/2J if i=1, x = {a..: 1lc].ji=l ,21 i=L,2,...}. i=2i hence P"(') is countably additive and lives on each i, j, and let the set X,r = {a..: i=l}. Let P,(a,.) d lJ = 0 for 1.1 ro(xr) = 0. rinally, let P(') = oP (.) + BPo(') (cr,B > 0, a +B= 1). Then p(.) is not conglomerable in n Fix n = {njrnj = {atj,"rj]}. 1 for each j. see fschervish as P(xr) ="c, ihereas-r(xrln:) et al, (1981+, p.2I7) for discussion of how neither conglomerability nor non-congLomerabiLity in a given partition is closed and = 0 if under convex combinations. ] Recently, Schervish et al. (1984) have shown that (subject to mild regularity conditions) every finitely but not countably additive probability admils failures of (denumerabLe) conglomeraThe least upper bound of the failures is given by the bility. coefficient of the p.f.a, component in a decomposition of a finitely additive probability into a convex combination of a countably additive probability and a p.f.a. probability--corresponding to B in exampLer. Subject to the translation (1), o-finite measures arising Hence, from "improper" densities convert to p.f.a. probabilities. they adrnit arbitrariLy large failures of conglomerability. (ff one adds clauses to (1) for using limits to translate u(')-measures into P(.)-measures, as is done by lleath and Sudderth (1978), the P(') which results is non-atomic as we11.) Here is a sirnple ilLustration of non-conglomerabiLity with Jeffreys' prior from example' where the failure exceeds .67. Example": Let I t N(0,1) and Let p(O) be partially defined as a p.f.a. probability according to (1) from the uniform "improper" COHERENCE density. Let I. = {x,0: i-l < l"-ol. i} and let J. = {x: i-1 Note that P(I:) > 0 and P(I.,) > .67 in : l"l . i] i=l ,2,... particul-ar. Moreover, P(r.) = P(r. le) = r(r. lx). (The first equality is by the confidence interval property of L and a The tacit assumption of conglomerability in the 0-partition, second equality is by Jeffreys' analysis, as in exarnpler,) Define a denurnerable partition n = {h.: hi = Ii*lg (r, n;.)}. Then, since P(J.)=0 (according to (1)), P(rlltt.l=o for each element of n, yet P(Il) > .67. Also, P(h,) > 0 so that, as in exampler, for non-atomic distributions non-congLomerability is present r^rithout conditioning upon events of probability 0' Is a merely finitely additive probability coherent? Does non-conglomerability entail a "Dutch Book" (sure loss) for anyone posEing (conditional) odds that fail conglomerability? liOl At least according to the positions defended by Savage (1954) and deFinetti (1974) finiteLy additive probability is coherent. (See lseidenfe].d and Schervish (1983)] for details.) Savage's postulate system Pl-P7 adrnits all non-atornic finitely DeFinettits criterion for additive personal probability. (against many cal1ed-off bets), finitely avoiding a sure-l-oss or equivalentLy his criterion of admissibility of "previsions" against his (proper) squared-error score' establishes coherence of all finitely additive probabil-ity. Ttrere are, however, rival accounts of coherence, even for In their important (1978) paper' finitely additive probability. Eeath and Sudderth introduce conditions of coherence that (in effect) require conglomerability simul-taneously in two specific the partitioo by the unknown (and unobserveil) partitions: parameter 0, and the partition by the random variable(s) to be observed. (Typically, for textbook statistical problems, these are partitions with cardinality of the continuum. ) Thus "H-S coherence" is a stricter slandard than (deFinettits) coherence' additive prior fails to carry a posterior disYhen a finitely tribution that is conglomerable in the partition by the observed 85 86 SEIDENFELD then that prior is "H-S incoherent'" The authors accept the H-S standard for, in this situation, they say the prior fails to yield a posterior (p. 28) and should not be used (p. 31). Of course, such priors have posteriors that are coherent in the sense of deFinetti or Savage. I queslion the appropriateness of the added restrictions imposed by "H-S coherence" beyond what is required by coherence (in the deFinetti sense). For one, a "H-S coherent" distribution can be "H-S incoherent" conditional on an event defined This is so1el-y in terms of the observed random variable(s). random variable, illustrated by the following. Example4: Let (x|xr) be i.i.d N(0,o2), $tith both parameters "unknown." It is straightforward to shos, that (z) (e,o2)' P(x 'min- < 0 < xmax'I (0,o2)) ='5 for each pair As Heath and Sudderth show (1978, P' 341) there is a "H-s coherent" finitely additive prior, corresponding to Jeffreys' "improper" ignorance density (uniform over 0 and independently uniform over 1n o). The posterior satisfies (31 (*,,x,)) =.5 for each observation (xl,xt)' P(x-.'mln- < 0 < x*--max'I L' I Define the random variable t = (xr+ x) / (x, -x), Buehler and Feddersen (t903) show that (0'o2)' tt'" ,(*,nir, I01*r"* | {0,o2), l.l '1'5) > '518 for each (xr, xr) pairs But, as \^re can partition the event ltl < 1.5 by those (x' xr), in (in t), given congl-omerability satisfying the inequality it follows frorn (3) that' conditionally, (5) P(xmln <o<x max I ltl.1.5)=.s but given conglomerability in (0,o2), it follows from (4) that, conditionaLly, (6) P(xmln < o < xmax I ltl . 1.5) > .5]8 -(5) and (6) are contradictory, it cannot be that the Thus, since "H-S coherentt' distribution associated with (3) is also a "H-S coherent" distribution given ltl . f.S. (As shown in [Kadane et al. (1981)], such conditional "H-S incoherence"is restricted to COHERENCE 87 events of (unconditional) probabiLity 0. As the observation, likewise, carries an unconditionaL probabiLity 0, iE can be that for each possible observation there is some containing event of probability 0 that, conditionally, establishes "H-S incoherence.") It is open, I believe, when such condirional "li-S incoherence" can arise. (The problern is equivalerrt to what Buehler (1959) called the question of "relevant subsets.") In light of these findings: that all merely finitely addiEive probabilities suf fer non-conglomerability in a denumerable partition ; that a "H-S coherent" distribution can be conditionally "H-S incoherent"; and that this may be possible for each point in lhe sample space, why do the authors subscribe to Ehe more resErictive notion of "H-S coherence"? If coherence is to serve as a norm for inductive logic, if incoherence is a mark of the irrational analogous to the objection of deductive inconsistency, then one nust justify the serious charge thac non-conglomerability is I susPect irrational in order that "H-S coherence" be justified, that the family of pragmatically bettable propositions cannot be insulared from the anomaly of non-conglomerability without Better' I reintroducing Ehe principle of countable additivity. think, to use deFinettirs standard rhan to make coherence inEo a procrustean bed. A short Postscript on ttl,ogic" The authors are intent on connecting problems in the foundafions of statistics with questions abouE logic and lhe foundations of mathemalics. As one who teaches courses in the philosophy of science and in mathematical logic, of course I am pleased to see economisEs concerned about the "logic" of probability (in Unfortunately, though the path aPPears evident lo statistics). at least three economists, Ehere is one philosopher who does not follow the trail from Aristotle through G6'de1 to Birnbaum. For example, what is the economic relevance of Gddel's celebrated incompleteness Lheorern? Should economists be more concerned than, say, physicists or even mathematicians about SEIDENFELD chis fact regarding formalized theories? Do Ehe authors know of a single case where Godelean-incompleteness matters in economics? What axiomatized economic theory and what economic hypothesis are involved? I am nonpLussed by the repeated appeal to G6de1's importanr result. I'trere is a substantial literature dealing with Lhe idea tha! probability theory gives a generalization of deductive 1ogic. The theme is hardly surprising since it is natural lo think of logic algebraically [H"lto" O96D 1 thro,,gh the pair (A,M), where A is a Boolean algebra and M is a Boolean ideal in A (generaEed by the "anEi-axioms"). Then a measure algebra over the quotient algebra A/M (assuming M is a Proper ideal) provides a quick link where exactly the deductive consequences of the axioms carry probability 1. For a clear sufinary of the Program to use probability for generalizing deductive to inductive 1ogic, see carnap's (1950,9 43B). For a view of probability as multivalued logic, see Reichenbach's (1949, chapter 10)' However, what I find surprising and rather puzzling is the authors' assertion that the Aristotelean principle of excluded middle does noE hold for the "standard" interpretations of probability, by which they mean the "frequencyt' and "subjective" interpretations (see their fn. 5 and AppenaixS4 and55)' This is a much stronger clairn than the modesE resulr that (formal) probabilicy theory carries an interPretation as a non-classical logic' Let us consider the reasons they offer on behalf of cheir claim. In the Appendix 94, "orrcet"ing the "frequency" interpretation they write, Even in Ehose situations where the conditions of a law of large numbers do hold, lte may not know exactly the limiting frequencies in infinite reference classesl thus our probability statements entail assertions about Itris means that probability unknown relative friquencies. sEatements can at best be approximations to long-run relative frequencies, and hence Aristotelian logic does noE apply under a frequency interpretation' In the Appendix 55, concerning the "subjective" interpretation COHERENCE rhey write, Consequently, if the distribuEions constructed in DeGroot's .manner are construed as yielding subjective probabilities--i.e,, degrees of belief--this means that subjective probability assignments do not require a unique degree-of-belief function.l8 Thus, approximatione aiE necessary Lo represent someone's opinions by a distribution, thereby ruling out any application of Aristotelian logic to Ehis siEuation. As best I can make out, these arguments resE on a lacic assump:ion (which I reject) which amounts to the presumption that approximations are at odds with Ariscotelean 1ogic. (I apologize, in advance, if this is not what the authors intend.) Of course, we are well aware of the problem posed by Ehe i.nevitable vagueness of ordinary language, Just how few hairs nake a given head bald? (If one more hair makes no difference, then by mathemaEical induction the head is bald regardless the Sr-rt I reject the assumption auober of hairs growing on it!) that approximations must involve the kind of vagueness that makes by standards of classical 1ogic. The familiar rhem unintelligible terms "palm" and "back of my handt' are infected with vagueness: lust where is the dividing line? But that does not preclude an epplicarion of classical logic to statements 1ike, "I clap wiEh tae palrns ofmy hands." And it is understood by all what. the traffic cops means when it is reported that I was driving :pproximately 45 mph in a 25 mph zone. I do not believe this uaderstanding involves application of some non-standard logic! Htrat is sorety rnissing from the authors' arguments (above) is . demonstralion that the sense of "approximation" used in the 'sEandard" interpretations of probability is of the sort Ehat cannot be captured by classical logic. This is not to deny that I, too, find serious deficiencies I think the ia the frequency interpretation of probability. *dispositional" theory is more reasonable: chance is a useful (see hacking. (rgo:)] ana r€nse of "empirical" probability. (1980, chapter 11) ] for discussion of "chance.") J do noE ILevi r SEIDENFELD 90 find classical logic to be a hindrance in the philosophically difficulty task of producing defensible explications of "probabNor do I see where the authors have shown us good reason ility." Lo suspect classical logic as one of the sources of the conflicts in the ongoing debates over the foundations of statisEics. Teddy Seidenfeld Washington University ADDITIONAL REFERENCES Buehler, R,J., (1959). Some Validity Criteria For Statistical Inferences. Annals Math. Statis! ' , 30, 845-863. Buehler, R.J. & Feddersen, A.P., (1963). NoEe on conditional property of Student's t. Annals Math. Statist., 34' 1098-1100' Carnap, R., (1950). Logical Foundations of Probability. Chicago: University of Chicago Press. Dawid, A.P., Stone, M. & Zidek, J.V., (1973). Marginaliza!ion paradoxes in Bayesian and structural inference' J. Roy. statisc, Soc. 35, 189-233 (with discussion). deFinetti, B., (I97D. New York: Wi1ey. Probability, Induction and Statistics' ( lg7, . Finitely additive conditional probabilities, Ann. Probability, 3' 89-99' conglomerability and disinlegrations' Dubins, L. , Hacking, I., (1965). Logic of Statistical Inference. New York: Cambridge Universi.ty Press. Halmos, P.R. , 0962). Algebraic Logic' New York: Chelsea' Jeffreys, H., (Lg4O. An Invariant Form for the Prior Probability in Estimation Problems. Proc. Roy' Soc., A, 186, 453. Kadane, J.8., Schervish, M. & Seidenfeld, T., (1981) ' Statistical Imp1ications of Finitely Additive Probability, In: Bayesian Inference and Decision Techniques with Applications: Essays in Honor of Bruno deFinetti, Coel, P.K. & Zellner, A. (eds.), forthcoming. Kolmogorov, A.N., (1956). Foundations of the Theory of Prob- abi1.ity. New York: Chelsea. 91 COHERENCE Levi, r., (1980) The Enterprise of Knowledge. Cambridge, Mass': MIT Press. Reichenbach, H., (1949). The Iheory of Probability. Los Angeles: Savage, L.J. (1954). Foundations of Statistics. York: wiley' University of California Press' New Schervish, M., Seidenfeld, T. & Kadane, J.B., (1984). fn9 Extent of Non-Conglomerability of Finitely Additive Probabilities. Z. Wahrscheinl ichkei ts theorie verw. ' 66 , 205-226 ' Seidenfeld, T., (1979). Why I am not an objective Bayesian. Theory and Decision, 11, 413-440, Seidenfeld, T,, (1982). Paradoxes of Congl-omerability and Fiducial Inference. In: Logic, Methodology and Philosophy of Science VI, Cohen, L.J.r Los, J., Pfeiffer, H. & Podewski, K' (eds.) Amsterdam: North Hol1and, 395-412. Seidenfeld, T. & Schervish, M., (1983). A Conflict Between Finite Additivity and Avoiding Dutch Book. Phil' Science, 50, 398-472' ECONOMETRTC REVTEWS, 4(1), 93-99 (1985) COMMENT that logical lnference plays a large part ln sclence and thaE it ls ioportant to get lt rlghc. Even lnstrunentallsts, for whom sclence is magic, would agree Ehat it ls lmportant to find the rlght - 1.e. successful - spells. The authors under discusston - Swamy, Conway and von zur Muehlen - pick out a cerEalo dominant conceptlon of loglc whlch they call "Arlstotellan". Slnce -Aristotellan 1oglc" norrnally neans a partlcular system conflned to sylloglsms, I sha1l foLlow the customary usage of loglcians and philosophers and talk of "classical" logtc. The authors I thesls 1s that ecooometrtc practlce does not conform to the prlnciples of classlcal loglc and calls for a rlval loglc, namely some brand of many-valued loglc. I sha11 argue that vtrtually everythlng they say about classlcal loglc is mlstaken and thelr advocacy of many-valued 1oglc ls mlsconcelved. The authors see classlcal logtc as characterised by three principles. The Law of Excluded Middle says that every statement is elther true or faLse. The Law of Cont.radlctlon (to glve lt lts customary name) says that a statement and lts negation cannot both be true. And the Axioo of Identity (as the authors call lt for want. of an accepted termtnology) says that a term must oot be used in dlfferent senses ln the same argument. I shall deal wlth these Ln reverse order, Everyone agrees 93 Copyrigbt @ 1985 by Marcol Dekker, lnc. 07 47 -4938 | 85/040 l 4093 $3.s0/0 ECONOMETRTC REVTEWS, 4(1), 93-99 (1985) COMMENT that logical loference plays a large part in sclence aod that tt ls important to get it rtght. Even lnstrumentallsts, for whom sclence 1s maglc, would agree Ehat it 1s important to find the right - i.e. successful - spells. The authors under discusslon - Swamy, Conway and von zur Muehlen - pick out a cerEaln dominant conceptlon of logtc whtch they call "Arlstotelian". Since -Aristot.elian 1oglc" norrnally neans a partlcular system conflned to syllogisms, I sha1l follow the customary usage of loglclans and philosophers and talk of "class1cal" 1oglc. The authors I t.hesis 1s that econoneErlc practlce does oot conform to the prlnclples of classical loglc and calls for a rlval logic, narnely some brand of many-valued loglc. I shal1 argue that vlrtua1ly everythlng they say about classleal loglc ls mlstaken and thelr advocacy of nany-valued loglc is misconcelved. The authors see classlcal loglc as characterlsed by three prlnclples. The Law of Excluded l,tiddle says that every statement is either true or faLse. The Law of Contradletlon (to glve 1t lts customary name) says that a statement and lts negatlon cannot both be true. And the Axiom of Identity (as the authors call tt for want of an accepted termtnology) says that a tentr rnrst oot be used in different senses ln the same argument. I shall cleal wlth these ln reverse order, Everyone agrees 93 Copyright O 1985 by Marcel Dekker, Inc. o7 47 4938 I 8s/040 l -0093 $3.50/0 SMILEY 94 followed by some remarks on nany-valued loglc. The Axion of Identlty. It is unfortunate that the one economlst whom the authors cit.e as violatlng thls rule actually provides a textbook example of obedlence co tt! For thelr os,n quotatlon from Muth ln Exanple 3 shows hlm uslng trro terms, namely "objectlve probabillty" and "subjective probabi1lty", lnstead of uslng the same term "probabtlity" ln t\to senses. But ln any case the l-ssue ls a spurious one. The purpose of Ehe loglclanst rule is to avoid fallacles due to amblgulty. Ustng a symbol to stand for dlfferent things ln the same formula ls asking for trouble. Outslde formal loglc, howeverr the fact that a term stands for more than one thtng does not necessarlty create arnblguity. The difference is that a fu11y formaL argument leaves oo scope for the play of context' whereas in an lnformal argument the context can make lt perfectty clear which sense of a word ls ln question at (rhlch momentr so there need be no genuine conflict wlth Ehe spirit of the loglclansr house ru1e. The Law of Contradlctlon. The authors say that cerEain theorles of ratlooal expectatlons have been found to be lnconslstent (1.e. self-contradictorY), and thls 1s supposed t.o vlolate the law of contradictlon (Example 4). In itself it does oothing of the sort: lt all depends oo what happens next. If the dlscovery of lnconslstency ls taken as grounds for abandonlng or anending the theory ln question, then far from vlolating the law of contradictlon such behaviour will vlndicate lt. For the 1aw of cootradictlon doesntt say EhaE there cannot be an lnconslstent theory: i.t says that an FOUNDATIONS OF ECONOMETRICS-COMMENT 95 inconsistenE theory cannot be true; and this is what leads usr as seekers after truth, to reject a theory once it ls known to be lnconslstent. The only way an economlst could violate the law of contradictlon would be by adoittlng that his pet theory rtas lnconslsEent but clalming that thls didnft matter. The same nlsunderstandlng runs through the sectlon on G6delts theorem, wlth lts extraordinary heading "Giidelrs challenge to the axiom of noncontradictlon". Giidel showed that any sufflclentLy strong theory must be lncooslstent or incomplete, but he could only be seen as challenglng the law of contradictlon 1f he had lnvited theorlsts to sacrlfice consistency in order to achleve completeness, and of course he atltl no such thing. I agree with the authorsr warnlng about the danger of creatlng lnconslstency when rival hypotheses are comblned into a more general nodel, but I cannot see whab Ehey think thls has got to do wlth Giidel|s theorem. Nor can I nake out the alleged impllcatlons of G6delts theorem for Dempst.errs or Tverskyrs crlterla of consistency, or for the clalm that "coherence cannot be achleved ln sufflclently large worlds". 0n the contrary the whole appeal to Giidel rs theorem appears to be a red herring. of Excluded Mlddle. This ls sald to be vlolated by economlstsr reltance oo approxlmations, for example in demand systems (Example 1) or Ehe lnterpreEation of probabtllty (pp. 1In.r 51, 54). This Ls not so' If someone puts forward an approximatlon as an approxiroatlon, e'9. by assertlng that'il = 22/7 +.002, then what he says ls The Law 96 SMILEY straightforerardly true. On the other hand tf he puts it forward wlthout quallficatlon, asserting baldly thaE T( = 22/7, then what he says 1s stralghtforwardly false. Neither way 1s there any violatlon of the law of excluded ntddle. The ldea that there is a status "approxlrnately true" genulnely located somewhere between true and false 1s a mlstake, llke that ldea that there ls a person called "the average plumber" who genuinely has 1.3 chlldren and 1.9 Legs etc. The rnlstake ln each case ls to take at face value an idiom that onty nakes sense when construed as shorthand for something else. Thus the clalm that the average plumber has 1.3 chlldren needs to be seen as shorthand for an assertion about the ratlo of plumbers to chlldren of plunbers, and llkewlse the clalm that Tf = 22/7 is "approxlmately true" needs to be seen as shorthand for an assertlon about the relative smallness of the difference between the two numbers. When chis ls done the nystery disappears along with the phrase that caused lt. A common mlstake ln dlscussions of the law of excluded rnlddle ls to confuse a statementrs being true wlth 1ts being knowo to be true. The authors do thts when they equate our inablllty Eo tell whether statementg abouE the future are true or false with thelr actualty not betng elther true or false. The same mistake lles behind thelr clain that probable sEatements are nelther true nor false (Example 2). Thelr authorlty here ls Boland fs "The Foundatlons of Economlc Method", and what Boland says ls "If \ile adopt the stochastic-conventlonallst vlew that identlfies absolute truth wlth a probablltty of I and absolute falslty with 0 then o.. a stochastlc statement with a probablllty of 0.6 ls not absolutely true, oor ls lt I tI FOUNDATIONS OF ECONOMETRICS-COMMENT 97 absolutely false". I canrt speak for "stochastic conventLonalists", but any reputable wrlter on probablllty w111 say that what corresponds to a statementrs having probabllity I ls not its being true but lts belng known to be true (or lts betng certaln, confirrned, etc. See for exarnple Carnap, "The Logical Foundations of Probabtlity", p.177). Probabtllty is not a continuum between truth and falsity; it is a conEinuum in a different dlmenslon of assessment r Less actually turns on the issue of excluded mlddle than the authors thtnk. They say that the use of logic Eo establlsh concLusl-ons (rnodus ponens), and to refute theortes (modus tollens), both depend on the law of excluded otddle (pp.7, 19). In fact neLther does so, as nay be conflrned by observlng that both hold for lntultlonist loglc desplce its rejectlon of excluded niddle. A11 modus ponens requlres 1s Ehat valld arguments are truth-preservlng, plus the presence of a set of true premlsses to work from. The lack of true premlsses, not any supposed failure of excluded niddle, ts the reason why "approxlmate modus ponens" doesnrt work (p.16). Modus tollens llkewlse only requlres that valid arguments are truth-preserving, plus the presence of an untrue conclusion to work back from. The authors seem to have been led astray agaln by Boland, who describes refutation as lf it rtere a two-step process: (a) work back from the untTuth of the concluslon to the untruth of the assumptloos taken collectively; (b) argue from the collective untruth of the assumptlons to the falslty of at least one of them. The second step does indeed depend on he law of excluded rnlddle, but lt is a polntless addition: the flrst step is already an adequate account both of modus SMILEY 98 tollens and of refutatlon. logic. I have tried to show that Ehere ls no reason for econometrlclans to reJect classlcal logic, but in case anyone 1s unconvl-nced let IE cooclude with three l,Iany-valued cautions about rnany-valued logic. The flrst ls that, contrary to the i.mpresslon given by the authors (p.2), not all the aLternatives to classical logic wtll be found under Ehe heading of nany-vaLued logic. As logiclans use Ehe idea, lt is not enough for a system to posit more than two truth-values; lt also needs to be truthfunctional, where this rreans that the truth-values ascrlbed to compound sentences are to depend solely on the truthvalues of their coilponents. Thls ls why probabiltty theory will never make a mrny-valued 1ogic, for although negation is Eruth-functlonal conJunctlon 1s not: the probabtLtty of not-A ls a function of the probablllty of A, but the probabtllty of A-and-B depends on other things than the separate probablllt1es of A and B. Stnl1arly the fact that lntultlonlsts replace the cLasslcal- dichotomy of "true" and "fal-se" by a trlchotomy of "proved", "refuted" and "undecided" does not that lntuitlonist logic ls many'Dean valued. The second cautlon ls that an apparently many-valued logic may t.urn out to be rrerely a sysEem of classlcal logic in disgulse. Thls emerges very clearly frorn the most common net.hod of constructlng many-valued loglcs, whereby some of the truth-values are "deslgnaEed" and an argument ls counted as valld if it always leads from deslgnated values to deslgnated values (cf. p.37). For all thls tells usr "designated" nay turn out to be just another name for FOUNDATIONS OF ECONOUETRICS-COMMENT 99 -true", with the designated values representlng various subcases of Eruth and the undeslgnated ones subcases of In other words, what looks l-lke a rival to falsity. classical logic rnay slmply be a fine-gralned versloo of it. There are nonet,heless some genulne rlvals to classical loglc. Fttzzy logic ls one of them, not because lt is deslgned to handle fuzzy concepts but because 1t treats truth as ltself a ftzzy concept. But - and thls ls the last point - an econometrlclan who follows up the authors t recommendatLon of ftzzy logtc as a tool for dealing wlth -inexact concepts such as approxlmation" (p.37) is llable to be disappolnted. The reason ls that "inexact" can mean at least three things. The statement " 7[' i" . small number" is inexact because it ls Eg. - lt lacks clear-cut truthcondltions owing to the ftzzy nature of the coocept "sroall". The approxlmatlons " Tl = 22/7" aod " It = 22/7 + .002" are not like this at all. One is lnexact because untrue; the other 1s lnexact because tt is unspeclfic; but both have perfectly cl-ear-cut trrrth-coodltions: there ts nothlng vague or fuzzy about either of them. Even lf fuzzy logtc should be the right tool for dealing with fuzzlness and vagueness, wouldnrt lt be fatr to deflne econometrlcs as the part of economLcs that avoids fuzzy concepts and vague statements? T. J. Clare College, Smlley Cambrldge ECoNoMETRIC REVTEWS, 4(1), 101-119 (1985) REPLY are honored by professor Dale poirlerrs offer to publlsh a compressed verslon of our paper along wLth commentarles by such leadlng researchers 1n phllosophy of sclence and econometrlcs as Professors L.A. Boland, I.J. Good, M.H. pesaran, T. Seldenfeld, and r.J. sniley. Thelr comments and those offered earller by our former colleague, Dr. Edward J. Green, have helped us rethlnk sone of the materlal we have presented. Even where crltlclsn has been well founded, falrness diccaces that we present our responses here rather than ln an altered text. Where, ln our subJective vlew, conments were illfounded, we wl1l so lndlcate. As our followlng replles show, our posltlon remalns firm. We Analytlc-Synthetic Distlnction It was good to read professor Bolandrs klnd remarks. Our dlscusslon of Arlstotlers axloms for adrnlsslblllty of statements lnto loglcal arguments orres so much to hls very inportant contrrbu:ions to the foundatlons of eeonomlcs. Ife concur ln the oplnlon that the "analytlcal-synthetlc' dlstinctloo is susplcious. The offendlng foocnote 2 containing Ehls distlnctlon was added to paclfy sorne crltlcs nentloned in our authorsr footnote. In any case' this dlstlnctlon ls not cruclal for our dlscussion, provlded readers understand the word "true" in the same way as Boland does. Arlstotellan Loslc clearly lndlcated wldely known sources for our c€rnlnology ln referrlng to Arlstotlers axlons, Sniley appears to tave the curlous lrnpresslon that "The Law of contradlctlon'. ls a c'lstomary name and that rre have re-chrlstened one of Arlstotlers Even though we 101 Corrisht @ 1985 by Marccl Dckkcr, Inc. 07 47 4938 1 85 1 040 1 -0 1 0 1 $3.s0/0 ECoNoMETRTC REVTEWS, 4(t), 101-119 (1985) REPLY we are honored by professor Dare poirierrs offer to publlsh a compressed versLon of our paper along wlth connentarles by such leadlng researchers ln phllosophy of sclence and econometrlcs as Professors L.A. Boland, I.J. Good, M.H. pesaran, T. Seldenfeld, and r.J. sniley. Thelr comments and those offered earlier by our former colleague, Dr. Edward J. Green, have helped us rethlnk some of the materlal we have presented. Even where crltlclsm has been well founded, falrness dictates that \re present our responses here rather than ln an altered text. where, ln our subJectlve view, comnents rrere lllfounded, we wl11 so lndlcate. As our followlng replles show, our posltlon remalns flrm. Analytlc-Synthetic Dlstlnctlon It nas good to read professor Bolandrs klnd remarks. Our dlscusslon of Arlstotlers axloms for adnnissibllity of statements lnto loglcal arguments owes so much to hls very inportant contributions to the foundatlons of econoralcs. we concur in the oplnlon Shat the "analytlcal-synthetic" dlstlnction ls susplclous. The offending foocnote 2 concainlng Ehls distlnctlon was added to paclfy some crltlcs nentloned in our authorsr footnote. In any case' this dlstlnctlon ls not cruclal for our dlscusslon, provlded readers understand the word "true" ln the same way as Boland does. Arlstotellan Loglc clearly lndlcated wldely known sources for our lernlnol0gy in referrlng to Arlstotlers axlons, snlley appears to have the curlous lmpressLon that "The Law of contradlctlon" is a cuscomary name and that we have re-chrlstened one of Arlscotlers Even though we 101 Cog''tixbt @ 1985 by Marccl Dckkcr, Inc. 0747 4938185104014t 0l $3.s0/0 702 SWAMY, CONWAY, AND VON ZUR MUEHLEN axloms as "The Axlom of ldentity" "for lack of an accepted termlnology." Smlley and Boland nay settle between themselves the assertion that we "have probably been 1ed astray by Bo1and." But that the use of modus ponens and 99dus-!e11eng does not depend on the axlom of the excluded-mlddle is absolutely false. Bolandrs Ereatmenc of the lnterrelatlonshlp between modus ponens (or modus tollens) and Arlstotlets axioms ls nothiog short of excellent. Smlleyrs claim that modus ponens or modus tollens only requlres that valid arguments are Eruth-preservlng lacks a Program guaranteeing that valid arguments are truth-preservlng, thereby asklng readers to act b1lnd1y and mechanlcally. The Axlom of Identlty Snlleyts defense of one economlsE, some of whose work we had found to be 1n vlolation of the axlon of ldentlty, ls based on the somewhat objectlonable assertlon that "economists do not-thank God!--make use of formallzed argunents." Well! Whatever economlsEs do or don't do, the axlom of ldenElty neans that loglcal falslty of the sEatement 11ke "2=4" follows fron the very definltlons of the terms. In an analogous manner' loglcal falslty of the ratlonal expectatlons hypothesls (nnn) as adopted by one "perfect example of obedience to lthe axlorn of ldentlty] " follows from the very deflnltlons of the terms Slnce "subjectlve and objectlve probabllity dlstrlbutions." subjectlve and ob-'iective Probabllitles are entlrely dlfferent concepcs, their lnterchangeable use 1n a theory therefore vlolates The same must be said of Pesaran's statethe axlom of ldentlty. "under the REH che two concepts [subjectlve and objecment that tlve probabllitles] are by definltlon lndisElngulshable from one another as they w111 be one and the same thlng." Two different deflnltlons cannot be made equlvalent by deflnltion. The Axlom of Noncontradlctlon Although Boland agrees wlEh our reconmendatlon Ehat the "axlon of noncontradlctlon" should not be abandoned, Sml1ey FOUNDATIONS OF ECONOMETRICS-REPLY 103 evidently belleves Ehat there cannoE be any vlolatlons of the axlom unless an economist acknowledges--without conErlElon--that hls theory ls lnconslstent. Thls 1s just misgulded. There are no a11bls for vlolatlons of t.hls axiom, acknowledged or not. Incidentally, scientlfic progress is made by accumulaElng evldence agalnst loglcally valld theorles' not by uncoverlng thlnklng mlstakes. Our reasons for not abandonlng the axlom of nonconcradlcclon are the sane as Bolandfs. The Axlorn of the Excluded-Mlddle Bolandrs polnt that "every lndirect proof lnvolves both the axlom of noncontradictlon and axlom of the excluded-mlddle" 1s rell taken. In fact, thls corrects an error ln an earller verslon of the paper. We agree wlth Boland that "avoidlng the use of indirect proofs does not require abandonnent of che axlom of the excluded-mlddle!" Any scatemenE co Ehe contrary that nlght be found ln the paper ls accidental. Our reason for abandoning the axlom of che excluded-nlddle ls not that lndlrect proofs are invalld, although they are so 1n the lnflnite case and ln other cases menEloned above, but only that economeEric work cannot proceed, and may be lmposslble, 1f we lnslst on satlsfylng the axLon of the excluded-nlddle. Let us clarify thls polnt. The real alm of lnference 1s usually to generate a predictlon about che value of sorne future observables. Suppose that a predlctive dlstributlon, say p(yl*), derl-ved from a 1oglca1ly conslstent econometrlc roodel (the truth status of whlch is unknown) ls used to produce an lnterval predlction, say 10.5 to 30.8, for the value of the random variable f tn a future perlod, say T*s. Then both sraremenrs, "pr(10.5!]1a"!30.4;=6.95" and "the probablllty that the reallzed value of if+sr saY Y14g, lles between 10.5 and 30'8 ls either 0 or 1r" are correct because the random varlable i1.r" must be distlngulshed fron the value ylas taken by that random varlable. The statement that 10.5jiT+s!30.8 dlffers from the statement that the L0.59T+s<30.8. I.le have therefore thls lmportant distlnctlon: j1*" asslgnnent of a truth value of 0.95 to a statement about 104 SWAMY, CONWAY, AND VON ZUR MUEHLEN violates the axlon of the excluded-rnlddle' wh1le the asslgnment of a cruth value of 0 or 1 to a statemeot about yT+s does not. Clearly, Bolandrs remark that "any conslderatlon of statenents about the future does noc necessltate a rejectlon of the raxlom of the excluded-nlddle' slnce an adnlsslble staternent ls stl11 elther Erue or false even chough we may not yet know lts truth status" must refer to reallzatLons, not future events consldered as random events. Indeed, when we lncroduce random varlables of Ehe type iT+"r r. cannot even assign a truth value of 1 or 0 to the statemenc, 10.5$1+s!30.4, and hence caonot avold violating the axlom of the excluded-nlddle unless the lnterval 10.5-30.8 covers the entire range of posslble values for Ir*. or ls outslde the This dlfference ln Ehe asslgnnent of truth values range of ]r*". to the stateDents about I1*" and Yr*", lncldentally, applles whether \re use Bayes theoren or not and cannot be bhanged by avoldlng the use of lndlrect proofs. Good's remark thac "[1]f a proposltlon is now only probable for us, then we donrE know now whether lt ls true or false, though we mlght know later, and lt can be crue or false even lf we never flnd out whlch" l-s lnconsequential here. Further, Goodfs stacement chat "... lprobabtllcy theory] also obeys the usual laws of ArlsEotelean logle" ls correct ln the speclal case when by probablllty theory ls meant an unlnc.erpreted calculus of probabillty as deduced, e.8., by Kolmogorov. In any case, we do not know of any exanpLe where "pr(10.5ET+J30.8)=p, 01!n$," does not vlolate the axl.om of the excluded-middle. Pesaran mlsses Ehe polnt that '10.5$1*1!30.8" and "10.5(y1.r"(30.8" are two dlfferent statements, and nlthout reallzlng that we are asslgning a truth value p, Olpll, to the former and noc to the latter, he lncorrectly critlclzes our argument. Smlley says that a common mlstake ln dlscusslons of Ehe axlom of the excluded-nlddle ls to confuse a statementrs belng true wlth lts belng known to be true. Throughouc our dlscusslon of Arlstotellan loglc we only refer to statements whlch are elEher true or false, saylng speclflcally that the truth of assunptions cannot be known SWAMY, CONWAY, AND VON ZUR MUEHLEN the axlorn of the excluded-mlddle. Since the concept of lnfinlty ls used extenslvely in econometrlc work, our objective of dlscoverlng the foundatlons as manlfested in the nature of stochastlc econometrlc theorles cannot be served by rejecclng the use of thls concept, as suggested by Boland. Non-Bayesians as well as some Bayeslans use countablyadditlve dlstributlons where the very deflnltlon of countableaddltivity lnvolves the concept of lnflnlty. Vagueness of Language Both Good and Seldenfeld refer to vagueness and amblgulty of ordlnary language to crltlclze some of our arguments. Good belleves Ehat ArlscoEellan loglc can be used even when the concepts ate fuzzy, but hls arguments are not clear to us. Hls statement thaE "Ehe quesElon whether a rnan has a beard ... ls a clear and sfunp1e exarnple showlng the need ln prlnclple for non-Arlstotelean fr,zzy ].oglc." seems to contradlet another statement, "Arlstotelean loglc could be largely restored to the questlon of beardedness..." We do not deny thaE ordlnary language rnay be vague. Indeed we took care noC to use vague language ourselves. What we have sald ls that slnce there 1s no approxlmate modus ponens, modus ponens or Arlstotelian loglc cannot be used 1f any of our assumptlons are only approxlmately crue. Of course, Seldenfeld rejects "a Eacit assumptlon ... whlch amounts to the presumptlon EhaC approxlmatlons are at odds wlCh Arlscotelean loglcr" and no apology 1s requlred. He also rejects the assumptlon that "approxlmatlons must lnvolve the klnd of vagueness chat makes thern unlncelllglble by standards of classlcal loglc." Thls criticlsm ls beslde the polnt because we do not use Arlstotellan loglc to dlstlngulsh "unlnrelllglble" fron "1ntelliglble" scatenenrs buE to pass along known truths (or falsltles) fron assumptlons (or concluslons) to concluslons (or assunptlons). Let us conslder Seldenfeldrs own example. He says, "... lt is understood by all what the trafflc cop means when lt ls reported that I was drlvlng FOUNDATIONS OF ECONOMETRICS-REPLY 107 approxlmately at 45 mph In a 25 mph zone. I do not belleve thls understandlng lnvolves appllcatlon of some non-scandard loglc!" A personrs understanding of any staEement depends on how well che terms appearlng ln that scatement are defined and how nel1 chat person understands those deflnitlons. Our concern here 1s dlfferent; what can a trafflc courc conclude from the premlse that Professor Seldenfeld was drlvlng approxlnately at 45 nph in a 25 nph zone? No judge ln his rlghc mlnd would allow such evldence ln court because any number less than or greaEer than 25 ls stlll approxlnaEely 45. The evldence ls legally inconcluslve. A semantlc abuse llke Smlleyrs phrase "the average plumber has 1.3 chlldrenr" serving as shorthand for an assertlon about the ratlo of plunbers to chlldren of plumbers would probably not pass an edLtorrs scrutiny. Slmllar1y, no one who has studled nathemaEl-cs w111 use Smlleyts phrase, "Il=22/7 ls rapproxlmately truer," as shorthand for an assertlon about the relatl-ve smallness of the dlfference between Ehe Ewo numbers. We repeat: the nain focus ln our paper is on the connectlon between assumptions and concluslons and not on the usefulness of phrases as shorthands for sentences. The Problen of InductLon It has always been our understandlng that outslde of used-car lots "solutlon" means "complete solutlonr" not partlal solutlon. In Sectlon 4.1 and foocnote 7, we have clarifled what we rnean by "the problern of lnducElon." Good agrees wlth the statemenE that thls problem has no complete solutlon. I{ow do we overstate our case by repeatlng Chls correct scatement whenever necessary? Indeed, ln the paper we never say or funp1y that partlal solutlons cannot be valuable. Concepts such as locally coherent procedures, Blrnbaumrs confldence concept, and Kleferts suggestlons discussed in Sectlons 7.3 and 7.6 would be useful for gettlng partlal solutlons. Boland and Good agree that the problem of lnductlon has no (conplete) solutlon. Therefore ne cannoc verlfy the Eruth of Ehe SIIAMY, CONWAY, AND VON ZUR MUEHLEN condltlons of Ehe laws of large nurnbers or the condltlons under whlch subJectlve probablllty distributlons exlst. Because of thls dlfflculty we are unable to pass along truchs to frequentlstsf or subjectivlstst specl.ficatlon of probabillty dlstributlons slnce these truths are always unknown to ua. It ls prudent to a1low for the possiblllty that the Arlscotellan prlnciple of the excludedniddle does noc hold for frequency or subjectlve interpretaclons of probabillty. What, then' ls so "surprlsing and rather puzzlTng" (Seldenfeld) about thls assertlon? For exarnple, Cram6r (1946, p. 143), a leadlng frequentlst, has sald that to any event connecEed wlth a random experlment, we should be able to ascrlbe a number p such EhaE, ln a long serles of repetltlons of the experlnent, the frequency of the event would be approxlnately equal to p. Slnllar1y' H111 (1975, p. 559), a leading subjectlvisE' has pointed out that the notlon of a "true" denslty ls dlstasteful to a subjectlvist such as hlmself. There are no approxlmately true frequentist or subjectlve probabllitles from whlch unarnblguously true concluslons w111 valldly fo1low r{lth the assurance of modus ponens. If our presentatlon of Bolandrs demonstratlon that there 1s no approxlmate modus ponens or aPProxlmate rnodus tollens ls not "a demonstraElon that the sense of rapproximatlonr used ln the rstandardr lnterpreEatlons of probablllty ls of che sort that cannot be captured by classlcal loglcr" then what is it? It ls not classlcal loglc but the lack of a (complete) solutlon to the problem of lnductlon that ls a hlndrance ln che phllosophlcally dlfflculc task of produclng defenslble expllcations of "probab11ity." It ls also the source of confllcts ln the If we can ongolng debates over Ehe foundatlons of statlstlcs. verlfy the truth of the condltlons of a law of large numbers, then we can say that the frequency lncerpretaclon of probablllty ls also true. Alternatlvely, lf lte can verify the truth of the condiclons under whlch subjectlve probabllltles exist, then we can pass along that known truth to the subjectlve lnterpretaElon In elther case' controversles of the type that of probablllty. FOUNDATIONS OF ECONO},IETRICS-REPLY 109 appeared ln Kiefer (L977a, pp. 822-827 ) surroundlng the question: I{hat 1s the correct lnterpretatlon of probablllty? should end. Slnpllclty or Parsinony Returnlng to Goodrs comments, we do not agree that the concepts of slmpllclEy or parsimony and conplexlty, are absolutely essentLal for model cholce. Conslder, for exarnple, Goodrs own exanple of flttlng a polynonial to n observatlons of a dependent and an lndependent varlable. Though Good belleves Ehat a polynonlal of lower degree ls nore parslmonious than a polynomlal of che (n-1)th degree Ehat fits the data exactly, the latter polynonlal may not very often be the better predl-ctor of the dependent varlable outslde the saurple than polynornlals of lower degree. If it is, then lt 1s useful, but the operatlonal meanlng Good attaches Eo the word "parsLmony" ls nlsleadlng. If our purpose ls not to predlct but to provlde a relative evldentlal evaluatlon of models, then we cannot work wlEh exact flts because an equatlon thac flts the glven data exactly will not enable us to estlmate the error varlance, and hence staclstlcal evLdence 1o the form of Birnbaumrs df and d! f" not possible. Exact fits are analogous to sltuatlons presented by a slngle observatlon y fron a normal dlstrlbutlon wlth unknown mean u and unknown sEandard devlatlon o. The 1lkellhood here 1s inflnlte at the polnt o=0, U=y, suggestlng very strongly that small values of o are more plauslble than larger values, no natter what the value of j that 1s observed. What you get fron thls ls not parsimony, but mlsleading evldence wlth probablllty 1! The confldence concept of Blrnbaun lllunlnates the reason, ocher than slnpllclty or parslmony, for preferrlng ldentlflable rnodels to nonl.dentlflable models whlch nay flt the data exactly. Identlfiabl1lcy is a necessary conditlon for statlstlcal conslstency. It should also be noted that there ls no necessary connectlon between estlmatlon efflclency and predlctlve efflclency and Ehat estlmatlon efflclency applles on1-y to rare sltuatlons. To see thLs, recall the three posslble lnterpretatlons labeled (a), (b), SWAMY, CONWAY, AND VON ZUR MUEHLEN and (c) of the parameters ln the (fl,S,p)-paradigrn presented in applles under all three inEerpretatlons (a), (b), and (c), estlmation efficlency applles only under lnterpretatton (c). Loosely speaking, one can guess estimatlon efflclency, whenever 1c applies, by conputlng the observatlon/parameter ratlo. When this ratlo ls not sufficlently large, parameter estimates wl11 be lmpreclse or Ehe efficiency of the parameter estlmates w111 be low. Also, lf we follow Blrnbaum to provlde evldentlal interpretatlon, then thls evidence nay be worthless ln thls case. However, the predictlons lmp1led by these lmpreclse paramecer estlmates rnay be successful Isee Lad and Swany (1985)1. Moreover, lt ls dlffLcult to opt for lnterprecatlon (c) 1n nonexperlrnental sltuatlons llke econometrlcs because there may not be any model-free physlcal quantlcies standing behind each model paramecer. If lnterpretati"on (c) does not apply' then In any case, 1f we reject estlmatlon efflciency ls irrelevant. a roodel on the ground that lt has too many parameters' then we Therefore, rnay be rejectlng a best predlctor of our varlables. the relevant conslderatlon for predlctlve purposes ls not the number of parameters 1n a nodel but a nodelrs predlctive efflciency. Thus, the prlnclple of parsimony based on the number of unknown parameters can be quite mlsleading. Sectlon 7.6. Though predlctlve efficlency I-tany-Valued Loglc Goodts openlng remark, "Slnce the enphasls was on nultlvalued loglc, perhaps it ls nelther true nor false that they answered the ltltle] questlon," is an amuslng pun. Let us establlsh here that we dld not propose dropplng any of Arlstotlers axloms merely for the purpose of understandlng our paper. More to che polntr whlle we concede the lmportance of language and of avoiding lts amblgultles (exarnples of anblguities ln language rnay be found even among the commentaries here), our Lntent was Eo dellneaEe a mlnlmal set of axloms for a foundatlon of econometrlcs' not physlcs' llngulsElcs or posslbly psychology. We belleve we achieved our purpose. FOUNDATIONS OF ECONOMETRICS-REPLY Contrary to Che lmpresslon glven by Sm11ey, we do not say thaE all the alCernatives to classlcal loglc w111 be found under the headlng of "many-valued" loglc. We treat probablllty theory as a verslon of many-valued logic as suggesEed by Lyndon (L966, p. 33) under the headlng of "Many-valued 1oglcs." He proposes that probablllty theory, ln 1ts most prlmltlve form, resembles many-valued loglc 1n Chat lE atEaches to each formula f as probabtllty a number p ln the lnterval [0,1]. We are arrrare that probabllity dlffers from loglc 1n that pr(AUB) does noL depend on only pr(A) and pr(B) [see Lyndon (1966, p. 33)]. For rhls reason, a systein of rules of lnference that flEs the probability theory ls dlfferent fron the system that flts Arlscotelian loglc Isee McCawley (1981, p. 368)1. Good agrees that probablllty theory certalnly can be regarded as a verslon of many-valued loglc. In our dlscusslon of a verslon of many-valued loglc rJe are, by the way, not dealing wlth degrees of meanlng but wlth degrees of certalnty. The analogy bet$reen ftzzy set theory and probablllty theory 1s merely senantlc. We lncluded a discusslon of flzzy sets 1n Sectlon 7.5 because of 1ts analogles wlth probablllCy theory. The act of asslgnlng truth values to proposltlons ln fuzzy sets theory para11e1s the act of asslgnlng probablllties to sets ln probablllEy theory. We do noE mean to say that the theory of tuzzy sets ls the same as a theory of (part1a11y-ordered) probabilitles. The arguments glven 1n McCawley (1981, Chapter 12) dlsprove Snlleyts claim that fuzzy log|e cannot handle lnexact concepts such as approxlmatlon. Our answer to Snl1ey's closlng questlon 1s no lf by "flzzy concepts" and "vague sEatemenEs" he neans all those statenents whlch violate the axlorn of the excludedmlddle. Many-valued loglc can be used wlthout replaclng the axlorn of noncontradlctlon. The answer to Bolandts questlon, "If st.atements can be Inany-valuedr' what constitutes a contradlction?" depends on the type of many-valued log1c we use. Let us conslder probablllty theory whlch 1s a verslon of many-valued loglc. Glven the trlplet SI^IAMY, CONWAY, AND VON ZUR MUEHLEN (S,BrP)=(a sanple space, a Borel field, a probablllty set functlon), a set belonglng Eo B w111 have only one cruth value, vlz., a probablllEy. In thls sense, a sEatement cannot have ewo dlfferent truth values. Boland further asks, "But, can we say thaE a staternent whlch ls, by any measurement,0.5 Erue is also 0.5 false?" Our answer ls: 1f the probabtllty of a set AeB ls 0.5, then yes, the probablllty of the complernent of A 1s 0.5. Thls is not a conEradlctlon. Aga1n, ln Ehe case of probablllty theory, the axlom of noncontradictlon means coherence ln de Finettlts sense. Thls answers Bolandrs questlon: "Just what does the axlom of noncontradlctloo mean when we abandon chis axlom of the excludedmtddle?" Incoherent oplnlons that do not conform to probabillty calculus are noE free frorn contradlctlons. Smlley mlsrepresents lnstrumentaliscs by saylng that they conslder sclence as rnaglc. Whl1e we are not opposed Eo lnstrumenta11sm, we also do not want to relegate econometrlcs to lmmediate practlcal problems. As Blrnbaurn polnts out, hypothesis testing can be used for evldentlal lncerpretatlon. Contrary to what Boland has said, the questlon of testlng need not necessarlly lnvolve sErategies and preferences concernlng whether a type I or type II error 1s the least preferred on1y, but may lnstead concern relatlve evidentlal evaluatlons of trro or nore hypotheses. Idea11y, we should follow Blrnbaumfs confl.dence concepE: under no rnodel sha11 there be a hlgh probabllity of outcomes lnterpreted as strong evldence agalnst thac rnodel. Thls is not to deny the presence of practlcal dlfflcultles lnvolved 1n the enplrlcal lmplementatlon of Blrnbaumts confldence concept desplte lEs undenlable appeal. Regardlng plpe dreams, \{e are afraid that the econometrics of che founders w111 remaln 111usory 1f lc 1s based on pesaranrs prescrlptlon. A requlrement that lts foundatlon should rest on all of Aristotlers axl-ome includlng the axlom of the excluded-mlddle nay Just remaln the dream of a mldsunmer nlght. But there ls l1ght at the end of the tunnel lf Blrnbaunrs confldence concept 1s taken serlously. More Eo the pol-nt, loglcally conslstent econometrlc FOUNDATIONS OF ECONOMETRICS-REPLY of endogenous varlables, glven exogenous varlables, rarely, lf ever, produce degenerate predlctive distrlbuclons of endogenous varlables. Using nondegenerate predlcclve dlstrlbutlons' we can aL best make probabllistlc statements about Ehe yet-to-be observed future value of an endogenous varlable. The probablllties of such statements wl11 usually be nelthet zero nor one. Alternatlvely, uslng Blrnbaumrs confidence concept' we may glve an evldentlal evaluatlon of two or more competlng models, as long as they are 1oglcal1y conslstenc, of course. This procedure nay not glve a strong evldence agalnst a nodel unless the observaclon/parameter ratlo is sufficlently high. A Ehlrd approach ls to conpare hypoEheses concernlng real events by uslng posterlor odds ln favor of one hypothesls relaElve to another, as ln Zellner (1971). However, the atteinpt to do thls 1s often thwarted by sample lnformatlon that ls weak and hence incapable of providlng the econometrlclan wlth satlsfactory odds ln favor of elther hypothesls. Furthermore, evaluatlon of posterior odds ls hlghly sensltive to the prlor distrlbutlon of parameters under che nalntalned hypothesls. In these cases, one ls not really comparlng hypotheses, but rather To establlsh comblnatlons of hypotheses wlth prlor dlstrlbutlons. of competing hypotheses we may conslder bounds on plauslblllty this: As suggested by H111 (1984a), a sample could reveal how extreme prlor dlstrlbutlons would have to be (wlthin a cercain class of prlors) l-n order to produce declslve evidence for one hypothesls versus another. We would then be in a posltlon to say that such prlor dlstrlbutlons are not ln our "ball-parkr" and therefore that such daEa are lncapable of glving a clear lndlcatlon as to whlch rnodel is preferable. Thls program 1s conslsEenE with Klefer's suggestlon summarized at che end of Sectlon 7.6. Thus' che app1lcabl1lEy of "fvzzy" econometric cheory based on BLrnbaumrs, Hl11ts and Klefer's suggesELons would not be lfunlted to bulldlng economeEric models representlng merely "ftzzy" econonlc theory as suggested by Boland. models that specify Ehe condltlonal dlstrlbutlons swAMY, CONWAY, AND VON ZUR MUEHLEN Achievlng Coherence Returnl-ng to some technlcal comments, the accuracy of Seldenfeldrs clalm that "nany fanlllar (textbook) rorchodoxt statlsclcal procedures have a Bayeslan model under an rimproperr prlor" seems s11ghtly exaggerated. Example 1 does not represenc very many fanlllar (textbook) "orthodox" st.aCistical procedures. For lnstance, frequentlst admissible procedures that are equally famlliar are omltced. Thelr Bayeslan descrlptlons do not always use "inproper" prlors! The "examples" he glves referrlng to Jeffreysr prlors, flnite addlclvity and non-conglomerablllty, are perhaps unnecessary 1n vlew of our references to Zellner's (1971), Vlllegast (1977a, b, 1981) and H111's (1981,1982,1984) work on those subjects. We hope that our st.atement, "[1]f an inproper prlor does noc lead to a posterlor whlch could have been obtalned from a proper, flnitely addltlve prlor, then such a prlor results 1n incoherence and should not be used." ls not really as outrageous as Seidenfeldts crltlcism suggests. Because we hesltated co Eake a chance on obtalnlng lncoherent lnferences, we felt obllged to err on the slde of cautlon 1n reconmendlng the use of lnproper prlors. It ls true that lf Che chosen lmproper prlor has a posterlor whlch could not be obtalned from a proper, flnit.ely additive prlor, then the Heath and Sudderth condltlon for coherency cannoE be used to show that the chosen lmproper prlor 1s lncoherenc. It may be posslble to show, as 1n Hill (1984), that the apparenc 111 consequences of uslng Ehe chosen lmproper prlor cannot really be nade operational. For thls reason, our cautlon may be excesslve. In any case, ao lmproper prlor should not be used if 1c could not be shown by any nethod that the bad consequences of uslng such a prlor cannot be rnade operatlonal. Uslng a new condltlonal frequency lnterpretation of statlstlcal Lnferences, V11legas (1977a, 1981) shows that the posted condltlonal odds based on a posterlor for a posslbly lmproper prlor can be coherenE. The followlng are our responses to Seldenfeld's questlons on FOUNDATIONS OF ECONOMETRICS-REPLY 115 the seventh page of hls conments. Flrst, we conslder lleath and Sudderthts derivatlon of a necessary and sufflcient eonditlon for In referring coherency to be correct. That ls why we accept lt. co our so cal1ed "lnterestlng clalns" spanning lnporcant contrlbutlons fron A-Z--a11 of which are, lncldentally, referenced with proofs--Seldenfeld proposes Ewo proofs by assertlon: (1) the 'approprlateness of the added restrlctlons lmposed by 'H-S coherencet" is to be questloned, and (2) "when ... condltlonal 'H-S lncoherencer can arlse" is an open questlon. If lt ls not known when "11-S lncoherence" could arlse, how is lt chat one should reject "H-S coherence?" [,{e understand H111rs (1981, 1982, 1984) argurnent leading to the concluslon that the I{-S requlrement for coherence 1s too restrl-ctlve co the extent thac le goes beyond the de Flnettl form of coherence (whlch only requires avoldance of sure loss wlth a flnlte number of garnbles). Even Ehough the condltlon of coherency 1s fundarnental for de Finettlrs lnductlve 1oglc, the requirement of conglornerabillty ls not part of that loglc. As H111 (1981) has suggested, we should learn to undersEand aod llve wlth non-conglomerablllty. Subjectlvlsts who' 1lke ltlll' prefer the flnltely addltlve theory, regard Ehe counEably addltlve sltuatlon as rnerely a speclal case of lnteresE' to be justlfied or not ln each appllcatlon on lts own merlt. The congloneracl"ve property should be considered ln the same manner. Whenever we declde co work wlth a flnltely addlclve theory, the Il-S condltlon for coherency ls useful. We must justlfy the serlous charge that non-conglonerablllty ls l-rraEional, as Seldenfeld has suggested, only lf we apply the H-S condltion for coherency bllndly and mechanlcally. Gidelrs Incornpleteness Theorem Most practlcloners of the phllosophy of sclence and mathematlcal loglc wl11 appreci-ate attempts to develop the trall from Arlstotle through Codel to Blrnbaum as long as such efforts pass the tests that phllosophers of sclence have constructed. Thorough- l- 16 SI^rAMY ' CONWAY ' AND VON ZUR MUEHLEN ness of coverage \dou1d be one of then. But Seldenfeld appears to have sone mlsglvlngs concernlng our dlscussion of Godelts lncompleteness theorem, polntlng to a lack of plausible examples ln economlcs. A confllct bet\teen deductlve conpleteness and loglcal consistency ln econometrlcs necessarlly arlses 1f the econonetrlc theorles we are deallng wlth are consistent, axlomaLlzed extenslons of che theory of naeural numbers, as polnted out by Edward Green (Unlverslty of PtEtsburgh) who klndly drew our attentlon to an excellent book by Joseph R. Shoenfleld entltled Mathenatlcal Loglc. There it 1s scated wlth proof that a conslstent Eheory T ls also complete lf lt admlts ellnlnatlon of quantlfiers, if lt contalns a constant, and lf lts every varlable-free formula ls decidable ln T. In hls comments, Smlley neglected to conslder the posslblllty that these condltlons are vlolated when rlval hypotheses are combined into a more general model or when one tries to achleve Dempsterrs or Tversky's type of consi-stency. Likewlse, our reference to Barnard and Good notwlthstandlng, Srnlley apParently does not see the relatlonshlp between these conditlons and coherence ln sufficiently large worlds. Let us explaln why we think that G;de1rs theorem has relevance to a foundatlonal dlscusslon of ecooometrlcs. A forrnula f of a theory T 1s decldable ln T lf either f or not-f ls a theoren of T. The concept of completeness ls related to dectdablllcy: lf an axlomatlzable theory T is cornplete and consistent, then 1t 1s decidable. For exanple, both predlcate loglc and arlthnetlc are undecldable though the former ls axLomatlzable and 1s the latter ls not. As Lyndon (1966, p. 38) polnts out"'lt the exceptlon rather than che rule that a theory of any genulne nathematlcal cornplexlty 1s decldable." Llke nost decidabillty proofs for theorles ln a language with quantlflers, the decldabi11ty proofs for econometric theorles can use the method of ellmlnatlon of quantl-flers. If econometrlc theorles saclsfy the lsornorphlsm condltion and the submodel condltlonr then they adnlt ellnlnatlon of quantLflers [see Shoenfleld, op. cit.r p. 85]. It FOUNDATIONS OF ECONOMETRICS-REPLY tr7 ls not posslble to show that every econometric theory satlsfles these condltlons. One set of undecldabtltty results whose proofs are lndependent of Godelts proof of the undecidabtllcy of arlthnetlc Ls due to Post and Turlng who escabllsh the undecldablllty of the questlon of whether an ldeallzed compuElng machlne, glven a certaln lnput, will come to the end of lts conputation [see Lyndon (1966, p. 42)1. Thls shows that, contrary to what Good has sald, Turlngrs idea to base the foundations of nathematics (or econonetrlcs) on a carefully specifled computer was not so good. A result related to that of Post and Turlng ls the unsolvabllity of the word problern for senlgroups or groups [see Lyndon (1966, g.42)l whlch 1s relevant here because lt ls posslble chat some econometrlc theorles are the lsomorphlsms of the word problem for semlgroups or groups. Unacceptable resulEs can be easl1y obEalned lf che undecldabl1lty of formalized econometrlc theorles are lgnored. For appealing axloms of group rationexample, Ithen two intultively allty, viz., "external Bayeslanity" and "ltke1lhood prlnclple," are used to determlne an oplnlon pool ln a nultl-agent declslon problem, then a generally unacceptable consensus may arlse whlch lgnores alt but one of the oplnlons expressed [see Genest (1984)]. One posslble reason for M11Eon Frlednan's perennlal unhapplness wtth Federal Reserve pollcy 1s perhaps the recurrlng lncompatlblllty of hls opinlons with those of the pools used by the Federal Reserve. The lnpossibillty of obtalnlng a pollcy concluslon whlch ls acceptable to all (ratlonal) economists even wlth ldentical lnformatlon sets provldes one plauslble example of undecldablllty ln economlcs. The absence of consensus among economlsts even wlth ldentlcal lnformatlon sets ls the result of the undecidabillty of certaln formallzed econometrLc theorles. Thls means that such theorles can be elther lnconsistent or lncomplete. If we denand conslstency, then we may not be able to achl.eve completeness. But the axlom of noncontradlction cannot be satlsfled lf the theorl-es are lnconsls- SI^]AMY, CONWAY, AND VON ZUR MUEHLEN tent. Our scatement 1s consistent $71th Nagel and Newmants (1958' p. 6) observatlon thaL G;de1 "presenEed mathematiclanrs wlth the asEoundlng and melancholy concluslon that the axlonatlc method has certaln lnherent llnlcatlons ..." and Hofstadler's (1980, p, 25) observatlon chat G;delts paper "1n some ways utterly demollshed "noE only that there were Hilbertrs program" ... by reveallng... lrreparable rholesf ln the axlomatlc system proposed by Russell and Whitehead, but more generally' that no axlonatic systen whatsoever could produce all number-theoretlcal truths, unless lt were an lnconslstent system!" If these statements donrt represent a challange, then what does?! Thus, Godel poses a challenge to che axlom of noncontradlctlon. This ls not the same as Smlleyrs clalm that'[Coaef] could only be seen as challenglng the Law of Contradictlon lf he had urged mathematlclans to sacriflce conslstency for the sake of completeness." The concluslon part of thls sentence would not even follow from 1ts assumptlon part' Srnileyrs suggestlon that the whole buslness of Godelts Eheorem ls sonethlng of a red herrlng ls a red herrlng. Tn conclusion, to errors and omlsslons catalogued by our klnd dlscussants we plead not gul1ty. Any remalnlng lmpresslon that our lnEerpretatlons of classical logic are lncorrect or Ehat our advocacy of nany-valued logic ls mlsconcelved should, 1n Slr Wlnston Churchlllrs fellcltous phrase, perhaps be wrltten off as "gross termlnologlcal lnexactltude." As best as we can judge' those crltlcisms are not based on a clear, correct, or even fair understandlng of our paper. Nevertheless' we are wl111ng to asslgn a truth value greater than zero (but less than one) to the prospect that some readers w111 not agree wlth all we have to say. P.A.V.B. Swany Federal Reserve Board Roger K. ConwaY DeparEment of Commerce P. von zur Muehlen Federal Reserve Board FOUNDATIONS OF ECONOMETRICS-REPLY ADDITIONAL REFERENCES Crarn6r, H., (1946). Machernatlcal Methods of Statlsclcs. Prlnceton Unlverslty Press. Princeton: Genest, C., (1984). A confllct subjectlve dlst.ributlons. between two axloms for cornblnlng J.R. Staclst. Soc. B, 46, 403-405. Hi1l, 8.M., (1982). Conglomerablllty for distributlons that are only finltely additlve. Technlcal Report /il13, Unlverslty of Mlchlgan, Ann Arbor. H111, 8.M., (1984a;. Discusslon of che papers by DeGroot and Erlksson, Rossl, and Lltterman. Proceedlngs of the Soclal and Economlc Sectlon, Anerlcan Statlstlcal Assoclation. Ilofstadter, D.R., (1980). Codel, Escher, Bach: An ELernal Golden Brald. New York: Vintage Books. Lad, F. and Swamy, P.A.V.B., (1985). Enplrical assessments of the efflclent rLarkets hypothesls: A subjectlvlst analysls of the variance bounds approach, Special Studles Paper, Federal Reserve Board, Washlngton, DC. Nagel, E. and Newman, J.R., (1958). Godelts Proof. New York Unlverslty Press. New York: FOUNDATIONS OF ECONOMETRICS-REPLY 105 unless the problern of lnduction 1s solvable. Smileyrs dlstlnction ls therefore not germane. In Sectlon 7.5 we dlscuss sEat,ements llke, pr(10.5<iT+s<30.8)=p, 0311, and pr(10.5(y1*;!30.8)=0 or 1. Both these statements, though dlfferent, are true. Therefore, Snlleyrs asserElon Ehat ne equate people's present 1nabl1ity Co tell whether fuEure statements are true or false wlth thelr actually belng nelther true nor false is lncorrect. Furthermore, he seerns t.o be lmplylng that in our Chlnklng probablllcy ls a contl-nuum between truth and falslty. Following Dempster, we lnterpret probablllties as degrees of certalnty, and thls 1s what Sniley may have meant. In a related context, Goodrs renark that "... the statement thac sonethlng ls approxlnately true can 1tse1f be absolutely Erue lf anythlng can be." appears to be a semantlc puzzle Lnvolvlng a contradlctlon. Indeed, Popper (1965), who ls quoted ln Bolandrs comments, has shown that anythlng can be true if the axiom of noncontradlcElon 1s vlolaced. Popper put great emphasls on the fact that ln standard logic a contradlctlon leads to loglcal anarchy, as for example, in Pesaranrs statenent that "[1]f the orlglnal denand system ls caken to be the truth, then from a strlccly Loglcal vlewpolnt both approxlrnatlons are false." Goodfs Justlflcatlon of the sEaEement that no vlolatlon of the law of the excluded-mlddle follows fron the statenent thac an event occurs wlch probablllty p, whether Ehe probablllty ls lnterpreted ln the long-run frequency sense or 1n the sub-jectlve (personal) sense ls couched ln some conslderable quantlty of fuzzy semantics, thereby undermlnlng itself: "ntght notr" "somewhat "mlght "some unclearr" meanr" er" "mlght be true or falser" 'provlded long run 1s not too 1ong." If che parameter "p" 1n the statement, "the probablllty of an event 1s pr" ls characterl-zed ln such semantlc hedges, then lt ls, of courser funposslble to determlne lf a specifled value of p ls true or false. Therefore ee are justlfled ln assumlng that the probablllCy of an evenc ls approxl.nately p. But thls assunptlon lnvolves the vlolatlon of