Time series

Australian & New …, 2000

Australian Economic Papers, 1993

first-order autoregressive (AR(1)) error process has played an important role in econometric applications of the linear regression model. Lately, increased attention has been paid to the first-order moving average (M4(1)) error process as an alternative (see for example Nicholls, Pagan and Terrell, 1975; King, 1983a; King and McAleer, 1987; and Silvapulle and King, 1991). There is also the disturbance process based on the sum of independent white noise and AR(1) components proposed by Revankar (1980). When one examines the usual reasons for including an error term in the regression model; these being to account for the effects of omitted regressors, errors in the measurement of the dependent variable, arbitrary human behaviour and functional approximation; Revankar's proposal is appealing because some of the effects are typically autocorrelated while others are independent (see King, 1986). All three error processes have one thing in commonthey are special cases ofARMA(1,l) processes. In the AR(1) case, the correlation between disturbances k periods apart is pk while for an MA(1) process it is p (with IpI s 0.5) for k = 1 and zero for k > 1. The ARMA(1,l) process provides a compromise between these two extremes, having a correlation of rpk-1 which allows for non-zero correlations between disturbances more than one period apart and also allows a separate rate of decay from the correlation between adjacent disturbances. The latter may be particularly appealing for models in which the errors must take account of a learning process that for most agents lasts at least one period. This all suggests we need to have confidence in the power (or the ability to reject the null hypothesis when it is false) of tests for independence of regression errors against ARMA(1,l) alternatives rather than just AR(1) or possibly MA(1) alternatives. The most popular test, the DW test, is approximately locally best invariant (LBI) against AR(1) alternatives (Durbin and Watson, 1971) and MA(1) alternatives (King, 1983b) and is approximately uniformly LBI against ARMI(1,l) alternatives (King and Evans, 1988). While this property ensures close to optimum power in the neighbourhood of the null hypothesis (H,) of independent errors, it guarantees nothing about the test's power some distance from H, when wrongly accepting H, might have damaging consequences. King (1983b, 1985a, 1987) has demonstrated that an attractive alternative is to use a point-optimal invariant (POI) test, i.e. a test that optimises power at a predetermined point under the alternative hypothesis.

T he great workhorse of applied econometrics is the least squares model. This is a natural choice, because applied econometricians are typically called upon to determine how much one variable will change in response to a change in some other variable. Increasingly however, econometricians are being asked to forecast and analyze the size of the errors of the model. In this case, the questions are about volatility, and the standard tools have become the ARCH/ GARCH models. The basic version of the least squares model assumes that the expected value of all error terms, when squared, is the same at any given point. This assumption is called homoskedasticity, and it is this assumption that is the focus of ARCH/ GARCH models. Data in which the variances of the error terms are not equal, in which the error terms may reasonably be expected to be larger for some points or ranges of the data than for others, are said to suffer from heteroskedasticity. The standard warning is that in the presence of heteroskedasticity, the regression coefficients for an ordinary least squares regression are still unbiased, but the standard errors and confidence intervals estimated by conventional procedures will be too narrow, giving a false sense of precision. Instead of considering this as a problem to be corrected, ARCH and GARCH models treat heteroskedasticity as a variance to be modeled. As a result, not only are the deficiencies of least squares corrected, but a prediction is computed for the variance of each error term. This prediction turns out often to be of interest, particularly in applications in finance. The warnings about heteroskedasticity have usually been applied only to cross-section models, not to time series models. For example, if one looked at the

and several NSF referees for helpful comments and suggestions. All remaining errors and omissions remain the responsibility of the author.

Journal of Business & Economic Statistics, 1993

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.

Journal of Time Series Analysis, 1983

Squared-residual autocorrelations have been found useful in detecting nonlinear types of statistical dependence in the residuals of fitted autoregressive-moving average (ARMA) models (Granger and Andersen, 1978; Miller, 1979). In this note it is shown that the normalized squared-residual autocorrelations are asymptotically unit multivariate normal. The results of a simulation experiment confirming the small-sample validity of the proposed tests is reported.

1996

ARCH models are used widely in analyzing economic and financial tme series data. Many tests are available to detect the presence of ARCH; however, there is no acceptable procedure available for testing an estimated ARCH model. In this paper we develop a test for a linear regression model with ARCH disturbances using the framework of the information matrix (IM) test. For the ARCH specification, the covariance matrix of the indicator vector is not block diagonal, and the IM test is turned out to be a test for variation in the fourth moment, i.e, a test for heterokurtosis. An illustrative example is provided to demonstrate the usefulness of the proposed test.

The portmanteau statistic for testing the adequacy of an autoregressive moving average (ARMA) model is based on the first m autocorrelations of the residuals from the fitted model. We consider some of portmanteau tests for univariate linear time series such as Box and Pierce [2], Ljung and Box [9], Monti [12], Peña and Rodríguez [13 and 14], Generalized Variance Test (Gvtest) by Mahdi and McLeod [11] and Fisher [4]

The double autoregressive model finds its use in the modelling of conditional heteroscedasticity of time series data. In view of its growing popularity, the goodness-of-fit of the model is examined. The asymptotic distributions of the residual and squared residual autocorrelations are derived. Two test statistics are then constructed which can be used to measure the adequacy of the conditional mean and conditional variance components of a fitted model. Our goodness-of-fit tests out-perform other benchmark tests such as the Ljung–Box test in simulation studies. To illustrate the testing procedure, the model is fitted to the weekly log-return series of the Hang Seng Index.

2017

Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence.