Forecasting water consumption in Spain using univariate time series models

1 Forecasting water consumption in Spain using univariate time series models Jorge Caiadoa a Department of Economics and Management, School of Business Administration/Polytechnic Institute of Setúbal, and CEMAPRE/Technical University of Lisbon, Campus do IPS, Estefanilha, 2914-503 Setúbal, Portugal. Tel.: +351 265 709 438. Fax: +351 265 709 301. E-mail: jcaiado@esce.ips.pt Abstract: In this paper, we examine the daily water demand forecasting performance of double seasonal univariate time series models (Exponential Smoothing, ARIMA and GARCH) based on multi-step ahead forecast mean squared errors. We investigate whether combining forecasts from di¤erent methods and from di¤erent origins and horizons could improve forecast accuracy. We use daily data for water consumption in Spain from 1 January 2001 to 30 June 2006. Keywords: ARIMA; Combined forecasts; Double seasonality; Exponential Smoothing; Forecasting; GARCH; Water demand. 1. Introduction Water demand forecasting is of great economic and environmental importance. Many factors can in‡uence directly or indirectly water consumption. These include rainfall, temperature, demography, pricing and regulation. Weather conditions have been widely used as inputs of multivariate statistical models (regression, transfer function, vector autoregression, and arti…cial neural networks) for water demand modelling and forecasting. See Maidment and Miaou (1986), Fildes, Randall and Stubbs (1997), Zhou et al. (2000), Jain, Varshney and Joshi (2001), Bougadis, Adamowski and Diduch (2005), and Gato, Jayasuriya and Roberts (2007). These approaches have drawbacks in water demand prediction as a result of weather conditions variability and changes. Water demand is highly dominated by daily, weekly and yearly seasonal cycles. The univariate time series models based on the historical data series can be quite useful for short-term demand forecasting as we accommodate the various periodic and seasonal cycles in the model speci…cations and forecasts. To improve forecast accuracy, we may then combine forecasts derived from the various univariate methods and from di¤er- ent forecast horizons. Combining forecasts can reduce errors by averaging of independent forecasts, and is particularly useful when we are uncertain about which forecasting method is better for future prediction. Some relevant empirical studies using combined forecasts are summarized in Clemen (1989) and Armstrong (2001). In this paper, we examine the daily water demand forecasting performance of double seasonal univariate time series models based on multi-step ahead forecast mean squared errors. We investigate whether combining forecasts from di¤erent methods and from di¤erent origins and horizons could improve forecast accuracy. The most accuracy forecasting methods are then used for outof-sample daily and weekly average forecasting of water consumption in Spain. Our interest in this problem arose from time series competition organized by Spanish IEEE Computational Intelligence Society at the SICO’2007 Conference. The remainder of the paper is organized as follows. Section 2 discusses the methodology used in time series modelling and forecasting. Section 3 describes the dataset used in the study. Section 4 presents the empirical results. Section 5 o¤ers some concluding remarks. 2 2. Methodology 2.1. Forecast evaluation Denote the actual observation for time period t by Yt and the forecasted value for the same period by Ft . The mean squared error (MSE) statistic for the post-sample period t = m+1; m+2; :::; m+ h is de…ned as follows: M SE = 1 h m+h X 1 t=m+1 (Yt 2 Ft ) . (1) This statistic is used to evaluate the out-ofsample forecast accuracy using a training sample of observations of size m < n (where n is the sample size) to estimate the model, and then computing recursively the one-step ahead forecasts for time periods m + 1, m + 2, ... by increasing the training sample by one. For k-step ahead forecasts, we begin at the start of the training sample and we compute the forecast errors for time periods t = m + k, m + k + 1, ... using the same recursive procedure. 2.2. Random walk The naïve version of the random walk model is de…ned as Ft+1 = Yt . (2) This purely deterministic method uses the most recent observation as a forecast, and is used as a basis for evaluating of time series models described below. 2.3. Exponential smoothing Exponential smoothing is a simple but very useful technique of adaptive time series forecasting. Standard seasonal methods of exponential smoothing includes the Holt-Winters’ addtive trend, multiplicative trend, damped aditive trend and damped multiplicative trend (see Gardner, 2006). We implemented the double seasonal versions of the Holt-Winters’ exponential smoothing (Taylor, 2003) in order to take into account the two seasonal cycle periods in the water consumption (daily and weekly). The double seasonal additive methods outperformed the double seasonal multiplicative methods. Within the double seasonal additive methods, the additive trend was found to be the best for one-step ahead forecasting. The forecasts for Taylor’s exponential smoothing for double seasonal additive method with additive trend are determined by the following expressions: Lt = (Yt St 7 Dt 365 ) + (1 )(Lt 1 + Tt 1 ) Tt = (Lt Lt 1 ) + (1 )Tt 1 St = (Yt Lt Dt 365 ) + (1 )St 7 Dt = (Yt Lt St 7 ) + (1 )Dt 365 (3) (4) (5) (6) Ft+h = Lt + Tt h + St+h 7 + Dt+h 365 + h [Yt (Lt 1 Tt 1 St 7 Dt 365 )] (7) where Lt is the smoothed level of the series; Tt is the smoothed additive trend; St is the smoothed seasonal index for weekly period s1 = 7; Dt is the smoothed seasonal index for daily period s1 = 365; and are the smoothing parameters for the level and trend; and are the seasonal smoothing parameters; is an adjustment for …rst-order autcorrelation; and Ft+h is the forecast for h periods ahead, with h 7. We initialize the values for the level, trend and seasonal periods as follows: 365 L365 T365 S1 D1 = = 1 X Yt 365 t=1 1 3652 = Y1 = Y1 730 X t=366 Yt 365 X t=1 Yt ! L7 ; :::; S7 = Y7 L7 L365 ; :::; D365 = Y365 L365 The smoothing parameters , , , and are chosen by minimizing the MSE statistic for one-step-ahead in-sample forecasting using a linear optimization algorithm. 2.4. ARIMA model We implemented a double seasonal multiplicative ARIMA model (see Box, Jenkins and Reinsel, 1994) of the form: 3 p (B) P1 (B s1 ) P2 (B s2 )(1 B)d (1 B s1 )D1 (1 B s2 )D2 (Yt c) = q (B) Q1 (B s1 ) Q2 (B s2 )"t (8) where c is a constant term; B is the lag operator such that B k Yt = Yt k ; p (B) and q (B) are regular autoregressive and moving average polynomials of orders p and q; P1 (B s1 ), s2 s1 s2 P2 (B ), Q1 (B ) and Q2 (B ) are seasonal autoregressive and moving average polynomials of orders P1 , P2 , Q1 and Q2 ; s1 and s2 are the seasonal periods; d, D1 and D2 are the orders of integration; and "t is a white noise process with zero mean and constant variance. The roots of the polynomials p (B) = 0, s2 s1 q (B) = 0, P2 (B ) = 0, P1 (B ) = 0, s2 s1 Q2 (B ) = 0 should lie outQ1 (B ) = 0 and side the unit circle. This model is often denoted as ARIMA(p,d,q) (P1 ,D1 ,Q1 )s1 (P2 ,D2 ,Q2 )s2 . We examine the sample autocorrelations and the partial autocorrelations of the di¤erenced series in order to identify the integer’s p, q, P1 , Q1 , P2 and Q2 . After identifying a tentative ARIMA model, we estimate the parameters by Marquardt nonlinear least squares algorithm (for details, see Davison and MacKinnon, 1993). We check the adequacy of the model by using suitable …tted residuals tests. We use the Schwarz Bayesian Criterion (SBC) for model selection. 2.5. GARCH model In many practical applications to time series modelling and forecasting, the assumption of nonconstant variance may be not reliable. The models with nonconstant variance are referred to as conditional heteroscedasticity or volatility models. To deal with the problem of heteroscedasticity in the errors, Engle (1982) and Bollerslev (1986) proposed the autoregressive conditional heteroskedasticity (ARCH) and the generalized ARCH (or GARCH) to model and forecast the conditional variance (or volatility). The GARCH(p,q) model assumes the form: 2 t =!+ p X j=1 2 j t j + q X i=1 2 i "t i , (9) where p is the order of the GARCH terms and q is the order of the ARCH terms. The necessary conditions for the model (9) to be variance and covariance stationary are: ! > 0; = 1; :::; p; i 0, j = 1; :::; q; and Pj p 0, j P q j=1 j + i=1 i < 1. Last summation quanti…es the shock persistence to volatility. A higher persistence indicates that periods of high (slow) volatility in the process will last longer. In most economical and …nancial applications, the simple GARCH(1,1) model has been found to provide a good representation of a wide variety of volatility processes as discussed in Bollerslev, Chou and Kroner (1992). In order to capture seasonal and cyclical components in the volatility dynamics, we implemented a seasonal-periodic GARCH model of the form: 2 t = !+ + 2 1 t 1 + 2 365 "t 365 + 2 1 "t 1 M X m=1 + m 2 7 "t 7 cos 2 mSt 7 2 mSt 2 mDt +'m sin + m cos 7 365 2 mSt 2 mDt + 0m "2t 7 cos + m sin 365 7 2 mDt 2 mSt + 0m "2t 365 cos +'0m "2t 7 sin 7 365 2 mDt + 0m "2t 365 sin , (10) 365 where St and Dt are repeating step functions with the days numerated from 1 to 7 within each week, and from 1 to 365 within each year, respectively. This approach was used by Campbell and Diebold (2005) to model conditional variance in daily average temperature data, and by Taylor (2006) to forecast electricity consumption. We set M = 3 for the Fourier series. We estimate the model by the method of maximum likelihood, assuming a generalized error distribution (GED) for the innovations series (see Nelson, 1991). 2.6. Combining forecasts We examine whether combining forecasts from the various univariate methods and from di¤erent 4 forecast origins and horizons could provide more accurate forecasts than the individual methods being combined. We consider all possible combinations of the forecast methods Holt-Winters (HW), ARIMA (A) and GARCH (G), and we compute the simple (unweighted) average of the forecasts, = 8 = 0, and 1 = 2 = 0. 3 = 0, 1 = The estimated results are shown in Table 1. We …tted a signi…cant parameter ARIMA-GARCH model of the form: (1 (1 = (1 Ft = (HW ) Ft (A) Ft + + (G) Ft 3 , (11) where is the forecasted value of method ( ) in time period t. We drop the random walk (the worst method) of the combination. 3. Data We analyze the daily water consumption series in Spain from 1 January 2001 to 30 June 2006 (2006 observations). We have drop February 29 in the leap year 2004 in order to maintain 365 days in each year. This series is plotted in Figure 1. The dataset was obtained from the Spanish IEEE Computational Intelligence Society (http://www.congresocedi.es/2007/). We use the …rst 1976 observations from 1 January 2001 to 31 May 2006 as training sample for model estimation, and the remaining 30 observations from 1 June 2006 to 30 June 2006 as postsample for forecast evaluation. 4. Empirical study 4.1. Estimation results The implementation of the double seasonal Holt-Winters method to the water demand series Yt gives the values: = 0:000, = 0:755, = 0:303, = 0:294 and = 0:607. After evaluating di¤erent ARIMA formulations, we apply the following multiplicative double seasonal ARIMA model: (1 (1 = (1 1B 7 2B B )(1 9B 9 2 B )(1 This model ARIMA(4; 0; 9) 4B 365 4 )(Yt 3B 21 )(1 1B 7 2B 14 ) c) )(1 can be (2; 1; 3)7 1B 365 )"t represented as (0; 1; 1)365 , with 2B B )(1 9B 9 2 B )(1 4B 365 4 )(Yt 3B )(1 1B 7 2B 14 ) c) 21 )(1 1B 365 )"t and 2 t () Ft 1B 7 = !+ 2 1 t 1 +'1 sin + 2 Dt 365 2 1 "t 1 + + '03 "2t 2 365 "t 365 365 sin 6 Dt 365 The model estimates are given in Table 2. 4.2. Forecasting results The performance of the estimated univariate methods were evaluated by computing MSE statistics for multi-step forecasts from 1 to 7 days ahead. Table 3 gives the forecasts results for the post-sample period from 1 June 2006 to 30 June 2006. Table 4 gives the forecast results for the weekly 7-days of the same post-sample period. The ARIMA and GARCH models appear to have the same forecast performance for all the forecast horizons. The Holt-Winters outperformed the ARIMA and GARCH models in long horizons. In contrast, for one to four steps ahead forecasting the ARIMA and GARCH models performed better than the Holt-Winters. The random walk model ranked last for any of the forecast horizons considered. For the 7-days of week, the ARIMA appear to perform well for Monday and Tuesday forecasting, the simple combinations Holt-ARIMA and Holt-GARCH appear to be most useful for Wednesday forecasting, the Holt appears to be the most appropriate method for Thursday, Friday and Sunday forecasting, and the GARCH appears to be the best method for Sunday forecasting. In Table 5 we present the out-of-sample forecasts for water demand series. Our forecasts are based on the most accuracy forecasting method used for multi-step ahead average forecasting for the 7-days cycle. We consider the periods from 1 July to 31 July 2006 (31 daily forecasts), from . 5 14 12 10 8 6 4 2 0 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Figure 1. Daily water demand in Spain for the period 1 January 2001 to 30 June 2006 Jan-06 6 1 July to 29 December 2006 (26 weekly average forecasts) and from 3 July to 31 December 2006 (26 weekly average forecasts). 5. Concluding remarks In this paper, we compared the forecast accuracy of individual and combined univariate time series models for multi-step-ahead water demand forecasting. We implemented double seasonal versions of the Holt-Winters, ARIMA and GARCH models in order to accommodate the two seasonal cycles periods in water consumption (daily and weekly). The empirical results suggest that all the univariate time series models can be quite useful for short-term forecasting. Moreover, the examination of the multi-step-ahead forecasting performance for each day of week suggest the use of different methods and di¤erent combined forecasts to improve forecasting accuracy. REFERENCES 1. Armstrong, J. (2001). "Combining forecasts", in Principles of Forecasting: A Handbook for Researchers and Practitioners, J. S. Armstrong (ed.), Kluwer Academic Publishers. 2. Bollerslev, T. (1986). "Generalized autoregressive conditional heteroskedasticity", Journal of Econometrics, 31, 307-327. 3. Bollerslev, T., Chou, R. and Kroner, K. (1992). "ARCH modeling in Finance", Journal of Econometrics, 52, 5-59. 4. Bougadis, J., Adamowski, K. and Diduch, R. (2005). "Short-term municipal water deamand forecasting", Hydrological Processes, 19, 137-148. 5. Box, G., Jenkins, G. and Reinsel, G. (1994). Time Series Analysis: Forecasting and Control, 3rd ed., Prentice-Hall, New Jersey. 6. Campbell, S. and Diebold, F. (2005). "Weather forecasting for weather derivatives", Journal of the American Statistical Association, 100, 6-16. 7. Clemen, R. (1989). "Combining forecasts: a review and annoted bibliography", International Journal of Forecasting, 5, 559-584. 8. Davison, R. and MacKinnon, J. (1993). Estimation and Inference in Econometrics, Oxford University Press, Oxford. 9. Engle, R. (1982). "Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom in‡ation", Econometrica, 50, 987-1008. 10. Fildes, R., Randall, A. and Stubbs, P. (1997). "One-day ahead demand forecasting in the utility industries: Two case studies", Journal of the Operational Research Society, 48, 15-24. 11. Gardner Jr., E. (2006). "Exponential smoothing: The state of the art - Part II", International Journal of Forecasting, 22, 637-666. 12. Gato, S., Jayasuriya, N. and Roberts, P. (2007). "Temperature and rainfall thresholds for base use urban water demand modelling", Journal of Hydrology, 337, 364-376. 13. Jain, A., Varshney, A. and Joshi, U. (2001). "Short-term water demand forecast modeling at IIT Kanpur using arti…cial neural networks", Water Resources Management, 15, 299-231. 14. Maidment, D. and Miaou, S. (1986). "Daily water use in nine cities", Water Resources Research, 22, 845-851. 15. Nelson, D. (1991). "Conditional heteroskedasticity in asset returns: a new approach", Econometrica, 59, 347-370. 16. Taylor, J. (2003). "Short-term electricity demand forecasting using double seasonal exponential smoothing", Journal of the Operational Research Society, 54, 799-805. 17. Taylor, J. (2006). "Density forecasting for the e¢cient balancing of the generation and consumption of electricity", International Journal of Forecasting, 22, 707-724. 18. Zhou, S., McMahon, T., Walton, A. and Lewis, J. (2000). "Forecasting daily urban water demand: a case study of Melbourne", Journal of Hidrology, 236, 153-164. 7 Table 1 Seasonal ARIMA model estimates for water demand series Model: ARIMA(4,0,9) (2,1,3)7 (0,1,1)365 Parameter Lag Estimate Standard error c -0.004 0.007 1 0.592 0.025 1 2 0.134 0.027 2 4 0.061 0.023 4 9 0.053 0.024 9 7 -0.757 0.023 1 14 -0.561 0.029 2 21 -0.366 0.032 3 365 -0.644 0.023 1 Residual ACF Lag Estimate 1 0.004 2 0.009 3 -0.020 4 0.001 5 -0.026 6 0.015 7 -0.010 Residual PACF Lag Estimate 1 0.004 2 0.009 3 -0.020 4 0.001 5 -0.025 6 0.015 7 -0.010 R2 adjusted = 0.662 Q(20) = 18:31 (0.11) Q2 (20) = 71:62 (0.00) Notes: Q(20) (Q2 (20)) is the Ljung-Box statistic for serial correlation in the residuals (squared residuals) up to order 20; p-value in parentheses. Table 2 Seasonal-periodic GARCH model estimates for water demand series Model: ARIMA(4,0,9) (2,1,3)7 (0,1,1)365 -GARCH(1,1) (0,1)365 Residual ACF Parameter Lag c 1 2 4 9 1 2 3 1 1 2 4 9 7 14 21 365 ! 1 1 365 '1 '03 GED 1 1 365 365 Estimate -0.011 0.502 0.137 0.075 -0.064 -0.747 -0.534 -0.346 -0.640 0.107 0.103 0.483 0.109 0.026 0.062 1.361 Standard error 0.008 0.029 0.030 0.024 0.023 0.023 0.028 0.031 0.025 0.028 0.037 0.108 0.032 0.011 0.035 0.055 Lag 1 2 3 4 5 6 7 Estimate -0.007 0.023 -0.028 -0.026 -0.042 0.026 -0.006 Squared residual ACF Lag Estimate 1 0.012 2 -0.030 3 0.028 4 0.018 5 0.008 6 -0.023 7 0.015 Lag 1 2 3 4 5 6 7 Residual PACF Estimate 0.007 0.023 -0.028 -0.026 -0.040 0.027 -0.006 Squared residual PACF Lag Estimate 1 0.012 2 -0.031 3 0.029 4 0.016 5 0.009 6 -0.023 7 0.015 R2 adjusted = 0.657 Q(20)=19.20 (0.08) Q2 (20)=13.61 (0.33) Notes: As Table 1. 8 Table 3 MSE for multi-step-ahead forecasts for post-sample period Forecast horizon 1-step 2-step 3-step 4-step 5-step 6-step 7-step Average RW 0.96 1.55 1.82 2.09 2.23 1.91 1.33 1.70 HW 0.38 0.51 0.49 0.48 0.43 0.42 0.40 0.44 ARIMA 0.35 0.45 0.47 0.45 0.44 0.45 0.44 0.44 GARCH 0.35 0.45 0.45 0.46 0.46 0.47 0.46 0.44 HW-A 0.35 0.46 0.45 0.46 0.43 0.43 0.41 0.43 Combined HW-G 0.35 0.45 0.45 0.46 0.43 0.43 0.42 0.43 forecasts A-G HW-A-G 0.35 0.35 0.45 0.45 0.45 0.45 0.46 0.46 0.45 0.44 0.46 0.44 0.45 0.43 0.44 0.43 Table 4 MSE for multi-step ahead forecasts for weekly 7-days in post-sample period Forecast horizon 1-step Days of week Monday Tuesday Wednesday Thursday Friday Saturday Sunday RW 16.18 0.28 0.18 3.15 0.47 3.00 1.20 HW 2.33 0.53 0.14 4.19 0.37 0.23 1.26 ARIMA 1.18 0.20 0.25 5.26 0.54 0.64 0.40 GARCH 1.25 0.19 0.26 5.40 0.54 0.58 0.33 HW-A 1.71 0.34 0.19 4.71 0.45 0.39 0.70 Combined HW-G 1.75 0.34 0.20 4.78 0.45 0.37 0.61 forecasts A-G HW-A-G 1.21 1.55 0.19 0.29 0.26 0.21 5.33 4.93 0.54 0.48 0.61 0.45 0.36 0.53 4-step Monday Tuesday Wednesday Thursday Friday Saturday Sunday 3.86 2.66 8.39 11.27 1.83 4.14 10.23 0.42 0.15 0.48 3.63 1.78 1.29 3.23 0.43 0.16 0.69 3.79 1.88 1.21 1.10 0.54 0.17 0.77 4.14 1.94 1.26 0.81 0.42 0.15 0.58 3.71 1.83 1.25 2.03 0.48 0.15 0.62 3.88 1.86 1.28 1.82 0.48 0.16 0.73 3.96 1.91 1.24 0.95 0.46 0.16 0.64 3.85 1.87 1.25 1.56 7-step Monday Tuesday Wednesday Thursday Friday Saturday Sunday 0.30 0.15 1.09 13.60 7.91 4.19 0.70 0.19 0.07 0.27 2.54 2.14 1.43 1.14 0.24 0.06 0.39 3.33 2.25 1.48 0.29 0.38 0.08 0.29 3.42 2.38 1.59 0.22 0.21 0.06 0.33 2.92 2.19 1.46 0.63 0.28 0.06 0.28 2.96 2.26 1.51 0.51 0.30 0.06 0.34 3.38 2.32 1.54 0.26 0.26 0.06 0.31 3.08 2.26 1.50 0.42 Average Monday Tuesday Wednesday Thursday Friday Saturday Sunday 4.79 4.13 4.89 7.52 5.21 4.02 6.10 0.61 0.44 0.43 3.01 1.99 1.06 2.35 0.55 0.16 0.46 3.71 2.25 1.24 0.68 0.65 0.17 0.48 3.91 2.32 1.26 0.50 0.57 0.25 0.41 3.33 2.12 1.13 1.38 0.62 0.26 0.41 3.43 2.15 1.14 1.22 0.59 0.16 0.47 3.80 2.28 1.25 0.59 0.59 0.21 0.42 3.51 2.18 1.17 1.02 9 Table 5 Out-of-sample daily and weekly average forecasts for water demand series Forecast period 1 July 2006 2 July 2006 3 July 2006 4 July 2006 5 July 2006 6 July 2006 7 July 2006 8 July 2006 9 July 2006 10 July 2006 11 July 2006 12 July 2006 13 July 2006 14 July 2006 15 July 2006 16 July 2006 17 July 2006 18 July 2006 19 July 2006 20 July 2006 21 July 2006 22 July 2006 23 July 2006 24 July 2006 25 July 2006 26 July 2006 27 July 2006 28 July 2006 29 July 2006 30 July 2006 31 July 2006 Daily forecasts 9.3393 9.4614 8.6222 10.3095 9.9691 9.6744 9.4109 8.0849 8.4503 7.6937 9.5814 9.3338 9.2385 9.1268 7.7251 8.3192 7.7194 9.2434 9.1702 9.1609 8.6665 7.4185 8.2409 7.7286 9.2441 9.0913 8.9348 8.6137 7.2653 8.0448 7.5470 Forecast period 1 Jul -7 Jul 8 Jul-14 Jul 15 Jul -21 Jul 22 Jul -28 Jul 29 Jul - 4 Aug 5 Aug - 11 Aug 12 Aug - 18 Aug 19 Aug - 25 Aug 26 Aug - 1 Sep 2 Sep - 8 Sep 9 Sep - 15 Sep 16 Sep - 22 Sep 23 Sep - 29 Sep 30 Sep - 6 Oct 7 Oct - 13 Oct 14 Oct - 20 Oct 21 Oct - 27 Oct 28 Oct - 3 Nov 4 Nov - 10 Nov 11 Nov - 17 Nov 18 Nov - 24 Nov 25 Nov - 1 Dec 2 Dec - 8 Dec 9 Dec- 15 Dec 16 Dec - 22 Dec 23 Dec - 29 Dec 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 Average forecasts 9.5410 8.7871 8.5721 8.4674 8.1969 7.8059 7.3662 7.6512 8.0755 8.3625 8.4238 8.4734 8.3959 8.3155 7.8275 7.5310 7.6197 7.5537 7.9359 7.9195 7.7142 7.6954 7.2456 7.6863 7.5248 7.0362 Forecast period 3 Jul -9 Jul 10 Jul-16 Jul 17 Jul -23 Jul 24 Jul -30 Jul 31 Jul - 6 Aug 7 Aug - 13 Aug 14 Aug - 20 Aug 21 Aug - 27 Aug 28 Aug - 3 Sep 4 Sep - 10 Sep 11 Sep - 17 Sep 18 Sep - 24 Sep 25 Sep - 1 Oct 2 Oct - 8 Oct 11 Oct - 15 Oct 16 Oct - 22 Oct 21 Oct - 29 Oct 30 Oct - 5 Nov 6 Nov - 12 Nov 13 Nov - 19 Nov 20 Nov - 26 Nov 27 Nov - 3 Dec 4 Dec - 10 Dec 11 Dec- 17 Dec 18 Dec - 24 Dec 25 Dec - 31 Dec 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 2006 Average forecasts 9.2173 8.7170 8.5171 8.4175 8.0443 7.7455 7.3417 7.7199 8.2972 8.2373 8.4191 8.5735 8.3159 8.3421 7.5624 7.5278 7.6764 7.6166 7.9908 7.8022 7.8061 7.5147 7.3100 7.7958 7.3708 6.9743