Chris Brooks Real Estate Modelling and Forecasting_8

Vector autoregressive models 363 Table 11.11 Dynamic VAR forecasts Coefficients used in the forecast equation Constant ARPRETt−1 ARPRETt−2 1SPYt−1 1SPYt−2 110Yt−1 110Yt−2 1AAAt−1 1AAAt−2 ARPRETt −0.0025 0.0548 0.0543 0.0223 0.0136 −0.0257 0.0494 −0.0070 −0.0619 1SPYt −0.0036 −0.9120 0.2825 0.1092 −0.0263 0.0770 −0.0698 −0.0003 0.1158 110Yt −0.0040 0.0985 −0.2192 −0.2280 −0.3501 0.4401 −0.2612 −0.0706 0.1325 1AAAt −0.0058 −0.3003 −0.3176 −0.1792 −0.2720 0.2644 −0.1739 0.1266 0.0202 Forecasts ARPRETt May 07 −0.0087 Jun. 07 −0.1015 Jul. 07 −0.0958 Aug. 07 −0.0130 Sep. 07 −0.0062 Oct. 07 −0.0049 Nov. 07 −0.0044 Dec. 07 −0.0035 Jan. 08 −0.0029 1SPYt −0.0300 0.0000 −0.0100 0.0589 −0.0180 −0.0039 0.0007 0.0000 −0.0015 110Yt 0.0600 0.3500 −0.1000 −0.0777 −0.0080 −0.0066 0.0050 0.0015 −0.0039 1AAAt 0.0000 0.3200 −0.0600 −0.0314 0.0123 −0.0003 0.0031 0.0009 −0.0038 the system. Table 11.11 shows six months of forecasts and explains how we obtained them. The top panel of the table shows the VAR coefficients estimated over the whole-sample period (presented to four decimal points so that the forecasts can be calculated with more accuracy). The lower panel shows the VAR forecasts for the six months August 2007 to January 2008. The forecast for ARPRET for August 2007 (−0.0130 or −1.3 per cent monthly return) is given by the following equation: −0.0025 +[0.0548 ×−0.0958 +0.0543 ×−0.1015] +[0.0223 ×−0.0100 +0.0136 ×0.0000] +[−0.0257 ×−0.1000 +0.0494 ×0.3500] +[−0.0070 ×−0.0600 −0.0619 ×0.3200] 364 Real Estate Modelling and Forecasting The forecast for 1SPYt for August 2007 – that is, the change between July 2007 and August 2007 (0.0589 or 5.89 basis points) – is given by the following equation: −0.0036 +[−0.9120 ×−0.0958 +0.2825 ×−0.1015] +[0.1092 × −0.0100 −0.0263 ×0.0000] +[0.0770 ×−0.1000 −0.0698 ×0.3500] +[−0.0003 ×−0.0600 +0.1158 ×0.3200] The forecasts for August 2007 will enter the calculation of the September 2007 figure. This version of the VAR model is therefore a truly dynamic one, as the forecasts moving forward are generated within the system and are not conditioned by the future values of any of the variables. These are sometimes called unconditional forecasts (see box 11.1). In table 11.11, the VAR forecasts suggest continuously negative monthly REIT price returns for the six months following the last observation in July 2007. The negative growth is forecast to get smaller every month and to reach −0.29 per cent in January 2008 from −1.3 per cent in August 2007. Box 11.1 Forecasting with VARs ● One of the main advantages of the VAR approach to modelling and forecasting is that, since only lagged variables are used on the right-hand side, forecasts of the future values of the dependent variables can be calculated using only information from within the system. ● We could term these unconditional forecasts, since they are not constructed conditional on a particular set of assumed values. ● Conversely, however, it may be useful to produce forecasts of the future values of some variables conditional upon known values of other variables in the system. ● For example, it may be the case that the values of some variables become known before the values of the others. ● If the known values of the former are employed, we would anticipate that the forecasts should be more accurate than if estimated values were used unnecessarily, thus throwing known information away. ● Alternatively, conditional forecasts can be employed for counterfactual analysis based on examining the impact of certain scenarios. ● For example, in a trivariate VAR system incorporating monthly REIT returns, inflation and GDP, we could answer the question ‘What is the likely impact on the REIT index over the next one to six months of a two percentage point increase in inflation and a one percentage point rise in GDP?’. Within the VAR, the three yield series are also predicted. It can be argued, however, that series such as the Treasury bond yield cannot be effectively forecast within this system, as they are determined exogenously. Hence we can make use of alternative forecasts for Treasury bond yields (from the conditional VAR forecasting methodology outlined in box 11.1). Assuming Vector autoregressive models 365 Table 11.12 VAR forecasts conditioned on future values of 110Y ARPRETt May 07 −0.0087 Jun. 07 −0.1015 Jul. 07 −0.0958 Aug. 07 −0.0130 Sep. 07 −0.0139 Oct. 07 0.0006 Nov. 07 −0.0028 Dec. 07 0.0144 Jan. 08 −0.0049 1SPYt −0.0300 0.0000 −0.0100 0.0589 0.0049 0.0108 0.0112 −0.0225 −0.0143 110Yt 0.0600 0.3500 −0.1000 0.2200 0.3300 0.4000 0.0000 0.0000 −0.1000 1AAAt 0.0000 0.3200 −0.0600 −0.0314 0.0911 0.0455 0.0511 −0.0723 −0.0163 that we accept this argument, we then obtain forecasts from a different sourcefortheten-yearTreasurybondyield.InourVARforecast,theTreasury bond yield was falling throughout the prediction period. Assume, however, that we have a forecast (from an economic forecasting house) of the bond yieldrisingandfollowingthepatternshownintable11.12.Weestimatethe forecasts again, although, for the future values of the Treasury bond yield, we do not use the VAR’s forecasts but our own. By imposing our own assumptions for the future values of the move-ments in the Treasury bill rate, we affect the forecasts across the board. With the unconditional forecasts, the Treasury bill rate was forecast to fall in the first three months of the forecast period and then rise, whereas, according to our own assumptions, the Treasury Bill rate rises immediately and it then levels off (in November 2007). The forecasts conditioned on the Treasury bill rate are given in table 11.12. The forecasts for August 2007 have not changed, since they use the actual values of the previous two months. 11.11.1 Ex post forecasting and evaluation We now conduct an evaluation of the VAR forecasts. We estimate the VAR over the sample period March 1972 to January 2007, reserving the last six months for forecast assessment. We evaluate two sets of forecasts: dynamic VAR forecasts and forecasts conditioned by the future values of the Trea-sury and corporate bond yields. The parameter estimates are shown in table 11.13. The forecast for ARPRET for February 2007 is produced in the same way as in table 11.11, although we are now computing genuine out-of-sample 366 Real Estate Modelling and Forecasting Table 11.13 Coefficients for VAR forecasts estimated using data for March 1972 to January 2007 Constant ARPRETt−1 ARPRETt−2 1SPYt−1 1SPYt−2 110Yt−1 110Yt−2 1AAAt−1 1AAAt−2 ARPRETt 0.0442 0.0552 0.0203 0.013 −0.0251 0.0492 −0.0072 −0.0609 −0.0019 1SPYt −0.9405 0.2721 0.1037 −0.0264 0.0744 −0.0696 0.0035 0.1145 −0.0033 110Yt 0.0955 −0.205 −0.2305 −0.3431 0.4375 −0.2545 −0.0626 0.1208 −0.0042 1AAAt −0.3128 −0.3119 −0.1853 −0.2646 0.2599 −0.1682 0.1374 0.0086 −0.0062 Table 11.14 Ex post VAR dynamic forecasts ARPRETt 1SPY 110Y 1CBY Actual Dec. 06 −0.0227 Jan. 07 0.0718 Feb. 07 −0.0355 Mar. 07 −0.0359 Apr. 07 −0.0057 May. 07 −0.0087 Jun. 07 −0.1015 Jul. 07 −0.0958 Forecast −0.0067 0.0030 0.0000 −0.0006 −0.0013 −0.0018 Actual −0.0100 0.0200 0.0100 0.0700 −0.0500 −0.0300 0.0000 −0.0100 Forecast −0.0579 0.0186 −0.0071 −0.0061 −0.0052 −0.0036 Actual −0.0400 0.2000 −0.0400 −0.1600 0.1300 0.0600 0.3500 −0.1000 Forecast 0.0976 −0.0146 −0.0111 −0.0124 −0.0041 −0.0008 Actual −0.0100 0.0800 −0.0100 −0.0900 0.1700 0.0000 0.3200 −0.0600 Forecast 0.0470 −0.0222 −0.0161 −0.0136 −0.0064 −0.0030 forecasts as we would in real time. The forecasts for all series are compared to the actual values, shown in table 11.14. In the six-month period February 2007 to July 2007, REIT returns were negative every single month. The VAR correctly predicts the direction for four of the six months. In these four months, however, the prediction for negative monthly returns is quite short of what actually happened. We argued earlier that the Treasury bond yield is unlikely to be deter-mined within the VAR in our example. For the purpose of illustration, we taketheactualvaluesoftheTreasuryyieldandrecalculatetheVARforecasts. We should expectan improvement in this conditional forecast, since we are Vector autoregressive models 367 Table 11.15 Conditional VAR forecasts ARPRETt 1SPY 110Y 1CBY Actual Dec. 06 −0.0227 Jan. 07 0.0718 Feb. 07 −0.0355 Mar. 07 −0.0359 Apr. 07 −0.0057 May. 07 −0.0087 Jun. 07 −0.1015 Jul. 07 −0.0958 Forecast −0.0067 0.0065 −0.0030 −0.0092 0.0043 −0.0108 Actual −0.0100 0.0200 0.0100 0.0700 −0.0500 −0.0300 0.0000 −0.0100 Forecast −0.0579 0.0084 −0.0128 0.0138 −0.0021 0.0170 Actual −0.0400 0.2000 −0.0400 −0.1600 0.1300 0.0600 0.3500 −0.1000 Actual −0.0100 0.0800 −0.0100 −0.0900 0.1700 0.0000 0.3200 −0.0600 Forecast 0.0470 −0.0580 −0.0348 0.0483 −0.0015 0.0731 Table 11.16 VAR forecast evaluation Mean forecast error Mean absolute error RMSE Theil’s U1 Dynamic −0.05 0.05 0.06 0.93 Conditional −0.04 0.04 0.06 0.87 now effectively assuming perfect foresight for one variable. The results are reported in table 11.15. TheARPRET forecastshavenotchangedsignificantlyand,insomemonths, theforecastsareworsethantheunconditionalones.Theformalevaluations of the dynamic and the conditional forecasts are presented in table 11.16. The mean forecast error points to an under-prediction (error defined as the actual values minus the forecasted values) of 5 per cent on average per month. The mean absolute error confirms the level of under-prediction. WhenweuseactualvaluesfortheTreasurybillrate,thesestatisticsimprove but only slightly. Both VAR forecasts have a similar RMSE but the Theil statistic is better for the conditional VAR. On both occasions, however, the Theil statistics indicate poor forecasts. To an extent, this is not surprising, giventhelowexplanatorypoweroftheindependentvariablesintheARPRET equation in the VAR. Moreover, the results both of the variance decompo-sition and the impulse response analysis did not demonstrate strong influ-ences from any of the yield series we examined. Of course, these forecast ... - tailieumienphi.vn