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Cointegration in real estate markets 395 There are similarities but also differences between the two error correc-tion equations above. In both equations, the error correction term takes a negative sign, indicating the presence of forces to move the relationship back to equilibrium, and it is significant at the 1 per cent level. For the rent-GDP equation (12.56), the adjustment to equilibrium is 6.5 per cent every quarter – a moderate adjustment speed. This is seen in figure 12.8, where disequilibriumsituationspersistforlongperiods.Fortherent–employment error correction equation (12.57), the adjustment is higher at 11.8 per cent every quarter – a rather speedy adjustment (nearly 50 per cent every year). An interesting finding is that 1GDP is highly significant in equation (12.56), whereas 1EMP in equation (12.57) is significant only at the 10 per cent level. Equation (12.56) has a notably higher explanatory power with an adjusted R2 of 0.68, compared with 0.30 for equation (12.57). The results of the diagnostic checks are broadly similar. Both equations have residuals that are normally distributed, but they fail the serial correlation tests badly. Serial correlation seems to be a problem, as the tests show the presence of serial correlation for orders 1, 2, 3 and 4 (results for orders 1 and 4 only are reported here). Both equations fail the heteroscedasticity and RESET tests. Anoptionavailabletotheanalystistoaugmenttheerrorcorrectionequa-tions and attempt to rectify the misspecification in the equations (12.56) and (12.57) in this way. We do so by specifying general models containing four lags of 1GDP in equation (12.56) and four lags of 1EMP in equation (12.57). We expect this number of lags to be sufficient to identify the impact of past GDP or employment changes on rental growth. We subsequently removeregressorsusingasthecriteriontheminimisationofAIC.The1GDP and 1EMP terms in the final model should also take the expected positive signs. For brevity, we now focus on the GDP equation. 1RENTt =−3.437 −0.089RESGDPt−1 +1.6421GDPt−1 +2.4661GDPt−4 (−10.07) (−4.48) (2.23) (3.32) (12.58) Adj. R2 = 0.69; DW = 0.43; number of observations = 66 (3Q1991–4Q2007). Diagnostics: normality BJ test value: 2.81 (p = 0.25); LM test for serial correlation (first order): 41.18 (p = 0.00); LM test for serial correlation (fourth order): 45.57 (p = 0.00); heteroscedasticity with cross-terms: 23.43 (p = 0.01); RESET: 1.65 (p = 0.20). Equation(12.58)isthenewrent-GDPerrorcorrectionequation.The1GDP term has lost some of its significance compared with the original equation, andtheinfluenceofchangesinGDPonchangesinrealrentsinthepresence of the error correction term is best represented by the first and fourth lags of 1GDP. The error correction term retains its significance and now points 396 Real Estate Modelling and Forecasting to a 9 per cent quarterly adjustment to equilibrium. In terms of diagnostics, the only improvement made is that the model now passes the RESET test. We use the above specification to forecast real rents in Sydney. We carry out two forecasting exercises – ex post and ex ante – based on our own assumptions for GDP growth. For the ex post (out-of-sample) forecasts, we estimatethemodelsupto4Q2005andforecasttheremainingeightquarters of the sample. Therefore the forecasts for 1Q2006 to 4Q2007 are produced by the coefficients estimated using the shorter sample period (ending in 4Q2005). This error correction model is 1RENTt = −3.892 − 0.097RESGDPt−1 +1.2951GDPt−1 (−11.40) (−5.10) (1.87) + 3.0431GDPt−4 (12.59) (4.31) Adj. R2 = 0.76; DW = 0.50; number of observations = 58 (3Q1991–4Q2005). Wecanhighlightthefactthatallthevariablesarestatisticallysignificant, with 1GDPt−1 at the 10 per cent level and not at the 5 per cent level, which was the case in (12.58). The explanatory power is higher over this sample period, which is not surprising given the fact that the full-sample model did not replicate the changes in rents satisfactorily towards the end of the sample. Table 12.4 contains the forecasts from the error correction model. The forecast for 1Q2006 using equation (12.59) is given by 1RENT1Q2006 = −3.892 −0.097 ×(−7.06) +1.295 ×0.5 +3.043 ×0.2 = −1.951 (12.60) This is the predicted change in real rent between 4Q2005 and 1Q2006, from whichwegettheforecastforrealrentfor1Q2006of82.0(column(ii))andthe growth rate of −2.32 per cent (quarter-on-quarter [qoq] percentage change), shown in column (vii). The value of the error correction term in 4Q2005 is produced by the long-run equation estimated for the shorter sample period (2Q1990 to 4Q2005): RENTt = −7.167 + 0.642GDPt (12.61) (−0.65) (7.42) Adj. R2 = 0.47; DW = 0.04; number of observations = 63 (2Q1990–4Q2005). Again, we perform unit root tests on the residuals of the above equation. The findings reject the presence of a unit root, and we therefore proceed to estimate the error correction term for 4Q2005. In equation (12.61), the fitted values are given by the expression (−7.167 +0.642 ×GDPt). The error Cointegration in real estate markets 397 Table 12.4 Ex post forecasts from error correction model (i) (ii) RENT 1Q05 83.8 2Q05 83.9 3Q05 84.1 4Q05 84.0 1Q06 82.0 2Q06 81.1 3Q06 80.2 4Q06 79.8 1Q07 80.0 2Q07 80.3 3Q07 80.5 4Q07 81.7 (iii) (iv) (v) (vi) GDP ECT 1GDP 1RENT 151.7 0.2 152.1 0.4 152.5 0.4 153.0 −7.06 0.5 −0.100 153.6 −9.40 0.6 −1.951 154.2 −10.77 0.6 −0.986 154.8 −12.01 0.6 −0.853 155.6 −12.95 0.8 −0.429 156.4 −13.24 0.8 0.226 157.2 −13.50 0.8 0.254 158.2 −13.86 1.0 0.279 159.2 1.0 1.182 (vii) RENT(qoq%) −2.32 −1.20 −1.05 −0.53 0.28 0.32 0.35 1.47 Notes: Bold numbers indicate model-based forecasts. ECT is the value of the error correction term (the residual). correction term is ECTt = actual rent – fitted rent = RENTt −(−7.167 +0.642GDPt) = RENTt +7.167 −0.642GDPt Hence the value of ECT4Q2005, which is required for the forecast of changes in rents for 1Q2006, is ECT1Q2006 = 84.0 +7.167 −0.642 ×153.6 = −7.06 (12.62) and for 1Q2006 to be used for the forecast of 1rent2Q2006 is ECT1Q2006 = 82.0 +7.167 −0.642 ×153.6 = −9.4 Now, using the ECM, we can make the forecast for 2Q2006: 1RENT2Q2006 = −3.892 −0.097 ×(−9.44) +1.295 ×0.6 +3.043 ×0.4 = −0.986 (12.63) Thisforecastchangeinrenttranslatesintoafallintheindexto81.1–thatis, rent ‘growth’ of −1.20 per cent on the previous quarter. Using the forecast value of 81.1 for rent in 2Q2006, we forecast again the error correction term using equation (12.61), and the process continues. Table 12.5 provides an evaluation of the GDP error correction model’s forecasts. 398 Real Estate Modelling and Forecasting Table 12.5 Forecast evaluation Measure Value Mean error 1.18 Absolute error 1.37 RMSE 1.49 Theil’s U1 statistic 0.61 Table 12.6 Ex ante forecasts from the error correction model (i) (ii) RENT 1Q07 85.5 2Q07 87.5 3Q07 89.1 4Q07 89.7 1Q08 90.1 2Q08 90.3 3Q08 90.9 4Q08 91.5 1Q09 91.6 2Q09 91.8 3Q09 92.0 4Q09 92.3 (iii) (iv) (v) (vi) GDP ECT 1GDP 1RENT 156.4 0.8 157.2 0.8 158.2 1.0 159.2 −2.95 1.0 160.0 −2.98 0.8 0.440 160.8 −3.34 0.8 0.115 161.6 −3.17 0.8 0.640 162.4 −3.02 0.8 0.625 163.2 −3.39 0.8 0.118 164.0 −3.72 0.8 0.151 164.9 −4.02 0.9 0.180 165.7 0.8 0.371 (vii) RENT(qoq %) 0.83 2.34 1.83 0.67 0.49 0.13 0.71 0.69 0.13 0.16 0.20 0.40 Notes: Bold numbers indicate forecasts. The forecast assumption is that GDP grows at 0.5 per cent per quarter. In 2007 the forecasts improved significantly in terms of average error. The ECM predicts average growth of 0.60, which is quite short of the actual figureof1.4percentperquarter.Wenowusethemodeltoforecastouteight quarters from the original sample period. We need exogenous forecasts for GDP, and we therefore assume quarterly GDP growth of 0.5 per cent for the period 1Q2008 to 4Q2009. Table 12.6 presents these forecasts. For the ECM forecasts given in table 12.6, the coefficients obtained from the error correction term represented by equation (12.61) and the short-run equation (12.59) are used. The ECM predicts a modest acceleration in real rentsin2008followedbyaslowdownin2009.Theseforecastsare,ofcourse, based on our own somewhat arbitrary assumptions for GDP growth. Cointegration in real estate markets 399 12.7 The Engle and Yoo three-step method The Engle and Yoo (1987) three-step procedure takes its first two steps from Engle–Granger (EG). Engle and Yoo then add a third step, giving updated estimates of the cointegrating vector and its standard errors. The Engle and Yoo (EY) third step is algebraically technical and, additionally, EY suffers fromalltheremainingproblemsoftheEGapproach.Thereis,arguably,afar superior procedure available to remedy the lack of testability of hypotheses concerning the cointegrating relationship: the Johansen (1988) procedure. For these reasons, the Engle–Yoo procedure is rarely employed in empirical applications and is not considered further here. 12.8 Testing for and estimating cointegrating systems using the Johansen technique TheJohansenapproachisbasedonthespecificationofaVARmodel.Suppose that a set of g variables (g ≥ 2) are under consideration that are I(1) and that it is thought may be cointegrated. A VAR with k lags containing these variables can be set up: yt = β1yt−1 + β2yt−2 +··· + βkyt−k + ut g ×1 g ×g g ×1 g ×g g ×1 g ×g g ×1 g ×1 (12.64) In order to use the Johansen test, the VAR in (12.64) needs to be turned into a vector error correction model of the form 1yt = 5yt −k +011yt−1 +021yt−2 +···+0k −11yt −(k −1) +ut (12.65) where 5 = (Pi=1 βi) −Ig and 0i = (Pj=1 βj) −Ig. This VAR contains g variables in first-differenced form on the LHS, and k −1 lags of the dependent variables (differences) on the RHS, each with a 0 coefficient matrix attached to it. In fact, the Johansen test can be affected by the lag length employed in the VECM, and so it is useful to attempt to select the lag length optimally. The Johansen test centres around an examination ofthe5matrix.5canbeinterpretedasalong-runcoefficientmatrix,since, in equilibrium, all the 1yt −i will be zero, and setting the error terms, ut, to their expected value of zero will leave 5yt −k = 0. Notice the comparability between this set of equations and the testing equation for an ADF test, which has a first-differenced term as the dependent variable, together with a lagged levels term and lagged differences on the RHS. ... - tailieumienphi.vn
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