Property Estate Modelling and Forecasting_12 doc

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 disequilibrium situations persist for long periods. For the rent–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 GDP is highly significant in equation (12.56), whereas EMP 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 R 2 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. An option available to the analyst is to augment the error correction equa- 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 GDP in equation (12.56) and four lags of EMP 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 remove regressors using as the criterion the minimisation of AIC. The GDP and EMP terms in the final model should also take the expected positive signs. For brevity, we now focus on the GDP equation. RENT t =−3.437 − 0.089RESGDP t−1 + 1.642GDP t−1 + 2.466GDP t−4 (−10.07) (−4.48) (2.23) (3.32) (12.58) Adj. R 2 = 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) is the new rent-GDP error correction equation. The GDP term has lost some of its significance compared with the original equation, and the influence of changes in GDP on changes in real rents in the presence of the error correction term is best represented by the first and fourth lags of GDP. 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 estimate the models up to 4Q2005 and forecast the remaining eight quarters 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  ˆ RENT t =−3.892 − 0.097RESGDP t−1 + 1.295GDP t−1 (−11.40) (−5.10) (1.87) + 3.043GDP t−4 (12.59) (4.31) Adj. R 2 = 0.76; DW = 0.50; number of observations = 58 (3Q1991–4Q2005). We can highlight the fact that all the variables are statistically significant, with GDP t−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  ˆ RENT 1Q2006 =−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 which we get the forecast for real rent for 1Q2006 of 82.0 (column (ii)) and the 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): ˆ RENT t =−7.167 + 0.642GDP t (12.61) (−0.65) (7.42) Adj. R 2 = 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 × GDP t ). The error Cointegration in real estate markets 397 Table 12.4 Ex post forecasts from error correction model (i) (ii) (iii) (iv) (v) (vi) (vii) RENT GDP ECT GDP RENT RENT(qoq%) 1Q05 83.8 151.7 0.2 2Q05 83.9 152.1 0.4 3Q05 84.1 152.5 0.4 4Q05 84.0 153.0 −7.06 0.5 −0.100 1Q06 82.0 153.6 −9.40 0.6 −1.951 −2.32 2Q06 81.1 154.2 −10.77 0.6 −0.986 −1.20 3Q06 80.2 154.8 −12.01 0.6 −0.853 −1.05 4Q06 79.8 155.6 −12.95 0.8 −0.429 −0.53 1Q07 80.0 156.4 −13.24 0.8 0.226 0.28 2Q07 80.3 157.2 −13.50 0.8 0.254 0.32 3Q07 80.5 158.2 −13.86 1.0 0.279 0.35 4Q07 81.7 159.2 1.0 1.182 1.47 Notes: Bold numbers indicate model-based forecasts. ECT is the value of the error correction term (the residual). correction term is ECT t = actual rent – fitted rent = RENT t − (−7.167 +0.642GDP t ) = RENT t + 7.167 −0.642GDP t Hence the value of ECT 4Q2005 , which is required for the forecast of changes in rents for 1Q2006, is ECT 1Q2006 = 84.0 +7.167 − 0.642 × 153.6 =−7.06 (12.62) and for 1Q2006 to be used for the forecast of rent 2Q2006 is ECT 1Q2006 = 82.0 +7.167 − 0.642 × 153.6 =−9.4 Now, using the ECM, we can make the forecast for 2Q2006: RENT 2Q2006 =−3.892 − 0.097 × (−9.44) + 1.295 × 0.6 +3.043 × 0.4 =−0.986 (12.63) This forecast change in rent translates into a fall in the index to 81.1 – that is, 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) (iii) (iv) (v) (vi) (vii) RENT GDP ECT GDP RENT RENT(qoq %) 1Q07 85.5 156.4 0.8 0.83 2Q07 87.5 157.2 0.8 2.34 3Q07 89.1 158.2 1.0 1.83 4Q07 89.7 159.2 −2.95 1.0 0.67 1Q08 90.1 160.0 −2.98 0.8 0.440 0.49 2Q08 90.3 160.8 −3.34 0.8 0.115 0.13 3Q08 90.9 161.6 −3.17 0.8 0.640 0.71 4Q08 91.5 162.4 −3.02 0.8 0.625 0.69 1Q09 91.6 163.2 −3.39 0.8 0.118 0.13 2Q09 91.8 164.0 −3.72 0.8 0.151 0.16 3Q09 92.0 164.9 −4.02 0.9 0.180 0.20 4Q09 92.3 165.7 0.8 0.371 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 figure of 1.4 per cent per quarter. We now use the model to forecast out eight 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 rents in 2008 followed by a slowdown in 2009. These forecasts are, of course, 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 from all the remaining problems of the EG approach. There is, arguably, a far 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 The Johansen approach is based on the specification of a VAR model. 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: y t = β 1 y t−1 + β 2 y t−2 +···+ β k y t−k + u t g ×1 g × gg×1 g ×gg×1 g × gg× 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 y t = y t −k +  1 y t−1 +  2 y t−2 +···+ k −1 y t −(k −1) + u t (12.65) where  = (  k i=1 β i ) − I g and  i = (  i j=1 β j ) − I g . 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  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 of the  matrix.  can be interpreted as a long-run coefficient matrix, since, in equilibrium, all the y t −i will be zero, and setting the error terms, u t ,to their expected value of zero will leave y t −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. 400 Real Estate Modelling and Forecasting The test for cointegration between the ysiscalculatedbylookingatthe rank of the  matrix via its eigenvalues. 3 The rank of a matrix is equal to the number of its characteristic roots (eigenvalues) that are different from zero (see section 2.7). The eigenvalues, denoted λ i , are put in ascending order: λ 1 ≥ λ 2 ≥···≥λ g .Iftheλs are roots, in this context they must be less than one in absolute value and positive, and λ 1 will be the largest (i.e. the closest to one), while λ g will be the smallest (i.e. the closest to zero). If the variables are not cointegrated, the rank of  will not be significantly different from zero, so λ i ≈ 0 ∀i. The test statistics actually incorporate ln(1 − λ i ), rather than the λ i themselves, but, all the same, when λ i = 0, ln(1 −λ i ) = 0. Suppose now that rank () = 1,thenln(1 −λ 1 ) will be negative and ln(1 − λ i ) = 0 ∀i>1. If the eigenvalue i is non-zero, then ln(1 −λ i ) < 0 ∀i>1.That is, for  to have a rank of one, the largest eigenvalue must be significantly non-zero, while others will not be significantly different from zero. There are two test statistics for cointegration under the Johansen approach, which are formulated as λ trace (r) =−T g  i=r+1 ln(1 − ˆ λ i ) (12.66) and λ max (r, r + 1) =−T ln(1 − ˆ λ r+1 ) (12.67) where r is the number of cointegrating vectors under the null hypothesis and ˆ λ i is the estimated value for the ith ordered eigenvalue from the  matrix. Intuitively, the larger ˆ λ i is, the more large and negative ln(1 − ˆ λ i ) will be, and hence the larger the test statistic will be. Each eigenvalue will have associated with it a different cointegrating vector, which will be eigenvectors. A significantly non-zero eigenvalue indictates a significant cointegrating vector. λ trace is a joint test in which the null is that the number of cointegrating vectors is smaller than or equal to r against an unspecified or general alternative that there are more than r.Itstartswithp eigenvalues, and then, successively, the largest is removed. λ trace = 0 when all the λ i = 0,for i = 1, ,g. λ max conducts separate tests on each eigenvalue, and has as its null hypoth- esis that the number of cointegrating vectors is r against an alternative of r + 1. 3 Strictly, the eigenvalues used in the test statistics are taken from rank-restricted product moment matrices and not from  itself. Cointegration in real estate markets 401 Johansen and Juselius (1990) provide critical values for the two statistics. The distribution of the test statistics is non-standard, and the critical values depend on the value of g − r, the number of non-stationary components and whether constants are included in each of the equations. Intercepts can be included either in the cointegrating vectors themselves or as additional terms in the VAR. The latter is equivalent to including a trend in the data- generating processes for the levels of the series. Osterwald-Lenum (1992) and, more recently, MacKinnon, Haug and Michelis (1999) provide a more complete set of critical values for the Johansen test. If the test statistic is greater than the critical value from Johansen’s tables, reject the null hypothesis that there are r cointegrating vectors in favour of the alternative, that there are r + 1 (for λ trace ) or more than r (for λ max ). The testing is conducted in a sequence and, under the null, r = 0, 1, ,g−1, so that the hypotheses for λ max are H 0 : r = 0 versus H 1 :0<r≤ g H 0 : r = 1 versus H 1 :1<r≤ g H 0 : r = 2 versus H 1 :2<r≤ g . . . . . . . . . H 0 : r = g − 1 versus H 1 : r = g The first test involves a null hypothesis of no cointegrating vectors (corre- sponding to  having zero rank). If this null is not rejected, it would be concluded that there are no cointegrating vectors and the testing would be completed. If H 0 : r = 0 is rejected, however, the null that there is one cointegrating vector (i.e. H 0 : r = 1) would be tested, and so on. Thus the value of r is continually increased until the null is no longer rejected. How does this correspond to a test of the rank of the  matrix, though? r is the rank of .  cannot be of full rank (g) since this would correspond to the original y t being stationary. If  has zero rank then, by analogy to the univariate case, y t depends only on y t −j and not on y t−1 , so that there is no long-run relationship between the elements of y t−1 . Hence there is no cointegration. For 1 < rank() <g, there are r cointegrating vectors.  is then defined as the product of two matrices, α and β  , of dimension (g × r) and (r × g), respectively – i.e.  = αβ  (12.68) The matrix β gives the cointegrating vectors, while α gives the amount of each cointegrating vector entering each equation of the VECM, also known as the ‘adjustment parameters’. 402 Real Estate Modelling and Forecasting For example, suppose that g = 4, so that the system contains four vari- ables. The elements of the  matrix would be written  = ⎛ ⎜ ⎜ ⎝ π 11 π 12 π 13 π 14 π 21 π 22 π 23 π 24 π 31 π 32 π 33 π 34 π 41 π 42 π 43 π 44 ⎞ ⎟ ⎟ ⎠ (12.69) If r = 1, so that there is one cointegrating vector, then α and β will be (4 × 1):  = αβ  = ⎛ ⎜ ⎜ ⎝ α 11 α 12 α 13 α 14 ⎞ ⎟ ⎟ ⎠  β 11 β 12 β 13 β 14  (12.70) If r = 2, so that there are two cointegrating vectors, then α and β will be (4 × 2):  = αβ  = ⎛ ⎜ ⎜ ⎝ α 11 α 21 α 12 α 22 α 13 α 23 α 14 α 24 ⎞ ⎟ ⎟ ⎠  β 11 β 12 β 13 β 14 β 21 β 22 β 23 β 24  (12.71) and so on for r = 3, Suppose now that g = 4, and r = 1, as in (12.70) above, so that there are four variables in the system, y 1 , y 2 , y 3 and y 4 , that exhibit one cointegrating vector. Then y t−k will be given by y t−k = ⎛ ⎜ ⎜ ⎝ α 11 α 12 α 13 α 14 ⎞ ⎟ ⎟ ⎠  β 11 β 12 β 13 β 14  ⎛ ⎜ ⎜ ⎝ y 1 y 2 y 3 y 4 ⎞ ⎟ ⎟ ⎠ t−k (12.72) Equation (12.72) can also be written y t−k = ⎛ ⎜ ⎜ ⎝ α 11 α 12 α 13 α 14 ⎞ ⎟ ⎟ ⎠  β 11 y 1 + β 12 y 2 + β 13 y 3 + β 14 y 4  t−k (12.73) Given (12.73), it is possible to write out the separate equations for each variable y t . It is also common to ‘normalise’ on a particular variable, so that the coefficient on that variable in the cointegrating vector is one. For example, normalising on y 1 would make the cointegrating term in the Cointegration in real estate markets 403 equation for y 1 α 11  y 1 + β 12 β 11 y 2 + β 13 β 11 y 3 + β 14 β 11 y 4  t−k etc. Finally, it must be noted that the above description is not exactly how the Johansen procedure works, but is an intuitive approximation to it. 12.8.1 Hypothesis testing using Johansen The Engle–Granger approach does not permit the testing of hypotheses on the cointegrating relationships themselves, but the Johansen set-up does permit the testing of hypotheses about the equilibrium relationships between the variables. Johansen allows a researcher to test a hypothesis about one or more coefficients in the cointegrating relationship by viewing the hypothesis as a restriction on the  matrix. If there exist r cointegrating vectors, only those linear combinations or linear transformations of them, or combinations of the cointegrating vectors, will be stationary. In fact, the matrix of cointegrating vectors β can be multiplied by any non-singular conformable matrix to obtain a new set of cointegrating vectors. A set of required long-run coefficient values or relationships between the coefficients does not necessarily imply that the cointegrating vectors have to be restricted. This is because any combination of cointegrating vectors is also a cointegrating vector. It may therefore be possible to combine the cointegrating vectors thus far obtained to provide a new one, or, in general, a new set, having the required properties. The simpler and fewer the required properties are, the more likely it is that this recombination process (called renormalisation) will automatically yield cointegrating vectors with the required properties. As the restrictions become more numerous or involve more of the coefficients of the vectors, however, it will eventually become impossible to satisfy all of them by renormalisation. After this point, all other linear combinations of the variables will be non-stationary. If the restriction does not affect the model much – i.e. if the restriction is not binding – then the eigenvectors should not change much following the imposition of the restriction. A statistic to test this hypothesis is given by test statistic =−T r  i=1 [ln(1 − λ i ) − ln(1 − λ i ∗ )] ∼ χ 2 (m) (12.74) where λ ∗ i are the characteristic roots of the restricted model, λ i are the characteristic roots of the unrestricted model, r is the number of non- zero characteristic roots in the unrestricted model and m is the number of restrictions. 404 Real Estate Modelling and Forecasting Restrictions are actually imposed by substituting them into the relevant α or β matrices as appropriate, so that tests can be conducted on either the cointegrating vectors or their loadings in each equation in the system (or both). For example, considering (12.69) to (12.71) above, it may be that theory suggests that the coefficients on the loadings of the cointegrating vector(s) in each equation should take on certain values, in which case it would be relevant to test restrictions on the elements of α –e.g.α 11 = 1, α 23 =−1, etc. Equally, it may be of interest to examine whether only a subset of the variables in y t is actually required to obtain a stationary linear combination. In that case, it would be appropriate to test restrictions of elements of β.For example, to test the hypothesis that y 4 is not necessary to form a long-run relationship, set β 14 = 0, β 24 = 0, etc. For an excellent detailed treatment of cointegration in the context of both single-equation and multiple-equation models, see Harris (1995). 12.9 An application of the Johansen technique to securitised real estate Real estate analysts expect that greater economic and financial market link- ages between regions will be reflected in closer relationships between mar- kets. The increasing global movements of capital targeting real estate fur- ther emphasise the connections among real estate markets. Investors, in their search for better returns away from home and for greater diversifica- tion, have sought opportunities in international markets, particularly in the more transparent markets (see Bardhan and Kroll, 2007, for an account of the globalisation of the US real estate industry). The question is, of course, whether the stronger economic and financial market dependencies and global capital flows result in greater integration between real estate mar- kets and, therefore, stronger long-run relationships. We apply the Johansen technique to test for cointegration between three continental securitised real estate price indices for the United States, Asia and Europe. For the global investor, these indices could represent oppor- tunities for investment and diversification. They give exposure to different regional economic environments and property market fundamentals (for example, trends in the underlying occupier markets). Given that these are publicly traded indices, investors can enter and exit rapidly, so it is a liq- uid market. This market may therefore present arbitrage opportunities to investors who can trade them as expectations change. Figure 12.9 plots the three indices. [...]... performance of all real estate firms listed on the Singapore Stock Exchange This index is also converted into real terms using the consumer price index ● Gross domestic product (FCGDP) GDP is expected to have an influence on the demand for commercial space Since commercial real estate prices 412 Real Estate Modelling and Forecasting cover offices and shops, the author aggregates financial and business services... forecasting meetings; and ● make the forecast process more effective Having reviewed econometric techniques for real estate modelling and forecasting, it is interesting to consider how these methodologies are applied in reality Accordingly, this chapter focuses on how forecasting is actually conducted in the real estate field We address key aspects of real estate forecasting in practice and provide useful... regions whereas the fall of the Asian index in 1998 and 1999 reflected the regional turbulence (the currency crisis) Figure 12.10 plots the returns series 4 The data are the FTSE EPRA/NAREIT indices and can be obtained from those sites or from online databases 406 Real Estate Modelling and Forecasting It is clear from plotting the series in levels and in first differences that they will have a unit root... the users of office space are financial institutions and business service organisations) and commerce sector GDP (which is a proxy for demand for shop space) A positive impact on commercial real estate prices is implied The author also takes the natural logs of these first three series (denoted LPPIC, LSESP and LFCGDP) to stabilise variations in the series and induce normality ● Interest rates (PRMINT) The... commercial real estate prices, real estate stock prices, gross domestic product in financial and business services and commerce, interest rates and the supply of commercial space in Singapore over the period 1980 to 1997 He uses the following framework to examine whether the variables are cointegrated, cppt = a + b(pspt ) + c(GDPt ) + d(irt ) + e(sost ) + ut (12.75) where cpp is the commercial real estate price,... coefficients is above one, suggesting that real estate prices are elastic with regard to all explanatory variables Cointegration in real estate markets 413 ● More specifically, the long-term real estate stock price elasticity coeffi- cient of 1.4 per cent implies that a 1 per cent increase per quarter in the real estate stock index leads to a commercial real estate price increase of 1.4 per cent on average... are a wealth of studies in the business economics and finance fields on judgemental forecasting and on the means to adjust quantitative forecasts In this chapter, we draw upon selected research from this literature to provide a context for judgemental intervention in the forecast process in the real estate industry and highlight key issues, for forecasters and users alike This discussion also benefits from... forecasts and increasing ownership of the forecasts Following on from the previous point, not everyone is familiar with econometric models and forecast processes, and hence it is more difficult to accept model-based forecasts Even if the forecaster demonstrates the validity of the model, the management or the experts may not feel comfortable because of a lack of understanding of the technical aspects, and. .. individual becomes more familiar with the final forecast and controls the process, in the sense that it thereby incorporates own beliefs and today’s information The forecast is ‘owned’ and can be communicated more easily As we highlight later, communicating models and forecasts in a simple way increases their acceptability 13.2 How do we intervene in and adjust model-based forecasts? When judgement is deployed,... market in question (a term for it in the United Kingdom is active demand), that can act as a harbinger for absorption or take-up The forecaster will, of course, incorporate these suggestions and assess their contributions to the model and to forecast accuracy In both points (1) and (2), a degree of interaction between the forecaster and an expert with good domain knowledge, helping the forecaster in . influence on the demand for commercial space. Since commercial real estate prices 412 Real Estate Modelling and Forecasting cover offices and shops, the author aggregates financial and business services. (−1.62) + 0 .126 7 ×(−1.09) −0.1138 × (−1.40) =−0.13 410 Real Estate Modelling and Forecasting Table 12. 10 VECM ex ante forecasts Asia Europe United States AS EU US ECT Apr. 07 114.43 112. 08 126 .81 May 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