Chris Brooks Real Estate Modelling and Forecasting_6 pdf

Forecast evaluation 299 Table 9.15 Mean forecast errors for the changes in rents series Steps ahead 123456 7 8 (a) LaSalle Investment Management rents series VAR(1) −1.141 −2.844 −3.908 −4.729 −5.407 −5.912 −6.158 −6.586 VAR(2) −0.799 −1.556 −2.652 −3.388 −4.155 −4.663 −4.895 −5.505 AR(2) −0.595 −0.960 −1.310 −1.563 −1.720 −1.819 −1.748 −1.876 Long-term mean −2.398 −3.137 −3.843 −4.573 −5.093 −5.520 −5.677 −6.049 Random walk 0.466 −0.246 −0.923 −1.625 −2.113 −2.505 −2.624 −2.955 (b) CB Hillier Parker rents series VAR(1) −1.447 −3.584 −5.458 −7.031 −8.445 −9.902 −11.146 −12.657 AR(2) −1.845 −2.548 −2.534 −1.979 −1.642 −1.425 −1.204 −1.239 Long-term mean −3.725 −5.000 −6.036 −6.728 −7.280 −7.772 −8.050 −8.481 Random walk 1.126 −0.108 −1.102 −1.748 −2.254 −2.696 −2.920 −3.292 forecast is made in 1Q97 for the period 2Q97 to 1Q99). In this way, forty- four one-quarter forecasts, forty-four two-quarter forecasts, and so forth are calculated. The forty-four one-quarter forecasts are compared with the realised data for each of the four methodologies. This is repeated for the two-quarter-, three-quarter-, . . . , and eight-quarter-ahead computed values. This compar- ison reveals how closely rent predictions track the corresponding historical rent changes over the different lengths of the forecast horizon (one to eight quarters). The mean forecast error, the mean squared forecast error and the percentage of correct sign predictions are the criteria employed to select the best performing models. Ex ante forecasts of retail rents based on all methods are also made for eight quarters from the last available observation at the time that the study was written. Forecasts of real retail rents are therefore made for the peri- ods 1999 quarter two to 2001 quarter one. An evaluation of the forecasts obtained fromthe different methodologies is presented in tables 9.15 to 9.17. Table 9.15 reports the MFE. As noted earlier, a good forecasting model should have a mean forecasting error of zero. The first observation that can be made is that, on average, all mean errors are negative for all models and forecast horizons. This means that all models over-predict, except for the one-quarter-ahead CBHP forecast using the random walk. This bias could reflect non-economic influences 300 Real Estate Modelling and Forecasting Table 9.16 Mean squared forecast errors for the changes in rents series Steps ahead 12345678 (a) LaSalle Investment Management rents series VAR(1) 111.30 112.92 112.59 106.86 106.00 108.91 114.13 115.88 VAR(2) 67.04 69.69 75.39 71.22 87.04 96.64 103.89 115.39 AR(2) 77.16 84.10 86.17 76.80 79.27 86.63 84.65 86.12 Long-term mean 159.55 163.42 139.88 137.20 139.98 143.91 150.20 154.84 Random walk 138.16 132.86 162.95 178.34 184.43 196.55 202.22 198.42 (b) CB Hillier Parker rents series VAR(1) 78.69 117.28 170.41 236.70 360.34 467.90 658.41 867.72 AR(1) 75.39 88.24 84.32 92.18 88.44 89.15 80.03 87.44 Long-term mean 209.55 163.42 139.88 137.20 139.98 143.91 150.20 154.84 Random walk 198.16 132.86 123.71 149.78 132.94 148.79 149.62 158.13 during the forecast period. The continuous fall in rents in the period 1990 to 1995, which constitutes much of the out-of-sample period, may to some extent explain this over-prediction, however. Reasons that the authors put forward include the contention that supply increases had greater effects during this period when retailers were struggling than in the overall sample period and the fact that retailers benefited less than the growth in GDP at that time suggested, as people were indebted and seeking to save more to reduce indebtedness. Of the two VAR models used for LIM rents, the VAR(2) model – i.e. a VAR with a lag length of two – produces more accurate forecasts. This is not surprising, given that the VAR(1) model of changes in LIM rents is a poor performer compared with the VAR(2) model. The forecasts produced by the random walk model appear to be the most successful when forecasts up to three quarters ahead are considered, however. Then the AR model becomes the best performer. The same conclusion can be reached for CBHP rents, but here the random walk model is superior to the AR(2) model for the first four quarter-ahead forecasts. Table 9.16 shows the results based on the MSFE, an overall accuracy mea- sure. The computations of the MSFE for all eight time horizons in the CBHP case show that the AR(2) model has the smallest MSFEs. The VAR model appears to be the second-best-performing methodology when forecasts up Forecast evaluation 301 Table 9.17 Percentage of correct sign predictions for the changes in rents series Steps ahead 12345678 (a) LaSalle Investment Management rents series VAR(1) 6245404034333129 VAR(2) 8075726761635647 AR(2) 8080798173757471 Long-termmean4039403834333132 (b) CB Hillier Parker rents series VAR(1) 7666676949434147 AR(2) 7880817973787774 Long-termmean4241424034353334 Note: The random walk in levels model cannot, by definition, produce sign predictions, since the predicted change is always zero. to two quarters ahead are considered, but, as the forecast time horizon lengthens, the performance of the VAR deteriorates. In the case of LIM retail rents, the VAR(2) model performs best up to four quarters ahead, but when longer-term forecasts are considered the AR process appears to generate the most accurate forecasts. Overall, the long-term mean procedure out- performs the random walk model in the first two quarters of the forecast period for both series, but this is reversed when the forecast period extends beyond four quarters. Therefore, based on the MSFE criterion, the VAR(2) is the most appropriate model to forecast changes in LIM rents up to four quar- ters but then the AR(2) model performs better. This criterion also suggests that changes in CBHP rents are best forecast using a pure autoregressive model across all forecasting horizons. Table 9.17 displays the percentage of correct predictions of the sign for changes in rent from each model for forecasts up to eight periods ahead. While the VAR model’s performance can almost match that of the AR speci- fication for the shortest horizon, the latter model dominates as the models forecast further into the future. From these results, the authors conclude that rent changes have substantial memory for (at least) two periods. Hence useful information for predicting rents is contained in their own lags. The predictive capacity of the other aggregates within the VAR model is limited. There is some predictive ability for one period, but it quickly disappears thereafter. Overall, then, the autoregressive approach is to be preferred. 302 Real Estate Modelling and Forecasting Key concepts The key terms to be able to define and explain from this chapter are ● forecast error ● mean error ● mean absolute error ● mean squared error ● root mean squared error ● Theil’s U1 statistic ● bias, variance and covariance proportions ● Theil’s U2 statistic ● forecast efficiency ● forecast improvement ● rolling forecasts ● in-sample forecasts ● out-of-sample forecasts ● forecast encompassing 10 Multi-equation structural models Learning outcomes In this chapter, you will learn how to ● compare and contrast single-equation and systems-based approaches to building models; ● discuss the cause, consequence and solution to simultaneous equations bias; ● derive the reduced-form equations from a structural model; ● describe and apply several methods for estimating simultaneous equations models; and ● conduct a test for exogeneity. All the structural models we have considered thus far are single-equation models of the general form y = Xβ + u (10.1) In chapter 7, we constructed a single-equation model for rents. The rent equation could instead be one of several equations in a more general model built to describe the market, however. In the context of figure 7.1, one could specify four equations – for demand (absorption or take-up), vacancy, rent and construction. Rent variation is then explained within this system of equations. Multi-equation models represent alternative and competitive methodologies to single-equation specifications, which have been the main empirical frameworks in existing studies and in practice. It should be noted that, even if single equations fit the historical data very well, they can still be combined to construct multi-equation models when theory suggests that causal relationships should be bidirectional or multidirectional. Such systems are also used by private practices even though their performance may be poorer. This is because the dynamic structure of a multi-equation 303 304 Real Estate Modelling and Forecasting system may affect the ability of an individual equation to reproduce the properties of an historical series. Multi-equation systems are frameworks of importance to real estate forecasters. Multi-equation frameworks usually take the form of simultaneous- equation structures. These simultaneous models come with particular conditions that need to be satisfied for their estimation and, in general, their treatment and estimation require the study of specific econometric issues. There is also another family of models that, although they resemble simultaneous-equations models, are actually not. These models, which are termed recursive or triangular systems, are also commonly encountered in the real estate field. This chapter has four objectives. First, to explain the nature of simultaneous-equations models and to study the conditions that need to be fulfilled for their estimation. Second, to describe the available estima- tion techniques for these models. Third, to draw a distinction between simultaneous and recursive multi-equation models. Fourth, to illustrate the estimation of a systems model. 10.1 Simultaneous-equation models Systems of equations constitute one of the important circumstances under which the assumption of non-stochastic explanatory variables can be vio- lated. Remember that this is one of the assumptions of the classical linear regression model. There are various ways of stating this condition, differing slightly in terms of strictness, but they all have the same broad implica- tion. It can also be stated that all the variables contained in the X matrix are assumed to be exogenous – that is, their values are determined outside the equation. This is a rather simplistic working definition of exogeneity, although several alternatives are possible; this issue is revisited later in this chapter. Another way to state this is that the model is ‘conditioned on’ the variables in X, or that the variables in the X matrix are assumed not to have a probability distribution. Note also that causality in this model runs from X to y, and not vice versa – i.e. changes in the values of the explanatory variables cause changes in the values of y, but changes in the value of y will not impact upon the explanatory variables. On the other hand, y is an endogenous variable – that is, its value is determined by (10.1). To illustrate a situation in which this assumption is not satisfied, con- sider the following two equations, which describe a possible model for the Multi-equation structural models 305 demand and supply of new office space in a metropolitan area: Q dt = α + βR t + γ EMP t + u t (10.2) Q st = λ + µR t + κINT t + v t (10.3) Q dt = Q st (10.4) where Q dt = quantity of new office space demanded at time t,Q st = quan- tity of new office space supplied (newly completed) at time t,R t = rent level prevailing at time time t,EMP t =office-using employment at time t,INT t = interest rate at time t, and u t and v t are the error terms. Equation (10.2) is an equation for modelling the demand for new office space, and (10.3) is a specification for the supply of new office space. (10.4) is an equilibrium condition for there to be no excess demand (firms requiring more new space to let but they cannot) and no excess supply (empty office space due to lack of demand for a given structural vacancy rate in the market). 1 Assuming that the market always clears – that is, that the market is always in equilibrium – (10.2) to (10.4) can be written Q t = α + βR t + γ EMP t + u t (10.5) Q t = λ + µR t + κINT t + v t (10.6) Equations (10.5) and (10.6) together comprise a simultaneous structural form of the model, or a set of structural equations. These are the equa- tions incorporating the variables that real estate theory suggests should be related to one another in a relationship of this form. The researcher may, of course, adopt different specifications that are consistent with theory, but any structure that resembles equations (10.5) and (10.6) represents a simul- taneous multi-equation model. The point to emphasise here is that price and quantity are determined simultaneously: rent affects the quantity of office space and office space affects rent. Thus, in order to construct and rent more office space, everything else equal, the developers will have to lower the price. Equally, in order to achieve higher rents per square metre, developers need to construct and place in the market less floor space. R and Q are endogenous variables, while EMP and INT are exogenous. 1 Of course, one could argue here that such contemporaneous relationships are unrealistic. For example, interest rates will have affected supply in the past when developers were making plans for development. This is true, although on several occasions the contemporaneous term appears more important even if theory supports a lag structure. To an extent, this owes to the linkages of economic and monetary data in successive periods. Hence the current interest rate gives an idea of the interest rate in the recent past. For the sake of illustrating simultaneous-equations models, however, let us assume the presence of relationships such as (10.2) and (10.3). 306 Real Estate Modelling and Forecasting A set of reduced-form equations corresponding to (10.5) and (10.6) can be obtained by solving (10.5) and (10.6) for R and Q separately. There will be a reduced-form equation for each endogenous variable in the system, which will contain only exogenous variables. Solving for Q, α +βR t + γ EMP t + u t = λ + µR t + κINT t + v t (10.7) Solving for R, Q t β − α β − γ EMP t β − u t β = Q t µ − λ µ − γ INT t µ − v t µ (10.8) Rearranging (10.7), βR t − µR t = λ − α +κINT t − γ EMP t + ν t − u t (10.9) (β − µ)R t = (λ − α) + κINT t − γ EMP t + (ν t − u t ) (10.10) R t = λ − α β − µ + κ β − µ INT t − γ β − µ EMP t + v t − u t β − µ (10.11) Multiplying (10.8) through by βµ and rearranging, µQ t − µα − µγ EMP t − µu t = βQ t − βλ − βκINT t − βv t (10.12) µQ t − βQ t = µα − βλ −βκINT t + µγ EMP t + µu t − βv t (10.13) (µ − β)Q t = (µα − βλ) − βκINT t + µγ EMP t + (µu t − βv t ) (10.14) Q t = µa −βλ µ − β − βκ µ − β INT t + µγ µ − β EMP t + µu t − βv t µ − β (10.15) (10.11) and (10.15) are the reduced-form equations for R t and Q t . They are the equations that result from solving the simultaneous structural equations given by (10.5) and (10.6). Notice that these reduced form equations have only exogenous variables on the RHS. 10.2 Simultaneous equations bias It would not be possible to estimate (10.5) and (10.6) validly using OLS, as they are related to one another because they both contain R and Q, and OLS would require them to be estimated separately. What would have happened, however, if a researcher had estimated them separately using OLS? Both equations depend on R. One of the CLRM assumptions was that X and u are independent (when X is a matrix containing all the variables on the RHS of the equation), and, given the additional assumption that E(u) = 0,then E(X  u) = 0 (i.e. the errors are uncorrelated with the explanatory variables) It is clear from (10.11), however, that R is related to the errors in (10.5) and (10.6) – i.e. it is stochastic. This assumption has therefore been violated. Multi-equation structural models 307 What would the consequences be for the OLS estimator, ˆ β, if the simul- taneity were ignored? Recall that ˆ β = (X  X) −1 X  y (10.16) and that y = Xβ + u (10.17) Replacing y in (10.16) with the RHS of (10.17), ˆ β = (X  X) −1 X  (Xβ + u) (10.18) so that ˆ β = (X  X) −1 X  Xβ + (X  X) −1 X  u (10.19) ˆ β = β + (X  X) −1 X  u (10.20) Taking expectations, E( ˆ β) = E(β) + E((X  X) −1 X  u) (10.21) E( ˆ β) = β +E((X  X) −1 X  u) (10.22) If the Xs are non-stochastic (i.e. if the assumption had not been violated), E[(X  X) −1 X  u] = (X  X) −1 X  E[u] = 0, which would be the case in a single- equation system, so that E( ˆ β) = β in (10.22). The implication is that the OLS estimator, ˆ β, would be unbiased. If the equation is part of a system, however, then E[(X  X) −1 X  u] = 0, in general, so the last term in (10.22) will not drop out, and it can therefore be concluded that the application of OLS to structural equations that are part of a simultaneous system will lead to biased coefficient estimates. This is known as simultaneity bias or simultaneous equations bias. Is the OLS estimator still consistent, even though it is biased? No, in fact, the estimator is inconsistent as well, so that the coefficient estimates would still be biased even if an infinite amount of data were available, although proving this would require a level of algebra beyond the scope of this book. 10.3 How can simultaneous-equation models be estimated? Taking (10.11) and (10.15) – i.e. the reduced-form equations – they can be rewritten as R t = π 10 + π 11 INT t + π 12 EMP t + ε 1t (10.23) Q t = π 20 + π 21 INT t + π 22 EMP t + ε 2t (10.24) 308 Real Estate Modelling and Forecasting where the π coefficients in the reduced form are simply combinations of the original coefficients, so that π 10 = λ − α β − µ ,π 11 = κ β − µ ,π 12 = −γ β − µ ,ε 1t = v t − u t β − µ π 20 = µα −βλ µ − β ,π 21 = −βκ µ − β ,π 22 = µγ µ − β ,ε 2t = µu t − βv t µ − β Equations (10.23) and (10.24) can be estimated using OLS as all the RHS variables are exogenous, so the usual requirements for consistency and unbiasedness of the OLS estimator will hold(provided that there are no other misspecifications). Estimates of the π ij coefficients will thus be obtained. The values of the π coefficients are probably not of much interest, however; what we wanted were the original parameters in the structural equations – α, β, γ , λ, µ and κ. The latter are the parameters whose values determine how the variables are related to one another according to economic and real estate theory. 10.4 Can the original coefficients be retrieved from the πs? The short answer to this question is ‘Sometimes’, depending upon whether the equations are identified. Identification is the issue of whether there is enough information in the reduced-form equations to enable the structural- form coefficients to be calculated. Consider the following demand and sup- ply equations: Q t = α + βR t supply equation (10.25) Q t = λ + µR t demand equation (10.26) It is impossible to say which equation is which, so, if a real estate analyst simply observed some space rented and the price at which it was rented, it would not be possible to obtain the estimates of α, β, λ and µ. This arises because there is insufficient information from the equations to estimate four parameters. Only two parameters can be estimated here, although each would be some combination of demand and supply parameters, and so neither would be of any use. In this case, it would be stated that both equations are unidentified (or not identified or under-identified). Notice that this problem would not have arisen with (10.5) and (10.6), since they have different exogenous variables. 10.4.1 What determines whether an equation is identified or not? Any one of three possible situations could arise, as shown in box 10.1. [...]... is not statistically significant, and the explanatory power is a mere 6 per cent 328 Real Estate Modelling and Forecasting Another option is to check for a long-term relationship between demand and employment only (so that the error correction term is based on this bivariate relationship), and then we could still add the lagged rent growth term Both the completions and absorption equations are far... 4Q01 1Q01 2Q00 3Q99 4Q98 1Q98 2Q97 3Q96 60,000 1Q95 Figure 10.4 Actual and equilibrium real office rents in Tokyo Real Estate Modelling and Forecasting 4Q95 326 The risk premium is set at 1.5 per cent, the operating expense ratio at 2 per cent and the depreciation rate also at 2 per cent The bond yield in 1Q1996 was 4.44 per cent and the annual inflation rate that quarter was −0.15 per cent Hence the... 0 and λ3 = 0 If the null hypothesis is rejected, Rt and Qst should be treated as endogenous If λ2 and λ3 are significantly different from zero, there is extra important information for modelling ABSt from the reduced-form equations On the other hand, if the null is not rejected, Rt and Qst can be treated as exogenous for ABSt , and there is no useful additional information available for ABSt from modelling. .. obtain ˆ ABSt = 48.55 + 0.554 × 225 − 1.929 × (−0.91) − 0.00479 × 252.1 = 174 From identities (10.60) and (10.61), we obtain the demand and vacancy for 1Q2004: respectively, 19,445 and 6.7 per cent Table 10.4 compares the simulations for the main real estate variables, real rent growth, vacancy, absorption and completions, with the actual figures If we start with completions, it is clear that the variation... RENT is real industrial rents, CC is the construction cost, AVFS is the availability of industrial floor space (a measure of physical vacancy and not as a percentage of stock) and GDP is gross domestic product The αs, βs and γ s are the structural parameters to be estimated, and ut , et and εt are the stochastic disturbances Therefore, in this system, the three endogenous variables NIBSUPt , RENT t and. .. equations are similar to those in 320 Real Estate Modelling and Forecasting Figure 10.1 Actual values and historical simulation of new industrial building supply 7,000 Actual 6,000 5,000 4,000 Simulated 3,000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1,000 1978 2,000 table 10.1 The explanatory power of the NIBSUP and RENT equations shows a marginal fall On the other hand, the adjusted R 2 is higher... Hendershott (1996), we believe that the expected inflation proxy 324 Real Estate Modelling and Forecasting used is a moving average of annualised percentage changes in the GDP deflator Because of implausibly negative real interest rates during the 1975 to 1981 period, the authors make an amendment to the resulting real interest rate series The real risk-free interest rate (the twenty-year government bond yield... 1980s and the beginning of the 1990s) and has fallen in periods of economic expansion (the second half of the 1980s and after 322 Real Estate Modelling and Forecasting 1993) The simulated series tracks the actual series very well The simulation fit has improved considerably since 1990 and reproduces the last cycle of available industrial space very accurately 10.8 A special case: recursive models Consider... contained in the endogenous variables of other equations) 316 Real Estate Modelling and Forecasting Full-information maximum likelihood involves estimating all the equations in the system simultaneously using maximum likelihood.2 Thus, under FIML, all the parameters in all equations are treated jointly, and an appropriate likelihood function is formed and maximised Finally, limitedinformation maximum likelihood... above estimate and 1 per cent Hence the lowest permissible value of the real risk-free rate is 1 per cent The expression for the cash flow return is r + dep + oper = real risk-free rate + premium + dep + oper = real risk-free rate + 2% + 2% + 1.5% = real risk-free rate + 5.5% Finally, an estimate is needed for the replacement cost (RC) The authors observe that, over the period 1977 to 1985, real effective . 112.92 112.59 1 06. 86 1 06. 00 108.91 114.13 115.88 VAR(2) 67 .04 69 .69 75.39 71.22 87.04 96. 64 103.89 115.39 AR(2) 77. 16 84.10 86. 17 76. 80 79.27 86. 63 84 .65 86. 12 Long-term mean 159.55 163 .42 139.88. as (10.2) and (10.3). 3 06 Real Estate Modelling and Forecasting A set of reduced-form equations corresponding to (10.5) and (10 .6) can be obtained by solving (10.5) and (10 .6) for R and Q separately 143.91 150.20 154.84 Random walk 138. 16 132. 86 162 .95 178.34 184.43 1 96. 55 202.22 198.42 (b) CB Hillier Parker rents series VAR(1) 78 .69 117.28 170.41 2 36. 70 360 .34 467 .90 65 8.41 867 .72 AR(1) 75.39