Vector autoregressive models 363 Table 11.11 Dynamic VAR forecasts Coefficients used in the forecast equation ARPRET t SPY t 10Y t AAA t Constant −0.0025 −0.0036 −0.0040 −0.0058 ARPRET t−1 0.0548 −0.9120 0.0985 −0.3003 ARPRET t−2 0.0543 0.2825 −0.2192 −0.3176 SPY t−1 0.0223 0.1092 −0.2280 −0.1792 SPY t−2 0.0136 −0.0263 −0.3501 −0.2720 10Y t−1 −0.0257 0.0770 0.4401 0.2644 10Y t−2 0.0494 −0.0698 −0.2612 −0.1739 AAA t−1 −0.0070 −0.0003 −0.0706 0.1266 AAA t−2 −0.0619 0.1158 0.1325 0.0202 Forecasts ARPRET t SPY t 10Y t AAA t May 07 −0.0087 −0.0300 0.0600 0.0000 Jun. 07 −0.1015 0.0000 0.3500 0.3200 Jul. 07 −0.0958 −0.0100 −0.1000 −0.0600 Aug. 07 −0.0130 0.0589 −0.0777 −0.0314 Sep. 07 −0.0062 −0.0180 −0.0080 0.0123 Oct. 07 −0.0049 −0.0039 −0.0066 −0.0003 Nov. 07 −0.0044 0.0007 0.0050 0.0031 Dec. 07 −0.0035 0.0000 0.0015 0.0009 Jan. 08 −0.0029 −0.0015 −0.0039 −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 SPY t 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 10Y ARPRET t SPY t 10Y t AAA t May 07 −0.0087 −0.0300 0.0600 0.0000 Jun. 07 −0.1015 0.0000 0.3500 0.3200 Jul. 07 −0.0958 −0.0100 −0.1000 −0.0600 Aug. 07 −0.0130 0.0589 0.2200 −0.0314 Sep. 07 −0.0139 0.0049 0.3300 0.0911 Oct. 07 0.0006 0.0108 0.4000 0.0455 Nov. 07 −0.0028 0.0112 0.0000 0.0511 Dec. 07 0.0144 −0.0225 0.0000 −0.0723 Jan. 08 −0.0049 −0.0143 −0.1000 −0.0163 that we accept this argument, we then obtain forecasts from a different source for the ten-year Treasury bond yield. In our VAR forecast, the Treasury bond yield was falling throughout the prediction period. Assume, however, that we have a forecast (from an economic forecasting house) of the bond yield rising and following the pattern shown in table 11.12. We estimate the 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 ARPRET t SPY t 10Y t AAA t Constant 0.0442 −0.9405 0.0955 −0.3128 ARPRET t−1 0.0552 0.2721 −0.205 −0.3119 ARPRET t−2 0.0203 0.1037 −0.2305 −0.1853 SPY t−1 0.013 −0.0264 −0.3431 −0.2646 SPY t−2 −0.0251 0.0744 0.4375 0.2599 10Y t−1 0.0492 −0.0696 −0.2545 −0.1682 10Y t−2 −0.0072 0.0035 −0.0626 0.1374 AAA t−1 −0.0609 0.1145 0.1208 0.0086 AAA t−2 −0.0019 −0.0033 −0.0042 −0.0062 Table 11.14 Ex post VAR dynamic forecasts ARPRET t SPY 10YCBY Actual Forecast Actual Forecast Actual Forecast Actual Forecast Dec. 06 −0.0227 −0.0100 −0.0400 −0.0100 Jan. 07 0.0718 0.0200 0.2000 0.0800 Feb. 07 −0.0355 −0.0067 0.0100 −0.0579 −0.0400 0.0976 −0.0100 0.0470 Mar. 07 −0.0359 0.0030 0.0700 0.0186 −0.1600 −0.0146 −0.0900 −0.0222 Apr. 07 −0.0057 0.0000 −0.0500 −0.0071 0.1300 −0.0111 0.1700 −0.0161 May. 07 −0.0087 −0.0006 −0.0300 −0.0061 0.0600 −0.0124 0.0000 −0.0136 Jun. 07 −0.1015 −0.0013 0.0000 −0.0052 0.3500 −0.0041 0.3200 −0.0064 Jul. 07 −0.0958 −0.0018 −0.0100 −0.0036 −0.1000 −0.0008 −0.0600 −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 take the actual values of the Treasury yield and recalculate the VAR forecasts. We should expect an improvement in this conditional forecast, since we are Vector autoregressive models 367 Table 11.15 Conditional VAR forecasts ARPRET t SPY 10YCBY Actual Forecast Actual Forecast Actual Actual Forecast Dec. 06 −0.0227 −0.0100 −0.0400 −0.0100 Jan. 07 0.0718 0.0200 0.2000 0.0800 Feb. 07 −0.0355 −0.0067 0.0100 −0.0579 −0.0400 −0.0100 0.0470 Mar. 07 −0.0359 0.0065 0.0700 0.0084 −0.1600 −0.0900 −0.0580 Apr. 07 −0.0057 −0.0030 −0.0500 −0.0128 0.1300 0.1700 −0.0348 May. 07 −0.0087 −0.0092 −0.0300 0.0138 0.0600 0.0000 0.0483 Jun. 07 −0.1015 0.0043 0.0000 −0.0021 0.3500 0.3200 −0.0015 Jul. 07 −0.0958 −0.0108 −0.0100 0.0170 −0.1000 −0.0600 0.0731 Table 11.16 VAR forecast evaluation Dynamic Conditional Mean forecast error −0.05 −0.04 Mean absolute error 0.05 0.04 RMSE 0.06 0.06 Theil’s U1 0.93 0.87 now effectively assuming perfect foresight for one variable. The results are reported in table 11.15. The ARPRET forecasts have not changed significantly and, in some months, the forecasts are worse than the unconditional ones. The formal evaluations 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. When we use actual values for the Treasury bill rate, these statistics improve 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, given the low explanatory power of the independent variables in the ARPRET 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 368 Real Estate Modelling and Forecasting evaluation results refer to a single period of six months during which REIT prices showed large falls. A better forecast assessment would involve con- ducting this analysis over a longer period or rolling six-month periods; see chapter 9. Key concepts The key terms to be able to define and explain from this chapter are ● VAR system ● contemporaneous VAR terms ● likelihood ratio test ● multivariate information criteria ● optimal lag length ● exogenous VAR terms (VARX) ● variable ordering ● Granger causality ● impulse response ● variance decomposition ● VAR forecasting ● conditional and unconditional VAR forecasts 12 Cointegration in real estate markets Learning outcomes In this chapter, you will learn how to ● highlight the problems that may occur if non-stationary data are used in their levels forms: ● distinguish between types of non-stationarity; ● run unit root and stationarity tests; ● test for cointegration; ● specify error correction models; ● implement the Engle–Granger procedure; ● apply the Johansen technique; and ● forecast with cointegrated variables and error correction models. 12.1 Stationarity and unit root testing 12.1.1 Why are tests for non-stationarity necessary? There are several reasons why the concept of non-stationarity is important and why it is essential that variables that are non-stationary be treated dif- ferently from those that are stationary. Two definitions of non-stationarity were presented at the start of chapter 8. For the purpose of the analysis in this chapter, a stationary series can be defined as one with a constant mean, constant variance and constant autocovariances for each given lag. The discus- sion in this chapter therefore relates to the concept of weak stationarity. An examination of whether a series can be viewed as stationary or not is essential for the following reasons. ● The stationarity or otherwise of a series can strongly influence its behaviour and properties. To offer one illustration, the word ‘shock’ is usually used 369 370 Real Estate Modelling and Forecasting 0.00 0.25 0.50 0.75 200 160 120 80 40 0 Frequency R 2 Figure 12.1 ValueofR 2 for 1,000 sets of regressionsofa non-stationary variable on another independent non-stationary variable to denote a change or an unexpected change in a variable, or perhaps simply the value of the error term during a particular time period. For a stationary series, ‘shocks’ to the system will gradually die away. That is, a shock during time t will have a smaller effect in time t + 1, a smaller effect still in time t + 2, and so on. This can be contrasted with the case of non-stationary data, in which the persistence of shocks will always be infinite, so that, for a non-stationary series, the effect of a shock during time t will not have a smaller effect in time t + 1, and in time t + 2, etc. ● The use of non-stationary data can lead to spurious regressions. If two stationary variables are generated as independent random series, when one of those variables is regressed on the other the t-ratio on the slope coefficient would be expected not to be significantly different from zero, and the value of R 2 would be expected to be very low. This seems obvi- ous, for the variables are not related to one another. If two variables are trending over time, however, a regression of one on the other could have a high R 2 even if the two are totally unrelated. If standard regression techniques are applied to non-stationary data, therefore, the end result could be a regression that ‘looks’ good under standard measures (signif- icant coefficient estimates and a high R 2 ) but that is actually valueless. Such a model would be termed a ‘spurious regression’. To give an illustration of this, two independent sets of non-stationary variables, y and x, were generated with sample size 500, one was regressed on the other and the R 2 was noted. This was repeated 1,000 times to obtain 1,000R 2 values. A histogram of these values is given in figure 12.1. As the figure shows, although one would have expected the R 2 values for each regression to be close to zero, since the explained and explanatory Cointegration in real estate markets 371 –750 –250 0 250 500 750–500 120 100 80 60 40 20 0 Frequency t-ratio Figure 12.2 Valueoft-ratio of slope coefficientfor 1,000 sets of regressionsofa non-stationary variable on another independent non-stationary variable variables in each case are independent of one another, in fact R 2 takes on values across the whole range. For one set of data, R 2 is bigger than 0.9, while it is bigger than 0.5 over 16 per cent of the time! ● If the variables employed in a regression model are not stationary then it can be proved that the standard assumptions for asymptotic analysis will not be valid. In other words, the usual ‘t-ratios’ will not follow a t-distribution, and the F-statistic will not follow an F -distribution, and so on. Using the same simulated data as used to produce figure 12.1, figure 12.2 plots a histogram of the estimated t-ratio on the slope coeffi- cient for each set of data. In general, if one variable is regressed on another unrelated variable, the t-ratio on the slope coefficient will follow a t-distribution. For a sam- ple of size 500, this implies that, 95 per cent of the time, the t-ratio will lie between +2 and −2. As the figure shows quite dramatically, however, the standard t-ratio in a regression of non-stationary variables can take on enormously large values. In fact, in the above example, the t-ratio is bigger than two in absolute value over 98 per cent of the time, when it should be bigger than two in absolute value only around 5 per cent of the time! Clearly, it is therefore not possible to undertake hypoth- esis tests validly about the regression parameters if the data are non- stationary. 12.1.2 Two types of non-stationarity There are two models that have been frequently used to characterise the non-stationarity: the random walk model with drift, y t = µ + y t−1 + u t (12.1) 372 Real Estate Modelling and Forecasting and the trend-stationary process, so-called because it is stationary around a linear trend, y t = α + βt + u t (12.2) where u t is a white noise disturbance term in both cases. Note that the model (12.1) can be generalised to the case in which y t is an explosive process, y t = µ + φy t−1 + u t (12.3) where φ>1. Typically, this case is ignored, and φ = 1 is used to characterise the non-stationarity because φ>1 does not describe many data series in economics, finance or real estate, but φ = 1 has been found to describe accurately many financial, economic and real estate time series. Moreover, φ>1 has an intuitively unappealing property: not only are shocks to the system persistent through time, they are propagated, so that a given shock will have an increasingly large influence. In other words, the effect of a shock during time t will have a larger effect in time t + 1, a larger effect still in time t +2, and so on. To see this, consider the general case of an AR(1) with no drift: y t = φy t−1 + u t (12.4) Let φ take any value for now. Lagging (12.4) one and then two periods, y t−1 = φy t−2 + u t−1 (12.5) y t−2 = φy t−3 + u t−2 (12.6) Substituting into (12.4) from (12.5) for y t−1 yields y t = φ(φy t−2 + u t−1 ) + u t (12.7) y t = φ 2 y t−2 + φu t−1 + u t (12.8) Substituting again for y t−2 from (12.6), y t = φ 2 (φy t−3 + u t−2 ) + φu t−1 + u t (12.9) y t = φ 3 y t−3 + φ 2 u t−2 + φu t−1 + u t (12.10) T successive substitutions of this type lead to y t = φ T +1 y t−(T +1) + φu t−1 + φ 2 u t−2 + φ 3 u t−3 +···+φ T u t−T + u t (12.11) There are three possible cases. (1) φ<1 ⇒ φ T → 0 as T →∞ The shocks to the system gradually die away; this is the stationary case. [...]... white noise (pure random) process, while figures 12.4 and 12.5 plot a random walk versus a random walk with drift and a deterministic trend process, respectively Comparing these three figures gives a good idea of the differences between the properties of a stationary, a stochastic trend and a deterministic trend process In figure 12.3, a white noise process visibly has no trending behaviour, and it frequently... the random walk yt = yt−1 + ut (12.24) An I(2) series contains two unit roots and so would require differencing twice to induce stationarity I(1) and I(2) series can wander a long way from their mean value and cross this mean value rarely, while I(0) series should cross the mean frequently The majority of financial and economic time series contain a single unit root, although some are stationary and. .. real estate market and its linkages with the overall stock market Tuluca, Myer and Webb (2000) use cointegration analysis to find that the capital values of treasury bills, bonds, stocks, securitised real estate and direct real estate are cointegrated, forming two long-run relationships Liow (2000) also finds evidence of a long-run relationship between direct property, property stocks and macroeconomic... the ADF/PP procedure to see if the same conclusion is obtained The null and alternative hypotheses under each testing approach are as follows: ADF/PP H0 : yt ∼ I (1) H1 : yt ∼ I (0) KPSS H0 : yt ∼ I (0) H1 : yt ∼ I (1) There are four possible outcomes (1) (2) (3) (4) Reject H0 Do not reject H0 Reject H0 Do not reject H0 and and and and do not reject H0 reject H0 reject H0 do not reject H0 For the... best way to understand the ideas discussed above is to consider some diagrams showing the typical properties of certain relevant types Cointegration in real estate markets Figure 12.3 Example of a white noise process 375 4 3 2 1 0 –1 1 40 79 118 157 196 235 274 313 352 391 430 469 –2 –3 –4 Figure 12.4 Time series plot of a random walk versus a random walk with drift 70 60 Random walk Random walk with... analysis of Liow and Yang (2005) establishes a contemporaneous linear long-run relationship between securitised real estate, the stock market and selected macroeconomic series in Japan, Hong Kong, Singapore and Malaysia, showing that the series interact and move together in the long run Moreover, these authors conclude that securitised real estate stocks are substitutable in Hong Kong and Singapore,... stochastic non-stationarity of the three component series, we now test whether the property of cointegration is established between office rents and GDP and between office rents and total employment using the Engle–Granger procedure We examine the series on a pair-wise basis rather than together, mainly for ease of illustration and interpretation In a research context, however, if theory suggests that a number... relationships and cointegration in real estate The concept of cointegration and the implications of cointegrating relationships are very relevant in the real estate market Real estate economic and investment theory often suggests that two or more variables would be expected to hold some long-run relationship with one another Such relationships may hold both in the occupier (leasing) and investment... developers and others, the effects of globalisation and 384 Real Estate Modelling and Forecasting international movements in capital in the real estate markets should lead to greater linkages between markets through investors seeking to exploit arbitrage opportunities Markets in which investor arbitrage is taking place may cointegrate – for example, international office markets such as London, Paris and New... formal hypothesis-testing procedure that answers the question ‘Given the sample of data to hand, is it plausible that the true data-generating process for y contains one or more unit roots?’ The early and pioneering work on testing for a unit root in time series was done by Fuller and Dickey (Fuller, 1976; Dickey and Fuller, 1979) The basic objective of the test is to examine the null hypothesis that . 677 784 83 3 88 5157 365 469 625 729 937 989 Figure 12.6 Autoregressive processes with differingvalues of φ (0, 0 .8, 1) to fall; obviously, the effect of the drift on the series becomes greater and greater. 433 451 469 487 Random walk Random walk with drift Figure 12.4 Time series plot of a random walk versus a random walk with drift of processes. Figure 12.3 plots a white noise (pure random) process,. −0.01 28 0.1300 0.1700 −0.03 48 May. 07 −0.0 087 −0.0092 −0.0300 0.01 38 0.0600 0.0000 0.0 483 Jun. 07 −0.1015 0.0043 0.0000 −0.0021 0.3500 0.3200 −0.0015 Jul. 07 −0.09 58 −0.01 08 −0.0100 0.0170 −0.1000