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Applications of regression analysis 203 Figure 7.3 Actual, fitted and 30 residual values of 20 rent growth 10 regressions 0 −10 −20 −30 Actual Fitted Actual Fitted 30 20 10 0 −10 −20 −30 (a) Actual and fitted – model A 30 20 10 0 −10 −20 (b) Actual and fitted – model B Model A Model B (c) Residuals – models A and B unchanged. The DW statistic in both models takes a value that denotes the absence of first-order correlation in the residuals of the equations. The coefficient on OFSgt suggests that, if we use model A, a 1 per cent rise inOFSgt willonaveragepushrealrentgrowthupby4.55percent,whereas, according to model B, it would rise by 5.16 per cent. One may ask what the real sensitivity of RRg to OFSg is. In reality, OFSg is not the only variable affecting real rent growth in Frankfurt. By accounting for other effects, in our case for vacancy and changes in vacancy, the sensitivity of RRg to OFSg changes. If we run a regression of RRg on OFSg only, the sensitivity is 6.48 per cent. OFSg on its own will certainly encompass influences from othervariables,however–thatis,theinfluenceofothervariablesonrentsis occurring indirectly through OFSg. This happens because the variables that have an influence on rent growth are to a degree correlated. The presence of other statistically significant variables takes away from OFSg and affects the size of its coefficient. The coefficient on vacancy in model B implies that, if vacancy rises by 1 per cent, it will push real rent growth down by 0.74 per cent in the same year.Theinterpretationofthecoefficienton1VACt−1 islessstraightforward. If the vacancy change declines by one percentage point – that is, from, say, a fall of 0.5 per cent to a fall of 1.5 per cent – rent growth will respond by rising 2.4 per cent after a year (due to the one-year lag). The actual and fitted values are plotted along with the residuals in figure 7.3. 204 Real Estate Modelling and Forecasting Thefittedvaluesreplicatetoadegreetheupwardtrendofrealrentgrowth in the 1980s, but certainly not the volatility of the series; the models com-pletely miss the two spikes. Since 1993 the fit of the models has improved considerably. Their performance is also illustrated in the residuals graph (panel (c)). The larger errors are recorded in the second half of the 1980s. After 1993 we discern an upward trend in the absolute values of the resid-uals of both models, which is not a welcome feature, although this was corrected after 1998. 7.1.3 Diagnostics This section computes the key diagnostics we described in the previous chapters. Normality Model A Model B Skewness 0.89 0.54 Kurtosis 3.78 2.71 The Bera–Jarque test statistic for the normality of the residuals of each model is BJA = 26·0.892 + (3.78 −3)2 ¸ = 4.09 BJB = 27·0.542 + (2.71 −3)2 ¸ = 1.41 The computed values of 4.09 and 1.41 for models A and B, respectively, are lowerthan5.99,theχ2(2)criticalvalueatthe5percentlevelofsignificance. Hence both these equations pass the normality test. Interestingly, despite the two misses of the actual values in the 1980s, which resulted in two large errors, and the small sample period, the models produce approximately normally distributed residuals. Serial correlation Table 7.5 presents the results of a Breusch–Godfrey test for autocorrelation in the model residuals. The tests confirm the findings of the DW test that the residuals do not exhibit first-order serial correlation. Similarly, the tests do not detect second-order serial correlation. In all cases, the computed Applications of regression analysis 205 Table 7.5 Tests for first- and second-order serial correlation Model A Model B Order Constant 1VACt−1 VACt OFSgt RESIDt−1 RESIDt−2 R2 T r T −r Computed test stat. χ2(r) First −1.02 0.19 – 0.31 0.07 – 0.009 25 1 24 χ2(1) = 0.22 Second −2.07 0.29 – 0.54 0.00 0.20 0.062 24 2 22 χ2(2) = 1.36 First 1.04 – −0.03 −0.35 0.10 – 0.009 25 1 χ2(1) = 0.23 Second −0.08 – 0.04 −0.18 0.09 0.17 0.035 25 χ2(2) = 0.81 Critical χ2(r) χ2(1) = 3.84 χ2(2) = 5.99 Notes: The dependent variable is RESIDt; T is the number of observations in the main equation; r is the number of lagged residuals (order of serial correlation) in the test equation; the computed χ2 statistics are derived from (T −r)R2 ∼ χ2. χ2 statistic is lower than the critical value, and hence the null of no auto-correlation in the disturbances is not rejected at the 5 per cent level of significance. When we model in growth rates or in first differences, we tend to remove serial correlation unless the data are still smoothed and trending or impor-tant variables are omitted. In levels, with trended and highly smoothed variables, serial correlation would certainly have been a likely source of misspecification. Heteroscedasticity test WeruntheWhitetestwithcross-terms,althoughweacknowledgethesmall number of observations for this version of the test. The test is illustrated and the results presented in table 7.6.2 All computed test statistics take a valuelowerthantheχ2 criticalvalueatthe5percentsignificancelevel,and hence no heteroscedasticity is detected in the residuals of either equation. 2 The results do not change if we run White’s test without the cross-terms, however. 206 Real Estate Modelling and Forecasting Table 7.6 White’s test for heteroscedasticity Model A Constant 0.44 Model B Constant 63.30 1VACt−1 1VACt−12 OFSgt OFSgt2 1VACt−1 × OFSgt R2 T r Computed χ2(r) Critical at 5% 47.70 −8.05 84.18 −19.35 −22.34 0.164 26 5 χ2(5) = 4.26 VACt VACt2 OFSgt OFSgt2 VACt × OFSgt χ2(5) = 11.07 −12.03 0.53 69.41 −15.21 −1.73 0.207 27 5 χ2(5) = 5.59 Notes: The dependent variable is RESIDt2; the computed χ2 statistics are derived from: T ∗R2 ∼ χ2. The RESET test Table7.4givestherestrictedformsformodelsAandB.Table7.7containsthe unrestrictedequations.ThemodelscleartheRESETtest,sincethecomputed values of the test statistic are lower than the critical values, suggesting that our assumption about a linear relationship linking the variables is the correct specification according to this test. Structural stability tests WenextapplytheChowbreakpointteststoexaminewhetherthemodelsare stable over two sub-sample periods. In the previous chapter, we noted that events in the market will guide the analyst to establish the date (or dates) and generate two (or more) sub-samples in which the equation is tested for parameter stability. In our example, due to the small number of observa-tions, we simply split the sample in half, giving us thirteen observations in each of the sub-samples. The results are presented in table 7.8. The calculations are as follows. 1383.86 −(992.91 +209.81) 26 −6 (992.91 +209.81) 3 1460.02 −(904.87 +289.66) 27 −6 (904.87 +289.66) 3 Applications of regression analysis 207 Table 7.7 RESET results Model A (unrestricted) Model B (unrestricted) Constant 1VACt−1 VACt OFSgt Fitted2 Coefficient −6.55 −2.83 – 3.88 0.02 p-value 0.07 0.01 – 0.02 0.32 Coefficient −3.67 – −0.72 5.45 0.01 p-value 0.41 – 0.03 0.00 0.71 URSS RRSS F-statistic F-critical (5%) 1,322.14 1,383.86 1.03 F(1,22) = 4.30 1,450.95 1,460.02 0.14 F(1,23) = 4.28 Note: The dependent variable is RRgt. Table 7.8 Chow test results for regression models Model A Model B (i) Variables Full (ii) (iii) First half Second half (i) (ii) (iii) Full First half Second half Constant −6.39 (0.08) −3.32 −7.72 (0.60) (0.03) Constant −3.53 (0.42) 13.44 −4.91 (0.25) (0.37) 1VACt−1 OFSgt Adj. R2 DW RSS −2.19 (0.01) 4.55 (0.00) 0.59 1.81 1,383.86 −4.05 −1.78 (0.11) (0.01) 4.19 4.13 (0.10) (0.01) 0.44 0.80 2.08 1.82 992.91 209.81 VACt −0.74 (0.02) OFSgt 5.16 (0.00) 0.57 1.82 1,460.02 −3.84 −0.60 (0.05) (0.07) 2.38 5.29 (0.36) (0.00) 0.51 0.72 2.12 2.01 904.87 289.66 Sample 1982–2007 1982–94 1995–2007 1981–2007 1981–94 1995–2007 T 26 13 13 27 14 13 F-statistic 1.00 1.56 Crit. F(5%) F(3,20) at 5% ≈ 3.10 F(3,21) at 5% ≈ 3.07 Notes: The dependent variable is RRgt; cell entries are coefficients (p-values). ... - tailieumienphi.vn
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