2.7 Empirical Results
2.7.3 Evidence for Bubbles Semi-parametric Estimation
We described in detail three semi-parametric methods to estimate the value of the long memory parameter d, namely the Local Whittle Estimator, ˆdLW E, the Exact
Local Whittle Estimator ˆdELW and the 2-step Exact Local Whittle Estimator,
ˆ
d2ELW. While ˆdLW E is not consistent when d > 0.5, the ˆdELW and ˆd2ELW in
contrast provide good estimates even in the non-stationary region. Importantly, all the three Whittle based estimation methods are robust to conditional het- eroskedasticity and non-normality, see Robinson and Henry(1999),Henry(2001)
and Nielsen and Frederiksen (2005). However, a drawback of using these semi-
parametric procedures is that the estimate depends on the value of the bandwidth parameter or Fourier frequency m, see Baillie (1996).23
23Semi-parametric estimates have a slower rate of convergence than parametric ones but better robustness properties. However, parametric estimates are consistent under short samples, see Robinson and Henry(1999).
To address this issue, we follow Kumar and Okimoto (2007) and choose m based on simulations. We simulate Yt = (1 − L)−dt, where t is a Gaussian
white noise process, with sample size 125.24 The optimal m is the one which minimizes the sample Mean Squared Error for several choices of m, that is m = {n0.60, n0.65, n0.70, n0.75, n0.80}. Based on the simulation results, we select m = 0.75
for both 2ELW and ELW and m = 0.80 for the LWE. The simulation results also showed that the 2-step ELW estimator gave consistent estimates which were closer to the true value of d. Armed with the optimal m, we estimate the memory parameter using the three Whittle procedures. The results are reported in Table
2.6. We include the parametric results for comparison.
In general, it is seen in Table 2.6 that the Exact Local Whittle methods, ˆ
dELW and ˆd2ELW, report a higher persistence value compared to the Local Whittle
procedure. This is expected as the Local Whittle Estimate, is less preferred and ˆ
dLW E, gives over-differenced values when the series is non-stationary, seeShimotsu
and Phillips (2005).
Unit root testing on the frequency domain Whittle estimates is implemented by the application of the Efficient Fractional Dicky-Fuller Test (EFDF) ofLobato
and Velasco (2007). The EFDF tests the null of a unit root (d = 1) against the
alternative of a fractional root (d = ˆd < 1). The numbers in bold indicate that the null of a unit root was not rejected.
It is observed that only three aggregate or disaggregate series exhibit unit root behaviour consistent with price bubbles. These are the log rent-price ratio’s of the aggregate Census and the Southern MSA’s of Dallas and Houston. This is despite the fact that the 2-step ELW estimates are higher than unity for most of the series. This is because the EFDF that examines unit root behaviour can only accommodate stationary values of d in the alternative, i.e. d < 1. To the best of our knowledge, right-tailed unit root tests in the frequency domain do not exist in the literature. In order to overcome this limitation, we impose a Linear Restriction of d = 1.5, i.e. an explosive root in the frequency domain. I(d > 1) in column 5 indicates that a null of an explosive root was not rejected or in other words a bubble exists.
Table 2.6. Semi-parametric and Parametric Estimates of National and Regional δt’s
Housing Market Semi-Parametric Estimates Parametric Estimates
ˆ
dLW E dˆELW dˆ2ELW Conclusion dˆARF IM A ρARF IM A dˆARF IM A−EGARCH ρARF IM A−EGARCH Conclusion
FHFA 0.965 1.077 1.368 I(d > 1) 1.290 0.517 0.692 1.042 I(0)
Case-Shiller 0.951 1.059 1.615 I(d > 1) 1.276 0.794 1.591 1.000 I(1)
Census 0.924 1.031 1.192 I(1) 0.074 0.983 0.484 0.980 I(0)
Midwest
Chicago 0.962 1.039 1.343 I(d > 1) 0.764 0.941 0.841 0.906 I(1)
Cleveland 0.955 1.034 1.195 I(d > 1) 0.088 0.976 0.378 0.997 I(0)
Detroit 0.930 1.037 1.309 I(d > 1) 0.577 0.988 0.276 0.995 I(0)
Northeast
Boston 0.934 1.034 1.487 I(d > 1) 0.996 0.933 0.295 0.969 I(0)
New York 0.989 1.050 1.568 I(d > 1) 1.000 0.892 1.039 0.876 I(1)
Philadelphia 1.096 1.118 1.376 I(d > 1) 0.685 0.950 0.667 0.952 I(0)
South
Atlanta 0.960 1.042 1.244 I(d > 1) 0.517 1.074 0.334 1.002 I(0)
Dallas 0.941 1.053 1.074 I(1) 0.102 1.004 0.267 1.000 I(0)
Houston 1.069 1.203 1.221 I(1) 0.223 0.960 0.244 0.953 I(0)
West
Los Angeles 1.086 1.161 1.690 I(d > 1) 0.979 0.413 1.175 0.522 I(1)
San Francisco 1.120 1.147 1.577 I(d > 1) 0.830 0.742 0.567 0.972 I(0)
Seattle 1.139 1.177 1.458 I(d > 1) 0.648 0.933 1.345 0.354 I(1)
Notes: This table reports semi-parametric estimates of d for the log rent-price ratios of the three national HPI’s and 12 MSA’s. The sample spans the quarterly time period 1982Q4-2013Q4 except for Case-Shiller which is 1987Q1-2013Q4. The optimal frequency, m, selected by simulations is n0.80 for LWE and n0.75 for ELW and 2ELW (de-trended), where n is the sample size i.e. 125. The asymptotic standard error for LWE is 0.072 and for ELW and 2ELW is 0.081. Unit root test for the semi-parametric whittle estimates is done by implementing the Efficient Dicky-Fuller Test (EFDF). This tests for the null of a unit root (d = 1) against the alternative of fractional roots (d = ˆd < 1). For the parametric procedure, Linear Restrictions on d and ρ acts as unit root tests. Numbers in bold indicate that the null of a unit root cannot be rejected. Parametric estimates are extracted from Tables2.2,2.3,2.4and2.5.
We can summarize that the log rent-price ratio’s of all three aggregate and 12 MSA’s follow a process consistent with housing bubbles. These results contrast strongly with the parametric test. Barros et al.(2012) estimate Whittle and log pe- riodogram estimates on regional FHFA HPI’s of several U.S. States and find long memory values of d > 1 in most of them. They also implement semi-parametric Whittle estimate on FHFA national HPI in the quarterly span of 1975:1-2010:7 and get a value of ˆdLW E = 1.500 and a parametric value of 1.478. They reject the
null of a unit root. However, they did not consider either explosive alternatives or the rental series in their analysis.
The three MSA’s from the Northeast region (Boston, New York and Philadel- phia) on average reported higher than unity persistence in the long memory based on the 2ELW method (and also the parametric EGARCH one for New York). In general there is widespread consensus in the literature (see Case and Shiller
(2003); Case et al. (2012)) and anecdotal evidence for self-fulfilling price expec- tations or housing bubbles in the Boston and New York metropolises.
One reason for such high persistence in these cities, as argued byGyourko et al.
(2013), is that the marginal home buyers in ”superstar” cities are high income household who have moved from other parts of the city. This pattern would imply that the median homes in such cities are purchased by new residents whose income exceeds that of the median income. Furthermore, our result provide empirical validity to arguments by Green et al. (2005) and others who hypothesize that house price appreciation depends largely on elasticity of housing supply. They compute supply elasticities for 45 MSA’s and find that densely populated regions like New York and Los Angeles have highly inelastic housing supply. This explains why we obtain higher than average d values in the Northeast and West MSA’s.
Columns 6-9 in Table 2.6 report the parametric results extracted from the previous section. Comparisons can be made using the two most reliable esti- mators, the semi-parametric ˆd2ELW and the parametric ˆdARF IM A−EGARCH. It is
evident that the estimated values of long memory by the 2-step Whittle is sig- nificantly larger than the ARF IM A − EGARCH one. Most of the persistence for the parametric procedure is concentrated in the AR part reflected by near unity values of ρ. The Census series is a perfect example for this phenomenon. Here the 2-step ELW gave a unit root long memory value of 1.192 while the ARF IM A − EGARCH procedure estimated d as 0.484 which is stationary. This is primarily because of the short sample size we use, for small samples most of the persistence will be carried by the short run ARM A components resulting in low values of d. The persistence in the short run indicated by ρ for the Cen- sus was 0.980 with a unit root. Furthermore, our results from the parametric ARF IM A − EGARCH model is not completely reliable as the residual tests did indicate normal and autocorrelation errors in some of the series. Nevertheless, we caveat both the parametric and the non-parametric results to the presence of structural breaks which if present can induce spurious long memory. It is im- perative that we examine for any endogenous breaks and account for it in our estimation.