# The Markov Régime-Switching Model

Hamilton's model is first applied in its simplest form, to test the hypothesis that the data was generated by a mixture of two, three or four normal distributions. By comparing μ1 with μ2, ... ,μn and σ1 with σ2, ... , σn, a measure of the validity of the model for each value of n is obtained. If the null hypothesis that the parameters of the model are equal across régimes can be rejected, then the null hypothesis that there exist n-1 states of nature, must be rejected in favour of the alternative i. e. there being at least n. In this case n=2 is rejected, but n>2 is not.

For the two estimated régimes, the null hypothesis that the means are equal and that the variances are equal across régimes is tested, using a t-statistic and an f-statistic respectively. That is,

against a two-sided alternative in each case. The results are presented with corresponding p-values in table 6.3. In all cases the null hypothesis, that the means are equal across the two régimes at the 0.01 level and that the variances are equal for most of the series, was rejected. The statistical tests and individual parameter stationarity tests strongly support the discrete mixture of two distributions as a statistical model of real estate returns.

Table 6.4 on the next page shows the results of the normality tests applied to the data, sorted into régimes using the Markov régime-switching model. It is clearly evident that tests for normality are less likely to be rejected, but the bootstrapping results described above must be borne in mind. It is not possible to determine the extent to which these results are a consequence of the reduction in sample size, leading to a loss in the power of the tests.

Figures 6.4 and 6.5 plot the USQTP and UKMTP returns series respectively against q, the probability of state one occurring. As shown,

Hamilton's E-M algorithm efficiently sorts the data in the UK. However, it fails to capture important cyclical shifts in the US data series. This is also found to be the case for each of the individual sectors. One possible explanation for this may be that the UK data is superior in both informational content and depth to that of the US data. The NCREIF index is based on a similar number of properties (approximately 1,500)8 to that of the IPD index, but is quarterly as opposed to monthly.9 The NCREIF index, however, is based on a weighted average of properties, only some of which have been reappraised in any quarter; returns on the IPD index are all based on reappraisals. The relative geographical area covered by the indices is also an important factor. The NCREIF index covers the whole of the US, whereas the IPD index covers the UK, an area the size of California.

A summary of the test statistics on the discrete mixture of two distributions, split according to Hamilton's model, are presented in table 6.3. The discrete mixture of two distributions is considerably more descriptive of the data generating process than the stationary normal model.

Each mixture of two normal distributions model exhibits significantly different means and variances, rejecting the mean stationarity null hypothesis at the 0.01 probability level. The relative strength of rejection of mean non-stationarity, compared to the variance parameter, indicates its larger contribution to the discrete mixture model specification.

Although the empirical results in this chapter strongly support the two distributions hypotheses, they can be subjected to criticism. It may be that the aforementioned statistical properties are simply a result of time varying appraiser behaviour. Alternatively, they may be a consequence of the unique nature of the real estate market, in terms of its response to structural shifts within the economy and as an artefact of the 'true' returns series.

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8As at December, 1994.

9The Investment Property Databank also constructs an annual index that includes approximately 12,000 properties.