# Results

## The Normality of Commercial Real Estate Returns

The issues discussed in Chapter 2, and the results of previous studies which have been discussed above, prompt consideration of the normality of value indices in commercial real estate. In particular, the conclusions of those studies which point to the returns from real estate being non-normal will be examined.

### Tests for Normality

Under the condition of normality, the coefficient of skewness asymptotically follows a N(0,6/T) distribution, where T is the sample size, and the coefficient of kurtosis asymptotically follows a N(0,24/T) distribution. Three popular tests for normality are employed in this study: the Shapiro-Wilk W-statistic (Shapiro, Wilk & Chen ), the Kolmogorov-Smirnov test and the Jarque-Bera test (Jarque & Bera ). The Kolmogorov-Smirnov probability values that are reported are based on those tabulated by Lilliefors . The Jarque-Bera test is asymptotically distributed as an X2 variable with two degrees of freedom, under the null hypothesis of normality.

Figure 6.1 on the following page plots the normalised estimated unconditional distribution of the UKMTO index, together with a normal distribution of the same mean and variance, in which it displays a bimodal structure. Therefore, the series may not be drawn for a single distribution, but a mixture of two distributions. The resulting normalised estimated unconditional distributions found by the Markov régime-switching model are plotted in figure 6.2. As shown in table 6.1 the discrete mixture of distributions found exhibited significant differences in mean and variance. This is supported by figure 6.3, in which λ in plotted against UKMTP. As detailed above, λ is the probability that xt was drawn from state one and is not conditional upon whether the process is in state one or state two in period t-1.

As table 6.1 also illustrates, the tests overwhelmingly reject the normality of returns in all series. The stationary normal distribution is rejected at the 0.05 probability level by two or more tests in all cases, except USQTR and UKQTP: only for the latter is normality not rejected at all. These departures from normality are due predominantly to skewness rather than excess kurtosis. Faced with this clear rejection of normality, alternative distributional assumptions must be considered, and this issue will be examined below.

### The Effect of Sample Size on the Rejection of Normality

Shapiro, Wilk & Chen [1968, p. 1371] remark of the Shapiro-Wilk W-statistic that: 'Contrary to popular beliefs, sensitive assessment (approximately 50% power at 5% level) of even moderate non-normality (e. g. X2(4)) is possible in samples as small as n=20. Furthermore, extreme non-normality (e. g. X2(1)) can be detected with sample size less than 10.

Lilliefors  shows that the Kolmogorov-Smirnov test also has power against some alternatives with a sample size of 10, while Huang & Bolch  demonstrate that the test has good power against a range of alternatives with 50 observations. However, the small sample properties of the Jarque-Bera  test are not well known. In particular, this parametric test is by definition asymptotic. This possible lack of power in small samples may result in an inability to reject the null hypothesis of normality even if the true data generating process is non-normal. The Jarque-Bera tests for normality show that within a régime; see section 6.4.2, the null hypothesis that the observations are normally distributed, is much less likely to be rejected. However, this is by no means a sufficient condition for normality of the returns within a régime, splitting the sample in this manner the sample size is considerably reduced, and with

it the likelihood of rejection.

In order to investigate whether the non-rejection of normality within a régime is simply due to a sample-size effect, the following bootstrapping procedure was conducted; see for example, Efron . Three new series of length equal to the number of observations in the full series and the number in the sub-samples of the two regime as generated by the Markov régime-switching model; see section 6.4.2, are created by sampling with replacement from the original series. The Jarque-Bera test is then applied to each of the three series, noting whether or not rejection of normality occurs at the 0.05 level in each case. This procedure is repeated 10,000 times for each of the returns series, and the proportion of times rejection occurs for each series at each sample size is calculated. The results are presented in table 6.2 on page 186.

The results clearly demonstrate that the likelihood of rejection falls substantially as the number of observations is reduced. This is a pure sample size effect, as sampling is being undertaken with replacement from the actual data. ### Accommodating the Non-Normality of Commercial Real Estate Returns

To eliminate non-normality from most series, Myer & Webb [1994, p. 280] suggested the use of biannual or annual returns. However, the above observation indicates that studies suggesting the use of annual/biannual returns data, which are usually less likely to reject normality than quarterly or monthly series, may simply be picking up a small sample size effect caused by the asymptotic nature of the tests for normality which are frequently used. Also, given the length of time series available for real estate and the applications for which they may be used, such an approach is questionable, with which Myer & Webb  concur. They also show that a reduction in non-normality can be achieved by using real as opposed to nominal returns.

Using real estate returns series corrected for autocorrelation, as suggested in Firstenberg, Ross & Zisler  and Liu, Hartzell & Grissom [1992a], has also been shown to reduce the non-normality problem. However, a considerable degree of non-normality remains and, under some conditions, it can produce unrealistic return series (Myer & Webb ). 