# Presentation of the Data

# Presentation of the Data

The data and methodology used to estimate the probabilities of the next period's forecast returns, and to calculate the asset's actual return in each period, was identical in all respects to those described in section 3.4.2, except that which dealt with real estate.

### The Adjustment of Commercial Real Estate Returns

The foundation of Barkham & Geltner's [1994] work rested on the assumption that appraisers, when valuing at time *t*, rely on the information they have in the market at time *t* as it affects the value they have for the property at time *t*-l. This infers that the information available to them at time *t* is noisy, and that they are forced to rely on past valuations. This is effectively an AR(1) process that smoothes the evolution of prices. Under this assumption the 'real' past process of prices can be inferred. However, this simple exponential unsmoothing model assumes that the weight valuers place on past valuations is held constant through time, an assumption inconsistent with the findings contained in Chapter 6.

As a result, this methodology could not be used directly; reliance was instead made on its estimation of the underlying volatility. In common with the methodology employed in Webb & Rubens [1988] and Grauer & Hakansson [1995], and in line with a broad range of stochastic processes, the variance of capital returns was increased by *Ө*. This obtained a reasonable proxy of the unsmoothed Barkham & Geltner [1994] series. This is similar to the median 'unsmoothed-smoothed' variance method described above, insofar as it implies that the observed index is an unbiased prediction of the true value, but with its estimation of volatility is biased. Note that this adjustment makes the method mean preserving, and is consistent with a broad class of stochastic processes, including Brownian motion.

Allowance for this bias was therefore made by increasing the variance of the index. Using the median 'unsmoothed-smoothed' variance, 159.6% ("RE-M") suggested by Barkham & Geltner [1994], 16 the variance of the capital returns component of the series was increased. The sensitivity of the analysis was also tested by using 127.7% ("RE-L") and 191.5% ("RE-H"), ± 20% as the degree of variance.

As discussed in Chapter 6, the returns from real estate may be better described by a discrete mixture of two normal distributions. The Markov régime-switching model - detailed in section 6.4.2 - was therefore employed to provide estimates of next period distribution and probability, *q*; see section 6.2. Figure 7.3 plots *q*, the conditional, and *λ*, the unconditional probability estimates for the JLW total returns series. The figure illustrates that a high degree of continuity is observed when the JLW series was unconditionally sorted i. e. when observations are assigned a probability estimate, *λ*, without reference to the observations location within the series. This strengthens support for the use of a discrete mixture of a two distributions model as a description of the data generating process.

A box and whisker plot of the two unconditional distributions is shown in figure 7.4, with figure 7.5 plotting these against expected normal values.

The simple régime-switching forecast and historic moving average estimates are plotted against the JLW total returns series in figure 7.6. As illustrated, the simple forecasting model provides a better estimate of next period return than any of the historic moving average estimates. This is supported by table 7.3, which details their correlation with the actual JLW series. The correlation between the JLW total returns series and the simple regime-switching forecast is significant at the 0.05 level.

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^{16}RE-0.625 in Chapter 3 on page 45.