We will summarise the statistical reviews of the Wilkie model in five subsections which consider the methodology, model and parameter uncertainty and non-stationarity, non-normality of the residuals, heteroscedasticity, and non-linearity as in Rambaruth (2003).
Methodology
Huber (1997) reviews the Wilkie Model in both empirical and theoretical sense. He expresses his reservations about the methodology proposed in specifying the Wilkie model in the discussion of 1995 paper by raising the ‘data-mining’ issue. He criticises Wilkie’s approach in which he recommended that asset models should be developed by establishing a linear relationship based on economic theory (or ‘common sense’), fitting it to the data and then testing whether this relationship satisfies various goodness-of-fit tests. If the tests are not satisfied, then parameters should be added until the tests are satisfied or the results should be ignored on theoretical grounds. This methodology ignores the problems associated with multiple hypothesis testing (which can lead to data-mining). According to Huber (1997), it basically restricts the model to the Auto- regressive Integrated Moving Average (ARIMA) class and it does not allow ‘common sense’ to be influenced by the data which would allow us to improve our understanding of the economy.
Hardy (2003) points out the problem of ‘data-mining’ by which Huber means that a statistical time-series approach, which finds a model to match the available data, cannot then use the same data to test the model. Thus, with only one data series available, all non-theory-based time-series modelling is rejected. One way around the problem is to use part of the available data fit the model, and the rest to test the fit. She emphasises that the problem for a complex model with many parameters is that data are already scarce.
Model and Parameter Uncertainty and Non-stationarity
Kitts (1990) was the first to point out that the parameters of the Wilkie inflation model (Wilkie, 1986) may not be constant over time. If the mean rate of inflation is likely to change in the future, i.e. when the current stationary sub-period ends, then the model is inadequate as it does not necessarily describe the way in which appropriate investment variables will move over the future long-term.
Huber (1997) examined the parameter constancy of the original price inflation model by recursively estimating its parameters on incrementally larger data sets. He
drew the graphs of QM U and QA with 95% confidence intervals and concluded that
these parameters may not be constant. However, Huber had some reservations about interpreting these results because they might simply be due to the non-normality of the residuals or they could be due to the change in the calculation of the official UK price index.
For the dividend yield model, he emphasized the sensitivity of theY W parameter
to the years 1940 and 1974 as Figure 1.16 illustrates. He states that if they are excluded
from the regression, thenY W becomes insignificantly different from zero. The problem
with including Y W is that it results in a general tendency for changes in yields to be
correlated with changes in inflation, but this correlation only seems to be appropriate for large increases in yields and inflation.
For the consols yield model, Wilkie (1995) noted thatCY becomes insignificantly
different from zero when an intervention variable for 1974 was included. Huber argues
that CY appears to have a similar problem to Y W because the parameterCY seems
to describe mainly the event that the largest increase in interest rates coincided with
the largest residual from the share dividend yield model. However, ifCY is set to zero,
then the model implies that there is no relationship between equity returns and real interest rates. He concludes that as this does not appear to be a reasonable assumption,
it may explain why Wilkie (1995) included CY in the model.
Cairns in the discussion of Wilkie (1995) drew attention to the standard errors of parameters estimates which he found extremely important because not only is a model an approximation to reality, but it is not known what the ‘true’ set of parameters should
be for this model. It is, therefore, essential as part of any simulation exercise, to repeat the exercise many times using a range of parameter values which is consistent with the past data and with the standard errors of the parameters and their correlations. Non-normality and Non-independence of Residuals
In 1989, the Financial Management Group of the Institute and Faculty of Actuaries
was assigned the task of criticising the model from a statistical viewpoint (Geoheganet
al., 1992). This group performed some tests of simulations and examined the standard
deviations of returns and correlations between different asset classes at different time
horizons on the Wilkie (1986) model. In this review, Geohegan et al. identified three
areas of concern regarding the suitability of the model.
• The existence of burst of inflation, indicating that once an upward trend in in-
flation is established, there is a tendency for it to continue.
• The existence of large, irregular shocks, such as those in the mid-1970s.
• The possible skewness of residuals.
The only substantive criticism was of the inflation model. The AR(1) model ap-
peared too thin tailed, and did not reflect prolonged periods of high inflation.
In an early review of the model, Kitts (1990) reported that there is some evidence that the residuals are not independent, so that the model does not capture the frequency of the occurrence of sustained periods of extreme inflation and deflation. Moreover, the distribution of the residuals are not normal due to non-constant variance.
Finklestein, in the discussion of Wilkie (1995), expresses his concerns about the skewness of the data and the assumption of normality. He believes that the underly- ing probability distributions are stable non-Gaussian which are suggested for further research in Wilkie (1995).
The motivation behind introducing an ARCH model for the price inflation in Wilkie (1995) was mainly these criticisms.
Heteroscedasticity
Since it is the inflation process which drives the Wilkie model, it is crucial that this model has a good representation.
Geoghegan et al. (1992) reported the existence of bursts of inflation, indicating that once an upward trend in inflation was established, there is a tendency for it to continue as mentioned before. This leads to what was described in Engle (1982) as auto-regressive conditional heteroscedastic (ARCH) model (see also Mills, 1990). In
Appendix B of Geohegan et al. (1992) Wilkie demonstrates how his model could be
adopted to incorporate ARCH effects. In 1995 paper, he suggests using an ARCH model for inflation that would model the heavy tail. Although it has been seen that the ARCH model provided a better fit to the data for the period 1923-1995, updating the data to 2009 showed that it is not as good as it was. The recursive estimates of
the ARCH model parameter, QSB also indicates that since the value is very close to
zero up to early 1970s, the AR(1) process is enough to model the price inflation. As
mentioned in Section 1.4, the ARCH model is a useful description for the periods that include 1960s and 1970s.
A further problem that is fundamental to all ARCH models is their complex struc- ture. Also, with small data sets the parameters are unstable which we show in Fig- ures 1.8 and 1.9.
Non-linearity
In the discussion of Wilkie (1995), Tong comments on the several aspects of linear models which limits one’s horizon and the need to use non-linear models. First of all, he criticises the linear models as not respecting the current position while making a forecast and giving exactly the same prediction interval regardless of the current position. Second, since the current position is not always known precisely, because of information delay, there is always some relevance in looking at the sensitivity of the model to the initial value (current position) which might be a trivial exercise for a linear model. As a final aspect, introducing the model with some exogenous variables, then some non-linearity may be required. In the written contribution, Wilkie (1995)
refers to Tong’s argument and besides stating that he found non-linearity was very much worthy of further investigation, he pointed out that the many of the data series were rather too short to allow clear phases of different types to be distinguished.
Whitten and Thomas (1999) suggest that the economy behaves differently in times of hyperinflation, than it does in times of ‘normal’ inflation levels. By definition, this belief cannot be incorporated into linear models. Wilkie’s linear model is widely used and for the most part a good representation of its economic variables. Following the footsteps of Wilkie (1986, 1995), they thought it best to adapt his model to incorpo- rate this non-linearity, rather than fundamentally change its formulation. Thus, they proposed a threshold non-linear model which is discussed briefly in Section 1.12.