A Discussion of the Appropriateness of the Stochastic Eco-

A large number of papers have been written that investigate the characteristics of economic time series data and discuss whether stochastic asset models, such as the Wilkie model, are appropriate, based on the results of these investigations. Two recent papers that have evaluated stochastic asset models in the context of Australian data are Sherris (1997) and Sherris et al. (1999). One major result of these papers is that many of the time series were found to be non-normally distributed, with time varying volatility. Thus, the assumption of independent and identically distributed normal errors, which underlies many of the above mentioned time series models, is violated. This same point has also been raised in a number of papers with regards to financial data from other countries. In fact, Wilkie himself acknowledged this in Wilkie (1995), and it was for this reason that Wilkie proposed the GARCH variant of his model. Time varying volatility was also allowed for in the Share Price Index component of the CAS/SOA model through the regime-switching model.

The only difference between the basic Wilkie model and its GARCH variant is that the AR(1) process used to model inflation in the basic model is replaced by a GARCH(1,0) process in the GARCH variant. All three of the stochastic asset models considered in this thesis use the same AR(1) process to model inflation, thus a GARCH variant of the Kemp model and the CAS/SOA model can be devised by replacing the AR(1) inflation process with a GARCH inflation process in each case

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116 Stochastic Sub-Models (in the case of the CAS/SOA model, this would be in addition to the allowance for time varying volatility in the Share Price Index sub-model, and may potentially render this second allowance unnecessary). A second GARCH variant can also be devised by substituting a GARCH(1,1) process for the AR(1) inflation process. That is, assuming that σQ2 (t) varies with time in accordance with the process:

σQ2 (t) =βQ,1+βQ,2ε2Q(t−1) +βQ,3σ2Q(t−1), t = 1,2, . . . . (7.25)

All three of these versions of each of the three stochastic asset models are considered in this thesis.

There is much debate as to the exact nature of the relationships between the different economic variables and as to how these relationships should be allowed for in an asset model. In particular, there is debate as to whether a “cascade” structure should be used to model the variables or whether an alternative model structure, such as a Vector Autoregressive process, that allows for feedback relationships be- tween the various economic time series, should be used. The fact that this contention exists indicates that different statistical tests and different data sets have given rise to conflicting results in this matter, and it is for this reason that so many different stochastic asset models have been proposed.

Just as it is infeasible to consider, in this thesis, every stochastic asset model that has been proposed, so it is infeasible to repeat the large volume of tests that have been used in the past to determine which economic variables are related and how best to model these dependencies. Instead, it shall be assumed that a “cascade” structure is an appropriate means of describing the economic variables (all three of the models considered employ a “cascade” structure, and in spite of criticism, “cascade” structures are employed in a large number of stochastic asset models including recent models such as the CAS/SOA model), and we shall limit ourselves to determining the most appropriate of the three models previously described for the data we have. In order to determine the most appropriate of these models, we will employ some of the tests used by Harris (1995) in his comparison of stochastic asset models. These tests are described in Section 7.2.5 and the results of these tests are presented in Section 7.2.7.

In Sections 6.3 and 6.4, evidence was provided which suggested that significant relationships exist between unemployment rates and mortality, and between unem- ployment rates and lapsation. Thus, any stochastic economic sub-model used in the overall stochastic asset-liability model should be capable of forecasting unemploy- ment rates. Of the three models considered, only the CAS/SOA model does this (in the CAS/SOA model, unemployment was linked to the other model variables using

Stochastic Sub-Models 117 a relationship based on the Phillips curve11_{). Consequently, it is necessary for us to} extend both the Wilkie model and the Kemp model to include an unemployment sub-model.

In the case of the Kemp model, it was decided to model the unemployment rate using a geometric random walk with drift of the form:

ln (1 +U(t)) = ∆ lnQb(t) +ψU+εU(t). (7.26)

This is of the same form as the models used by Kemp to describe the evolution of asset returns.

In the case of the Wilkie model, there is no “obvious” sub-model to use to model the unemployment rate. It is uncommon for stochastic asset models to include the unemployment rate as one of the variables in the model and no alternatives to this model were found in the review of stochastic asset models. We decided, therefore, to use the process given in Equation (7.24) (Model 1), along with three slight variants on this model: Model 2: U(t) =φU(1−αU,1) +αU,1U(t−1) +αU,2∆ lnQb(t) +αU,3∆ lnQb(t−1) +εU(t) ;