a 3 (L)' * ai(L)

a i2(L) ^ a12(L)

* *

Wide-sense stationarity of yt requires that xt and zt be wide-sense

* *

stationary, and further depends on the nature of zt . If zt is strongly

exogenous then ai(l) - a2(l) > 0 is still the necessary and sufficient condition,

and Weiss [1986b] has given conditions for the wide-sense stationarity of yt in

*

the GARCHZ model in some cases in which zt includes squared lagged dependent variables.

*

The elements of zt must also be strongly exogenous to preserve the block diagonal structure of the information matrix between ß and a of pure

GARCH processes. The GAUCHZ model is attractive under these

circumstances both because of its generality and tractability. However, in a pure time series context Weiss [1984] has found significant relationships for the

2

conditional variance of many economic variables with yt-i , which argues against the empirical plausibility of simple combinations of strongly exogenous variables and GARCH effects. In such case <jaß * 0 in general, and the same occurs when functions of (it affect h t.

For further reference in this Thesis the terms 'ARCH class' or 'ARCH model' denote the more general family of GARCHZ parameterizations that preserve a block diagonal information matrix, as this turns out to be a central issue for the properties of this class.

§2.4 Objective and design of Monte Carlo experiments

The theoretical results derived in the Thesis are based on asymptotic theory. In practical situations with economic data the sample sizes are typically small, but asymptotic theory makes complex general problems tractable and thus provides an approximation to the sampling properties of estimators and tests. In the absence of operative exact results, this

approximation makes possible empirical work in a wide range of situations and provides a benchmark for further theoretical developments. The cost of using asymptotic results in small sample situations is, of course, that we are dealing only with approximations whose quality may differ substantially from one problem to another.

At the other extreme we have Monte Carlo methods, which permit a more precise assessment of the sampling properties of estimators and test-statistics in specific situations, but are limited in scope to the specific forms of data generating processes and models used and thus lack generality. In some cases, the availability of results on invariant properties of the exact distribution of some statistics gives more general validity to simulation results. For

example, the distribution of least squares residuals in the classical linear model does not depend on the true value of the parameter vector , amplifying

the scope of simulation experiments in this context (see for example White and MacDonald [1980]). In generalized regression models, a topic of particular relevance for our purposes, Breusch [1980] proved a similar result and thus established the invariance of various statistics to the true value of the mean parameters. Unfortunately, these invariance properties do not extend to the type of models that we consider because generally the dependence of the conditional variance on mean parameters and the nonlinear nature of the problem cause the distribution of the residuals of both the mean and the variance equations to depend on the full parameter vector 9 .

A different approach to extend the conclusions obtained from Monte Carlo experimentation has been put forward by Hendry [1979,1984], who advocates the use of response surfaces. These surfaces represent attempts to estimate the distribution of statistics conditionally on the key parameters of the problem, using a specification coherent with the theoretical asymptotic distribution. Maasoumi and Phillips [1982] have expressed a skeptical view on the

possibilities of response surfaces, but nevertheless the approach seems to be gaining popularity, and Engle et al [1985] provide an application of this approach to the study of the ARCH model.

As a complement to the theory presented in the next few chapters, we have performed some Monte Carlo experiments. The nature of this evidence is

rather limited because we cannot even claim the applicability of Breusch's results in our framework, and therefore the aim of our experiments is simply to analyze in a few models whether the asymptotic results provide reasonable approximations to the properties of estimators and test-statistics in small to moderate samples, or whether there are obvious departures from the

asymptotic theory which might suggest situations where special care is

required in the application of the theoretical body of the Thesis. The results of the simulations are presented and discussed in Chapters 3 to 6 .

The experiments consider the estimation and testing of two different models, the Poisson-N model,

yt I ~ N [ xt' ß , xt' ß ], and the ARCH(l) model,

yt I y t ~ N [ xt' ß , ao + ai ut-i ] , where xt = (1 , xi t )'.

We have conducted two types of experiments. The first type looks at the behavior of estimators, and the second type looks at the size and power of tests. The DGP's are designed to provide either correct specification or a misspecified model. Specification error may take the form of a misspecified conditional mean, a misspecified conditional variance, or a misspecified distributional form. The regressor xn was generated as an independent uniform variate in ( - 1 , 0 ) , and an additional regressor X2t generated as independent uniform in

( 0 , 1 ) was also included in some experiments to induce mis specification in the conditional mean. The correlation between xit and X2t was set to 0.5 , the

regressors were fixed in repeated samples, and all pseudo-random numbers were generated using NAG Library routines. Most experiments consider four sample sizes ( T = 2 0 , 5 0 , 1 0 0 , 2 0 0 ) , and in order to reduce experiment variance the smaller samples are always taken as subsamples of the larger ones. The number of replications for each simulation experiment was 500 .

It was thought convenient to have the simplest possible DGP's. The true mean parameter vector ßo was set to ßo = ( 0 , 0 )' for the ARCH model, and to ßo = ( 1 , 1 ) ' for the Poisson-N simulations. However some of the departures from the null hypothesis analyzed for the ARCH model have no consequence if ßo = 0 (e.g. ht = ao + ai yt_i and ht = ao + ai ut-i are identical) , and a true parameter vector ßo = ( 1 , 1 ) was also considered for some ARCH estimation experiments, and for the simulations to assess the power of the consistency tests presented in

Chapter 5. The additional regressor X2t , whenever included, was assigned a