Preliminary-test estimation of a mis-specified linear model with spherically symmetric disturbances
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis considers some finite sample properties of preliminary test (pre-test) estimators of the unknown parameters of a (possibly) mis-specified linear regression model. We investigate two types of misspecification which may or may not occur simultaneously. The first relates to the distribution of the regression disturbances, which is assumed to be normal, when in fact, the error distribution belongs to a broader family of spherically symmetric distributions. The second mis-specification is that the model's design matrix may exclude relevant regressors. We analyse some finite sample properties of three pre-test estimators. The first is an estimator of the prediction vector after a pre-test for exact linear restrictions on the location vector. Secondly, we consider an estimator of the error variance after the same pre-test. Finally, we analyse an estimator of the error variance after a pre-test for homogeneity of the variances in the two-sample linear regression model. In each case we extend the existing literature by generalising the model's error distribution and allowing for model mis-specification through the omission of regressors. To provide a setting for this research, we survey the relevant pretesting literature in Chapter Two. This discussion assumes that the errors are normally distributed. There is a body of research, however, which proposes that some economic data series may be generated by processes whose underlying distributions have thicker tails than that which would result from a normality assumption. We briefly examine this literature in Chapter Three. One alternative family of distributions, which has received considerable attention, is the spherically symmetric family of distributions. Well known members of this family include the normal and the multivariate Student-t distributions. So, we include in Chapter Three a rationale for investigating spherically symmetric regression disturbances as an alternative to the usual normality assumption. We also discuss several studies which consider the linear regression model under a spherically symmetric disturbance assumption. Having provided a setting and rationale for our research in Chapters Two and Three, Chapters Four, Five and Six present the finite sample properties of the aforementioned pre-test estimators. In each of these chapters we derive the exact bias and the exact risk functions (under quadratic loss) of the estimators under the mis-specified regression model. We also give the non-null distributions of the commonly used test-statistics for the investigated pre-tests, and we generalise many of the results reported in the existing literature. In particular, we derive the critical values of the test which result in a minimum of the bias and of the risk of the pre-test estimators of the error variance. To illustrate the results we assume multivariate Student-t regression disturbances, rather than the general spherically symmetric family, and numerically evaluate the derived expressions for various cases. Our results suggest, when estimating the prediction vector, that the mis-specification of the distribution of the regression disturbances has little impact on the qualitative properties of the predictor pre-test estimator, though there are quantitative effects. However, when estimating the error variance, after either a pre-test for linear restrictions or for homogeneity of the error variances, we find that mis-specifying the error distribution can have a substantial qualitative, and quantitative, impact on the bias and the risk functions of the estimators. Imposing the linear restrictions, even if they are valid, or always pooling the samples, even if the error variances are identical, may often be inappropriate strategies. The final chapter, Chapter Seven, contains some concluding remarks. In particular, we consider some possible future research topics.