Pre-test estimation in a regression model with a mis-specified error covariance matrix

This thesis considers some finite sample properties of a number of preliminary test (pre-test) estimators of the unknown parameters of a linear regression model that may have been mis-specified as a result of incorrectly assuming that the disturbance term has a scalar covariance matrix, and/or as a result of the exclusion of relevant regressors. The pre-test itself is a test for exact linear restrictions and is conducted using the usual Wald statistic, which provides a Uniformly Most Powerful Invariant test of the restrictions in a well specified model. The parameters to be estimated are the coefficient vector, the prediction vector (i.e. the expectation of the dependent variable conditional on the regressors), and the regression scale parameter. Note that while the problem of estimating the prediction vector is merely a special case of estimating the coefficient vector when the model is well specified, this is not the case when the model is mis-specified. The properties of each of these estimators in a well specified regression model have been examined in the literature, as have the effects of a number of different model mis-specifications, and we survey these results in Chapter Two. We will extend the existing literature by generalising the error covariance matrix in conjunction with allowing for possibly excluded regressors. To motivate the consideration of a nonscalar error covariance matrix in the context of a pre-test situation we briefly examine the literature on autoregressive and heteroscedastic error processes in Chapter Three. In Chapters Four, Five, Six, and Seven we derive the cumulative distribution function of the test statistic, and exact formulae for the bias and risk (under quadratic loss) of the unrestricted, restricted and pre-test estimators, in a model with a general error covariance matrix and possibly excluded relevant regressors. These formulae are data dependent and, to illustrate the results, are evaluated for a number of regression models and forms of error covariance matrix. In particular we determine the effects of autoregressive errors and heteroscedastic errors on each of the regression models under consideration. Our evaluations confirm the known result that the presence of a non scalar error covariance matrix introduces a distortion into the pre-test power function and we show the effects of this on the pre-test estimators. In addition to this we show that one effect of the mis-specification may be that the pre-test and restricted estimators may be strictly dominated by the corresponding unrestricted estimator even if there are no relevant regressors excluded from the model. If there are relevant regressors excluded from the model it appears that the additional mis-specification of the error covariance matrix has little qualitative impact unless the coefficients on the excluded regressors are small in magnitude or the excluded regressors are not correlated with the included regressors. As one of the effects of the mis-specification is to introduce a distortion into the pre-test power function, in Chapter Eight we consider the problem of determining the optimal critical value (under the criterion of minimax regret) for the pre-test when estimating the regression coefficient vector. We show that the mis-specification of the error covariance matrix may have a substantial impact on the optimal critical value chosen for the pre-test under this criterion, although, generally, the actual size of the pre-test is relatively unaffected by increasing degrees of mis-specification. Chapter Nine concludes this thesis and provides a summary of the results obtained in the earlier chapters. In addition, we outline some possible future research topics in this general area