This thesis is concerned with the finite sample properties of some estimators of the unknown parameters in a linear model which is (possibly) mis-specified through the exclusion of relevant regressors. We assume that in addition to sample information, prior information regarding the unknown parameters is available in the form of a linear inequality constraint imposed on the regression coefficients. The combination of this type of prior information and sample information in specifying the corresponding statistical model leads to what has been identified in the literature as the inequality restricted estimator. If the statistical significance of the inequality constraint is tested prior to the estimation process, then the estimator thereby generated is called the inequality pre-test estimator.

The properties of these estimators of the coefficient vector in a properly specified model have been examined rather thoroughly in the literature. In this thesis, we extend the results reported in the literature to the case where the underlying regression model is underfitted. We also investigate the sampling performance of the corresponding estimators for the model's disturbance variance, as well as the choice of an optimal size for the pre-test. The general background and motivation for this study are given in Chapter 1.

Much of the earlier research on inequality restricted and pre-test estimation are built on results from studies that assume that the prior information is in the form of linear equality restrictions. We survey the relevant literature in this area in Chapter 2. Chapter 3 reviews the literature on inequality restricted and pre-test estimation. We focus on this problem in the context of the standard linear model with a single linear inequality constraint on the coefficient vector, as this is directly related to the theme of this thesis.

In Chapter 4, we derive and evaluate the risk, under quadratic loss, of the inequality restricted and pre-test estimators for the regression prediction vector in an underfitted model. This analysis takes the established literature further by allowing for mis-specification in the regressor matrix. We consider the risk of the prediction vector, rather than the coefficient vector itself, so that our results are data independent. The risk functions of the corresponding estimators for the regression disturbance variance in the properly specified and underfitted models are derived in Chapters 6 and 7 respectively.

As in the case where the prior information exists as linear equality restrictions, our results show that when the model is underfitted, the use of valid prior information does not necessarily guarantee a reduction in risk. This result holds for the estimation of both the prediction vector and the scale parameter. When one is estimating the regression disturbance variance, with an appropriate choice of test size, the inequality pre-test estimator can uniformly dominate the estimator that uses sample information only. We also find that the risk functions of the estimators of the error variance are affected more by mis-specification than are the corresponding predictive risks.

In the case where no strictly dominating estimator exists, the question of the choice of an optimal critical value of the pre-test remains. Chapters 5 and 8 explore this issue when one is estimating the prediction vector and scale parameter respectively. We find that most of our results concur qualitatively with those reported in the literature when the prior information exists as exact equality restrictions.

Chapter 9 contains some concluding remarks and a summary of the major results obtained in earlier chapters. We also outline some possible future research topics in this general area.