On the use of 'improved' estimators in econometrics

Abstract

This thesis carries a title that might appear to be too extensive as a topic. However, those familiar with the literature on biased estimators may agree that there is a well defined class of estimation procedures of interest to both mathematical statisticians and econometricians. Efforts to introduce ideas which deviate from the traditional classical notion of unbiasedness have encountered enormous resistance. Admittedly, results relating to biased estimators are not as wellestablished as those relating to unbiased estimators, but unbiasedness is an arbitrary and unnecessarily stringent criterion. One should not therefore neglect the usefulness of biased estimators. With this background in mind, the thesis was written to synthesize the many differently motivated contributions which aim at improved estimation of unknown economic linear relationships. Apart from highlighting the author’s own contributions in the area, the author has also attempted to make the thesis a self-contained one. Chapter 1 motivates the study and defines the framework in which new estimators are developed. The fundamentals of Bayesian inference are discussed and the relation between formal and empirical Bayes procedures is examined. Chapter 2 provides a synthesis of different attempts to improve upon the traditional unbiased estimator. This chapter is necessary because it is not generally acknowledged that the differently motivated efforts can lead to the same result - namely, some sort of shrinkage must be introduced to improve estimation and that all the improved estimators are basically generalised Bayes rules. Chapter 3 introduces the controversial ridge estimator and provides a comprehensive survey. A new contribution made in this chapter is the introduction of a recursive algorithm for generating the ridge trace. Chapter 4, 5 and 6 form the core of the thesis where new ideas are developed. Specifically, Chapter 4 attempts theoretical and Monte Carlo studies of the potential and realised reduction in risk of the biased estimators. A number of good adaptive ridge estimators are identified. As an illustration these are applied to re-estimating an investment function. Significantly more accurate predictions are achieved by the biased estimators than by conventional ordinary least squares estimator and the preliminary test estimators. Two new contributions are made in Chapter 5. Firstly, an analysis of seasonal variability in the distributed lag model sets the stage for the introduction of various estimators which can incorporate bi-dimensional prior information in the form of exchangeability and smoothness. Secondly, estimation of distributed lag model in the frequency domain is justified and the Spectral Ridge Estimator is introduced as an extension of Hannan’s Efficient Estimator. The estimator’s performance is compared to other well-known estimators using Almon’s data. Chapter 6 works out the small sample bias and mean square error of a Generalised Ridge Instrumental Variable estimator for a structural equation in the context of a simultaneous equation system. The problem of undersized sample is tackled and the traditional optimism about 2SPC questioned. A new estimator which involves the application of ridge regression instead of the traditional least square regression at both stages of a 2SLS procedure is proposed and its statistical properties analysed (both asymptotically and in finite sample). Some further results concerning ridge regression are presented in the last chapter, i.e. 7. The robustness of ridge regression under misspecification is analysed. Problems of testing stochastic hypotheses and the construction of confidence sets are also discussed. Some of the criticisms of the technique are reviewed and a personal view is expressed.