Lee, Benny Man-Shan
Description
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...[Show more] 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.
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