Multicollinearity and the Mean Square Error of Alternative Estimators

The problem of collinearity suggests the search for an alternative to ordinary least
squares which, although biased, might reduce the mean square error of the coefficient of
interest. Two types of estimators are examined, and the corresponding mean square error
loss functions are calculated.
1. INTRODUCTION
THEPROBLEM OF MULTICOLLINEARITY in regression analysis is essentially a lack of
sufficient information in the sample to permit accurate estimation of the individual
parameters. In many situations, however, the researcher may not be interested in
the values of all the parameters. Moreover, in small samples it may seem appropriate
to forgo the unbiasedness of ordinary least squares estimators to reduce the
mean square error of the parameters of interest. These ideas provide the motivation
for the current paper.
More specifically, let
where x and z are nonstochastic, each variable has mean zero, and u is a serially
independent random disturbance with mean zero and constant variance. We
consider the problem of estimating p when x and z are highly correlated and the
mean square error (MSE) of the estimate of fi is the measure of the goodness of the
estimator.