COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 786 "Best Median Unbiased Estimation in Linear Regression Donald W.K. Andrews and Peter C.B. Phillips March 1986 We first show that the Generalized Least Squares estimator is the best median unbiased
estimator of the regression parameters for quite general loss functions, when the
parameter space is unrestricted. Of note is the fact that this result holds without moment
restrictions. Thus, the errors may have multivariate Cauchy distribution. Next, we show
that a restricted GLS estimator is best median unbiased for a linear combination of the
regression parameters, when that linear combination is restricted to lie in an interval.
Certain other linear combinations of the parameter vector may be subject to arbitrary
additional restrictions. The paper then presents best median unbiased estimators of the
error variance sigma-squared, as well as monotone functions of sigma-squared, when the
errors are normally distributed. If sigma-squared is constrained to lie in a finite
interval, the best estimator is a censored version of its unconstrained counterpart. When
sigma-square is constrained only to be positive, the best median unbiased estimator is
always larger than the best mean unbiased estimator s-squared, and is approximately equal
to s-squared calculated with its degrees of freedom reduced by .66. |
