# SFB 303 Discussion Paper No. A - 170

**Author**: Carroll, Raymond J., and Wolfgang Härdle

**Title**: Second Order Effects in Semiparametric Weighted Least Squares Regression

**Abstract**: We consider a heteroscedastic linear regression model with normally distributed errors in which the
variances depend on an exogenous variable. Suppose that the variance function can be parameterized
as* psi (zi,
theta)* with *theta* unknown. It is well known that, under mild regularity conditions, the weighted least squares
estimate with consistently estimated weights has the same limit distribution as if theta were known. The
covariance of this estimate can be expanded to terms of order
*n*^{-1}. If the variance function
is unknown but smooth, the problem is adaptable, i.e., one can estimate the variance function nonparametrically
in such a way that the resulting generalized least squares estimate has the same first order normal limit
distribution as if the variance function were completely specified. We compute an expansion for the covariance
in this semiparametric context, and find that the rate of convergence is slower than for its parametric counterpart.
More importantly, we find that there is an effect due to how well one estimates the variance function. For kernel
regression, we find that the optimal bandwidth is of the usual order, but that the constant depends on the variance
function as well as the particular linear combination being estimated.

**Keywords**:

**JEL-Classification-Number**:

**Creation-Date**: April 1988

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