Note on Second Order Effects in a Semiparametric Context

摘要

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(z sub i, theta) with theta unknown. If theta is any root-N consistent estimate of theta based on squared residuals, it is well known that the resulting generalized (weighted) least squares estimate with estimated weights has the same limit distribution as if theta were known. The covariance of this estimate can be expanded to terms of order 1/N-sq. 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 square estimate has the same first order normal limit distribution as if the variance function were completely specified. A special case is computed for an expansion for the covariance in this semiparametric context, and the rate of convergence is found to be slower for this estimate than for its parametric counterpart. More importantly, there is an effect due to how well one estimates the variance function. A kernel regression estimator is used to find that the optimal bandwidth in the problem is of the usual order, but the constant depends on the variance function as well as the particular linear combination being estimated. (jhd)