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Second order effects in semiparametric weighted least squares regression. (English) Zbl 0669.62020
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 $$\psi (z_ i,\vartheta)$$ with $$\vartheta$$ 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 $$\vartheta$$ 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.

##### MSC:
 62G05 Nonparametric estimation 62J05 Linear regression; mixed models
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