Estimating the conditional variance of Y, given X, in a simple regression model

Consider the regression model [image omitted] . In a variety of situations, an estimate of VAR (Y &7C X) = λ (X) is needed. The paper compares the small-sample accuracy of five estimators of λ (X). The results suggest that the optimal estimator is a somewhat complex function of the underlying distributions. In terms of mean squared error, one of the estimators, which is based in part on a non-robust version of Cleveland's smoother, performed about as well as a bagged version of the so-called running interval smoother, but the running interval smoother was found to be preferable in terms of bias. A modification of Cleveland's smoother, stemming from Ruppert et al. (1997), achieves its intended goal of reducing bias when the error term is homoscedastic, but under heteroscedasticity, bias can be high, and in terms of mean squared error it does not compete well with the kernel method considered in the paper. When ε has a heavy-tailed distribution, a robust version of Cleveland's smoother performed particularly well except in some situations where X has a heavy-tailed distribution as well. A negative feature of using Cleveland's robust smoother is relatively high bias, and when there is heteroscedasticity and X has a heavy-tailed distribution, a kernel-type method and the running interval smoother give superior results in terms of both mean squared error and bias.