Quantile inference for heteroscedastic regression models

Consider the nonparametric heteroscedastic regression model Y=m(X)+σ(X)ɛY=m(X)+σ(X)ɛ, where m(·)m(·) is an unknown conditional mean function and σ(·)σ(·) is an unknown conditional scale function. In this paper, the limit distribution of the quantile estimate for the scale function σ(X)σ(X) is derived. Since the limit distribution depends on the unknown density of the errors, an empirical likelihood ratio statistic based on quantile estimator is proposed. This statistics is used to construct confidence intervals for the variance function. Under certain regularity conditions, it is shown that the quantile estimate of the scale function converges to a Brownian motion and the empirical likelihood ratio statistic converges to a chi-squared random variable. Simulation results demonstrate the superiority of the proposed method over the least squares procedure when the underlying errors have heavy tails.