Exact optimal and adaptive inference in regression models under heteroskedasticity and non-normality of unknown forms∗

In this paper, we derive simple point-optimal sign-based tests in the context of linear and nonlinear regression models with fixed regressors. These tests are exact, distribution-free, robust against heteroskedasticity of unknown form, and they may be inverted to obtain confidence regions for the vector of unknown parameters. Since the point-optimal sign tests depend on the alternative hypothesis, we propose an adaptive approach based on split-sample techniques in order to choose an alternative such that the power of point-optimal sign tests is close to the power envelope. The simulation results show that when using approximately 10% of sample to estimate the alternative and the rest to calculate the test statistic, the power of point-optimal sign test is typically close to the power envelope. We present a Monte Carlo study to assess the performance of the proposed “quasi”-point-optimal sign test by comparing its size and power to those of some common tests which are supposed to be robust against heteroskedasticity. The results show that our procedures are superior.