Hypothesis testing on high dimensional quantile regression

Quantile regression has been an important analytical tool in econometrics since it was proposed in 1970s. Many advantages make it still popular in the era of big data. This paper focuses on the testing problems of high-dimensional quantile regression, supplementing to robust methods in the literature of high-dimensional hypothesis testing. We first construct a new test statistic based on the quantile regression score function. The new test statistic avoids the inverse operation of the covariance matrix, and hence becomes applicable to high-dimensional or even ultrahigh-dimensional settings. The proposed method retains robustness for non-Gaussian and heavy-tailed distributions. We then derive the limiting distributions of the proposed test statistic under both the null and the alternative hypotheses. We further investigate the case where the design matrix follows an elliptical distribution. We examined the finite sample performance of our proposed method through Monte Carlo simulations. The numerical comparisons exhibit that our proposed tests outperform some existing methods in terms of controlling Type I error rate and power, when the data deviate from the Gaussian assumptions or are heavy-tailed. We illustrate our proposed high-dimensional quantile testing in financial econometrics, through an empirical analysis of Chinese stock market data.