Abstract: Asymmetric least squares (ALS) regression is a convenient and effective method of summarizing the conditional distribution of a response variable given the covariates. Recent years have seen a growing interest in ALS amongst statisticians, econometricians and financial analysts. However, existing work on ALS only considers the traditional low-dimension-and-large-sample setting. In this talk, we systematically explore the Sparse Asymmetric LEast Squares (SALES) regression under high dimensionality. We show the complete theory using penalties such as lasso, MCP and SCAD. A unified efficient algorithm for fitting SALES is proposed and is shown to have a guaranteed linear convergence.

An important application of SALES is to detect heteroscedasticity in high-dimensional data and from that perspective it provides a computationally friendlier alternative to sparse quantile regression (SQR). However, when the goal is to separate the set of significant variables for the mean and that for the standard deviation of the conditional distribution, SALES and SQR can fail when overlapping variables exist. To that end, we further propose a Coupled Sparse Asymmetric LEast Squares (COSALES) regression. We show that COSALES can consistently identify the two important sets of significant variables for the mean and standard deviation simultaneously, even when the two sets have overlaps.