Abstract

In recent years, an extensive literature has focused on the L1 penalized least squares (Lasso) estimators of high dimensional linear regression when the number of covariates p is considerably larger than the sample size n. However, there is limited attention paid to the properties of the estimators when the errors or/and the covariates are serially dependent.

In this study, we investigate the theoretical properties of the Lasso estimators for linear regression with random design under serially dependent and/or non-sub-Gaussian errors and covariates. In contrast to the traditional case where the errors are i.i.d. and have finite exponential moments, we allow for p to be at most a power of n if the errors have only polynomial moments. In addition, the rate of convergence becomes slower due to the serial dependencies in errors and the covariates. We also consider sign consistency for model selection based on Lasso. Adopting the framework of functional dependence measure, we provide a detailed description on how the rates of convergence and the selection consistency of the estimators depend on the dependence measures and moment conditions of the errors and the covariates.

In applications, many important macroeconomic variables are not sampled at the same frequency. For example, GDP data are sampled quarterly, industrial production data are sampled monthly, and most interest rate data are available daily. We apply the results obtained to nowcasting in which serially correlated errors and a large number of covariates are common.

This talk is based on joint works with Yuefeng Han.