On Lasso with many time series predictors and moderately dense coefficients

Abstract It has been shown that the error bound of the Lasso estimator attains the minimax optimal rate when the corresponding regression coefficients are sparse. The existing theory, however, is developed under fixed-design predictors with i.i.d. errors. Subsequent extensions to time-series predictors primarily focus on hard sparsity. This paper provides a unified treatment of the Lasso with time-series predictors under both hard and soft sparsity, with the latter allowing all regression coefficients to be nonzero. We first show that the key results continue to hold in the time-series setting. We then derive the convergence rate of the mean squared post-sample prediction error (MSPPE) for the Lasso predictor. Finally, for moderately dense coefficients, we show that the Lasso may require substantially fewer candidate variables to achieve the desired convergence rate. Monte Carlo simulations are conducted to evaluate the performance of the Lasso with time-series predictors.