High-Dimensional Time Series Regressions With HAC and HAR Penalty Loadings

This paper deals with LASSO regression in high-dimensional sparse linear models with time series data. We propose heteroskedasticity and autocorrelation consistent (HAC) and heteroskedasticity and autocorrelation robust (HAR) estimates for the penalty loadings and evaluate the in-sample fitting performance. Based on our results, we make precise recommendations for the implementation of LASSO estimation regarding tuning parameters, bandwidth selection procedures and kernels. We employ the algorithm for HAC/HAR estimation in Heberle and Sattarhoff (2017) using the fast Fourier transform, which is particularly fast. Moreover, the bandwidth parameter has no impact on its computational performance. This algorithm plays a key role in high-dimensional applications making possible iterative estimation procedures and the use of very large bandwidths to ensure robustness with reasonable computing times. Numerical simulations show gains in estimation accuracy over the standard LASSO estimator both in terms of prediction norm and Euclidean norm. We observe a trade-off between sample size and the no. of covariates, results improving the larger the no. of covariates and the smaller the no. of observations which is natural, because the relevance of regularization procedures increases with the number of dimensions