  Recent research has focused on $\ell_1$ penalized least squares (Lasso)
estimators for high-dimensional linear regressions in which the number of
covariates $p$ is considerably larger than the sample size $n$. However, few
studies have examined the properties of the estimators when the errors and/or
the covariates are serially dependent. In this study, we investigate the
theoretical properties of the Lasso estimator for a linear regression with a
random design and weak sparsity under serially dependent and/or nonsubGaussian
errors and covariates. In contrast to the traditional case, in which the errors
are independent and identically distributed and have finite exponential
moments, we show that $p$ can be at most a power of $n$ if the errors have only
finite polynomial moments. In addition, the rate of convergence becomes slower
owing to the serial dependence in the errors and the covariates. We also
consider the sign consistency of the model selection using the Lasso estimator
when there are serial correlations in the errors or the covariates, or both.
Adopting the framework of a functional dependence measure, we describe 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.
Simulation results show that a Lasso regression can be significantly more
powerful than a mixed-frequency data sampling regression (MIDAS) and a Dantzig
selector in the presence of irrelevant variables. We apply the results obtained
for the Lasso method to nowcasting with mixed-frequency data, in which serially
correlated errors and a large number of covariates are common. The empirical
results show that the Lasso procedure outperforms the MIDAS regression and the
autoregressive model with exogenous variables in terms of both forecasting and
nowcasting.
