Title: High-dimensional Linear Regression for Dependent Observations with Application to Nowcasting,
Abstract: In the last few years, an extensive literature has been focused on the
$\ell_1$ 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 in
which the errors are i.i.d and have finite exponential moments, we show that
$p$ can at most be 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 via Lasso when there are serial correlations in the errors
or the covariates or both. Adopting the framework of functional dependence
measure, we provide a detailed description on how the rates of convergence and
the selection consistencies of the estimators depend on the dependence measures
and moment conditions of the errors and the covariates. Simulation results show
that Lasso regression can be substantially more powerful than the mixed
frequency data sampling regression (MIDAS) in the presence of irrelevant
variables. We apply the results obtained for the Lasso method to nowcasting
mixing frequency data in which serially correlated errors and a large number of
covariates are common. In real examples, the Lasso procedure outperforms the
MIDAS in both forecasting and nowcasting.