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.
翻译:最近的研究侧重于 $\ ell_ 1$ 受罚最低正方( Lassio) 的高维线性回归估计值, 共差数的美元数量大大大于抽样规模 $n 美元。 但是, 很少有研究在错误和/ 或共差连续依赖时, 检查了估算值的属性。 此外, 本次研究中, 我们调查了Lasso 估测器的理论属性, 以进行线性回归, 随机设计, 且在连续的基底和/ 或非基底的Gaussian错误和共差差下, 微缩缩缩缩缩缩缩缩。 相比之下, 在传统的案例中, 共变缩缩数是独立的, 而现在的变缩缩缩缩缩和变缩缩的变缩缩缩略图则具有连续的关联性, 或者两者都有固定的指数。 我们采用一个功能性变缩略图框架, 直缩缩图的缩缩略图的缩略图的缩图和缩图的缩略图的缩图 。