@article{Dimensional:1994,
      recid = {1994},
      author = {Han, Yuefeng},
      title = {Robust Estimation of High Dimensional Time Series},
      publisher = {The University of Chicago},
      school = {Ph.D.},
      address = {2019-08},
      pages = {126},
      abstract = {In recent years, extensive research has focused on the  $\ell_1$ penalized least squares (Lasso) estimators of  high-dimensional 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 and/or the covariates are  serially dependent and/or heavy tailed.

This thesis  concerns the theoretical properties of the Lasso estimators  for linear regression with random design and weak sparsity  under serially dependent and/or non-sub-Gaussian errors and  covariates. In contrast to the traditional case in which  the errors are independent and identically distributed  (i.i.d.) 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 due to the serial dependence 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 consistency of the  estimators depend on the dependence measures and moment  conditions of the errors and the covariates. We apply the  results obtained for the Lasso method to now-casting with  mixed-frequency data for which serially correlated errors  and a large number of covariates are common. The empirical  results show the superiority of Lasso procedure in both  forecasting and now-casting.

This thesis also proposes a  new robust $M$-estimator for generalized linear models. We  investigate properties of the proposed robust procedure and  the classical Lasso procedure both theoretically and  numerically. As an extension, we also introduce robust  estimator for linear regression. We show that the proposed  robust estimator for linear model will achieve the optimal  rate which is the same as the one for i.i.d sub-Gaussian  data. Simulation results show that the proposed method  performs well numerically in terms of heavy-tailed and  serially dependent covariates and/or errors, and it  significantly outperforms the classical Lasso method. For  applications, we demonstrate the regularized robust  procedure via analyzing high-frequency trading data in  finance. We also provide new Bousquet type inequalities for  high-dimensional time series, which could be quite useful  in empirical process of dependent data.},
      url = {http://knowledge.uchicago.edu/record/1994},
      doi = {https://doi.org/10.6082/uchicago.1994},
}