Lasso Inference for High-Dimensional Time Series

Abstract

The desparsified lasso is a high-dimensional estimation method which provides uniformly valid inference. We extend this method to a time series setting under Near-Epoch Dependence (NED) assumptions allowing for non-Gaussian, serially correlated and heteroskedastic processes, where the number of regressors can possibly grow faster than the time dimension. We first derive an oracle inequality for the (regular) lasso, relaxing the commonly made exact sparsity assumption to a weaker alternative, which permits many small but non-zero parameters. The weak sparsity coupled with the NED assumption means this inequality can also be applied to the (inherently misspecified) nodewise regressions performed in the desparsified lasso. This allows us to establish the uniform asymptotic normality of the desparsified lasso under general conditions. Additionally, we show consistency of a long-run variance estimator, thus providing a complete set of tools for performing inference in high-dimensional linear time series models. Finally, we perform a simulation exercise to demonstrate the small sample properties of the desparsified lasso in common time series settings.

Regularized estimation of high‐dimensional vector autoregressions with weakly dependent innovations

“…Effectively, one could use our results to provide theoretical justification for the concentration bounds under particular model specifications. On the other hand, Adamek et al (2020) provides theoretical justification for desparsified inference in some large dimensional models discussed in our article.…”

“…In Adamek et al (2020), authors consider a near epoch dependent time series. This condition covers misspecification and non-Gaussian, conditionally heteroskedastic models, such as GARCH innovations.…”

“…The importance of our results relies on the fact that our non-asymptotic guarantees serve as a fundamental ingredient for the derivation of asymptotic properties of penalized estimators in high-dimensional VAR models, as in Adamek et al (2020). In particular, our results apply with minimal restrictions on the conditional variance model, allowing, for instance, large-dimensional multivariate linear processes in the variance.…”

“…Authors assume the series is either 𝛽-mixing process with sub-Weibull marginal distributions or 𝛼-mixing Gaussian processes. Finally, Adamek et al (2020) develop theoretical results for point estimation and inference in near epoch dependent time series using desparsified lasso under high level conditions on the generating process.…”