[Paper Review] LASSO-Driven Inference in Time and Space

Victor Chernozhukov, Wolfgang Karl Härdle+2 more|arXiv (Cornell University)|Jun 13, 2018

Financial Risk and Volatility Modeling47 references3 citations

TL;DR

This paper proposes a LASSO-driven inference framework for high-dimensional systems of time series equations with cross-sectional and temporal dependence. By using a block multiplier bootstrap to select a unified penalty level and deriving a uniform Bahadur representation, it enables valid simultaneous inference on multiple parameters, achieving oracle-like efficiency and bootstrap consistency under weak dependence assumptions.

ABSTRACT

We consider the estimation and inference in a system of high-dimensional regression equations allowing for temporal and cross-sectional dependency in covariates and error processes, covering rather general forms of weak temporal dependence. A sequence of regressions with many regressors using LASSO (Least Absolute Shrinkage and Selection Operator) is applied for variable selection purpose, and an overall penalty level is carefully chosen by a block multiplier bootstrap procedure to account for multiplicity of the equations and dependencies in the data. Correspondingly, oracle properties with a jointly selected tuning parameter are derived. We further provide high-quality de-biased simultaneous inference on the many target parameters of the system. We provide bootstrap consistency results of the test procedure, which are based on a general Bahadur representation for the $Z$-estimators with dependent data. Simulations demonstrate good performance of the proposed inference procedure. Finally, we apply the method to quantify spillover effects of textual sentiment indices in a financial market and to test the connectedness among sectors.

Motivation & Objective

• To develop a unified penalty selection method for high-dimensional systems of regression equations with temporal and cross-sectional dependence.
• To enable simultaneous inference on multiple parameters in high-dimensional time series models with dependent data.
• To establish theoretical consistency of the inference procedure under weak dependence and high-dimensional settings.
• To provide a robust, post-regularization inference method that maintains validity despite model selection uncertainty.
• To apply the method to real-world financial data, such as sentiment spillover and sector connectedness.

Proposed method

• Applies LASSO to a system of high-dimensional regression equations for variable selection across multiple equations.
• Uses a block multiplier bootstrap procedure to select a global tuning parameter that controls for multiplicity and data dependence.
• Derives a uniform Bahadur representation for de-biased Z-estimators under dependent data, enabling valid inference.
• Establishes maximal inequalities for empirical processes in the context of Z-estimation with weakly dependent time series.
• Implements a uniform near-oracle bound for joint estimators, ensuring consistency under restricted eigenvalue conditions.
• Proposes an algorithm for local and global inference on parameters of interest after regularization.

Experimental results

Research questions

• RQ1 How can a single, globally optimal tuning parameter be selected for a system of high-dimensional regression equations with dependent errors and covariates?
• RQ2 What is the theoretical justification for simultaneous inference on multiple parameters after LASSO selection in weakly dependent time series?
• RQ3 Can a block multiplier bootstrap consistently estimate the sampling distribution of de-biased estimators in high-dimensional time series systems?
• RQ4 How do the proposed methods perform in finite samples compared to single-equation LASSO approaches?
• RQ5 To what extent can the method detect spillover effects and connectedness in financial sentiment networks?

Key findings

• The block multiplier bootstrap provides consistent estimation of the sampling distribution of de-biased estimators, ensuring valid simultaneous inference under weak dependence.
• The method achieves uniform near-oracle bounds for joint estimators, implying that estimation error is close to the ideal oracle case.
• The restricted eigenvalue condition holds at the population level, supporting consistent variable selection and estimation.
• The proposed inference procedure exhibits strong finite-sample performance, outperforming single-equation LASSO in terms of coverage and power.
• Empirical application to financial sentiment indices reveals significant spillover effects and connectedness across sectors, validating the method's practical utility.