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LASSO Inference for High Dimensional Predictive Regressions

Zhan Gao, Ji Hyung Lee, Ziwei Mei, Zhentao Shi

TL;DR

This paper tackles valid inference in high-dimensional predictive regressions where regressors can be mixed between local-to-unit-root and stationary processes. It introduces XDlasso, a two-stage debiasing approach that combines an IVX-based instrument with a desparsified LASSO to remove both shrinkage bias and Stambaugh bias, yielding asymptotically normal estimators and Wald tests without prior knowledge of which regressors are nonstationary. The authors establish RE/DB conditions, prove consistency of the Slasso workhorse, derive the auxiliary regression rates, and prove asymptotic normality of the XDlasso estimators as well as a Wald test for joint hypotheses. Monte Carlo simulations show XDlasso achieves accurate size and competitive power under mixed persistence and heteroskedasticity, while empirical applications to stock return predictability and inflation demonstrate robustness and the practical value of inference in data-rich macro-financial settings.

Abstract

LASSO inflicts shrinkage bias on estimated coefficients, which undermines asymptotic normality and invalidates standard inferential procedures based on the t-statistic. Given cross sectional data, the desparsified LASSO has emerged as a well-known remedy for correcting the shrinkage bias. In the context of high dimensional predictive regression, the desparsified LASSO faces an additional challenge: the Stambaugh bias arising from nonstationary regressors modeled as local unit roots. To restore standard inference, we propose a novel estimator called IVX-desparsified LASSO (XDlasso). XDlasso simultaneously eliminates both shrinkage bias and Stambaugh bias and does not require prior knowledge about the identities of nonstationary and stationary regressors. We establish the asymptotic properties of XDlasso for hypothesis testing, and our theoretical findings are supported by Monte Carlo simulations. Applying our method to real-world applications from the FRED-MD database, we investigate two important empirical questions: (i) the predictability of the U.S. stock returns based on the earnings-price ratio, and (ii) the predictability of the U.S. inflation using the unemployment.

LASSO Inference for High Dimensional Predictive Regressions

TL;DR

This paper tackles valid inference in high-dimensional predictive regressions where regressors can be mixed between local-to-unit-root and stationary processes. It introduces XDlasso, a two-stage debiasing approach that combines an IVX-based instrument with a desparsified LASSO to remove both shrinkage bias and Stambaugh bias, yielding asymptotically normal estimators and Wald tests without prior knowledge of which regressors are nonstationary. The authors establish RE/DB conditions, prove consistency of the Slasso workhorse, derive the auxiliary regression rates, and prove asymptotic normality of the XDlasso estimators as well as a Wald test for joint hypotheses. Monte Carlo simulations show XDlasso achieves accurate size and competitive power under mixed persistence and heteroskedasticity, while empirical applications to stock return predictability and inflation demonstrate robustness and the practical value of inference in data-rich macro-financial settings.

Abstract

LASSO inflicts shrinkage bias on estimated coefficients, which undermines asymptotic normality and invalidates standard inferential procedures based on the t-statistic. Given cross sectional data, the desparsified LASSO has emerged as a well-known remedy for correcting the shrinkage bias. In the context of high dimensional predictive regression, the desparsified LASSO faces an additional challenge: the Stambaugh bias arising from nonstationary regressors modeled as local unit roots. To restore standard inference, we propose a novel estimator called IVX-desparsified LASSO (XDlasso). XDlasso simultaneously eliminates both shrinkage bias and Stambaugh bias and does not require prior knowledge about the identities of nonstationary and stationary regressors. We establish the asymptotic properties of XDlasso for hypothesis testing, and our theoretical findings are supported by Monte Carlo simulations. Applying our method to real-world applications from the FRED-MD database, we investigate two important empirical questions: (i) the predictability of the U.S. stock returns based on the earnings-price ratio, and (ii) the predictability of the U.S. inflation using the unemployment.
Paper Structure (40 sections, 18 theorems, 293 equations, 6 figures, 24 tables)

This paper contains 40 sections, 18 theorems, 293 equations, 6 figures, 24 tables.

Table of Contents

  1. Introduction
  2. Model and Procedure
  3. High Dimensional Predictive Regression
  4. Two Types of Biases
  5. Necessity of IVX Transformation
  6. Joint Inference for Low-Dimensional Coefficients
  7. Asymptotic Theory
  8. Consistency of Slasso
  9. The Auxiliary Regression
  10. Asymptotic Distributions
  11. Monte Carlo Simulations
  12. Setup
  13. Results
  14. Empirical Applications
  15. Predictability of Stock Return Using Earnings-Price Ratio
  16. ...and 25 more sections

Key Result

Lemma 1

Under Assumptions assu:tail--assu:asym_n, there exists an absolute constant $c_{\kappa}$ such that w.p.a.1 as $n\to\infty.$ In addition, there exists some absolute constant $C_{{\rm DB}}$ sch that w.p.a.1 as $n\to\infty$, where $r$ is defined in Assumption assu:alpha.

Figures (6)

  • Figure 1: Histograms of $t$-statistics from XDlasso and Dlasso
  • Figure 2: Power curves of XDlasso inference
  • Figure 3: Power of XDlasso inference for nonstationary and stationary regressors
  • Figure 4: S&P 500 monthly Return and Log Earnings-price Ratio
  • Figure 5: Inflation and Unemployment Rate
  • ...and 1 more figures

Theorems & Definitions (50)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Lemma 1
  • ...and 40 more