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Unlocking the Regression Space

Liudas Giraitis, George Kapetanios, Yufei Li, Alexia Ventouri

TL;DR

This work develops a general regression framework that accommodates broad heterogeneity in regressors and disturbances, including nonstationarity and structural change, by embedding them in an extended regression space with $z_t=\mu_t+I_{gt}\eta_t$ and $u_t=h_t\varepsilon_t$. It proves consistency and asymptotic normality of both fixed- and time-varying-parameter OLS estimators under minimal conditions, and shows that White-type heteroskedasticity-robust standard errors remain valid, even with missing data. The authors provide extensive Monte Carlo evidence and an empirical application to asset returns, demonstrating reliable inference under complex forms of heterogeneity and intermittent data. The framework unifies robust inference in regression with time-varying coefficients, missingness, and MD-noise, broadening applicability in economics and finance. Overall, the paper delivers a flexible, computationally simple approach with strong finite-sample performance and broad empirical relevance.

Abstract

This paper introduces and analyzes a framework that accommodates general heterogeneity in regression modeling. It demonstrates that regression models with fixed or time-varying parameters can be estimated using the OLS and time-varying OLS methods, respectively, across a broad class of regressors and noise processes not covered by existing theory. The proposed setting facilitates the development of asymptotic theory and the estimation of robust standard errors. The robust confidence interval estimators accommodate substantial heterogeneity in both regressors and noise. The resulting robust standard error estimates coincide with White's (1980) heteroskedasticity-consistent estimator but are applicable to a broader range of conditions, including models with missing data. They are computationally simple and perform well in Monte Carlo simulations. Their robustness, generality, and ease of implementation make them highly suitable for empirical applications. Finally, the paper provides a brief empirical illustration.

Unlocking the Regression Space

TL;DR

This work develops a general regression framework that accommodates broad heterogeneity in regressors and disturbances, including nonstationarity and structural change, by embedding them in an extended regression space with and . It proves consistency and asymptotic normality of both fixed- and time-varying-parameter OLS estimators under minimal conditions, and shows that White-type heteroskedasticity-robust standard errors remain valid, even with missing data. The authors provide extensive Monte Carlo evidence and an empirical application to asset returns, demonstrating reliable inference under complex forms of heterogeneity and intermittent data. The framework unifies robust inference in regression with time-varying coefficients, missingness, and MD-noise, broadening applicability in economics and finance. Overall, the paper delivers a flexible, computationally simple approach with strong finite-sample performance and broad empirical relevance.

Abstract

This paper introduces and analyzes a framework that accommodates general heterogeneity in regression modeling. It demonstrates that regression models with fixed or time-varying parameters can be estimated using the OLS and time-varying OLS methods, respectively, across a broad class of regressors and noise processes not covered by existing theory. The proposed setting facilitates the development of asymptotic theory and the estimation of robust standard errors. The robust confidence interval estimators accommodate substantial heterogeneity in both regressors and noise. The resulting robust standard error estimates coincide with White's (1980) heteroskedasticity-consistent estimator but are applicable to a broader range of conditions, including models with missing data. They are computationally simple and perform well in Monte Carlo simulations. Their robustness, generality, and ease of implementation make them highly suitable for empirical applications. Finally, the paper provides a brief empirical illustration.

Paper Structure

This paper contains 17 sections, 16 theorems, 290 equations, 49 figures, 8 tables.

Table of Contents

  1. Introduction
  2. OLS estimation in general regression space
  3. Time-varying OLS estimation in extended regression space
  4. Regression with missing data
  5. Estimation of a stationary AR($p$) model with an m.d. noise
  6. Monte Carlo Simulations
  7. Estimation of a fixed parameter
  8. Estimation of a time-varying parameter
  9. Estimation of regression parameter with missing data
  10. Estimation of a stationary AR($p$) model
  11. Empirical experiment
  12. Conclusion
  13. Proofs of Theorems \ref{['t:r1']} and \ref{['t:r1-R']}, Corollaries \ref{['c:co1']} and \ref{['c:co1Power']}, and Lemma \ref{['l:le']}
  14. Proofs of Theorem \ref{['t:r1FT']} and Corollaries \ref{['c:co1TV']} and \ref{['c:co1TVPower']}
  15. Proofs of Theorems \ref{['t:prop1']}, \ref{['t:prop2']} and Theorem \ref{['t:AR']}
  16. ...and 2 more sections

Key Result

Theorem 2.1

Suppose that $(y_1, ..., y_n)$ is a sample from the regression model (e:r1) and Assumptions a:r0, a:ETA and a:r3 are satisfied. Then, the OLS estimator $\widehat{\beta}$ is consistent, i.e.

Figures (49)

  • Figure 1: $y_t$
  • Figure 2: $z_{2t}$
  • Figure 3: $u_t$
  • Figure 5: $y_t$
  • Figure 6: $z_{2t}$
  • ...and 44 more figures

Theorems & Definitions (21)

  • Definition 2.1
  • Theorem 2.1
  • Example 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Remark 2.1
  • Corollary 2.2
  • Lemma 2.1
  • Theorem 3.1
  • Corollary 3.1
  • ...and 11 more