Nuclear Norm Regularized Estimation of Panel Regression Models

TL;DR

This paper addresses panel regressions with interactive fixed effects by introducing two convex estimators based on nuclear-norm penalties: a regularized estimator $\hat{\beta}_\psi$ and a nuclear-norm minimization estimator $\hat{\beta}_*$, both targeting consistent estimation of the common coefficients $\beta_0$ while avoiding non-convexity in the factor structure $\Gamma_0 = \lambda_0 f_0'$. It establishes consistency and convergence rates for these estimators across low-rank and general regressor regimes, and shows how to use them as warm starts for finite-step iterations that reproduce the usual Bai-Moon-Weidner LS estimator without solving a non-convex problem. A key contribution is identifying global conditions under which nuclear-norm regularization resolves the identification problem when the true number of factors $R_0$ is unknown, especially with low-rank regressors. The practical side includes a data-driven procedure to choose the penalty and to estimate $R_0$, supported by Monte Carlo simulations that demonstrate the bias behavior of the convex estimators and the bias-reducing power of the post-estimation iterations. Overall, the work provides scalable, consistent methods for panel data with interactive fixed effects and offers guidance for extending to nonlinear and high-dimensional settings.

Abstract

In this paper we investigate panel regression models with interactive fixed effects. We propose two new estimation methods that are based on minimizing convex objective functions. The first method minimizes the sum of squared residuals with a nuclear (trace) norm regularization. The second method minimizes the nuclear norm of the residuals. We establish the consistency of the two resulting estimators. Those estimators have a very important computational advantage compared to the existing least squares (LS) estimator, in that they are defined as minimizers of a convex objective function. In addition, the nuclear norm penalization helps to resolve a potential identification problem for interactive fixed effect models, in particular when the regressors are low-rank and the number of the factors is unknown. We also show how to construct estimators that are asymptotically equivalent to the least squares (LS) estimator in Bai (2009) and Moon and Weidner (2017) by using our nuclear norm regularized or minimized estimators as initial values for a finite number of LS minimizing iteration steps. This iteration avoids any non-convex minimization, while the original LS estimation problem is generally non-convex, and can have multiple local minima.

Figures (2)

• Figure 1: Plot of $L_R(\beta)$ and $Q_{\psi}(\beta)$ for the example detailed in Appendix \ref{['subsec:ap.ex.nonconvex']}. The true parameter is $\beta_0=2$.
• Figure 2: Plot of the functions $q_\psi(s)$ and $\ell_\psi(s)$ for $\psi=1$.

Theorems & Definitions (40)

• Lemma 1
• Example 1
• Proposition 1
• Theorem 1
• Lemma 2: Convergence Rate of $\widehat{\Gamma}_{\psi}$
• Theorem 2
• Theorem 3
• Remark 1
• Remark 2
• Theorem 4