[Paper Review] Square-root nuclear norm penalized estimator for panel data models with approximately low-rank unobserved heterogeneity

Jad Beyhum, Éric Gautier|arXiv (Cornell University)|Apr 19, 2019

Spatial and Panel Data Analysis4 citations

TL;DR

This paper proposes a square-root nuclear norm penalized estimator for panel data models with approximately low-rank unobserved heterogeneity, enabling consistent estimation without prior knowledge of error variance. The method solves a convex optimization problem, achieves asymptotic normality via a two-stage procedure, and provides robust rank estimation and convergence rates under growing sample sizes.

ABSTRACT

This paper considers a nuclear norm penalized estimator for panel data models with interactive effects. The low-rank interactive effects can be an approximate model and the rank of the best approximation unknown and grow with sample size. The estimator is solution of a well-structured convex optimization problem and can be solved in polynomial-time. We derive rates of convergence, study the low-rank properties of the estimator, estimation of the rank and of annihilator matrices when the number of time periods grows with the sample size. Two-stage estimators can be asymptotically normal. None of the procedures require knowledge of the variance of the errors.

Motivation & Objective

• To develop a convex, variance-free estimator for panel data models with approximately low-rank unobserved heterogeneity.
• To establish convergence rates and low-rank properties of the estimator under unknown and growing rank.
• To propose a two-stage estimator that achieves asymptotic normality without requiring error variance knowledge.
• To enable the use of the baseline estimator as initialization for iterative algorithms.
• To provide consistent estimation of the rank and annihilator matrices in large-dimensional settings.

Proposed method

• The estimator minimizes a square-root nuclear norm penalty on the unobserved heterogeneity matrix, formulated as a convex optimization problem.
• The method uses a two-stage procedure: first estimate the low-rank component via nuclear norm penalization, then refine the regression coefficient estimator.
• The estimator is initialized using the least-squares estimator, which is robust even when inconsistent.
• The approach handles unknown error variance by using a square-root loss, ensuring scale-invariant estimation.
• The method incorporates within-transforms to remove individual and time effects, improving finite-sample performance.
• Hard-thresholding is applied to the singular values of the estimated matrix to improve rank estimation.

Experimental results

Research questions

• RQ1 Can a convex, variance-free estimator be developed for panel data models with approximately low-rank unobserved heterogeneity?
• RQ2 What are the convergence rates and low-rank properties of the square-root nuclear norm penalized estimator under growing sample size?
• RQ3 Does the two-stage estimator achieve asymptotic normality without knowledge of the error variance?
• RQ4 How well does the estimator recover the true rank of the unobserved heterogeneity matrix in finite samples?
• RQ5 Can the baseline estimator serve as a reliable initialization for iterative algorithms in this context?

Key findings

• For N=T=150, the MSE of the two-stage estimator $\widetilde{\beta}^{(2)}$ reaches 1×10⁻⁵, indicating high precision.
• The coverage probability of 95% confidence intervals for $\widetilde{\beta}^{(2)}$ reaches 95% in large samples (N=T=150), confirming asymptotic normality.
• In small samples (N=T=50), the MSE of $\widehat{\beta}_{pt}$ is 5×10⁻⁴, showing improved performance over the least-squares estimator (MSE=0.053).
• The within-transformed version improves rank recovery: in N=T=150, $\mathrm{rank}(\widehat{\Pi}_1^t)$ is correctly estimated as 2 in all cases.
• The estimator consistently estimates the true rank of the unobserved heterogeneity matrix, with 89% accuracy at N=T=50 and 100% at N=T=150 for the within-transformed case.
• The two-stage estimator $\widetilde{\beta}^{(2)}$ achieves bias below 10⁻⁴ in large samples, indicating high consistency.