[Paper Review] Varying Coefficient Panel Data Model with Interactive Fixed Effects

Sanying Feng, Gaorong Li+2 more|arXiv (Cornell University)|Mar 7, 2018

Spatial and Panel Data Analysis22 references3 citations

TL;DR

This paper proposes a varying coefficient panel data model with interactive fixed effects, where coefficient functions vary with a covariate and are estimated via B-spline approximation and a robust nonlinear iteration scheme. The authors establish asymptotic theory for consistency, convergence rates, and distributional properties, and develop a residual-based block bootstrap for confidence intervals, offering a flexible, nonparametric approach to modeling heterogeneous effects in panel data with endogenous unobserved factors.

ABSTRACT

In this paper, we propose a varying coefficient panel data model with unobservable multiple interactive fixed effects that are correlated with the regressors. We approximate each coefficient function by B-spline, and propose a robust nonlinear iteration scheme based on the least squares method to estimate the coefficient functions of interest. We also establish the asymptotic theory of the resulting estimators under certain regularity assumptions, including the consistency, the convergence rate and the asymptotic distribution. Furthermore, we develop a least squares dummy variable method to study an important special case of the proposed model: the varying coefficient panel data model with additive fixed effects. To construct the pointwise confidence intervals for the coefficient functions, a residual-based block bootstrap method is proposed to reduce the computational burden as well as to avoid the accumulative errors. Simulation studies and a real data analysis are also carried out to assess the performance of our proposed methods.

Motivation & Objective

• To develop a flexible semiparametric panel data model that allows coefficient functions to vary with observed covariates while accounting for unobserved interactive fixed effects.
• To address the challenge of endogeneity between regressors and unobserved interactive fixed effects in panel data models.
• To provide a consistent, asymptotically normal estimation framework for coefficient functions using B-spline approximation and nonlinear least squares.
• To construct valid pointwise confidence intervals for coefficient functions using a residual-based block bootstrap method to reduce computational burden and avoid error accumulation.

Proposed method

• Approximates unknown coefficient functions using B-spline basis expansions to enable nonparametric estimation within a linear model framework.
• Proposes a robust nonlinear iteration scheme based on least squares to estimate the spline coefficients, handling the dependence between regressors and interactive fixed effects.
• Derives asymptotic properties—consistency, convergence rate, and asymptotic normality—under regularity conditions on the error structure and design.
• Applies a least squares dummy variable (LSDV) method to the special case of additive fixed effects, simplifying estimation in that setting.
• Develops a residual-based block bootstrap procedure to construct pointwise confidence intervals for coefficient functions, improving computational efficiency and reducing cumulative estimation errors.
• Uses principal component analysis to estimate the unobserved common factors and factor loadings, with theoretical justification for consistency and convergence rates.

Experimental results

Research questions

• RQ1 How can a varying coefficient panel data model be extended to include interactive fixed effects that are correlated with regressors?
• RQ2 What estimation method ensures consistency and asymptotic normality when both coefficient functions and interactive fixed effects are unknown and endogenous?
• RQ3 Can a residual-based block bootstrap method effectively construct confidence intervals for nonparametric coefficient functions without incurring high computational costs or error accumulation?
• RQ4 How does the proposed B-spline-based nonlinear iteration scheme compare to existing methods in terms of convergence and robustness under endogeneity?
• RQ5 What are the theoretical properties—consistency, convergence rate, and asymptotic distribution—of the proposed estimators under general regularity conditions?

Key findings

• The proposed estimator for the coefficient functions achieves consistency and attains a convergence rate of Op((NT)^(-1/2)) under regularity conditions.
• The asymptotic distribution of the coefficient function estimator is normal, enabling valid inference via confidence intervals.
• The residual-based block bootstrap method effectively reduces computational burden and avoids error accumulation in confidence interval construction.
• The LSDV method provides a consistent and efficient estimation approach for the special case of additive fixed effects.
• The convergence rate of the estimated factor loadings and common factors is Op(∥γ̂ − γ̃∥ + δ⁻¹_NT + ζ^(1/2)_Ld), with δ_NT and ζ_Ld related to estimation error in the factor structure.
• The theoretical results are validated through simulation studies and a real data analysis, demonstrating the method’s robustness and practical utility.