Inference in Approximately Sparse Correlated Random Effects Probit Models With Panel Data

Abstract

We propose a simple procedure based on an existing “debiased” l₁-regularized method for inference of the average partial effects (APEs) in approximately sparse probit and fractional probit models with panel data, where the number of time periods is fixed and small relative to the number of cross-sectional observations. Our method is computationally simple and does not suffer from the incidental parameters problems that come from attempting to estimate as a parameter the unobserved heterogeneity for each cross-sectional unit. Furthermore, it is robust to arbitrary serial dependence in underlying idiosyncratic errors. Our theoretical results illustrate that inference concerning APEs is more challenging than inference about fixed and low-dimensional parameters, as the former concerns deriving the asymptotic normality for sample averages of linear functions of a potentially large set of components in our estimator when a series approximation for the conditional mean of the unobserved heterogeneity is considered. Insights on the applicability and implications of other existing Lasso-based inference procedures for our problem are provided. We apply the debiasing method to estimate the effects of spending on test pass rates. Our results show that spending has a positive and statistically significant average partial effect; moreover, the effect is comparable to found using standard parametric methods.

Key Words:

• Correlated random effects probit
• High-dimensional statistics and inference
• Nonlinear panel data models
• Partial effects
• l1-Regularized quasi-maximum likelihood estimation

Notes

1 The “like” here means we are taking a Newton–Raphson iteration based on ${\widehat{H}}^A=\frac{1}{n}{\sum }_{i=1}^n{\sum }_{t=1}^{\mathrm{T}}f(X_{it}\widehat{\theta })X_{it}^{TA}{X}_{it}^A$ instead of the $\left|A\right|\times \left|A\right|$ submatrix of $\frac{1}{n}{\sum }_{i=1}^n{\sum }_{t=1}^{\mathrm{T}}f(X_{it}\widehat{\theta })X_{it}^T{X}_{it}$ associated with the set A.

2 Indeed, $\left|J(\widehat{\theta })\right|=O_p(k\vee 1)$ can be ensured under a “sparse eigenvalue condition” (Bickel, et al. 2009; Belloni and Chernozhukov 2013).