Panel Probit Models with Time-Varying Individual Effects: Reestimating the Effects of Fertility on Female Labour Participation*

This paper considers a probit model for panel data in which the individual effects vary over time by interacting with unobserved factors. In estimation we adopt a correlated random effects approach for individual effects to get around the incidental parameter problem. This allows us to construct (asymptotically) unbiased estimators for average marginal effects (AMEs), which are often the ultimate quantities of interest in many empirical studies. We derive the asymptotic distributions for the AME estimators as well as provide the consistent estimators for their asymptotic variances. Next, we design a specification test for detecting whether individual effects are time-varying or not, and establish the asymptotic distribution for the proposed test statistic under the null hypothesis of no time variation of individual effects. Monte Carlo simulations demonstrate satisfactory finite sample performance of our proposed method. An empirical application to study the effect of fertility on labour force participation (LFP) is provided. We find that fertility has a larger impact on female LFP in Germany than in the US during the 1980s. We also provide some new empirical evidence of a even stronger effect of fertility on LFP during the 2010s in Germany, which might call for a reconsideration of relevant policies recently enacted such as the subsidized child care programme.

Automated summary

This paper introduces a probit model for panel data with individual effects that vary over time and addresses the incidental parameter problem using a correlated random effects approach. It aims to construct unbiased estimators for average marginal effects (AMEs) and provides empirical evidence on the impact of fertility on female labor force participation, showing that fertility has a larger effect in Germany compared to the US and even stronger in the 2010s, potentially impacting relevant policies.