Principal Stratification with Bayesian Additive Regression Trees for Count-Valued Intermediate Variables: Estimating the Effect of Fertility on Women's Employment

Abstract

Estimating the causal effect of fertility on women's employment is challenging because fertility and labour-market decisions are jointly determined. Instrumental-variable strategies are widely used, but credible instruments are rare and their validity often depends on covariates. Two-stage least squares, the dominant implementation, does not flexibly accommodate covariate-dependent instrument validity and is poorly suited to count-valued treatment and effect heterogeneity more broadly. We extend an existing framework that combines principal stratification with Bayesian Additive Regression Trees (BART) to settings with count-valued intermediate variables, such as number of children. The approach defines a causal estimand that respects the count structure of the intermediate variable, while BART enables flexible, covariate-dependent modelling of principal strata and outcomes, accommodating covariate-dependent instrument validity and producing estimates of effect heterogeneity. Simulations demonstrate that our approach outperforms conventional and flexible instrumental-variable estimators under nonlinear confounding and heterogeneous treatment effects. Applied to Demographic and Health Survey data from Nigeria, Senegal, and Kenya, the approach suggests a negative average effect in Nigeria but no clear average effect elsewhere, with employment penalties concentrated among younger, less-educated women, heterogeneity that standard approaches would obscure. The approach is available in the R package PrinceBART.