Continuous Endogenous Variables in Partially Linear Regression Models

This paper studies the estimation of Engel curves for rural China using the dataset of Gong, van Soest and Zhang (2005), extending their partially linear specification in two directions. First, we replace the parametric first stage — which imposes linearity of the relationship between log disposable income and log total expenditure — with a nonparametric kernel regression, motivated by the pronounced threshold nonlinearity documented in their own data. Second, we replace the linear parametric control function correction with a nonparametric counterpart, following the semiparametric triangular system framework of Newey, Powell and Vella (1999), Su and Ullah (2008), and Delgado and Parmeter (2014). The resulting three-stage estimator is √N-consistent and asymptotically normal for the parametric component β, and converges at the standard kernel rate for the nonparametric component. Monte Carlo simulations confirm the finite-sample reliability of the proposed estimator across a range of sample sizes and endogeneity levels. We assess whether the gender-of-children effects on expenditure shares documented by Gong et al. (2005) are robust to these two relaxations.