Inferences For Partially Functional Coefficients Models With Endogeneity

This paper proposes a semiparametric method to estimate the partially functional coefficient model when the endogeneity arises.The model has a partially linear component and allows the effect of endogenous variables vary over different levels of selected exogenous variables.Addressing the endogeneity by our method in such a model can be particularly useful in practice because it solves the problem when not enough exogenous variables are available to satisfy the order condition.Our proposed semiparametric profile estimation procedure is based on the control variables approach.We suggest a computationally straightforward profile least squares method to estimate functional and constant coefficients jointly.To satisfy the order condition when the number of exogenous variables is inadequate,we employ the series estimation approach which capture the nonlinearity between endogenous variables.The consistency and asymptotic normality property of the proposed estimator is established.We re-correct the asymptotic variance to account for the sampling error arising from the series estimation step.Our Monte Carlo study suggests that in the finite sample,the proposed estimator is less biased and more efficient compared to another existing estimator proposed by Cai et al.(2017).Furthermore,we apply our approach to Oreopoulos(2006)data to study the impact of the years of education on income.In this application,our model controls for the functional coefficients of the endogenous education variable and its squared form.By using only one instrument variable,we are still able to obtain the marginal effect of schooling,which is similar to the conventional IV results while it reveals different effects at different levels of schooling.We also find that the classic IV model might underestimate the effect due to the ignoring the interactive relation between the black male and the working experience.