Nonparametric Control Function Estimation Of Nonlinear Models With Endogenuity And Its Application

In econometric model,people control a series of variables to explain the real data generation process behind the model.However,there are always some unobservable variables or variables that researchers did not expect to be missed in the error term.If these variables are related to the control variables in the model,the model will not meet the Gauss Markov theoem,so that the estimation of the model will be biased.In econometrics,this problem is called endogenous problem(Wooldridge,2002).In empirical research,most of the data used by econometric economists are non experimental data,so they are more or less affected by endogenous.Solving the endogenuity problem is the same as solving the problem of model identification with endogenuity.Common methods to solve endogenous problems include instrumental variables(IV),fixed effect model(Fe),propensity matching(PSM),experiments and quasi experiments.In comparison,for the quantitative analysis based on survey data,instrumental variable method has unique advantages.In recent years,the attitude of economists towards instrumental variables is changing from unfamiliar and hesitant to gradually accepted.More and more papers using instrumental variable method have been published by the world's top journals.There is no doubt that the process that instrumental variable method is gradually paid attention to and accepted by the academic community fully demonstrates the evolution of quantitative analysis method of international econometrics and the further integration with other disciplines in methodology.In the linear model,the two most commonly used methods are two stage least square(2SLS)and control function(CF).The two methods have the same starting point.Firstly,the endogenous variables are decomposed into two parts by using the instrumental variables: one is the linear projection of the endogenous variables on the instrumental variables,which represents the components unrelated to the regression error term;the other is the residual of the endogenous variables after the projection on the instrumental variables,which represents the parts related to the regression error term.The two-stage least square method uses the part which is not related to the regression error term instead of the endogenous variable to estimate.The control function method uses the residual as the proxy variable of the missing variable to add to the model for estimation.Both of these methods can effectively solve the endogenous problem in linear model,and are widely used by empirical researchers.But it is much more difficult to solve the endogeneity problem in the non parametric or semi parametric model with endogeneity.If the two-stage least square method is adopted,it will encounter the problem of ill conditioned regression(hall and Horowitz,2005),which makes the estimators inconsistent.Therefore,many econometric economists have to choose more control function method to solve the endogenous problem in nonparametric models.However,the control function method has its disadvantages,that is,the proxy variable must be consistent with the missing variable as much as possible.The traditional control function method takes the residual as the proxy variable directly,which is applicable in the linear model,but in the nonparametric model,the elimination of endogeneity is not thorough because the proxy variable can not fully replace the missing variable.In this paper,the linear form of traditional control function is extended to nonparametric form,which is recorded as nonparametric control function(NCF)and used to deal with the problem of variable endogeneity in nonlinear model.As the missing variables in the model are likely to be unobservable,the specific correlation form is also unknown.Therefore,in order to be more general,it is advisable to assume that the relationship between the error term and the omitted variable is an unknown functional relationship,and then the residual function form is used as the proxy variable to solve the endogeneity in the model.This setting has the following points: 1.The non parameter setting increases the ability of the control function method to solve the endogeneity.Because of the more flexible assumption of nonparametric function,nonparametric control function method is more flexible and stable in solving endogeneity.2.The non parameter setting satisfies the diversity of the actual data.The data-driven nonparametric estimation method ensures that the control function method can deal with different types of data.3.The estimation method is simple and easy to operate.Because the nonparametric part only appears as the proxy variable,we pay more attention to the estimation of the former part,so we can choose the spline function method to approach the nonparametric part effectively.The main work of this paper is to study the application of nonparametric control function method in solving endogenous model in longitudinal data and panel data.Firstly,the non parametric control function method is used to solve the variable endogeneity problem of the nonlinear model in the cross section data,and the effect of the non parametric control function method and the traditional control function method in the model estimation is compared.The results show that the nonparametric control function method is more stable than the traditional control function method in limited samples.Then this method is applied to study the relationship among housing debt,education and intergenerational liquidity.Based on the survey data of China Household Income Survey(chip)project in 2013,the non parametric control function method is used to solve the endogenous variables in the model.The regression results show that the housing debt will weaken the intergenerational liquidity of low-income families and have no significant impact on high-income families.Secondly,the nonparametric control function method is extended to the fixed effect partial linear panel data model and the interactive effect partial linear panel data model.Through strict theoretical derivation,it is found that the estimators processed by the nonparametric control function method are consistent.In the panel data,the fixed effect partial linear model is constructed to study the influence of family background on children's achievement,and the non parametric control function method is used to solve the endogeneity of income variables in the model.The results show that children of low-income families are more easily restricted by family background.In addition,a partial linear model of interaction effect is built in panel data to study the effect of infrastructure investment on China's economic development,and the non parametric control function method is used to solve the endogeneity of investment variables in the model.The results show that the impact of infrastructure investment on China's economy does not appear inflection point,so infrastructure construction is still an effective means to promote economic development.