Quadratic Inference Function Estimation And Application In Semi-parametric Instrumental Variable Models With Longitudinal Data

With the development of society and the application of statistics in various fields, the physical issue we analysis is more and more complex. When establishing statistical models, the linear regression model cannot satisfy the actual demand any more. The statistical models have developed from linear regression model to semi-parametric regression model. Semi-parametric regression models combine both parametric and non-parametric component, and can better find the inherent law of the data. Considering the situation when explanatory variables are the exogenous variables, many literatures have discussed the statistical methods and theory of semi-parametric regression models, and then extend this model into longitudinal data. However, when explanatory variables are the endogenous variables, the existing statistical methods and theory are no longer suitable. For semi-parametric models with longitudinal data, how to solve the problem of endogenous variables and within-subject correlation is the core of this paper.In this paper, we consider the interest’s parameter of estimation in semi-parametric instrumental variable models with longitudinal data, and propose the three step estimation procedure. First, we approximate the non-parametric part of the semi-parametric model based on the B-splines, so that we can converse this model to the parametric model. Second, in order to deal with the endogenous variables, we introduce instrumental variable to decompose the endogenous variables, and then estimate model by exogenous variables. Assuming that the parametric is given, we estimate the non-parametric part. Finally, in order to obtain the effective estimation of the parametric, we establish the goal function of the interest’s parameter by the quadratic inference function method. Under some regularity conditions, we show that the proposed estimator is consistent and asymptotically normal. In order to discuss the finite sample properties of proposed estimator, we made simulation studies for the proposed method. Simulation studies show that the proposed method eliminates the effect of endogenous variables, and improves the efficiency whether the work correlation matrix is correctly specified or not. Finally, the proposed method is applied to explore the relationship between trade openness and economic growth, using the foreign markets close degree as an instrumental variable. The results show that trade openness and economic growth has the significant positive correlation.