Research On Solving And Applying Of Quantile Regression For Panel Data

Panel data model is the crucial constituent portion of modern econometrics,as the fast development of econometrics,the theory-research and applying research based on panel data is increasing day by day,whenever in developing and developed countries.Traditional methods for panel data exists certain limitation.On the one hand,most traditional panel data models base on the assumption of mean regression,the results only can capture the structural relationship around the mean value,but not exactly capture the tail relationship of variables;on the other hand,traditional panel data suppose error term obeys normal distribution,sometimes the result from it is not efficient or robust when the leptokurtic data and the fat tail data exist the sample obtained can not satisfy the classical assumptions.Quantile regression can remedy the defect of traditional panel data model exactly.Koenker(2004)first proposed quantile regression for longitudinal data,it was the important complement of traditional panel data model.The new method not only makes use of the large sample of panel data,but also exactly describes the effects of explaining variables as the conditional distribution of covaribles changing.Meanwhile,the assumption of error term is broaden and the explaining ability of model is enhanced.The estimators is more stable and efficient.Recently,the research on quantile regression for panel data has developed gradually in and abroad.The research direction includes:model constructing,model solving,parametric tests and asymptotic properties of quantile regression;dynamic panel data model;nonlinear panel data model;nonparametric and semi-parametric estimation on quantile regression for panel data;censored quantile regression,lasso quantile regression and autoregression quantile for panel data.According to analyze the development and the present situation of quantile regression for panel data,we can find that:for one thing,the computational method of estimators is not unique for quantile regression model whenever fixed effects or random effects,improving of present methods or exploring new methods may simplify the process of computation and enhance the explaining ability of model;for another,the research about nonlinear quantile regression for panel data is lacking,compared with the nonlinear quantile regession for time series,the former is on the initial stage to further development.Firstly,this paper recommends the developing progress,current situation and application status of quantile regression for panel data,orderly explain the existing research and the one which is to be researched in and abroad,that is the foundation to light the research direction.Secondly,this paper describes the theories from model construction and parameter estimation to parameter tests and properties of quantile regression for panel data,and introduces three kinds of quantile regression models for panel data:penalized quantile regression,two-stage quantile regression and dynamic quantile regression with instrumental variables.Finally,we explore the problems of model construction and parameter estimation from three aspects.The main content of this paper includes:(1)Considering the computational method of estimating parameters for fixed effects panel data model using quantile regression,as the present methods are computationally complicated or can not obtain the individual effects estimation.According to the Hooke-Jeeves algorithm in solving multidimensional unconstrained extremum problems,we can obtain the numerical solution of the parameters to be estimated.Monte-Carlo simulation is carried out to show the performance of the estimators compared with the other quantile regression methods.The nonlinear effects of financial development on economic growth are explored using the quantile regression with fixed effects panel data.(2)In order to deal with the correlation within the cross sections in random effects panel data,we use Copula functions to explore the solving method for quantile regression model with random effects panel data.By means of the relationship between quantile regression and ALD distribution,we propose the maximum likelihood function with Copula structure to compute the quantile parameters in random effects panel data model.Monte-Carlo simulation is conducted to test the unbiasedness and efficiency of estimators.The new proposed method is applied to analyze the effects of inflation of economic growth in China.(3)To make up the limitation of linear quantile regression,Copula quantile regression curves are applied to panel data model.We research on modeling and solving of nonlinear quantile regression for panel data model.By generating random panel data with Clayton Copula correlation,we conduct Monte-Carlo simulation and confirm that when there are nonlinear correlation exists between variables,Copula quantile regression fit more precisely.Using the proposed model,we analyze the nonlinear correlation between house price index and consuming price index with panel data from 35 cities.The innovation of this paper contains:(1)Considering the existing problems in current quantile regression models for panel data with fixed effects,we propose a new solution method——Hooke-Jeeves algorithm.We write the programming code according to algorithm,and compute the numerical solution with Matlab.Compared with the existing method,the algorithm is comparatively simple to implement,we can also get the coefficients of independent variable and individual effects at the same time.(2)Based on the relation between quantile regression and ALD,we propose the maximum likelihood estimation method for quantile regression model with fixed effects panel data.Maximum likelihood function with Copula function is structured,and coordinate rotation method is used to carry out the iterative computation.The new method can not only solve the internal correlation within the cross sections,but also shrink the mean root square error.(3)By combining Copula quantile regression curves with panel data model,we propose the nonlinear quantile method for panel data model.We can solve the minimum problem by enabling the optimization tool box in Matlab and calling the function order ’fmincon’.It is showed from the Monte-Carlo simulation that when the variables have nonlinear correlations,Copula quantile regression fits more closely,and forecasts more accurately.In this paper,we study the quantile regression for panel data complementing the theory from model constructing and parameter estimating.However there still exits certain parts unfinished to be explored in the future.Considering the estimators computed by the new methods,parameter tests and asymptotic properties should be researched in detail as the complements to the theory system.