Empirical Likelihood-Based Unified Confidence Region For A Predictive Regression Model And The Related Empirical Tests

With the continuous advancement of economic globalization,financial markets have been paid pay more and more attention.The research on the predictability of economic variables is of great interest to the investors' investment decisions and the maintenance of financial market stability.Predictive regression model is one of the most widely used models in economic and financial fields.In hypothesis testing problem,the predictability refers to the examination about whether the regression parameter of regression model equals to zero.However,if we consider the point estimation of the parameters by using least squares method directly,the credibility will be not high.Therefore,one can consider to construct the confidence interval first,then to point estimation,to some extent,it will help to improve its reliability.Bearing this in mind,we first show the asymptotic distribution of the least squares estimator of joint confidence region for model parameters in the case where the predictor sequence under the case of moderate deviations from a unit root.According to the results,we know that the least squares estimator of joint confidence domain for model parameters is not robust.and its asymptotic distribution will vary with the different nature of the predictor sequence,which will bring great challenges to practical applications.This motivates us to consider a more robust method,which can be used to determine whether the benefit variable is predictable without distinguishing the nature of the predictor sequence.Therefore,this paper chooses to use the empirical likelihood method.The empirical likelihood method is a nonparametric statistical method proposed by Owen in 1988.Unlike the parameter estimation method,the shape of the confidence region is determined completely by the data,and the confidence region has the advantage of likelihood invariance.According to the non-parametric version of Wilks' theorem as proved by Owen(1988),under mild regular conditions,has an asymptotic limit as the sample size tends to infinity.Qin & Lawless(1994)combined the empirical likelihood method with the generalized estimating equation,which greatly extended the scope of the empirical likelihood method.On the basis of this,the empirical likelihood method is applied to the problem of constructing the confidence region of the parameters of the predicted regression model.It can be proved that the statistics that constructed in this paper converge to the chi-square distribution asymptotically,regardless of whether the sequence of the predictor variables is stable or not.In order to test the validity of the proposed empirical likelihood method,the Monte Carlo method is used in this paper.To cover the different settings of the parameters,the simulation part of this paper includes the following cases: whether the AR(1)model has an intercept term,whether the predictor is stable,and whether the null hypothesis ? equals to 0.According to the simulation results,the empirical likelihood method proposed in this paper performs well regardless of the sample size of 400 or 2000.Furthermore,in order to illustrate the practical value of the method,this paper applies the method to CRSP data and CSI 300 Index,the result of empirical analysis shows that the economic variables that can predict the return rate of American stocks,such as the earnings-price ratio and the dividend-price ratio,but do not have the ability to predict the return rate of Chinese stocks..There are three innovations in this paper,the first one is perfecting the asymptotic distribution of the least squares estimator of joint confidence region for model parameters in the case where the predictor under the case of moderate deviations from a unit root;the second one is constructing the joint confidence region of joint confidence region for model parameters,by doing this,we can not only know whether the predictor variable is predictable,but also obtains the correlation between the regression parameter the intercept;the last one is using Monte Carlo method to verify the effectiveness of the proposed method,and applying to the data of the stock market index of the United States and the stock market index of China,respectively,to analyze the difference in predictability between the stock markets of China and the United States.