Bayesian Estimation For First-order Autoregressive Model With Explanatory Variables

Many economic relationships in nature are dynamic, some dynamic relationships arenot only related with certain variables, but also have a delayed explanatory variable. Al-though the ordinary regression model and the autoregression model are widely used in theeconomy, neither an ordinary regression model nor an autoregressive model can characterizethese relationships very well. The time dependence is not taken into account in the ordinaryregression model, the influence of the outer factors is not taken into account in the autore-gression model, this paper considers a generalization of both the ordinary regression modeland the autoregressive model, the autoregression model with explanatory variables.Many statistical scholars have already studied the autoregression model with explana-tory variables, However, the Bayesian estimation of the model has not been studied. TheBayesian method is a powerful tool for time series models, it has many advantages that theclassic methods don’t have. Therefore, the Bayesian estimation is introduced into this model,we study the Bayesian estimation of the autoregression model with explanatory variables.First we introduce a condition when it is stationary and its maximum likelihood estima-tion, secondly both knownσ2and unknownσ2are discussed in the Bayesian estimation ofthe model. Whenσ2is known, We consider a noninformative prior and a normal prior andgive two corresponding Bayesian estimation for the regression coefcitents of the model.For the caseσ2is unknown, another Bayesian estimation is given for all unknown paramtersof the model under a normal-inverse-gamma prior. Finally, simulate all bayes estimators,the simulation results show that the Bayesian estimation is not strong depended on a prior, isrobust. We compare the simulation results of Bayesian method and the maximum likelihoodmethod, when the sample size is small, the Bayesian method is superior to the maximumlikelihood method, When the sample size increases, the diference between the two estima-tion methods gradually disappear.