Statistical Inference For A Class Of Random Variable-driven Autoregressive Model

Multivariate time series analysis has drawn much attention for statistical re-searchers in recent years.In practice,we often encounter such time series,for example,in the economy,we need to study the relationship between monthly unemployment rate changes in different regions;records of rainfall,temperature and pressure in the region should be considered in the weather forecast analysis.The difference from the one-dimensional time series is that the dependence between multivariate time series consists of not only the dependency of each marginal sequence,and also the cross de-pendence between different components of multivariate series.The prediction accuracy of the component can be improved by using the information contained in the vector.On the other hand,many financial time series are conditional heteroscedasticity,there have been many multivariate volatility models have modeled this kind of vector time series,and multivariate random coefficient autoregressive model is one of them.At present,there are few studies on this model,the non-parametric method is used to make statistical inference on the parameters of the model in the chapter 2.The change rules of many time series in real life can be affected by other sequences.For example,the per capita consumption of residents will be affected by income,and the growth rate of national GDP will be affected by unemployment rate.Autoregressive model with covariates has limitations in analyzing actual economic data,which often fail to accurately depict the changing characteristics of actual data,from the perspec-tives of economic interpretation and model prediction.In the chapter 3,we introduce random variables into the autoregressive model with covariables,and propose a random coefficient autoregressive model with the covariates,the parameter estimation of the model is discussed.Furthermore,considering that the observations of covariables often contain measurement errors,we propose a class of random coefficient autoregressive models with observation errors in covariables in the chapter 4.Based on the above discussion,the main results of this thesis are as follow:First,for multidimensional random coefficients autoregressive model,empirical likelihood method based on a modified conditional least square estimation equation is used to study the parameter estimation and hypothesis testing of the model.We obtain the asymptotic distribution of the empirical likelihood ratio statistic and the asymptotic property of the maximum empirical likelihood estimator,and use the em-pirical likelihood method to test whether the autoregressive coefficient is constant.The effect of parameter estimation and the power of the test are verified by numerical sim-ulation.The simulation results show that although the maximum empirical likelihood estimation is similar to the modified conditional least squares estimation and has no obvious advantages,the empirical likelihood method shows advantages in the test.Second,in order to better depict the economic data with the change of time,we propose the random coefficient autoregressive(RCAR-X)model with covariables,and give Bayesian estimation of model parameters via MCMC algorithm.At the same time,two test problems of whether the autoregressive coefficient is constant and whether the model has covariate are regarded as comparison of two models.The Bayesian factor is used as the criterion to compare the two models.When calculating the Bayesian factor,the Laplace method is used to approximate the marginal likelihood.We also study the accuracy of MCMC algorithm and its robustness for prior information via numerical simulations,and evaluate the performance of Bayesian factor test to correctly identify models.Finally,we propose a class of random coefficient autoregressive models with ob-servation errors in covariables.The conditional least square and weighted conditional least square methods are used to estimate the model parameters,and the consistency and asymptotic normality of the two kinds of estimators are proved.Furthermore,the empirical likelihood method based on the weighted conditional least square estimation equation is used to investigate the test of the randomness of autoregressive coefficients.We compare the performance of the estimators by numerical simulation.The simula-tion results show that the weighted conditional least squares estimators are superior to the conditional least squares estimators.The results of numerical simulation also show that it is reasonable to use the empirical likelihood method to test whether the autoregressive coefficient is constant.