Asymptotic Theory For RCA(1) Model With Time-Varying Variances

The classical autoregressive model is the foundation of time series analysis and plays a very important role in the development of time series analysis.However,the classical autoregressive models often fail to capture some certain important characteristics of the data due to overly strict assumptions on the coefficients and the innovation variance,such as the peak and fat tails in financial data.Therefore,a number of extensions of the classical autoregressive model have been developed to enhance it’s utility and improve its performance of fitting real data.The first extension is to relax the restriction on the coefficients by replacing the constant coefficients with fixed functions of time-varying form,which enable the coefficients to vary slowly over time,or just like the random coefficient autoregressive model that allows its coefficients to be random variables.The second extension is the autoregressive conditional heteroscedasticity model,which fits the actual data with heteroscedasticity such as heavy-tails by relaxing the restrictions on the random error’s variance.The third extension through combining with the ideas of random coefficient autoregressive models and the autoregressive conditional heteroscedasticity models to build more generalized models,so that both of them are included as special cases.So,in order to expand the application of autoregressive model in heteroscedasticity data field with unknown time-varying function form and further enhance the theoretical and practical value of the autoregressive models,this paper proposes a generalized autoregressive model—the first-order time-varying heteroscedasticity random coefficient autoregressive model(THRCA(1)).Under certain assumptions,we first derive the conditional least squares estimator of the model’s constant coefficient and the asymptotic properties of the corresponding estimator.Then,according to the special form of model’s conditional heteroscedasticity,the model’s time-varying function of heteroscedastic variance and the variance of random coefficients are estimated based on the semi-parametric theory.Finally,the theoretical results of the model are simulated numerically.