Asymptotics For The Conditional Self-weighted M Estimator Of GRCA Models And Their Applications

It is well known that the coefficients of autoregressive models are nonrandom,while in practical problems,the parameters of time series are not necessarily fixed.Therefore,we study the generalized random coefficient autoregressive(GRCA,for short)model,which allows the coefficients to be random,and the dependence between random coefficients and random errors is possibly existed.This model is of great significance in economy,finance and other fields.Firstly,due to the robustness of M estimation,we study the conditional self-weighted M(SM,for short)estimator on the mean of the random coefficient vector of the GRCA model.By taking different loss function,we can get different estimators,including but not limited to the conditional self-weighted least-squares(SLS,for short)estimator,the conditional self-weighted least absolute deviation(SLAD,for short)estimator,the conditional self-weighted Huber(SH,for short)estimator and the conditional self-weighted quantile regression(SQR,for short)estimator.Next,we give the asymptotic normality of conditional SM estimator for GRCA model.What’s more,when studying the asymptotic normality of the estimator,the errors can be of infinite variance.Then,based on its asymptotic normality,we propose the Wald test,and define a Wald test statistic.It can be seen that the Wald test statistic follows the chi-square distribution,and the form of rejection region is given.Then,we give the strong consistency of conditional SM estimator for GRCA model.Moreover,under some appropriate conditions,for different conditional SM estimators,we can obtain the strong consistency of different estimators.Finally,under GRCA(1)model and GRCA(2)model respectively,we conduct data simulation on the asymptotic properties of conditional SM estimator,and verify the accuracy of theories by observing the bias,the mean squared error(MSE,for short)and the mean absolute error(MAE,for short)of in finite samples.In addition,the sizes and powers of Wald test are obtained by testing Wald test statistics.These simulations are reasonable and valid.