Empirical Euclidean Likelihood Inference For Autoregressive Models With Time-Varying Variance

The autoregressive model is a commonly used time series model,which generally assumes that the observations at the current moment are related to the observations at several past moments and that the variance of the data is constant(homoscedasticity).However,in practical applications,the variance of time series data tends to change with time,which is often said that the data has timevarying characteristics.Failure to consider time-varying characteristics may lead to inefficient estimation and unreliable inference.Many statisticians have noticed this problem and introduced the autoregressive model with time-varying variance(ARTV)to portray this situation.This model can better characterize time series data and has a wide range of applications in finance,medicine,geology,economics,and meteorology,especially in financial risk investment,such as in the study of risk management problems of different assets,where time-varying variance can reflect the risk volatility of the stock market,and thus can make more accurate predictions of the risk of different assets.For autoregressive models with time-varying variance,this paper uses the empirical euclidean likelihood(EEL)method to make statistical inferences about the model parameters.In this paper,we first estimate the ARTV model parameters using the least squares method,and then give the asymptotic distribution of this estimate.Because there is unknown time-varying variance in the given distribution,this paper uses three existing anti-heteroskedasticity methods to estimate the covariance matrix Λ;according to the time-varying characteristics of the ARTV model variance,this paper uses the nonparametric methods of empirical likelihood and empirical euclidean likelihood to ARTV model parameters for statistical inference and prove the strong consistency and asymptotic chi-squared distribution of the empirical euclidean likelihood estimators under the null hypothesis;subsequent numerical simulations are conducted to assess the test effects of the test statistics corresponding to the six methods by rejection rates.The error variance of the ARTV model in this paper is deflated by an unknown nonparametric time-varying function,and the use of the empirical euclidean likelihood ratio test statistic avoids estimating the variance function in the presence of heteroskedasticity compared to existing parametric and nonparametric estimation methods.The results of numerical simulations show that the empirical euclidean likelihood method is more computationally efficient and provides better estimates of the ARTV model parameters and relatively robust tests compared with the existing test statistics.Combining the estimation method and numerical simulation results proposed in this paper,the empirical euclidean likelihood method can perform the estimation of model parameters more effectively,which in turn provides new ideas and methods for research in related fields.