Robust Statistical Inference For AR Models With Heavy-Tailed Heteroscedastic Noises

The distribution of noises is an important part of statistical inference.Traditional LSE and other methods usually require normal distribution of noises,or they require the mean or median is 0.However,in the financial market,asset pricing and other fields,the positive and negative effects of data are often asymmetric.People usually react more quickly to extreme situations,which is easy to cause a large amount of data information to accumulate at the tail,making the time series show a heavy-tailed heteroscedasticity.At this time,the disturbance term no longer meets the conditions that required for general statistical inference,and the parameter estimation will have a great deviation,leading to the failure of prediction.The quantile regression method has a very robust prediction effect for the heavy-tailed data since it does not make any assumptions about the disturbance term.On the other hand,the Adaptive Lasso method compresses the small coefficients and enlarges the large ones by modifying one of the defects of Lasso’s constant weight,so as to obtain a more accurate sparse model and realize the selection of variables.Based on the above situations,this paper considers extending the hypothesis of disturbance term in the time series so that one weaker condition can be met,and then we establish a systematic statistical inference method to realize the parameter estimation and variable selection of the model,thus providing a more general statistical inference method for the heavy-tailed heteroscedastic noises time series model,and making the model more consistent with the needs of real life and more widely used.In order to make statistical inferences about the model,this paper first uses self-weighted quantile regression(SWQR)to estimate the parameters of AR(p)model,which successfully proves that the SWQR method is asymptotically normal,and its convergence rate is n,which is faster than that of LSE in the heavy-tailed heteroscedastic noises model.Secondly,in order to further realize variable selection,the SWQR method with Adaptive Lasso penalty is used to fit the model again,and it is proved that the SWQR method with Adaptive Lasso penalty has Oracle property,and its convergence rate is also n.This presents that the SWQR method with Adaptive Lasso penalty can achieve parameter estimation and variable selection at the same time.Therefore,compared with the traditional Wald test method,the SWQR method with Adaptive Lasso penalty does not need to test all parameter estimates,which can improve the statistical inference efficiency.At the same time,the experiment shows that its variable selection effect is also better than that of SWQR,especially when the autoregressive order of p is higher,the SWQR method with Adaptive Lasso penalty is better.In addition,in order to determine the quantile point and the key condition of the disturbance term,this paper uses the autocorrelation function to compare the quantile points through the Portmanteau test,so as to select the optimal quantile.Finally,this paper tests the parameter estimation and variable selection effects of the SWQR method with Adaptive Lasso penalty by selecting two actual cases,the daily closing price of CSI 300 Index,and the daily closing price of silver futures.