Online Monitoring Of Parameter Changes In A Linear Regression Model With Long-memory Errors

Online monitoring for the data observed in real-time helps to detect possible changes in the data as early as possible.This study extends the modified moving sum statistic(m MOSUM)method to online monitoring coefficient and variance change and the case which are changed at the same time in a linear regression model with longmemory time series errors.This study mainly focuses on whether the m MOSUM method still can improve the monitoring effect when the linear regression model has long-memory time series errors.To begin with,the m MOSUM method defined by the residual of the least square estimation is used to online monitor regression coefficients changes in the model.Under the null hypothesis,the limit distribution of the monitoring statistics is obtained by modifying the boundary function,and the consistency of the method is proved under the alternative hypothesis.The results of numerical simulation show that when the linear regression model has long memory time series errors,the m MOSUM method is still effective except for the case where the long memory parameter value is larger.As the location of the change point moves further back,the effect of the modified method on the increase of the power and the reduction of the run length is more obvious.Finally,the feasibility of this method is demonstrated by an empirical analysis of a set of macroeconomic data for the United States.Furthermore,the m MOSUM method defined by the residual squares is used for online monitoring variance changes in the model.The null distribution and consistency are established by modifying the boundary function.The results of numerical simulation show that the m MOSUM method can still improve the monitoring effect for the variance change points in the model.Moreover,the feasibility of the proposed method is demonstrated by modeling a set of PM2.5 concentration and SO2 concentration data in Xining City and monitoring the variance change point.Finally,the two m MOSUM statistics mentioned above are used to further study the online monitoring problem if the variance and coefficients have changed in the model at the same time.Through deriving the asymptotic distribution of the two statistics and analyzing the numerical simulation results,which are given discrimination of between mean and variance change-point.The results of numerical simulation also show that the m MOSUM method is still effective to monitor the changes of the coefficient and variance at the same time.