Theory And Application Of Threshold Autoregressive Model With Exogenous Variables For Matrix Time Series

In fields such as finance,meteorology,and industry,time series often exhibit non-linear dynamic characteristics.Compared to AR models,TAR models are better able to accurately capture these nonlinear features.However,when studying variables with a matrix structure,traditional TAR models cannot provide a good characterization.There-fore,this paper proposes the construction of a threshold autoregressive model with ex-ogenous variables for matrix time series MTARX,which can maintain the matrix struc-ture and achieve greater dimensionality reduction.This paper introduces the MTARX model.A statistic that asymptotically follows aχ~2distribution is proposed,based on constructing prediction residuals,to test the thresh-old effects of the MTARX model.The Frobenius norm is used to define the objective function for parameter estimation in the MTARX model,and the least squares method is employed to estimate the coefficient matrix.The threshold and lag orders are de-termined through grid search,and the AIC and BIC formulas for determining the lag orders within each regime of the MTARX model are provided using the relevant theory of the vector threshold autoregressive model.Two methods are proposed to determine the number of regimes.Finally,the MTARX model is applied to study the impact of humidity on air pollutant concentrations in three cities.The model is evaluated by com-paring it with the time series autoregressive model with exogenous variables for matrix time series,model testing,and residual size.The results show that the MTARX model can significantly reduce residual variance and improve prediction accuracy.