In this paper, a time-varying first-order mixture integer-valued threshold autoregressive process driven by explanatory variables is introduced. The basic probabilistic and statistical properties of this model are studied in depth. We proceed to derive estimators using the conditional least squares (CLS) and conditional maximum likelihood (CML) methods, while also establishing the asymptotic properties of the CLS estimator. Furthermore, we employed the CLS and CML score functions to infer the threshold parameter. Additionally, three test statistics to detect the existence of the piecewise structure and explanatory variables were utilized. To support our findings, we conducted simulation studies and applied our model to two applications concerning the daily stock trading volumes of VOW.
Keywords: Wald test; explanatory variables; mixture thinning operator; parameter estimation; threshold integer-valued autoregressive models.