Testing and modelling for the structural change in covariance matrix time series with multiplicative form

We first construct a new generalized Hausman test for detecting the structural change in a multiplicative form of covariance matrix time series model. This generalized Hausman test is asymptotically pivotal, and has nontrivial power in detecting a broad class of alternatives. Moreover, we propose a new semiparametric covariance matrix time series model. The proposed model has a time-varying long-run component that takes the structural change into account, and a BEKK-type short-run component that captures the temporal dependence. We propose a two-step estimation procedure to estimate this semiparametric model, and establish the asymptotics of the related estimators. Finally, the importance of the generalized Hausman test and the semiparametric model is illustrated by means of simulations and an application to realized covariance matrix data. Copyright © 2023 Institute of Statistical Science, Academia Sinica.