  This study focuses on the problem of testing for normality of innovations in
stationary time series models.To achieve this, we introduce an information
matrix (IM) based test. While the IM test was originally developed to test for
model misspecification, our study addresses that the test can also be used to
test for the normality of innovations in various time series models. We provide
sufficient conditions under which the limiting null distribution of the test
statistics exists. As applications, a first-order threshold moving average
model, GARCH model and double autoregressive model are considered. We conduct
simulations to evaluate the performance of the proposed test and compare with
other tests, and provide a real data analysis.
