A note on Jarque-Bera normality test for ARMA-GARCH innovations

In this paper, we study the Jarque-Bera (JB) normality test for the innovations of ARMA-GARCH models, whose construction is based on the residuals. The validity of the JB test for ARMA-GARCH innovations should be carefully investigated in advance of actual practice, since the residual-based test may behave differently, depending upon the structure of the time series models and the form of the test statistic (cf. Chen & Kuan, 2003; Hwang & Baek, 2009; Lee & Wei, 1999). In order to demonstrate the validity of the JB test, we prove that the asymptotic distribution of the original form of the JB test is identical to that of the test statistic based on true errors under mild conditions. Simulation results are provided for illustration.