We propose a semi-parametric coupled component GARCH model for intraday and overnight volatility that allows the two periods to have different properties. To capture the very heavy tails of overnight returns, we adopt a dynamic conditional score model with t innovations. We propose a several step estimation procedure that captures the nonparametric slowly moving components by kernel estimation and the dynamic parameters by t maximum likelihood. We establish the consistency and asymptotic normality of our estimation procedures. We extend the modelling to the multivariate case. We apply our model to the study of the component stocks of the Dow Jones industrial average over the period 1991-2016. We show that actually overnight volatility has increased in importance during this period. In addition, our model provides better intraday volatility forecast since it takes account of the full dynamic consequences of the overnight shock and previous ones.