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 Dow Jones industrial average component stocks over the period 1991-2016 and the CRSP cap based portfolios over the period of 1992-2015. We show that actually the ratio of overnight to intraday volatility has increased in importance for big stocks in the last 20 years. 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.