We propose a semi-parametric coupled component GARCH model for intraday and overnight volatility that allows the two return series to have different properties. We adopt a dynamic conditional score model with t-distributed innovations that captures the very heavy tails of overnight returns. We propose a several-step estimation procedure that captures the nonparametric slowly moving components by kernel estimation and the dynamic parameters by estimated maximum likelihood. We establish the consistency, asymptotic normality, and semiparametric efficiency of our semiparametric estimation procedures. We extend the modelling to the multivariate case where we allow time varying correlation between stocks. We apply our model to the study of Dow Jones industrial average component stocks, CRSP size-based portfolios, and size-based portfolios in four large international markets over the period 1993-2017. We show that the ratio of overnight to intraday volatility has actually increased in importance for Dow Jones stocks during the last two decades. This ratio has also increased for large stocks in the CRSP database, but decreased for small stocks in CRSP. Notably, the slope increases monotonically from the smallest-cap decile to the largest-cap decile. This pattern also exists in other international markets. The multivariate model shows that overnight and intraday correlations have both increased, but overnight correlations have increased more substantially during recent crises than intraday correlations.