We propose a semi-parametric coupled component exponential GARCH model for intraday and overnight returns that allows the two series to have different dynamical 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 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 and CRSP size-based portfolios 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.