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This article challenges Fixed Effects (FE) modelling as the ‘default’ for time-series-cross-sectional and panel data. Understanding differences between within- and between-effects is crucial when choosing modelling strategies. The downside of Random Effects (RE) modelling – correlated lower-level covariates and higher-level residuals – is omitted-variable bias, solvable with Mundlak’s (1978a) formulation. Consequently, RE can provide everything FE promises and more, and this is confirmed by Monte-Carlo simulations, which additionally show problems with another alternative, Plümper and Troeger’s Fixed Effects Vector Decomposition method, when data are unbalanced. As well as being able to model time-invariant variables, RE is readily extendable, with random coefficients, cross-level interactions, and complex variance functions. An empirical example shows that disregarding these extensions can produce misleading results. We argue not simply for technical solutions to endogeneity, but for the substantive importance of context and heterogeneity, modelled using RE. The implications extend beyond political science, to all multilevel datasets.