This article compares various specifications of the gravity model of trade as nested versions of a general specification that uses bilateral country-pair fixed effects to control for heterogeneity. For each specification, the author show that the atheoretical restrictions used to obtain them from the general model are not supported statistically. Because the gravity model has become the "workhorse" baseline model for estimating the effects of international integration, this has important empirical implications. In particular, they show that, unless heterogeneity is accounted for correctly, gravity models can greatly overestimate the effects of integration on the volume of trade.