We set up two rival Computable General Equilibrium (CGE) models of world trade, one based on classical theories of comparative advantage, the other based on recent gravity theories. We have tested them by indirect inference on the time-series of trade facts for four major countries or country blocs: the UK, the US, China and the EU. The UK is a small enough economy for the rest of the world's behaviour to be treated as exogenous, so we test the UK model with this held constant; the other countries/blocs are large so we test their model by a 'part of model' test in which the other world variables are simulated by a reduced form VAR of the unknown true world model. We show by Monte Carlo experiments that these tests have high power. Our findings are that the Gravity version of the world model is rejected strongly for two of these country cases, but passes the test for the other two. By contrast the Classical model is comfortably accepted in all cases; our power experiment implies that this world model is very likely to be close to the truth and should therefore be used for policy analysis. The policy message of the classical model is that protection is damaging to welfare; this includes protection by customs union, where even though some members may gain, general welfare is reduced.