The CEA's Impossible Job
First, there are model simulations. That is, the CEA took a conventional Keynesian-style macroeconomic model and used those set of equations to estimate the effect the stimulus should have had. Essentially, the model offers an estimate of the policy's effect, conditional on the model being a correct description of the world. But notice that this exercise is not really a measurement based on what actually occurred. Rather, the exercise is premised on the belief that the model is true, so no matter how bad the economy got, the inference is that it would have been even worse without the stimulus. Why? Because that is what the model says. The validity of the model itself is never questioned.
(Moreover, the fact that other organizations simulating similar models come to similar conclusions is no evidence about the validity of the model's simulations. It only tells you the CEA staff did not commit egregious programming errors when running their computer simulations.)
Second, the CEA offers some statistical evidence that things got better after the stimulus passed. Some of this evidence comes early in the document in the form of simple graphs. Some comes later by examining deviations from forecasts based on a two-variable vector autoregression. But the nature of the evidence is basically the same: Post hoc ergo propter hoc.
Of course, there were a lot of other things going on in the economy at this time. Monetary policy, for example, has gone to extraordinary measures to get the economy going. TARP was also an unusual intervention that seems to have done its job of returning the economy to some degree of financial normalcy (even if leaving the bad taste of increased moral hazard). Giving credit for the economic improvement to the fiscal stimulus is a large leap.
In the end, I do not find this CEA document very persuasive. At the same time, I feel the CEA's pain. The stimulus act instructed them to do the (nearly) impossible. Perhaps someday someone will conduct a study that credibly measures the macroeconomic effects of this particular fiscal stimulus. But it won't be easy. And it won't look much like the study released today.