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In the past I have argued that there is no reason to be swayed by CBO or private-sector reports that the Obama stimulus created or saved more than 1.4 million jobs if you had been previously skeptical of the bill. My argument was that these reports didn't reflect any new evidence, but instead simply re-ran the econometric models that had been used to forecast the effects of the stimulus before it was signed into law. 

Obviously the Obama administration doesn't think much of this logic, nor does mainstream economic commentary. See Menzie Chinn's analogy between the stimulus and aspirin to get a sense of that viewpoint. 

The Cato Institutes's Daniel Mitchell noted a March 5th speech in which the CBO's director, Douglas Elmendorf, was asked about his methodology for estimating the stimulus's impact. He answered by fairly straightforwardly saying that his estimates do not provide new evidence, and that the CBO has not been able to distinguish the effects of the ARRA from simultaneously occurring macroeconomic movements. The video of the speech is here, with the relevant questions in between the 39 and 41 minute marks. I've transcribed the relevant passage, emphasizing the key parts:

QUESTIONER: You mentioned that with the ARRA as it was moving through Congress you provided projections based on a set of multipliers and econometric models, and then later, when evaluating the effects you also used multipliers and econometric models...and that gives the impression of assuming what one's trying to prove in terms of measuring the effects.... How would that be different if you compared the initial projections both baseline and with the stimulus bill to the actual experience? 

ELMENDORF: We try to be very clear in our reports on ARRA that we don't think that we can learn much, from watching the evolution of particular components of GDP over the last few quarters, about the effects of the stimulus. We think the best evidence about the effects of past policies comes from detailed studies, done often several years later, about the behavior of particular that households got tax rebates sooner or later or what have you. So we don't think you can learn much from that, and therefore we are, we fall back on repeating the sort of analysis we did before. And we try to be very explicit about that. That is essentially repeating the same exercise we did rather than any independent check on it. The part that's a check is that we watch how the money has flowed of the government budget and update that. We're reading new evidence, and if we thought that we saw any evidence that substantially shifted the body of work in this area then we would shift our views. We haven't seen that at this point. 

QUESTION: If the stimulus did not do what it was originally forecast to do, then that would not have been detected by the subsequent analysis, is that correct?

ELMENDORF: That's right. That's right. And in terms of what we would have found otherwise, I mean I don't remember each of our forecasts, certainly by last March our economic forecast took on board a very large decline in employment , a run-up in the unemployment rate, weak GDP growth over the recovery in the second half of last year. Our January forecast - last January's forecast - did not have that - we marked down the forecast considerably from January to March. Our first estimate of the effects of the stimulus package I think were coming out between those benchmarks. So it's hard for me to go back and disentangle those pieces entirely.

If you thought that the assumptions that went into the CBO's initial forecast were appropriate, then you still think the stimulus worked. But if you didn't, the recent reports from the CBO and private forecasters showing that the bill increased employment relative to what it would have been without the stimulus should not make you change your mind.