The recession of 2007 to 2009 was brutal. The economy contracted by 4 percent. Unemployment peaked at 10 percent. About 8.5 million jobs were lost. It's no wonder that a year later, nearly two-thirds of Americans thought President Obama's stimulus package hadn't worked.

But a new study suggests that without the stimulus — and, more crucially, without bank bailouts and the Federal Reserve's intervention — things would have been much, much worse. Princeton economist Alan Blinder and Moody's Analytics' Mark Zandi estimate, in a paper for the Center on Budget and Policy Priorities, that without these policies:

The recession would have lasted twice as long.

The economy would have shrunk by nearly 14 percent, not 4 percent.

Unemployment would have peaked at nearly 16 percent, not 10 percent.

More than 17 million jobs would have been lost, around twice the actual number.

In 2015, there would still be 3.6 million fewer jobs and 7.6 percent unemployment.

Government policies saved millions of jobs

Blinder and Zandi arrive at these estimates through economic modeling. Zandi and Moody's have one of the most cited private sector models of how the economy responds to changes in public policy, and regularly produce "multiplier" estimates of how much the economy would grow in response to $1 more funding for a given policy. (For example, the current paper estimates that a $1 temporary increase in food stamps at the start of 2009 would have grown the economy by $1.74.)

So Blinder and Zandi used the Moody's model to simulate how the economy would've looked if no special measures had been taken in the wake of the financial crisis and recession. They assume that the government's "automatic stabilizers" — programs like food stamps and progressive taxes that help people more when the economy's suffering — took effect, and that the Federal Reserve took interest rates down to zero. They then compared the economy in that counterfactual to the actual history.

They did better than that, though. They also modeled the contribution of each individual policy:

The 2009 stimulus package cut unemployment by 1.4 points and increased GDP by 3.3 percent in 2010.

The Fed's quantitative easing added 1.1 percent to GDP and cut unemployment by 0.6 points in 2012.

The bank bailouts — specifically the TARP program and the Fed's "stress tests" — cut unemployment in 2011 by 2.2 points and increased GDP by 4.2 percent.

The auto bailout cut unemployment by 0.4 points and increased GDP by 1 percent in 2010.

Can we trust this estimate?

It's important to keep in mind that this is an estimation based on a model. That has its advantages, and it's a valid economic technique. It's a lot easier to construct a precise counterfactual when you can use a model than it is to try to statistically isolate the effects of a given policy. But some critics have argued that these models assume the stimulus worked rather than proving it.

One reason you might want to trust Blinder and Zandi, however, is that the empirical evidence on the stimulus backs them up. I did a roundup of the best studies back in 2011 on just the fiscal side, but here are a few of the empirical findings on the 2008-'09 interventions in general:

The empirical evidence suggests these policies worked — which provides some basis to trust modeling estimates of the effects. What's more, a 2014 poll of eminent economists found that 37 thought the 2009 stimulus lowered the unemployment rate, and only one disagreed.

The counterfactuals this paper doesn't consider

Blinder and Zandi are fairly compelling in arguing that the policies chosen by the Bush/Obama administration, Congress, and the Federal Reserve led to a shallower recession and quicker recovery that we would've had otherwise. But it's doubtful that any administration would have done literally nothing. And there's a wide array of alternative policies that at least potentially could have bested the ones chosen:

So Blinder and Zandi can tell us that the policy response was better than nothing. "If my only options are what actually happened — even including the resulting moral hazard — and the Blinder-Zandi scenario, then bail, bail, bail, stimulate, stimulate, stimulate," as AEI's Jim Pethokoukis writes.

But that doesn't mean what happened was the best possible policy response. Maybe we could have done better.