The first rule of empirical social science is that correlation isn’t causation: Purchases of baby strollers predict childbirth but few of us believe baby strollers cause childbirth. We’re all trained in the post hoc fallacy, we’re often looking for solutions to it. 

One problem with estimating the effect of government purchases on the short-run economy is that government spending is responding to the economy in real time: government dollars aren’t dropped in randomly, like in a lab experiment.  This could make government spending look worse than it really is: maybe governments decide to spend more when the economy is just plain worse, so it looks like government spending is awful for the economy.  Or it could make government spending look better than it really is: maybe governments are more likely to go on a spending spree when a free-market recovery is “just around the corner” all on its own.   Hard to say. 
Unless you have a set of natural experiments, that is: A set of times when government spending rises or falls for reasons other than the state of the economy–because of a war overseas, perhaps caused by an unpredictable dictator who decides to invade for her own reasons.  Then you’d be in the world of petri dishes where the drug goes in one set of petri dishes but not the other.  
The conventional Keynesian (and New Keynesian) story has been that government spending should get more bang for the buck–a bigger multiplier effect–when the spending happens during a deep recession.  A new paper by Owyang, Ramey, Zubairy uses news about changes in future military spending as a kind of natural experiment* to test this idea: 
[W]e find no evidence that multipliers are greater during periods of high unemployment in the U.S.

This result conflicts with a widely-publicized result by Auerbach and Gorodnichenko who, just looking at historical U.S. patterns without any recourse to natural experiments, found that extra government spending during recessions predicted high GDP growth in the near future.  

On Twitter, Noah asks, “Can these be reconciled?”
Of course they can: With enough effort any uncomfortable research result can be explained away.  But can they be reconciled without doing violence to good science?  At least partly, I suspect.  
Here are three channels, each might matter, maybe none do:
1. Government spending takes a while (there’s a fiscal lag in textbook jargon), so by the time politicians see a recession, pass a stimulus bill, and get the workers hired, the economy is recovering on its own (or recovering because of loose monetary policy).  You get to the pharmacist to fill your prescription right around the time you were getting better already, and you credit your recovery to the prescription.
2.  Government spending is especially likely to be enacted just when politicians see a natural, market-based (or monetary-driven) recovery just around the corner: they pass stimulus for crude credit-taking purposes.  This is like your doctor handing you a few free drug samples on the sixth day of your week-long cold.  She knows you’ll feel better in a couple of days no matter what, and you’ll probably give some credit to the drugs she gave you.  Remember: As David might say, this isn’t a conspiracy, it’s just self interest. 
3.  A less nefarious version of #2: State and local governments find it hard to borrow cash when the future looks grim, so state and local governments genuinely find it hard to spend a lot of money until a recovery is already just around the corner.  You’re just too sick to go to the pharmacist until you’re halfway better, but you still credit the drugs with your final recovery.  
You can imagine #2 and #3 blurring together in practice: Politicians would like to spend more in a recession, but are reluctant to go on a spending spree until the Congressional Budget Office or other outside experts tell them the coast is almost clear.  Rating agencies, credit markets, economic advisers: All are in the business of trying to predict recovery, all have some influence on when the money gets spent.  And to the extent they are successful in delaying spending until the very end of a slump, they will make it look like government spending helps more than it really does. 
Banks start lending money before a boom because they start hearing about good business ideas.  A naive observer might think the money growth is causing the private economy to grow even when it’s just predicting the private economy will grow.  That “endogenous money” process is fairly well understood (PDF); the same process on the fiscal side deserves more attention.  
I put together a toy model of the economy to illustrate these points.  I’ll run through it tomorrow.  
*Wonkish: Ramey prefers the term “instrument‘, there’s a difference, but she uses the term “natural experiment” with some caveats on her EconTalk interview