The biggest problem here is controlling for monetary policy; because theory says that whether or not fiscal policy works depends on how monetary policy responds to fiscal policy.

This sounds sensible to me. It all depends on whether or not there is monetary policy offset. (Though, strictly speaking, it also depends on what exact goods the government buys; if for example the government buys goods that are perfect substitutes for private consumption or investment, which is like the government doing your shopping for you with your credit card, fiscal policy should have no effect.)

New Keynesian macroeconomics says that a (temporary) increase in government spending will cause an increase in the natural real interest rate. And unless the central bank increases the actual real interest rate by an equal amount, the result will be an increase in Aggregate Demand, which will cause an increase in real output and/or inflation. There is a fiscal policy multiplier. Plus, if the central bank holds the nominal interest rate constant, any increase in inflation and expected inflation will cause the real interest rate to fall, which will cause an additional increase in Aggregate Demand, multiplying the original fiscal policy multiplier. (In an open economy part of the effect works through the real exchange rate.)

Simon Wren-Lewis, who understands these problems at least as well as I do, cites an IMF working paper by Vincent Belinga and Constant Lonkeng Ngouana (pdf), which makes a brave attempt to deal with these problems, and finds a positive fiscal policy multiplier for real output, but only in periods when monetary policy is "accommodative". That empirical result seems to confirm the New Keynesian theory. But when we look more closely at what "accommodative" actually means in the paper, it doesn't.

"This conjecture is confirmed ex-post: we find that the Fed funds rate falls following a positive federal spending shock under accommodative monetary policy, though inflation rises. As a consequence, the real interest rate falls even sharper (see Section III.C)." (page 14)

Take any policy variable X. Suppose we find that in part of our data sample, a positive shock to X is followed by a fall in the Fed funds rate, and by a rise in the inflation rate. And suppose we also find, in the same part of our data sample, that a positive shock to X is followed by a rise in real GDP. We might be puzzled why a shock to X causes the Fed to respond by lowering the Fed funds rate, but we would not be puzzled by the rest of the results. Yep, a negative shock to the Fed funds rate will cause a rise in inflation and real GDP. That's what monetary shocks are supposed to do.

And the estimated Impulse Response Functions in Figures 5 and 6 show that when monetary policy is "accommodative" a 1% positive shock to government spending "causes" an approximately 50 basis point fall in the nominal Fed funds rate for 4.5 years, and an approximately 50 basis point fall in the real Fed funds rate for 5 years (or more). If that 50 basis point fall in the Fed funds rate caused a (say) 1% rise in real GDP, that would be the right order of magnitude to explain the "fiscal policy multiplier" in Figure 3.

It gets worse.

A "non-accommodative" monetary policy means that the central bank follows a "Taylor-type rule"; the central bank raises the nominal interest rate in response to the gap between actual and target inflation, and the gap between actual and potential output, but does so slowly, because the lagged nominal interest rate also appears in the (estimated) Taylor-type rule.

Now theory says that fiscal policy should in fact work, temporarily, if the central bank follows a Taylor-type rule like that. Because a (temporary) increase in government spending will raise the natural rate immediately, but the central bank will raise the actual rate to match the natural rate only slowly, in response to the rise in output and inflation caused by the fiscal policy shock. So there will be a period of time when fiscal policy causes the natural rate to rise above the actual rate of interest, which increases Aggregate Demand, which increases output and/or inflation.

But the Impulse Response Function in Figure 3 shows a fiscal policy multiplier that is basically zero for the non-accommodative monetary policy.

The bottom line, for me, is that I am less confident that fiscal policy works as advertised after reading the IMF paper than I was before I read it.

[Disclaimer. This is a sophisticated and difficult paper to read, especially for a non-econometrician like me. Their technique for distinguishing between "accommodative" vs "non-accommodative" monetary policy regimes is quite complicated, and I am not 100% sure I have understood it correctly.]