Well you often come across research between money and inflation (lately on whether money matters at all for inflation or not).
Here is a short noteby St Lois Fed economist Yi Wen who looks at the relation between money stock and growth in GDP.
The Fed has taken unconventional measures over the past 18 months to contain the financial crisis and limit ramifications for the broader economy. These measures have resulted in an extraordinary increase in reserve balances at commercial banks—which is a key component, along with currency, of the monetary base. …..
Many analysts have raised concerns that the increased reserves will ultimately increase inflation and the price level. One might also expect such an enormous increase in reserves to stimulate aggregate output, thereby mitigating the adverse effects of the financial crisis on the economy. But can such an impact be estimated quantitatively?
However, the idea is not to prove causality:
Historically, we can look at postwar U.S. data and see how much gross domestic product (GDP) growth can be associated with or forecasted by the growth rate of the monetary base. Note that such a statistical association is not “causal.” We merely want to see whether, historically, fast growth of the monetary base has been associated with faster growth of real output. One approach is to use an analysis that captures the impact of current and past increases of the monetary base on current GDP growth, taking into consideration the influence of the history of GDP on its own future growth. This estimation can be done at different horizons using statistical tools
The results (via a chart):
The chart indicates that in the very short run (say at the 2-quarter horizon), money base growth is slightly negatively associated with GDP growth. How ever, around the typical business cycle horizon (say within the horizon of 8 to 16 quarters or 2 to 4 years), money base growth has a significant positive relation with GDP growth. In particular, at the 12-quarter horizon, for every 1 percent increase in money base growth, there is about 0.4 percent corresponding increase in GDP growth. Such a positive relation disappears again in the very long run beyond the typical business cycle, perhaps because in the long run money growth is inflationary, which leads to higher prices and lower output.
Therefore, historical data tell us that if there is any positive association between money growth and GDP growth, the impact comes about 3 years after an initial acceleration of base growth.
So far monetary base has not had an impact on output. However, the author adds there has never been a monetary expansaion like the current one and we will see what the results would be.