Importantly, Williamson and I agree that simple raw correlations tell us almost nothing especially when there are very few data points. As always, I stress that, to me, the interesting thing is the difference between statistical calculations however simple and crude (people agree on what they are if not what they mean) and impressions based on reading newspapers where people perceive very different supposed facts.
Of course even if one believed the conventionally calculated OLS standard errors (which is crazy) and assume that there are no omitted variables, which is crazy, one would get an +/-2 standard deviation interval from 0.248 to 3.184 so basically I totally misled the computer about the data generatingprocess but it still understands that 19 data points can’t prove or disprove anything.Here 19 not 21 as FRED hadn't yet reported 2014q3 GDP and I used only changes t to t+1 where t as well as t+1 are during the recovery so didn't use 2009q2-2009q3.
I just want to explain more thoroughly why a simple correlation of 21 growth rates tells us little. First, and most importantly, many things other than G affect GDP. It is not possible to consider many variables with only 21 observations, so there is no solution to the problem. One way to see this is to look at the correlation starting at the peak 2007q4 not the trough 2008q2 (as I forget who on the web did -- sorry for no link tell me who you are in comments). For this sample, the correlation is strongly negative. Now this excludes the deflation of a housing bubble and a financial crisis, so it isn't convincing evidence that G causes lower GDP. But it does focus the mind on the risk that there are less obvious ommitted variables which explain GDP changes during the recovery.
Second and less importantly, there can be reverse causation. It is possible that high GDP growth causes high G growth. G is generally modified as exogenous and shifting due to the mysterious whims of policy makers, but state and local governments have limited authority to run deficits so low GDP growth causes pressure for low state and local spending growth. I don't think this is a big deal in quarterly data, but it is an issue.
Finally the standard errors and t-statistics are definitely nonsense. They are calculated under the assumption that disturbances are identically normally distributed and uncorrelated. They aren't, so the conventional standard errors are nonsense. This isn't always such a huge issue if large samples of data are used, but a set of 21 observations is small.