Let me put forth a hypothetical. Suppose we a (overly simplified) system as follows:
Obviously, the point I am trying to make is that in such a (grossly oversimplified) system, the Fed could fufill its objective because at any time t it can control spending totally independently of the price level. Any recorded GDP numbers would, therefore, be totally under Fed control in the pressent. The Fed could always meet its goal regardless of the GDP function because it has the direct ability to manipulate the statistic it is trying to alter, not because its action are having the desired effect on the economy itself.
Now I realize things are much more complex in reality, but my question is this: What statistical mechanisms are used to prevent this sort of error from creeping into GDP reporting? You often hear Austrians talk about the Fed "paving over" recessions with printed money, implying that reported GDP growth during recessions is inflated by new money, but I've never heard any specific criticisms. We know inflation lags a bit as price are "sticky", and that newly-created money tends to be used rather quickly. However, these facts seem obvious, and this seems like the sort of thing economists would already include in their statistics. Am I wrong?