If a house has a good thermostat, we should observe a strong negative correlation between the amount of oil burned in the furnace (M), and the outside temperature (V). But we should observe no correlation between the amount of oil burned in the furnace (M) and the inside temperature (P). And we should observe no correlation between the outside temperature (V) and the inside temperature (P).
An econometrician, observing the data, concludes that the amount of oil burned had no effect on the inside temperature. Neither did the outside temperature. The only effect of burning oil seemed to be that it reduced the outside temperature. An increase in M will cause a decline in V, and have no effect on P.
A second econometrician, observing the same data, concludes that causality runs in the opposite direction. The only effect of an increase in outside temperature is to reduce the amount of oil burned. An increase in V will cause a decline in M, and have no effect on P.
But both agree that M and V are irrelevant for P. They switch off the furnace, and stop wasting their money on oil.
This is based on Milton Friedman's thermostat analogy (pdf). The central bank is the thermostat. This post is just another way of saying what David Beckworth is saying about James Hamilton's post on velocity. And it's another way of saying what Adam P. is saying about Menzie Chinn's post on the monetary base. And it's another way of saying what I said about why there's so little good evidence to tell if fiscal or monetary policy work. And the basic argument pre-dates Milton Friedman. Some old Keynesians (Maurice Peston?) figured out the basics back in the early 1970's.
If a policymaker is looking at information I to set a control variable C to target some variable T, then deviations of T around the target T* should be uncorrelated with C and with anything in I. The policymaker's forecast errors T-T* must be uncorrelated with the control variable and anything else in his information set, if the policymaker has rational expectations.
It's much more general than fiscal or monetary policy.
It is no surprise, for example, that Phillips Curves look flat when there's an inflation targeting central bank. Almost everything should be flat, if the target variable is on the vertical axis, and the central bank is doing its job right. It's only when the econometrician has got data on the horizontal axis that the central bank didn't have, and couldn't forecast, that there's a chance that the curve won't be flat. The only thing the econometrician is estimating is the central bank's forecast errors.
Update. Yep. Peston, M.H. 1972. “The Correlation between Targets and Instruments.” Economica 29(156): 427–31. Bottom line of that paper: there ought not be any.