First differencing a nonstationary process
This is something that I can't quite understand about using first differences to turn a nonstationary process into a stationary one. If we regress dY on dX, aren't we just going to end up with the second derivative of Y to X? If so, why do people interpret it in levels? For example, in the example in Baby Wooldridge, he does a regression of a differenced Y on a differenced X, but interprets the resultant coefficient as the effect of an increase of one of level X on level y. Shouldn't it be the effect of an increase of one in change in X on change in Y, i.e. an acceleration?
Also, in the same vein, why is it that I see regressions of a differenced dependent variable on a level explanatory variable? How can you explain a change in something by an explanatory variable in levels? Shouldn't all the explanatory variables in that regression equation be differenced as well?