Ive got a regression in the form log(y) = a +b1x1 + b2x1^2 + b3x2 + b4x3 + b5* log(x4).
I've interpreted my coefficients as they should be interpreted.I figured out the quadratic and e^b3 and e^b4 are percent changes in y for unit changes in x. I even tested this by changing x by a unit holding all else constant in both cases and saw it was true.
However, for x4 I know b5 should be the elasticity that is (dx4/x4)/(dy/y) however I am not seeing that result when plugging in new values of x4. For example, I found y using all variable averages, and then did the same but timesed x4 by 2: log(2*x4)(kept everything else constant). I expected the percent change in y to be that of b5. (As dx/x) = 1. Instead, I got a number far below the elasticity.
I did the same for a 1% change in x(of course this time dx/x was 0.01 so I had to divide through it) and I received a number much closer to b4 but still not exactly it. Isn't the partial derivative of y with respect to x4 in a log-log function meant to give you elasticity (dy/dx4)* (x4/y) and wouldn't this elasticity be constant?
Why is it not? Thanks.
