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A reader asks me to post my solution here. Please note that this solution is
arrived at with a specific assumption.
The elasticity of X w.r.t Z = b*z y/x where z, y,x are the samples means of
Z, Y, and X respectively, and d is the partial derivative symbol.
It is derived at by totally differentiating the regression equation X/Y = a
+ bZ + cK + error, and assume dY/dZ=0, that is to assume Z affects X only,
If Z affects Y also, it may not be possible to identify the elasticity of X
w.r.t. Z without imposing other restrictions.
I do not need the standard error of the elasticity for my purpose, therefore
gave no though about it.
My friend (a economics professor) pointed out the solution, errors here are
mine. I'll validate the solution with some data, and will report here if it
needs any modification.
Ontario Tobacco Research Unit
For a regression equation: X/Y = a + bZ + cK + error where X, Y, Z and K
are variables. Given estimates of the coefficients b and c, what is the
formula for the marginal effect of Z on X ie. dX/dZ, and the formula for the
elasticity of X with respect to Z.
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