Elasticity interpretation in linear regression models with powers of logarithms [Question]
Hi everyone, thanks for reading this! Here is my question, for a TL:DR feel free to skip to the question at the very end.
In a linear regression of the form
log(y) = a + b log(x) + u
b can be interpreted as the constant elasticity of y with respect to x. In models which do not involve logarithms, like
y = a + b x + c x^2 + d x z + u
quadratic, cubic... and mixed terms allow the partial effect of x to depend on the value of x and/or other regressors. I am trying to put these two notions together, to allow for models of the form, for instance,
(*) log(y) = a + b log(x) + c log^2(x) + d log(x)z + u
A little bit of calculus shows
D log(y)/D x = (b + 2c log(x) + z)/x
So that for small Delta x it holds approximately
Delta log(y) = (Delta y)/y = (b + 2c log(x) + z) * (Delta x)/x
Would it be correct to state that the model (*) allows for a variable elasticity, with the elasticity at given values of x, z, given by b + 2c log(x) + z?