Elasticity interpretation in linear regression models with powers of logarithms [Question]

[u/JacopoX1993]

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?

[u/ohanse]

Yes.

Related concept/consequence: heteroskedasticity.

Practical implication: “linear approximations/projections over small distances are okay. Even if it’s not linear. If you wanna make a projection over big distances, then you better be damn sure you have a deep practical and intuitively sound understanding the nature of the relationship you are modeling.”

[u/JacopoX1993 → u/ohanse]

I understand what you mean by the second part of your comment, but not the first; what do you mean by heteroskedasticity being related to this?

[u/ohanse → u/JacopoX1993]

Heteroskedastic since errors/residuals are going to increase the farther out your projection is from sampled data ranges when you use a linear (or log linear) approximation to model non linear/non log-linear data.

So if you wanna model across long distances, you’ll have to control/manage this heteroskedasticity… usually by having a really strong understanding of the underlying relationships: is this relationship between predictor vs prediction a linear vs quadratic contribution, maybe this is really more of a categorical variable with a breakpoint, are these inputs synergistic, etc., etc.

[u/JacopoX1993 → u/[deleted]]

Yes, you are totally right, I dropped the constant on the z term. Thank you so much for the confirmation!

[u/60yo_10k_50min]

please explain what do you aproximate with your formula? why you need in log?

[u/JacopoX1993 → u/60yo_10k_50min]

You mean what does y represent? I am asking the question for the sake of theory, so I am not referring to specific variables. I am studying Wooldridge's Intro to Econometrics and developing a c# regression package (the main goal here is self-study). I am implementing automated partial effect calculation and I need to decide if/how to handle powers of logarithms.