# How to test that a particular parameter in a nonlinear model does not vary across subpopulations?

My original model is: $Y=\alpha(\delta*X_1^{-\rho}+(1-\delta)*X_2^{-\rho})^{-\eta/\rho}exp(e_t)$

Which I partially linearize to: $ln(Y)=ln(\alpha)-(\eta/\rho)*ln(\delta*X_1^{-\rho}+(1-\delta)*X_2^{-\rho})+e_t$

I've used a nonlinear least squares algorithm to estimate the parameters.

Now, I am trying to determine if the "intercept" in this partly linearized model (in this case $ln(\alpha)$) is different for the different years of data that I have. I have 3 years of data, with about 24 observations in each year.

My first thought is to just add a dummy variable for each year as $ln(\beta*Year)$ and just test if the two coefficients that would result (three years minus one comparison years) are jointly statistically different from zero using an F-test. But I doubt that this would be legit since I am trying to test if one parameter ($ln(\alpha)$) varies over years rather than if all the parameters vary over the years.

My second thought is to just split up the sample by year, run three different regressions, and somehow test if all the $ln(\alpha)$ parameters are different. I don't what test I would use or how I would use it, though, to do this. I'm thinking maybe a t-test with unequal variance on each pair of the coefficients, but I feel like pairwise tests on each parameter does not get me a legitimate "joint" test. I have also read about the Chow test, but again I want to test if just one parameter is different across the subpopulations, not all the parameters, which is what the Chow test does.

Any thoughts on what the right procedure is?

Thanks!

• One approach to this kind of problem is pretty much covered - with a partly worked example - in my answer here: nls curve fitting of nested/shared parameters. The alternative - if the data are all completely independent - is to compute the standard error of the difference in the estimates from the separate regressions and do a form of t-test based on that. – Glen_b -Reinstate Monica Mar 11 '14 at 1:13