If I have understood the question well, you have a system of regression equations, not a single one; unless you replace the alpha coefficients in the main equation which yields a new one in terms of new exogenous variables (Z'n x yn, etc.). Then alpha coeffs will disappear and the single regression can be solved for the other coeffs, i.e. betas and gammas. Alphas can be calculated easily.

This is a common situation in some areas eg econometrics. See the attached search for more details. The CRAN documentation describes an R package that does this computation. The outline is given essentially by Dr Shakouri above.. Best wishes David Booth

This is the whole idea of multilevel modelling / Random coefficient - so you are perfectly fine to do so if you use appropriate methods. Look at e.g ML, HLM etc.

Sincerely

Jonas

yes of course, as Jonas Månsson pointed out, it's the main idea of multilevel modeling and random coefficients. If you need references please ask for it. I do no which software you are using, but STATA have a whole reference guide on multilel models (HLM).

Thank you very much for your helpful answers!

As I understood now the alphas are random coefficients. The issue I am facing now is that the authors mention that the model's specification is equivalent to a conditional logit model. And I cannot find a way to run a multilevel conditional logit model where observations are grouped by id, only regular multilevel logit models without grouping observations by id. Does anyone know how to handle this problem?

Or, in general, does anyone know how to implement this computation in STATA (or R)?

Hamed Shakouri G. how do I replace the alpha coefficients in the main equation in order to obtain a new equation in terms of new exogenous variables (Z'n x yn, etc.)?

And assuming it works the same way for conditional logit models as for regular logit models, can I use melogit or xtlogit in STATA? And how do I write the syntax?

E.g., the random coefficients belong to variables y, m and f, would the following be correct (I assume not)?

melogit V y² m² f² y*m y*f m*y || y: z1 z2 z3 || m: z1 z2 z3 || f: z1 z2 z3

Or how do I tell the model that y, m and f are supposed to be random coefficients and at the same time the dependent variable in another regression with z1, z2 and z3 as independent variables?

Using xtlogit on the other hand, I do not see where I could tell the model which ones are the random coefficient variables and that at the same time they are a dependent variable in another regression with z1, z2 and z3 as independent variables.

Sorry for late coming back to your question.

If Z is independent of y, m, and f (low correlations), I would simply replace alpha's with (beta's + Z' . gamma's) leading to: V = beta_y . y's + gamma_y . (Z'y) + ... .

Now we have a regression with beta's and gamma's (and delta) as the coefficients, rather than alphas, where (Z'y), (Z'm), and (Z'f) are also added as new exogenous variables independent enough than the other exogenous variables, with gamma's as their unknown coefficients.

Then, after solving the regression for all the coefficients, the estimates of alpha's can be obtained. Hopefully, the number of data samples is large enough to estimate all the coefficients successfully.

I was unable to see the image of equation for no reason. I am not sure if understand your question well! If you have three equations with same coefficients, then maybe you have a simultaneous equations system and need to solve them together.