# SUR and interaction terms

Suppose I want to determine if a simultaneous model (A) was identified:

$$y_1 = \beta_{10} + \beta_{11} x_1 + \beta_{12} y_2 + \epsilon_1$$

$$y_2 = \beta_{20} + \beta_{21} y_1 + \beta_{22} x_2 + \epsilon_2$$

Where the y's are endogenous and the x's are exogenous. I'd be able to estimate this and find the rank order conditions and determine if it passes and can be properly identified.

However, what about equations of the sort (B):

$$y_1 = \beta_{10} + \beta_{11} x_1 + \beta_{12} y_2 x_3 + \epsilon_1$$

$$y_2 = \beta_{20} + \beta_{21} y_1 x_3 + \beta_{22} x_2 + \epsilon_2$$

My intuition is not sure about the interaction terms as I run through the procedure in Maddala and other sources. The interaction terms are endogenous, but are they new endogenous variables (eg. do I have 2 endogenous variables or 4)? How do I check the rank-order conditions? Is there some good reading material on interaction terms in SUR?

Work done so far: I've done some pretty extensive article searching and read my textbook(s) for an explicit example but did not find one. I note that STATA will let me run interaction terms, which implies that there is some econometrics supporting interaction terms in SUR, though I'd like to see an official example of it. I can run estimations of the sort:

sureg (y1 = x1 c.y2#c.x3) (y2 = c.y1#c.x3 x2)


My instincts say perhaps I am overthinking this, and it could simply be substituted for and solved in reduced form, call it model type (C). Something straightforward, starting with:

$$y_1/x_3 = \beta_{10}/x_3 + \beta_{11} x_1/x_3 + \beta_{12} y_2 + \epsilon_1/x_3$$

$$y_2 = \beta_{20} + \beta_{21} y_1 x_3 + \beta_{22} x_2 + \epsilon_2$$

and therefore:

$$y_1/x_3 = \beta_{10}/x_3 + \beta_{11} x_1/x_3 + \beta_{12} (\beta_{20} + \beta_{21} y_1 x_3 + \beta_{22} x_2 + \epsilon_2) + \epsilon_1/x_3$$

...algebra exercise continues

Anyway, my question is: How do I check rank/order conditions for SUR with interaction terms, in models of type B? Are there any examples of this being done I can reference/reading on the subject, or are models of type B unsupported?

Just because STATA allows running of the following: sureg (y1 = x1 c.y2#c.x3) (y2 = c.y1#c.x3 x2) does not necessitate that STATA actually does the algebra to solve the coefficients of interest in the backend. I have ran a test withl functional forms cannot be estimated and are indeterminate, but STATA still estimated them fine.