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Consider the following simultaneous system

$$ y_{1} =\beta _{1}y_{2}+\alpha _{1}z+u_{1} \\ y_{2} =\beta _{2}y_{1}+\alpha _{2}z+u_{2} $$

where $y_{1}$, $y_{2}$ and $z$ are vectors of random variables each of length $n$. $u_{1}$ and $u_{2}$ are disturbances and all greek letters are coefficients to be estimated.

The Wikipedia entry on the simultaneous equation model mentions the possibility to estimate all simultaneous equations at once. For large systems, this will be computationally costly but for smaller ones this seems a reasonable alternative.

Question: How can we estimate the system above at once?

I had a look at a couple of econometrics books (e.g. Wooldridge, Introductory Econometrics; or Greene, Econometric Analysis) but I couldn't find any information about jointly determining all coefficients in the system. But maybe I have been looking just at the wrong places.

Bert Breitenfelder
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    Hi: You'd have to re-write the model so that everything is on one side ( y1 and y2 can't be on both ). Then you can estimate the reduced form. But then, when you want to go back and estimate the original model, whether you can do that or not depends on whether its identified. I can't go into it all here but any decent book that has a chapter on simultaneous equations will explain it. It's pretty complicated so I don't think I could do it justice even if I tried. – mlofton Jan 25 '21 at 17:57
  • @mlofton, thanks, yes. Estimating the reduced form is discussed in the books mentioned in the post. The Wikipedia article suggests that there is another possibility involving a joint estimation of the equations. My question is about how such a joint estimation would look like. – Bert Breitenfelder Jan 27 '21 at 07:42
  • Hi: Do you use R ? If so, I would check out the references-examples in the systemfit package by Arnie Heniingsen because it's been too long since I dealt with systems to remember much. His package is quite nicely documented so, even if you don't use R, I bet it would still be helpful for methodology. There are lots of issues with identification of systems so you need to be quite careful and familiar with the theory before you start estimating models. – mlofton Jan 28 '21 at 15:20
  • @mlofton, the systemfit package looks promising. Thanks. – Bert Breitenfelder Jan 29 '21 at 14:30

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