# How do I identify this simultaneous equations model?

I have the following model (which is in this form is not identifiable if the $y$'s are indeed endogenous):

(1) $y_1 = a_0 + a_1y_2 + a_2y_3 + \boldsymbol{Xa} + \boldsymbol{u}$

(2) $y_2 = b_0 + b_1y_1 + b_2y_3 + \boldsymbol{Xb} + \boldsymbol{v}$

(3) $y_3 = c_0 + c_1y_1 + c_2y_2 + \boldsymbol{Xc} + \boldsymbol{w}$

where $\boldsymbol{X}$ is a $n\times k$ matrix of exogenous variables $\boldsymbol{a, b, c}$ which are each $k\times 1$ vectors and the $\boldsymbol{u,v,w}$ are the random noise.

Since every endogenous variable (i.e., $y_i$ for $i=1,2,3$) is a function of the other $y$'s and each equation contains the whole set of exogenous variables (i.e., $X$) one cannot identify the equations given above.

So I checked my empirical data and the theory I have in mind, and I cannot relax the dependency on which of the endogenous variables are related to each other.

My next guess would be that I could check the exogenous variables to find a way to exclude at least 2 of them in each equation and each of these 2 should in the best case not be excluded in any of the other equations.

This would be just amazing, but I won't get out 6 different exogenous variables anyway.

How should I deal with this problem?

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## migrated from stackoverflow.comAug 27 '12 at 16:21

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Should be over on CrossValidated... – Charlie Aug 27 '12 at 16:15
Identifiability won't fall out of a blue sky, so don't expect there will be any kind of a magic trick to render your structural form identified. You do need the excluded exogenous variables here to make any sense out of this. May be assuming zero correlations of $u$, $v$ and $w$ could help a tiny bit, but most of the identification should come from the $X$'s. – StasK Aug 27 '12 at 23:00
True. Unfortunatly i cannot assume any 0 correlation between the residuals of the reduced form i.e. $y_i = \boldsymbol{\pi}_i + \boldsymbol{e}_i i=1,2,3$ and the errors of the structual form $\boldsymbol{u, v, w}$ is it enough to exclude 3 exogenous variables or do i need 6? in my understanding for each endogenous variable in each equation i need at least 1 variable exlcuded from another equation. For that reason if i would say i exclude $x_j$ in (2) and $x_k$ from (3) i could then identify (1). in the same manner i exclude $x_i$ from (1), so i can identify the equation (2) by $x_i$ and $x_k$? – Druss2k Aug 28 '12 at 14:39

The most common way to create identifiability is to put constraints on the model coefficients such as requiring a$_0$+a$_1$+a$_2$=0.
Hey, thx for your reply. But how does one justify for this kind of restriction? I mean it does surely come from the theory behind it but if i go for this common solution i need to bend the theory behind this model to adjust for the argumented model? is there maybe some reference where this is like "the state of the art" way of dealing with such a problem? another thing is that i still need to exklude at least 3 exogenous variables to account for the endogenous variables $y_1,y_3$ in (2) and $y_1,y_2$ in (3) dont i? – Druss2k Aug 27 '12 at 18:48
Are you aware of Rank-Order-Instrument Variables? Im not quite sure but as published by Verbeek and Vella could i not estimate the reduced form for any $y_i, i=1,2,3$ i.e. $y_i = \boldsymbol{X\pi_i} + \boldsymbol{v}_i$. I would get three additional rank-order based variables. Im not sure if 3 are already enough. Letz say we denote the 3 variables we will get by $\boldsymbol{a}_i, i=1,2,3$. Then i could include $\boldsymbol{a_2,a_3}$ in (1), $\boldsymbol{a_1,a_3}$ in (2) etc.. But dont we need at least 6 excluded variables to identify the given problem above? Im very greatful for your help ! – Druss2k Aug 27 '12 at 20:09
another thing im not quite sure of is if i really need at least 6 excluded variables for (1), (2), (3) being identifiable or are 3 enough as well? i mean in the reduced forms of $y_1,y_2,y_3$ the endogenous explanatory variables are excluded anyway and to estimate the instrument in a 2SLS manner i add exact one exlcuded variable to each reduced form. then i can replace every endogenous variable by its 2SLS estimate? – Druss2k Aug 27 '12 at 23:59