Different Instruments for Different Equations
Some simultaneous equations models require different instrument variables for different equations. In those cases, identification might require checking each equation separately. Moreover, the traditional Three-Stage Least Squares (3SLS) estimator (for definition, see Cameron and Trivedi [1] p. 214) might not be consistent.[2] In addition, the General Method of Moments (GMM) version of 3SLS (GMM with weight matrix I⊗Ω ̂, Ω ̂ being the estimate of variance of system errors, see [1] p. 214) is likely to be not efficient. Thus, extra care might be warranted in the estimation of such models.
Examples in which the same set of instruments does not apply to every equation include fully recursive systems (y1’s equation contains only exogenous variables and yi’s right-hand side equation includes y1,…,y(i-1) for i=2, … , G, and error terms of different equations are uncorrelated ).[2] To illustrate the issue, consider the following example:
y1 = α1 * y2 + β11 * z1 + β12 * z2 + u1
y2 = α2 * y1 + β21 * z1 + β23 * z3 + u2
where y’s are endogenous and E[zi uj]=0 for i=1,2 and j=1,2. Now we assume E[z3 u1]=0 but E[z3 u2]≠0. The first equation is identified and can be estimated by Two-Stage Least Squares (2SLS) using (z1,z2,z3) as instruments. On the other hand, in the second equation, z3 is not a valid instrument and thus cannot be included in the set of IV. To estimate the second equation, we need extra instruments from outside of the current model. For example, suppose we have z4,z5 such that E[ziu2] = 0 for I = 4,5. Then, using (z1,z2,z4,z5) as instruments, 2SLS can estimate the second equation.
In the above illustration, parameters are estimated equation by equation. Using GMM, system estimation is feasible. However, the use of traditional 3SLS should be avoided because it is generally consistent only when all the instruments are uncorrelated with all the errors.[2] Also, assumptions necessary for GMM 3SLS to be asymptotically efficient, particularly the “homoscedastic error” assumption, are likely to be violated in this setting.[2] Thus, in light of efficiency, optimal GMM should be used.
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