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Some simultaneous equations models require different instrument variables for different equations. In those cases, identification might require checking each equation separately. Moreover, traditional Three-Stage Least Squares (3SLS) estimator (for definition, see Cameron and Trivedi 〔Cameron, C. and P. K. Trivedi (2005), "Microeconometrics: Methods and Applications," Cambridge University Press, New York.〕 p. 214) might not be consistent.〔Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.〕 In addition, General Method of Moments (GMM) version of 3SLS (GMM with weight matrix I⊗Ω ̂, Ω ̂ being estimate of variance of system errors, see 〔 p. 214) is likely to be not efficient. Thus, extra care might be warranted in estimation of such models. Examples in which the same set of instruments do 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 ).〔 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(uj'' )=0 for i=1,2 and j=1,2. Now we assume ''E(u1 )''=0 but ''E(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()'' = 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.〔 Also, assumptions necessary for GMM 3SLS to be asymptotically efficient, particularly “homoscedastic error” assumption, are likely to be violated in this setting.〔 Thus, in light of efficiency, optimal GMM should be used. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Different Instruments for Different Equations」の詳細全文を読む スポンサード リンク
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