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Econometrics

Econometrics. Lecture Notes Hayashi, Chapter 4d Multiple Equation GMM: Common Coefficients. Assumptions. Linearity: y im = z im ’ d + e im d is L x 1vector of common coefficients. y i = z i ’ d + e i. Assumptions.

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Econometrics

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  1. Econometrics Lecture Notes Hayashi, Chapter 4d Multiple Equation GMM: Common Coefficients

  2. Assumptions • Linearity: yim = zim’d + eim • dis Lx1vector of common coefficients. • yi = zi’d + ei

  3. Assumptions • Ergodic Stationarity: {wi} = {wi1,wi2,…,wiM} is jointly stationary and erogodic, where wim = {yim,zim,xim}. • xim is Kmx1 instruments for the m-th equation.

  4. Assumptions • Orthogonality Condition: • E(ximeim) = 0 (m=1,2,…M). That is, xim is predetermined for each equation m. • E(gi) = 0, where

  5. Assumptions • E(gi) = 0 Sxzd = sxy

  6. Assumptions • Rank Condition: For each equation m (m=1,2,…,M), Sxz(mKmxL) is of full column rank. • Asymptotic Normality:gi is a m.d.s. with finite 2nd moments. That is, E(gigi’) is nonsingular.

  7. Assumptions

  8. Generalized Method of Moments • Let gn(d) = 1/n igi = sxy-Sxzd

  9. Generalized Method of Moments • The consistent GMM estimator of d isdGMMW = (Sxz’WSxz)-1Sxz’W sxyfor any symmetric positive denitie matrix W • Bias (Sampling Error): dGMMW-d = (Sxz’WSxz)-1Sxz’Wgn(d) • J Statistic:J(dGMMW,W) = n gn(dGMMW)Wgn(dGMMW)d 2(mKm-L)

  10. Generalized Method of Moments • Consistent estimate of Avar(dGMMW):(Sxz’WSxz)-1Sxz’W Ŝ WSxz (Sxz’WSxz)-1

  11. Generalized Method of Moments • Efficient GMM estimator of d is obtained by setting W = Ŝ-1 : • dGMM = (Sxz’Ŝ-1Sxz)-1 Sxz’Ŝ-1sxy • Est(Avar(dGMM)) = (Sxz’Ŝ-1Sxz)-1 • J(dGMM,Ŝ-1) = n gn(dGMM) Ŝ-1gn(dGMM) d 2(mKm-L)

  12. Special Case: FIVE • Under conditional homoscedasticity, dFIVE = dGMM with Ŝ is defined by

  13. Special Case: 3SLS • If xi = xi1= xi2=…= xiM (common instruments) and under conditional homoscedasticity, d3SLS = dFIVE with Ŝ is defined by

  14. Special Case: SUR or RE • If, in addition, all regressors are predetermined and satisfy cross orthogonalities: xi = union of (zi1, zi2,…, ziM), Then dSUR = d3SLS. • This is also called Random-Effects Estimator: dRE = dSUR = d3SLS.

  15. Special Case: SUR or RE

  16. Special Case: Pooled OLS • For the SUR estimator, we obtain Ŝ using the estimated smh from the equation-by-equation OLS:

  17. Special Case: Pooled OLS • If instead the following Ŝ is used, then dSUR is the pooled OLS estimator: • This is a GMM estimator with a non-optimal choice of W.

  18. Special Case: Pooled OLS

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