Econometrics

1
Econometrics

• Lecture Notes Hayashi, Chapter 4a
• Single Equation GMM Review

2
Assumptions

• Linearity yi zid ei
• Ergodic Stationarity wi yi,zi,xi is
  jointly stationary and erogodic. zi is Lx1 xi is
  Kx1.
• Orthogonality ConditionE(gi) E(xiei)
  Exi(yi-zid ) 0. That is, xi is
  predetermined.

3
Assumptions

• E(gi) 0 ? Sxzd sxy, where Sxz E(xizi) and
  sxy E(xiyi)
• Rank Condition E(xi zi) (KxL) is of full
  column rank.
• Asymptotic Normalitygi is a m.d.s. with finite
  2nd moments. That is, is nonsingular.

4
Generalized Method of Moments

• Define gn(d) 1/n ?igi sxy-Sxzd
• Let W be a KxK symmetric positive definite
  matrix, and J(d,W) n gn(d)Wgn(d).
• The consistent GMM estimator of d isdGMMW
  argmind J(d,W) dGMMW (SxzWSxz)-1SxzW sxy
• Bias dGMMW-d (SxzWSxz)-1SxzWgn(d)
• J(dGMMW,W) ?d ?2(K-L)

5
Large Sample Properties of GMM

• ?n (dGMMW - d) ?d N(0,Avar(dGMMW))
• Consistent estimate of Avar(dGMMW)(SxzWSxz)-1Sx
  zW S WSxz (SxzWSxz)-1where

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Large Sample Properties of GMM

• Efficient GMM estimator of d is obtained by
  setting W S-1
• dGMM (SxzS-1Sxz)-1 SxzS-1sxy
• Est(Avar(dGMM)) (SxzS-1Sxz)-1
• J(dGMM,S-1) n gn(dGMM) S-1gn(dGMM) ?d ?2(K-L)
  

7
Special Cases

• Under conditional homoscedasticity, d2SLS dIV
  dGMM.
• If endogenous regressors satisfy orthogonalities
  (that is, predetermined regressors), dOLS dIV.