Single-Equation GMM Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) Single-Equation GMM 1 / 36
Generalized Method of Moments Estimator Generalized Method of Moments Estimator 1 Distribution of the GMM Estimator 2 Estimation of the Optimal Weight Matrix 3 Nonlinear GMM 4 Hypothesis Testing 5 Conditional Moment Restrictions 6 7 Alternative Inference Procedures and Extensions Ping Yu (HKU) Single-Equation GMM 2 / 36
Generalized Method of Moments Estimator GMM Estimator Ping Yu (HKU) Single-Equation GMM 2 / 36
Generalized Method of Moments Estimator Linear GMM Estimator Suppose x 0 y i = i β + u i E [ x i u i ] 6 = 0 , E [ z i u i ] = 0 , then the moment conditions are � � �� = 0 , y i � x 0 E [ g ( w i , β )] = E z i i β (1) � � 0 . y i , x 0 i , z 0 where g ( � , � ) is a set of moment conditions, and w i = i Define the sample analog of (1) n n � � = 1 � � g n ( β ) = 1 g i ( β ) = 1 ∑ ∑ y i � x 0 Z 0 y � Z 0 X β z i i β . n n n i = 1 i = 1 When l > k , we cannot solve g n ( β ) = 0 exactly as intuitively shown in Figure 1. The idea of the GMM is to define an estimator which sets g n ( β ) "close" to zero. Ping Yu (HKU) Single-Equation GMM 3 / 36
Generalized Method of Moments Estimator 0 Figure: g n ( β ) = 0 Can Not Hold Exactly for Any β : k = 1 , l = 2 Ping Yu (HKU) Single-Equation GMM 4 / 36
Generalized Method of Moments Estimator continue... For some l � l weight matrix W n > 0, let J n ( β ) = n � g n ( β ) 0 W n g n ( β ) . This is a non-negative measure of the "length" of the vector g n ( β ) under the inner product h� , �i W n . - If W n = I l , then, J n ( β ) = n � g n ( β ) 0 g n ( β ) = n k g n ( β ) k 2 , the square of the Euclidean length. The GMM estimator minimizes J n ( β ) . The first order conditions for the GMM estimator are �b � ∂ = 2 n ∂ n ( b β ) W n g n ( b ∂β g 0 0 ∂β J n β β ) = � 1 � � 1 �� � Z 0 y � Z 0 X b n X 0 Z = � 2 n W n β , n so �� � � �� � 1 �� � � �� b X 0 Z Z 0 X X 0 Z Z 0 y β GMM = W n W n . (2) Ping Yu (HKU) Single-Equation GMM 5 / 36
Generalized Method of Moments Estimator More on W n and the GMM Estimator If l = k , then g n ( β ) = 0 . The GMM estimator reduces to the MoM estimator (the IV estimator) and W n is not required. While the estimator depends on W n , the dependence is only up to scale, for if W n is replaced by c W n for some c > 0, b β GMM does not change. In Section 4 of Chapter 7, β is identified as ( Γ 0 A Γ ) � 1 Γ 0 A λ = � �� � 1 � � � � � x i z 0 E [ z i z 0 i ] � 1 A E [ z i z 0 i ] � 1 E z i x 0 x i z 0 E [ z i z 0 i ] � 1 A E [ z i z 0 i ] � 1 E [ z i y i ] , so E E i i i there, W n is the sample analog of E [ z i z i ] � 1 A E [ z i z i ] � 1 . When A = E [ z i z i ] , we obtain the 2SLS estimator, that is, W n = ( Z 0 Z ) � 1 . From the FOCs of GMM estimation, we can see that although we cannot make g n ( β ) = 0 exactly, we could let some of its linear combinations, say B n g n ( β ) , be zero, where B n is a k � l matrix. � � p 1 n X 0 Z For a weight matrix W n , B n = W n . If W n � ! W > 0, and � � = G 0 , B n converges to B = G 0 W . So b p n X 0 Z 1 x i z 0 � ! E β is as if defined by a MoM i estimator such that B g n ( b β ) = 0 . Ping Yu (HKU) Single-Equation GMM 6 / 36
Distribution of the GMM Estimator Distribution of the GMM Estimator Ping Yu (HKU) Single-Equation GMM 7 / 36
Distribution of the GMM Estimator Distribution of the GMM Estimator Note that � 1 � � 1 � p n X 0 Z n Z 0 X ! G 0 WG W n � and � 1 � � 1 � d n X 0 Z p n Z 0 u ! G 0 W N ( 0 , Ω ) , W n � h i � � z i z 0 i u 2 g i g 0 where Ω = E = E with g i = z i u i . i i So �b � p d n β GMM � β � ! N ( 0 , V ) , where � � � 1 � �� � � 1 . G 0 WG G 0 W Ω WG G 0 WG V = (3) In general, GMM estimators are asymptotically normal with "sandwich form" asymptotic variances. It is easy to check this asymptotic distribution is the same as the MoM estimator defined by B g n ( b β ) = 0 . Ping Yu (HKU) Single-Equation GMM 8 / 36
Distribution of the GMM Estimator Optimal Weight Matrix A natural question is what is the optimal weight matrix W 0 that minimizes V . This turns out to be Ω � 1 (exercise). This yields the efficient GMM estimator: � � � 1 b X 0 Z Ω � 1 Z 0 X X 0 Z Ω � 1 Z 0 y , β = � � � 1 G 0 Ω � 1 G which has the asymptotic variance V 0 = . This corresponds to the linear combination matrix B = G 0 Ω � 1 . W 0 = Ω � 1 is usually unknown in practice, but it can be estimated consistently. h i � � � � σ 2 ∝ E u 2 = σ 2 , then Ω = E z i z 0 z i z 0 In the homoskedastic case, E i j z i i i suggesting the weight matrix W n = ( Z 0 Z ) � 1 , which generates the 2SLS estimator. So the 2SLS estimator is the efficient GMM estimator under homoskedasticity Ping Yu (HKU) Single-Equation GMM 9 / 36
Distribution of the GMM Estimator Optimal Weight Matrix - An Illustration Suppose E [ x i ] = E [ y i ] = µ and Cov ( x i , y i ) = 0. We try to find an efficient GMM estimator for µ - the common mean of x and y . The moment conditions are E [ g ( w i , µ )] = 0 , where w i = ( x i , y i ) 0 : � x i � µ � g ( w i , µ ) = . y i � µ Since µ appears in both moment conditions, we hope to find a better estimator than x or y which uses only one moment condition. Suppose b µ = ω x + ( 1 � ω ) y ; then the asymptotic distribution of b µ is � � p x + ( 1 � ω ) 2 σ 2 d 0 , ω 2 σ 2 n ( b µ � µ ) � ! N . y Minimizing the asymptotic variance, we have σ 2 y ω = . σ 2 x + σ 2 y The sample (of x and y ) with a larger variance is given a smaller weight, and the sample with a smaller variance is given a larger weight. Ping Yu (HKU) Single-Equation GMM 10 / 36
Distribution of the GMM Estimator continue... n o σ 2 x σ 2 σ 2 x , σ 2 y The asymptotic variance under this optimal weight is y � min . y σ 2 x + σ 2 Note that E [ g ( w i , µ ) g ( w i , µ ) 0 ] � 1 W 0 = ! � 1 ! E [( x i � µ ) 2 ] σ � 2 E [( x i � µ )( y i � µ )] 0 x = = . E [( y i � µ ) 2 ] σ � 2 0 E [( x i � µ )( y i � µ )] y So ! ( x � µ ) 2 + ( y � µ ) 2 J n ( µ ) = n � g n ( µ ) 0 W 0 g n ( µ ) = n , σ 2 σ 2 x y and b µ = ω x + ( 1 � ω ) y is the same as the weighted average above. In practice, σ 2 x and σ 2 y are unknown. In this simple example, they can be substituted by their sample analog. The next section deals with the general case. Ping Yu (HKU) Single-Equation GMM 11 / 36
Estimation of the Optimal Weight Matrix Estimation of the Optimal Weight Matrix Ping Yu (HKU) Single-Equation GMM 12 / 36
Estimation of the Optimal Weight Matrix Estimation of the Optimal Weight Matrix Given any weight matrix W n > 0, the GMM estimator b β GMM is consistent yet inefficient. For example, we can set W n = I l . In the linear model, a better choice is W n = ( Z 0 Z ) � 1 which corresponds to the 2SLS estimator. i b Given any such fist-step estimator, we can define the residuals b u i = y i � x 0 β GMM � � w i , b and moment equations b g i = z i b u i = g β GMM . Construct n β GMM ) = 1 ∑ g n ( b b g n = g i , n i = 1 g � b b = g i � g n , i and define ! � 1 ! � 1 n n 1 1 ∑ ∑ g � g �0 g 0 i � g n g 0 b i b g i b b W n = = . (4) i n n n i = 1 i = 1 p ! Ω � 1 , and GMM using W n as the weight matrix is asymptotically efficient. W n � Ping Yu (HKU) Single-Equation GMM 13 / 36
Estimation of the Optimal Weight Matrix An Alternative Estimator A common alternative choice is to set ! � 1 n 1 ∑ g 0 b g i b W n = , (5) i n i = 1 which uses the uncentered moment conditions. Since E [ g i ] = 0 , these two estimators are asymptotically equivalent under the hypothesis of correct specification. However, Alastair Hall (2000) has shown that the uncentered estimator is a poor choice. When constructing hypothesis tests, under the alternative hypothesis the moment conditions are violated, i.e. E [ g i ] 6 = 0 , so the uncentered estimator will contain an undesirable bias term and the power of the test will be adversely affected. Ping Yu (HKU) Single-Equation GMM 14 / 36
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