Generalized Method of Moments (I)

GENERALIZED METHOD OF MOMENTS I THE GMM ESTIMATOR: The idea is to choose estimates of the parameters by setting sample moments to be close to population counterparts. To describe the underlying moment model and the GMM estimator, let β denote a p×1 parameter vector, wi a data observation with i = 1, ..., n, where n is the sample size. Let gi (β) = g(wi , β) be a m × 1 vector of functions of the data and parameters. The GMM estimator is based on a model where, for the true parameter value β0 the moment conditions E[gi (β0 )] = 0 are satisfied. The estimator is formed by choosing β so that the sample average of gi (β) is close to its zero population value. Let def ĝ(β) = n 1X gi (β) n i=1 denote the sample average of gi (β). Let  denote an m×m positive semi-definite matrix. The GMM estimator is given by β̂ = arg min ĝ(β)0 Âĝ(β). β That is β̂ is the parameter vector that minimizes the quadratic form ĝ(β)0 Âĝ(β). The GMM estimator chooses β̂ so the sample average ĝ(β) is close to zero. To see this let kgkAˆ = q ˆ which is a well defined norm as long as Aˆ is positive definite. g 0 Ag, Then since taking the square root is a strictly monotonic transformation, and since the minimand of a function does not change after it is transformed, we also have β̂ = arg min kĝ(β) − 0k . β Thus, in a norm corresponding to  the estimator β̂ is being chosen so that the distance between ĝ(β) and 0 is as small as possible. As we discuss further below, when m = p, so there are the same number of parameters as moment functions, β̂ will be invariant to  asymptotically. When m > p the choice of  will affect β̂. The acronym GMM is an abreviation for ”generalized method of moments,” refering to GMM being a generalization of the classical method moments. The method of moments is based on knowing the form of up to p moments of a variable y as functions of the parameters, i.e. on E[y j ] = hj (β0 ), (1 ≤ j ≤ p). The method of moments estimator β̂ of β0 is obtained by replacing the population mo­ ments by sample moments and solving for β̂, i.e. by solving n 1X (yi )j = hj (β̂), (1 ≤ j ≤ p). n i=1 Alternatively, for gi (β) = (yi − h1 (β), ..., yip − hp (β))0 , method of moments solves ĝ(β̂) = 0. This also means that β̂ minimizes ĝ(β)0 Âĝ(β) for any Â, so that it is a GMM estimator. GMM is more general in allowing moment functions of different form than yij − hj (β) and in allowing for more moment functions than parameters. One important setting where GMM applies is instrumental variables (IV) estimation. Here the model is yi = Xi0 β0 + εi , E[Zi εi ] = 0, where Zi is an m × 1 vector of instrumental variables and Xi a p × 1 vector of right-hand side variables. The condition E[Zi εi ] = 0 is often called a population ”orthogonality condition” or ”moment condition. ”Orthogonality” refers to the elements of Zi and εi being orthogonal in the expectation sense. The moment condition refers to the fact that the product of Zi and yi − Xi0 β has expectation zero at the true parameter. This moment condition motivates a GMM estimator where the moment functions are the vector of products of instrumental variables and residuals, as in gi (β) = Zi (yi − Xi0 β). The GMM estimator can then be obtained by minimizing ĝ(β)0 Âĝ(β). Because the moment function is linear in parameters there is an explicit, closed form for the estimator. To describe it let Z = [Z1 , ..., Zn ]0 , X = [X1 , ..., Xn ]0 , and y = (y1 , ..., yn )0 . In this example the sample moments are given by ĝ(β) = n X i=1 Zi (yi − Xi0 β)/n = Z 0 (y − Xβ)/n. The first-order conditions for minimization of ĝ(β)0 Âĝ(β) can be written as ˆ = X 0 Z AZ ˆ ˆ 0 (y − X β) ˆ 0 y − X 0 Z AZ ˆ 0 X β. 0 = X 0 Z AZ These assuming that X 0 Z ÂZ 0 X is nonsingular, this equation can be solved to obtain ˆ 0 X)−1 X 0 Z AZ ˆ 0 y. βˆ = (X 0 Z AZ This is sometimes referred to as a generalized IV estimator. It generalizes the usual two stage least squares estimator, where  = (Z 0 Z)−1 . Another example is provided by the intertemporal CAPM. Let ci be consumption at time i, Ri is asset return between i and i + 1, α0 is time discount factor, u(c, γ0 ) utility function, Zi observations on variables available at time i. First-order conditions for utility maximization imply that moment restrictions satisfied for gi (β) = Zi {Ri · α · uc (ci+1 , γ)/uc (ci , γ) − 1}. Here GMM is nonlinear IV; residual is term in brackets. No autocorrelation because of one-step ahead decisions (ci+1 and Ri known at time i + 1). Empirical Example: Hansen and Singleton (1982, Econometrica), u(c, γ) = cγ /γ (constant relative risk aversion), ci monthly, seasonally adjusted nondurables (or plus services), Ri from stock returns. Instrumental variables are 1, 2, 4, 6 lags of ci+1 and Ri . Find γ not significantly different than one, marginal rejection from overidentification test. Stock and Wright (2001) find weak identification. Another example is dynamic panel data. It is a simple model that is important starting point for microeconomic (e.g. firm investment) and macroeconomic (e.g. crosscountry growth) applications is E ∗ (yit |yi,t−1 , yi,t−2 , ..., yi0 , αi ) = β0 yi,t−1 + αi , where αi is unobserved individual effect and E ∗ (·) denotes a population regression. Let ηit = yit − E ∗ (yit |yi,t−1 , ..., yi0 , αi ). By orthogonality of residuals and regressors, E[yi,t−j ηit ] = 0, (1 ≤ j ≤ t, t = 1, ..., T ), E[αi ηit ] = 0, (t = 1, ..., T ). Let ∆ denote the first difference, i.e. ∆yit = yit −yi,t−1 . Note that ∆yit = β0 ∆yi,t−1 +∆ηit . Then, by orthogonality of lagged y with current η we have E[yi,t−j (∆yit − β0 ∆yi,t−1 )] = 0, (2 ≤ j ≤ t, t = 1, ..., T ). These are instrumental variable type moment conditions. Levels of yit lagged at least two period can be used as instruments for the differences. Note that there are different instruments for different residuals. There are also additional moment conditions that come from orthogonality of αi and ηit . They are E[(yiT − β0 yi,T −1 )(∆yit − β0 ∆yi,t−1 )] = 0, (t = 2, ..., T − 1). These are nonlinear. Both sets of moment conditions can be combined. To form big moment vector by ”stacking”. Let ⎛ ⎞ yi0 ⎜ .. ⎟ t gi (β) = ⎜ . ⎟ ⎝ ⎠ (∆yit − β∆yi,t−1 ), (t = 2, ..., T ), yi,t−2 ⎛ ⎜ giα (β) = ⎜ ⎝ ∆yi2 − β∆yi1 .. . ∆yi,T −1 − β∆yi,T −2 ⎞ ⎟ ⎟ (yiT ⎠ − βyi,T −1 ). These moment functions can be combined as gi (β) = (gi2 (β)0 , ..., giT (β)0 , giα (β)0 )0 . Here there are T (T −1)/2+(T −2) moment restrictions. Ahn and Schmidt (1995, Journal of Econometrics) show that the addition of the nonlinear moment condition giα (β) to the IV ones often gives substantial asymptotic efficiency improvements. Arellano and Bond approach: gi (β) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ∆yi,1 ∆yi,2 .. . ∆yi,t−1 ⎞ ⎟ ⎟ ⎟ (yit ⎟ ⎠ − βyi,t−1 ) Assumes that have representation yit = ∞ X atj yi,t−j + bαi . j=1 Hahn, Hausman, Kuersteiner approach: Long differences gi (β) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ yi0 yi2 − βyi1 .. . yi,T −1 − βyi,T −2 ⎞ ⎟ ⎟ ⎟ (yiT ⎟ ⎠ − yi1 − β(yi,T −1 − yi0 )] Has better small sample properties by getting most of the information with fewer moment conditions. IDENTIFICATION: Identification is essential for understanding any estimator. Unless parameters are identified, no consistent estimator will exist. Here, since GMM estimators are based on moment conditions, we focus on identification based on the moment functions. The parameter value β0 will be identified if there is a unique solution to ḡ(β) = 0, ḡ(β) = E[gi (β)]. If there is more than one solution to these moment conditions then the parameter is not identified from the moment conditions. One important necessary order condition for identification is that m ≥ p. When m < p, i.e. there are fewer equations to solve than parameters. there will typically be multiple solutions to the moment conditions, so that β0 is not identified from the moment conditions. In the instrumental variables case, this is the well known order condition that there be more instrumental variables than right hand side variables. When the moments are linear in the parameters then there is a simple rank condition that is necessary and sufficient for identification. Suppose that gi (β) is linear in β and let Gi = ∂gi (β)/∂β (which does not depend on β by linearity in β). Note that by linearity gi (β) = gi (β0 ) + Gi (β − β0 ). The moment condition is 0 = ḡ(β) = G(β − β0 ), G = E[Gi ] The solution to this moment condtion occurs only at β0 if and only if rank(G) = p. If rank(G) = p then the only solution to this equation is β − β0 = 0, i.e. β = β0 . If rank(G) < p then there is c 6= 0 such that Gc = 0, so that for β = β0 + c 6= β0 , ḡ(β) = Gc = 0. For IV G = −E[Zi Xi0 ] so that rank(G) = p is one form of the usual rank condition for identification in the linear IV seeting, that the expected cross-product matrix of instrumental variables and right-hand side variables have rank equal to the number of right-hand side variables. In the general nonlinear case it is difficult to specify conditions for uniqueness of the solution to ḡ(β) = 0. Global conditions for unique solutions to nonlinear equations are not well developed, although there has been some progress recently. Conditions for local identification are more straightforward. In general let G = E[∂gi (β0 )/∂β]. Then, assuming ḡ(β) is continuously differentiable in a neighborhood of β0 the condition rank(G) = p will be sufficient for local identification. That is, rank(G) = p implies that there exists a neighborhood of β0 such that β0 is the unique solution to ḡ(β) for all β in that neighborhood. Exact identification refers the case where there are exactly as many moment conditions as parameters, i.e. m = p. For IV there would be exactly as many instruments as righthand side variables. Here the GMM estimator will satisfy ĝ(β̂) = 0 asymptotically. When there is the same number of equations as unknowns, one can generally solve the equations, so a solution to ĝ(β) = 0 will exist asymptotically. The proof of this statement (due to McFadden) makes use of the first-order conditions for GMM, which are h i0 0 = ∂ĝ(β̂)/∂β Âĝ(β̂). The regularity conditions will require that both ∂ĝ(β̂)/∂β and  are nonsingular with probability approaching one (w.p.a.1), so the first-order conditions imply ĝ(β̂) = 0 w.p.a.1. This will be true whatever the weight matrix, so that β̂ will be invariant to ˆ the form of A. Overidentification refers to the case where there are more moment conditions than parameters, i.e. m > p. For IV this will mean more instruments than right-hand side variables. Here a solution to ĝ(β) = 0 generally will not exist, because this would solve √ more equations than parameters. Also, it can be shown that nĝ(β̂) has a nondegenerate asymptotically normal distribution, so that the probabability of ĝ(β̂) = 0 goes to zero. When m > p all that can be done is set sample moments close to zero. Here the choice of  matters for the estimator, affecting its limiting distribution. TWO STEP OPTIMAL GMM ESTIMATOR: When m > p the GMM esti­ mator will depend on the choice of weighting matrix Â. An important question is how to choose  optimally, to minimize the asymptotic variance of the GMM estimator. It turns p out that an optimal choice of Aˆ is any such that Aˆ −→ Ω−1 , where Ω is the asymptotic √ √ P variance of nĝ(β0 ) = ni=1 gi (β0 )/ n. Choosing Aˆ = Ω̂−1 to be the inverse of a con­ sistent estimator Ω̂ of Ω will minimize the asymptotic variance of the GMM estimator. This leads to a two-step optimal GMM estimator, where the first step is construction of Ω̂ and the second step is GMM with  = Ω̂−1 . The optimal  depends on the form of Ω. In general a central limit theorem will lead to Ω = n−→∞ lim E[nĝ(β0 )ĝ(β0 )0 ], when the limit exists. Throughout these notes we will focus on the stationary case where E[gi (β0 )gi+ (β0 )0 ] does not depend on i. We begin by assuming that E[gi (β0 )gi+ (β0 )0 ] = 0 for all positive integers . Then Ω = E[gi (β0 )gi (β0 )0 ]. In this case Ω can be estimated by replacing the expectation by a sample average and β0 by an estimator β̃, leading to Ω̂ = n 1X gi (β̃)gi (β̃)0 . n i=1 The β̃ could be obtained by GMM estimator by using a choice of  that does not depend on parameter estimates. For example, for IV β̃ could be the 2SLS estimator where  = (Z 0 Z)−1 . In the IV setting this Ω̂ has a heteroskedasticity consistent form. Note that for ε̃i = yi − Xi0 β̃, n 1X Zi Zi0 ε̃2i . n i=1 The optimal two step GMM (or generalized IV) estimator is then Ω̂ = β̂ = (X 0 ZΩ̂−1 Z 0 X)−1 X 0 ZΩ̂−1 Z 0 y. Because the 2SLS corresponds to a non optimal weighting matrix this estimator will generally have smaller asymptotic variance than 2SLS (when m > p). However, when homoskedasticity prevails, Ω̂ = σ̂ε2 Z 0 Z/n is a consistent estimator of Ω, and the 2SLS estimator will be optimal. The 2SLS estimator appears to have better small sample properties also, as shown by a number of Monte Carlo studies, which may occur because using a heteroskedasticity consistent Ω̂ adds noise to the estimator. When moment conditions are correlated across observations, an autocorrelation con­ sistent variance estimator estmator can be used, as in Ω̂ = Λ̂0 + L X =1 w L (Λ̂ + Λ̂0 ), Λ̂ = n− X i=1 gi (β̃)gi+ (β̃)0 /n. where L is the number of lags that are included and the weights w L are used to ensure Ω̂ is positive semi-definite. A common example is Bartlett weights w L = 1 − /(L + 1), as in Newey and West (1987). It is beyond the scope of these notes to suggest choices of L. √ A consistent estimator Vˆ of the asymptotic variance of n(β̂ − β0 ) is needed for asymptotic inference. For the optimal  = Ω̂−1 a consistent estimator is given by V̂ = (Ĝ0 Ω̂−1 Ĝ)−1 , Ĝ = ∂ĝ(β̂)/∂β. One could also update the Ω̂ by using the two step optimal GMM estimator in place of β̃ in its computation. The value of this updating is not clear. One could also update the  in the GMM estimator and calculate a new GMM estimator based on the update. This iteration on Ω̂ appears to not improve the properties of the GMM estimator very much. A related idea that is important is to simultaneously minimize over β in Ω̂ and in the moment functions. This is called the continuously updated GMM estimator (CUE). For example, when there is no autocorrelation, for Ω̂(β) = Pn i=1 gi (β)gi (β)0 /n the CUE is β̂ = arg min ĝ(β)0 Ω̂(β)−1 ĝ(β). β The asymptotic distribution of this estimator is the same as the two step optimal GMM estimator but it tends to have smaller bias in the IV setting, as will be discussed below. It is generally harder to compute than the two-step optimal GMM. ASYMPTOTIC THEORY FOR GMM: We mention precise results for the i.i.d. case and give intuition for the general case. We begin with a consistency result: If the data are i.i.d. and i) E[gi (β)] = 0 if and only if β = β0 (identification); ii) the GMM minimization takes place over a compact set B containing β0 ; iii) gi (β) is p continuous at each β with probability one and E[supβ∈B kgi (β)k] is finite; iv)  → A p positive definite; then β̂ → β0 . See Newey and McFadden (1994) for the proof. The idea is that, for g(β) = E[gi (β)], by the identification hypothesis and the continuity conditions g(β)0 Ag(β) will be bounded away from zero outside any neighborhood N of β0 . Then by the law of large numbers p ˆ 0 Aˆ ˆg(β). But, g( ˆg(β̂) ≤ g(β ˆ 0) → and iv), so will ĝ(β)0 Aˆ 0 from the definition of ˆ β) ˆ 0 )0 Ag(β β̂ and the law of large numbers, so β̂ must be inside N with probability approaching one. The compact parameter set is not needed if gi (β) is linear, like for IV. Next we give an asymptotic normality result: p If the data are i.i.d., β̂ → β0 and i) β0 is in the interior of the parameter set over which minimization occurs; ii) gi (β) is continuously differentiable on a neighborhood N p of β0 iii) E[supβ∈N k∂gi (β)/∂βk] is finite; iv)  → A and G0 AG is nonsingular, for G = E[∂gi (β0 )/∂β]; v) Ω = E[gi (β0 )gi (β0 )0 ] exists, then √ d n(β̂ − β0 ) −→ N(0, V ), V = (G0 AG)−1 G0 AΩAG(G0 AG)−1 . See Newey and McFadden (1994) for the proof. Here we give a derivation of the asymptotic variance that is correct even if the data are not i.i.d.. By consistency of β̂ and β0 in the interior of the parameter set, with probability approaching (w.p.a.1) the first order condition 0 = Ĝ0 Âĝ(β̂), is satisfied, where Ĝ = ∂ĝ(β̂)/∂β. Expand ĝ(β̂) around β0 to obtain 0 = Ĝ0 Âĝ(β0 ) + Ĝ0 ÂḠ(β̂ − β0 ), where Ḡ = ∂ĝ(β̄)/∂β and β̄ lies on the line joining β̂ and β0 , and actually differs from row to row of Ḡ. Under regularity conditions like those above Ĝ0 ÂḠ will be nonsingular √ w.p.a.1. Then multiplying through by n and solving gives ³ ´−1 √ √ n(β̂ − β0 ) = − Ĝ0 ÂḠ Ĝ0  nĝ(β0 ). By an appropriate central limit theorem p p √ p d nĝ(β0 ) −→ N(0, Ω). Also we have Aˆ −→ ³ ´−1 A, Ĝ −→ G, Ḡ −→ G, so by the continuous mapping theorem, Ĝ0 ÂḠ (G0 AG)−1 G0 A. Then by the Slutzky lemma, √ d −1 n(β̂ − β0 ) −→ − (G0 AG) G0 AN(0, Ω) = N(0, V ). p Ĝ0  −→ The fact that A = Ω−1 minimizes the asymptotic varince follows from the Gauss Markov Theorem. Consider a linear model. E[Y ] = Gδ, V ar(Y ) = Ω. The asymptotic variance of the GMM estimator with A = Ω−1 is (G0 Ω−1 G)−1 . This is also the variance of generalized least squares (GLS) in this model. Consider an estmator δ̂ = (G0 AG)−1 G0 AY . It is linear and unbiased and has variance V . Then by the GaussMarkov Theorem, V − (G0 Ω−1 G)−1 is p.s.d.. We can also derive a condition for A to be efficient. The Gauss-Markov theorem says that GLS is the the unique minimum variance estimator, so that A is efficient if and only if (G0 AG)−1 G0 A = (G0 Ω−1 G)−1 G0 Ω−1 . Transposing and multiplying gives ΩAG = GB, where B is a nonsingular matrix. This is the condition for A to be optimal. TESTING IN GMM: An important test statistic for GMM is the test of overiden­ tifying restrictions that is given by T = nĝ(β̂)0 Ω̂−1 ĝ(β̂). When the moment conditions E[gi (β0 )] = 0 are satisfied then as the sample size grows we will have d T −→ χ2 (m − p). Thus, a test with asymptotic level α consists of rejecting if T ≥ q where q is the the 1 − α quantile of a χ2 (m − p) distribution. p Under the conditions for asymptotic normality, Ω nonsinglar, and Ω̂ → Ω, for an d efficient GMM estimator it follows that T −→ χ2 (m − p). See Newey and McFadden (1994) for a precise proof. Here we outline the proof. Let R denote a symmetric square root of the matrix Ω (i.e. RR = Ω) and let H = R−1 G. Using expansions similarly to the proof of asymptotic normality, we have √ √ √ √ ¯ n(β̂ − β0 ) = [I − G(Ĝ ¯ 0Ω ˆ −1 Ḡ)−1 Ĝ0 Ω̂−1 ] nĝ(β0 ) nĝ(β̂) = nĝ(β0 ) + G √ = [I − G(G0 Ω−1 G)−1 G0 Ω−1 ] nĝ(β0 ) + op (1), √ = R[I − H(H 0 H)−1 H 0 ]R−1 nĝ(β0 ) + op (1), ¯ is given above in the discussion of asymptotic normality and op (1) is a ran­ where G dom variable that converges in probability to zero. Then by R−1 ΩR−1 = I we have √ d R−1 nĝ(β0 ) −→ N(0, I) so that √ d nĝ(β̂) −→ R[I − H(H 0 H)−1 H 0 ]U, U ∼ N(0, I). p By consistency of Ω̂ and nonsingularity of Ω it follows that Ω̂−1 −→ Ω−1 . Then by the p continuous mapping theorem RΩ̂−1 R −→ I so that by I − H(H 0 H)−1 H 0 idempotent with rank m − p, d T −→ U 0 (I − H(H 0 H)−1 H 0 )U. By a standard result in multivariate normal theory, U 0 (I − H(H 0 H)−1 H 0 )U is distributed as χ2 (m − p), giving the result. This statistic will not detect all misspecification. It is only a test of overidentifying restrictions. Intuitively, p moments are ”used up” in estimating β. In the exactly iden­ tified case where m = p note that ĝ(β̂) = 0, so there is no way to test any moment conditions. An example is in IV estimation, where gi (β) = Zi (y − Xi0 β). There the test stastistic is ˆ ˆ −1 Z 0 ε/n, ˆ εˆ = y − X β. T = ε̂0 Z Ω One could also use an updated Ω̂, based on ε̂. This takes particular form in the indepen­ ˆ = εˆ0 ε/T ˆ the test statistic dent observations case. When homoskedasticity holds and Ω is ˆ ε0 ε, ˆ T = n · εˆ0 Z(Z 0 Z)−1 Z 0 ε/ˆ that is nR2 from regression of ε̂ on Z. When β̂ is the generalized IV estimator and the heteroskedasticity consistent Ω̂ = Pn i=1 Zi Zi0 ε̂2i /n is used in forming the test statistic it is T = e0 r̂(r̂0 r̂)−1 r̂0 e, the nR2 from regressing e = (1, ..., 1)0 on r̂ = [ε̂1 Z1 , ..., ε̂n Zn ]0 . It is also possible to test subsets of moment restrictions. Partition gi (β) = (gi1 (β)0 , gi2 (β)0 )0 and Ω conformably. One simple test is 2 ˆ − min nĝ 2 (β)0 Ω̂−1 Tˆ1 = min nĝ(β)0 Ω̂−1 g(β) 22 ĝ (β). β β The asymptotic distribution of this is χ2 (m1 ) where m1 is the dimension of ĝ 1 (β). Another version can be formed as ng̃10 Ω̃−1 g̃ 1 , 2 ˆ ˆ − Ω̂12 Ω̂−1 where g̃ 1 = ĝ 1 (β) 22 ĝ (β) and Ω̃ is an estimator of the asymptotic variance of √ 1 ng̃ . If m1 ≤ p then, except in any degenerate cases, a Hausman test based on the difference of the optimal two-step GMM estimator using all the moment conditions and using just ĝ2 (β) will be asymptotically equivalent to this test, for any m1 parameters. See Newey (1984, GMM Specification Testing, Journal of Econometrics.) We can also consider tests of a null hypothesis of the form H0 : s(β0 ) = 0, where s(β) is a q × 1 vector of functions. Let s(β) by a q × 1 vector of functions with q < p and rank(∂s(β0 )/∂β) = q, β̃ be a restricted GMM estimator β̃ = arg mins(β)=0 ĝ(β)0 Ω̂−1 ĝ(β), and G̃ = ∂ĝ(β̃)/∂β. Under the null hypotheses H0 : s(β) = 0 and the same conditions as for the overidentification test, d W ald : W = ns(β̂)0 [∂s(β̂)/∂β(Ĝ0 Ω̂−1 Ĝ)−1 ∂s(β̂)/∂β]−1 s(β̂) → χ2 (q), p ˆ −1 ĝ(β̂) − W → 0, SSR : nĝ(β̃)0 Ω̂−1 ĝ(β̃) − nĝ(β̂)0 Ω p LM : nĝ(β̃)0 Ω̂−1 G̃(G̃0 Ω̂−1 G̃)−1 G̃0 Ω̂−1 ĝ(β̃) − W → 0. Here SSR is like sum of squared residuals, LM is Lagrange Multiplier, and W is a Wald statistic. The asymptotic approximation often more accurate for SSR and LM than W. Equivalence also holds under sequence of local alternatives. ADDING MOMENT CONDITIONS: The optimality of the two step GMM estimator has interesting implications. One simple but useful implication is that adding moment conditions will also decrease (or at least not decrease) the asymptotic variance of the optimal GMM estimator. This occurs because the optimal weighting matrix for fewer moment conditions is not optimal for all the moment conditions. To explain further, suppose that gi (β) = (gi1 (β)0 , gi2 (β)0 )0 . Then the optimal GMM estimator for just the first set of moment conditions gi1 (β) is uses  = à (Ω̂1 )−1 0 0 0 ! , where Ω̂1 is a consistent estimator of the asymptotic variance of Pn i=1 √ gi1 (β0 )/ n. This Aˆ is not generally optimal for the entire moment function vector gi (β). For example, consider the linear regression model E[yi |Xi ] = Xi0 β0 . The least squares estimator is a GMM estimator with moment functions gi1 (β) = Xi (yi − Xi0 β). The conditional moment restriction implies that E[εi |Xi ] = 0 for εi = yi − Xi0 β0 . We can add to these moment conditions by using nonlinear functions of Xi as additional ”instrumental variables.” Let gi2 (β) = a(Xi )(yi − Xi0 β) for some (m − p) × 1 vector of functions of Xi . Then the optimal two-step estimator based on gi (β) = à Xi a(Xi ) ! (yi − Xi0 β) will be more efficient than least squares when there is heteroskedasticity. This estimator has the form of the generalized IV estimator described above where Zi = (Xi0 , a(Xi )0 )0 . It will provide no efficiency gain when homoskedasticity prevails. Also, the asymptotic variance estimator V̂ = (Ĝ0 Ω̂−1 Ĝ)−1 tends to provide a poor approximation to the vari­ ance of β̂. See Cragg (1982, Econometrica). Interesting questions here are what and how many functions to include in a(X) and how to improve the variance estimator. Some of these issues will be further discussed below. Another example is provided by missing data. Consider again the linear regression model, but now just assume that E[Xi εi ] = 0, i.e. Xi0 β0 may not be the conditional mean. Suppose that some of the variables are sometimes missing and Wi denote the variables that are always observed. Let ∆i denote a complete data indicator, equal to 1 if (yi , Xi ) are observed and equal to 0 if only Wi is observed. Suppose that the data is missing completely at random, so that ∆i is independent of Wi . Then there are two types of moment conditions available. One is E[∆i Xi εi ] = 0, leading to a moment function of the form gi1 (β) = ∆i Xi (yi − Xi0 β). GMM for this moment condition is just least squares on the complete data. The other type of moment condition is based on Cov(∆i , a(Wi )) = 0 for any vector of functions a(W ), leading to a moment function of the form gi2 (η) = (∆i − η)a(Wi ). One can form a GMM estimator by combining these two moment conditions. This will generally be asymptotically more efficient than least squares on the complete data when Yi is included in Wi . Also, it turns out to be an approximately efficient estimator in the presence of missing data. As in the previous example, the choice of a(W ) is an interesting question. Although adding moment conditions often lowers the asymptotic variance it may not improve the small sample properties of estimators. When endogeneity is present adding moment conditions generally increases bias. Also, it can raise the small sample variance. Below we discuss criteria that can be used to evaluate these tradeoffs. One setting where adding moment conditions does not lower asymptotic efficiency is when those the same number of additional parameters are also added. That is, if the second vector of moment functions takes the form gi2 (β, γ) where γ has the same dimension as gi2 (β, γ) then there will be no efficiency gain for the estimator of β. This situation is analogous to that in the linear simultaneous equations model where adding exactly identified equations does not improve efficiency of IV estimates. Here adding exactly identified moment functions does not improve efficiency of GMM. Another thing GMM can be used for is derive the variance of two step estimators. Consider a two step estimator β̂ that is formed by solving n 1X gi2 (β, γ̂) = 0, n i=1 where γ̂ is some first step estimator. If γ̂ is a GMM estimator solving Pn 1 i=1 gi (γ)/n =0 ˆ γ̂) is a (joint) GMM estimator for the triangular moment conditions then (β, gi (β, γ) = The asymptotic variance of à gi1 (γ) gi2 (β, γ) ! . √ n(β̂ − β0 ) can be calculated by applying the general GMM formula to this triangular moment condition. When E[∂gi2 (β0 , γ0 )/∂γ] = 0 the asymptotic variance of β̂ will not depend on esti­ mation of γ, i.e. will the same as for GMM based on gi (β) = gi2 (β, γ0 ). A sufficient condition for this is that E[gi2 (β0 , γ)] = 0 for all γ in some neighborhood of γ0 . Differentiating this identity with respect to γ, and assuming that differentiation inside the expectation is allowed, gives E[∂gi2 (β0 , γ0 )/∂γ] = 0. The interpretation of this is that if consistency of the first step estimator does not affect consistency of the second step estimator, the second step asymptotic variance does not need to account for the first step. CONDITIONAL MOMENT RESTRICTIONS: Often times the moment re­ strictions on which GMM is based arise from conditional moment restrictions. Let ρi (β) = ρ(wi , β) be a r × 1 residual vector. Suppose that there are some instruments zi such that the conditional moment restrictions E[ρi (β0 )|zi ] = 0 are satisfied. Let F (zi ) be an m × r matrix of instrumental variables that are functions of zi . Let gi (β) = F (zi )ρi (β). Then by iterated expectations, E[gi (β0 )] = E[F (zi )E[ρi (β0 )|zβi ]] = 0. Thus gi (β) satisfies the GMM moment restrictions, so that one can form a GMM esti­ mator as described above. For moment functions of the form gi (β) = F (zi )ρi (β) we can think of GMM as a nonlinear instrumental variables estimator. The optimal choice of F (z) can be described as follows. Let D(z) = E[∂ρi (β0 )/∂β|zi = z] and Σ(z) = E[ρi (β0 )ρi (β0 )0 |zi = z]. The optimal choice of instrumental variables F (z) is F ∗ (z) = D(z)0 Σ(z)−1 . This F ∗ (z) is optimal in the sense that it minimizes the asymptotic variance of a GMM estimator with moment functions gi (β) = F (zi )ρi (β) and a weighting matrix A. To show this optimality let Fi = F (zi ), Fi∗ = F ∗ (zi ), and ρi = ρi (β0 ). Then by iterated expectations, for a GMM estimator with moment conditions gi (β) = F (zi )ρi (β), G = E[Fi ∂ρi (β0 )/∂β] = E[Fi D(zi )] = E[Fi Σ(zi )Fi∗0 ] = E[Fi ρi ρ0i Fi∗0 ]. Let hi = G0 AFi ρi and h∗i = Fi∗ ρi , so that ∗0 0 0 G0 AG = G0 AE[Fi ρi h∗0 i ] = E[hi hi ], G AΩAG = E[hi hi ]. Note that for Fi = Fi∗ we have G = Ω = E[h∗i h∗i 0 ]. Then the difference of the asymptotic variance for gi (β) = Fi ρi (β) and some A and the asymptotic variance for gi (β) = Fi∗ ρi (β) is −1 (G0 AG)−1 G0 AΩAG(G0 AG)−1 − (E[h∗i hi∗0 ]) −1 = (E[hi h∗i 0 ]) −1 {E[hi h0i ] − E[hi h∗i 0 ] (E[h∗i h∗i 0 ]) −1 E[h∗i h0i ]} (E[h∗i h0i ]) . The matrix in brackets is the second moment matrix of the population least squares projection of hi on h∗i and is thus positive semidefinite, so the whole matrix is positive semi-definite. Some examples help explain the form of the optimal instruments. Consider the linear regression model E[yi |Xi ] = Xi0 β0 and let ρi (β) = yi − Xi0 β, εi = ρi (β0 ), and σi2 = E[ε2i |Xi ] = Σ(zi ). Here the instruments zi = Xi . A GMM estimator with mo­ ment conditions F (zi )ρi (β) = F (Xi )(yi − Xi0 β) is the estimator described above that will be asymptotically more efficient than least squares when F (Xi ) includes Xi . Here ∂ρi (β)/∂β = −Xi0 , so that the optimal instruments are Fi∗ = −Xi . σi2 Here the GMM estimator with the optimal instruments in the heteroskedasticity corrected generalized least squares. Another example is a homoskedastic linear structural equation. Here again ρi (β) = yi − Xi0 β but now zi is not Xi and E[ε2i |zi ] = σ 2 is constant. Here D(zi ) = −E[Xi |zi ] is the reduced form for the right-hand side variables. The optimal instruments in this example are Fi∗ = −D(zi ) . σ2 Here the reduced form may be linear in zi or nonlinear. For a given F (z) the GMM estimator with optimal A = Ω−1 corresponds to an approximation to the optimal estimator. For simplicity we describe this interpretation for r = p = 1. Note that for gi = Fi ρi it follows similarly to above that G = E[gi h∗0 i ], so that G0 Ω−1 = E[h∗i gi0 ](E[gi gi0 ])−1 . That is G0 Ω−1 are the coefficients of the population projection of h∗i on gi . Thus we can interpret the first order conditions for GMM 0 = Ĝ0 Ω̂−1 ĝ(β̂) = Ĝ0 Ω̂−1 n X Fi ρi (β)/n, i=1 can be interpreted as an estimated mean square approximation to the first order condi­ tions for the optimal estmator 0= n X i=1 Fi∗ ρi (β)/n. (This holds for GMM in other models too). One implication of this interpretation is that if the number and variety of the elements of F increases in such a way that linear combinations of F can approximate any function arbitrarily well then the asymptotic variance for GMM with optimal A will approach the optimal asymptotic variance. To show this, recall that m is the dimension of Fi and let the notation Fim indicate dependence on m. Suppose that for any a(z) with E[Σ(zi )a(zi )2 ] finite there exists m × 1 vectors π m such that as m −→ ∞ E[Σ(zi ){a(zi ) − π m0 Fim }2 ] −→ 0. For example, when zi is a scalar the nonnegative integer powers of a bounded monotonic m0 −1 transformation of zi will have this property. Then it follows that for hm i = ρi Fi Ω G 2 ∗ m0 m 2 2 ∗ m0 m 2 E[{h∗i − hm i } ] ≤ E[{hi − ρi Fi π } ] = E[ρi {Fi − Fi π } ] = E[Σ(zi ){Fi∗ − Fim0 π m }2 ] −→ 0. ∗ m m0 ∗ ∗0 Since hm i converges in mean square to hi , E[hi hi ] −→ E[hi hi ], and hence m0 −1 −1 −→ (E[h∗i h∗0 (G0 Ω−1 G)−1 = (G0 Ω−1 E[gi gi0 ]Ω−1 G)−1 = (E[hm i hi ]) i ]) . Because the asymptotic variance is minimzed at h∗i the asymptotic variance will ap­ ∗ proach the lower bound more rapidly as m grows than hm i approaches hi . In practice this may mean that it is possible to obtain quite low asymptotic variance with relatively few approximating functions in Fim . An important issue for practice is the choice of m.