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2.2. GENERALIZED METHOD OF MOMENTS 37where z t is a vector of instrumental variables which may be differentfrom x t and ² t (q t ,x t , β) may be a nonlinear function of the underlyingk-dimensional parameter vector β and observations on q t and x t . 8 Toestimate β by GMM, let w t ≡ z t ² t (q t ,x t , β) where we now write the ⇐(17)vector of orthogonality conditions as E(w t )=0. Mimickingthestepsabove for GMM estimation of the linear regression coefficients, you’llwant to choose the parameter vector β to minimizeÃ1TTXt=1w t! 0Ŵ −1 Ã1T!TXw tt=1, (2.37)where Ŵ is a consistent estimator of the asymptotic covariance matrix1 Pof √ wt T. It is sometimes called the long-run covariance matrix. Youcannot guarantee that w t is iid in the generalized environment. It maybe serially correlated and conditionally heteroskedastic. To allow forthese possibilities, the formula for the weighting matrix isW = Ω 0 +∞Xj=1(Ω j + Ω 0 j ), (2.38)where Ω 0 =E(w t w 0 t)andΩ j =E(w t w 0 t−j). A popular choice for estimatingŴ is the method of Newey and West [114]Ŵ = ˆΩ 0 + 1 TmXµj=11 − j +1T ³ˆΩ j + ˆΩ 0 j´, (2.39)where ˆΩ 0 = 1 P Tt=1wT t w 0 t,andˆΩ j = 1 P Tt=j+1wTt w 0 t−j. The weightingfunction 1 − (j+1) is called the Bartlett window. WhenŴ Tconstructedby Newey and West, it is guaranteed to be positive deÞnite which isa good thing since you need to invert it to do GMM. To guaranteeconsistency, the Newey-West lag length (m) needsgotoinÞnity, but ataslowerratethanT . 9 You might try values such as m = T 1/4 .Totest8 Alternatively, you may be interested in a multiple equation system in which thetheory imposes parameter restrictions across equations so not only may the modelbe nonlinear, ² t could be a vector of error terms.9 Andrews [2] and Newey and West [115] offer recommendations for letting thedata determine m.⇐(18)(eq. 2.37)