nonlineargmmslides

# nonlineargmmslides - Nonlinear GMM Eric Zivot November 2...

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Nonlinear GMM Eric Zivot November 2, 2009

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Nonlinear GMM estimation occurs when the K GMM moment conditions g ( w t , θ ) a renon l inea rfunct ionso fthe p model parameters θ . The moment conditions g ( w t , θ ) may be K p nonlinear functions sat- isfying E [ g ( w t , θ 0 )] = 0 Alternatively, for a response variable y t ,L explanatory variables z t , and K instruments x t , the model may de f neanon l inea rerro rterm ε t a ( y t , z t ; θ 0 )= ε t such that E [ ε t ]= E [ a
Note: if z t is endogenous then cannot use nonlinear least squares to esti- mate θ. Given x t orthogonal to ε t , de f ne g ( w t , θ 0 )= x t ε t = x t a ( y t , z t ; θ 0 ) so that E [ g ( w t , θ 0 )] = E [ x t ε t ]= E [ x t a ( y t , z t ; θ 0 )] = 0 de f nes the GMM orthogonality conditions.

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In general, the GMM moment equations produce a system of K nonlinear equations in p unknowns. Global identi f cation of θ 0 requires that E [ g ( w t , θ 0 )] = 0 E [ g ( w t , θ )] 6 = 0 for θ 6 = θ 0 Local Identi f cation requires that the K × p matrix G = E " g ( w t , θ 0 ) θ 0 # has full column rank p . Remark Global identi f cation does not require di f erentiability of g ( w t , θ 0 )
The sample moment condition for an arbitrary θ is g n ( θ )= n 1 n X t =1 g ( w t , θ ) If K = p, then θ 0 is apparently just identi f ed and the GMM objective function is J ( θ )= n g n ( θ ) 0 g n ( θ ) which does not depend on a weight matrix. The corresponding GMM estimator is then ˆ θ =argm in θ J ( θ ) and solves g n ( ˆ θ )=

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If K>p , then θ 0 is apparently overidenti f ed. Let ˆ W denote a K × K symmetric and p.d. weight matrix, possibly dependent on the data, such that ˆ W p W as n →∞ with W symmetric and p.d. The GMM estimator of θ 0 , denoted ˆ θ ( ˆ W ) , is de f ned as ˆ θ ( ˆ W )=argm in θ J ( θ , ˆ W )= n g n ( θ ) 0 ˆ Wg n ( θ ) The f rst order conditions are ∂J ( ˆ θ ( ˆ W ) , ˆ W ) θ =2 G n ( ˆ θ ( ˆ W )) 0 ˆ Wg n ( ˆ θ ( ˆ W )) = 0 G n ( ˆ θ ( ˆ W )) = g n ( ˆ θ ( ˆ W ))
The e cient GMM estimator uses ˆ W = ˆ S 1 such that ˆ S p S =avar( n g n ( θ 0 )) . If { g ( w t , θ 0 ) } is an ergodic-stationary MDS then S = E [ g ( w t , θ 0 ) g ( w t , θ 0 ) 0 ] If { g ( w t , θ 0 ) } is a serially correlated linear process then S =L R V = Γ 0 + X j =1 ( Γ j + Γ 0 j )= Ψ (1) ΣΨ (1) 0 Γ 0 = E [ g ( w t , θ 0 ) g 0 t ( w t , θ 0 )] , Γ j = E [ g ( w t , θ 0 ) g ( w t j , θ 0 ) 0 ] As with e cient GMM estimation of linear models, the e cient GMM estimator of nonlinear models may be computed using a two-step, iterated, or continuous updating estimator.

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Computation Since the GMM objective function is a quadratic form, the Gauss-Newton (GN) algorithm is well suited for f nding the minimum. The GN algorithm starts from a
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## This note was uploaded on 08/23/2010 for the course ECON 583 taught by Professor Zivot during the Fall '09 term at University of West Alabama-Livingston.

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nonlineargmmslides - Nonlinear GMM Eric Zivot November 2...

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