Generalized Method of Moments

1 This is page iPrinter: Opaque this1 Generalized Method of IntroductionThis chapter describes Generalized Method of Moments (GMM) estima-tion for linear and non-linear models with applications in economics andfinance. GMM estimation was formalized by Hansen (1982), and since hasbecome one of the most widely used methods of estimation for modelsin economics andfinance. Unlike maximum likelihood estimation (MLE),GMM does not require complete knowledge of the distribution of the specified Moments derived from an underlying model are needed forGMM estimation . In some cases in which the distribution of the data isknown, MLE can be computationally very burdensome whereas GMM canbe computationally very easy. The log-normal stochastic volatility model isone example.

2 In models for which there are more moment conditions thanmodel parameters, GMM estimation provides a straightforward way to testthe specification of the proposed model. This is an important feature thatis unique to GMM chapter is organized as follows. GMM estimation for linear modelsis described in Section Section describes methods for estimating theefficient weight matrix. Sections and give examples of estimation andinference using theS+FinmetricsfunctionGMM. Section describes GMMestimation and inference for nonlinear models. Section provides numer-ous examples of GMM estimation of nonlinear models infinance includ-ing Euler equation asset pricing models, discrete-time stochastic volatilitymodels, and continous-time interest rate diffusion Generalized Method of MomentsThe theory and notation for GMM presented herein follows the excel-lent treatment given in Hayashi (2000).

3 Other good textbook treatments ofGMM at an intermediate level are given in Hamilton (1994), Ruud (2000),Davidson and MacKinnon (2004), and Greene (2004). The most compre-hensive textbook treatment of GMM is Hall (2005). Excellent surveys ofrecent developments in GMM are given in the special issues of theJournalof Business and Economic Statistics(1996, 2002). Discussions of GMMapplied to problems infinance are given in Ogaki (1992), Ferson (1995),Andersen and Sorensen (1996), Campbell, Lo and MacKinlay (1997), Jamesand Webber (2000), Cochrane (2001), Jagannathan and Skoulakis (2002),and Hall (2005). Single Equation Linear GMMC onsider the linear regression modelyt=z0t 0+ t,t=1,..,n( )whereztis anL 1 vector of explanatory variables, 0is a vector ofunknown coefficients and tis a random error term.

4 The model ( ) allowsfor the possibility that some or all of the elements ofztmay be correlatedwith the error term t, ,E[ztk t]6= [ztk i]6=0thenztkis called anendogenous variables then the least squares estimator of 0in ( ) is biasedand with the model ( ), it is assumed that there exists aK 1vector ofinstrumental variablesxtwhich may contain some or all of theelements the vector of unique and non-constantelements of{yt,zt,xt}.It is assumed that{wt}is a stationary and ergodicstochastic instrumental variablesxtsatisfy the set ofKorthogonality condi-tionsE[gt(wt, 0)] =E[xt t]=E[xt(yt z0t 0)] =0( )wheregt(wt, 0)=xt t=xt(yt z0t 0).Expanding ( ), gives the relation xy= xz 0where xy=E[xtyt]and xz=E[xtz0t].For identification of 0,itisrequired that theK LmatrixE[xtz0t]= xzbe of full that 0is the unique solution to ( ).

5 Note, ifK=L,then xzis invertible and 0may be determined using 0= 1xz Single Equation Linear GMM iiiA necessary condition for the identification of 0is theorder conditionK L( )which simply states that the number of instrumental variables must begreater than or equal to the number of explanatory variables in ( ). IfK=Lthen 0is said to be (apparently) just identified; ifK>Lthen 0is said to be (apparently) over-identified; ifK<Lthen 0is not word apparently in parentheses is used to remind the reader thatthe rank conditionrank( xz)=L( )must also be satisfied for the regression model ( ), the error terms are allowed to be condi-tionally heteroskedastic as well as serially correlated. For the case in which tis conditionally heteroskedastic, it is assumed that{gt}={xt t}is astationary and ergodic martingale difference sequence (MDS) satisfyingE[gtg0t]=E[xtx0t 2t]=SwhereSis a non-singularK Kmatrix.

6 The matrixSis the asymptoticvariance-covariance matrix of the sample Moments g=n 1 Pnt=1gt(wt, 0).This follows from the central limit theorem for ergodic stationary martin-gale difference sequences (see Hayashi page 106) n g=1 nnXt=1xt td N(0,S)whereavar( g)=Sdenotes the variance-covariance matrix of the limitingdistribution of n the case in which tis serially correlated and possibly conditionallyheteroskedastic as well, it is assumed that{gt}={xt t}is a stationaryand ergodic stochastic process that satisfies n g=1 nnXt=1xt td N(0,S)S= Xj= j= 0+ Xj=1( j+ 0j)where j=E[gtg0t j]=E[xtx0t j t t j].In the above,avar( g)=Sisalso referred to as thelong-run varianceof Definition of the GMM EstimatorThegeneralized Method of Moments (GMM) estimator of in ( ) is con-structed by exploiting the orthogonality conditions ( ).

7 The idea is to cre-ate a set of estimating equations for by making sample Moments matchiv1. Generalized Method of Momentsthe population Moments defined by ( ). The sample Moments based on( ) for an arbitrary value aregn( )=1nnXt=1g(wt, )=1nnXt=1xt(y z0t )= 1nPnt=1x1t(y z0t )..1nPnt=1xKt(y z0t ) These moment conditions are a set ofKlinear equations these sample Moments to the population momentE[xt t]=0gives the estimating equationsSxy Sxz =0( )whereSxy=n 1 Pnt=1xtytandSxz=n 1 Pnt=1xtz0tare the sample ( 0is just identified) andSxzis invertible then the GMMestimator of is =S 1xzSxywhich is also known as theindirect least squaresestimator. IfK>Lthen there may not be a solution to the estimating equations ( ). In thiscase, the idea is to try tofind that makesSxy Sxz as close to zero aspossible.

8 To do this, let Wdenote aK Ksymmetric and positive definite( ) weight matrix, possibly dependent on the data, such that Wp Wasn withWsymmetric and Then the GMM estimator of ,denoted ( W),is defined as ( W)=argmin J( , W)whereJ( , W)=ngn( )0 Wgn( )( )=n(Sxy Sxz )0 W(Sxy Sxz )SinceJ( , W) is a simple quadratic form in ,straightforward calculusmay be used to determine the analytic solution for ( W): ( W)=(S0xz WSxz) 1S0xz WSxy( )Asymptotic PropertiesUnder standard regularity conditions (see Hayashi Chapter 3), it can beshown that ( W)p 0 n ( W) 0 d N(0,avar( ( W))) Single Equation Linear GMM vwhereavar( ( W)) = ( 0xzW xz) 1 0xzWSW xz( 0xzW xz) 1( )A consistent estimate ofavar( ( W)),denoted[avar( ( W)),may be com-puted using[avar( ( W)) = (S0xz WSxz) 1S0xz W S WSxz(S0xz WSxz) 1( )where Sis a consistent estimate forS=avar( g).]]

9 The Efficient GMM EstimatorFor a given set of instrumentsxt,the GMM estimator ( W)isdefinefor an arbitrary positive definite and symmetric weight matrix variance of ( W) in ( ) depends on the chosen weight matrix natural question to ask is: What weight matrixWproduces thesmallest value ofavar( ( W))? The GMM estimator constructed with thisweight matrix is called theefficient GMM estimator. Hansen (1982) showedthat efficient GMM estimator results from setting W= S 1such that Sp this choice of W,the asymptotic variance formula ( ) reducestoavar( ( S 1)) = ( 0xzS 1 xz) 1( )of which a consistent estimate is[avar( ( S 1)) = (S0xz S 1 Sxz) 1( )The efficient GMM estimator is defined as ( S 1)=argmin ngn( )0 S 1gn( )which requires a consistent estimate , consistent estimation ofS, in turn, requires a consistent estimate of.]

10 To see this, consider the casein which tin ( ) is conditionally heteroskedastic so thatS=E[gtg0t]=E[xtx0t 2t].A consistent estimate ofShas the form1 S=1nnXt=1xtx0t 2t=1nnXt=1xtx0t(yt z0t )2such that p .Similar arguments hold for the case in whichgt=xt tisa serially correlated and heteroskedastic and MacKinnon (1993, section ) suggest using a simple degrees-of-freedom corrected estimate ofSthat replacesn 1in ( ) with (n k)toimprovethefinite sample performance of tests based on ( ).vi1. Generalized Method of MomentsTwo Step Efficient GMMThe two-step efficient GMM estimator utilizes the result that a consistentestimate of may be computed by GMM with an arbitrary positive definiteand symmetric weight matrix Wsuch that Wp ( W)denotesuch an estimate.