2.2. GENERALIZED METHOD OF MOMENTS

where V = σ2Q−1. If you let D = E(∂(zt²t)/∂β0) = Q, the GMM covariance matrix V can be expressed as V = σ2Q−1 = [D0W−1D]−1.

The Þrst equality is the standard OLS calculation for the covariance matrix and the second equality follows from the properties of (2.35).

You would never do OLS by minimizing (2.35) since to get the

ˆ −1

weighting matrix W , you need an estimate of β which is what you want in the Þrst place. But this is what you do in the generalized environment.

Generalized environment. Suppose you have an economic theory that relates qt to a vector xt. The theory predicts the set of orthogonality conditions

E[zt²t(qt, xt, β)] = 0,

Let R be a k×q restriction matrix and r is a q dimensional vector of constants. Consider the q linear restrictions Rβ = r on the coe cient vector. The Wald statistic has an asymptotic chi-square distribution

(19)under the null hypothesis that the restrictions are true

It follows that the linear restrictions can be tested by comparing the Wald statistic against the chi-square distribution with q degrees of freedom.

GMM also allows you to conduct a generic test of a set of overidentifying restrictions. The theory predicts that there are as many orthogonality conditions, n, as is the dimensionality of wt. The parameter vector β is of dimension k < n so actually only k linear combinations of the orthogonality conditions are set to zero in estimation. If the theoretical restrictions are true, however, the remaining n − k orthogonality conditions should di er from zero only by chance. The

minimized value of the GMM objective function, obtained by evaluat-

ˆ 2 ing the objective function at β, turns out to be asymptotically χn−k

under the null hypothesis that the model is correctly speciÞed.

2.3Simulated Method of Moments

Under GMM, you chose β to match the theoretical moments to sample moments computed from the data. In applications where it is di cult or impossible to obtain analytical expressions for the moment conditions E(wt) they can be generated by numerical simulation. This is the simulated method of moments (SMM) proposed by Lee and Ingram [92] and Du e and Singleton [40].

In SMM, we match computer simulated moments to the sample moments. We use the following notation.

β is the vector of parameters to be estimated.

{qt}Tt=1 is the actual time-series data of length T. Let q0 = (q1, q2, . . . , qT ) denote the collection of the observations.

{q˜i(β)}Mi=1 is a computer simulated time-series of length M which is generated according to the underlying economic theory. Let q˜0(β) = (˜q1(β), q˜2(β), . . . , q˜M (β)) denote the collection of these M observations.

h(qt) is some vector function of the data from which to simulate the moments. For example, setting h(qt) = (qt, qt2, qt3)0 will pick o the Þrst three moments of qt.

HT (q) = T1 PTt=1 h(qt) is the vector of sample moments of qt.

ˆ ˆ 1 Pm j+1 ˆ ˆ0

WT = Ω0 + T j=1(1 − T )(Ωj + Ωj) is the Newey-West estimate of the long-run covariance matrix of ut.

gT,M (β) = HT (q) − HM (q˜(β)) is the deviation of the sample moments from the simulated moments.

The SMM estimator is that value of β that minimizes the quadratic distance between the simulated moments and the sample moments

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