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2.2. GENERALIZED METHOD OF MOMENTS 352.2 Generalized Method of MomentsOLS can be viewed as a special case of the generalized method of moments(GMM) estimator studied by Hansen [70]. Since you are presumablyfamiliar with OLS, you can build your intuition about GMMby Þrst thinking about using it to estimate a linear regression. Aftergetting that under your belt, thinking about GMM estimation in morecomplicated and possibly nonlinear environments is straightforward.OLS and GMM. Suppose you want to estimate the coefficients inthe regressionq t = z 0 t β + ² t, (2.31)where β is the k-dimensional vector of coefficients, z t is a k-dimensionaliidvector of regressors and ² t ∼ (0, σ 2 )and(q t ,z t ) are jointly covariancestationary. The OLS estimator of β is chosen to minimize1TTXt=1² 2 t = 1 T= 1 TTXt=1TX(q t − β 0 z t )(q t − z 0 tβ)t=1q 2 t − 2β 1 TTXt=1z t q t + β 0 1TTXt=1(z t z 0 t )β. (2.32)When you differentiate (2.32) with respect to β and set the result tozero, you get the Þrst-order conditions,− 2 TTX|t=1{z }(a)TXz t ² t = −2 1 T t q t )+2βt=1(z 1 TTX| {zt=1}(b)(z t z 0 t ) =0. (2.33)If the regression is correctly speciÞed, the Þrst-order conditions form aset of k orthogonality or ‘zero’ conditions that you used to estimate β.These orthogonality conditions are labeled (a) in (2.33). OLS estimationis straightforward because the Þrst-order conditions are the set ofk linear equations in k unknowns labeled (b) in (2.33) which are solvedby matrix inversion. 7 Solving (2.33) for the minimizer ˆβ, you get, ⇐(16) (last7 In matrix notation, we usually write the regression as q = Zβ + ² where qis the T-dimensional vector of observations on q t , Z is the T × kdimensionalmatrix of observations on the independent variables whose t-th row is z 0 t, β is thek-dimensional vector of parameters that we want to estimate, ² is the T-dimensionalvector of regression errors, and ˆβ =(Z 0 Z) −1 Z 0 q.line of footnote)