Abstract
For both consistency and asymptotic normality of the GMM estimator it is not necessary to assume that \(\hat \theta \) precisely minimizes the GMM objective function (2.1.6). Andrews (1997) points out that for Theorem 2 (consistency) \(\hat \theta \) is required to be within Op(1) of the global minimum and for Theorem 3 (asymptotic normality), \(\hat \theta \) is required to be within Op(n-0.5), where Xn = Op(an) conveniently abbreviates plimXn/an =0 (cf. Amemiya, 1985, p. 89). The estimator \(\hat \theta \) is usually obtained by iterative numerical optimization methods like the Newton-Raphson algorithm (cf. Amemiya, 1985, ch. 4.4). Starting from any value of the parameter space this procedure produces a sequence of estimates \(\hat \theta \)j ( j = 0,1,2,…) which hopefully converges to the global minimum of the objective function. A typical Newton-Raphson iteration to the solution of the minimization problem (2.1.6) has the form
Convergence to a global minimum is ensured by this algorithm if the objective function is convex which, however, should be the exception for many nonlinear models encountered in microeconometric applications as discussed in the previous chapter. Otherwise the iteration routines could run into a local minimum which renders the parameter estimators inconsistent and alters their asymptotic distribution. To circumvent this problem Andrews (1997) proposes an optimization algorithm which guarantees consistency and asymptotic normality of the resulting GMM estimators provided that r > q holds. Andrews’ method is described in detail in the next section.
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© 2001 Springer-Verlag Berlin Heidelberg
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Inkmann, J. (2001). Computation of GMM Estimators. In: Conditional Moment Estimation of Nonlinear Equation Systems. Lecture Notes in Economics and Mathematical Systems, vol 497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56571-7_4
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DOI: https://doi.org/10.1007/978-3-642-56571-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41207-6
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