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2.3. SIMULATED METHOD OF MOMENTS 39β is the vector of parameters to be estimated.{q t } T t=1 is the actual time-series data of length T .Letq0 =(q 1 ,q 2 ,...,q T )denote the collection of the observations.{˜q i (β)} M i=1 is a computer simulated time-series of length M which isgenerated according to the underlying economic theory. Let˜q 0 (β) =(˜q 1 (β), ˜q 2 (β),...,˜q M (β)) denote the collection of theseM observations.h(q t ) is some vector function of the data from which to simulate themoments. For example, setting h(q t )=(q t ,q 2 t ,q 3 t ) 0 will pick offthe Þrst three moments of q t .H T (q) = 1 TP Tt=1h(q t ) is the vector of sample moments of q t .H M (˜q(β)) = 1 MP Mi=1h(˜q i (β)) is the corresponding vector of simulatedmoments where the length of the simulated series is M.u t = h(q t ) − H T (q) ish in deviation from the mean form.ˆΩ 0 = 1 TP Tt=1u t u 0 t is the sample short-run variance of u t .ˆΩ j = 1 TP Tt=1u t u 0 t−j is the sample cross-covariance matrix of u t .Ŵ T = ˆΩ 0 + 1 P mj=1(1 − j+1TT)( ˆΩ j + ˆΩ 0 j) is the Newey-West estimate ofthe long-run covariance matrix of u t .g T,M(β) =H T (q) − H M (˜q(β)) is the deviation of the sample momentsfrom the simulated moments.The SMM estimator is that value of β that minimizes the quadraticdistance between the simulated moments and the sample momentsg T,M (β) 0 h W −1T,MigT,M (β), (2.42)where W T,M = h³ 1+ T M´WTi.LetˆβS be SMM estimator. It is asymptoticallynormally distributed with√T (ˆβS − β) D → N(0, V S ),