Efficiency Bounds Implied by Multiperiod Conditional Moment Restrictions

Asymptotic properties of Monte Carlo estimators of diffusion processes

“…, d}. This establishes (36) since the continuity of the pathwise integral with respect to the Lebesgue measure gives…”

“…If we apply the Doss transformation and sample Gaussian increments (W j t+(n+1)h − W j t+nh ) then the error UX N =X N −X of the Euler approximation with transformation can be expanded as 35 From Prohorov's theorem we know that tightness is sufficient for relative compactness (a family of probability measures is relatively compact if every sequence of probability measures contains a weakly convergent subsequence), which in turn, together with convergence of finite dimensional distributions, implies weak convergence of probability measures (Billingsley (1968) pp. [35][36][37][38][39][40]. A family of probability measures Π on a metric space is tight if for every > 0 there exist a compact set K such that P(K) ≥ 1 − for all P ∈ Π.…”

Optimal Inference in Dynamic Models with Conditional Moment Restrictions

“…In effect, we consider a wider class of estimators that contains GMM as a special case, and we determine the optimal estimator within this generalized class. The optimal estimating equation is quite explicit and readily computable, and we show that the estimator achieves the general efficiency bound derived by Hansen (1985) and Hansen, Heaton & Ogaki (1988).…”

“…Thus, our method is strictly more efficient than the usual GMM estimator with optimal weight matrix (or norm) of Hansen (1982), which is commonly used in econometrics and obtains as the special case where our time-varying instruments are made constant by replacing them by their unconditional mean. Our approach provides the means of computing estimators that reach the lower variance bound of Hansen (1985) and Hansen, Heaton & Ogaki (1988) in general, including in dynamic heteroskedastic models. The optimal instruments resemble those known from the i.i.d.…”

“…Firstly, in the restricted class where w t (θ) = w t , i.e., instruments do not depend on parameters, efficiency bounds were derived by Hansen (1985) (Lemma 4.3) and Hansen, Heaton & Ogaki (1988) (Theorem 4.2) for general settings not necessarily restricted to the martingale difference case. We return to the non-martingale case in Section 4 below.…”