3.4
Consistency
Recall from Chapter 1 that consistency proofs for the general sieve extremum esti-
mator exist under mild regularity conditions that allow for great generality in the
choice of sieves and for a variety of forms of dependence and heterogeneity to be present in the data; see e.g. Gallant (1987), White and Wooldrige (1991) and Chen (2007). This section establishes the consistency of the SNPII estimator. In particu- lar, the convergence in probability (and almost surely) of ˆθT,S, as defined in either
(3.2) or (3.3), to the parameter θ0 ∈ Θ. As explained in Chapter 2, we can make use of the existing consistency theorems and reduce our task to the verification of its assumptions. Basically, under appropriate regularity conditions, we will proceed to obtain the consistency of the SNPII estimator from (i) the uniform convergence of the criterion function QT,Sacross the sieves ΘT∀T ∈ N, and (ii) the identifiable
uniqueness ofθ0∈ Θ.
Note that in Section 3.1 we have not been precise as to which metric δB is
defined onB, requiring only that it induces Tychonoff’s topology on the set.12 One way of obtaining simpler proofs for the theorems that follow, consists of further restricting the class of metrics that are allowed to equipB. In particular, we impose the seemingly mild regularity condition (satisfied e.g. by both metrics in (3.1); see Proposition A.39) that the product metric δBbe Lipschitz weaker than the uniform
product metric (see Definitions A.36 and A.38).13
Assumption 3.4.1.∃k ∈ R+such that δB(β, β) ≤ k·supi∈N βi−βi Bi∀(β, β)∈ B × B where βi:= πi(β) ∈ Biandβi:= πi(β)∈ Bi are projections for every i∈ N.
In what follows we shall make use of some simplifying assumptions. Namely, we assume that the vectors of auxiliary estimators converge to their singleton limits uniformly over i ∈ N and across sieves {ΘT}T∈N ⊆ Θ. This assumption is useful
because it simplifies considerably the proofs and allows us to avoid a number of technical details that can easily distract us from the main consistency argument being conveyed. Nonetheless, it is important to keep in mind the following. First, uniform convergence of auxiliary estimators over i∈ N is not a necessary condition for the consistency results derived here. Second, albeit unnecessarily restrictive, uniform convergence of auxiliary estimators over i ∈ N can be easily derived (see Section 3.8 for a discussion of alternative sufficient conditions). Third, uniform convergence of auxiliary estimators over θ ∈ Θ is a typical assumption in indirect
12Convergence results are nonetheless meaningful. Given metrics{ · B
i}i∈Non{Bi}i∈N, any pair
of product metrics δBinducing Tychonoff’s topology onB is, by definition, topologically equivalent,
and convergence in one implies convergence in the other (see Definition A.34 and Remark A.35)).
13It is possible that all metrics inducing the product topology satisfy this requirement, in which
3 A T -Consistent and Asymptotically Gaussian SNPII Estimator
inference. Hence, it does not really carry any new elements. Furthermore, for the
reasons covered in Chapter 1, here we actually work under the weaker requirement of uniform convergence across sieves. Finally, uniform convergence jointly over i∈ N and across sieves{ΘT}T∈Ncan also be obtained (see again Section 3.8 for details).
Having emphasized the unnecessarily restrictive nature of this assumption, let us now proceed to make deliberate use of it.14
Assumption 3.4.2. (i) supi∈NβˆiT− β∗i(θ0)B
i
p
→ 0 as T → ∞;
(ii) supθ∈ΘTsupi∈Nβ˜T,si (θ) − β∗i(θ)B
i
p
→ 0 as T → ∞ ∀s ∈ {1, ..., S}.
In the context of indirect inference, identification ofθ0requires the fundamental condition that the product binding function β∗ be injective. This is ensured by having, for every pair (θ, θ) ∈ Θ × Θ, at least one i ∈ N such that the limit β∗i
of ˜βiT,s satisfies β∗i(θ) = β∗i(θ). Furthermore, to ensure the “transfer” of some
topological structure from Θ to the factor spacesBi (and ultimately toB), we shall
assume that the factor binding function β∗i is an open map ∀i ∈ N. Finally, to
guarantee the continuity of the limit criterion function Q∞ we also impose thatβ∗
be continuous on Θ ∀i ∈ N. Together, these conditions imply that the product binding functionβ∗is a homeomorphism on its range (see proof of Theorem 3.4.1). The parameter space Θ is thus homeomorphic (topologically equivalent) to a subset ofB. This conveys a natural sense in which inference on Θ can be conducted through inference onB.
Assumption 3.4.3. β∗i : Θ→ Bi is (i) an open map ∀i ∈ N; (ii) continuous on
Θ∀i ∈ N; and (iii) for every (θ, θ)∈ Θ × Θ, ∃ i ∈ N : β∗i(θ) = β∗i(θ).
Finally, as we shall see, given Assumption 3.4.3, a sufficient condition forθ0to be an identifiably unique minimizer (Definition A.52) of the limit criterion function
Q∞, is that μ∞ have a well-separated minimum at the origin. In particular, we
now require the uniform convergence of the deterministic sequence of criterion di- vergences {μT}T∈N to a limit criterion divergence μ∞ that satisfies an identifiable
uniqueness condition w.r.t. 0B∈ B where 0Bdenotes the origin ofB.
Assumption 3.4.4. The sequence{μT}T∈Nsatisfies supβ∈B|μT(β) − μ∞(β)|→0 as
T → ∞ for some continuous μ∞:B → R.
Assumption 3.4.5. infβ∈Sc(0
B,)⊂Bμ∞(β) − μ∞(0B)> 0 ∀ > 0.
Note here that Assumption 3.4.4 is concerned with the sure convergence of a sequence of well defined deterministic divergences {μT}T∈N. This assumption does
not address the probabilistic convergence of the random sequence{μT(ΔT,S(θ))}T∈N 14In Assumption 3.4.2 recall thatβ∗
3.5 Convergence Rate and Asymptotic Normality