DESCRIPCIÓN DE LA MICROSERIE

ond countable (Lemma A.25). By Lemma A.24 this implies that B(_{B) = ⊗}i∈NB(Bi)

where _⊗i∈NB(Bi) denotes the product σ-algebra. These algebras are thus used

interchangeably. Most importantly, the projection mappings πi : B → Bi are

B(B)/B(Bi)-measurable ∀i ∈ N (Corollary A.17).7 In Section 3.5 below, the fac-

tor spaces Bi are assumed to satisfy Bi ⊆ Rn, n ∈ N for every i ∈ N, so that

√

T -convergence and asymptotic normality results are specific to SNPII estimators

making use of infinitely many parametric auxiliary estimators.

3.2

The SNPII Estimator

Before attempting to analyze the properties of the SNPII estimator in detail, let us first define it more rigorously. Recall from Chapter 2 that the SNPII estimator is a map ˆθT,S: Ω→ ΘT satisfying, for fixed S∈ N,8

ˆ

θT,S∈ arg min

θ∈ΘTQT,S(θ) a.s. ∀T ∈ N, (3.2)

where QT,S : Ω× Θ → R is called the criterion function and ΘT ⊆ Θ as imposed

by Assumption 3.1.2. In what follows, conditions shall be imposed on the criterion functions QT,S: Θ× Ω → R and the sieves ΘT so as to guarantee that the arg min

set exists and that QT,Sconverges in some appropriate sense to a limit deterministic

criterion function Q∞: Θ→ R.9 When such conditions are too restrictive, then the

above definition can easily be relaxed to that of an approximate extremum estimator ˆ

θT,S satisfying, for fixed S∈ N,

QT,S(ˆθT,S)≤ inf

θ∈ΘTQT,S(θ) + Op(ηT), (3.3)

with ηT → 0 as T → ∞. Clearly, setting Op(ηT) = 0∀T ∈ N yields an exact sieve

extremum estimator. When furthermore the arg min set exists, then the extremum estimator is given by (3.2) above. Now, recall also that a fundamental feature of

and translation invariance which is sometimes unnecessary. More importantly, a norm inducing the product topology onB might not exist. The fact that δB inherits translation invariance from

the norms · Bi ∀ i will be used in the proof of Theorem 3.4.1.

7_{B(B)/B(A)-measurability of maps from a measurable space (A, B(A)) into (B, B(B)) is thus}

implied by the B(Bi)/B(A)-measurability of the projections πi◦ f : A → Bifor every i ∈ N (see

Corollary A.27).

8_{Please note the change in notation. For completeness, the SNPII estimator (and its criterion}

function) is now indexed by S. In Chapter 2 this notation was avoided for the sake of simplicity and to highlight the generality of the results which applied to sieve extremum estimators in general.

9_{Nothing is gained by letting Q}

T,Sbe defined only on the sieves ΘT, i.e. by letting QT,S: ΘT×

Ω→ R+

0, since an agreeing measurable extension is guaranteed to exist on Θ (see e.g. Stinchcombe

3 A T -Consistent and Asymptotically Gaussian SNPII Estimator the SNPII estimator in (3.2) or (3.3) is its appropriate definition as a minimizer of a divergence defined on the auxiliary parameter space_{B. In particular, let us define} the maps ˆ_βT : Ω→ B and ˜βT,S(θ) : Ω → B, ∀θ ∈ Θ. Each of these consists of a

vector of random variables called auxiliary estimators or auxiliary statistics indexed by i∈ N and taking values on the factor-spaces Bi. The first vector,

ˆ βT :=  ˆ β1T, ˆβ 2 T, ˆβ 3 T, ... 

collects those estimators ˆβiT : Ω → Bi that are functions of observed data xT.

Auxiliary estimators of interest should be simple to work with in applications and designed so as to possess desirable convergence properties. In particular, they should take values on well chosen (possibly finite dimensional compact) factor spacesBiso

that they do not suffer from the complications of estimation on large complex spaces. The second vector of auxiliary estimators,

˜ βT,S(θ) := ₁ S S  s=1 ˜ β1T,s(θ) , 1 S S  s=1 ˜ β2T,s(θ) , 1 S S  s=1 ˜ β3T,s(θ) , ... 

collects (for any givenθ ∈ Θ) averages of those estimators ˜βiT(θ) : Ω → Bi that are

functions of the “artificial” sequence of data ˜xs

T(θ) drawn from D(θ). These estima-

tors should have desirable properties ∀θ ∈ ΘT, T ∈ N. In particular, ˆβT should be

measurable and converge in suitable manner to a limit pointβ∗₀:=β∗_0,1, β∗_0,2, ...

in_{B. The random map ˜β}T,S: Ω× Θ → B, called empirical binding function, should

be measurable and converge in an appropriate fashion to a limit deterministic map β∗ _{: Θ → B, called the binding function β}∗ _:=_β∗

1, β∗2, ...



.10 Making use of the two vectors of auxiliary estimators ˆβT and ˜βT,S(θ), recall that the SNPII estimator’s

criterion function was defined in Chapter 2 as,

QT,S(θ) := μT  ˆ βT, ˜βT,S(θ)  and Q∞(θ) = μ∞  β∗ 0,β∗(θ)  , ∀ θ ∈ Θ,

where μT is a criterion divergence and the sequence{μT}T∈Nconverges in a suitable

manner to a limit criterion divergence μ∞. Much notational simplicity in proofs can

however be achieved by adopting an alternative more restrictive ‘norm-like’ criterion divergence μT that minimizes the difference ˆβT− ˜βT,S(θ).11 Hence, here we shall

define the SNPII estimator’s criterion function QT,Sand its limit Q∞as the following

real-valued maps, QT,S(θ) := μT  ˆ βT− ˜βT,S(θ)  and Q∞(θ) = μ∞  β∗ 0− β∗(θ)  , ∀ θ ∈ Θ, (3.4)

10_{Under Tychonoff’s topology onB this shall be obtained by the appropriate convergence of the}

projectionsβ∗

i inBifor every i = 1, 2, ....

11_{Note that Assumption 3.1.3 implies thatB is a subset of a vector space. Te difference ˆβ}

T−

˜

3.2 The SNPII Estimator where the criterion divergence is now defined as μT:B → R for every T ∈ N, and its

limit as μ∞:B → R. Further notational simplification is obtained by defining also

the centered empirical binding function ΔT,S(θ) := ˆβT− ˜βT,S(θ) which can be seen

as the natural estimator of the centered binding function Δ∞(θ) := b(θ0)− b(θ). Since∃θ0∈ Θ : D(θ0) = D0, i.e. D0∈ DΘ, theB-valued centered binding function Δ∞ : Θ→ B crosses the origin of B at θ0, i.e. Δ∞(θ0) = 0. Its estimator ΔT,S :

Ω× Θ → B does not necessarily cross the origin.

Recall from Chapter 2 that the justification for the use of a sequence of criterion divergences _{μT}T∈N instead of a single fixed μ is a practical one. In particular,

we note that in applications it is not possible to make use of an infinite number of auxiliary estimators. Hence, μT can then be appropriately chosen to be a divergence

that gives ‘positive weight’ only to a finite subset of the vectors ˆβT and ˜βT,S(θ) for

every T ∈ N, yet converges to a divergence μ∞ that gives ‘positive weight’ to the

entire vector of auxiliary estimators. For concreteness, let us consider as an example the use of a criterion divergence that extends naturally the classical indirect inference setting of Gourieroux et al. (1993). This shall constitute our example of reference and we shall return to it whenever an illustration seems convenient.

Example 3.2.1. Let μT consist of a weighted sum of squared distances between

auxiliary estimators that assigns positive weights wT,i to an increasing number kT of

auxiliary estimators, QT,S(θ) = μT  ˆ βT − ˜βT,S(θ)  = i∈N wT,i  ˆ βiT− ˜β i T,S(θ) 2 (3.5)

where wT,i> 0 for every i≤ kT and kT→ ∞ as T → ∞. Several aspects play a role

in ensuring that QT,S(θ) converges to a well defined (finite) limit Q∞(θ) for every

θ ∈ Θ, Q∞(θ) = μ∞  β∗ 0− β∗(θ)  = i∈N wi  β∗ i(θ0)− β∗i(θ) 2 . (3.6)

Namely, appropriate regularity conditions must be imposed on (i) the ‘speed’ at which kT diverges, (ii) the ‘decay’ of wT,iover i and T , and (iii) the stochastic properties

of ˆβ_T and ˜β_T,S(θ), e.g. the speed of convergence to their limits β∗₀ and β∗(θ). Indeed, in the context of Example 3.2.1 above, let kT = T and wi = 1∀i < kT.

Then QT,S(θ) will diverge for every θ = θ0 and QT,S(θ0) will converge only under strict conditions on the convergence rate of ˆβT and ˜βT,S(θ0). On the contrary,

if wT,i → 0 along T and i in an appropriate fashion, then QT,S(θ) might remain

bounded and converge to a well defined limit Q∞(θ) for every θ ∈ Θ, even under

very weak conditions on the stochastic behavior, dependence and heterogeneity of the auxiliary instruments ˆβT and ˜βT,S(θ0). These conditions are discussed in the

3 A T -Consistent and Asymptotically Gaussian SNPII Estimator with obtaining a well-defined Q∞by providing appropriate conditions under which

weighted sums of converging statistics are themselves well defined and converging to a finite limit. Until then, we simply assume this to be the case. In the context of Example 3.2.1, this can naturally be seen as an implicit restriction on the behavior of kT, wT,i, ˆβT and ˜βT,S(θ0).