# On the Existence of Strongly Consistent Indirect Estimators When the Binding Function Is Compact Valued

Authors: Stelios Arvanitis
Journal: Journal of Mathematics
 (2013)

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## Abstract

We provide sufficient conditions for the definition and the existence of strongly consistent indirect estimators when the binding function is a compact valued correspondence. We use conditions that concern the asymptotic behavior of the epigraphs of the criteria involved, a relevant notion of continuity for the binding correspondence as well as an indirect identification condition that restricts the behavior of the aforementioned correspondence. These are generalizations of the analogous results in the relevant literature and hence permit a broader scope of statistical models. We examine simple examples involving Levy and ergodic conditionally heteroskedastic processes.


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## Body

## 1. Introduction

Indirect estimators (henceforthIE) are multistep M-estimators defined in the context of (semi-) parametric inference. They are minimizers of criteria (inversion criterion) that are functions of an auxiliary estimator, itself derived as an extremum estimator. The latter minimizes a criterion function (auxiliary criterion) that partially reflects the structure of a possibly misspecified auxiliary statistical model. The inversion criterion is usually a (possibly stochastic) distance function evaluated on the auxiliary estimator as well as on some functional approximation of a mapping between the statistical models involved that is termed binding function. This is constructed by some limiting argument concerning the auxiliary criterion. The IE is finally defined by minimization of the inversion criterion. This definition is conceptually justified by properties of the binding function that guarantee indirect identification and the subsequent use of the analogy principle. Given the auxiliary criterion, differences between IEs hinge on differences on the distance functions, and/or the approximations of the binding function and/or the optimization errors involved.Indirect inference algorithms were initially employed in [1], formally introduced by [2], complemented by [3], and extended by [4]. Furthermore econometric applications of these estimators have become increasingly popular. They have been applied to stochastic volatility and equity return models (e.g., [5–7]), exchange rate models (e.g., [8, 9]), commodity price and storage models (e.g., [10]), dynamic panel data (e.g., [11]), stochastic dif and only iferential equation models (e.g., [12, 13]), and in ARMA models (e.g., [14–17]).In the present paper, we are concerned with the issue of theexistence of strongly consistent IE allowing for cases where the binding function is compact valued (hence possibly multivalued). Therefore we perform our study in a more general framework than the ones employed in the relevant literature.Our motivation lies in cases where the auxiliary criterion is a quasilikelihood function involving a class of stationary-ergodic volatility processes defined by someGARCH or SV type models that represent the statistical model at hand and an auxiliary class of stationary-ergodic invertible processes living in the premises of a possibly misspecified analogous model. In such frameworks the limit criterion could assume extended real values due to the possible existence of parameter values that imply nonexistence of relevant moments. Furthermore since the limit criterion is a statistical divergence between the two classes of processes the binding function can in principle be multivalued due to the geometry involved. Notice that even when this is true, it is possible to study single valued reductions of it via measurable selections. This could however imply stricter conditions for indirect identification. Furthermore since these frameworks are generally suboptimal with respect to asymptotic efficiency, actual multivaluedness could lead to efficiency gains.This study has the form of a calculus of escalating weak sufficient conditions that enable thedefinition of IE in this framework and the proof of existence of strongly consistent ones. First, using mild assumptions on the structure of the auxiliary criterion functions, we are occupied with a weaker than the uniform notion of convergence of the relevant sequence of criterion functions. This is termed epiconvergence and essentially concerns the almost sure asymptotic behavior of their epigraphs. By construction it is suitable for the study of the asymptotic behavior of their minimizers. This form of convergence has been extensively studied in the statistical literature (see among others [18–20]) and it enables the definition of the binding function and the determination of the limiting relation between this function and the auxiliary estimator.Then we strengthen our assumptions in order to obtain a form of continuity of the binding function that enables the definition ofIE derived from this function and the verification of their existence. Finally the imposition of a relatively weak condition of indirect identification on the behavior of the binding function along with the limiting relation already established enables the proof of the existence of strongly consistent IE via the use of the same limit arguments also used for the pseudoconsistency of the auxiliary estimator and the subsequent definition of the binding function. This framework readily enables the description of conditions concerning the behavior of any approximation of the binding function that could also be used for the definition of IE in a similar manner.Hence we manage to extend the framework for the definition ofIE in a threefold manner. We allow for the auxiliary and/or the inversion criteria and/or their appropriate limits to assume extended real values. We study their asymptotic behavior via the use of the weaker known topology associated with convergence of minimizers and we allow for the binding function to be a correspondence with values on the collection of nonempty and compact subsets of the relevant parameter space. This incorporates the definitions used in the existing literature but simultaneously generalizes the set of the statistical models that are in accordance with these conventions.The structure of the paper is as follows. We formulate our setup and define and study the asymptotic behavior of the auxiliary estimator, the binding correspondence, and finally of theIE. We then exhibit some of our results by a set of simple examples. We conclude posing some questions for future research. In the appendix we briefly describe some general notions that are essentially used in main body.


## 2. Assumptions and Main Results

### 2.1. General Setup

We construct our framework and describe the underlying statistical problem. Let the triad(Ω,𝒥,P) denote a complete probability space. Let also (Θ,dΘ) and (B,dB) denote two compact separable metric spaces. Let also ℬ(Θ) and ℬ(B) denote the corresponding Borel algebras. ℝ¯ denotes the extended real line. Again ℬ(ℝ¯) denotes the corresponding Borel algebra.Theauxiliary criterion is a function cn(ω,θ,β):Ω×Θ×B→ℝ¯, which is of the form qn(yn,β) with qn:Kn×B→ℝ¯, for yn:Ω×Θ→Kn, for Kn some appropriate space, usually homeomorphic to ℝmn for some m>0. yn represents the sample. qn reflects part of the structure of an auxiliary model, a statistical model defined on the measurable space (Kn,ℬ(Kn)), with B as its parameter space (e.g., it can be a likelihood function or a GMM type criterion; see Section 3) (which in general is a correspondence B⇉𝒫(Kn), with 𝒫(Kn) the set of probability measures on Kn). yn(·,θ) is measurable for any θ and thereby represents the underlying statistical model which is essentially the set {P∘yn-1(·,θ),θ∈Θ}. These two models need not coincide.We abbreviate withP a.s. any statement that concerns elements of 𝒥 of unit probability. When not nessesary we avoid notating the potential dependence of those elements on the parameters.In the following we provide with an escalating description of a set of sufficient conditions that enablefirst the existence of the auxiliary estimator, second the construction of the binding function and the description of the asymptotic relation between the two, third an appropriate form of continuity of the binding function which along with the previous enables the definition of the IE, and finally consistency.


### 2.2. Definition and Existence of the Auxiliary Estimator

We begin with a sufficient weak assumption on the behavior ofcn that enables the definition and the existence of the auxiliary estimator. It comprises of a joint measurability condition along with a pointwise with respect to θ and P a.s. with respect to ω continuity and some condition concerning the facilitation of minimization. All these conditions are weak enough so that their verification is easy in many cases. Remember that a function with values in the extended real line is called proper, if it does not take the value -∞ and its image contains at least a real number. It is called inf-compact, if its level sets (for a∈ℝ, the level set of c:E→ℝ¯ with respect to a (Level≤a(c)) is defined by {x∈E:c(x)≤a}) are compact. inf-compactness follows trivially when the function is lsc andits domain is compact.Assumption 1.
Let the following holds.(1)
cn is ℬ(ℝ¯)/𝒥⊗ℬ(Θ)⊗ℬ(B) measurable.(2)
cn(ω,θ,·):B→ℝ¯ is lower semicontinuous (lsc) and proper P a.s., for all θ∈Θ. (It is obvious that the element of 𝒥 of unit probability with respect to which Assumption 1 holds can depend on θ).In our examples presented in Section3, the issue of joint measurability is handled easily due to the fact that the cn’s considered are in fact Caratheodory functions, that is, jointly continuous (with respect to (θ,β)) and pointwise measurable. Separability of Θ×B and Lemma 4.51 of Aliprantis and Border [21] implies the required measurability. Properness is an ad hoc consideration that is easily established in many cases. For instance, when cn has the form of a quasilikelihood function it P a.s. does not attain extended values. This is also the case in instances where cn has the form of a hemimetric (see Appendix) as in Section 2 due to the compactness of its arguments.Remark 2.
The joint measurability and the pointwise semicontinuity imply thatcn(·,θ,·) is a normal integrand (see Definition 3.5 and Proposition 3.6 in Chapter 5 of [22]). The compactness of B then implies that cn is inf-compact P a.s. for all θ∈Θ.We are now ready to define and prove existence for the auxiliary estimator. We remind the reader that the Appendix provides with the definition of a measurable compact valued correspondence.Definition 3.
The auxiliary correspondenceβn#(ω,θ,εn) satisfies
(1)βn#(ω,θ,εn)=εn-argminBqn(yn,β)=εn-argminBcn(ω,θ,β)≑{β∈B:cn(ω,θ,β)≤infBcn(ω,θ,·)+εn},
where εn is a P a.s. nonnegative random variable defined on Ω.Notice that the dependence ofβn# on θ is due to the assumed dependence of ynon θ. In practice βn# is evaluated by the minimization appearing in the first equality of the previous display.Proposition 4.
Under Assumption1  βn#(ω,θ,εn) is ℬ(BF)/𝒥-measurable and P a.s. non emptycompact valued for all θ∈Θ.Proof.
For anyθ∈Θβn#(ω,θ,εn) is non empty and compact due to Assumption 1(2) and the compactness of B. (For the ω for which εn(ω)≠0 nonemptiness of βn#(ω,θ,εn(ω)) does not necessitate Assumption 1(2) due to the properties of the inf). Then the joint measurability of cn(·,θ,·) due to Assumption 1 along with Proposition 3.10(iii) in Chapter 5 of [22] guarantees the measurability for infBcn(ω,β)+εn. The result follows from the separability of B.Obviouslyβn#(ω,θ,0)=argminBcn(ω,θ,β). In the following, dependence on Ω will henceforth be suppressed (where possible) for notational simplicity. Dependence on B, Θ, and the “optimization error” εn will be kept. (The fundamental selection theorem (Theorem 2.13 of [22]) implies the existence of a measurable selection, i.e., a ℬ(B)/𝒥⊗ℬ(Θ)-measurable random element βn*:Ω×Θ→ℝ termed as auxiliary selection, defined by cn(ω,θ,βn*(θ))≤infBcn(ω,θ,β)+εn. We will not use selections to define and explore the subsequent definition of the IE since this would imply stricter conditions for identification.)


### 2.3. Epilimits and Existence of a Fell Consistent Auxiliary Correspondence

Assumption1 is not sufficient for the construction of the binding function as an appropriate limit of the auxiliary correspondence. The following assumption facilitates the investigation of the issue of (pseudo-) consistency for the auxiliary correspondence and thereby permits this construction. It indicates the almost sure epiconvergence of the auxiliary criterion to a proper, semicontinuous asymptotic counterpart. The first part is essentially the sequential characterization of this form of convergence. For the topological definition of epiconvergence along with some properties please see the Appendix. For the equivalence between the topological and the sequential definitions see among others [20]. The second part enforces to the limit criterion a property that are is not generally invariant with respect to epiconvergence.Assumption 5.
There exists a functionc:Θ×B→ℝ¯ such that(1)
∀θ∈Θ, P a.s. for any β:
(i)
liminfn→∞cn(ω,θ,βn)≥c(θ,β) for all βn such that βn→β, and(ii)
limsupn→∞cn(ω,θ,βn)≤c(θ,β) for some βn such that βn→β,(2)
c(θ,·) is proper ∀θ∈Θ.Remark 6.
For anyθ, c(θ,·):B→ℝ¯ is lower semicontinuous since the local compactness of B×ℝ and the subsequent coincidence of convergence with respect to the Fell topology with the Painleve-Kuratowski one (see Appendix), the epigraph of c(θ,·) is closed. In the case that cn=(1/n)∑i=1n‍mi(ω,θ,β) and (mi(θ,β))i∈ℤ is ergodic for any (θ,β), the assumed epiconvergence would follow if for any θ there exists an open cover of B, so that E|infβ∈Am0(ω,θ,β)|<+∞ for any A in the cover (condition 𝒞0 and Theorem 2.3 of [23]). In the considered cases c is proper since it is either an expectation that cannot assume the value -∞, and there exists at least some parameter value for which it is finite or it is defined by composition with a (hemi-) metric and assumes the value 0 for at least one parameter value (see, for example, [24] Part 1, (ii) in association with Part 2 of the proof of Theorem 5.3.1, where cn is a quasilikelihood function and Θ coincides with B for the former or Section 2 for the latter case). inf-compactness follows from the compactness of Θ.The assumption essentially enables the use of the fact that theargmin correspondence is upper continuous as a function defined on the relevant space of lsc functions equiped with the topology of epiconvergence. Analogous assumptions have been used for the establishment of strong consistency of various estimators. See among others [18–20, 25]. Hence it makes possible the definition of the binding function as
(2)b(θ)≑argminBc(θ,β).
The following proposition provides its existence and is essentially similar to Proposition 4.Proposition 7.
Under Assumptions1 and 5 the binding correspondence b is nonempty compact valued.Proof.
It follows from Remark6.Both the auxiliary and the binding correspondence will be used for the definition of theIE via some intuition that utilizes an analogy principle. The following result explores their asymptotic relation. For its construction the hemimetric δu is needed that is defined by δu(A1,A2)=inf{ε>0:A2⊂Nε(A1)} where Nε(A1)={x∈E:d(x,A1)<ε}, where d(x,A1)≑infy∈A1d(x,y) and A1, A2 are nonempty closed subsets of some topological space. For several continuity and measurability properties of δu see the Appendix.The first and last implications of the following proposition are already well known. Its second implication is a partial generalization of Theorems 7.30, 7.32 of [26] in our setting. Given the definitions of the upper and the Fell topology in the Appendix, the first essentially establishes the upper pseudoconsistency and the other two the Fell pseudoconsistency of βn# with respect to to b.Proposition 8.
Under Assumptions1 and 5 the following holds:(1)
for any  εn  such that  εn→0P  a.s. then  δu(b(θ),βn#(θ,εn))→0P  a.s.,(2)
there exists a nonnegative random variable,  εn*, such that  εn*→0P  a.s.  and  δ(b(θ),βn#(θ,εn))→0P  a.s.,(3)
if  b(θ)  is singleton then for any  εn  such that  εn→0P  a.s.  then  δ(b(θ),βn#(θ,εn))→0P  a.s.For its proof we will use the following lemmata. Letaθ≑infBc(θ,β) which is well defined due to Remark 6 and the compactness of B. Remember that for (Ϝn) a sequence of non empty sets LiϜn is the set comprised of the limit points of any possible sequence (xn) such thatxn∈Ϝn, and LsϜn is the one comprised of the analogous cluster points. Also Level≤a(cn(θ,·))≑{x∈B:c(θ,x)≤a} for a∈ℝ.Lemma 9.
Under Assumptions1 and 5,
(3)limsupninfBcn(θ,β)≤infBc(θ,β)Pa.s.Proof.
Consider the family ofθ-parameterized correspondences epin(ω,θ)≑epi(cn(ω,θ,·)). Due to the fact that B is locally compact, epin(ω,θ) is a random closed set in the sense of the previous paragraph, that is, a ℬ(BF)/𝒥⊗ℬ(Θ)-measurable correspondence. Hence epin(ω,·) is an ℬ(BF)/𝒥-measurable correspondence due to the measurability of the relevant projection. Now due to Assumption 5 (see Section 4) we have that for large n and for all ω in an element of 𝒥 of unit probability epin(ω,θ)∩B×(-∞,aθ)≠∅ since B×(-∞,aθ) is open in the relevant product topology. Hence infBcn(ω,θ,β)≤infBc(θ,β) for all ωdescribed previously.The next result will be used for the proof of Proposition8(2)-(3).Lemma 10.
Under Assumptions1 and 5 there exists a sequence of random variables defined on Ω, say an* (It is obvious from the proof that an* depends also on θ). Such that an*→infB(c(θ,β))Pa.s. and
(4)Li(Level≤an*(cn(θ,·)))⊇b(θ)Pa.s.Proof.
From the sequential implication of epiconvergence in Remark6, we have that, for any x∈b(θ), there exists a measurable xn such that xn→xPa.s. Obviously for an,x=cn(θ,xn) which is measurable, we have that xn∈Level≤an,x(cn(θ,·)). Since b(θ) is compact, it is totally bounded and therefore, for any ε>0, there exist an m(ε)∈ℕ and {yi,i=1,…,m(ε)}⊂b(θ), such that the collection of balls (in B) 𝒪(yi,(ε/2)) covers b(θ). For some real sequence qn→0, consider {yi,i=1,…,m(qn)} and extract analogously random sequences (yi,n) such that yi,n→yiPa.s. Define an,yi=cn(θ,yi,n) and n*(n)=min{n′:|an′,yi-aθ|≤(qn/2),Pa.s.foralli=1,…,m(qn)} which is well defined due to Egoroff’s theorem and the fact that m(qn) is finite. Obviously n* is nondecreasing in n. Then define an*=max{an*,yi,i=1,…,m(qn)} which is measurable and an*→aθPa.s. Then for x∈b(θ), it follows that, for any ε>0, there exist an n and a measurable xn∈Level≤an*(cn(θ,·)) such that d(xn,x)<εPa.s.Proof of Proposition8.
For (1) we have first that for any measurable nonnegativeεn that need not converge to zero if xn∈βn#(θ,εn) measurable, such that a subsequence xnk→xPa.s. 									(5)c(θ,x)≤liminfnkcnk(θ,xnk)Pa.s.=liminfnk(cnk(θ,xnk)+εnk)Pa.s.≤limsupnk(cnk(θ,xnk)+εnk)Pa.s.=limsupnk(infBcnk(θ,β)+εn)Pa.s.≤infBc(θ,x)+εnPa.s.,
where that last inequality follows from Lemma 9. This establishes that for any nonnegative random variable εn(6)Ls(βn#(θ,εn))⊆εn-argminBc(θ,β)Pa.s.
Now (1) follows from the fact that ε-argminB⊆ε′-argminB if ε≤ε′. For (2) notice that, from the definition of the Fell topology in Section 4 for any ε>0, we have that for large n, epin(ω,θ)∩B×[aθ-2ε,aθ-ε]=∅Pa.s. since B×[aθ-2ε,aθ-ε] is compact in the relevant product topology. This implies that liminfninfBcn(θ,β)≥aθPa.s. and in conjunction with Lemma 9 that infBcn(θ,β)→aθPa.s. Then using Lemma 10 set εn*=an*-aθ which is obviously measurable and converges to zero Pa.s. (3) follows from (1) the single valuedness of b(θ) and the compactness of B.The proof of Lemma10 implies that the sequence (εn*)n∈ℕ that appears in Proposition 8(2) is nonunique. However the fact that this implication does not hold for any sequence of nonnegative random variables that P a.s. converge to zero is the cause of that in what follows we can only prove the existence of strongly consistent indirect estimators among the set of the ones to be defined.


### 2.4. Upper Hemicontinuity of the Binding Correspondence

Proposition8 enables the use of δu(b(θ),βn#) as the inversion criterion. The following assumption concerns the upper continuity of the binding correspondence which along with the relevant properties of δu would imply the analogous continuity property for the particular inversion criterion and thereby facilitate the issue of existence and consistency of the IE to be defined.Assumption 11.
b is upper hemicontinuous that is, for any θ and θ*→θ, δu(b(θ),b(θ*))→0.The following proposition provides with sufficient conditions for this to hold. It essentially strengthens Assumption5 in that it requires that the relevant P  a.s. epiconvergence be continuous on Θ. Notice that its requirements are also stricter with respect to measurability compared to the ones in Assumption 5 since the former requires that for any θ the relevant set of P unit probability does not depend on the sequence that converges to θ.Proposition 12.
If, for anyθ, P  a.s. for any ω, any  θn→θ, and any β,(1)
liminfn→∞cn(ω,θn,βn)≥c(θ,β),   for all  βn  such that  βn→β,(2)
limsupn→∞cn(ω,θn,βn)≤c(θ,β),   for some  βn  such that  βn→β,
then Assumption11 holds.Proof.
Suppose that𝒟F0 metrizes 𝒯F on F0(B) (see Section 4). Then it is obvious that Proposition 12(1)-(2) is equivalent to the requirement that, for any θ and any θn→θ, 𝒟F0(epin(ω,θn),epi(θ)) converges to zero P a.s. Then, for any θ and any θn→θ, c(θn,·) epiconverges to c(θ,·) (𝒟F0(epi(θn),epi(θ))→0), that is, c(θ,·) is epicontinuous on Θ. This is due to the following standard argument: for an arbitrary θ and ε>0, we have that 𝒟F0(epin(ω,θ′),epi(θ))<(ε/2)P a.s. for any θ′ in some open neighborhood of θ and large enough n, due to the assumed form of convergence and Egoroff’s theorem. By the same reasoning 𝒟F0(epin(ω,θ′),epi(θ′))<(ε/2)P a.s. for any such θ′. The result follows from the fact that epi(θ) is independent of ω. This along with 3.1 of Theorem 5.3.4 and proposition Appendix D.2 of [22] implies that the composite mapping θ→c(θ,·)→argminBc(θ,β) is appropriately continuous.An examination of the proof of Proposition8 shows that if the conditions (1)-(2) of Proposition 12 hold then Proposition 8 could be restated with δu(b(θ),βn#(θn,εn)) replacing δu(b(θ),βn#(θ,εn)) when this occurs. This means that the premises of Proposition 12 render βn# strongly continuously (upper in the first case or Fell in the second and third cases of Proposition 8) b(θ)-consistent. Furthermore the compactness of Θ then implies that these notions are also uniform with respect to Θ.Remark 13.
Proposition12(1)-(2) would obviously be implied if cn(ω,θ,β) is P a.s. jointly continuous and converges jointly uniformly P a.s. to c(θ,β). Since we allow cn and/or c to assume extended values, the relevant notion of uniform convergence must also be extended as in Definition 7.12 of [26]. The following lemma provides with a set of even weaker sufficient conditions than extended jointly uniform P a.s. convergence when cn has the form of an arithmetic mean with respect to stationary and ergodic processes.Lemma 14.
Suppose thatcn(ω,θ,β)=(1/n)∑i=1n‍mi(ω,θ,β), (mi(θ,β))i∈ℤ is ergodic for any (θ,β), cn(ω,·,·):Θ×B→ℝ¯ is jointly continuous P a.s., there exists a finite open cover of Θ×B, such that E|inf(θ,β)∈Am0(ω,θ,β)|<+∞, for any A in the cover, E(m0(ω,θ,β*)) assumes values in ℝ¯ for any θ∈Θ, and β* in a countable dense subset of B. Then Proposition 12 holds.Proof.
Proposition12(1) follows from the fact that the assumption framework of the lemma implies condition 𝒞0 and thereby Theorem 2.3 of [23], which implies the joint P a.s. epiconvergence of cn to Em0. For Proposition 12(2) notice that the separability of B and the P a.s. continuity of m0 implies the existence of a countable dense B* such that for any β and any δ>0 there exists a β*∈B* such that
(7)limsupn→∞cn(ω,θn,β)≤limsupn→∞(cn(ω,θn,β)-cn(ω,θn,β*))+limsupn→∞cn(ω,θn,β*)Pa.s.≤δ+limsupn→∞cn(ω,θn,β*)Pa.s.
By assumption the subset of Ω of P unit probability can be chosen independent of δ and δ can be chosen arbitrarily small. Hence, Proposition 12(2) would be implied for βn=β, if, for any θ, β*∈B*, Pa.s. and any θn→θ,
(i)limsupn→∞cn(ω,θn,β*)≤Em0(θ,β*)
due to the countability of B*. Notice that ((mi(θ,β*)∧-ρ)∨ρ)i∈ℤ is also stationary-ergodic for any ρ>0 (see, e.g., Proposition 2.1.1. of [24]), and hence the uniform version of Birkhoff’s LLN implies that
(8)1n∑i=1n((mi(ω,θn,β*)∧-ρ)∨ρ)
converges P a.s. to ∫-ρρ‍m0(ω,θ,β*)dP(ω) for any ρ>0. Due to the separability of ℝ the subset of Ω of P unit probability can be chosen independent of ρ. Hence from Definition 7.12 of [26] we obtain (i).This lemma explores sufficient conditions for the required continuity ofb solely via restrictions on the behavior of cn which in applications is generally more analytically tractable than c. Furthermore it combines joint epi-onvergence with pointwise (on B) extended uniform (with respect to θ) almost sure convergence. Finally notice that analogous result would also hold if ergodicity is replaced by any kind of mixing condition that would justify the LLNs used in the previous proof or implied in Remark 13.


### 2.5. Definition, Existence, and Consistency of the Indirect Estimator

We are now ready to define theIE and explore the issues of its existence and consistency. Proposition 8 along with the measurability of the auxiliary correspondence and the upper hemicontinuity of b facilitate the use of δu(b(θ),βn#(θ0)) for some distinguished θ0∈Θ, for the definition of the IE and the subsequent existence argument. Again an almost surely nonnegative random variable will assume the role of the “optimization error” in this second step of the estimation procedure.Definition 15.
The indirect correspondenceθn#(ω,θ0,εn,εn#) satisfies
(9)θn#(ω,θ0,εn,εn#)=εn#-argminΘδu(b(θ),βn#(θ0,εn))≑{infΘδu(b(θ),βn#(θ0,εn))+εn#θ*∈Θ:δu(b(θ*),βn#(θ0,εn))≤infΘδu(b(θ),βn#(θ0,εn))+εn#},
where εn# is a nonnegative random variable defined on Ω.The dependence ofθn# on θ0 follows form the analogous dependence of βn# which in turn follows from the dependence of yn on θ0. Obviously in practice this dependence is not “visible.” We are initially concerned with the question of existence of the IE. We again suppress the dependence of θn on ω when there is not a risk of confusion.Proposition 16.
Under Assumptions1 and 11  θn# is ℬ(ΘF)/𝒥-measurable, P a.s. nonempty, compact valued correspondence.Proof.
First, notice that due to LemmaA.9, Proposition 4 (implied by Assumption 1), Assumption 11, and the facts that βn#(θ0,εn) is independent of θ and b(θ) is independent of ω, we obtain that δu(b(θ),βn#(θ0,εn)) is ℬ(ℝ¯)/ℬ(ΘF)⊗𝒥-measurable. Due to Lemma A.4 and Proposition 4  δu(b(θ),βn#(θ0,εn)) is P a.s. lsc and therefore a normal integrand. It is also P a.s. proper due to the fact that b and βn# are P a.s. compact valued. Hence the result follows from Proposition 4 where cn=δu(b(θ),βn#(θ0,εn)), when we consider B=Θ and Θ={θ0} (the left-hand sides correspond to the notation of the latter lemma).The fundamental selection theorem (Theorem 2.13 of [22]) would also enable the definition of the IE as a measurable function with values in Θ. Having established existence we turn to the issue of consistency. We need an assumption of indirect identification that is essentially derived from the form of the roots of the hemimetric used. Notice that this assumption along with the proof of the following proposition justifies thedefinition of the Definition 15 by an analogy principle. Mathematically both the definition and the existence argument do not require the following assumption in order to be valid.Assumption 17.
Ifθ≠θ0⇒b(θ0)-b(θ)≠∅.Remark 18.
This condition is weaker than a condition of the form “Ifθ≠θ0⇒b(θ0)∩b(θ)=∅” and stronger that a condition of the form “If θ≠θ0⇒b(θ0)≠b(θ).” The latter cannot be used due to the properties of δu upon which the definition of the IE is based. In the case that the binding correspondence is single valued, these become equivalent. This also makes evident the claim that if the auxiliary estimator is defined by a measurable selection of βn# the corresponding identification condition cannot be weaker than the one above.The main result of the current section follows. It merely concerns theexistence of strongly consistent IE inside the established framework. Denote by θ(θ0) the argminΘδu(b(θ),b(θ0)) which is nonempty and compact due to the compactness of Θ, the properness of δu, the hemicontinuity of b, and Lemma A.3. Obviously, θ0∈θ(θ0), while {θ0}=θ(θ0)ifandonlyif Assumption 17 holds. Since we once again are dealing with compact valued correspondences, convergence is metrized by δu and/or δ.Proposition 19.
Under Assumptions1, 5, and 11 and if εn#→0P a.s. then(1)
δu(θ(θ0),θn#(ω,θ0,εn*,εn#))→0P  a.s. where  εn*  is defined in Proposition8(2),(2)
if   b(θ0)  is singleton then  δu(θ(θ0),θn#(ω,θ0,εn,εn#))→0P  a.s. for any  εn→0P  a.s.,
if furthermore Assumption17 holds then(1*)
δ({θ0},θn#(ω,θ0,εn*,εn#))→0P  a.s. where  εn*  is defined in Proposition8(2),(2*)
if   b(θ0)  is singleton then  δ({θ0},θn#(ω,θ0,εn,εn#))→0P  a.s. for anyεn→0Pa.s.Proof.
First notice that due to Proposition8(2), Assumption 11, and Lemma A.6 we have that for any θ and θn→θ(10)liminfnδu(b(θn),βn#(θ0,εn*))≥δu(b(θ),b(θ0))Pa.s.
and that, for any θ and θn=θ, due to Lemma A.4(11)limsupnδu(b(θn),βn#(θ0,εn*))≤δu(b(θ),b(θ0))Pa.s.
Hence (1) follows from Proposition 8(1) for cn=δu(b(θn),βn#(θ0,εn*)) and if we denote by B (in the notation of this lemma) the Θ space and by Θ (in the notation of this lemma) {θ0}. (2) follows in the same manner if we replace any invocation of Proposition 8(2) with Proposition 8(3). Finally, notice that if Assumption 17 holds, then θ(θ0)={θ0} establishing (1*) and (2*) via another use of Proposition 8(3).If Assumption17 does not hold then the implications of Proposition 19(1)-(2) correspond to the fact that the statistical model is only indirectly set identifiedgiven this framework. They are trivial when θ(θ0)=Θ and the closer to zero δu({θ0},θ(θ0)) is the more informative they become. We once again point out that the implications Proposition 19(1) and (1*) merely explore the issue of the existence of strongly consistent estimators among those that comply with Definition 15. The properties of the δu function along with Proposition 8 do not permit for a stronger result without strengthening the assumption framework. Finally notice that this framework enables both the definition and the result on consistency of the IE to be derived via the use of exact same notions that were used for the analogous results concerning the auxiliary one.


### 2.6. Extension

In most casesb is analytically unknown even if several of its properties, such as some of the ones discussed above, can be established. In these cases the estimators defined in Definition 15 are infeasible. However, it may be the case that (possibly) stochastic and algorithmically feasible approximations of the unknown b can be used for the construction of several other classes of feasible IE. In such a context the results derived previously could be used to describe properties of such approximations that would imply that these estimators are well defined and among them strongly consistent ones exist. Let κn(ω,θ) denote such an approximation. We readily obtain the following result for the indirect estimator defined by the substitution of b with κn in Definition 15.Proposition 20.
Consider theIE defined by
(12)ϑn#(ω,θ0,εn,εn#)=εn#-argminΘδu(κn(ω,θ),βn#(θ0,εn)),
where εn,εn# as before.(i)
Under Assumption1 and if  κn(ω,θ)  is  ℬ(BF)/𝒥⊗ℬ(ΘF)-measurable,  P  a.s. compact valued and upper hemicontinuous, then  ϑn#  is  ℬ(ΘF)/𝒥-measurable,  P  a.s. nonempty, compact valued correspondence.(ii)
If moreover Assumption5 holds and  P  a.s. for any  θ  and any  θn→θ, δ(b(θ),κn(ω,θ))  and  δu(b(θ),κn(ω,θn))  converge to zero then the implications of Proposition19(1)-(2) hold also for  ϑn#. If furthermore Assumption17 holds of then the implications  Proposition 19(1*)-(2*)hold also for  ϑn#.Proof.
(i) As in the proof of Proposition16 from Lemma A.9, Proposition 4 the P a.s. upper hemicontinuity of κn the facts that βn#(θ0,εn) is independent of θ, and κn is jointly measurable, we obtain that δu(βn#(θ0,εn),b(θ)) is ℬ(ℝ¯)/ℬ(ΘF)⊗𝒥-measurable. Due to Lemma A.4 and Proposition 4  δu(κn(ω,θn),βn#(θ0,εn)) is P a.s. lsc and therefore a normal integrand. It is also P a.s. proper due to the fact that κn and βn# are P a.s. compact valued. Hence the result follows from Proposition 4. (ii) It suffices to prove that δu(κn(ω,θ),βn#(θ0,εn))→eδu(b(θ),b(θ0))P a.s. when βn#(θ0,εn)  satisfies the implications (2) or (3) of Proposition8. The rest would then follow as in the proof of Proposition 20. Notice that for any θ and any θn→θ(13)liminfnδu(κn(ω,θn),βn#(θ0,εn*))≥δu(b(θ),b(θ0)),Pa.s.
due to Lemma A.6, the definition of βn# and the P a.s. continuous w.r.t θ upper convergence of κn to b. Moreover for θn=θ(14)limsupnδu(κn(ω,θ),βn#(θ0,εn*))≤limsupnδu(κn(ω,θ),b(θ))+limsupnδu(b(θ),βn#(θ0,εn*)),Pa.s.≤limsupnδ(κn(ω,θ),b(θ))+δu(b(θ),b(θ0)),Pa.s.,
where the first inequality follows from the triangle inequality and the second from the definition of δ and the P a.s. Fell convergence of βn# to b(θ0) and Lemma A.4. Due to the P a.s. pointwise with respect to θ Fell convergence of κn to b we have that limsupnδ(κn(ω,θ),b(θ))=0Pa.s. and therefore we obtain the needed result.Notice that this proposition generalizes the results of Propositions16 and 20. For a simple example consider the case where κn=βn#. This is possible when by some sort of resampling technique (e.g., bootstrap or Monte Carlo) realizations of the yn random elements are available to the practitioner for any θ and thereby so is βn#(θ,ϵn) for any θ and some optimization error ϵn  independent of θ. Then a feasible IE can be defined by the approximate minimization of δu(βn#(θ,ϵn),βn#(θ0,εn)) with respect to θ. In this case the joint measurability of βn# would follow from the joint measurability of cn and ϵn, the separability of B, and the subsequent joint measurability of the relevant projection. The P a.s. upper hemicontinuity of βn# would follow from an easy extension of the implication Proposition 8(1) if Assumption 1 is strengthened so that the mapping θ→epin(ω,θ) is P a.s. Fell continuous. The P a.s. joint continuity of cn would suffice. Then the P a.s. pointwise Fell convergence to b would follow as in Proposition 8(2) or (3) and the P a.s. continuous w.r.t θ upper convergence to b would follow if Proposition 12 holds with the set of unit probability independent of θ. It would suffice that cn is P a.s. jointly continuous and converges to c jointly uniformly.


## 2.1. General Setup

We construct our framework and describe the underlying statistical problem. Let the triad(Ω,𝒥,P) denote a complete probability space. Let also (Θ,dΘ) and (B,dB) denote two compact separable metric spaces. Let also ℬ(Θ) and ℬ(B) denote the corresponding Borel algebras. ℝ¯ denotes the extended real line. Again ℬ(ℝ¯) denotes the corresponding Borel algebra.Theauxiliary criterion is a function cn(ω,θ,β):Ω×Θ×B→ℝ¯, which is of the form qn(yn,β) with qn:Kn×B→ℝ¯, for yn:Ω×Θ→Kn, for Kn some appropriate space, usually homeomorphic to ℝmn for some m>0. yn represents the sample. qn reflects part of the structure of an auxiliary model, a statistical model defined on the measurable space (Kn,ℬ(Kn)), with B as its parameter space (e.g., it can be a likelihood function or a GMM type criterion; see Section 3) (which in general is a correspondence B⇉𝒫(Kn), with 𝒫(Kn) the set of probability measures on Kn). yn(·,θ) is measurable for any θ and thereby represents the underlying statistical model which is essentially the set {P∘yn-1(·,θ),θ∈Θ}. These two models need not coincide.We abbreviate withP a.s. any statement that concerns elements of 𝒥 of unit probability. When not nessesary we avoid notating the potential dependence of those elements on the parameters.In the following we provide with an escalating description of a set of sufficient conditions that enablefirst the existence of the auxiliary estimator, second the construction of the binding function and the description of the asymptotic relation between the two, third an appropriate form of continuity of the binding function which along with the previous enables the definition of the IE, and finally consistency.


## 2.2. Definition and Existence of the Auxiliary Estimator

We begin with a sufficient weak assumption on the behavior ofcn that enables the definition and the existence of the auxiliary estimator. It comprises of a joint measurability condition along with a pointwise with respect to θ and P a.s. with respect to ω continuity and some condition concerning the facilitation of minimization. All these conditions are weak enough so that their verification is easy in many cases. Remember that a function with values in the extended real line is called proper, if it does not take the value -∞ and its image contains at least a real number. It is called inf-compact, if its level sets (for a∈ℝ, the level set of c:E→ℝ¯ with respect to a (Level≤a(c)) is defined by {x∈E:c(x)≤a}) are compact. inf-compactness follows trivially when the function is lsc andits domain is compact.Assumption 1.
Let the following holds.(1)
cn is ℬ(ℝ¯)/𝒥⊗ℬ(Θ)⊗ℬ(B) measurable.(2)
cn(ω,θ,·):B→ℝ¯ is lower semicontinuous (lsc) and proper P a.s., for all θ∈Θ. (It is obvious that the element of 𝒥 of unit probability with respect to which Assumption 1 holds can depend on θ).In our examples presented in Section3, the issue of joint measurability is handled easily due to the fact that the cn’s considered are in fact Caratheodory functions, that is, jointly continuous (with respect to (θ,β)) and pointwise measurable. Separability of Θ×B and Lemma 4.51 of Aliprantis and Border [21] implies the required measurability. Properness is an ad hoc consideration that is easily established in many cases. For instance, when cn has the form of a quasilikelihood function it P a.s. does not attain extended values. This is also the case in instances where cn has the form of a hemimetric (see Appendix) as in Section 2 due to the compactness of its arguments.Remark 2.
The joint measurability and the pointwise semicontinuity imply thatcn(·,θ,·) is a normal integrand (see Definition 3.5 and Proposition 3.6 in Chapter 5 of [22]). The compactness of B then implies that cn is inf-compact P a.s. for all θ∈Θ.We are now ready to define and prove existence for the auxiliary estimator. We remind the reader that the Appendix provides with the definition of a measurable compact valued correspondence.Definition 3.
The auxiliary correspondenceβn#(ω,θ,εn) satisfies
(1)βn#(ω,θ,εn)=εn-argminBqn(yn,β)=εn-argminBcn(ω,θ,β)≑{β∈B:cn(ω,θ,β)≤infBcn(ω,θ,·)+εn},
where εn is a P a.s. nonnegative random variable defined on Ω.Notice that the dependence ofβn# on θ is due to the assumed dependence of ynon θ. In practice βn# is evaluated by the minimization appearing in the first equality of the previous display.Proposition 4.
Under Assumption1  βn#(ω,θ,εn) is ℬ(BF)/𝒥-measurable and P a.s. non emptycompact valued for all θ∈Θ.Proof.
For anyθ∈Θβn#(ω,θ,εn) is non empty and compact due to Assumption 1(2) and the compactness of B. (For the ω for which εn(ω)≠0 nonemptiness of βn#(ω,θ,εn(ω)) does not necessitate Assumption 1(2) due to the properties of the inf). Then the joint measurability of cn(·,θ,·) due to Assumption 1 along with Proposition 3.10(iii) in Chapter 5 of [22] guarantees the measurability for infBcn(ω,β)+εn. The result follows from the separability of B.Obviouslyβn#(ω,θ,0)=argminBcn(ω,θ,β). In the following, dependence on Ω will henceforth be suppressed (where possible) for notational simplicity. Dependence on B, Θ, and the “optimization error” εn will be kept. (The fundamental selection theorem (Theorem 2.13 of [22]) implies the existence of a measurable selection, i.e., a ℬ(B)/𝒥⊗ℬ(Θ)-measurable random element βn*:Ω×Θ→ℝ termed as auxiliary selection, defined by cn(ω,θ,βn*(θ))≤infBcn(ω,θ,β)+εn. We will not use selections to define and explore the subsequent definition of the IE since this would imply stricter conditions for identification.)


## 2.3. Epilimits and Existence of a Fell Consistent Auxiliary Correspondence

Assumption1 is not sufficient for the construction of the binding function as an appropriate limit of the auxiliary correspondence. The following assumption facilitates the investigation of the issue of (pseudo-) consistency for the auxiliary correspondence and thereby permits this construction. It indicates the almost sure epiconvergence of the auxiliary criterion to a proper, semicontinuous asymptotic counterpart. The first part is essentially the sequential characterization of this form of convergence. For the topological definition of epiconvergence along with some properties please see the Appendix. For the equivalence between the topological and the sequential definitions see among others [20]. The second part enforces to the limit criterion a property that are is not generally invariant with respect to epiconvergence.Assumption 5.
There exists a functionc:Θ×B→ℝ¯ such that(1)
∀θ∈Θ, P a.s. for any β:
(i)
liminfn→∞cn(ω,θ,βn)≥c(θ,β) for all βn such that βn→β, and(ii)
limsupn→∞cn(ω,θ,βn)≤c(θ,β) for some βn such that βn→β,(2)
c(θ,·) is proper ∀θ∈Θ.Remark 6.
For anyθ, c(θ,·):B→ℝ¯ is lower semicontinuous since the local compactness of B×ℝ and the subsequent coincidence of convergence with respect to the Fell topology with the Painleve-Kuratowski one (see Appendix), the epigraph of c(θ,·) is closed. In the case that cn=(1/n)∑i=1n‍mi(ω,θ,β) and (mi(θ,β))i∈ℤ is ergodic for any (θ,β), the assumed epiconvergence would follow if for any θ there exists an open cover of B, so that E|infβ∈Am0(ω,θ,β)|<+∞ for any A in the cover (condition 𝒞0 and Theorem 2.3 of [23]). In the considered cases c is proper since it is either an expectation that cannot assume the value -∞, and there exists at least some parameter value for which it is finite or it is defined by composition with a (hemi-) metric and assumes the value 0 for at least one parameter value (see, for example, [24] Part 1, (ii) in association with Part 2 of the proof of Theorem 5.3.1, where cn is a quasilikelihood function and Θ coincides with B for the former or Section 2 for the latter case). inf-compactness follows from the compactness of Θ.The assumption essentially enables the use of the fact that theargmin correspondence is upper continuous as a function defined on the relevant space of lsc functions equiped with the topology of epiconvergence. Analogous assumptions have been used for the establishment of strong consistency of various estimators. See among others [18–20, 25]. Hence it makes possible the definition of the binding function as
(2)b(θ)≑argminBc(θ,β).
The following proposition provides its existence and is essentially similar to Proposition 4.Proposition 7.
Under Assumptions1 and 5 the binding correspondence b is nonempty compact valued.Proof.
It follows from Remark6.Both the auxiliary and the binding correspondence will be used for the definition of theIE via some intuition that utilizes an analogy principle. The following result explores their asymptotic relation. For its construction the hemimetric δu is needed that is defined by δu(A1,A2)=inf{ε>0:A2⊂Nε(A1)} where Nε(A1)={x∈E:d(x,A1)<ε}, where d(x,A1)≑infy∈A1d(x,y) and A1, A2 are nonempty closed subsets of some topological space. For several continuity and measurability properties of δu see the Appendix.The first and last implications of the following proposition are already well known. Its second implication is a partial generalization of Theorems 7.30, 7.32 of [26] in our setting. Given the definitions of the upper and the Fell topology in the Appendix, the first essentially establishes the upper pseudoconsistency and the other two the Fell pseudoconsistency of βn# with respect to to b.Proposition 8.
Under Assumptions1 and 5 the following holds:(1)
for any  εn  such that  εn→0P  a.s. then  δu(b(θ),βn#(θ,εn))→0P  a.s.,(2)
there exists a nonnegative random variable,  εn*, such that  εn*→0P  a.s.  and  δ(b(θ),βn#(θ,εn))→0P  a.s.,(3)
if  b(θ)  is singleton then for any  εn  such that  εn→0P  a.s.  then  δ(b(θ),βn#(θ,εn))→0P  a.s.For its proof we will use the following lemmata. Letaθ≑infBc(θ,β) which is well defined due to Remark 6 and the compactness of B. Remember that for (Ϝn) a sequence of non empty sets LiϜn is the set comprised of the limit points of any possible sequence (xn) such thatxn∈Ϝn, and LsϜn is the one comprised of the analogous cluster points. Also Level≤a(cn(θ,·))≑{x∈B:c(θ,x)≤a} for a∈ℝ.Lemma 9.
Under Assumptions1 and 5,
(3)limsupninfBcn(θ,β)≤infBc(θ,β)Pa.s.Proof.
Consider the family ofθ-parameterized correspondences epin(ω,θ)≑epi(cn(ω,θ,·)). Due to the fact that B is locally compact, epin(ω,θ) is a random closed set in the sense of the previous paragraph, that is, a ℬ(BF)/𝒥⊗ℬ(Θ)-measurable correspondence. Hence epin(ω,·) is an ℬ(BF)/𝒥-measurable correspondence due to the measurability of the relevant projection. Now due to Assumption 5 (see Section 4) we have that for large n and for all ω in an element of 𝒥 of unit probability epin(ω,θ)∩B×(-∞,aθ)≠∅ since B×(-∞,aθ) is open in the relevant product topology. Hence infBcn(ω,θ,β)≤infBc(θ,β) for all ωdescribed previously.The next result will be used for the proof of Proposition8(2)-(3).Lemma 10.
Under Assumptions1 and 5 there exists a sequence of random variables defined on Ω, say an* (It is obvious from the proof that an* depends also on θ). Such that an*→infB(c(θ,β))Pa.s. and
(4)Li(Level≤an*(cn(θ,·)))⊇b(θ)Pa.s.Proof.
From the sequential implication of epiconvergence in Remark6, we have that, for any x∈b(θ), there exists a measurable xn such that xn→xPa.s. Obviously for an,x=cn(θ,xn) which is measurable, we have that xn∈Level≤an,x(cn(θ,·)). Since b(θ) is compact, it is totally bounded and therefore, for any ε>0, there exist an m(ε)∈ℕ and {yi,i=1,…,m(ε)}⊂b(θ), such that the collection of balls (in B) 𝒪(yi,(ε/2)) covers b(θ). For some real sequence qn→0, consider {yi,i=1,…,m(qn)} and extract analogously random sequences (yi,n) such that yi,n→yiPa.s. Define an,yi=cn(θ,yi,n) and n*(n)=min{n′:|an′,yi-aθ|≤(qn/2),Pa.s.foralli=1,…,m(qn)} which is well defined due to Egoroff’s theorem and the fact that m(qn) is finite. Obviously n* is nondecreasing in n. Then define an*=max{an*,yi,i=1,…,m(qn)} which is measurable and an*→aθPa.s. Then for x∈b(θ), it follows that, for any ε>0, there exist an n and a measurable xn∈Level≤an*(cn(θ,·)) such that d(xn,x)<εPa.s.Proof of Proposition8.
For (1) we have first that for any measurable nonnegativeεn that need not converge to zero if xn∈βn#(θ,εn) measurable, such that a subsequence xnk→xPa.s. 									(5)c(θ,x)≤liminfnkcnk(θ,xnk)Pa.s.=liminfnk(cnk(θ,xnk)+εnk)Pa.s.≤limsupnk(cnk(θ,xnk)+εnk)Pa.s.=limsupnk(infBcnk(θ,β)+εn)Pa.s.≤infBc(θ,x)+εnPa.s.,
where that last inequality follows from Lemma 9. This establishes that for any nonnegative random variable εn(6)Ls(βn#(θ,εn))⊆εn-argminBc(θ,β)Pa.s.
Now (1) follows from the fact that ε-argminB⊆ε′-argminB if ε≤ε′. For (2) notice that, from the definition of the Fell topology in Section 4 for any ε>0, we have that for large n, epin(ω,θ)∩B×[aθ-2ε,aθ-ε]=∅Pa.s. since B×[aθ-2ε,aθ-ε] is compact in the relevant product topology. This implies that liminfninfBcn(θ,β)≥aθPa.s. and in conjunction with Lemma 9 that infBcn(θ,β)→aθPa.s. Then using Lemma 10 set εn*=an*-aθ which is obviously measurable and converges to zero Pa.s. (3) follows from (1) the single valuedness of b(θ) and the compactness of B.The proof of Lemma10 implies that the sequence (εn*)n∈ℕ that appears in Proposition 8(2) is nonunique. However the fact that this implication does not hold for any sequence of nonnegative random variables that P a.s. converge to zero is the cause of that in what follows we can only prove the existence of strongly consistent indirect estimators among the set of the ones to be defined.


## 2.4. Upper Hemicontinuity of the Binding Correspondence

Proposition8 enables the use of δu(b(θ),βn#) as the inversion criterion. The following assumption concerns the upper continuity of the binding correspondence which along with the relevant properties of δu would imply the analogous continuity property for the particular inversion criterion and thereby facilitate the issue of existence and consistency of the IE to be defined.Assumption 11.
b is upper hemicontinuous that is, for any θ and θ*→θ, δu(b(θ),b(θ*))→0.The following proposition provides with sufficient conditions for this to hold. It essentially strengthens Assumption5 in that it requires that the relevant P  a.s. epiconvergence be continuous on Θ. Notice that its requirements are also stricter with respect to measurability compared to the ones in Assumption 5 since the former requires that for any θ the relevant set of P unit probability does not depend on the sequence that converges to θ.Proposition 12.
If, for anyθ, P  a.s. for any ω, any  θn→θ, and any β,(1)
liminfn→∞cn(ω,θn,βn)≥c(θ,β),   for all  βn  such that  βn→β,(2)
limsupn→∞cn(ω,θn,βn)≤c(θ,β),   for some  βn  such that  βn→β,
then Assumption11 holds.Proof.
Suppose that𝒟F0 metrizes 𝒯F on F0(B) (see Section 4). Then it is obvious that Proposition 12(1)-(2) is equivalent to the requirement that, for any θ and any θn→θ, 𝒟F0(epin(ω,θn),epi(θ)) converges to zero P a.s. Then, for any θ and any θn→θ, c(θn,·) epiconverges to c(θ,·) (𝒟F0(epi(θn),epi(θ))→0), that is, c(θ,·) is epicontinuous on Θ. This is due to the following standard argument: for an arbitrary θ and ε>0, we have that 𝒟F0(epin(ω,θ′),epi(θ))<(ε/2)P a.s. for any θ′ in some open neighborhood of θ and large enough n, due to the assumed form of convergence and Egoroff’s theorem. By the same reasoning 𝒟F0(epin(ω,θ′),epi(θ′))<(ε/2)P a.s. for any such θ′. The result follows from the fact that epi(θ) is independent of ω. This along with 3.1 of Theorem 5.3.4 and proposition Appendix D.2 of [22] implies that the composite mapping θ→c(θ,·)→argminBc(θ,β) is appropriately continuous.An examination of the proof of Proposition8 shows that if the conditions (1)-(2) of Proposition 12 hold then Proposition 8 could be restated with δu(b(θ),βn#(θn,εn)) replacing δu(b(θ),βn#(θ,εn)) when this occurs. This means that the premises of Proposition 12 render βn# strongly continuously (upper in the first case or Fell in the second and third cases of Proposition 8) b(θ)-consistent. Furthermore the compactness of Θ then implies that these notions are also uniform with respect to Θ.Remark 13.
Proposition12(1)-(2) would obviously be implied if cn(ω,θ,β) is P a.s. jointly continuous and converges jointly uniformly P a.s. to c(θ,β). Since we allow cn and/or c to assume extended values, the relevant notion of uniform convergence must also be extended as in Definition 7.12 of [26]. The following lemma provides with a set of even weaker sufficient conditions than extended jointly uniform P a.s. convergence when cn has the form of an arithmetic mean with respect to stationary and ergodic processes.Lemma 14.
Suppose thatcn(ω,θ,β)=(1/n)∑i=1n‍mi(ω,θ,β), (mi(θ,β))i∈ℤ is ergodic for any (θ,β), cn(ω,·,·):Θ×B→ℝ¯ is jointly continuous P a.s., there exists a finite open cover of Θ×B, such that E|inf(θ,β)∈Am0(ω,θ,β)|<+∞, for any A in the cover, E(m0(ω,θ,β*)) assumes values in ℝ¯ for any θ∈Θ, and β* in a countable dense subset of B. Then Proposition 12 holds.Proof.
Proposition12(1) follows from the fact that the assumption framework of the lemma implies condition 𝒞0 and thereby Theorem 2.3 of [23], which implies the joint P a.s. epiconvergence of cn to Em0. For Proposition 12(2) notice that the separability of B and the P a.s. continuity of m0 implies the existence of a countable dense B* such that for any β and any δ>0 there exists a β*∈B* such that
(7)limsupn→∞cn(ω,θn,β)≤limsupn→∞(cn(ω,θn,β)-cn(ω,θn,β*))+limsupn→∞cn(ω,θn,β*)Pa.s.≤δ+limsupn→∞cn(ω,θn,β*)Pa.s.
By assumption the subset of Ω of P unit probability can be chosen independent of δ and δ can be chosen arbitrarily small. Hence, Proposition 12(2) would be implied for βn=β, if, for any θ, β*∈B*, Pa.s. and any θn→θ,
(i)limsupn→∞cn(ω,θn,β*)≤Em0(θ,β*)
due to the countability of B*. Notice that ((mi(θ,β*)∧-ρ)∨ρ)i∈ℤ is also stationary-ergodic for any ρ>0 (see, e.g., Proposition 2.1.1. of [24]), and hence the uniform version of Birkhoff’s LLN implies that
(8)1n∑i=1n((mi(ω,θn,β*)∧-ρ)∨ρ)
converges P a.s. to ∫-ρρ‍m0(ω,θ,β*)dP(ω) for any ρ>0. Due to the separability of ℝ the subset of Ω of P unit probability can be chosen independent of ρ. Hence from Definition 7.12 of [26] we obtain (i).This lemma explores sufficient conditions for the required continuity ofb solely via restrictions on the behavior of cn which in applications is generally more analytically tractable than c. Furthermore it combines joint epi-onvergence with pointwise (on B) extended uniform (with respect to θ) almost sure convergence. Finally notice that analogous result would also hold if ergodicity is replaced by any kind of mixing condition that would justify the LLNs used in the previous proof or implied in Remark 13.


## 2.5. Definition, Existence, and Consistency of the Indirect Estimator

We are now ready to define theIE and explore the issues of its existence and consistency. Proposition 8 along with the measurability of the auxiliary correspondence and the upper hemicontinuity of b facilitate the use of δu(b(θ),βn#(θ0)) for some distinguished θ0∈Θ, for the definition of the IE and the subsequent existence argument. Again an almost surely nonnegative random variable will assume the role of the “optimization error” in this second step of the estimation procedure.Definition 15.
The indirect correspondenceθn#(ω,θ0,εn,εn#) satisfies
(9)θn#(ω,θ0,εn,εn#)=εn#-argminΘδu(b(θ),βn#(θ0,εn))≑{infΘδu(b(θ),βn#(θ0,εn))+εn#θ*∈Θ:δu(b(θ*),βn#(θ0,εn))≤infΘδu(b(θ),βn#(θ0,εn))+εn#},
where εn# is a nonnegative random variable defined on Ω.The dependence ofθn# on θ0 follows form the analogous dependence of βn# which in turn follows from the dependence of yn on θ0. Obviously in practice this dependence is not “visible.” We are initially concerned with the question of existence of the IE. We again suppress the dependence of θn on ω when there is not a risk of confusion.Proposition 16.
Under Assumptions1 and 11  θn# is ℬ(ΘF)/𝒥-measurable, P a.s. nonempty, compact valued correspondence.Proof.
First, notice that due to LemmaA.9, Proposition 4 (implied by Assumption 1), Assumption 11, and the facts that βn#(θ0,εn) is independent of θ and b(θ) is independent of ω, we obtain that δu(b(θ),βn#(θ0,εn)) is ℬ(ℝ¯)/ℬ(ΘF)⊗𝒥-measurable. Due to Lemma A.4 and Proposition 4  δu(b(θ),βn#(θ0,εn)) is P a.s. lsc and therefore a normal integrand. It is also P a.s. proper due to the fact that b and βn# are P a.s. compact valued. Hence the result follows from Proposition 4 where cn=δu(b(θ),βn#(θ0,εn)), when we consider B=Θ and Θ={θ0} (the left-hand sides correspond to the notation of the latter lemma).The fundamental selection theorem (Theorem 2.13 of [22]) would also enable the definition of the IE as a measurable function with values in Θ. Having established existence we turn to the issue of consistency. We need an assumption of indirect identification that is essentially derived from the form of the roots of the hemimetric used. Notice that this assumption along with the proof of the following proposition justifies thedefinition of the Definition 15 by an analogy principle. Mathematically both the definition and the existence argument do not require the following assumption in order to be valid.Assumption 17.
Ifθ≠θ0⇒b(θ0)-b(θ)≠∅.Remark 18.
This condition is weaker than a condition of the form “Ifθ≠θ0⇒b(θ0)∩b(θ)=∅” and stronger that a condition of the form “If θ≠θ0⇒b(θ0)≠b(θ).” The latter cannot be used due to the properties of δu upon which the definition of the IE is based. In the case that the binding correspondence is single valued, these become equivalent. This also makes evident the claim that if the auxiliary estimator is defined by a measurable selection of βn# the corresponding identification condition cannot be weaker than the one above.The main result of the current section follows. It merely concerns theexistence of strongly consistent IE inside the established framework. Denote by θ(θ0) the argminΘδu(b(θ),b(θ0)) which is nonempty and compact due to the compactness of Θ, the properness of δu, the hemicontinuity of b, and Lemma A.3. Obviously, θ0∈θ(θ0), while {θ0}=θ(θ0)ifandonlyif Assumption 17 holds. Since we once again are dealing with compact valued correspondences, convergence is metrized by δu and/or δ.Proposition 19.
Under Assumptions1, 5, and 11 and if εn#→0P a.s. then(1)
δu(θ(θ0),θn#(ω,θ0,εn*,εn#))→0P  a.s. where  εn*  is defined in Proposition8(2),(2)
if   b(θ0)  is singleton then  δu(θ(θ0),θn#(ω,θ0,εn,εn#))→0P  a.s. for any  εn→0P  a.s.,
if furthermore Assumption17 holds then(1*)
δ({θ0},θn#(ω,θ0,εn*,εn#))→0P  a.s. where  εn*  is defined in Proposition8(2),(2*)
if   b(θ0)  is singleton then  δ({θ0},θn#(ω,θ0,εn,εn#))→0P  a.s. for anyεn→0Pa.s.Proof.
First notice that due to Proposition8(2), Assumption 11, and Lemma A.6 we have that for any θ and θn→θ(10)liminfnδu(b(θn),βn#(θ0,εn*))≥δu(b(θ),b(θ0))Pa.s.
and that, for any θ and θn=θ, due to Lemma A.4(11)limsupnδu(b(θn),βn#(θ0,εn*))≤δu(b(θ),b(θ0))Pa.s.
Hence (1) follows from Proposition 8(1) for cn=δu(b(θn),βn#(θ0,εn*)) and if we denote by B (in the notation of this lemma) the Θ space and by Θ (in the notation of this lemma) {θ0}. (2) follows in the same manner if we replace any invocation of Proposition 8(2) with Proposition 8(3). Finally, notice that if Assumption 17 holds, then θ(θ0)={θ0} establishing (1*) and (2*) via another use of Proposition 8(3).If Assumption17 does not hold then the implications of Proposition 19(1)-(2) correspond to the fact that the statistical model is only indirectly set identifiedgiven this framework. They are trivial when θ(θ0)=Θ and the closer to zero δu({θ0},θ(θ0)) is the more informative they become. We once again point out that the implications Proposition 19(1) and (1*) merely explore the issue of the existence of strongly consistent estimators among those that comply with Definition 15. The properties of the δu function along with Proposition 8 do not permit for a stronger result without strengthening the assumption framework. Finally notice that this framework enables both the definition and the result on consistency of the IE to be derived via the use of exact same notions that were used for the analogous results concerning the auxiliary one.


## 2.6. Extension

In most casesb is analytically unknown even if several of its properties, such as some of the ones discussed above, can be established. In these cases the estimators defined in Definition 15 are infeasible. However, it may be the case that (possibly) stochastic and algorithmically feasible approximations of the unknown b can be used for the construction of several other classes of feasible IE. In such a context the results derived previously could be used to describe properties of such approximations that would imply that these estimators are well defined and among them strongly consistent ones exist. Let κn(ω,θ) denote such an approximation. We readily obtain the following result for the indirect estimator defined by the substitution of b with κn in Definition 15.Proposition 20.
Consider theIE defined by
(12)ϑn#(ω,θ0,εn,εn#)=εn#-argminΘδu(κn(ω,θ),βn#(θ0,εn)),
where εn,εn# as before.(i)
Under Assumption1 and if  κn(ω,θ)  is  ℬ(BF)/𝒥⊗ℬ(ΘF)-measurable,  P  a.s. compact valued and upper hemicontinuous, then  ϑn#  is  ℬ(ΘF)/𝒥-measurable,  P  a.s. nonempty, compact valued correspondence.(ii)
If moreover Assumption5 holds and  P  a.s. for any  θ  and any  θn→θ, δ(b(θ),κn(ω,θ))  and  δu(b(θ),κn(ω,θn))  converge to zero then the implications of Proposition19(1)-(2) hold also for  ϑn#. If furthermore Assumption17 holds of then the implications  Proposition 19(1*)-(2*)hold also for  ϑn#.Proof.
(i) As in the proof of Proposition16 from Lemma A.9, Proposition 4 the P a.s. upper hemicontinuity of κn the facts that βn#(θ0,εn) is independent of θ, and κn is jointly measurable, we obtain that δu(βn#(θ0,εn),b(θ)) is ℬ(ℝ¯)/ℬ(ΘF)⊗𝒥-measurable. Due to Lemma A.4 and Proposition 4  δu(κn(ω,θn),βn#(θ0,εn)) is P a.s. lsc and therefore a normal integrand. It is also P a.s. proper due to the fact that κn and βn# are P a.s. compact valued. Hence the result follows from Proposition 4. (ii) It suffices to prove that δu(κn(ω,θ),βn#(θ0,εn))→eδu(b(θ),b(θ0))P a.s. when βn#(θ0,εn)  satisfies the implications (2) or (3) of Proposition8. The rest would then follow as in the proof of Proposition 20. Notice that for any θ and any θn→θ(13)liminfnδu(κn(ω,θn),βn#(θ0,εn*))≥δu(b(θ),b(θ0)),Pa.s.
due to Lemma A.6, the definition of βn# and the P a.s. continuous w.r.t θ upper convergence of κn to b. Moreover for θn=θ(14)limsupnδu(κn(ω,θ),βn#(θ0,εn*))≤limsupnδu(κn(ω,θ),b(θ))+limsupnδu(b(θ),βn#(θ0,εn*)),Pa.s.≤limsupnδ(κn(ω,θ),b(θ))+δu(b(θ),b(θ0)),Pa.s.,
where the first inequality follows from the triangle inequality and the second from the definition of δ and the P a.s. Fell convergence of βn# to b(θ0) and Lemma A.4. Due to the P a.s. pointwise with respect to θ Fell convergence of κn to b we have that limsupnδ(κn(ω,θ),b(θ))=0Pa.s. and therefore we obtain the needed result.Notice that this proposition generalizes the results of Propositions16 and 20. For a simple example consider the case where κn=βn#. This is possible when by some sort of resampling technique (e.g., bootstrap or Monte Carlo) realizations of the yn random elements are available to the practitioner for any θ and thereby so is βn#(θ,ϵn) for any θ and some optimization error ϵn  independent of θ. Then a feasible IE can be defined by the approximate minimization of δu(βn#(θ,ϵn),βn#(θ0,εn)) with respect to θ. In this case the joint measurability of βn# would follow from the joint measurability of cn and ϵn, the separability of B, and the subsequent joint measurability of the relevant projection. The P a.s. upper hemicontinuity of βn# would follow from an easy extension of the implication Proposition 8(1) if Assumption 1 is strengthened so that the mapping θ→epin(ω,θ) is P a.s. Fell continuous. The P a.s. joint continuity of cn would suffice. Then the P a.s. pointwise Fell convergence to b would follow as in Proposition 8(2) or (3) and the P a.s. continuous w.r.t θ upper convergence to b would follow if Proposition 12 holds with the set of unit probability independent of θ. It would suffice that cn is P a.s. jointly continuous and converges to c jointly uniformly.


## 3. Examples

In this section we consider four simple examples that represent some of the previous results. The first concerns the case of a linear semiparametric model, the second a model comprised of Levy processes, and the final two emerge in the context of conditionally heteroskedastic ones. In any of these,Θ is a compact subset of ℝp and Ba compact subset of ℝq. In the second and the fourth ones the binding function is actually single valued (hence a fortiori compact valued) and 1-1 enabling the direct application of Proposition 19(2*). The first and second examples include cases in which the IE can be interpreted as performing “inconsistency” correction to the auxiliary one.Example 21 (semi-parametric linear model with linear auxiliary).
Consider then×p and n×q dimensional random matrices X(ω) and Z(ω), respectively, where n≥q≥p. Suppose that (X′X/n)→MX′X, (Z′Z/n)→MZ′Z, (Z′X/n)→MZ′XP a.s., where rank(MX′X)=rank(MZ′X)=p and p≤l≑rank(MZ′Z)≤q. For u an n×1 random vector, let the underlying statistical model be the set of “regressions” Y(ω,θ)=Xθ+u, θ∈Θ. For B a large enough compact and convex subset of ℝq and any β∈B, let cn(ω,θ,β)=(1/n)(Y-Zβ)′(Y-Zβ), which clearly satisfies Assumption 1 due to continuity with respect to β and the compactness of B. Obviously, cn is constructed by the auxiliary set of regression with respect to Z. Proposition 4 ensures the existence of βn which in the light of the previous can be interpreted as an OLSE in the context of the auxiliary model. Let 𝒫:ℝq→MZ′ZB be the (generally nonlinear) projection defined by the optimization problem
(15)argminx∈MZ′ZB∥x-y∥
for y in ℝq. 𝒫 is well defined due to the compactness and the convexity of B and the linearity and continuity of MZ′Z and continuous. Furthermore, for any y∈col(MZ′Z), consider the linear system MZ′Zx=y, which is always satisfied by any member of the coset Ky+Hq-l, where K is a matrix of rank l and Hq-l is a q-l-dimensional subspace of ℝq, which is trivial if and only if l=q whereas K=MZ′Z-1 and maximal in the case that l=p. For MX′u∈ℝp, MZ′u∈ℝq, Mu′u∈ℝ assume that (X′u/n)→MX′u, (u′u/n)→Mu′u and (Z′u/n)→MZ′uP a.s. The previous imply the joint uniform P a.s. convergence of cn to
(16)c(θ,β)=θ′(MX′Xθ+2MX′u)+β′(MZ′Zβ-2(MZ′Xθ+MZ′u))+Mu′u
which implies both Assumptions 5 and 11 (via Proposition 12 and Remark 13). Notice that
(17)b(θ)=B∩(K𝒫(MZ′Xθ+MZ′u)+Hq-l)
due to the convexity of c(θ,β) with respect to β for any θ and the definition of B. If 𝒫(MZ′Xθ0+MZ′u)-𝒫(MZ′Xθ+MZ′u)∉K-1Hq-l for any θ≠θ0 then Assumption 17 applies. (More precisely we have that b(θ)∩b(θ0)=∅.) Hence, Proposition 19(1*) implies the existence of a consistent IE for θ0∈Θ. In the special case where X=Z and MX′u∈MX′X(B-θ) then b(θ)={θ-MX′X-1MX′u} and Proposition 19(2) implies that any IE defined by Definition 15 can be perceived as an “inconsistency corrector” of the underlying OLSE for θ.We know consider the case of the estimation of the drift of a continuous time cadlag process.Example 22 (the drift of a levy process with bounded jumps).
LetW denote a standard Bownian motion and v a finite measure on the Borel algebra of ℝ-{0}, such that v(A)=0 when A⊆(-∞,-C2)∪(-C1,C1)∪(C2,+∞) for 0<C1<C2. Obviously v is a Levy measure (see paragraph 1.2.4 of [27]). For p=1 consider the stochastic process on ℝ+ defined by the following Levy-Ito decomposition (see Theorem 2.4.16 of [27])
(18)Xt(ω)=θt+Wt(ω)+∫|x|∈[C1,C2]‍xN(t,dx)(ω),
where N denotes the independence to W Poisson random measure of ℝ+×([-C2,C1]∪[C1,C2]) the existence of which is established by Theorem 2.3.6 of [27]. Let the underlying statistical model be the set of the previous stochastic processes and for B a large enough compact subset of ℝ and any β∈B, let cn(ω,θ,β)=(1/n)∑t=1n‍(yt-β)2, where yt=exp(Xt-Xt-1)-1. This can be perceived to emerge as an approximate likelihood function of the auxiliary model that contains the relevant discretizations of the processes that satisfy the SDE(19)dyt=βytdt+ytdWt
for each β∈B. Obviously Assumption 1 is satisfied, due to continuity with respect to β and the compactness of B. Proposition 4 ensures the existenceof βn which in the light of the previous can be interpreted as an (approximate) MLE in the context of the auxiliary model. Furthermore since
(20)|∫|x|∈[C1,C2]‍x(N(t,dx)-N(t-1,dx))|≤∫|x|∈[C1,C2]‍|x|(N(t,dx)-N(t-1,dx))≤C2(N(t,[C1,C2])-N(t-1,[C1,C2]))
and N(t,[C1,C2])-N(t-1,[C1,C2])~i.i.d.Poiss(v([C1,C2])) independent of W, we have that
(21)Eexp(Xt-Xt-1)=exp(θ+12)C
for
(22)0<C≤exp(-v([C1,C2])(1-exp(C2)))E(exp(Xt-Xt-1))2=exp(2(θ+1))C*
for
(23)0<C*≤exp(-v([C1,C2])(1-exp(2C2))).
Due to the definition of X the process y is i.i.d. and this along with the compactness of Θ, and B and the existence of the previous moments imply the joint uniform P a.s. convergence of cn to
(24)c(θ,β)=exp(2(θ+1))C*-2exp(θ+12)C(1+β)+(1+β)2
which implies both Assumptions 5 and 11 (via Proposition 12 and Remark 13). If B⊇exp(Θ+(1/2))C-1 then
(25)b(θ)={(exp(θ+12)C-1)}.
In this case Assumption 17 applies and therefore Proposition 19(2*) implies that any IE defined by Definition 15 is consistent for any θ0∈Θ. When v([C1,C2])=0 (and therefore v=0) whereas C=1 the IE can be perceived as an “inconsistency corrector” of the underlying MLE for the estimation of the drift of a geometric Brownian motion (see, e.g., paragraph 6.1.1 of [13]).For the last pair of examples, letz:Ω→ℝℤ be an i.i.d. (double infinite) sequence of random variables, with Ez0=0 and Ez02=1. Consider a random element σ2:Θ×Ω→(ℝ+)ℤ, with the product space Θ×Ω equipped with ℬ(Θ)⊗𝒥 with σt2(θ) independent of (zi)i≥t, forallt∈ℤ, forallθ∈Θ. Analogously, define the random element y:Θ×Ω→(ℝ)ℤ as
(26)(yt(ω)(θ))t∈ℤ,θ∈Θ=(zt(ω)σt2(ω)(θ))t∈ℤ,θ∈Θ.Then for allθ∈Θ, (yt(θ))t∈ℤ is called a conditionally heteroskedastic process, while the random element (yt(ω)(θ))t∈ℤ,θ∈Θ a conditionally heteroskedastic model. Our examples will solely concern ergodic heteroskedastic models. (The establishment of the ergodicity is initiated by the analogous establishment for (σt2(θ))t∈ℤforallθ∈Θ. Sufficient conditions for that are described and employed in a variety of heteroskedastic models in Chapter 5 of [24] via Theorem 5.2.1. Then the ergodicity of (yt(θ))t∈ℤ and (yt2(θ))t∈ℤforallθ∈Θ follows from the definition of z,y, the previous assumption and Proposition 2.2.1 of [24]).Example 23 (instrumental variables estimation in regressions on squared ARCH(1) processes).
Letσ4=Ez04<+∞ and 0<δ<(1/σ4). Suppose that a∈Θ=[0,δ] and consider the stochastic dif and only iference equation
(27)σt2(a)=1+azt-12σt2(a).
Due to the fact that σ4>1 Theorem 5.2.1. of [24] implies that for any α∈Θ the equation admits a unique stationary and ergodic solution defining the analogous ARCH(1) process. Consider the random vector Y(a)=(yt2(a))t∈{1,…,n} and the n×2 dimensional random matrices
(28)Z(a)=(1y-12(a)⋮⋮1yn-22(a)),X(a)=(1y02(a)⋮⋮1yn-12(a))
jointly measurable with respect to 𝒥⊗ℬ(Θ), where n>2 and ergodic for any a∈Θ. For β=(β1β2)∈B=[1,1+δ]×[-δ,δ], let cn(ω,a,β)=∥(1/n)Z′(a)(Y(a)-X(a)β)∥ which clearly satisfies Assumption 1 due to joint continuity with respect to (a,β) the compactness of B, the joint measurability, and the fact that cn is defined via composition with a norm. This consideration is motivated from the AR(1) representation of the ARCH(1) process with respect to the martingale dif and only iference noise vt=(zt2-1)σt2(a) (see, e.g., [28]) and cn can be perceived to emerge from an auxiliary model that is consisted of the set of “auxiliary” regression functions of Y on Xβ, along with the instrumental variables appearing in the columns of Z where obviously the ith element in any column is clearly orthogonal to vt for i≤t. Proposition 4 ensures the existenceof βn which in the light of the previous sentence can be interpreted as an IV estimator in the context of the auxiliary model. Due to the compactness of B, the definition of the ARCH(1) model, and the definitions of Θ and σ4 we have that
(29)Esupa,β∥1nZ′(a)(Y(a)-X(a)β)∥≤C1∥(111-δ11-δ1(1-δ)(1-δ2σ4))∥+C2∥(11-δ1+δ(1-δ)(1-δ2σ4))∥<+∞
for C1,C2>0 which along with (the uniform version of) Birkhoff’s Ergodic theorem (see, e.g., [24], Theorem 2.2.1) implies both Assumptions 5 and 11 (via Proposition 12 and Remark 13) for
(30)c(a,β)=∥(111-a11-a1(1-a)(1-a2σ4))(β1β2)-(11-a1+a(1-a)(1-a2σ4))∥.
In fact a simple calculation shows that
(31)b(a)={{(1a)},a∈(0,δ],{(β1,1-β1),β1∈[1,1+δ]},a=0
which clearly implies Assumption 17. Hence Proposition 19(2*) implies that any IE defined by Definition 15 is consistent if a0∈(0,δ], and Proposition 19(1*) implies the existence of an analogously consistent IE when a0=0. It is easy to see that, given βn,
(32)θn#(ω,θ0,εn*,0)={{a}ifβn⊆b(a)mid(projβ2(βn))ifβn⊈b(a),mid(projβ2(βn))≥0,{0}ifβn⊈b(a),mid(projβ2(βn))<0,
where projβ2 denotes projection to the β2-axis and mid(A) denotes the midpoint of the smallest interval that contains A. Notice that in our case mid(projβ2(βn)) is well defined due to the fact that βn is P a.s. compact valued hence its projβ2(βn) is a P a.s. compact subset of the real line. Finally and due to the fact that bootstrap resampling techniques are readily available in the context of this model, Proposition 20 implies also the analogous properties for IE defined by κn when this equals the auxiliary estimator derived from bootstrap resampling for any a.The final example is about an asymmetric heteroskedastic process.Example 24 (-cn is the quasilikelihood function of an approximate to QARCH(1) model).
LetE|z0|<1 and δ>0, and consider for γ∈Θ=[-δ,0]ϖ0,a0>0, ϖ=ϖ0+(δ2/4a0), and a0<exp(-2Eln|z0|) the stochastic dif and only iference equation
(33)σt2(γ)=ϖ+a0zt-12σt2(γ)+γzt-1σt(γ).
For any γ∈Θ, the previous defines a unique stationary and ergodic QARCH(1) volatility process with existing log moments that is uniformly bounded from below away from zero (see Lemmas 2.1 and 3.4 and Remark R.2 of Arvanitis and Louka [29]). Notice that Jensen’s inequality allows a0≥1 which in turn implies that Eσt2(γ)=+∞. For γ*∈B=[0,δ] consider the process defined by
(34)ht(γ,γ*)=ϖ+a0yt-12+γ*|yt-1|1yt-1<0=Pa.s.ϖ+a0yt-12+γ*|yt-1|1zt-1<0,
where ht(γ,γ*) is well defined due to the definition of B and it is stationary and ergodic with existing log moments due to the previous and Proposition 2.1.1. of [24]. Now consider
(35)cn(ω,γ,γ*)≑1n∑i=1n‍(-lnyi2(γ)hi(γ,γ*)+yi2(γ)hi(γ,γ*)),
where -cn can be considered as an approximation of (a monotonic transformation of) the conditional quasilikelihood function of the auxiliary conditionally heteroskedastic model defined by 3 and B. Also the ergodicity of (cn) for any (γ,γ*) follows from the previous and Proposition 2.1.1. of [24]. (In practice cn(ω,γ,γ*) is unknown but approximated by an analogous c^n(ω,γ,γ*) dependent on nonergodic solutions of the stochastic dif and only iference equation that defines h based on arbitrary initial conditions. In this case, due to ergodicity, Proposition 5.2.12 of [24] can be employed in order to ensure that supB|cn(ω,θ,β)-c^n(ω,θ,β)| converges almost surely to zero for any θ∈Θ (see the first part of the proof of Theorem 5.3.1 of [24]), thereby facilitating the asymptotic analysis of minimizers of c^n(ω,θ,β) by the analogous analysis of minimizers of cn(ω,θ,β).) Assumption 1 follows readily from the form of cn and the Pa.s. continuity with respect to (γ,γ*). Hence βn is well defined and can be interpreted as an approximate QMLE in the context of the auxiliary model. Now, consider an arbitrary finite open cover of B and notice that
(36)EinfA∩B(-lnϖ+a0y02+γy0ϖ+a0y02+γ*|y0|1z0<0+z02ϖ+a0y02+γy0ϖ+a0y02+γ*|y0|1z0<0)≥-EsupA∩Blnϖ+a0y02+γy0ϖ+a0y02+γ*|y0|1z0<0+E(ϖ+a0y02+γy0ϖ+a0y02+δ|y0|1z0<0)≥-Eln((1+γy0ϖ+a0y02)1z0<0)+E((1+γy0ϖ+a0y02)1z0≥0)>-∞
and that
(37)EinfA∩B(-lnϖ+a0y02+γy0ϖ+a0y02+γ*|y0|1z0<0+z02ϖ+a0y02+γy0ϖ+a0y02+γ*|y0|1z0<0)≤1-E(ln(1+γy0ϖ+a0y02))1z0≥0+E(1+γy0ϖ+a0y02)<+∞
for A an arbitrary member of the partition. Notice also that -2Eln|z|− E((ϖ+a0y02+γy0)/(ϖ+a0y02+γ*|y0|1z0<0))+E((ϖ+a0y02+γy0)/(ϖ+a0y02+γ*|y0|1z0<0))>-∞ for all (γ,γ*) due to the fact that
(38)Elnϖ+a0y02+γy0ϖ+a0y02+γ*|y0|1z0<0≥Eln(1+γy0ϖ+a0y02)>-∞.
Hence, Remark 6 implies that Assumption 5 holds with
(39)c(γ,γ*)=-2Eln|z|-Elnϖ+a0y02+γy0ϖ+a0y02+γ*|y0|1z0<0+Eϖ+a0y02+γy0ϖ+a0y02+γ*|y0|1z0<0.
Notice that c(0,γ*) is uniquely minimized at γ*=0 (see, e.g., Part 1. of the proof of Theorem 5.3.1. of [24] to obtain the analogous arguments along with the fact that ((ϖ+a0y02)/(ϖ+a0y02+γ*|y0|1z0<0))=Pa.s.1 if and only if γ*=0). When γ≠0 then c(γ,-γ)<c(γ,0) due to the fact that
(40)-Elnϖ+a0y02+γy0ϖ+a0y02-γ|y0|1z0<0+Elnϖ+a0y02+γy0ϖ+a0y02=E(ln(1+γy0ϖ+a0y02)1z0<0),Eϖ+a0y02+γy0ϖ+a0y02-γ|y0|1z0<0-Eϖ+a0y02+γy0ϖ+a0y02=-E(γy0ϖ+a0y021z0<0)<0
and that when x>0, then ln(1+x)<x. Furthermore, using the fact that by 3 h is P a.s. two times dif and only iferentiable with respect to γ* for γ*≠0 and since
(41)Esupγ*|(ϖ+a0y02+γ*|y0|1z0<0ϖ+a0y02+γy0-1)×ϖ+a0y02+γy0(ϖ+a0y02+γ*|y0|1z0<0)2|y0|1z0<0|≤(δ|γ|-1)×E|ϖ|y0|+a0|y0|3+γy02(ϖ+a0y02)21z0<0|<+∞,Esupγ*|(ϖ+a0y02+γy0(ϖ+a0y02+γ*|y0|1z0<0)2)z021z0<0|≤1ϖE|(1+γy0ϖ+a0y02)1z0<0|<+∞,Esupγ*|(ϖ+a0y02+γ*|y0|1z0<0ϖ+a0y02+γy0-1)×(ϖ+a0y02+γy0)2(ϖ+a0y02+γ*|y0|1z0<0)3z021z0<0|≤1ϖ(δ|γ|-1)E|(1+γy0ϖ+a0y02)21z0<0|<+∞
as well as dominated convergence, we have that
(42)∂c(γ,γ*)∂γ*=E(ϖ+a0y02+γ*|y0|1z0<0ϖ+a0y02+γy0-1)×ϖ+a0y02+γy0(ϖ+a0y02+γ*|y0|1z0<0)2|y0|1z0<0
which is zero if and only if γ*=-γ, and
(43)∂2c(γ,γ*)∂(γ*)2|γ*=-γ=E((ϖ+a0y02+γy0(ϖ+a0y02-γ|y0|1z0<0)2)z021z0<0)>0
establishing along with the previous that b(γ)={-γ}. This validates simultaneously both Assumptions 11 and 17. Notice that Assumption 11 could also be verified by the use of Lemma 14 due to the continuity of σt2 and ht with respect to the parameters, the existence of log moments, and the fact that ht is uniformly bounded from below away from zero. Then Proposition 19(2*) implies that any IE defined by Definition 15 is consistent for any γ0∈[-δ,0].


## 4. Conclusions

In this paper we generalize the definition ofIE and are occupied with the questions of existence and strong consistency. We allow for cases where the binding function is a compact valued correspondence. We have used conditions that concern the asymptotic behavior of the epigraphs of the criteria involved in the relevant procedures, a relevant notion of continuity for the binding correspondence and an indirect identification condition that restricts the behavior of the aforementioned correspondence. These results are generalizations of the analogous ones in the relevant literature and hence permit a broader scope of statistical models.First, notice that our framework could still be extended in the following manner. The established results would remain almost intact if the underlying parameter spaces were only locally compact under more restrictive assumptions on the behavior of the criteria involved. In such a case Proposition 4.2.1.(i) of [30] would permit the validity of the results, except for the compactness of the auxiliary and the binding correspondences, under the additional condition that P a.s. cn(ω,θ,·), c(θ,·) and δu(b(·),βn#(θ0,εn*)) have totally bounded level sets and nonempty argmins.Second, the present generalization is certainlynonunique. Again under stricter conditions on the behavior of the auxiliary criteria, possibly relevant to the ones in Proposition 3.42 of [31], the implication Proposition 8(2) could be strengthened to hold for any asymptotically null sequence of optimization errors. In this case in the definition of the IEδu could be replaced by δ and this would initially allow the identification condition in Assumption 17 to be replaced by the weaker “if θ≠θ0⇒b(θ0)≠b(θ).” If Assumption 11 was also strengthened to require Fell continuity then the strong consistency result would be valid for any  IE defined in this framework. We leave this for future research.We also leave for future research the questions of the definition and consistency ofIE when the κn appearing in Proposition 20 is some sort of integral of βn# (see, e.g., [32]) or some (possibly) stochastic approximation of it. The same holds for the issues of the establishment of the rates of convergence and the asymptotic distribution of the IE in this general framework. Notice that this limit theory could in principle be quite complex due to complexities in the analogous theory for βn# and/or to dif and only iferent properties of potential polynomial approximations for dif and only iferent selections of b around θ and so forth. The establishment of primitive conditions that guarantee the indirect identification Assumption 17 is also of separate interest.



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