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

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.

1. Introduction

Indirect estimators (henceforth IE) 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 the existence 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 some GARCH 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 the definition 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 of IE 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 of IE 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 the IE. 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,𝑑_Θ) and (𝐵,𝑑_𝐵) 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.

The auxiliary criterion is a function 𝑐_𝑛(𝜔,𝜃,𝛽)∶Ω×Θ×𝐵→ℝ, which is of the form 𝑞_𝑛(𝑦_𝑛,𝛽) with 𝑞_𝑛∶𝐾_𝑛×𝐵→ℝ, for 𝑦_𝑛∶Ω×Θ→𝐾_𝑛, for 𝐾_𝑛 some appropriate space, usually homeomorphic to ℝ^𝑚𝑛 for some 𝑚>0. 𝑦_𝑛 represents the sample. 𝑞_𝑛 reflects part of the structure of an auxiliary model, a statistical model defined on the measurable space (𝐾_𝑛,ℬ(𝐾_𝑛)), 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(𝐾_𝑛), with 𝒫(𝐾_𝑛) the set of probability measures on 𝐾_𝑛). 𝑦_𝑛(⋅,𝜃) is measurable for any 𝜃 and thereby represents the underlying statistical model which is essentially the set {𝑃∘𝑦_𝑛⁻¹(⋅,𝜃),𝜃∈Θ}. These two models need not coincide.

We abbreviate with 𝑃 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 enable first 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 of 𝑐_𝑛 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)𝑐_𝑛 is ℬ(ℝ)/𝒥⊗ℬ(Θ)⊗ℬ(𝐵) measurable.(2)𝑐_𝑛(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 Section 3, the issue of joint measurability is handled easily due to the fact that the 𝑐_𝑛’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 𝑐_𝑛 has the form of a quasilikelihood function it 𝑃 a.s. does not attain extended values. This is also the case in instances where 𝑐_𝑛 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 that 𝑐_𝑛(⋅,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 𝑐_𝑛 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 correspondence 𝛽^#_𝑛(𝜔,𝜃,𝜀_𝑛) satisfies 𝛽^#_𝑛𝜔,𝜃,𝜀_𝑛=𝜀_𝑛-argmin_𝐵𝑞_𝑛𝑦_𝑛,𝛽=𝜀_𝑛-argmin_𝐵𝑐_𝑛≑𝜔,𝜃,𝛽𝛽∈𝐵∶𝑐_𝑛𝜔,𝜃,𝛽≤inf_𝐵𝑐_𝑛(𝜔,𝜃,⋅)+𝜀_𝑛, (1) where 𝜀_𝑛 is a 𝑃 a.s. nonnegative random variable defined on Ω.

Notice that the dependence of 𝛽^#_𝑛 on 𝜃 is due to the assumed dependence of 𝑦_𝑛on 𝜃. In practice 𝛽^#_𝑛 is evaluated by the minimization appearing in the first equality of the previous display.

Proposition 4. Under Assumption 1  𝛽^#_𝑛(𝜔,𝜃,𝜀_𝑛) is ℬ(𝐵_𝐹)/𝒥-measurable and 𝑃 a.s. non emptycompact valued for all 𝜃∈Θ.

Proof. For any 𝜃∈Θ𝛽^#_𝑛(𝜔,𝜃,𝜀_𝑛) is non empty and compact due to Assumption 1(2) and the compactness of 𝐵. (For the 𝜔 for which 𝜀_𝑛(𝜔)≠0 nonemptiness of 𝛽^#_𝑛(𝜔,𝜃,𝜀_𝑛(𝜔)) does not necessitate Assumption 1(2) due to the properties of the inf). Then the joint measurability of 𝑐_𝑛(⋅,𝜃,⋅) due to Assumption 1 along with Proposition 3.10(iii) in Chapter 5 of [22] guarantees the measurability for inf_𝐵𝑐_𝑛(𝜔,𝛽)+𝜀_𝑛. The result follows from the separability of 𝐵.

Obviously 𝛽^#_𝑛(𝜔,𝜃,0)=argmin_𝐵𝑐_𝑛(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” 𝜀_𝑛 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 𝛽^∗_𝑛∶Ω×Θ→ℝ termed as auxiliary selection, defined by 𝑐_𝑛(𝜔,𝜃,𝛽^∗_𝑛(𝜃))≤inf_𝐵𝑐_𝑛(𝜔,𝜃,𝛽)+𝜀_𝑛. 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

Assumption 1 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 function 𝑐∶Θ×𝐵→ℝ such that(1)∀𝜃∈Θ, 𝑃 a.s. for any 𝛽:(i)liminf_𝑛→∞𝑐_𝑛(𝜔,𝜃,𝛽_𝑛)≥𝑐(𝜃,𝛽) for all 𝛽_𝑛 such that 𝛽_𝑛→𝛽, and(ii)limsup_𝑛→∞𝑐_𝑛(𝜔,𝜃,𝛽_𝑛)≤𝑐(𝜃,𝛽) for some 𝛽_𝑛 such that 𝛽_𝑛→𝛽,(2)𝑐(𝜃,⋅) is proper ∀𝜃∈Θ.

Remark 6. For any 𝜃, 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 𝑐_𝑛=(1/𝑛)∑^𝑛_𝑖=1𝑚_𝑖(𝜔,𝜃,𝛽) and (𝑚_𝑖(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|inf_𝛽∈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 𝑐_𝑛 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 the argmin 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 𝑏(𝜃)≑argmin_𝐵𝑐𝜃,𝛽. (2) The following proposition provides its existence and is essentially similar to Proposition 4.

Proposition 7. Under Assumptions 1 and 5 the binding correspondence 𝑏 is nonempty compact valued.

Proof. It follows from Remark 6.

Both the auxiliary and the binding correspondence will be used for the definition of the IE via some intuition that utilizes an analogy principle. The following result explores their asymptotic relation. For its construction the hemimetric 𝛿_𝑢 is needed that is defined by 𝛿_𝑢(𝐴₁,𝐴₂)=inf{𝜀>0∶𝐴₂⊂𝑁_𝜀(𝐴₁)} where 𝑁_𝜀(𝐴₁)={𝑥∈𝐸∶𝑑(𝑥,𝐴₁)<𝜀}, where 𝑑(𝑥,𝐴₁)≑inf_𝑦∈𝐴1𝑑(𝑥,𝑦) and 𝐴₁, 𝐴₂ are nonempty closed subsets of some topological space. For several continuity and measurability properties of 𝛿_𝑢 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 𝛽^#_𝑛 with respect to to 𝑏.

Proposition 8. Under Assumptions 1 and 5 the following holds:(1)for any  𝜀_𝑛  such that  𝜀_𝑛→0𝑃  a.s. then  𝛿_𝑢(𝑏(𝜃),𝛽^#_𝑛(𝜃,𝜀_𝑛))→0𝑃  a.s.,(2)there exists a nonnegative random variable,  𝜀^∗_𝑛, such that  𝜀^∗_𝑛→0𝑃  a.s. and  𝛿(𝑏(𝜃),𝛽^#_𝑛(𝜃,𝜀_𝑛))→0𝑃  a.s.,(3)if  𝑏(𝜃)  is singleton then for any  𝜀_𝑛  such that  𝜀_𝑛→0𝑃  a.s. then  𝛿(𝑏(𝜃),𝛽^#_𝑛(𝜃,𝜀_𝑛))→0𝑃  a.s.

For its proof we will use the following lemmata. Let 𝑎_𝜃≑inf_𝐵𝑐(𝜃,𝛽) which is well defined due to Remark 6 and the compactness of 𝐵. Remember that for (ϝ_𝑛) a sequence of non empty sets Liϝ_𝑛 is the set comprised of the limit points of any possible sequence (𝑥_𝑛) such that𝑥_𝑛∈ϝ_𝑛, and Lsϝ_𝑛 is the one comprised of the analogous cluster points. Also Level_≤𝑎(𝑐_𝑛(𝜃,⋅))≑{𝑥∈𝐵∶𝑐(𝜃,𝑥)≤𝑎} for 𝑎∈ℝ.

Lemma 9. Under Assumptions 1 and 5, limsup_𝑛inf_𝐵𝑐_𝑛𝜃,𝛽≤inf_𝐵𝑐𝜃,𝛽𝑃a.s. (3)

Proof. Consider the family of 𝜃-parameterized correspondences epi_𝑛(𝜔,𝜃)≑epi(𝑐_𝑛(𝜔,𝜃,⋅)). Due to the fact that 𝐵 is locally compact, epi_𝑛(𝜔,𝜃) is a random closed set in the sense of the previous paragraph, that is, a ℬ(𝐵_𝐹)/𝒥⊗ℬ(Θ)-measurable correspondence. Hence epi_𝑛(𝜔,⋅) is an ℬ(𝐵_𝐹)/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 epi_𝑛(𝜔,𝜃)∩𝐵×(−∞,𝑎_𝜃)≠∅ since 𝐵×(−∞,𝑎_𝜃) is open in the relevant product topology. Hence inf_𝐵𝑐_𝑛(𝜔,𝜃,𝛽)≤inf_𝐵𝑐(𝜃,𝛽) for all 𝜔described previously.

The next result will be used for the proof of Proposition 8(2)-(3).

Lemma 10. Under Assumptions 1 and 5 there exists a sequence of random variables defined on Ω, say 𝑎^∗_𝑛 (It is obvious from the proof that 𝑎^∗_𝑛 depends also on 𝜃). Such that 𝑎^∗_𝑛→inf_𝐵(𝑐(𝜃,𝛽))𝑃a.s. and LiLevel_{≤𝑎∗𝑛}𝑐_𝑛(𝜃,⋅)⊇𝑏(𝜃)𝑃a.s. (4)

Proof. From the sequential implication of epiconvergence in Remark 6, we have that, for any 𝑥∈𝑏(𝜃), there exists a measurable 𝑥_𝑛 such that 𝑥_𝑛→𝑥𝑃a.s. Obviously for 𝑎_𝑛,𝑥=𝑐_𝑛(𝜃,𝑥_𝑛) which is measurable, we have that 𝑥_𝑛∈Level_{≤𝑎𝑛,𝑥}(𝑐_𝑛(𝜃,⋅)). Since 𝑏(𝜃) is compact, it is totally bounded and therefore, for any 𝜀>0, there exist an 𝑚(𝜀)∈ℕ and {𝑦_𝑖,𝑖=1,…,𝑚(𝜀)}⊂𝑏(𝜃), such that the collection of balls (in 𝐵) 𝒪(𝑦_𝑖,(𝜀/2)) covers 𝑏(𝜃). For some real sequence 𝑞_𝑛→0, consider {𝑦_𝑖,𝑖=1,…,𝑚(𝑞_𝑛)} and extract analogously random sequences (𝑦_𝑖,𝑛) such that 𝑦_𝑖,𝑛→𝑦_𝑖𝑃a.s. Define 𝑎_{𝑛,𝑦𝑖}=𝑐_𝑛(𝜃,𝑦_𝑖,𝑛) and 𝑛^∗(𝑛)=min{𝑛^′∶|𝑎_{𝑛′,𝑦𝑖}−𝑎_𝜃|≤(𝑞_𝑛/2),𝑃a.s.forall𝑖=1,…,𝑚(𝑞_𝑛)} which is well defined due to Egoroff’s theorem and the fact that 𝑚(𝑞_𝑛) is finite. Obviously 𝑛^∗ is nondecreasing in 𝑛. Then define 𝑎^∗_𝑛=max{𝑎_{𝑛∗,𝑦𝑖},𝑖=1,…,𝑚(𝑞_𝑛)} which is measurable and 𝑎^∗_𝑛→𝑎_𝜃𝑃a.s. Then for 𝑥∈𝑏(𝜃), it follows that, for any 𝜀>0, there exist an 𝑛 and a measurable 𝑥_𝑛∈Level_{≤𝑎∗𝑛}(𝑐_𝑛(𝜃,⋅)) such that 𝑑(𝑥_𝑛,𝑥)<𝜀𝑃a.s.

Proof of Proposition 8. For (1) we have first that for any measurable nonnegative 𝜀_𝑛 that need not converge to zero if 𝑥_𝑛∈𝛽^#_𝑛(𝜃,𝜀_𝑛) measurable, such that a subsequence 𝑥_𝑛𝑘→𝑥𝑃a.s. 𝑐(𝜃,𝑥)≤liminf_𝑛𝑘𝑐_𝑛𝑘𝜃,𝑥_𝑛𝑘𝑃a.s.=liminf_𝑛𝑘𝑐_𝑛𝑘𝜃,𝑥_𝑛𝑘+𝜀_𝑛𝑘𝑃a.s.≤limsup_𝑛𝑘𝑐_𝑛𝑘𝜃,𝑥_𝑛𝑘+𝜀_𝑛𝑘𝑃a.s.=limsup_𝑛𝑘inf_𝐵𝑐_𝑛𝑘𝜃,𝛽+𝜀_𝑛𝑃a.s.≤inf_𝐵𝑐(𝜃,𝑥)+𝜀_𝑛𝑃a.s., (5) where that last inequality follows from Lemma 9. This establishes that for any nonnegative random variable 𝜀_𝑛Ls𝛽^#_𝑛𝜃,𝜀_𝑛⊆𝜀_𝑛-argmin_𝐵𝑐𝜃,𝛽𝑃a.s. (6) Now (1) follows from the fact that 𝜀-argmin_𝐵⊆𝜀^′-argmin_𝐵 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, epi_𝑛(𝜔,𝜃)∩𝐵×[𝑎_𝜃−2𝜀,𝑎_𝜃−𝜀]=∅𝑃a.s. since 𝐵×[𝑎_𝜃−2𝜀,𝑎_𝜃−𝜀] is compact in the relevant product topology. This implies that liminf_𝑛inf_𝐵𝑐_𝑛(𝜃,𝛽)≥𝑎_𝜃𝑃a.s. and in conjunction with Lemma 9 that inf_𝐵𝑐_𝑛(𝜃,𝛽)→𝑎_𝜃𝑃a.s. Then using Lemma 10 set 𝜀^∗_𝑛=𝑎^∗_𝑛−𝑎_𝜃 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 Lemma 10 implies that the sequence (𝜀^∗_𝑛)_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

Proposition 8 enables the use of 𝛿_𝑢(𝑏(𝜃),𝛽^#_𝑛) as the inversion criterion. The following assumption concerns the upper continuity of the binding correspondence which along with the relevant properties of 𝛿_𝑢 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, 𝛿_𝑢(𝑏(𝜃),𝑏(𝜃^∗))→0.

The following proposition provides with sufficient conditions for this to hold. It essentially strengthens Assumption 5 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 any 𝜃, 𝑃  a.s. for any 𝜔, any  𝜃_𝑛→𝜃, and any 𝛽,(1)liminf_𝑛→∞𝑐_𝑛(𝜔,𝜃_𝑛,𝛽_𝑛)≥𝑐(𝜃,𝛽),   for all  𝛽_𝑛  such that  𝛽_𝑛→𝛽,(2)limsup_𝑛→∞𝑐_𝑛(𝜔,𝜃_𝑛,𝛽_𝑛)≤𝑐(𝜃,𝛽),   for some  𝛽_𝑛  such that  𝛽_𝑛→𝛽,
then Assumption 11 holds.

Proof. Suppose that 𝒟_𝐹0 metrizes 𝒯_𝐹 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 𝜃_𝑛→𝜃, 𝒟_𝐹0(epi_𝑛(𝜔,𝜃_𝑛),epi(𝜃)) converges to zero 𝑃 a.s. Then, for any 𝜃 and any 𝜃_𝑛→𝜃, 𝑐(𝜃_𝑛,⋅) epiconverges to 𝑐(𝜃,⋅) (𝒟_𝐹0(epi(𝜃_𝑛),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 𝒟_𝐹0(epi_𝑛(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 𝒟_𝐹0(epi_𝑛(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,⋅)→argmin_𝐵𝑐(𝜃,𝛽) is appropriately continuous.

An examination of the proof of Proposition 8 shows that if the conditions (1)-(2) of Proposition 12 hold then Proposition 8 could be restated with 𝛿_𝑢(𝑏(𝜃),𝛽^#_𝑛(𝜃_𝑛,𝜀_𝑛)) replacing 𝛿_𝑢(𝑏(𝜃),𝛽^#_𝑛(𝜃,𝜀_𝑛)) when this occurs. This means that the premises of Proposition 12 render 𝛽^#_𝑛 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. Proposition 12(1)-(2) would obviously be implied if 𝑐_𝑛(𝜔,𝜃,𝛽) is 𝑃 a.s. jointly continuous and converges jointly uniformly 𝑃 a.s. to 𝑐(𝜃,𝛽). Since we allow 𝑐_𝑛 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 𝑐_𝑛 has the form of an arithmetic mean with respect to stationary and ergodic processes.

Lemma 14. Suppose that 𝑐_𝑛(𝜔,𝜃,𝛽)=(1/𝑛)∑^𝑛_𝑖=1𝑚_𝑖(𝜔,𝜃,𝛽), (𝑚_𝑖(𝜃,𝛽))_𝑖∈ℤ is ergodic for any (𝜃,𝛽), 𝑐_𝑛(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)∈𝐴}𝑚₀(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. Proposition 12(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 𝑐_𝑛 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 limsup_𝑛→∞𝑐_𝑛𝜔,𝜃_𝑛,𝛽≤limsup_𝑛→∞𝑐_𝑛𝜔,𝜃_𝑛,𝛽−𝑐_𝑛𝜔,𝜃_𝑛,𝛽^∗+limsup_𝑛→∞𝑐_𝑛𝜔,𝜃_𝑛,𝛽^∗𝑃a.s.≤𝛿+limsup_𝑛→∞𝑐_𝑛𝜔,𝜃_𝑛,𝛽^∗𝑃a.s. (7) 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 𝛽_𝑛=𝛽, if, for any 𝜃, 𝛽^∗∈𝐵^∗, 𝑃a.s. and any 𝜃_𝑛→𝜃, limsup_𝑛→∞𝑐_𝑛𝜔,𝜃_𝑛,𝛽^∗≤𝐸𝑚₀𝜃,𝛽^∗ (i) due to the countability of 𝐵^∗. Notice that ((𝑚_𝑖(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 1𝑛^𝑛_𝑖=1𝑚_𝑖𝜔,𝜃_𝑛,𝛽^∗∧−𝜌∨𝜌 (8) converges 𝑃 a.s. to ∫^𝜌_−𝜌𝑚₀(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 of 𝑏 solely via restrictions on the behavior of 𝑐_𝑛 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 the IE 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 𝛿_𝑢(𝑏(𝜃),𝛽^#_𝑛(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 correspondence 𝜃^#_𝑛(𝜔,𝜃₀,𝜀_𝑛,𝜀^#_𝑛) satisfies 𝜃^#_𝑛𝜔,𝜃₀,𝜀_𝑛,𝜀^#_𝑛=𝜀^#_𝑛-argmin_Θ𝛿_𝑢𝑏(𝜃),𝛽^#_𝑛𝜃₀,𝜀_𝑛≑𝜃^∗∈Θ∶𝛿_𝑢𝑏𝜃^∗,𝛽^#_𝑛𝜃₀,𝜀_𝑛≤inf_Θ𝛿_𝑢𝑏(𝜃),𝛽^#_𝑛𝜃₀,𝜀_𝑛+𝜀^#_𝑛, (9) where 𝜀^#_𝑛 is a nonnegative random variable defined on Ω.

The dependence of 𝜃^#_𝑛 on 𝜃₀ follows form the analogous dependence of 𝛽^#_𝑛 which in turn follows from the dependence of 𝑦_𝑛 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 𝜃_𝑛 on 𝜔 when there is not a risk of confusion.

Proposition 16. Under Assumptions 1 and 11  𝜃^#_𝑛 is ℬ(Θ_𝐹)/𝒥-measurable, 𝑃 a.s. nonempty, compact valued correspondence.

Proof. First, notice that due to Lemma A.9, Proposition 4 (implied by Assumption 1), Assumption 11, and the facts that 𝛽^#_𝑛(𝜃₀,𝜀_𝑛) is independent of 𝜃 and 𝑏(𝜃) is independent of 𝜔, we obtain that 𝛿_𝑢(𝑏(𝜃),𝛽^#_𝑛(𝜃₀,𝜀_𝑛)) is ℬ(ℝ)/ℬ(Θ_𝐹)⊗𝒥-measurable. Due to Lemma A.4 and Proposition 4  𝛿_𝑢(𝑏(𝜃),𝛽^#_𝑛(𝜃₀,𝜀_𝑛)) is 𝑃 a.s. lsc and therefore a normal integrand. It is also 𝑃 a.s. proper due to the fact that 𝑏 and 𝛽^#_𝑛 are 𝑃 a.s. compact valued. Hence the result follows from Proposition 4 where 𝑐_𝑛=𝛿_𝑢(𝑏(𝜃),𝛽^#_𝑛(𝜃₀,𝜀_𝑛)), 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 the definition 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. If 𝜃≠𝜃₀⇒𝑏(𝜃₀)−𝑏(𝜃)≠∅.

Remark 18. This condition is weaker than a condition of the form “If 𝜃≠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 𝛿_𝑢 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 𝛽^#_𝑛 the corresponding identification condition cannot be weaker than the one above.

The main result of the current section follows. It merely concerns the existence of strongly consistent IE inside the established framework. Denote by 𝜃(𝜃₀) the argmin_Θ𝛿_𝑢(𝑏(𝜃),𝑏(𝜃₀)) which is nonempty and compact due to the compactness of Θ, the properness of 𝛿_𝑢, 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 𝛿_𝑢 and/or 𝛿.

Proposition 19. Under Assumptions 1, 5, and 11 and if 𝜀^#_𝑛→0𝑃 a.s. then(1)𝛿_𝑢(𝜃(𝜃₀),𝜃^#_𝑛(𝜔,𝜃₀,𝜀^∗_𝑛,𝜀^#_𝑛))→0𝑃  a.s. where  𝜀^∗_𝑛  is defined in Proposition 8(2),(2)if   𝑏(𝜃₀)  is singleton then  𝛿_𝑢(𝜃(𝜃₀),𝜃^#_𝑛(𝜔,𝜃₀,𝜀_𝑛,𝜀^#_𝑛))→0𝑃  a.s. for any  𝜀_𝑛→0𝑃  a.s.,
if furthermore Assumption 17 holds then(1^∗)𝛿({𝜃₀},𝜃^#_𝑛(𝜔,𝜃₀,𝜀^∗_𝑛,𝜀^#_𝑛))→0𝑃  a.s. where  𝜀^∗_𝑛  is defined in Proposition 8(2), (2^∗)if   𝑏(𝜃₀)  is singleton then  𝛿({𝜃₀},𝜃^#_𝑛(𝜔,𝜃₀,𝜀_𝑛,𝜀^#_𝑛))→0𝑃  a.s. for any 𝜀_𝑛→0𝑃  a.s.

Proof. First notice that due to Proposition 8(2), Assumption 11, and Lemma A.6 we have that for any 𝜃 and 𝜃_𝑛→𝜃liminf_𝑛𝛿_𝑢𝑏𝜃_𝑛,𝛽^#_𝑛𝜃₀,𝜀^∗_𝑛≥𝛿_𝑢𝑏(𝜃),𝑏𝜃₀𝑃a.s. (10) and that, for any 𝜃 and 𝜃_𝑛=𝜃, due to Lemma A.4limsup_𝑛𝛿_𝑢𝑏𝜃_𝑛,𝛽^#_𝑛𝜃₀,𝜀^∗_𝑛≤𝛿_𝑢𝑏(𝜃),𝑏𝜃₀𝑃a.s. (11) Hence (1) follows from Proposition 8(1) for 𝑐_𝑛=𝛿_𝑢(𝑏(𝜃_𝑛),𝛽^#_𝑛(𝜃₀,𝜀^∗_𝑛)) 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 Assumption 17 does not hold then the implications of Proposition 19(1)-(2) correspond to the fact that the statistical model is only indirectly set identified given this framework. They are trivial when 𝜃(𝜃₀)=Θ and the closer to zero 𝛿_𝑢({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 𝛿_𝑢 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 cases 𝑏 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 𝜅_𝑛(𝜔,𝜃) denote such an approximation. We readily obtain the following result for the indirect estimator defined by the substitution of 𝑏 with 𝜅_𝑛 in Definition 15.

Proposition 20. Consider the 𝐼𝐸 defined by 𝜗^#_𝑛𝜔,𝜃₀,𝜀_𝑛,𝜀^#_𝑛=𝜀^#_𝑛-argmin_Θ𝛿_𝑢𝜅_𝑛(𝜔,𝜃),𝛽^#_𝑛𝜃₀,𝜀_𝑛, (12) where 𝜀_𝑛,𝜀^#_𝑛 as before.(i)Under Assumption 1 and if  𝜅_𝑛(𝜔,𝜃)  is  ℬ(𝐵_𝐹)/𝒥⊗ℬ(Θ_𝐹)-measurable,  𝑃  a.s. compact valued and upper hemicontinuous, then  𝜗^#_𝑛  is  ℬ(Θ_𝐹)/𝒥-measurable,  𝑃  a.s. nonempty, compact valued correspondence.(ii)If moreover Assumption 5 holds and  𝑃  a.s. for any  𝜃  and any  𝜃_𝑛→𝜃, 𝛿(𝑏(𝜃),𝜅_𝑛(𝜔,𝜃))  and  𝛿_𝑢(𝑏(𝜃),𝜅_𝑛(𝜔,𝜃_𝑛))  converge to zero then the implications of Proposition 19(1)-(2) hold also for  𝜗^#_𝑛. If furthermore Assumption 17 holds of then the implications Proposition 19(1^∗)-(2^∗) hold also for  𝜗^#_𝑛.

Proof. (i) As in the proof of Proposition 16 from Lemma A.9, Proposition 4 the 𝑃 a.s. upper hemicontinuity of 𝜅_𝑛 the facts that 𝛽^#_𝑛(𝜃₀,𝜀_𝑛) is independent of 𝜃, and 𝜅_𝑛 is jointly measurable, we obtain that 𝛿_𝑢(𝛽^#_𝑛(𝜃₀,𝜀_𝑛),𝑏(𝜃)) is ℬ(ℝ)/ℬ(Θ_𝐹)⊗𝒥-measurable. Due to Lemma A.4 and Proposition 4  𝛿_𝑢(𝜅_𝑛(𝜔,𝜃_𝑛),𝛽^#_𝑛(𝜃₀,𝜀_𝑛)) is 𝑃 a.s. lsc and therefore a normal integrand. It is also 𝑃 a.s. proper due to the fact that 𝜅_𝑛 and 𝛽^#_𝑛 are 𝑃 a.s. compact valued. Hence the result follows from Proposition 4. (ii) It suffices to prove that 𝛿_𝑢(𝜅_𝑛(𝜔,𝜃),𝛽^#_𝑛(𝜃₀,𝜀_𝑛))→−^𝑒𝛿_𝑢(𝑏(𝜃),𝑏(𝜃₀))𝑃 a.s. when 𝛽^#_𝑛(𝜃₀,𝜀_𝑛)  satisfies the implications (2) or (3) of Proposition 8. The rest would then follow as in the proof of Proposition 20. Notice that for any 𝜃 and any 𝜃_𝑛→𝜃liminf_𝑛𝛿_𝑢𝜅_𝑛𝜔,𝜃_𝑛,𝛽^#_𝑛𝜃₀,𝜀^∗_𝑛≥𝛿_𝑢𝑏(𝜃),𝑏𝜃₀,𝑃a.s. (13) due to Lemma A.6, the definition of 𝛽^#_𝑛 and the 𝑃 a.s. continuous w.r.t 𝜃 upper convergence of 𝜅_𝑛 to 𝑏. Moreover for 𝜃_𝑛=𝜃limsup_𝑛𝛿_𝑢𝜅_𝑛(𝜔,𝜃),𝛽^#_𝑛𝜃₀,𝜀^∗_𝑛≤limsup_𝑛𝛿_𝑢𝜅_𝑛(𝜔,𝜃),𝑏(𝜃)+limsup_𝑛𝛿_𝑢𝑏(𝜃),𝛽^#_𝑛𝜃₀,𝜀^∗_𝑛,𝑃a.s.≤limsup_𝑛𝛿𝜅_𝑛(𝜔,𝜃),𝑏(𝜃)+𝛿_𝑢𝑏(𝜃),𝑏𝜃₀,𝑃a.s., (14) 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 𝛽^#_𝑛 to 𝑏(𝜃₀) and Lemma A.4. Due to the 𝑃 a.s. pointwise with respect to 𝜃 Fell convergence of 𝜅_𝑛 to 𝑏 we have that limsup_𝑛𝛿(𝜅_𝑛(𝜔,𝜃),𝑏(𝜃))=0𝑃a.s. and therefore we obtain the needed result.

Notice that this proposition generalizes the results of Propositions 16 and 20. For a simple example consider the case where 𝜅_𝑛=𝛽^#_𝑛. This is possible when by some sort of resampling technique (e.g., bootstrap or Monte Carlo) realizations of the 𝑦_𝑛 random elements are available to the practitioner for any 𝜃 and thereby so is 𝛽^#_𝑛(𝜃,𝜖_𝑛) for any 𝜃 and some optimization error 𝜖_𝑛  independent of 𝜃. Then a feasible IE can be defined by the approximate minimization of 𝛿_𝑢(𝛽^#_𝑛(𝜃,𝜖_𝑛),𝛽^#_𝑛(𝜃₀,𝜀_𝑛)) with respect to 𝜃. In this case the joint measurability of 𝛽^#_𝑛 would follow from the joint measurability of 𝑐_𝑛 and 𝜖_𝑛, the separability of 𝐵, and the subsequent joint measurability of the relevant projection. The 𝑃 a.s. upper hemicontinuity of 𝛽^#_𝑛 would follow from an easy extension of the implication Proposition 8(1) if Assumption 1 is strengthened so that the mapping 𝜃→epi_𝑛(𝜔,𝜃) is 𝑃 a.s. Fell continuous. The 𝑃 a.s. joint continuity of 𝑐_𝑛 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 𝑐_𝑛 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 ℝ^𝑝 and 𝐵a compact subset of ℝ^𝑞. 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 the 𝑛×𝑝 and 𝑛×𝑞 dimensional random matrices 𝑋(𝜔) and 𝑍(𝜔), respectively, where 𝑛≥𝑞≥𝑝. Suppose that (𝑋^′𝑋/𝑛)→𝑀_𝑋′𝑋, (𝑍^′𝑍/𝑛)→𝑀_𝑍′𝑍, (𝑍^′𝑋/𝑛)→𝑀_𝑍′𝑋𝑃 a.s., where rank(𝑀_𝑋′𝑋)=rank(𝑀_𝑍′𝑋)=𝑝 and 𝑝≤𝑙≑rank(𝑀_𝑍′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 ℝ^𝑞 and any 𝛽∈𝐵, let 𝑐_𝑛(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, 𝑐_𝑛 is constructed by the auxiliary set of regression with respect to 𝑍. Proposition 4 ensures the existence of 𝛽_𝑛 which in the light of the previous can be interpreted as an OLSE in the context of the auxiliary model. Let 𝒫∶ℝ^𝑞→𝑀_𝑍′𝑍𝐵 be the (generally nonlinear) projection defined by the optimization problem argmin_{𝑥∈𝑀𝑍′𝑍𝐵}𝑥−𝑦 (15) for 𝑦 in ℝ^𝑞. 𝒫 is well defined due to the compactness and the convexity of 𝐵 and the linearity and continuity of 𝑀_𝑍′𝑍 and continuous. Furthermore, for any 𝑦∈col(𝑀_𝑍′𝑍), consider the linear system 𝑀_𝑍′𝑍𝑥=𝑦, which is always satisfied by any member of the coset 𝐾𝑦+𝐻_𝑞−𝑙, where 𝐾 is a matrix of rank 𝑙 and 𝐻_𝑞−𝑙 is a 𝑞−𝑙-dimensional subspace of ℝ^𝑞, which is trivial if and only if 𝑙=𝑞 whereas 𝐾=𝑀_𝑍⁻¹_′𝑍 and maximal in the case that 𝑙=𝑝. For 𝑀_𝑋′𝑢∈ℝ^𝑝, 𝑀_𝑍′𝑢∈ℝ^𝑞, 𝑀_𝑢′𝑢∈ℝ assume that (𝑋^′𝑢/𝑛)→𝑀_𝑋′𝑢, (𝑢^′𝑢/𝑛)→𝑀_𝑢′𝑢 and (𝑍^′𝑢/𝑛)→𝑀_𝑍′𝑢𝑃 a.s. The previous imply the joint uniform 𝑃 a.s. convergence of 𝑐_𝑛 to 𝑐𝜃,𝛽=𝜃^′𝑀_𝑋′𝑋𝜃+2𝑀_𝑋′𝑢+𝛽^′𝑀_𝑍′𝑍𝛽−2𝑀_𝑍′𝑋𝜃+𝑀_𝑍′𝑢+𝑀_𝑢′𝑢 (16) which implies both Assumptions 5 and 11 (via Proposition 12 and Remark 13). Notice that 𝑏(𝜃)=𝐵∩𝐾𝒫𝑀_𝑍′𝑋𝜃+𝑀_𝑍′𝑢+𝐻_𝑞−𝑙 (17) due to the convexity of 𝑐(𝜃,𝛽) with respect to 𝛽 for any 𝜃 and the definition of 𝐵. If 𝒫(𝑀_𝑍′𝑋𝜃₀+𝑀_𝑍′𝑢)−𝒫(𝑀_𝑍′𝑋𝜃+𝑀_𝑍′𝑢)∉𝐾⁻¹𝐻_𝑞−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 𝑀_𝑋′𝑢∈𝑀_𝑋′𝑋(𝐵−𝜃) then 𝑏(𝜃)={𝜃−𝑀_𝑋⁻¹_′𝑋𝑀_𝑋′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). Let 𝑊 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]) 𝑋_𝑡(𝜔)=𝜃𝑡+𝑊_𝑡(𝜔)+_{|𝑥|∈[𝐶1,𝐶2]}𝑥𝑁(𝑡,𝑑𝑥)(𝜔), (18) 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 𝑐_𝑛(𝜔,𝜃,𝛽)=(1/𝑛)∑^𝑛_𝑡=1(𝑦_𝑡−𝛽)², where 𝑦_𝑡=exp(𝑋_𝑡−𝑋_𝑡−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 SDE𝑑𝑦_𝑡=𝛽𝑦_𝑡𝑑𝑡+𝑦_𝑡𝑑𝑊_𝑡 (19) 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 𝛽_𝑛 which in the light of the previous can be interpreted as an (approximate) MLE in the context of the auxiliary model. Furthermore since _{|𝑥|∈[𝐶1,𝐶2]}𝑥(𝑁()(≤𝑡,𝑑𝑥−𝑁𝑡−1,𝑑𝑥))_{|𝑥|∈[𝐶1,𝐶2]}|𝑥|(𝑁(𝑡,𝑑𝑥)−𝑁(𝑡−1,𝑑𝑥))≤𝐶₂𝑁𝑡,𝐶₁,𝐶₂−𝑁𝑡−1,𝐶₁,𝐶₂ (20) and 𝑁(𝑡,[𝐶₁,𝐶₂])−𝑁(𝑡−1,[𝐶₁,𝐶₂])∼^i.i.d.Poiss(𝑣([𝐶₁,𝐶₂])) independent of 𝑊, we have that 𝐸exp𝑋_𝑡−𝑋_𝑡−11=exp𝜃+2𝐶 (21) for 0<𝐶≤exp−𝑣𝐶₁,𝐶₂1−exp𝐶₂𝐸exp𝑋_𝑡−𝑋_𝑡−1²=exp(2(𝜃+1))𝐶^∗ (22) for 0<𝐶^∗≤exp−𝑣𝐶₁,𝐶₂1−exp2𝐶₂. (23) 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 𝑐_𝑛 to 𝑐𝜃,𝛽=exp(2(𝜃+1))𝐶^∗1−2exp𝜃+2𝐶1+𝛽+1+𝛽² (24) which implies both Assumptions 5 and 11 (via Proposition 12 and Remark 13). If 𝐵⊇exp(Θ+(1/2))𝐶−1 then 𝑏(𝜃)=1exp𝜃+2𝐶−1. (25) 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, let 𝑧∶Ω→ℝ^ℤ be an i.i.d. (double infinite) sequence of random variables, with 𝐸𝑧₀=0 and 𝐸𝑧²₀=1. Consider a random element 𝜎²∶Θ×Ω→(ℝ⁺)^ℤ, with the product space Θ×Ω equipped with ℬ(Θ)⊗𝒥 with 𝜎²_𝑡(𝜃) independent of (𝑧_𝑖)_𝑖≥𝑡, forall𝑡∈ℤ, forall𝜃∈Θ. Analogously, define the random element 𝑦∶Θ×Ω→(ℝ)^ℤ as 𝑦_𝑡(𝜔)(𝜃)_{𝑡∈ℤ,𝜃∈Θ}=𝑧_𝑡(𝜔)𝜎²_𝑡(𝜔)(𝜃)_{𝑡∈ℤ,𝜃∈Θ}. (26)

Then for all 𝜃∈Θ, (𝑦_𝑡(𝜃))_𝑡∈ℤ is called a conditionally heteroskedastic process, while the random element (𝑦_𝑡(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 (𝜎²_𝑡(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 (𝑦_𝑡(𝜃))_𝑡∈ℤ and (𝑦²_𝑡(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). Let 𝜎₄=𝐸𝑧⁴₀<+∞ and 0<𝛿<(1/√𝜎₄). Suppose that 𝑎∈Θ=[0,𝛿] and consider the stochastic dif and only iference equation 𝜎²_𝑡(𝑎)=1+𝑎𝑧²_𝑡−1𝜎²_𝑡(𝑎). (27) 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)=(𝑦²_𝑡(𝑎))_{𝑡∈{1,…,𝑛}} and the 𝑛×2 dimensional random matrices 𝑍(𝑎)=1𝑦²₋₁(𝑎)⋮⋮1𝑦²_𝑛−2(𝑎),𝑋(𝑎)=1𝑦²₀(𝑎)⋮⋮1𝑦²_𝑛−1(𝑎) (28) jointly measurable with respect to 𝒥⊗ℬ(Θ), where 𝑛>2 and ergodic for any 𝑎∈Θ. For 𝛽=^𝛽1_𝛽2∈𝐵=[1,1+𝛿]×[−𝛿,𝛿], let 𝑐_𝑛(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 𝑐_𝑛 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 𝑣_𝑡=(𝑧²_𝑡−1)𝜎²_𝑡(𝑎) (see, e.g., [28]) and 𝑐_𝑛 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 𝑣_𝑡 for 𝑖≤𝑡. Proposition 4 ensures the existenceof 𝛽_𝑛 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 𝐸sup_𝑎,𝛽1𝑛𝑍^′(𝑎)𝑌(𝑎)−𝑋(𝑎)𝛽≤𝐶₁1111−𝛿11−𝛿(1−𝛿)1−𝛿²𝜎₄+𝐶₂11−𝛿1+𝛿(1−𝛿)1−𝛿²𝜎₄<+∞ (29) 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 𝑐=11𝑎,𝛽11−𝑎11−𝑎(1−𝑎)1−𝑎²𝜎₄𝛽₁𝛽₂−11−𝑎1+𝑎(1−𝑎)1−𝑎²𝜎₄. (30) In fact a simple calculation shows that 𝑏(𝑎)=1𝑎(],𝛽,𝑎∈0,𝛿₁,1−𝛽₁,𝛽₁∈[1,1+𝛿],𝑎=0 (31) 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 𝛽_𝑛, 𝜃^#_𝑛𝜔,𝜃₀,𝜀^∗_𝑛={𝑎},0if𝛽_𝑛⊆𝑏(𝑎)midproj_𝛽2𝛽_𝑛if𝛽⊆_𝑛^/𝑏(𝑎),midproj_𝛽2𝛽_𝑛{0}≥0,if𝛽⊆_𝑛^/𝑏(𝑎),midproj_𝛽2𝛽_𝑛<0, (32) where proj_𝛽2 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(proj_𝛽2(𝛽_𝑛)) is well defined due to the fact that 𝛽_𝑛 is 𝑃 a.s. compact valued hence its proj_𝛽2(𝛽_𝑛) 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 𝜅_𝑛 when this equals the auxiliary estimator derived from bootstrap resampling for any 𝑎.

The final example is about an asymmetric heteroskedastic process.

Example 24 (−𝑐_𝑛 is the quasilikelihood function of an approximate to QARCH(1) model). Let 𝐸|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 𝜎²_𝑡𝛾=𝜛+𝑎₀𝑧²_𝑡−1𝜎²_𝑡𝛾+𝛾𝑧_𝑡−1𝜎_𝑡𝛾. (33) 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 𝐸𝜎²_𝑡(𝛾)=+∞. For 𝛾^∗∈𝐵=[0,𝛿] consider the process defined by ℎ_𝑡𝛾,𝛾^∗=𝜛+𝑎₀𝑦²_𝑡−1+𝛾^∗𝑦_𝑡−11_{𝑦𝑡−1𝑃<0a.s.}=𝜛+𝑎₀𝑦²_𝑡−1+𝛾^∗𝑦_𝑡−11_{𝑧𝑡−1<0}, (34) where ℎ_𝑡(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 𝑐_𝑛𝜔,𝛾,𝛾^∗≑1𝑛^𝑛_𝑖=1𝑦−ln^2𝑖𝛾ℎ_𝑖𝛾,𝛾^∗+𝑦²_𝑖𝛾ℎ_𝑖𝛾,𝛾^∗, (35) where −𝑐_𝑛 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 (𝑐_𝑛) for any (𝛾,𝛾^∗) follows from the previous and Proposition 2.1.1. of [24]. (In practice 𝑐_𝑛(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 sup_𝐵|𝑐_𝑛(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 𝑐_𝑛(𝜔,𝜃,𝛽).) Assumption 1 follows readily from the form of 𝑐_𝑛 and the 𝑃a.s. continuity with respect to (𝛾,𝛾^∗). Hence 𝛽_𝑛 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 𝐸inf_𝐴∩𝐵−ln𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0+𝑧²₀𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0≥−𝐸sup_𝐴∩𝐵ln𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0+𝐸𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛿𝑦₀1_𝑧0<0≥−𝐸ln1+𝛾𝑦₀𝜛+𝑎₀𝑦²₀1_𝑧0<0+𝐸1+𝛾𝑦₀𝜛+𝑎₀𝑦²₀1_𝑧0≥0>−∞ (36) and that 𝐸inf_𝐴∩𝐵−ln𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0+𝑧²₀𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0≤1−𝐸ln1+𝛾𝑦₀𝜛+𝑎₀𝑦²₀1_𝑧0≥0+𝐸1+𝛾𝑦₀𝜛+𝑎₀𝑦²₀<+∞ (37) for 𝐴 an arbitrary member of the partition. Notice also that −2𝐸ln|𝑧|− 𝐸((𝜛+𝑎₀𝑦²₀+𝛾𝑦₀)/(𝜛+𝑎₀𝑦²₀+𝛾^∗|𝑦₀|1_𝑧0<0))+𝐸((𝜛+𝑎₀𝑦²₀+𝛾𝑦₀)/(𝜛+𝑎₀𝑦²₀+𝛾^∗|𝑦₀|1_𝑧0<0))>−∞ for all (𝛾,𝛾^∗) due to the fact that 𝐸ln𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0≥𝐸ln1+𝛾𝑦₀𝜛+𝑎₀𝑦²₀>−∞. (38) Hence, Remark 6 implies that Assumption 5 holds with 𝑐𝛾,𝛾^∗=−2𝐸ln|𝑧|−𝐸ln𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0+𝐸𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0. (39) 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+𝑎₀𝑦²₀)/(𝜛+𝑎₀𝑦²₀+𝛾^∗|𝑦₀|1_𝑧0<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 −𝐸ln𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀−𝛾𝑦₀1_𝑧0<0+𝐸ln𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀=𝐸ln1+𝛾𝑦₀𝜛+𝑎₀𝑦²₀1_𝑧0<0,𝐸𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀−𝛾𝑦₀1_𝑧0<0−𝐸𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀=−𝐸𝛾𝑦₀𝜛+𝑎₀𝑦²₀1_𝑧0<0<0 (40) 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 𝐸sup_𝛾∗𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0𝜛+𝑎₀𝑦²₀+𝛾𝑦₀×−1𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0²𝑦₀1_𝑧0<0≤𝛿𝛾𝜛𝑦−1×𝐸⁰+𝑎₀𝑦₀³+𝛾𝑦²₀𝜛+𝑎₀𝑦²₀²1_𝑧0<0<+∞,𝐸sup_𝛾∗𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0²𝑧²₀1_𝑧0<0≤1𝜛𝐸1+𝛾𝑦₀𝜛+𝑎₀𝑦²₀1_𝑧0<0<+∞,𝐸sup_𝛾∗𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0𝜛+𝑎₀𝑦²₀+𝛾𝑦₀×−1𝜛+𝑎₀𝑦²₀+𝛾𝑦₀²𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0³𝑧²₀1_𝑧0<0≤1𝜛𝛿𝛾−1𝐸1+𝛾𝑦₀𝜛+𝑎₀𝑦²₀²1_𝑧0<0<+∞ (41) as well as dominated convergence, we have that 𝜕𝑐𝛾,𝛾^∗𝜕𝛾^∗=𝐸𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0𝜛+𝑎₀𝑦²₀+𝛾𝑦₀×−1𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀+𝛾^∗𝑦₀1_𝑧0<0²𝑦₀1_𝑧0<0 (42) which is zero if and only if 𝛾^∗=−𝛾, and 𝜕²𝑐𝛾,𝛾^∗𝜕𝛾^∗2_{𝛾∗=−𝛾}=𝐸𝜛+𝑎₀𝑦²₀+𝛾𝑦₀𝜛+𝑎₀𝑦²₀−𝛾𝑦₀1_𝑧0<0²𝑧²₀1_𝑧0<0>0 (43) 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 𝜎²_𝑡 and ℎ_𝑡 with respect to the parameters, the existence of log moments, and the fact that ℎ_𝑡 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 of IE 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. 𝑐_𝑛(𝜔,𝜃,⋅), 𝑐(𝜃,⋅) and 𝛿_𝑢(𝑏(⋅),𝛽^#_𝑛(𝜃₀,𝜀^∗_𝑛)) have totally bounded level sets and nonempty argmins.

Second, the present generalization is certainly nonunique. 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 IE𝛿_𝑢 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 of IE when the 𝜅_𝑛 appearing in Proposition 20 is some sort of integral of 𝛽^#_𝑛 (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 𝛽^#_𝑛 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.

Appendices

A. Some General Notions

In this section we briefly describe some general notions that are essentially used in the main body.

A.1. Fell and Upper Topology

Let (𝐸,𝜏_𝐸) denote a topological space. We identify it with 𝐸 when there is no risk of confusion. We denote by 𝐹₀(𝐸) the set of closed nonempty subsets of 𝐸. We will use two topologies on 𝐹₀(𝐸) cosntructed via 𝜏_𝐸 and the inclusion partial order on 2^𝐸. The first is entitled upper topology and we define it as the restriction of the topology in Definition 1.3.1 of [30] on 𝐹₀(𝐸).

Definition A.1. The upper topology 𝒯_𝑈 on 𝐹₀(𝐸) is generated by the subbase consisting of [⋅,𝐺]=ϝclosed∶ϝ⊂𝐺,∀𝐺∈𝜏_𝐸,nonempty. (A.1)

The upper topology is extremely useful for the analysis of the asymptotic behavior of sequences of sets of minimizers. If 𝜏_𝐸 is generated by a metric (say 𝑑) with respect to which 𝐸 is compact then 𝒯_𝑈 is hemimetrizable (see Definition 4.1.3 for the term hemimetric and Proposition 4.2.2 of [30]) by 𝛿_𝑢∶𝐹₀(𝐸)×𝐹₀(𝐸)→ℝ, defined by 𝛿_𝑢(𝐴,𝐵)=inf{𝜀>0∶𝐵⊂𝑁_𝜀(𝐴)} where 𝑁_𝜀(𝐴)={𝑥∈𝐸∶𝑑(𝑥,𝐴)<𝜀}. (where 𝑑(𝑥,𝐴)≑inf_𝑦∈𝐴𝑑(𝑥,𝑦)). Obviously, when 𝐵⊆𝐴 then 𝛿_𝑢(𝐴,𝐵)=0. The following properties of 𝛿_𝑢 will be useful in what follows.

Lemma A.2. consider 𝛿_𝑢(𝐴,𝐵)=0𝑖𝑓𝑎𝑛𝑑𝑜𝑛𝑙𝑦𝑖𝑓𝐵⊆𝐴. (A.2)

Proof. Since 𝐴 is closed if 𝑥∈𝐵 and 𝑥∉𝐴, then 𝑑(𝑥,𝐴)=𝛿>0. But then 𝐵⊆^/𝑁_𝛿/2(𝐴) and therefore 𝛿_𝑢(𝐴,𝐵)>(𝛿/2).

Lemma A.3. 𝛿_𝑢 is a lower semicontinuous (lsc) real function with respect to the first argument.

Proof. If 𝐴_𝑛→𝐴 with respect to the upper topology on 𝐹₀(𝐸), then 𝛿_𝑢(𝐴,𝐵)≤𝛿_𝑢(𝐴,𝐴_𝑛)+𝛿_𝑢(𝐴_𝑛,𝐵), and hence liminf_𝑛𝛿_𝑢(𝐴_𝑛,𝐵)≥𝛿_𝑢(𝐴,𝐵).

Lemma A.4. 𝛿_𝑢 is an upper semicontinuous (usc) real function with respect to the second argument.

Proof. Suppose that 𝐵_𝑛→𝐵 with respect to the upper topology on 𝐹₀(𝐸); then by the triangle inequality we have that 𝛿_𝑢(𝐴,𝐵_𝑛)≤𝛿_𝑢(𝐴,𝐵)+𝛿_𝑢(𝐵,𝐵_𝑛) establishing that limsup_𝑛𝛿_𝑢(𝐴,𝐵_𝑛)≤𝛿_𝑢(𝐴,𝐵).

The second topology on 𝐹₀(𝐸), known as the Fell topology, is defined by the use of the following subbase (see [22], paragraph 1.1, and [30], Definition 4.5.1).

Definition A.5. The Fell topology, say 𝒯_𝐹, is the smallest topology on 𝐹₀(𝐸) consisting of both(1)𝐹_𝐺={ϝclosed∶ϝ∩𝐺≠∅}, ∀𝐺∈𝜏_𝐸, nonempty,(2)𝐹^𝐾={ϝclosed∶ϝ∩𝐾=∅}, ∀𝐾⊂𝐸 nonempty and compact.

From Theorems 4.5.3-5 of [30] we have that when 𝐸 is locally compact and Hausdorff then (𝐹₀(𝐸),𝒯_𝐹) is locally compact and ϝ_𝑛→ϝ with respect to the Fell topology if and only if ϝ=Liϝ_𝑛=Lsϝ_𝑛 where Liϝ_𝑛 is the set comprised of the limit points of any possible sequence (𝑥_𝑛) such that𝑥_𝑛∈ϝ_𝑛, and Lsϝ_𝑛 is the one comprised of the analogous cluster points. Hence, in this case this type of convergence coincides with the Painleve-Kuratowski convergence (see among others, Appendix B of [22], or Definition 3.1.4. of [30]). If 𝐸 is also separable then the Fell topology is metrizable. If furthermore 𝐸 is compact and metrized by 𝑑, then the Fell topology is actually metrized by the Hausdorff extended metric defined via a symmetrization of 𝛿_𝑢; that is, 𝛿(𝐴,𝐵)=max{𝛿_𝑢(𝐴,𝐵),𝛿_𝑙(𝐴,𝐵)} where 𝛿_𝑙(𝐴,𝐵)≑𝛿_𝑢(𝐵,𝐴). In this case we can prove the following lemma.

Lemma A.6. If 𝐸 is compact and metrized by 𝑑, then 𝛿_𝑢 is a lower semicontinuous (lsc) real function with respect to the product topology on 𝐹₀(𝐸)×𝐹₀(𝐸), when the first factor is endowed with 𝒯_𝑈 and the second with 𝒯_𝐹.

Proof. If (𝐴_𝑛,𝐵_𝑛)→(𝐴,𝐵) with respect to the aforementioned product topology on 𝐹₀(𝐸)×𝐹₀(𝐸), then 𝛿_𝑢(𝐴,𝐵)≤𝛿_𝑢(𝐴,𝐴_𝑛)+𝛿_𝑢(𝐴_𝑛,𝐵_𝑛)+𝛿_𝑙(𝐵,𝐵_𝑛) establishing that liminf_𝑛𝛿_𝑢(𝐴_𝑛,𝐵_𝑛)≥𝛿_𝑢(𝐴,𝐵).

A.2. Epigraphs of Semicontinuous Functions and Epiconvergence

Consider now the case where 𝐸 is locally compact and Hausdorff and 𝑐∶𝐸→ℝ.

Definition A.7. The epigraph of 𝑐 is epi(𝑐)={(𝑥,𝑡)∈𝐸×ℝ∶𝑐(𝑥)≤𝑡}. (A.3)

Note that despite the fact that the image of 𝑐 may include infinities, epi(𝑐) is by definition a subset of 𝐸×ℝ. If and only if 𝑐 is lower semicontinuous (lsc) we have that, due to Proposition A.2 of [22], epi(𝑐)∈𝐹(𝐸×ℝ) with respect to the obvious product topology. Hence any relevant lsc function can be identified with its epigraph, which in turn lies in a space endowed with Fell topology, which in turn implies a notion of convergence.

Definition A.8. A sequence (𝑐_𝑛) of lsc functions epiconverges to 𝑐(𝑐→−_𝑛^𝑒𝑐) if and only if epi(𝑐_𝑛)→epi(𝑐) with respect to the Fell topology.

It is easy to see that uniform convergence implies epiconvergence (see, e.g., Remark 6 above). Furthermore the relevant set of lsc is closed with respect to the Fell topology. This notion is particularly suitable for the description of the asymptotic behavior of the set of minimizers of sequences of lsc functions (see Theorem 3.4 of [22] along with Theorem 7.1.4 of [30], Definition D.1 and Proposition D.2 of [22]).

A.3. Closed and Compact Valued Correspondences-Random Closed Sets

A closed valued correspondence is by definition a representation of an underlying function 𝑐 from a set Ω to 𝐹₀(𝐸) (i.e., a closed valued multifunction with domain the set Ω), when this is considered as a relation in Ω×𝐸. The benefit of not directly working with the underlying function is the fact that we can consider the graph of the correspondence as the set {(𝑥,𝑦)∶𝑦∈𝑐(𝑥)} which resides in Ω×𝐸 instead of the set {(𝜔,ϝ)∶ϝ=𝑐(𝜔)} inside Ω×𝐹₀(𝐸). When 𝑐(𝑥) is compact for any 𝑥, then the correspondence in obviously termed as compact valued. In the following we do not make explicit distinction between the correspondence and the underlying multifunction.

The Borel 𝜎-algebra on 𝐹₀(𝐸) generated by 𝒯_𝐹 will be abbreviated by ℬ(𝐸_𝐹) and is usually termed Effros algebra (see Paragraph 1.1 of [22]). If (Ω,𝒥) is a measurable space, then 𝑐 is a random closed set if and only if {𝜔∈Ω∶𝑐(𝜔)∈ϝ}∈𝒥 for any ϝ∈ℬ(𝐸_𝐹). Analogously we abbreviate by ℬ(𝐸_𝑈) the Borel 𝜎-algebra on 𝐹₀(𝐸) generated by 𝒯_𝑈 and by ℬ(𝐸_𝑈×𝐸_𝐹) the Borel 𝜎-algebra on 𝐹₀(𝐸)×𝐹₀(𝐸) generated by the product topology described in Lemma A.6. Finally denote by ℬ(ℝ) the Borel 𝜎-algebra of the extended real numbers with respect to the usual topology.

Lemma A.9. If 𝐸 is compact, separable, and metrized by 𝑑, then 𝛿_𝑢 is ℬ(ℝ)/ℬ(𝐸_𝑈)⊗ℬ(𝐸_𝐹) measurable.

Proof. The separability of 𝐸 implies the separability of (𝐹₀(𝐸),𝒯_𝑈) and (𝐹₀(𝐸),𝒯_𝐹) for if {𝑥_𝑛,𝑛=0,1,…} is dense in 𝐸 then the countable subset of 𝐹₀(𝐸), {{𝑥_𝑛},𝑛=0,1,…} intersects any basic open set with respect to to either topology. This implies the separability of 𝐹₀(𝐸)×𝐹₀(𝐸) when equipped with the topology discussed in Lemma A.6. This in turn implies that the Borel 𝜎-algebra with respect to to the product topology on 𝐹₀(𝐸)×𝐹₀(𝐸) coincides with ℬ(𝐸_𝑈)⊗ℬ(𝐸_𝐹) by Lemma 1.4.1. of [33]. The rest follows by Lemma A.6 along with the fact that the sets in the subbase of the upper topology of ℝ generate ℬ(ℝ). (It is also possible to prove that in the context of separability ℬ(𝐸_𝑈)=ℬ(𝐸_𝐹)).