# On the Validity of Edgeworth Expansions and Moment Approximations for Three Indirect Inference Estimators

Stelios Arvanitis and Antonis Demos

# Abstract

This paper deals with higher order asymptotic properties for three indirect inference estimators. We provide conditions that ensure the validity of locally uniform, with respect to the parameter, Edgeworth approximations. When these are of sufficiently high order they also form integrability conditions that validate locally uniform moment approximations. We derive the relevant second order bias and MSE approximations for the three estimators as functions of the respective approximations for the auxiliary estimator. We confirm that in the special case of deterministic weighting and affinity of the binding function, one of them is second order unbiased. The other two estimators do not have this property under the same conditions. Moreover, in this case, the second order approximate MSEs imply the superiority of the first estimator. We generalize to multistep procedures that provide recursive indirect inference estimators which are locally uniformly unbiased at any given order. Furthermore, in a particular case, we manage to validate locally uniform Edgeworth expansions for one of the estimators without any differentiability requirements for the estimating equations. We examine the bias-MSE results in a small Monte Carlo exercise.

JEL: C10; C13

# Acknowledgments

We would like to thank, without implicating them, Gabriele Fiorentini, Enrique Sentana, Eric Renault, Peter Phillips, the seminar participants at the University of Piraeus, University of Cyprus, and AUEB, the conference participants at SSEE and at EEA-ESEM 2012. The comments of two anonymous referees and the associate editor have greatly improve the exposition of the paper. Financial support of BRFP II AUEB is gratefully acknowledged.

## Appendix

Proofs

Proof of Lemma 2:

1. Due to the triangle inequality and Assumption 3 we have that for ε>0

supθΘPθ(supθΘ|||βnb(θ)||||b(θ)b(θ)|||>ε)supθΘPθ(||βnb(θ)||>ε)=o(na)

Hence for qn(θ)=βnb(θ), q(θ*, θ)=b(θ*)–b(θ) and by Assumption 4.a) Lemma AL.3 applies. Hence for γ(θ)=θ due to Assumption 1.3 Lemma AL.1 also applies implying the result.

2. Given i), we have that θn∈𝒪ϵ(θ) with Pθ-probability 1o(na) that is locally independent of θ for any ϵ>0. For some ϵ small enough, such that Oϵ(θ)Oε0(θ0) (which exists due to the fact that ε0>ε) due to Assumption 6, we have that condition FOC of the appendix Lemmas AL.4 and AL.5 is satisfied by the GMR1 estimator with Qnbθ. Furthermore Assumption 6 implies conditions HUB (γ (θ)=θ hence set δ=ε0) and RANK of the same lemma. Condition TIGHT follows from 5, as under this assumption there is C*>0 locally independent of θ such that

(5)supθO¯ε(θ0)Pθ(n||βnb(θ)||>Cln1/2n)=o(na) (5)

(see Lemma AL.2 of Arvanitis and Demos 2014). Hence Lemma AL.4 applies ensuring that

supθO¯ε(θ0)Pθ(n||GMR1θ||>Cln1/2n)

for some C>0 locally independent of θ. Hence condition UTIGHT of Lemma AL.5 holds. Moreover Assumption 5 implies condition UEDGE of the same lemma for Mn(θ)=nmn(θ). Due to Assumption 6 for any θ∈𝒪̅ε (θ0) and any θ* sufficiently close to θ, bθ(θ) admits a Taylor expansion of order s*–1 around θ of the form

bθ(θ)=i=0s11i!Dibθ(θ)((θθ)i)+1(s1)!(Ds1bθ(θ+)Ds1bθ(θ))((θθ)s1)

where θ+ lies between θ* and θ. This implies that for any θn=GMR1 due to condition UTIGHT we have that with Pθ-probability 1o(na) locally independent of θ

bθ(θn)=i=0s11i!1ni/2Dibθ(θ)((n(θnθ))i)+Rn(θn,θ)

where Rn(θn,θ)=1(s1)!1n(s1)/2(Ds1bθ(θn+)Ds1bθ(θ))((n(θnθ))s1), and θn+ lies between θn and θ. Now by Assumption 6 bθ(θ) has full rank for any θ∈𝒪̅ε (θ0) and by submultiplicativity, the relation of θn+ to θn and condition UTIGHT

supθO¯ε(θ0)Pθ(1(s1)!1n(s1)/2(Ds1bθ(θn+)Ds1bθ(θ))×((n(θnθ))s1)>γn)supθO¯ε(θ0)Pθ(1(s1)!1n(s1)/2supθO¯ε0(θ0)Dsbθ(θ)||θn+θ||||n(θnθ)||s1>γn)supθO¯ε(θ0)Pθ(M(s1)!Csns/2lns/2n>γn)+o(na)

which is of order o(na) for γn=M(s1)!Csns/2lns/2n=o(na) and locally independent of θ. Analogously, due to Assumption 6 for any θ∈𝒪̅ε (θ0) and any θ* sufficiently close to θ, b(θ*) admits a Taylor expansion of order s*–1 around θ of the form

qn=βnb(θ)=βnb(θ)i=1s1i!Dib(θ)((θθ)i)1s!(Dsb(θ+)Dsb(θ))((θθ)s)

where θ+ lies between θ* and θ. This implies that for θn we have that with Pθ-probability 1o(na)

n(βnb(θn))=n(βnb(θ))+i=0s11(i+1)!1ni/2Di+1b(θ)((n(θnθ))i+1)+Rn#(θn,θ)

where Rn#(θn,θ)=1s!1n(s1)/2(Dsb(θn+)Dsb(θ))((n(θnθ))s), and θn+ lies between θn and θ. Now by Assumption 6 bθ(θ) has full rank for any θ∈𝒪̅ε (θ0) and so does the identity matrix in front of n(βnb(θ)), and thereby due to submultiplicativity, the relation of θn+ to θn and condition UTIGHT

supθO¯ε(θ0)Pθ(1s!1n(s1)/2(Dsb(θn+)Dsb(θ))×((n(θnθ))s)>γn#)supθO¯ε(θ0)Pθ(1s!1n(s1)/2supθO¯ε0(θ0)||Ds+1b(θ)||||θn+θ||×||n(θnθ)||s>γn#)supθO¯ε(θ0)Pθ(Ms!Cs+1ns/2ln(s+1)/2n>γn#)+o(na)

which is of order o(na) for γn#=Ms!Cs+1ns/2ln(s+1)/2n=o(na) and locally independent of θ. Finally due to Lemma AL.6 which applies by Assumptions 5 and 4 condition EXPAND holds and the result follows by the same lemma.   ■

Proof of Lemma 3: The proof is in the same spirit as the proof of Lemma 2. For ε>0, let E(ε,θ)={ωΩ:||βnb(θ)||>ε2}, then

supθΘ||Eθβnb(θ)||supθΘEθ||βnb(θ)||1E(ε,θ)+ε2.

As B is bounded, due to Assumption 1.2 and by Assumption 3 there exists an n* such that supθΘPθ(||βnb(θ)||>ε3)ε2M where M denotes the diameter of B. Hence

supθΘ||Eθβnb(θ)||ε for all nn

and since ε is arbitrary

(6)supθΘ||Eθβnb(θ)||=o(1) (6)

Due to the triangle inequality and Assumption 3 we have that for ε>0

supθΘPθ(supθΘ|||βnEθβn||||b(θ)b(θ)|||>ε)supθΘPθ(||βnb(θ)||+supθΘ||Eθβnb(θ)||>ε)=o(na)

For qn(θ)=Eθβn, q(θ*, θ)=b(θ*)–b(θ) and by Assumption 4.a) Lemma AL.3 applies. Hence for γ(θ)=θ due to Assumption 1.3 Lemma AL.1 also applies implying the result.

ii) Given i), we have that θn∈𝒪ϵ(θ) with Pθ-probability 1o(na) that is locally independent of θ for any ϵ>0. For some ϵ small enough, such that Oϵ(θ)Oϵ0(θ0) (which exists due to the fact that ε0>ε) due to Assumption 7, we have that condition FOC of the appendix Lemmas AL.4 and AL.5 is satisfied by the GMR2 estimator with QnEθ(βn)θ. Furthermore Assumption 7 and 6 imply conditions HUB (γ(θ)=θ hence set δ=ε0) and RANK of the same lemma due to the fact that since D2Eθβn is uniformly bounded on Oε0(θ0),DEθβn converges uniformly to Db(θ) due to Lemma AL.7 and therefore the rank condition is implied by 7 for large enough n. Condition TIGHT follows from 5 and Lemma AL.2 of Arvanitis and Demos (2014) and the fact that as a*>a≥0 we have that a*>0 and there exists a C2>0 locally independent of θ and for E={ωΩ:||βnb(θ)||>C2ln1/2nn1/2}

supθO¯ε(θ0)||Eθβnb(θ)||supθO¯ε(θ0)Eθ[||βnb(θ)||1E]+supθO¯ε(θ0)Eθ[||βnb(θ)||1E¯]MsupθO¯ε(θ0)Pθ(||βnb(θ)||>C2ln1/2nn1/2)+C2ln1/2nn1/2supθO¯ε(θ0)Pθ(||βnb(θ)||C2ln1/2nn1/2)=o(na)+C2ln1/2nn1/2(1o(na))=O(ln1/2nn1/2),

where the last line comes from equation 5, above, and therefore

supθO¯ε(θ0)Pθ(||βnEθβn||>C1ln1/2nn1/2)supθO¯ε(θ0)Pθ(||βnb(θ)||+||Eθβnb(θ)||>C1ln1/2nn1/2)supθO¯ε(θ0)Pθ(||βnb(θ)||+O(ln1/2nn1/2)>C1ln1/2nn1/2)=o(na)

Hence Lemma AL.4 applies ensuring that

supθO¯ε(θ0)Pθ(n||GMR2θ||>Cln1/2n)=o(na)

for some C>0 locally independent of θ, hence condition UTIGHT of Lemma AL.5 holds. Moreover Assumption 5 along with the fact that the support of βnb(θ) is uniformly bounded by 𝒪̅3η (0) for any η greater or equal than the diameter of B, and the fact that n(βnEθβn) admits a locally uniform Edgeworth expansion of order s*–1 (see Lemma 4.1 of Arvanitis and Demos 2014) imply condition UEDGE of Lemma 5 for Mn(θ)=n(mn(θ)βnEθβn) with order s*–1. Due to Assumption 7 for any θ∈𝒪̅ε (θ0) and any θ* sufficiently close to θ, Eθ(βn)θ(θ) admits a Taylor expansion of order s*–1 around θ and the rest follows as in the proof of Lemma 2 above with the difference that here Eθ(βn)θ is considered instead of bθ(θ) and Assumption 7 is employed instead of 6 (see Arvanitis and Demos 2013b, for details).■

Proof of Lemma 4: By Assumption 5, Lemma 8, below, adding subtracting n(b(θ)+i=1sV(kiβ(z,θ))ni2) and n(b(θn+)+i=1sV(kiβ(z,θn+))ni2), we get n(EθnβnEθβn)An(θ)=n(Eθnβnb(θn+)i=1sV(kjβ(z,θn+))ni2)n(Eθβnb(θ)i=1sV(kiβ(z,θ))ni2)+n(b(θn+)b(θ)i=1s1i!Dib(θ)((θn+θ)i))+Bn where Bn=i=1sV(kiβ(z,θn+))ni12i=1sV(kiβ(z,θ))ni12i=1s1i!j=1siDiV(kjβ(z,θ))nj12((θn+θ)i). Employing the mean value Theorem for V(kiβ(z,θn+)), and for θn++ such that ||θn++θ||<||θn+θ||, and collecting terms we get:

Bn=i=1s1ni121(si+1)!Dsi+1V(kiβ(z,θ))((θn++θ)si+1).

Taking into account that θn+O¯ε(θ0) with probability 1o(na) and employing the triangular inequality we have that, for s<s*,

supθO¯ε(θ0)Pθ(||n(Eθn+βnEθβn)An(θ)||>γn)supθO¯ε(θ0)Pθ(supθO¯ε(θ0)nEθβnb(θ)i=1sV(kiβ(z,θ))ni2>γn6)+i=1ssupθO¯ε(θ0)Pθ(1ni12||Bn||>γn3s)+supθO¯ε(θ0)Pθ(nb(θn+)b(θ)i=1s1i!Dib(θ)((θn+θ)i)>γn3)+o(na).

Now we have that

an=nEθβnb(θ)i=1sV(kiβ(z,θ))ni2=o(na)

independent of θ, due to Lemma 8.

Further, due to the continuity of Dsi+1V(kiβ(z,θ)), Assumption 8, and the assumption of the asymptotic behavior of θn+ we get

supθO¯ε(θ0)Pθ(1ni12||Bn||>γn3s)o(na)

provided that γnγn(1)=lnsi+12nns23ssupθO¯ε(θ0)sup||Dsi+1V(kiβ(z,θ))||(si+1)!.

Furthermore using the same reasoning as above

supθO¯ε(θ0)Pθ(n||b(θn+)b(θ)i=1s1i!Dib(θ)((θnθ)i)||>γn3)o(na)

when γnγn(2)=3supθO¯ε(θ0)||Ds+1b(θ)||(s+1)!lns+12nns2. Hence for γn=max(γn(2),6an,γn(1), i=1,,s) the result follows for large enough n.■

Proof of Lemma 5: Consider the random element θn defined by

θn=argminθO¯ε0(θ0)βnb(θ)i=1s11ni2V(kiβ(z,θ)),

where ε0<ε (defined in Assumption 5) Compactness and continuity imply the existence of θn. Compactness along with equicontinuity of V(kiβ(z,θ)) for all i along with Assumptions 1, 3 and Lemma AL.1 imply that

supθO¯ε0(θ0)Pθ(||θnθ||>δ)=o(na),δ>0.

Again the existence of equicontinuous derivatives implies that D(b(θ)+i=1s11ni2V(kiβ(z,θ))) converges uniformly over O¯ε0(θ0) which implies that rankD(b(θ)+i=1s11ni2V(kiβ(z,θ)))=p for all θ and large enough n. The last pair of assertions along with the equality of dimensions imply that βnb(θn)i=1s11ni2V(kiβ(z,θn))=0 with Pθ probability 1o(na) that is independent of θ∈𝒪̅ε (θ0). But then for any ε*<ε0 Lemma 8 also implies that ||βnEθnβn||ρn with supθO¯ε(θ0)Pθ(ρn>o(na))=o(na). But by definition

||βnEGMR2βn||||βnEθnβn||ρn

with Pθ probability 1o(na) that is independent of θO¯ε(θ0) and thereby there exists a random element vn such that supθO¯ε(θ0)Pθ(||vn||>o(na))=o(na) and βnEGMR2βn=vn with Pθ probability 1o(na) that is independent of θO¯ε(θ0). Notice that this necessarily holds even if the Euclidean norm is replaced by any other (possibly stochastic) norm. Consequently, this implies condition FOC of Lemma AL.5. Notice that the uniform consistency follow for the GMR1 and GMR2 as in the first parts of Lemmas 2, 3. Assumption 6 along with uniform consistency and as p=q we have that b(GMR1)=βn with probability 1o(na) uniformly on 𝒪̅ε (θ0). Furthermore by the second part of Lemma 2 we have that

(7)supθO¯ε(θ0)Pθ(n||GMR1θ||>Mln1/2n)=o(na) (7)

Hence with probability 1o(na) locally independent of θ, applying the mean value theorem we have that

b(GMR1)=b(GMR2)+b/(θn+)θ(GMR1GMR2),

where θn+ is such that ||θn+GMR2||<||GMR1GMR2||. It follows that with Pθ-probability 1o(na) locally independent of θ

GMR1GMR2=(b/(θn+)θ)1(b(GMR1)b(GMR2)).

Hence with probability 1o(na) uniformly on 𝒪̅ε (θ0)

||GMR1GMR2||M||βnb(GMR2)||M(||βnEGMR2βn||+||EGMR2βnb(GMR2)||)M||EGMR2βnb(GMR2)||=O(1n)

and the last equality is true (as βn has a uniform Edgeworth expansion on 𝒪̅ε (θ0), Assumption 5, and apply Lemma 8). Taking into account equation 7 we get that, for some C>0, locally independent of θ

supθO¯ε(θ0)Pθ(n||GMR2θ||>Cln1/2n)=o(na).

This implies condition UTIGHT of Lemma AL.5. It also, along with Lemmas 4 and 8, implies that for any ε*<ε

supθO¯ε(θ0)Pθ(||n(βnEGMR2βn)Γn(θ)||>γn)=o(na)

where γn=o(na) independent of θ and

Γn(θ)=n(βnEθβn)i=1s11ni2V(kiβ(z,θ))i=1s1ni12i!Di(b(θ)+j=1siV(kjβ(z,θ))nj2)(n(GMR2θ)i)

which validates condition EXPAND of lemma AL.5 for Gn=Wnj=Idp×p. Moreover Assumption 5 along with the fact that the support of βnb(θ) is uniformly bounded by 𝒪̅3η (0) for any η greater or equal than the diameter of B, and Lemma 4.1 of Arvanitis and Demos (2014) imply condition UEDGE of Lemma AL.5 for Mn(θ)=n(mn(θ)βnEθβn) with order s*–1. Hence the conditions of Lemma AL.5 are satisfied and the result follows.■

Proof of Lemma 6:A. It is easy to see that this special assumption implies that GMR1=GT with probability 1o(na) independent of θ. The rest are trivial consequences of Lemma 2. B. Similarly this special assumption implies that GMR2=GT (Pθ almost surely for all θ). The rest are trivial consequences of Lemmas 3 or 5 i) or ii).■

Proof of Lemma 7:

1. By Assumption 9.2–4., uniform integrability and via the triangle inequality we get that

supθΘPθ(supθΘ||Eθcn(βn)c(θ,b(θ))||>ε)=o(na)

Hence for qn(θ)=Eθcn(βn), q(θ*, θ)=c(θ, b(θ*)) and by Assumptions 4.a) Lemma AL.3 applies. Hence for γ(θ)=θ due to Assumptions 1.3, 9 Lemma AL.1 also applies proving the result.

2. Given i), we have that θn∈𝒪ϵ(θ) with Pθ-probability 1o(na) that is locally independent of θ for any ϵ>0. For some ϵ small enough, such that Oϵ(θ)Oε0(θ0) (which exists due to the fact that ε0>ε) due to Assumption 10, we have that condition FOC of Lemma AL.4 (in the Appendix) is satisfied by the GT estimator for Qn=Eθ(cn(βn))θ. Furthermore Assumption 10 implies conditions HUB (γ(θ)=θ hence set δ=ε0) and RANK of the same lemma. Condition TIGHT follows from 5 Lemma AL.2 of Arvanitis and Demos (2014) and as Eθcn(b(θ))=0 the fact that supθO¯ε(θ0)Pθ(||Eθcn(βn)||>C1ln1/2nn1/2)supθO¯ε(θ0)Pθ(||βnb(θ)||>C1supθO¯ε(θ0)Eθ(κn)ln1/2nn1/2) we get that there exist C1>0 large enough locally independent of θ for which the last term in the display is of order o(na). Hence Lemma AL.4 applies ensuring that

supθO¯ε(θ0)Pθ(n||GTθ||>Cln1/2n)=o(na)

for some C>0 independent of θ. Hence condition UTIGHT of Lemma AL.5 holds. Moreover Assumption 5 implies condition UEDGE of the same lemma for Mn(θ)=nmn(θ). Due to Assumption 10 for any φ=(θb(θ)) for any θO¯ε0(θ0) and as GT=θn is sufficiently close to φ, Eθncn(βn)θ admits with Pθ-probability 1o(na) that is independent of θ a Taylor expansion of order s*–1 around φ, with remainder

Rn(θn,θ)=1(s1)!(Ds1(Eθn+cn(βn+)θ)Ds1(Eθcn(b(θ))θ))((φnφ)s1),

and θn+,βn+ lie between θn and θ and βn and b(θ), respectively (see Arvanitis and Demos 2014, for details). Due to Assumptions 10, 5, Lemma AL.2 of Arvanitis and Demos (2014) and by submultiplicativity, the relation of θn+ to θn and condition UTIGHT

supθO¯ε(θ0)Pθ(1n(s1)/2Rn(θn,θ)>γn)supθO¯ε(θ0)Pθ(2s1M(s1)!maxs(C,C+)ns/2lns/2n>γn)+o(na)

which is of order o(na) for γn=2s1M(s1)!maxs(C,C+)ns/2lns/2n=o(na) and independent of θ. Furthermore, due to the same assumption and the fact that c(θ, b(θ))=0 we have that with Pθ-probability 1o(na) locally independent of θ the s*–1 Taylor expansion of nEθncn(βn) around φ has remainder

Rn#(θn,θ)=1s!1n(s1)/2(DsEθn+cn(βn+)DsEθcn(b(θ)))((n(φnφ))s),

and θn+ lies between θn and θ. Hence analogously to the previous

supθO¯ε(θ0)Pθ(||Rn#(θn,θ)||>γn#)supθO¯ε(θ0)Pθ(2sMs!maxs+1(C,C+)ns/2ln(s+1)/2n>γn#)+o(na)

which is of order o(na) for γn#=2sMs!maxs+1(C,C+)ns/2ln(s+1)/2n=o(na) and independent of θ (see Arvanitis and Demos 2013b, for details). Then due to Assumption 10 and the fact that Eθcn(β)=c(θ, β), Eθcn(b(θ))θ,Eθcn(b(θ))β are of full rank for any θ∈𝒪̅ε (θ0). Finally due to Lemma AL.6 which applies by Assumptions 5 and 4 condition EXPAND holds and the result follows by the same lemma.■

Proof of Lemma 8: Let Qn denote the measure with density (1+i=1sπi(z)ni2)φV(z). Since 2a+m+1>2a+1, we have that supAC|Pn(A)Qn(A)|=O(naη), where η>0. Hence

na|qK(xm)(dPndQn)|na|Oc(lnn)ϵ(0)K(xm)(dPndQn)|+na|q\Oc(lnn)ϵ(0)K(xm)dPn|+na|q\Oc(lnn)ϵ(0)K(xm)dQn|naM(lnn)mϵOc(lnn)ϵ(0)|dPndQn|+naq\Oc(lnn)ϵ(0)|K(xm)|(dPn+|dQn|)M(lnn)mϵsupACna|Pn(A)Qn(A)|+naq\Oc(lnn)ϵ(0)|K(xm)|(dPn+|dQn|)

Due to the hypothesis for the support of Pn

naq\Oc(lnn)ϵ(0)|K(xm)|dPn=na[q\Oc(lnn)ϵ(0)]Onρ(0)|K(xm)|dPnna+mβρmqmq1||x||>c(lnn)ϵdPn

Hence

na|qxm(dPndQn)|M(lnn)mϵsupACna|Pn(A)Qn(A)|+na+mβρmqmP(||ζn||>c(lnn)ϵ)+naq\Oc(lnn)ϵ(0)|K(xm)||dQn|.

As supACna|Pn(A)Qn(A)|=O(nη) for η>0, we have that

(lnn)2ϵsupACna|Pn(A)Qn(A)|=o(1)

and na+m2ρmqmP(||ζn||>c(lnn)ϵ)=o(1) if ϵ12 and c2a+m+1 by Lemma 2 of Magdalinos (1992). Finally naq\Oc(lnn)ϵ(0)|K(xm)||dQn|=o(1) due to Gradshteyn and Ryzhik (1994) formula 8.357. For the uniform case first notice that

supθO¯ε(θ0)Pθ(||ζn||>Mln1/2n)=o(na)

This is due to the fact that the set {x∈ℝq:‖x‖≤M ln1/2n} has boundary of Lebesgue measure zero and

supθO¯ϵ(θ0)||x||>Mln1/2n(1+i=1s1ni2|πi(x,θ)|)φV(θ)(x)dx||z||>Mλmax(θ)ln1/2n(1+i=1s1ni2|πi(V1/2(θi)z,θi)|)φ(x)dx

where λmax(θ) denotes the maximum absolute eigenvalue of V1/2(θ) and θiO¯ε(θ0) exist for all i=1, …, s* due to the continuity and are independent of z due to the positivity and the fact that πi are polynomials in x, and θ* exists due to continuity of V and the compactness of 𝒪̅ε (θ0). For Ms*λmax(θ*) the result follows from Lemma 2 of Magdalinos (1992). The rest follows in the same spirit of the first part.■

Proof of Lemma 9: Our Assumptions and Lemmas 2, 8 ensure the validity of the mean approximation. Then from Theorem 3.1 of Arvanitis and Demos (2014) we have that the relevant moment approximation can be obtained if we employ the appropriate approximations of n(βnb(θn)),Wn(θn), and b(θn)θ in terms of n(θnθ) and n(θnθ). Therefore an appropriate approximation for n(θnθ) is obtained by inverting the product of these terms. Finally, integrating with respect to (1+π1(z,θ)n)φV(θ)(z), noting that k1β(z,θ)=z,k2β(z,θ)=zπ1(z,θ) we obtain result (see Arvanitis and Demos 2013b, for details).■

Proof of Lemma 10: The Assumptions and Lemmas 2, 8 ensure the validity of the mean approximation uniformly over O¯ε(θ0). Furthermore from Lemma AL.7 we have that supθO¯ε(θ0)DEθβnDb(θ)1nDφV(k2β)(θ)=o(1) (recall that φV(k1β)=0). Then the proof proceeds as in the proof of Lemma 9 (see Arvanitis and Demos 2013b, for details).■

Proof of Lemma 11: First, notice that our Assumptions and Lemmas 2, 8 ensure the validity of the mean approximation. Then employing the implicit function Theorem, as c(θ, b(θ))=0l and c(θ,b(θ))θ=0lxp, appropriate order Taylor expansions of nc(θn,βn),Wn(θn) and c(θn,βn)θ, and inverting the product of these terms we get an appropriate approximation of n(θnθ). Integrating this approximation w.r.t. (1+π1(z,θ)n)φV(θ)(z) and employing Theorem 3.1 of Arvanitis and Demos (2014) we get the result (see Arvanitis and Demos 2013b, for details).■

Proof of Corollary 3: When W** is independent of x and θ and b(θ) is affine then 𝒥=0 and j=bθ2cj(θ,b)ββbθ. Substituting these expressions in the expansion of n(θnθ) and integrating w.r.t. (1+π1(z,θ)n)φV(θ)(z) we get the result. On the other hand, when l p=q=l and b(θ) is affine then D1=(c(θ,b)βbθ)1,J=0,q1β=(bθ)1k1β,q1β=(bθ)1k1β, and j=bθ2cj(θ,b)ββbθ. The result follows by integrating, again, w.r.t. (1+π1(z,θ)n)φV(θ)(z) (see Arvanitis and Demos 2013b, for details).■

Proof of Lemma 12:

i) The result follows from Lemmas 6.A and 9. Notice that as p=q=l, we have that C1bθW=(bθ)1.

ii) The result follows from Lemmas 6.B and 10.■

Proof of Lemma 13: For both estimators we have that due to Lemma 8, Theorem 3.1 of Arvanitis and Demos (2014) along with the approximations employed in Lemmas 9, 10 we have Eθ(n(θnθ)(θnθ))=qC1bθW(k1β(z,θ)+k2β(z,θ)n)(k1β(z,θ)+k2β(z,θ)n)WbθC1φV(θ)(z)dz, where k1β(z,θ)=z,k2β(z,θ)=zπ1(z,θ). Keeping the relevant order terms, the result follows.■

Proof of Lemma 14: Again we have that due to Lemma 8, Theorem 3.1 of Arvanitis and Demos (2014) along with the approximations used in Lemma 3 when W** is independent of x and θ and b(θ) is affine, we get the appropriate expression from the proof of Lemma 3. Integrating w.r.t. (1+π1(z,θ)n)φV(θ)(z) we get the result (see Arvanitis and Demos 2013b, for details).■

Proof of Lemma 15: First notice that in any step of the procedure the binding function is the identity. Next the o(na) uniform consistency of θn ensures the analogous for any step of the recursion. Then Lemma 5 applies and accordingly θn admits a locally uniform Edgeworth expansion of order s*. Given this the exact same reasoning, implies the same result for θn for any k. Moreover Assumption 8 follows for the expansions in every step of the procedure due to the previous. The proof for the moment approximations for the case k=1 follows easily. Using induction, let us assume that the result holds for some k=h, i.e. assume that the appropriate expression for n(θn(hθ) is given by:

Eθn(θn(hθ)=1n2h+12V(k2h+2)+1n2h+22V(k2h+3)+o(n2h+22).

uniformly over 𝒪̅ε (θ0). Hence for θ∈𝒪̅ε (θ0), by Lemma 4 it follows that n(Eθn(h+1θn(hEθθn(h)(Idp+1n2h+22V(k2h+2)θ)n(θn(h+1θ) is bounded by a real sequence of order o(n2h+32) that is independent of θ, with Pθ-probability 1o(n2h+32) independent of θ. The h+1st-step GMR2 estimator satisfies with Pθ-probability 1o(n2h+32) independent of θ, θn(h=Eθn(h+1θn(h. Hence Lemma 8 and Theorem 3.1 of Arvanitis and Demos (2014) imply that the required approximation would be given by the integration of the Edgeworth density in the hth step of the following approximation

n(θn(hθ)(1n2h+12V(k2h+2)+1n2h+22V(k2h+3)).

This integration gives an error of o(n2h+22) as p(1+i=12h+2πi(z,θ)ni/2)φV(θ)(z)dz=1+o(n2h+22) due to the validity of the Edgeworth approximation of the distribution of n(θnθ) and the result follows. For the MSE approximation the result follows analogously, by simply noticing that (Idp+1n2h+22V(k2h+2/)θ)1=Idp+o(1).

General Results

The following results essentially justify the methodologies used for the establishment of the validity of the locally uniform Edgeworth expansions in all the previous cases. They are cited without proofs. Their analytical proofs can be found in Arvanitis and Demos (2013b).

Lemma AL.1:Suppose that:

• UUC

supθΘPθ(supβB|cn(β)c(θ,β)|>ε)=o(na),ε>0

• AB c(θ, β) is jointly continuous and γ(θ)=arg minβBc(θ, β), then

supθΘPθ(||βnγ(θ)||>ε)=o(na),ε>0

where βn∈arg minβBcn(β).

Lemma AL.2:Let Assumptions 4.a) and 3 hold. Then for j=*,**

supθΘPθ(||Wnj(θn)EθWj(θ)||>ε)=o(na),ε>0

Furthermore, there exists K>0 for which

supθΘPθ(||Wnj(θn)||>K)=o(na)

Lemma AL.2:Suppose that

cn(β)=qn(β)Wnj(θn)qn(β)

for some appropriate random element qnwhere Wnj,θn satisfy Assumptions 4.a), 3 and for q an appropriate jointly continuous function on Θ×B

supθΘPθ(supβB||qn(β)q(θ,β)||>ε)=o(na),ε>0

Then AL.1.UUC holds for c(θ,β)=q(θ,β)EθWj(θ)q(θ,β) which is jointly continuous.

Lemma AL.4:Suppose Wnj,θn˜ satisfy Assumptions 4, 3, βnand γ(θ) are as in lemma AL.1, γ is continuous on 𝒪̅ε (θ0) and that:

• FOC βnsatisfies

qn(βn)βWnj(θn)qn(βn)=0

with Pθ-probability1o(na)that is independent of θ,

• HUB for some δ, M>0 independent of θ such thatγ(O¯ε(θ0))O¯δ(γ(θ0))and for all i, supθO¯ε(θ0)Pθ(supβO¯δ(γ(θ0))2qn(βn)ββi>M)=o(na),

• RANK for anyβO¯δ(γ(θ0)),qn(β)βis of full rank with Pθ-probability1o(na)that is independent of θ and,

• TIGHT for some C>0 independent of θ, supθO¯ε(θ0)Pθ(n||qn(γ(θ))||>Cln1/2n)=o(na), then

supθO¯ε(θ0)Pθ(n||βnγ(θ)||>C+ln1/2n)=o(na)

for some C+>0 independent of θ.

Lemma AL.5:Suppose that:

• FOC βnsatisfies

Gn(βn)Wnj(θn)qn(βn)=vn

with Pθ-probability1o(na)andPθ(||vn||>o(na))=o(na)that are independent of θ and,

• UTIGHT There exists a C+>0 independent of θ for which

supθO¯ε(θ0)Pθ(n||βnγ(θ)||>C+ln1/2n)=o(na)

• UEDGE There exists a random element Mn(θ) with values in an Euclidean space, containing the elements of n(θnθ), the distribution of which admits a uniform over 𝒪̅ε (θ0) Edgeworth expansion Ψn,s(θ). The ith polynomial, say, πi(z, θ) of Ψn,s(θ) is equicontinuous on 𝒪̅ε (θ0) ∀z∈ℝq, for i=1, …, s–2, and if Σ(θ) denotes the variance matrix in the density of Ψn,s(θ) then it is continuous on 𝒪̅ε (θ0) and positive definite.

• EXPAND The following hold with Pθ-probability 1o(na) that is independent of θ

Gn(βn)=i=0s11ni/2j=0iCi,j,n(θ)(Mn(θ)j,Sn(θ)ij)+Rn(βn,θ)Wnj(θn)=i=0s11ni/2Ci,n(θ)(Mn(θ)i)+Rn(θn,θ)nqn(βn)=i=0s11ni/2j=0i+1Ci,j,n#(θ)(Mn(θ)j,Sn(θ)i+1j)+Rn#(βn,θ)

whereSn(θ)=n(βnγ(θ)),Ci,j,n:O¯ε(θ0)×qip,Ci,n:O¯ε(θ0)×qipare i-linear, Ci,j,n:Θ×qi+1pis (i+1) -linearθO¯ε(θ0),C0,0,n,C0,n,C0,0,n#(θ),C0,1,n#(θ)are independent of n and have full rankθO¯ε(θ0),Ci,n,Ci,n,Ci,j,n#are equicontinuous on 𝒪̅ε (θ0), and

supθO¯ε(θ0)Pθ(||Rnl||>γnl)=o(na), l=,,#

for real sequence γnl=o(na) independent of θ, for, l=*, **, #.

Thenn(βnγ(θ))admits a locally uniform Edgeworth expansion,Ψn,s(θ),over 𝒪̅ε (θ0). The ith polynomial, say, πi(z,θ)of the density ofΨn,s(θ)is equicontinuous onO¯ε(θ0)zq,for i=1, …, s–2, and if Σ*(θ) denotes the variance matrix in the density ofΨn,s(θ)then it is continuous on 𝒪̅ε (θ0) and positive definite.

Lemma AL.6: Under Assumptions 4 and 5 condition EXPAND hold for Wnj(θn) where Mn(θ)=nmn(θ).

Lemma AL.7: For real valued functions fn, f defined on Θ′⊇Θ, suppose that: supθ∈Θ|fnf|=o(1), and supθ∈Θ D2fn‖, supθ∈ΘD2f‖<M. Then supθ∈ΘDfnDf‖=o(1).

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Published Online: 2016-8-3

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