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

Appendix

Proofs

Proof of Lemma 2:

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

   supθ∗∈ΘPθ∗(supθ∈Θ|||βn−b(θ)||−||b(θ∗)−b(θ)|||>ε)≤supθ∗∈ΘPθ∗(||βn−b(θ∗)||>ε)=o(n−a∗)

   Hence for q_n(θ)=β_n–b(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 θ_n∈𝒪_ϵ(θ) with P_θ-probability 1−o(n−a∗) 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 Qn≑∂b′∂θ. 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)supθ∈O¯ε(θ0)Pθ(n||βn−b(θ)||>C∗ln1/2n)=o(n−a∗) (5)

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

   supθ∈O¯ε(θ0)Pθ(n||GMR1−θ||>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∈𝒪̅_ε (θ₀) and any θ_* sufficiently close to θ, ∂b′∂θ(θ∗) admits a Taylor expansion of order s^*–1 around θ of the form

   ∂b′∂θ(θ∗)=∑i=0s∗−11i!Di∂b′∂θ(θ)((θ∗−θ)i)+1(s∗−1)!(Ds∗−1∂b′∂θ(θ+)−Ds∗−1∂b′∂θ(θ))((θ∗−θ)s∗−1)

   where θ⁺ lies between θ_* and θ. This implies that for any θ_n=GMR1 due to condition UTIGHT we have that with P_θ-probability 1−o(n−a∗) locally independent of θ

   ∂b′∂θ(θn)=∑i=0s∗−11i!1ni/2Di∂b′∂θ(θ)((n(θn−θ))i)+Rn∗(θn, θ)

   where Rn∗(θn, θ)=1(s∗−1)!1n(s∗−1)/2(Ds∗−1∂b′∂θ(θn+)−Ds∗−1∂b′∂θ(θ))((n(θn−θ))s∗−1), and θn+ lies between θ_n and θ. Now by Assumption 6 ∂b′∂θ(θ) has full rank for any θ∈𝒪̅_ε (θ₀) and by submultiplicativity, the relation of θn+ to θ_n and condition UTIGHT

   supθ∈O¯ε(θ0)Pθ(‖1(s∗−1)!1n(s∗−1)/2(Ds∗−1∂b′∂θ(θn+)−Ds∗−1∂b′∂θ(θ))×((n(θn−θ))s∗−1)‖>γn∗)≤supθ∈O¯ε(θ0)Pθ(1(s∗−1)!1n(s∗−1)/2supθ∈O¯ε0(θ0)‖Ds∗∂b′∂θ(θ)‖||θn+−θ||||n(θn−θ)||s∗−1>γn∗)≤supθ∈O¯ε(θ0)Pθ(M(s∗−1)!Cs∗ns∗/2lns∗/2n>γn∗)+o(n−a∗)

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

   qn=βn−b(θ∗)=βn−b(θ∗)−∑i=1s∗1i!Dib(θ)((θ∗−θ)i)−1s∗!(Ds∗b(θ+)−Ds∗b(θ))((θ∗−θ)s∗)

   where θ⁺ lies between θ_* and θ. This implies that for θ_n we have that with P_θ-probability 1−o(n−a∗)

   n(βn−b(θn))=n(βn−b(θ))+∑i=0s∗−11(i+1)!1ni/2Di+1b(θ)((n(θn−θ))i+1)+Rn#(θn, θ)

   where Rn#(θn, θ)=1s∗!1n(s∗−1)/2(Ds∗b(θn+)−Ds∗b(θ))((n(θn−θ))s∗), and θn+ lies between θ_n and θ. Now by Assumption 6 ∂b′∂θ(θ) has full rank for any θ∈𝒪̅_ε (θ₀) and so does the identity matrix in front of n(βn−b(θ)), and thereby due to submultiplicativity, the relation of θn+ to θ_n and condition UTIGHT

   supθ∈O¯ε(θ0)Pθ(‖1s∗!1n(s∗−1)/2(Ds∗b(θn+)−Ds∗b(θ))×((n(θn−θ))s∗)‖>γn#)≤supθ∈O¯ε(θ0)Pθ(1s∗!1n(s∗−1)/2supθ∈O¯ε0(θ0)||Ds∗+1b(θ)|| ||θn+−θ||×||n(θn−θ)||s∗>γn#)≤supθ∈O¯ε(θ0)Pθ(Ms∗!Cs∗+1ns∗/2ln(s∗+1)/2n>γn#)+o(n−a∗)

   which is of order o(n−a∗) for γn#=Ms∗!Cs∗+1ns∗/2ln(s∗+1)/2n=o(n−a∗) 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)={ω∈Ω:||βn−b(θ)||>ε2}∈ℱ, then

supθ∈Θ||Eθβn−b(θ)||≤supθ∈ΘEθ||βn−b(θ)||1E(ε,θ)+ε2.

As B is bounded, due to Assumption 1.2 and by Assumption 3 there exists an n^* such that supθ∈ΘPθ(||βn−b(θ)||>ε3)≤ε2M where M denotes the diameter of B. Hence

supθ∈Θ||Eθβn−b(θ)||≤ε for all n≥n∗

and since ε is arbitrary

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

supθ∗∈ΘPθ∗(supθ∈Θ| ||βn−Eθβn||−||b(θ∗)−b(θ)|| |>ε)≤supθ∗∈ΘPθ∗(||βn−b(θ∗)||+supθ∈Θ||Eθβn−b(θ)||>ε)=o(n−a∗)

For q_n(θ)=E_θβ_n, 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 θ_n∈𝒪_ϵ(θ) with P_θ-probability 1−o(n−a∗) 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 Qn≑∂Eθ(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 D²E_θβ_n is uniformly bounded on Oε0(θ0),DE_θβ_n 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∗={ω∈Ω:||βn−b(θ)||>C2ln1/2nn1/2}∈ℱ

supθ∈O¯ε(θ0)||Eθβn−b(θ)||≤supθ∈O¯ε(θ0)Eθ[||βn−b(θ)||1E∗]+supθ∈O¯ε(θ0)Eθ[||βn−b(θ)||1E∗¯]≤Msupθ∈O¯ε(θ0)Pθ(||βn−b(θ)||>C2ln1/2nn1/2)+C2ln1/2nn1/2supθ∈O¯ε(θ0)Pθ(||βn−b(θ)||≤C2ln1/2nn1/2)=o(n−a∗)+C2ln1/2nn1/2(1−o(n−a∗))=O(ln1/2nn1/2),

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

supθ∈O¯ε(θ0)Pθ(||βn−Eθβn||>C1ln1/2nn1/2)≤supθ∈O¯ε(θ0)Pθ(||βn−b(θ)||+||Eθβn−b(θ)||>C1ln1/2nn1/2)≤supθ∈O¯ε(θ0)Pθ(||βn−b(θ)||+O(ln1/2nn1/2)>C1ln1/2nn1/2)=o(n−a∗)

Hence Lemma AL.4 applies ensuring that

supθ∈O¯ε(θ0)Pθ(n||GMR2−θ||>Cln1/2n)=o(n−a∗)

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 β_n–b(θ) is uniformly bounded by 𝒪̅_3η (0) for any η greater or equal than the diameter of B, and the fact that n(βn−Eθβn) 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)βn−Eθβn) with order s^*–1. Due to Assumption 7 for any θ∈𝒪̅_ε (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 ∂b′∂θ(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=1sℐV(kiβ(z, θ))ni2) and n(b(θn+)+∑i=1sℐV(kiβ(z, θn+))ni2), we get n(Eθn∗βn−Eθβn)−An(θ)=n(Eθn∗βn−b(θn+)−∑i=1sℐV(kjβ(z, θn+))ni2)−n(Eθβn−b(θ)−∑i=1sℐV(kiβ(z, θ))ni2)+n(b(θn+)−b(θ)−∑i=1s1i!Dib(θ)((θn+−θ)i))+Bn where Bn=∑i=1sℐV(kiβ(z, θn+))ni−12−∑i=1sℐV(kiβ(z, θ))ni−12−∑i=1s1i!∑j=1s−iDiℐV(kjβ(z, θ))nj−12((θn+−θ)i). Employing the mean value Theorem for ℐV(kiβ(z, θn+)), and for θn++ such that ||θn++−θ||<||θn+−θ||, and collecting terms we get:

Bn=∑i=1s1ni−121(s−i+1)!Ds−i+1ℐV(kiβ(z, θ))((θn++−θ)s−i+1).

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

supθ∈O¯ε(θ0)Pθ(||n(Eθn+βn−Eθβn)−An(θ)||>γn)≤supθ∈O¯ε(θ0)Pθ(supθ∈O¯ε(θ0)n‖Eθβn−b(θ)−∑i=1sℐV(kiβ(z, θ))ni2‖>γn6)+∑i=1ssupθ∈O¯ε(θ0)Pθ(1ni−12||Bn||>γn3s)+supθ∈O¯ε(θ0)Pθ(n‖b(θn+)−b(θ)−∑i=1s1i!Dib(θ)((θn+−θ)i)‖>γn3)+o(n−a∗).

Now we have that

an=n‖Eθβn−b(θ)−∑i=1sℐV(kiβ(z, θ))ni2‖=o(n−a)

independent of θ, due to Lemma 8.

Further, due to the continuity of Ds−i+1ℐV(kiβ(z, θ)), Assumption 8, and the assumption of the asymptotic behavior of θn+ we get

supθ∈O¯ε(θ0)Pθ(1ni−12||Bn||>γn3s)≤o(n−a∗)

provided that γn≥γn(1)=lns−i+12nns23s supθ∈O¯ε(θ0)sup||Ds−i+1ℐV(kiβ(z, θ))||(s−i+1)!.

Furthermore using the same reasoning as above

supθ∈O¯ε(θ0)Pθ(n||b(θn+)−b(θ)−∑i=1s1i!Dib(θ)((θn∗−θ)i)||>γn3)≤o(n−a∗)

when γn≥γn(2)=3supθ∈O¯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∗=argminθ∈O¯ε0(θ0)‖βn−b(θ)−∑i=1s−11ni2ℐV(kiβ(z, θ))‖,

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

supθ∈O¯ε0(θ0)Pθ(||θn∗−θ||>δ)=o(n−a∗), ∀δ>0.

Again the existence of equicontinuous derivatives implies that D(b(θ)+∑i=1s−11ni2ℐV(kiβ(z, θ))) converges uniformly over O¯ε0(θ0) which implies that rankD(b(θ)+∑i=1s−11ni2ℐV(kiβ(z, θ)))=p for all θ and large enough n. The last pair of assertions along with the equality of dimensions imply that ‖βn−b(θn∗)−∑i=1s−11ni2ℐV(kiβ(z, θn∗))‖=0 with P_θ probability 1−o(n−a∗) that is independent of θ∈𝒪̅_ε (θ₀). But then for any ε*<ε₀ Lemma 8 also implies that ||βn−Eθn∗βn||≤ρn with supθ∈O¯ε∗(θ0)Pθ(ρn>o(n−a∗))=o(n−a∗). But by definition

||βn−EGMR2βn||≤||βn−Eθn∗βn||≤ρn

with P_θ probability 1−o(n−a∗) that is independent of θ∈O¯ε∗(θ0) and thereby there exists a random element v_n such that supθ∈O¯ε∗(θ0)Pθ(||vn||>o(n−a∗))=o(n−a∗) and β_n–E_GMR2β_n=v_n with P_θ probability 1−o(n−a∗) that is independent of θ∈O¯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)=β_n with probability 1−o(n−a∗) uniformly on 𝒪̅_ε (θ₀). Furthermore by the second part of Lemma 2 we have that

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

b(GMR1)=b(GMR2)+∂b/(θn+)∂θ(GMR1−GMR2),

where θn+ is such that ||θn+−GMR2||<||GMR1−GMR2||. It follows that with P_θ-probability 1−o(n−a∗) locally independent of θ

GMR1−GMR2=(∂b/(θn+)∂θ)−1(b(GMR1)−b(GMR2)).

Hence with probability 1−o(n−a∗) uniformly on 𝒪̅_ε (θ₀)

||GMR1−GMR2||≤M||βn−b(GMR2)||≤M(||βn−EGMR2βn||+||EGMR2βn−b(GMR2)||)≤M||EGMR2βn−b(GMR2)||=O(1n)

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

supθ∈O¯ε(θ0)Pθ(n||GMR2−θ||>Cln1/2n)=o(n−a∗).

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

supθ∈O¯ε∗(θ0)Pθ(||n(βn−EGMR2βn)−Γn(θ)||>γn)=o(n−a)

where γ_n=o(n^−a) independent of θ and

Γn(θ)=n(βn−Eθβn)−∑i=1s−11ni2ℐV(kiβ(z, θ))−∑i=1s1ni−12i!Di(b(θ)+∑j=1s−iℐV(kjβ(z, θ))nj2)(n(GMR2−θ)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 β_n–b(θ) is uniformly bounded by 𝒪̅_3η (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)βn−Eθβn) 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 1−o(n−a∗) independent of θ. The rest are trivial consequences of Lemma 2. B. Similarly this special assumption implies that GMR2=GT (P_θ 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

   supθ∗∈ΘPθ∗(supθ∈Θ||Eθcn(βn)−c(θ, b(θ∗))||>ε)=o(n−a∗)

   Hence for q_n(θ)=E_θc_n(β_n), 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 θ_n∈𝒪_ϵ(θ) with P_θ-probability 1−o(n−a∗) 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 E_θc_n(b(θ))=0 the fact that supθ∈O¯ε(θ0)Pθ(||Eθcn(βn)||>C1ln1/2nn1/2)≤supθ∈O¯ε(θ0)Pθ(||βn−b(θ)||>C1supθ∈O¯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(n−a∗). Hence Lemma AL.4 applies ensuring that

   supθ∈O¯ε(θ0)Pθ(n||GT−θ||>Cln1/2n)=o(n−a∗)

   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 θ∈O¯ε0(θ0) and as GT=θ_n is sufficiently close to φ, ∂Eθnc′n(βn)∂θ admits with P_θ-probability 1−o(n−a∗) that is independent of θ a Taylor expansion of order s^*–1 around φ, with remainder

   Rn∗(θn, θ)=1(s∗−1)!(Ds∗−1(∂Eθn+cn(βn+)′∂θ)−Ds∗−1(∂Eθcn(b(θ))′∂θ))((φn−φ)s∗−1),

   and θn+,βn+ lie between θ_n and θ and β_n 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 θ_n and condition UTIGHT

   supθ∈O¯ε(θ0)Pθ(‖1n(s∗−1)/2Rn∗(θn, θ)‖>γn∗)≤supθ∈O¯ε(θ0)Pθ(2s∗−1M(s∗−1)!maxs∗(C, C+)ns∗/2lns∗/2n>γn∗)+o(n−a∗)

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

   Rn#(θn, θ)=1s∗!1n(s∗−1)/2(Ds∗Eθn+cn(βn+)−Ds∗Eθcn(b(θ)))((n(φn−φ))s∗),

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

   supθ∈O¯ε(θ0)Pθ(||Rn#(θn, θ)||>γn#)≤supθ∈O¯ε(θ0)Pθ(2s∗Ms∗!maxs∗+1(C, C+)ns∗/2ln(s∗+1)/2n>γn#)+o(n−a∗)

   which is of order o(n−a∗) for γn#=2s∗Ms∗!maxs∗+1(C, C+)ns∗/2ln(s∗+1)/2n=o(n−a∗) and independent of θ (see Arvanitis and Demos 2013b, for details). Then due to Assumption 10 and the fact that E_θc_n(β)=c(θ, β), ∂Eθc′n(b(θ))∂θ,∂Eθc′n(b(θ))∂β are of full rank for any θ∈𝒪̅_ε (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 Q_n denote the measure with density (1+∑i=1sπi(z)ni2)φV(z). Since 2a+m+1>2a+1, we have that supA∈ℬC|Pn(A)−Qn(A)|=O(n−a−η), where η>0. Hence

na|∫ℝqK(xm)(dPn−dQn)|≤na|∫Oc(lnn)ϵ(0)K(xm)(dPn−dQn)|+na|∫ℝq\Oc(lnn)ϵ(0)K(xm)dPn|+na|∫ℝq\Oc(lnn)ϵ(0)K(xm)dQn|≤naM(lnn)mϵ∫Oc(lnn)ϵ(0)|dPn−dQn|+na∫ℝq\Oc(lnn)ϵ(0)|K(xm)|(dPn+|dQn|)≤M(lnn)mϵsupA∈ℬCna|Pn(A)−Qn(A)|+na∫ℝq\Oc(lnn)ϵ(0)|K(xm)|(dPn+|dQn|)