Proofs of Lemmas and Theorems from Main Text

2.8 Supplementary Appendix

2.8.1 Proofs of Lemmas and Theorems from Main Text

Proof of Lemma 2.2:The proof is divided into two steps. In the fisrt step, we show thatβb→^p β0.

Observe that the three-stage estimation procedure is equivalent to a GMM estimator with(d_γ+2d_λ+d_β)moment conditions. We collect the moment conditions by

m(a;θ,q_τ)≡







m₁(a;γ,q_τ) m₂(a;γ,λ_s,qτ) m3(a;γ,λ_a,qτ) m4(a;γ,λs,λa,β,qτ)





 ,

where

m₁(a;γ,q_τ)≡ r−L(k(z)⁰γ)

L(k(z)⁰γ)(1−L(k(z)⁰γ))L⁰(k(z)⁰γ)k(z), m2(a;γ,λs,qτ)≡

L(k(z)⁰γ+t(z)⁰λ_s)−1

L(k(z)⁰γ)t(z),

m₃(a;γ,λa,q_τ)≡

1−r

1−L(k(z)⁰γ+t(z)⁰λa)−1

L(k(z)⁰γ)t(z), m₄(a;θ,q_τ)≡

rL(k(z)⁰γ)e(z)

L(k(z)⁰γ+t(z)⁰λs)1(y≤q_τ)− (1−r)L(k(z)⁰γ)e(z)

1−L(k(z)⁰γ+t(z)⁰λa)Λ(w;β)

In addition, let

Lcn(θ) ≡ kEn[m(A;θ,qbτ)]k²

Ωe_n, Lfn(θ) ≡ kEn[m(A;θ,qbτ)]k²

Ωe, Ln(θ) ≡ kEn[m(A;θ,q_τ)]k²

Ωe, L^∗(θ) ≡ kE[m(A;θ,q_τ)]k²

Ωe,

whereΩe_n≡diag(I_d_γ_+2d

λ,Ω_n)andΩe≡diag(I_d_γ_+2d

λ,Ω).

First, by Assumptions 2.7(c)(iii), (d), (e)(iii), (f), Lemma 2.5, and the uniform LLN,

Lcn(θ)−Lfn(θ)

≤ kΩ_n−Ωk ·(kEn[m(A;θ,qbτ)]−E[m(A;θ,qτ)]k²+kE[m(A;θ,qτ)]k²)

=O_p(δ_ω,n)(o_p(1) +O_p(1)) =o_p(1). (2.8.1)

Next, by the definition ofLfn(θ)andLn(θ),

Lfn(θ)−Ln(θ)

≤ kEn[m₄(A;θ,qb_τ)−m₄(A;θ,q_τ)]k²_Ω+O_p(1)λ_max(Ω)kEn[m₄(A;θ,qb_τ)−m₄(A;θ,q_τ)]k

=o_p(1), (2.8.2)

where the last line follows by Assumption 2.7(d) and Lemma 2.5.

Again, by the definition ofLn(θ)andL^∗(θ),

kLn(θ)−L^∗(θ)k ≤λ_max(Ω)(kEn[m(A;θ,qτ)]−E[m(A;θ,qτ)]k²

+O_p(1)kEn[m(A;θ,q_τ)]−E[m(A;θ,q_τ)]k) =o_p(1), (2.8.3)

where the last equality follows by Assumption 2.7(d) and Lemma 2.5.

Letθb≡arg minθ∈ΘLcn(θ).To show the consistency ofθb, we note that

θb−θ₀

≥ε

≤P

θ∈Θ:kθ−θinf₀k≥εLcn(θ)≤Lcn(θ₀)

≤P

inf

θ∈Θ:kθ−θ₀k≥εLfn(θ)≤Lfn(θ₀) +2 sup

θ∈Θ

|Lcn(θ)−Lfn(θ)|

≤P

inf

θ∈Θ:kθ−θ₀k≥εLn(θ)≤Ln(θ₀) +2 sup

θ∈Θ

|Lfn(θ)−Ln(θ)|+o_p(1)

≤P

inf

θ∈Θ:kθ−θ₀k≥εL^∗(θ)≤L^∗(θ₀) +2 sup

θ∈Θ

|Ln(θ)−L^∗(θ)|+o_p(1)

≤P

θ∈Θ:kθ−θinf₀k≥εkE[m(A;θ,q_τ)]k²≤ o_p(1) λmin(Ω)

=o(1),

where the first inequality is obtained by the definition ofθb, and the third to fifth line follow by (2.8.1)–(2.8.3), respectively. By Assumptions 2.3, 2.7(a)(ii), (c)(i), and (e)(ii),θ=θ0is the unique solution tokE[m(A;θ,q_τ)]k=0. Using this fact, the last line is obtained by Exercise 5.27 in Van der Vaart (1998).

In the second step, we prove that√

n(bβ−β0)→^d N(0,Σ_β).

A first-order Taylor expansion yields that

o_p(1) =M_n⁰Ωen

√1 n

∑

i=1

m(A_i;θb,qb_τ)

=M_n⁰Ωen

√1 n

n i=1

∑

(m(A_i;θ0,bq_τ)−m(A_i;θ₀,q_τ) +m(A_i;θ0,q_τ)) +Me_n√

n(bθ−θ0)

! ,

whereM_n≡En[∇_θm(A;qb_τ,θb)], andMe_n≡En[∇_θm(A;qb_τ,θe)],for someθelying betweenθbandθ₀.Usingθb→^p θ₀, it

can be shown that bothM_nandMe_nconverge in probability toM≡E[∇_θm(A;θ₀,q_τ)]. The proof is similar to that of Lemma 2.5, we omit the details to avoid repetition. Thus,

√n(bθ−θ0)

=−M_Ω⁻¹M⁰Ωe 1

√n

n i=1

∑

m(A_i;θ0,qbτ)−m(A_i;θ0,qτ) +m(A_i;θ0,qτ)

! +o_p(1)

=−M⁻¹

Ω M⁰Ωe 1

√n

n i=1

∑

E[m(A;θ0,bqτ)−m(A;θ0,qτ)|Y_n] +m(A_i;θ₀,q_τ)

! +op(1)

= 1

√n

n i=1

∑

ψθ(A_i;θ₀,q_τ) +o_p(1), (2.8.4)

where

M_Ω≡M⁰ΩM,e (2.8.5)

ψ_θ(a;θ0,qτ)≡ −M⁻¹

Ω M⁰Ωe

m(a;θ0,qτ)−M_q_τr·(1(y≤qτ)−τ) Q₀f_Y|R(q_τ|1)

, (2.8.6)

andM_q_τ = (0⁰,0⁰,0⁰,E[e(Z)R f_Y_|ZR(q_τ|Z,1)]⁰)⁰. The second equality follows from Lemma 2.4. The third one is due to a first-order Taylor expansion, the uniform LLN, and the following fact,

qb_τ−q_τ=−En

R(1(Y≤q_τ)−τ) Q₀f_Y_|R(q_τ|1)

+o_p(n^−1/2). (2.8.7)

See, e.g. Firpo (2007).

The asymptotic linear representation of βb corresponds to the lastd_β-elements of (2.8.6). LetM_k,s denote the Jacobian matrix of thek-th subvector ofE[m(A;qτ,θ)]with respect to thes-th subvector ofθ evaluated atθ₀. It is straightforward to show that,

√n(βb−β0) = 1

√n

n i=1

∑

ψ_β(A_i;θ0,q_τ),

where

ψ_β(a;θ₀,q_τ)≡ − 1

√n(M₄₄⁰ ΩM₄₄)⁻¹











M₄₄⁰ ΩM4,−4S⁻¹M_−4,−4⁰







m₁(a;γ₀) m2(a;γ0,λs,0) m3(a;γ₀,λa,0)







−M₄₄⁰ Ω^1/2(I−Ω^1/2M4,−4S⁻¹M_4,−4⁰ Ω^1/2M_Ω1/2M4,4)Ω^1/2me₄(a;θ0,q_τ)o

, (2.8.8)

forM4,−4≡(M₄₁,M₄₂,M₄₃),S≡M_−4,−4⁰ M−4,−4−M_4,−4⁰ Ω^1/2M_Ω^1/2_M_4,4Ω^1/2M4,−4,MH≡I−H(H⁰H)⁻¹H⁰, and

me4(a;θ0,qτ)≡m4(a;θ0,qτ)−E[R f_Y|ZR(q_τ|Z,1)e(Z)]·r·(1(y≤q_τ)−τ) Q₀f_Y|R(q_τ|1) .

This concludes the proof of Lemma 2.2.

Proof of Theorem 2.3:Part I: Asymptotic result forU QE[_q. Decomposing the difference, we have

U QE[_q(τ,G)−U QE_q(τ,G)≤ − 1

bf_Y|R(qbτ|1)− 1 f_Y|R(q_τ|1)

dbq,n(bθ,G)

− 1

f_Y_|R(q_τ|1)(db_q,n(bθ,G)−db_q,n(θ₀,G))

− 1

f_Y|R(q_τ|1)db_q,n(θ₀,G)−U QE(τ,G)

≡∆q,1+∆q,2+∆_q,3, (2.8.9)

where

db_q,n(bθ,G)≡Ena[b`(Z)Λ_x(W;βb)bg_q(X)], (2.8.10)

db_q,n(θ₀,G)≡ 1

1−Q₀En[(1−R)`(Z)Λ_x(W;β₀)bg_q(X;θ0)],

andgb_q(x;θ₀)≡G⁻¹ 1

1−Q0En[(1−R)`(Z)1(X≤x)]

−x.

We proceed by deriving the asymptotic linear representation of each term.

For∆q,1, we focus on the inverse of the density estimate first, 1

bf_Y_|R(qb_τ|1)− 1

f_Y|R(q_τ|1)=−bf_Y|R(qbτ|1)−f_Y|R(q_τ|1)

f_Y|R(q_τ|1)² +ξ1, (2.8.11) where

ξ₁≡(bf_Y|R(qb_τ|1)−f_Y|R(q_τ|1))² f_Y|R(q_τ|1)²bf_Y|R(qbτ|1) .

Next, we establish the rate for fb_Y_|R(qb_τ|1)−f_Y|R(q_τ|1). Decomposing the difference,

bf_Y_|R(qbτ|1)−f_Y_|R(q_τ|1) =(bf_Y|R(bqτ|1)−ef_Y|R(qbτ|1)) + (ef_Y|R(bqτ|1)−E[ef_Y_|R(qbτ|1)]) + (E[fe_Y_|R(qb_τ|1)]−f_Y|R(qb_τ|1)) + (f_Y|R(bq_τ|1)−f_Y_|R(q_τ|1))

≡∆_f_,1+∆f,2+∆f,3+∆f,4,

wherefe_Y_|R(q_τ|1)≡En

RK_b_y(Y−q_τ) /Q₀.

We analyze each term in turn. For∆f,1, a first-order approximation and applying the uniform LLN yield,

bf_Y_|R(qb_τ|1)−ef_Y|R(bq_τ|1) =−En[R−Q₀]

Q₀ ·En[RK_b_y(Y−qb_τ)]

En[R]

=−En[R−Q₀] Q₀ ·

E[RK_b_y(Y−q_τ)]

Q₀ +En[RK_b_y(Y−qb_τ)]

En[R] −E[RK_b_y(Y−q_τ)]

Q₀

=−En[R−Q₀]

Q₀ ·E[RK_b_y(Y−q_τ)]

Q₀ +o_p(n^−1/2),

where the third line follows along a similar line of argument as in Section B.3.2 of Sasaki et al. (2022).

Likewise, we have that

∆f,2=1 n

n i=1

∑

1 Q₀

RiK_b_y(Y_i−q_τ)−E[RK_b_y(Y−qτ)] +o_p(n^−1/2).

Lastly, we bound the bias,

∆f,3=E

RK_b_y(Y−qb_τ) Q0

−f_Y|R(bq_τ|1)

=b²_y(f_Y|R⁰⁰ (q_τ|1) + (f_Y|R⁰⁰ (eqτ|1)−f_Y⁰⁰_|R(q_τ|1))) 2

y²Ky(y)dy

=Bs,y(q_τ,by) +op(b²_y), (2.8.12)

whereB_s,y(q_τ,b_y)≡0.5b²_yf_Y⁰⁰_|R(q_τ|1)^Ry²K_y(y)dyandqelies betweenbq_τandqb_τ+c_yb_y, for some|c_y|<y, where ¯¯ yis the maximum absolute value of Supp(K_y). The second line follows by changing variables and a second-order expansion of f_Y|R(·|1)aboutqb_τ. The rate of remainder iso_p(b²_y)because|f_Y|R⁰⁰ (eq_τ|1)−f_Y⁰⁰_|R(q_τ|1)| ≤ |f_Y⁰⁰_|R(qe_τ|1)−f_Y|R⁰⁰ (bq_τ|1)|+

|f_Y⁰⁰_|R(qb_τ|1)−f_Y|R⁰⁰ (q_τ|1)|=O_p(b_y+n^−1/2),and by Assumption 2.8(c).

In view of (2.8.7), it follows that

∆_f,4=1 n

n i=1

∑

R_if_Y|R⁰ (q_τ|1)(1(Y_i≤q_τ)−τ)

Q₀f_Y|R(q_τ|1) +o_p(n^−1/2).

Collecting the linear expansions of∆f,1–∆_f_,4, we get

fb_Y_|R(qb_τ|1)−f_Y|R(q_τ|1)

=En

R Q0

(

K_b_y(Y−q_τ)−E[RK_b_y(Y−qτ)]

−(1(Y ≤qτ)−τ)f_Y⁰_|R(q_τ|1) f_Y|R(q_τ|1)

+B_s,y(q_τ,b_y) +ξ₂, (2.8.13)

whereξ₂collects the remainder terms from each∆_f,j, for j=1, ...,4, andξ₂=o_p(n^−1/2b^−1/2y +b²_y).

By Lemma 2.6, we have

dbq,n(θb,G)→^p dq(θ₀,G). (2.8.14) Collecting the results in (2.8.11), (2.8.13), and (2.8.14), we deduce that

∆q,1=1 n

n i=1

∑

ψq₁(A_i,θ0,q_τ,b_y) +Bs,y(q_τ,b_y)d_q(θ₀,G)

f_Y|R² (q_τ|1) +o_p(n^−1/2_s b^−1/2_y +b²_y), where

ψ_q₁(a,θ₀,q_τ,b_y)≡ r

Q₀ K_b_y(y−q_τ)−E[K_b_y(Y−q_τ)|R=1]

−(1(y≤q_τ)−τ)f_Y⁰_|R(q_τ|1) f_Y|R(q_τ|1)

·d_q(θ₀,G) ),

f_Y|R² (q_τ|1). (2.8.15)

Next, we derive the asymptotically linear representation of∆_q,2. Decomposing the difference, we get

dbq,n(bθ,G)−dbq,n(θ₀,G) =

dbq,n(bθ,G)−deq,n(bθ,G) +

deq,n(bθ,G)−dbq,n(θ₀,G)

≡∆q,4+∆_q,5,

wherede_q,n(bθ,G)≡En

(1−R)L(k(Z)⁰γ+t(Z)b ⁰bλ_s)

Q0(1−L(k(Z)⁰bγ+t(Z)⁰bλa))Λx(W;βb)bg_q(X)

.By a second-order Taylor expansion with respect to En[R]aroundQ₀,

∆q,4=−En[R−Q₀]

Q₀ ·(d_q(θ₀,G) +de_q,n(bθ,G)−d_q(θ₀,G)) +O_p(n⁻¹)

=−En[R−Q₀] Q0

·d_q(θ₀,G) +o_p(n^−1/2),

where the second line follows by Lemma 2.2, Assumptions 2.7(c)(ii), (e)(ii), Assumption 2.8(d), and the uniform LLN.

Again, by expandingdeq,n(bθ,G)aroundθ₀, we have that

∆q,5=M_θ,q,n(eθ)⁰(bθ−θ0) +o_p θb−θ0

whereθeis some value betweenθbandθ0,

M_θ,q,n(θ)≡En





 1−R

Q₀







Λx(W;β) ∇L,θ(Z)·bg_q(X;θ) +G_q,θ(X) L_s(Z)

1−L_a(Z)Λ_x,β(W;β)gbq(X;θ)











,

and

G_q,θ(x)≡G⁰(G⁻¹(Fb_X|R(x|1)))⁻¹·Ena

∇_L,θ(Z)1(X≤x) .

Slightly modifying Theorem 8.2 in Newey and McFadden (1994) and applying it to M_θ,q,n, we can deduce that Mθ,q,n(eθ)→^p Mθ,q(θ₀), whereMθ,q(θ₀)is defined in (2.7.6). Therefore,∆q,5=Mθ,q(θ₀)⁰(bθ−θ0) +o_p(n^−1/2).

In view of (2.8.4) and (2.8.6),∆q,5=¹_n∑ⁿ_i=1Mθ,q(θ₀)⁰ψθ(A_i;θ0,qτ) +op(n^−1/2).Hence,∆q,2=¹_n∑ⁿ_i=1ψq₂(A_i;θ0, q_τ) +o_p(n^−1/2),where

ψq₂(a;θ0,q_τ)≡ − 1 f_Y|R(q_τ|1)

M_θ,q(θ₀)⁰ψθ(a;θ0,q_τ)−r−Q₀ Q0

d_q(θ₀,G)

. (2.8.16)

Next, we derive the asymptotic linear expansion of∆q,3. By definition,

dbq,n(θ₀,G)−d_q(θ₀,G) =En[∆φ_q(A;θ₀,G)]

+ 1

1−Q0En[(1−R)`(Z)Λ_x(W;β0)g_q(X)]−d_q(θ₀,G)

where

∆φq(a;θ₀,G)≡ 1−r

1−Q₀`(z)Λ_x(w;β0)(gb_q(x;θ₀)−g_q(x;θ0)). (2.8.17)

By Lemma 2.7, we deduce thatEn[∆φ_q(A;θ₀,G)] =¹_n∑ⁿi=1ψ_g,q(A_i;θ₀,G) +o_p(n^−1/2),whereψ_g,qis given in (2.7.9).

Using this result, by the definition ofdb_q,n(θ₀,G), and by Theorem 2.1, it follows that∆_q,3=¹_n∑ⁿi=1ψ_q₃(A_i;θ₀,q_τ), where

ψq3(a;θ0,q_τ)≡ − 1 f_Y_|R(q_τ|1)

ψg,q(a;θ0,G) + 1−r

1−Q₀`(z)Λ_x(w;β₀)g_q(x)

−d_q(θ₀,G)). (2.8.18)

Collecting the linear expansions associated with∆q,j, j=1,2,3,and by the Lyapunov CLT, we deduce that, with ψ_qgiven in (2.7.3),

pnby([U QE_q(τ,G)−U QE_q(τ,G))→^d N(0,lim

by→0byE[ψ_q(A;θ0,qτ,b_y)ψ_q(A;θ0,q_τ,by)⁰]).

To finish the proof, we need to verify the asymptotic variance ofU QE[_q(τ,G),

lim

by→0b_yE[ψ_q(A;θ0,q_τ,b_y)ψ_q(A;θ0,q_τ,b_y)⁰]

= D²(θ₀,G) f_Y|R⁴ (q_τ|1)· lim

by→0

b_y Q0

K_b²

y(Y−q_τ)|R=1

− b_y

Q0 E[K_b_y(Y−q_τ)|R=1]2

= D²(θ₀,G) f_Y|R⁴ (q_τ|1)Q₀· lim

by→0

Z 1 b_yK_y²

y−q_τ b_y

f_Y_|R(y|1)dy−b_y f_Y|R(q_τ) +o(b²_y)2

= D²(θ₀,G) f_Y|R⁴ (q_τ|1)Q₀· lim

by→0

K_y²(u)f_Y|R(q_τ|1)du+O(b_y)

= D²(θ₀,G) f_Y|R³ (q_τ|1)Q₀·

K_y²(u)du,

where the third line is due to (2.8.12), and the fourth one is obtained by changing variable, a first-order expansion of K_y, and by Assumption 2.8(b).

Part II: Asymptotic results forU QE[_p. Letdb_p,n(θ₀,G)≡_1−Q¹

0En[(1−R)`(Z)Λ_x(W;β₀)gb_p(X;θ₀)],we perform a decomposition analogous to (2.8.9), U QE[_p(τ,G)−U QE_p(τ,G)≤ − 1

bf_Y_|R(qbτ|1)− 1 f_Y|R(q_τ|1)

db_p,n(bθ,G)

− 1

f_Y_|R(q_τ|1)(dbp,n(bθ,G)−dbp,n(θ₀,G))

− 1

f_Y_|R(q_τ|1)dbp,n(θ₀,G)−U QE(τ,G)

≡∆_p,1+∆p,2+∆p,3, (2.8.19)

with influence functions associated with∆_p,_jdenoted byψ_{p j}, j=1,2,3,respectively. The first term represents the estimation error of f_Y_s_|R=1, the second term corresponds to the estimation effect ofθ, and the last term accounts for the contribution ofbg_p.

The first term can be treated in a similar fashion as in the previous part, and therefore, we give the linear expansion directly without a proof,

ψp₁(a;θ₀,q_τ)≡ r

K_b_y(y−q_τ)−E[K_b_y(Y−q_τ)|R=1]

−(1(y≤q_τ)−τ)f_Y|R⁰ (q_τ|1) f_Y_|R(q_τ|1)

·dp(θ₀,G) ),

f_Y²_|R(q_τ|1). (2.8.20)

The linear expansion of∆p,2also takes a similar form as in (2.8.16), with

ψp₂(a;θ₀,q_τ)≡ − 1 fY|R(q_τ|1)

M_θ,p(θ₀)⁰ψθ(a;θ₀,q_τ)−r−Q₀ Q0

·d_p(θ₀,G)

, (2.8.21)

whereM_θ,p(θ₀)is defined in (2.7.6). Assumption 2.9(c) allows us to ignore the bias in the approximation ofG_p,θ₀(x) and in turn,M_θ,p(θ₀).

Next, we apply Lemma 5.1 in Newey (1994) to derive the linear expansion of∆p,3. Defineρ(·) = (ρ₁(·),ρ₂(·), ρ3),

φp(a;ρ)≡ −(1−r)`(z)Λ_x(w;β0)

1−Q₀ ·(G(x)−ρ1(x)/ρ₃)

ρ₂(x)/ρ₃ −dp(θ₀,G), Φp,1(a;ρ₁)≡(1−r)`(z)Λ_x(w;β0)

1−Q₀ · ρ1(x) f_X|R(x|1), Φp,2(a;ρ₂)≡(1−r)`(z)Λ_x(w;β0)

1−Q₀ ·(G(x)−F_X|R(x|1))ρ₂(x) f_X|R² (x|1) , Φp,3(a;ρ3)≡ −(1−r)`(z)Λ_x(w;β0)

1−Q₀ · G(x)ρ₃ f_X|R(x|1),

andΦ_p(a;ρ)≡∑³j=1Φ_p,_j(a;ρ_j).With these notations in hand, we can rewrite∆_p,3as follows,

∆p,3=− 1

f_Y|R(q_τ|1)(En[φ_p(A;ρb_n)]−E[φ_p(A;ρ₀)]),

whereρbn(·)≡ En

h(1−R)`(Z)1(X≤·) 1−Q₀

i ,En

h_(1−R)`(Z)I

bxK_bx(X−·) 1−Q₀

i ,^E_Qⁿ^[R]

, andρ0(·)≡(F_X|R(·|1),f_X|R(·|1),1)⁰. We proceed to verify the conditions of Lemma 5.1 in Newey (1994).

The first set of conditions controls the linearization error. By directly bounding the difference, we can show that

φ_p(a;ρ)−φ_p(a;ρ₀)−Φ_p(a;ρ−ρ₀) ≤

∑

j=1,2

K_jsup

x∈X|ρ_j(x)−ρj,0(x)|²+K₃|ρ₃−ρ3,0|², (2.8.22) where the inequality holds under Assumption 2.1(d)(i), Assumptions 2.7(c), (e), Assumption 2.8(d), and Assumption 2.9(a). By (2.8.22), Assumption 5.1(ii) in Newey (1994) is satisfied if√

nkρb−ρ₀k_∞→^p 0. In what follows, we provide the proof of this convergence result.

First, by Theorem B in Section 2.1.4 in Serfling (2009), we have √

nkρb1−ρ_1,0k²

∞=O_p(n^−1/2log(log(n))) = o_p(1).

Next, we let

ρe₂(x)≡E

(1−R)`(Z)I_b_xK_b_x(X−x) 1−Q₀

By the triangular inequality, kρb₂−ρ2,0k

∞≤ kρb₂−ρe₂k_∞+kρe₂−ρ2,0k

∞. Under the rate condition in Assumption 2.9(c), Lemma 8.10 in Newey and McFadden (1994) yields thatkρb2−ρe2k_∞=Op(log(n)^1/2n^−1/2b^−1/2x ).

Next, we bound the bias,eρ2−ρ2,0,

ρe2(x) = Z x¯

I_b_xK_b_x(v−x)f_X|R(v|1)dv

= Z

K_x(u)f_X|R(x+ub_x|1)du

=f_X|R(x|1) Z

K_x(u)du+b_xf_X|R⁰ (x|1) Z

uK_x(u)du +b²_xf_X|R⁰⁰ (ex|1)

2 Z

u²K_x(u)du

=f_X|R(x|1) +b²_xf_X|R⁰⁰ (x|1) 2

Z ∞

−∞u²K_x(u)du+o_p(b²_x).

whereI_x≡[(x−x)/b_x+ρ_x/2,(x−x)/b¯ _x−ρ_x/2]andxeis some value betweenxandx+c_xb_x, with|c_x| ≤max{|x|,|x|}.¯ The first line is due to Assumptions 2.1(c) and (e), the second line follows by changing variable, and the last line is due to Assumptions 2.9(a) and (b). Using this fact, we deduce that√

nkρb2−ρ2,0k²

∞=Op(log(n)n^−1/2b⁻¹_x +√ nb⁴_x), which iso_p(1)under Assumption 2.9(c).

Lastly,√

nkρb3−ρ3,0k²=O_p(n^−1/2). In view of the above three results, the desired condition is verified.

In the next step, we verify Assumption 5.2. Using Lemma 8.4 in Newey and McFadden (1994), the condition is satisfied so long asE[kΦ(a;ρb−ρ0)k²]→^p 0. The latter condition follows bykΦ(a;ρ)k ≤K₄kρk_∞and DCT. The constantK₄is finite under Assumption 2.1(d)(i), Assumptions 2.7(c), (e), Assumption 2.8(d), and Assumption 2.9(a).

Finally, we need to show the mean-square continuity (Assumption 5.3 in Newey (1994)). Towards this end, we derive the asymptotic linear representation of^RΦ_p(a,ρb−ρ)dF_A(a). We focus onρ₂first. Let

ψp,x,2(a;θ0,G)≡(1−r)`(z)

1−Q₀ π(x)−E

(1−R)`(Z) 1−Q₀ π(X)

, (2.8.23)

whereπ(x)is defined in (2.7.11). Now we proceed to verify thatψp,x,2is indeed the influence function of^RΦp(a;ρ2− ρ2,0)dF_A(a). It amounts to show that

Φp,2(a;ρb2−ρ2,0)dF_A(a)−1 n

n i=1

∑

ψp,x,2(A_i;θ0,G)

=o_p(n^−1/2).

For this purpose, we bound the first two moments and apply Chebyshev’s inequality. For the first moment, we have

√nE

Φp,2(a;ρb₂−ρ2,0)dF_A(a)−1 n

∑

i=1

ψp,x,2(A_i;θ₀,G)

=√ n

Φp,2(a;ρb2−ρ2,0)dF_A(a)

=√ n

Z Z

I_bx

π(x)f_X|R(v|1)K_b_x(v−x)dxdv− Z Z

π(x)f_X|R(x|1)K_x(u)dudx

≤√ n

Z Z

π(x+b_xu)f_X|R(x|1)K_x(u)dudx− Z Z

π(x)f_X|R(x|1)K_x(u)dudx +√

Z Z

I_x^cπ(x)f_X|R(x|1)K_x(u)dudx

≤√ n

f_X|R(x|1) Z

I_x^c

(π⁰(x)b_xu+π⁰⁰(x)be ²_xu²/2)K_x(u)dudx +

π(x)fX|R(x|1) ∞·√

n Z

I^c_x

Kx(u)du

=O(n^1/2b²_x) +o(1) =o(1), (2.8.24)

whereI_x^c is the complement ofI_x relative toX, andexis some value betweenxandx+c_xb_x. The second equality follows because

Φp,2(a;ρ2)dF_A(a)

= Z Z

`(z)Λ_x(w;β0)(G(x)−F_X|R(x|1))ρ₂(x)

f_X|R² (x|1) fX|ZR(x|z,0)fZ|R(z|0)dzdx

= Z Z

Λx(w;β0)(G(x)−F_X|R(x|1))ρ₂(x)

f_X|R² (x|1) fX|ZR(x|z,1)fZ|R(z|1)dzdx

= Z _Z

Λ_x(w;β₀)f_{Z|X R}(z|x,1)dz

(G(x)−F_X|R(x|1))ρ₂(x) f_X|R(x|1) dx

= Z

E[Λ_x(W;β₀)|X=x,R=1](G(x)−F_X|R(x|1))ρ₂(x) f_X|R(x|1) dx

= Z

π(x)ρ₂(x)dx.

Therefore,

E Z

Φp,2(a;ρb2)dF_A(a)

(1−R)`(Z) 1−Q

π(x)I_b_xK_b_x(X−x)dx

= Z Z

π(x)I_b_xK_b_x(v−x)f_X|R(v|1)dxdv, E

Φp,2(a;ρ2,0)dF_A(a)

= Z

π(x)f_X|R(x|1)dx· Z

K_x(u)du

= Z Z

π(x)K_x(u)f_X|R(x|1)dudx.

The second inequality of (2.8.24) follows from a second-order Taylor expansion with respect toπ(·), which is valid under Assumption 2.7(e) and Assumption 2.9(a). The second-to-last equality of (2.8.24) is due to Assumptions 2.9(a) and (b).

Next, since π(·)is continuous and bounded, and K_x has a compact support, we have that, by DCT, ^R_I

xπ(x−

b_xu)K_x(u)du→π(x). As a consequence, for the second moment,





√n Z

Φp,2(a;ρb2−ρ2,0)dF_A(a)− 1

√n

n i=1

∑

ψp,x,2(A_i;θ0,G)

2



≤E

(1−R)`(Z) 1−Q₀

π(x)I_b_xK_b_x(X−x)dx−π(X)

≤E

(1−R)`(Z) 1−Q₀

4#1/2

·E

Z Ix

π(X−bxu)K_x(u)du−π(X)

4#1/2

=O(1)o(1) =o(1),

where the second to last equality follows by Assumption 2.1(d), Assumption 2.7(e)(iii), and by DCT. This concludes the derivation of the linear expansion for the second term.

Linear expansions for terms involvingρ₁andρ₃follow by standard (functional) delta method. Hence, we provide

the influence functions directly. Let

ψp,x,1(a;θ₀,G)≡E

(1−R)`(Z)Λ_x(W;β0) (1−Q₀)f_X|R(X|1) ·

(1−r)`(z)1(x≤X)

1−Q₀ −F_X|R(X|1)

, (2.8.25)

ψp,x,3(a;θ₀,G)≡ −r−Q₀ Q0

·E

(1−R)`(Z)Λ_x(W;β0) 1−Q0

· G(X) fX|R(X|1)

. (2.8.26)

Combining (2.8.25), (2.8.23), and (2.8.26), we deduce that

∆p,3=1 n

∑

i=1

ψp3,i=−1 n

∑

i=1

1 f_Y_|R(q_τ|1)

ψg,p,i+(1−R_i)`(Z_i)Λ_x(W_i;β₀)g_p(X_i) 1−Q₀

−d_p(θ₀,G)) +o_p(n^−1/2). (2.8.27)

whereψg,pis defined in (2.7.10). Collecting the results in (2.8.20), (2.8.21), and (2.8.27), it follows that the asymptotic linear representation forU QE[_pis given as in (2.7.3). This completes our proof.

Dalam dokumen PDF ESSAYS ON THE ECONOMETRICS OF CAUSAL INFERENCE By Qi Xu for the degree of (Halaman 115-127)