Proofs for Results from Main Text

3.8 Supplementary Appendix

3.8.1 Proofs for Results from Main Text

τ_or=E[Y|D=1,T =1]−E[m1,0(X) +m0,1(X)−m0,0(X)|D=1,T=1], wheremd,t(x) =E[Y|D=d,T =t,X=x], and

τipw=E[(w_1,1(D,T)−w_1,0(D,T,X)−w_0,1(D,T,X) +w_0,0(D,T,X))Y],

where, for(d,t)∈S−,

w_1,1(D,T) = DT E[DT],

wd,t(D,T,X) = 1{D=d,T=t}p(1,1,X) p(d,t,X)

E

1{D=d,T=t}p(1,1,X) p(d,t,X)

,

andp(d,t,x) =P(D=d,T=t|X=x)is a so-called generalized propensity score.

Lemma 3.2 Under Assumptions 3.1 and 3.2, it follows thatτ_or=τ_ipw=τ.

Proof of Lemma 3.2:

Outcome regression estimand: Usingmd,t(·) =E[Y_t(d)|D=d,T =t,X=·],(d,t)∈S−, we get

τor=E[Y₁(1)|D=1,T =1]−E[E[Y₀(1)|D=1,T=0,X=x]|D=1,T=1]

+

∑

t∈{0,1}

(−1)^tE[E[Y_t(0)|D=0,T =t,X=x]|D=1,T =1]

=E[Y₁(1)|D=1,T =1]−E[E[Y₀(0)|D=1,T=0,X=x]|D=1,T=1]

+

∑

t∈{0,1}

(−1)^tE[E[Y_t(0)|D=0,T =t,X=x]|D=1,T =1]

=E[Y₁(1)−Y₁(0)|D=1,T =1] =τ,

where the second equality follows from Assumptions 3.2 (ii) and the third holds under Assumptions 3.2 (i).

Propensity score estimand: Let p(1,1) =P(D=1,T=1). Under the overlapping conditions in Assumption 3.2(iii),w_d,t(d⁰,t⁰,x)are well defined for(d,t)∈S−,(d⁰,t⁰)∈ {0,1}², andx∈X almost everywhere. Additionally,

E[w_d,t(D,T,X)Y] =E

p(1,1,X)Y I_d,t p(d,t,X)

E

1{D=d,T =t}p(1,1,X) p(d,t,X)

=E

E

E[Y|D=d,T=t,X]· Id,t

p(d,t,X) X

·p(1,1,X) p(1,1)

=E

E[Y|D=d,T =t,X]·p(1,1,X) p(1,1)

=E[E[Y|D=d,T =t,X]|D=1,T=1]

=E

m_d,t(X)|D=1,T=1 ,

for(d,t)∈S−. The second line follows by the LIE, the third equality is by the definition of propensity scores, and the next to last line is by Bayes’ Law. Next, fromE[w_1,1(D,T)Y] =E[Y|D=1,T =1]and the same arguments for the

OR estimand, we conclude thatτ_ipw=τ.

Proof of Theorem 3.1:

We follow the steps in Hahn (1998) for the derivation of the efficient influence function. Let f(y|d,t,x) = f(y|D=

d,T =t,X=x).

Step 1: characterize the tangent space of the statistical model. The observed likelihood is given as

f(y,d,t,x) =f(y|1,1,x)^dtf(y|1,0,x)^d(1−t)f(y|0,1,x)^(1−d)tf(y|0,0,x)^{(1−d)(1−t)}

·p(1,1,x)^dtp(1,0,x)^d(1−t)p(0,1,x)^(1−d)tp(0,0,x)^{(1−d)(1−t)}·f(x).

Consider the regular sub-model parameterized byθ≥0, with the true model indexed byθ₀=0,

f_θ(y,d,t,x) =f_θ(y|1,1,x)^dtf_θ(y|1,0,x)^d(1−t)f_θ(y|0,1,x)^(1−d)tf_θ(y|0,0,x)^{(1−d)(1−t)}

·p_θ(1,1,x)^dtp_θ(1,0,x)^d(1−t)p_θ(0,1,x)^(1−d)tp_θ(0,0,x)^{(1−d)(1−t)}

·f_θ(x).

The score function of this sub-model is given by

s_θ(y,d,t,x) =dts_θ(y|1,1,x) +d(1−t)s_θ(y|1,0,x) + (1−d)ts_θ(y|0,1,x) + (1−d)(1−t)s_θ(y|0,0,x) +dtp˙_θ(1,1,x)

pθ(1,1,x)+d(1−t)p˙_θ(1,0,x)

pθ(1,0,x)+ (1−d)tp˙_θ(0,1,x)

pθ(0,1,x)+ (1−d)(1−t)p˙_θ(0,0,x) pθ(0,0,x) +t_θ(x),

wheres_θ(y|d,t,x) =∂log f_θ(y|d,t,x)/∂ θ, ˙p_θ(d,t,x) =∂p_θ(d,t,x)/∂ θ, andt_θ(x) =∂log f_θ(x)/∂ θ for(d,t)∈S. For notational simplicity, we suppress subscripts whenθ=θ₀.

Now, the tangent space of this model is characterized by

T ={dts₁₁(y,x) +d(1−t)s₁₀(y,x) + (1−d)ts₀₁(y,x) + (1−d)(1−t)s₀₀(y,x) +dt p₁₁(x) +d(1−t)p₁₀(x) + (1−d)t p₀₁(x) + (1−d)(1−t)p₀₀(x) +s(x)},

for any functions{s_dt(·,·),p_dt(·)}_(d,t)∈S, ands(·)such that, for(d,t)∈S

s_dt(·,·)∈L2(Y ⊗X), with Z

s_dt(y,x)f(y|d,t,x)dy=0,∀x∈X, (3.8.1) p_dt(·)∈L2(X),with

∑

(d,t)∈S Z

p_dt(x)f(x)dx=0, (3.8.2)

and

s(·)∈L₂(X), with Z

s(x)f(x)dx=0. (3.8.3)

In Step 2, we show that the target parameter associated with the parametric sub-model ispath-wise differentiable, as defined in Newey (1990).

From Lemma 3.2, we know the ATT can be identified by∑(d,t)∈S(−1)^d+tE[E[Y|D=d,T=t,X]|D=1,T=1]

under Assumptions 3.1 and 3.2. For the parameterized sub-model, we define

τ(θ) =(^{R R}pθ(1,1,x)y f_θ(y|1,1,x)fθ(x)dydx−^{R R} pθ(1,1,x)y f_θ(y|1,0,x)fθ(x)dydx) Rp_θ(1,1,x)f_θ(x)dx

−(^{R R}p_θ(1,1,x)y f_θ(y|0,1,x)f_θ(x)dydx−^{R R}p_θ(1,1,x)y f_θ(y|0,0,x)f_θ(x)dydx)

R p_θ(1,1,x)f_θ(x)dx . (3.8.4)

Note that the derivative ofτ(θ)with respect toθ, evaluated atθ=0, is given by dτ(θ)

dθ θ=0

=

∑

(d,t)∈S

(−1)^d+t

R Ryp(1,1,x)s(y|d,t,x)f(y|d,t,x)f(x)dydx p(1,1)

+

R(τ(x)−τ)p(1,˙ 1,x)f(x)dx p(1,1)

+

R(τ(x)−τ)p(1,1,x)t(x)f(x)dx

p(1,1) .

For anyw= (y,d,t,x)∈W, define

F_τ(w) =dt(y−m_1,1(x))

p(1,1) +p(1,1,x) p(1,1)

−d(1−t)(y−m_1,0(x)) p(1,0,x)

−(1−d)t(y−m0,1(x))

p(0,1,x) +(1−d)(1−t)(y−m0,0(x)) p(0,0,x)

+ dt p(1,1)

∑

(d,t)∈S

(−1)^d+t

m_d,t(x)− Z

X m_d,t(x)f(x)dx

.

It can be readily verified that ^dτ(θ_dθ⁾

θ=0=E[F_τ(W)s₀(Y,D,T,X)], thereby showingτ(θ)is path-wise differentiable.

In Step 3, we show thatF_τ(W)is the efficient influence function forτ, which we will accomplish by invoking Theorem 3.1 in Newey (1990). To apply this theorem, we need to verify thatF_τ(·)∈T. By setting

s11(y,x) =y−m_1,1(x) p(1,1) , p₁₁(x) =p(1,1)⁻¹

∑

(d,t)∈S

(−1)^d+t

m_d,t(x)− Z

X m_d,t(x)f(x)dx

,

s_dt(y,x) = (−1)^d+tp(1,1,x)(y−m_d,t(x)) p(d,t,x)p(1,1) , p_dt(x),s(x) =0,

for(d,t)∈S−, it is straightforward to show that (3.8.1)–(3.8.3) hold, which leads to the desired result.

Finally, since p(1,1) =E

I_d,tp(1,1,X)p(d,t,X)⁻¹

, for (d,t)∈S, direct manipulation yields that F_τ(W) = ηeff(W). Now, we take the expectation of η_eff² (W)and the semi-parametric efficiency bound follows by standard

manipulation. This completes the proof.

Proof of Proposition 3.1:The proof follows directly from the LIE as displayed in the main text.

Proof of Proposition 3.2:

It follows by Theorem 3.1 that

E[η_eff(W)²] = 1 E[DT]²E

"

DT(τ(Y,X)−τ)²+

∑

(d,t)∈S−

I_d,tp(1,1,X)²

p(d,t,X)² (Y−m_d,t(X))²

#

= 1

E[DT]²E[DT(τ(X)−τ)²] +E

"

w1,1(D,T)²(Y−m1,1(X))²+

∑

(d,t)∈S−

w_d,t(D,T,X,p)²(Y−m_d,t(X))²

#

≡V_1,dr+V_2,dr,

where the second equality follows from direct manipulations and the fact that

E[DT·(Y−m_1,1(X))·(m_d,t(X)−E[m_d,t(X)|D=1,T=1])]

=E[E[p(1,1,X)·(m_1,1(X)−m_1,1(X))·(m_d,t(X)−E[m_d,t(X)|D=1,T=1])|X]] =0,

for(d,t)∈S.

Meanwhile, from Part (b) of Proposition 1 in Sant’Anna and Zhao (2020), we have the following decomposition,

E[η_sz(W)²] =V1,sz+V_2,sz,

whereV1,sz≡E

D(τ(X)−τ)²

/p²,and

V_2,sz≡1 p²E

DT

2(Y−m_1,1(X))²+D(1−T)

(1−λ)²(Y−m_1,0(X))²

+(1−D)T p(X)²

(1−p(X))²λ²(Y−m_0,1(X))²+(1−D)(1−T)p(X)²

(1−p(X))²(1−λ)²(Y−m_0,0(X))²

. (3.8.5)

Under Assumption 3.3, we have thatE[1{T =t}g(X)] =P(T =t)E[g(X)],E[I_d,tY g(X)] =P(T =t)E[1{D= d}Y_tg(X)], andp(d,t,x) = (1{t=1}λ+1{t=0}(1−λ))p(d,x). It then follows that

V_1,dr= 1

λp²E[D(τ(X)−τ)²], (3.8.6)

and therefore,

V_1,dr−V_1,sz=1−λ

p²λ E[D(τ(X)−τ)²]. (3.8.7)

We now focus onV_2,dr. Observe that

V_2,dr= 1 λ²p²

E[DT(Y₁−m1,1(X))²] +E

D(1−T)λ²p(X)²

(1−λ)²p(X)² (Y₀−m1,0(X))²

+E

(1−D)Tλ²p(X)²

λ²(1−p(X))² (Y₁−m0,1(X))²

+E

(1−D)(1−T)λ²p(X)²

(1−λ)²(1−p(X))² (Y₀−m0,0(X))²

=1 p²E

DT

λ²(Y−m1,1(X))²+D(1−T)

(1−λ)²(Y−m1,0(X))² +(1−D)T p(X)²

(1−p(X))²λ²

(Y−m0,1(X))²+(1−D)(1−T)p(X)²

(1−p(X))²(1−λ)²(Y−m0,0(X))²

=V2,sz, (3.8.8)

where the first equality follows becausep(d,t,x) =P(D=d,X=x)·P(T =t)under Assumption 3.3. The desired

result then follows from (3.8.7) and (3.8.8).

Proof of Lemma 3.1:

Letψ_d,t(W;w,m) =1{dt=1}w1,1(D,T)Y+1{dt6=1}

w_d,t(D,T,X)(Y−m_d,t(X)) +w1,1(D,T)m_d,t(X) , and ˜τ_dr=

∑_(d,t)∈S(−1)^d+tψd,t(W;w,m). Using ˜τ_dr, we decomposebτ_dr as

τbdr−τ= (bτdr−τ˜dr) + (τ˜dr−τ). (3.8.9) Note first that the second term, ˜τdr−τ, hasi.i.d.centered summands with bounded variance; thus, it isO_p(n^−1/2).

Now we investigate the behavior ofτbdr−τ˜dr, for which we make the following decomposition

ψ_d,t(W;w,b m)b −ψ_d,t(W;w,m) =(Y−m_d,t(X)) wb_d,t−w_d,t

(W) +m_d,t(X) (wb1,1−w1,1) (W) + w_1,1−w_d,t

(W) mb_d,t−m_d,t (X) +

(wb1,1−w_1,1) (W)− wbd,t−w_d,t

(W) mbd,t−md,t (X)

≡∆^ψ,1_d,t (W) +∆^ψ,2_d,t (W) +∆^ψ,3_d,t (W),

for(d,t)∈S. Here, we use the unifying notationw_d,t(W)to denotew_d,t(D,T,X)when(d,t)∈S−andw1,1(D,T) otherwise. We proceed by establishing convergence rates for each component in the above decomposition.

We first analyze∆^ψ,1_d,t . A second-order Taylor expansion ofψ1,1(W;w,bm)b aroundE[DT]yields that

En

h

∆^ψ,1_1,1(W)i

=En

Y

DT

En[DT]− DT E[DT]

=−En[DTY]

E[DT]² ·(En[DT]−E[DT]) +O_p(|En[DT]−E[DT]|²)

=−E[DTY]

E[DT]² ·(En[DT]−E[DT]) +o_p(n^−1/2). (3.8.10) When(d,t)∈S−, similar analysis reveals that

En

h

∆^ψ,1_d,t (W)i

=En

(Y−m_d,t(X)) wb_d,t−w_d,t

(W) +m_d,t(X) (wb1,1−w1,1) (W)

=En

(Y−m_d,t(X)) wb_d,t−w_d,t (W)

−En

DT m_d,t(X)

E[DT]² (En[DT]−E[DT]) +O_p(|En[DT]−E[DT]|²)

=−E

DT md,t(X)

E[DT]² (En[DT]−E[DT]) +o_p(n^−1/2), (3.8.11) where the last equation holds under Assumption 3.4.2(i).

Next, note that∆^ψ,2_1,1(·) =0, and when(d,t)∈S−, we deduce from Assumption 3.4.2(ii) that

En

h

∆^ψ,2_d,t (W)i

=En

(w_1,1−w_d,t)(W) mb_d,t−m_d,t (X)

=o_p(n^−1/2). (3.8.12)

Analogously,∆^ψ,3_1,1(·)is identically zero, and therefore, we only need to focus the other three cases, for which we have

En

h

∆^ψ,3_d,t (W)i

=En

(wb_1,1−w_1,1) (W)− wb_d,t−w_d,t (W)

mb_d,t−m_d,t (X)

=En

DT

E[DT]² mb_d,t−m_d,t (X)

·(En[DT]−E[DT]) +O_p(|En[DT]−E[DT]|²) (3.8.13)

−En

wb_d,t−w_d,t

(W)· mb_d,t−m_d,t (X)

, (3.8.14)

where the second equality follows from a second-order Taylor expansion ofEn[DT]aroundE[DT].

Taking the fact thatE[DT]>0 under Assumption 3.2 (iii) and thatmb_d,t is uniformly convergent tom_d,t, we obtain

En

DT

E[DT]² mbd,t−md,t (X)

≤En

DT E[DT]²

·

mbd,t−md,t (X)

.

mbd,t−m_d,t

∞=op(1).

Combining this result withEn[DT]−E[DT] =O_p n^−1/2

, we conclude that (3.8.13) iso_p n^−1/2 . Next, we studyEn

wb_d,t−w_d,t

(W)· mb_d,t−m_d,t (X)

. Let

w^†_d,t(W) = I_d,tbp(1,1,X)

p(1,1)p(d,t,b X), (3.8.15)

based on which, we have the following decomposition

En

h

w^†_d,t−wd,t

(W)· mbd,t−md,t (X)i

+En

h

wbd,t−w^†_d,t

(W)· mbd,t−md,t (X)i

=∆^1,n_w,m+∆^2,n_w,m. (3.8.16)

We consider theL₂-norm first. Under Assumption 3.4.2(iii),

∆^1,n_w,m=E h

w^†_d,t−w_d,t

(W)· mb_d,t−m_d,t (X)i

| {z }

≡∆¹_w,m

+o_p n^−1/2

.

Since ˆa/bˆ−a/b= (aˆ−a)/b−a(bˆ−b)/b²−(aˆ−a)(bˆ−b)/(bb) +ˆ a(bˆ−b)²/(bbˆ ²), we have

∆¹_w,m=E

δd,t(W)

p(d,t,X)(bp(1,1,X)−p(1,1,X))

−E

δd,t(W)p(1,1,X)

p²(d,t,X) (bp(d,t,X)−p(d,t,X))

−E

δd,t(W)

p(db ,t,X)p(d,t,X)(p(1,b 1,X)−p(1,1,X)) (bp(d,t,X)−p(d,t,X))

+E

δ_d,t(W)p(1,1,X)

p(db ,t,X)p(d,t,X)²(p(d,t,b X)−p(d,t,X))²

≡∆^1,1_w,m+∆^1,2_w,m+∆^1,3_w,m+∆^1,4_w,m,

whereδd,t(W) =p(1,1)⁻¹I_d,t mb_d,t−m_d,t (X).

For∆^1,1w,m,

∆^1,1_w,m

≤p(1,1)⁻¹ p^min_d,t ⁻¹ E

(bp(1,1,X)−p(1,1,X)) (mb_d,t−m_d,t)(X)

≤O(1)· kp(1,b 1,·)−p(1,1,·)k_L

2·

mb_d,t−m_d,t L2

=O_p(r_nsn),

wherep^min_d,t =inf_x∈X |p(d,t,x)|. The first inequality holds under Assumption 3.2(iii), and the second one is due to the Cauchy-Schwarz inequality.

Likewise,

∆^1,2_w,m

≤p(1,1)⁻¹sup

x∈X|p(1,1,x)|

x∈infX|p(d,t,x)|

−2

E

(p(db ,t,X)−p(d,t,X)) (mb_d,t−m_d,t)(X)

≤O(1)· kbp(d,t,·)−p(d,t,·)k_L

2·

mb_d,t−m_d,t L2

=O_p(r_nsn).

To analyze the convergence of the remaining two terms, we can use a similar approach to the one used for the previous two terms. However, to complete the analysis, we need to show thatp(d,tb ,x)is uniformly bounded away from 0 acrossX, with high probability. Due to the uniform convergence, for any givenε∈(0,1/2), there isN_ε>0 such that sup_x∈X |p(db ,t,x)−p(d,t,x)| ≤p^min_d,t/2 with probability at least 1−ε, whenevern≥N_ε. Thus, whennis sufficiently large, we have

x∈Xinf |bp(d,t,x)| ≥ inf

x∈X|p(d,t,x)| −sup

x∈X

|bp(d,t,x)−p(d,t,x)| ≥p^min_d,t/2>0,

with probability 1−ε, leading to our desired claim.

The sup-norm case can be handled analogously. Different from theL₂-norm, it is now possible to work directly with the empirical measure, leading to the conclusion that∆^1,nw,m=O_p(r_ns_n), without the necessity of imposing As- sumption 3.4.2 (iii).

Next,we examine the estimation effect of the normalizing weight as given in∆^2,nw,m. Letp(1,b 1) =En

h

I_d,t^p(1,1,X)b

bp(d,t,X)

i . Again, we focus onL₂-norm first. By definition,

∆^2,n_w,m=−p(1,b 1)⁻¹·En

h

w^†_d,t(W)· mb_d,t−m_d,t (X)i

| {z }

∆^2,1,n_w,m

·(p(1,b 1)−p(1,1)).

We can further decompose∆^2,1,nw,m into

∆^2,1,n_w,m =∆^1,n_w,m (3.8.17)

+ (En−E)

w_d,t(W)· mb_d,t−m_d,t (X)

(3.8.18) +E

wd,t(W)· mbd,t−md,t (X)

. (3.8.19)

=O_p(r_n) +O_p(r_ns_n) +o_p n^−1/2

Under Assumptions 3.4.2 (iii, iv), (3.8.17) and (3.8.18) areO_p(r_ns_n)ando_p n^−1/2

, respectively. Since pd,t(·)is uniformly bounded overX, (3.8.19) isOp(r_n)by the Cauchy-Schwartz inequality.

Analogously, we have

bp(1,1)−p(1,1) =(En−E)

I_d,t

bp(1,1,X)

bp(d,t,X)−p(1,1,X) p(d,t,X)

(3.8.20) + (En−E)

I_d,tp(1,1,X) p(d,t,X)

(3.8.21) +E

I_d,t

p(1,b 1,X)

p(db ,t,X)−p(1,1,X) p(d,t,X)

(3.8.22)

=O_p(s_n) +O_p n^−1/2

+o_p n^−1/2

.

Under Assumption 3.4.2 (v), (3.8.20) iso_p n^−1/2

. Since (3.8.21) is a centeredi.i.d.summand, it isO_p n^−1/2 . Arguing along the same line as for∆¹_w,m, we get (3.8.22) isO_p(s_n). Collecting these results, we conclude that both

∆^1,nw,mand∆^2,nw,mareOp(r_nsn).

Once again, analysis under the sup-norm rely directly on empirical measure, thus eliminating the need for condi- tions on the empirical process. Further details are not provided here for brevity.

To finish the proof of this lemma, we gather the results in (3.8.9), (3.8.10), (3.8.11), (3.8.12), (3.8.14), and (3.8.16), which leads to

τb_dr−τ=En

"

∑

(d,t)∈S

(−1)^d+tψ_d,t(W;w,m)−τ

# +τ

1−En[DT] E[DT]

+O_p(r_ns_n) +o_p n^−1/2

=En[η_eff(W)] +O_p(r_ns_n) +o_p n^−1/2

.

Proof of Theorem 3.2:

We proceed by applying Lemma 3.1. As we are working with the sup-norm, we need to verify the first two conditions in Assumption 3.4.2. Lemmas 3.7 and 3.8 provide the required verification for these conditions. With the bandwidth rate conditions in Assumption 3.5.5 guaranteeing that the leading remainder term isO_p(r_ns_n) =o_p n^−1/2

, we can

then derive the asymptotic normality directly from the CLT.

Proof of Theorem 3.3:

Proof of Part (a): We have already shown in Theorem 3.2 thatτbdr−τ=En[η_eff(W)] +o_p n^−1/2

. Following a similar line of reasoning, one can easily demonstrate thatτbsz−τ=En[η_sz(W)] +o_p n^−1/2

, under Assumptions 3.1, 3.2, 3.5, Condition (i), and the null hypothesis,H₀. Now, by the CLT, we have

√n(bτ_dr−bτ_sz)→^d N 0,E

h

(η_eff(W)−η_sz(W))²i .

It remains to show that

Vbn

→p V, (3.8.23)

and

V =ρsz>0. (3.8.24)

First, it is implied from the proof of Theorem 3.2 thatηbe f f(w)→^p ηeff(w), uniformly inw∈W. In a similar vein, ηbsz(w)→^p ηsz(w)uniformly overW, underH₀. Combining these two results, (3.8.23) then follows by the CMT and the weak LLN.

From Proposition 1 in Sant’Anna and Zhao (2020), we know thatη_sz(·)is the efficient influence function for all reg- ular estimators ofτ_sz, which is equal toτunderH₀. Moreover, since bothbτ_drandτb_szare consistent forτ_szunderH₀, it follows from Lemma 2.1 in Hausman (1978) thatE[η_eff(W)η_sz(W)] =E[η_sz(W)²]. Hence,E

h

(η_eff(W)−η_sz(W))²i

=E

ηeff(W)²

−E

ηsz(W)²

. Given this result, (3.8.24) now follows by Proposition 3.2 and the condition that Var[τ(X)|D=1]>0.

Proof of Part (b): We proceed by establishing: (i)bτsz−τbdr

→p τsz−τdr6=0; (ii)Vb_n→^p V<∞, underH₁.

Under Assumption 3.5, and Condition (i) of the theorem,bp(d,t,x)→^p p(d,t,x)andmb_d,t(x)→^p m_d,t(x), uniformly in x, for(d,t)∈S. Now, applying the LLN, we getτbdr

→p τdrandτbsz

→p τsz. Result (i) then follows from the CMT. Next, we deduce from the uniform consistency ofbpandm, the CMT, and LLN, that (3.8.23) holds underb H₁. Furthermore, Assumptions 3.2(iii) and 3.5.3 ensure that bothηb_szand ηb_dr are uniformly bounded, which leads toV <∞. This

concludes the proof of part (b).