$\det\left[J\left(x^{\delta};\delta\right)\right]$ when converting the pdf of
$\left(X,W\right)$ into that of $\left(X_{\delta},W\right).$

Define the local change function as

$\kappa\left(x\right)=\frac{\partial\mathcal{G}\left(x;\delta\right)}{\partial\delta}\bigg{|}_{\delta=0}.$

######  Theorem S.1.

Let Assumption S.1 hold. Then

$\Pi_{\tau,C}=E\left[\frac{\partial
E\left[\psi\left(Y,\tau,F_{Y}\right)|X,W\right]}{\partial
X^{\prime}}\kappa\left(X\right)\right]-E\left[\psi\left(Y,\tau,F_{Y}\right)\frac{\partial\ln
f_{U|X,W}(U|X,W)}{\partial X^{\prime}}\kappa\left(X\right)\right],$ (S.3)

where, as before,

$\psi\left(y,\tau,F_{Y}\right)=\frac{\tau-1\left(y<Q_{\tau}[Y]\right)}{f_{Y}(Q_{\tau}[Y])}.$

The theorem takes the same form as Theorem 1. Under the assumption that
$X_{j\delta}$ is a function of $X_{j}$ only for $j=1$ and $2$,
$\kappa_{j}\left(x\right)$ depends on $x_{j}$ only, and the effect from
changing $X_{1}$ into $X_{1\delta}$ and that from changing $X_{2}$ into
$X_{2\delta}$ are additively separable.

######  Corollary S.1.

Let Assumptions 2 and S.1 hold. Then

$\Pi_{\tau,C}=E\left[\frac{\partial
E\left[\psi\left(Y,\tau,F_{Y}\right)|X,W\right]}{\partial
X^{\prime}}\kappa\left(X\right)\right].$

For the case of a simultaneous location shift
$X_{1\delta}=X_{1}+\ell_{1}\left(\delta\right)$ and
$X_{2\delta}=X_{2}+\ell_{2}\left(\delta\right)$, we have

$\kappa\left(x\right)=\left(\dot{\ell}_{1}\left(0\right),\dot{\ell}_{2}\left(0\right)\right)^{\prime}\text{,}$

and so

$\displaystyle\Pi_{\tau,C}$
$\displaystyle=\frac{\dot{\ell}_{1}\left(0\right)}{f_{Y}(Q_{\tau}[Y])}\int_{\mathcal{W}}\int_{\mathcal{X}}\frac{\partial\mathcal{S}_{Y|X,W}\left(Q_{\tau}[Y]|x,w\right)}{\partial
x_{1}}f_{X,W}(x,w)dxdw$
$\displaystyle+\frac{\dot{\ell}_{2}\left(0\right)}{f_{Y}(Q_{\tau}[Y])}\int_{\mathcal{W}}\int_{\mathcal{X}}\frac{\partial\mathcal{S}_{Y|X,W}\left(Q_{\tau}[Y]|x,w\right)}{\partial
x_{2}}f_{X,W}(x,w)dxdw.$ (S.4)

Corollary S.1 shows that the compensated effect from the simultaneous location
shift is a linear combination of two location effects: one where the target
variable is $X_{1}$ and the other where the target variable is $X_{2}$. Thus,
we can write: $\Pi_{\tau,C}=\Pi_{\tau,L,1}+\Pi_{\tau,L,2}$. This additive
result follows because we have two unrelated location shifts whose effects
are, in essence, captured by the sum of two partial derivatives. This is
convenient since it immediately allows us to obtain the bias if we omit the
possible simultaneous change in a covariate different from the target
variable.

Corollary 1 in Firpo, Fortin, and Lemieux (2009) considers the case of a
simultaneous location shift in $k$ covariates, and delivers a $k\times 1$
vector of marginal effects. Theorem S.1 and Corollary S.1 complement such a
result by showing how to interpret a linear combination of the entries of the
vector of marginal effects. Furthermore, Theorem S.1 and Corollary S.1 allow
for the intervention of a target covariate to depend on another target
covariate. Here we consider only two target covariates for ease of exposition.
Our results can be easily extended to the case with more than two target
covariates.

Our framework can accommodate more complicated policy interventions, such as
simultaneous location-scale shifts in two target variables. In a potential
application, a compensated change may substitute the mean of one target
variable with the variance of another target variable. Given the generality of
$\mathcal{G}\left(x;\delta\right)$, Corollary S.1 is general enough to
accommodate various compensating policies.

###  S.2 Estimation of Simultaneous Effects

In this section, we focus on the estimation of $\Pi_{\tau,C}$ given in (S.1).
We use the same estimators of the quantile, the density of $Y,$ and the
parameters in the probit/logit model. We only need to make some minor
notational changes. As before
$\theta_{\tau}=\left(\alpha_{\tau}^{\prime},\beta_{\tau}^{\prime}\right)^{\prime},$
$\hat{\theta}_{\tau}=(\hat{\alpha}_{\tau}^{\prime},\hat{\beta}_{\tau}^{\prime})^{\prime}$
and
$Z_{i}=(\phi_{\mathrm{x}}(X_{i})^{\prime},\phi_{\mathrm{w}}(W_{i})^{\prime})^{\prime}$
but now
$\alpha_{\tau}=\left(\alpha_{\tau,1}^{\prime},\alpha_{\tau,2}^{\prime}\right)^{\prime},\hat{\alpha}_{\tau}=\left(\hat{\alpha}_{\tau,1}^{\prime},\hat{\alpha}_{\tau,2}^{\prime}\right)^{\prime}$
and $X_{i}=(X_{1i}^{\prime},X_{2i}^{\prime})^{\prime}.$ As in the case with
the location-scale effect, we estimate $\Pi_{\tau,C}$ by

$\hat{\Pi}_{\tau,C}=\hat{\Pi}_{\tau,L,1}+\hat{\Pi}_{\tau,L,2}$

where

$\displaystyle\hat{\Pi}_{\tau,L,1}$
$\displaystyle=-\frac{\dot{\ell}_{1}\left(0\right)}{\hat{f}_{Y}\left(\hat{q}_{\tau}\right)}\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\hat{\theta}_{\tau})\frac{\partial\phi_{\mathrm{x}}(X_{i})^{\prime}}{\partial
X_{1i}}\hat{\alpha}_{\tau,1},$ (S.5) $\displaystyle\hat{\Pi}_{\tau,L,2}$
$\displaystyle=-\frac{\dot{\ell}_{2}\left(0\right)}{\hat{f}_{Y}\left(\hat{q}_{\tau}\right)}\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\hat{\theta}_{\tau})\frac{\partial\phi_{\mathrm{x}}(X_{i})^{\prime}}{\partial
X_{2i}}\hat{\alpha}_{\tau,2}.$ (S.6)

For the next theorem, we define the diagonal matrix:

$\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)=\left(\begin{array}[c]{cc}\frac{\partial\phi_{\mathrm{x}}(X_{i})}{\partial
X_{1i}}&O\\\ O&\frac{\partial\phi_{\mathrm{x}}(X_{i})}{\partial
X_{2i}}\end{array}\right).$

We need the following modification of Assumption 4.

######  Assumption S.2.

Logit/Probit II. Assumption 4 holds with (iv) replaced by the following:

$\displaystyle M_{1L}\left(\theta\right)$
$\displaystyle=E\left\\{\left[\dot{g}(Z_{i}^{\prime}\theta)\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha\right]Z_{i}^{\prime}\right\\}$
$\displaystyle M_{2L}\left(\theta\right)$
$\displaystyle=E\left[g(Z_{i}^{\prime}\theta)\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\right]$

are well defined for any $\theta\in\mathcal{N}_{\theta_{\tau}}$ and

$\displaystyle\sup_{\theta\in\mathcal{N}_{\theta_{\tau}}}\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}\left[\dot{g}(Z_{i}^{\prime}\theta)\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha\right]Z_{i}^{\prime}-M_{1L}\left(\theta\right)\bigg{\|}\overset{p}{\rightarrow}0,$
$\displaystyle\sup_{\theta\in\mathcal{N}_{\theta_{\tau}}}\bigg{\|}\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\theta)\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}-M_{2L}\left(\theta\right)\bigg{\|}\overset{p}{\rightarrow}0.$

######  Theorem S.2.

Under Assumptions 3, 5, and S.2, the estimators given in (S.5) and (S.6)
satisfy

$\begin{pmatrix}\hat{\Pi}_{\tau,L,1}\\\
\hat{\Pi}_{\tau,L,2}\end{pmatrix}-\begin{pmatrix}\Pi_{\tau,L,1}\\\
\Pi_{\tau,L,2}\end{pmatrix}=\frac{1}{n}\sum_{i=1}^{n}\Phi_{i,\tau}^{L}+O_{P}\left(h^{2}\right)+o_{p}(n^{-1/2})+o_{p}(n^{-1/2}h^{-1/2}),$

where

$\displaystyle\Phi_{i,\tau}^{L}$
$\displaystyle=\frac{1}{f_{Y}\left(Q_{\tau}[Y]\right)}D_{L}\left\\{g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}-E\left[g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}\right]\right\\}$
$\displaystyle-\frac{1}{f_{Y}\left(Q_{\tau}[Y]\right)}D_{L}M_{L}H^{-1}s_{i}(\theta_{\tau};Q_{\tau}[Y])$
$\displaystyle-\left[\begin{pmatrix}\Pi_{\tau,L,1}\\\
\Pi_{\tau,L,2}\end{pmatrix}\frac{\dot{f}_{Y}(Q_{\tau}[Y])}{f_{Y}\left(Q_{\tau}[Y]\right)}+\frac{1}{f_{Y}\left(Q_{\tau}[Y]\right)}D_{L}M_{L}H^{-1}H_{Q}\right]\psi(Y_{i},\tau,F_{Y})$
$\displaystyle-\begin{pmatrix}\Pi_{\tau,L,1}\\\
\Pi_{\tau,L,2}\end{pmatrix}\frac{1}{f_{Y}\left(Q_{\tau}[Y]\right)}\left\\{\mathcal{K}_{h}\left(Y_{i}-Q_{\tau}[Y]\right)-E\mathcal{K}_{h}\left(Y_{i}-Q_{\tau}[Y]\right)\right\\},$
$D_{L}=\begin{pmatrix}-\dot{\ell}_{1}(0)&0\\\
0&-\dot{\ell}_{2}(0)\end{pmatrix}$

and

$M_{L}=M_{1L}\left(\theta_{\tau}\right)+\begin{pmatrix}M_{2L}\left(\theta_{\tau}\right),&O_{2\times
d_{\phi_{\mathrm{w}}}}\end{pmatrix}.$

For the asymptotic normality, the discussions after Theorem 2 are still
applicable.

In the special case that $\ell_{1}\left(\delta\right)=\delta$ and
$\ell_{2}\left(\delta\right)=-p\delta$, it suffices to change $D_{L}$ to
$\mathrm{diag}(1,-p).$ It is possible that $p$, the relative price $X_{1}$ in
terms of $X_{2}$, has to be estimated by $\hat{p}$ based on an independent
sample. In that case, the estimator of the compensated effect would be

$\hat{\Pi}_{\tau,L}=\hat{\Pi}_{\tau,L,1}-{\hat{p}}\hat{\Pi}_{\tau,L,2}.$

If the sample size $\tilde{n}$ of the independent sample for estimating $p$ is
much larger than $n$ (i.e., $\tilde{n}/n\rightarrow\infty),$ then the
expansion in Theorem S.2 still holds.

###  S.3 Proof of Theorem S.1

The proof of this Theorem is very similar to the proof of Theorem 1. The
following decomposition still holds

$\frac{F_{Y_{\delta}}(y)-F_{Y}(y)}{\delta}:=G_{1,\delta}\left(y\right)+G_{2,\delta}\left(y\right),$

where

$\displaystyle G_{1,\delta}\left(y\right)$
$\displaystyle=\int_{\mathcal{W}}\int_{\mathcal{X}}F_{Y|X,W}\left(y|x,w\right)\frac{\left[f_{X_{\delta},W}(x,w)-f_{X,W}(x,w)\right]}{\delta}dxdw,$
$\displaystyle G_{2,\delta}\left(y\right)$
$\displaystyle=\int_{\mathcal{W}}\int_{\mathcal{X}}\int_{\mathcal{U}}\mathds{1}\left\\{h(x,w,u)\leq
y\right\\}\frac{\left[f_{U|X,W}(u|x^{\delta},w)-f_{U|X,W}(u|x,w)\right]}{\delta}f_{X_{\delta},W}(x,w)dudxdw.$

We first consider the term $G_{1,\delta}\left(y\right).$ Under the assumptions
given, we have

$f_{X_{\delta},W}(x,w)=\det\left[J(x^{\delta};\delta)\right]f_{X,W}(x^{\delta},w).$

Evaluated at $\delta=0,$ $f_{X_{\delta},W}(x,w)$ is $f_{X,W}(x,w)$. Given
this, we expand $f_{X_{\delta},W}(x,w)-f_{X,W}(x,w)$ around $\delta=0$, which
is possible under Assumptions S.1(i) and (iii). We have

$\displaystyle f_{X_{\delta},W}(x,w)-f_{X,W}(x,w)$
$\displaystyle=\delta\frac{\partial\left[\det\left[J\left(x^{\delta};\delta\right)\right]f_{X,W}(x^{\delta},w)\right]}{\partial\delta}\bigg{|}_{\delta=0}+\delta
R_{1}(x,w,\delta)$
$\displaystyle=\delta\frac{\partial\det\left[J\left(x^{\delta};\delta\right)\right]}{\partial\delta}\bigg{|}_{\delta=0}f_{X,W}(x,w)+\delta\det\left[J(x^{\delta};\delta)\right]\left(\frac{\partial
x^{\delta}}{\partial\delta}\right)^{\prime}\frac{\partial
f_{X,W}(x^{\delta},w)}{\partial x^{\delta}}\bigg{|}_{\delta=0}+\delta
R_{1}(x,w,\delta)$
$\displaystyle=\delta\frac{\partial\det\left[J\left(x^{\delta};\delta\right)\right]}{\partial\delta}\bigg{|}_{\delta=0}f_{X,W}(x,w)+\left(\frac{\partial
x^{\delta}}{\partial\delta}\right)\bigg{|}_{\delta=0}\frac{\partial
f_{X,W}(x,w)}{\partial x}+\delta R_{1}(x,w,\delta),$ (S.7)

where, for $\tilde{\delta}(x,w)$ between $0$ and $\delta$,

$R_{1}(x,w,\delta)=\left\\{\frac{\partial\left[\det\left[J\left(x^{\delta};\delta\right)\right]f_{X,W}(x^{\delta},w)\right]}{\partial\delta}\bigg{|}_{\delta=\tilde{\delta}(x,w)}-\frac{\partial\left[\det\left[J\left(x^{\delta};\delta\right)\right]f_{X,W}(x^{\delta},w)\right]}{\partial\delta}\bigg{|}_{\delta=0}\right\\}.$

Using the arguments similar to those in the proof of Theorem 1, we can show
that $G_{1,\delta}\left(y\right)$ converges to

$\displaystyle G_{1,0}\left(y\right)$
$\displaystyle:=\int_{\mathcal{W}}\int_{\mathcal{X}}\frac{\partial\det\left[J\left(x^{\delta};\delta\right)\right]}{\partial\delta}\bigg{|}_{\delta=0}F_{Y|X,W}(y|x,w)f_{X,W}(x,w)dxdw$
$\displaystyle+\int_{\mathcal{W}}\int_{\mathcal{X}}F_{Y|X,W}(y|x,w)\left(\frac{\partial
x^{\delta}}{\partial\delta}\right)^{\prime}\bigg{|}_{\delta=0}\frac{\partial
f_{X,W}(x,w)}{\partial x}dxdw$
$\displaystyle:=G_{1,0}^{\left(1\right)}\left(y\right)+G_{1,0}^{\left(2\right)}\left(y\right)$

uniformly in $y\in\mathcal{Y}$, as $\delta\rightarrow 0$.

Using Assumption S.1 and the fact that $\frac{\partial
x_{i}^{\delta}}{\partial x_{j}}|_{\delta=0}=\mathds{1}\left\\{i=j\right\\}$,
we have

$\displaystyle\frac{\partial\det\left[J\left(x^{\delta};\delta\right)\right]}{\partial\delta}\bigg{|}_{\delta=0}$
$\displaystyle=\frac{\partial}{\partial\delta}\left(\frac{\partial
x_{1}^{\delta}}{\partial x_{1}}\frac{\partial x_{2}^{\delta}}{\partial
x_{2}}-\frac{\partial x_{1}^{\delta}}{\partial x_{2}}\frac{\partial
x_{2}^{\delta}}{\partial x_{1}}\right)\bigg{|}_{\delta=0}$
$\displaystyle=-\left(\frac{\partial\kappa_{1}\left(x\right)}{\partial
x_{1}}\frac{\partial x_{2}^{\delta}}{\partial x_{2}}+\frac{\partial
x_{1}^{\delta}}{\partial
x_{1}}\frac{\partial\kappa_{2}\left(x\right)}{\partial
x_{2}}-\frac{\partial\kappa_{1}\left(x\right)}{\partial x_{2}}\frac{\partial
x_{2}^{\delta}}{\partial x_{1}}-\frac{\partial x_{1}^{\delta}}{\partial
x_{2}}\frac{\partial\kappa_{2}\left(x\right)}{\partial
x_{1}}\right)\bigg{|}_{\delta=0}$
$\displaystyle=-\left(\frac{\partial\kappa_{1}\left(x\right)}{\partial
x_{1}}+\frac{\partial\kappa_{2}\left(x\right)}{\partial x_{2}}\right).$

So

$G_{1,0}^{\left(1\right)}\left(y\right)=-\int_{\mathcal{W}}\int_{\mathcal{X}}\left(\frac{\partial\kappa_{1}\left(x\right)}{\partial
x_{1}}+\frac{\partial\kappa_{2}\left(x\right)}{\partial
x_{2}}\right)F_{Y|X,W}(y|x,w)f_{X,W}(x,w)dxdw.$

Next, note that

$\left(\frac{\partial
x^{\delta}}{\partial\delta}\right)^{\prime}\bigg{|}_{\delta=0}\frac{\partial
f_{X,W}(x,w)}{\partial x}=-\kappa\left(x\right)^{\prime}\frac{\partial
f_{X,W}(x,w)}{\partial x}.$

Using integration by parts, we can show that for $j=1$ and $2,$

$\displaystyle-\int_{\mathcal{W}}\int_{\mathcal{X}}F_{Y|X,W}(y|x,w)\left[\kappa_{j}\left(x\right)\frac{\partial
f_{X,W}(x,w)}{\partial x_{j}}\right]dxdw$
$\displaystyle=\int_{\mathcal{W}}\int_{\mathcal{X}}f_{X,W}(x,w)\frac{\partial\left[F_{Y|X,W}(y|x,w)\kappa_{j}\left(x\right)\right]}{\partial
x_{j}}dxdw.$

So

$\displaystyle G_{1,0}^{\left(2\right)}\left(y\right)$
$\displaystyle=\int_{\mathcal{W}}\int_{\mathcal{X}}f_{X,W}(x,w)\left(\frac{\partial\left[F_{Y|X,W}(y|x,w)\kappa_{1}\left(x\right)\right]}{\partial
x_{1}}+\frac{\partial\left[F_{Y|X,W}(y|x,w)\kappa_{2}\left(x\right)\right]}{\partial
x_{2}}\right)dxdw$
$\displaystyle=\int_{\mathcal{W}}\int_{\mathcal{X}}f_{X,W}(x,w)\left(\frac{\partial\left[F_{Y|X,W}(y|x,w)\right]}{\partial
x_{1}}\kappa_{1}\left(x\right)+\frac{\partial\left[F_{Y|X,W}(y|x,w)\right]}{\partial
x_{2}}\kappa_{2}\left(x\right)\right)dxdw$
$\displaystyle+\int_{\mathcal{W}}\int_{\mathcal{X}}f_{X,W}(x,w)F_{Y|X,W}(y|x,w)\left(\frac{\partial\kappa_{1}\left(x\right)}{\partial
x_{1}}+\frac{\partial\kappa_{2}\left(x\right)}{\partial x_{2}}\right)dxdw.$

Therefore,

$\displaystyle G_{1,0}\left(y\right)$
$\displaystyle=\int_{\mathcal{W}}\int_{\mathcal{X}}\left[\frac{\partial
F_{Y|X,W}(y|x,w)}{\partial x_{1}}\kappa_{1}\left(x\right)+\frac{\partial
F_{Y|X,W}(y|x,w)}{\partial
x_{2}}\kappa_{2}\left(x\right)\right]f_{X,W}(x,w)dxdw$
$\displaystyle=\int_{\mathcal{W}}\int_{\mathcal{X}}\left[\frac{\partial\left[F_{Y|X,W}(y|x,w)\right]}{\partial
x^{\prime}}\kappa\left(x\right)\right]f_{X,W}(x,w)dxdw$
$\displaystyle=E\left[\frac{\partial F_{Y|X,W}(y|X,W)}{\partial
X^{\prime}}\kappa\left(X\right)\right].$

For $G_{2,\delta}\left(y\right),$ the proof of Theorem 1 remains valid, and we
have that $G_{2,\delta}\left(y\right)$ converges to

$G_{2,0}\left(y\right):=-E\left[\mathds{1}\left\\{h(X,W,U)\leq
y\right\\}\frac{\partial\ln f_{U|X,W}(U|X,W)}{\partial
X^{\prime}}\kappa\left(X\right)\right]$

uniformly in $y\in\mathcal{Y}$, as $\delta\rightarrow 0$.

Invoking the same argument as that in the proof of Theorem 1, we obtain the
desired result.

###  S.4 Proof of Theorem S.2

The proof of this theorem is similar to that of Theorem 2. We outline the main
steps and omit the details here. We have

$\begin{pmatrix}\hat{\Pi}_{\tau,L,1}\\\
\hat{\Pi}_{\tau,L,2}\end{pmatrix}-\begin{pmatrix}\Pi_{\tau,L,1}\\\
\Pi_{\tau,L,2}\end{pmatrix}=D_{L}\left[\frac{n^{-1}\sum_{i=1}^{n}g(Z_{i}^{\prime}\hat{\theta}_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\hat{\alpha}_{\tau}}{\hat{f}_{Y}\left(\hat{q}_{\tau}\right)}-\frac{E\left[g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}\right]}{f_{Y}\left(Q_{\tau}[Y]\right)}\right].$

But

$\displaystyle\frac{n^{-1}\sum_{i=1}^{n}g(Z_{i}^{\prime}\hat{\theta}_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\hat{\alpha}_{\tau}}{\hat{f}_{Y}\left(\hat{q}_{\tau}\right)}-\frac{E\left[g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}\right]}{f_{Y}\left(Q_{\tau}[Y]\right)}$
$\displaystyle=\frac{n^{-1}\sum_{i=1}^{n}\left[g(Z_{i}^{\prime}\hat{\theta}_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\hat{\alpha}_{\tau}\right]-E\left[g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}\right]}{f_{Y}\left(Q_{\tau}[Y]\right)}$
$\displaystyle-\frac{E\left[g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}\right]}{f_{Y}\left(Q_{\tau}[Y]\right)^{2}}\left\\{\dot{f}_{Y}(Q_{\tau}[Y])\left(\hat{q}_{\tau}-Q_{\tau}[Y]\right)+\hat{f}_{Y}\left(Q_{\tau}[Y]\right)-f_{Y}\left(Q_{\tau}[Y]\right)\right\\}+o_{p}(n^{-1/2}h^{-1/2}).$

Now

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\hat{\theta}_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\hat{\alpha}_{\tau}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}$
$\displaystyle+\left(\frac{1}{n}\sum_{i=1}^{n}\dot{g}(Z_{i}^{\prime}\tilde{\theta}_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\tilde{\alpha}_{\tau}Z_{i}^{\prime}\right)(\hat{\theta}_{\tau}-\theta_{\tau})+\left(\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\tilde{\theta}_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\right)(\hat{\alpha}_{\tau}-\alpha_{\tau})$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}+M_{L}(\hat{\theta}_{\tau}-\theta_{\tau})+o_{p}\left(n^{-1/2}\right).$

Therefore,

$\displaystyle\begin{pmatrix}\hat{\Pi}_{\tau,L,1}\\\
\hat{\Pi}_{\tau,L,2}\end{pmatrix}-\begin{pmatrix}\Pi_{\tau,L,1}\\\
\Pi_{\tau,L,2}\end{pmatrix}$
$\displaystyle=\frac{1}{f_{Y}\left(Q_{\tau}[Y]\right)}D_{L}\left[\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}-E\left[g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(X_{i}\right)^{\prime}\alpha_{\tau}\right]\right]$
$\displaystyle-\frac{1}{f_{Y}\left(Q_{\tau}[Y]\right)}D_{L}M_{L}H^{-1}\frac{1}{n}\sum_{i=1}^{n}s_{i}(\theta_{\tau};Q_{\tau}[Y])$
$\displaystyle-\left[\begin{pmatrix}\Pi_{\tau,L,1}\\\
\Pi_{\tau,L,2}\end{pmatrix}\frac{\dot{f}_{Y}(Q_{\tau}[Y])}{f_{Y}\left(Q_{\tau}[Y]\right)}+\frac{1}{f_{Y}\left(Q_{\tau}[Y]\right)}D_{L}M_{L}H^{-1}H_{Q}\right]\frac{1}{n}\sum_{i=1}^{n}\psi(Y_{i},\tau,F_{Y})$
$\displaystyle-\begin{pmatrix}\Pi_{\tau,L,1}\\\
\Pi_{\tau,L,2}\end{pmatrix}\frac{\hat{f}_{Y}\left(Q_{\tau}[Y]\right)-f_{Y}\left(Q_{\tau}[Y]\right)}{f_{Y}\left(Q_{\tau}[Y]\right)}+o_{p}(n^{-1/2})+o_{p}(n^{-1/2}h^{-1/2}).$

Combining this with (A.9) leads to the desired result.

### Details of Example 7

Before the location-scale shift,

$Y=\lambda+X\gamma+U=\lambda+\mu_{X}\gamma+U^{\circ}$

and the unconditional $\tau$-quantile $Q_{\tau}[Y]$ of $Y$ is
$\lambda+\mu_{X}\gamma+Q_{\tau}[U^{\circ}]$. After the location-scale shift
with $\mu=\mu_{X}$, we have

$X_{\delta}=\left(X-\mu_{X}\right)s\left(\delta\right)+\mu_{X}+\ell\left(\delta\right),$

and so

$Y_{\delta}=\lambda+X_{\delta}\gamma+U=\lambda+\left[\mu_{X}+\ell\left(\delta\right)\right]\gamma+U_{\delta}^{\circ}$

where $U_{\delta}^{\circ}=U+\left(X-\mu_{X}\right)\gamma
s\left(\delta\right).$ The unconditional $\tau$-quantile
$Q_{\tau}[Y_{\delta}]$ of $Y_{\delta}$ is
$\lambda+\left[\mu_{X}+\ell\left(\delta\right)\right]\gamma+Q_{\tau}[U_{\delta}^{\circ}].$
Hence

$\displaystyle\Pi_{\tau}$ $\displaystyle=\lim_{\delta\rightarrow
0}\frac{\ell\left(\delta\right)-\ell\left(0\right)}{\delta}\gamma+\lim_{\delta\rightarrow
0}\frac{Q_{\tau}[U_{\delta}^{\circ}]-Q_{\tau}[U^{\circ}]}{\delta}$
$\displaystyle=\dot{\ell}\left(0\right)\gamma+\lim_{\delta\rightarrow
0}\frac{Q_{\tau}[U_{\delta}^{\circ}]-Q_{\tau}[U^{\circ}]}{\delta}.$

The first term is the location effect, and the second term is the scale
effect.

Now, we write

$Q_{\tau}\left(U_{\delta}^{\circ}\right)=\sqrt{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}s\left(\delta\right)}Q_{\tau}\left[\varepsilon_{\delta}^{\circ}\right],$

where
$\varepsilon_{\delta}^{\circ}:=U_{\delta}^{\circ}/\sqrt{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}s\left(\delta\right)}$
is a random variable with zero mean and unit variance. So

$\displaystyle\lim_{\delta\rightarrow
0}\frac{Q_{\tau}[U_{\delta}^{\circ}]-Q_{\tau}[U^{\circ}]}{\delta}$
$\displaystyle=\lim_{\delta\rightarrow
0}\frac{\sqrt{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}s\left(\delta\right)}-\sqrt{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}}}{\delta}Q_{\tau}\left[\varepsilon_{\delta}^{\circ}\right]+\sqrt{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}}\lim_{\delta\rightarrow
0}\frac{Q_{\tau}\left[\varepsilon_{\delta}^{\circ}\right]-Q_{\tau}\left[\varepsilon_{0}^{\circ}\right]}{\delta}$
$\displaystyle=\frac{\sigma_{X}^{2}\gamma^{2}}{\sqrt{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}}}\dot{s}\left(0\right)Q_{\tau}\left[\varepsilon_{0}^{\circ}\right]+\sqrt{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}}\lim_{\delta\rightarrow
0}\frac{Q_{\tau}\left[\varepsilon_{\delta}^{\circ}\right]-Q_{\tau}\left[\varepsilon_{0}^{\circ}\right]}{\delta}.$

If
$Q_{\tau}\left[\varepsilon_{\delta}^{\circ}\right]=Q_{\tau}\left[\varepsilon_{0}^{\circ}\right]$
for any $\delta\geq 0,$ then the second term is zero, and we obtain

$\displaystyle\Pi_{\tau,S}$ $\displaystyle:=\lim_{\delta\rightarrow
0}\frac{Q_{\tau}[U_{\delta}^{\circ}]-Q_{\tau}[U^{\circ}]}{\delta}=\frac{\sigma_{X}^{2}\gamma^{2}}{\sqrt{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}}}\dot{s}\left(0\right)\cdot
Q_{\tau}\left[\varepsilon_{0}^{\circ}\right]$
$\displaystyle=\frac{\sigma_{X}^{2}\gamma^{2}}{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}}\dot{s}\left(0\right)\cdot
Q_{\tau}\left[U^{\circ}\right].$

Furthermore, if
$Q_{\tau}\left[X_{\gamma}^{\circ}/\sqrt{\sigma_{X}^{2}\gamma^{2}}\right]=Q_{\tau}\left[\varepsilon_{0}^{\circ}\right],$
then

$\Pi_{\tau,S}=\sqrt{\frac{\sigma_{X}^{2}\gamma^{2}}{\sigma_{U}^{2}+\sigma_{X}^{2}\gamma^{2}}}\dot{s}\left(0\right)Q_{\tau}\left[X_{\gamma}^{\circ}\right]=\dot{s}\left(0\right)\sqrt{R_{YX}^{2}}Q_{\tau}\left[X_{\gamma}^{\circ}\right].$

Note that if $X$ and $U$ are normals, then
$\varepsilon_{\delta}^{\circ},\varepsilon_{0}^{\circ}$ and
$X_{\gamma}^{\circ}/\sqrt{\sigma_{X}^{2}\gamma^{2}}$ are all standard normals,
and hence they have the same quantiles. Therefore,
$\Pi_{\tau,S}=\dot{s}\left(0\right)\sqrt{R_{YX}^{2}}Q_{\tau}\left[X_{\gamma}^{\circ}\right]$,
as given in Example 7.