Slides ub-3

Arthur Charpentier, UB School of Economics Summer School 2018 - Big Data for Economics
Big Data for Economics # 3*
A. Charpentier (Universit´e de Rennes 1)
https://github.com/freakonometrics/ub
UB School of Economics Summer School, 2018.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 1

#3 Loss Functions : from OLS to Quantile Regression*

References
Motivation
Machado & Mata (2005). Counterfactual decomposition of changes in wage
distributions using quantile regression, JAE.
References
Givord & d’Haultfœuillle (2013) La r´egression quantile en pratique, INSEE
Koenker & Bassett (1978) Regression Quantiles, Econometrica.
Koenker (2005). Quantile Regression. Cambridge University Press.
Newey & Powell (1987) Asymmetric Least Squares Estimation and Testing,
Econometrica.

Quantiles
Let Y denote a random variable with cumulative distribution function F,
F(y) = P[Y ≤ y]. The quantile is
Q(u) = inf x ∈ R, F(x) > u .
qqq
DECESSURVIE
60 70 80 90 100 110 120
FRCAR
q
DECESSURVIE
1.0 1.5 2.0 2.5 3.0
INCAR
q
q q
DECESSURVIE
10 20 30 40 50
INSYS
q q
DECESSURVIE
10 15 20 25 30 35 40 45
PAPUL
DECESSURVIE
5 10 15 20
PVENT
DECESSURVIE
500 1000 1500 2000 2500 3000
REPUL
Box plot, from Tukey (1977, Exploratory Data Analysis).

Defining halfspace depth
Given y ∈ Rd
, and a direction u ∈ Rd
, define the closed half space
Hy,u = {x ∈ Rd
such that u x ≤ u y}
and define depth at point y by
depth(y) = inf
u,u=0
{P(Hy,u)}
i.e. the smallest probability of a closed half space containing y.
The empirical version is (see Tukey (1975))
depth(y) = min
u,u=0
1
n
n
i=1
1(Xi ∈ Hy,u)
For α > 0.5, define the depth set as
Dα = {y ∈ R ∈ Rd
such that ≥ 1 − α}.
The empirical version is can be related to the bagplot, Rousseeuw et al., 1999.

Empirical sets extremely sentive to the algorithm
−2 −1 0 1
−1.5−1.0−0.50.00.51.0
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−2 −1 0 1
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where the blue set is the empirical estimation for Dα, α = 0.5.

The bagplot tool
The depth function introduced here is the multivariate extension of standard
univariate depth measures, e.g.
depth(x) = min{F(x), 1 − F(x−
)}
which satisﬁes depth(Qα) = min{α, 1 − α}. But one can also consider
depth(x) = 2 · F(x) · [1 − F(x−
)] or depth(x) = 1 −
1
2
− F(x) .
Possible extensions to functional bagplot. Consider a set of functions fi(x),
i = 1, · · · , n, such that
fi(x) = µ(x) +
n−1
k=1
zi,kϕk(x)
(i.e. principal component decomposition) where ϕk(·) represents the
eigenfunctions. Rousseeuw et al., 1999 considered bivariate depth on the ﬁrst two
scores, xi = (zi,1, zi,2). See Ferraty & Vieu (2006).

Quantiles and Quantile Regressions
Quantiles are important quantities in many
areas (inequalities, risk, health, sports, etc).
Quantiles of the N(0, 1) distribution.
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
−3 0 1 2 3
−1.645
5%

A First Model for Conditional Quantiles
Consider a location model, y = β0 + xT
β + ε i.e.
E[Y |X = x] = xT
β
then one can consider
Q(τ|X = x) = β0 + Qε(τ) + xT
β
where Qε(·) is the quantile function of the residuals.
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5 10 15 20 25 30
020406080100120
speed
dist

OLS Regression, 2 norm and Expected Value
Let y ∈ Rd
, y = argmin
m∈R



n
i=1
1
n
yi − m
εi
2



. It is the empirical version of
E[Y ] = argmin
m∈R



y − m
ε
2
dF(y)



= argmin
m∈R



E Y − m
ε
2



where Y is a random variable.
Thus, argmin
m(·):Rk→R



n
i=1
1
n
yi − m(xi)
εi
2



is the empirical version of E[Y |X = x].
See Legendre (1805) Nouvelles méthodes pour la détermination des orbites des
comètes and Gauβ (1809) Theoria motus corporum coelestium in sectionibus conicis
solem ambientium.

OLS Regression, 2 norm and Expected Value
Sketch of proof: (1) Let h(x) =
d
i=1
(x − yi)2
, then
h (x) =
d
i=1
2(x − yi)
and the FOC yields x =
1
n
d
i=1
yi = y.
(2) If Y is continuous, let h(x) =
R
(x − y)f(y)dy and
h (x) =
∂
∂x R
(x − y)2
f(y)dy =
R
∂
∂x
(x − y)2
f(y)dy
i.e. x =
R
xf(y)dy =
R
yf(y)dy = E[Y ]
0.0 0.2 0.4 0.6 0.8 1.0
0.51.01.52.02.5
0.0 0.2 0.4 0.6 0.8 1.0
0.51.01.52.02.5

Median Regression, 1 norm and Median
Let y ∈ Rd
, median[y] ∈ argmin
m∈R



n
i=1
1
n
yi − m
εi



. It is the empirical version of
median[Y ] ∈ argmin
m∈R



y − m
ε
dF(y)



= argmin
m∈R



E Y − m
ε
1



where Y is a random variable, P[Y ≤ median[Y ]] ≥
1
2
and P[Y ≥ median[Y ]] ≥
1
2
.
argmin
m(·):Rk→R



n
i=1
1
n
yi − m(xi)
εi



is the empirical version of median[Y |X = x].
See Boscovich (1757) De Litteraria expeditione per pontiﬁciam ditionem ad
dimetiendos duos meridiani and Laplace (1793) Sur quelques points du syst`eme du
monde.

Median Regression, 1 norm and Median
Sketch of proof: (1) Let h(x) =
d
i=1
|x − yi|
(2) If F is absolutely continuous, dF(x) = f(x)dx, and the
median m is solution of
m
−∞
f(x)dx =
1
2
.
Set h(y) =
+∞
−∞
|x − y|f(x)dx
=
y
−∞
(−x + y)f(x)dx +
+∞
y
(x − y)f(x)dx
Then h (y) =
y
−∞
f(x)dx −
+∞
y
f(x)dx, and FOC yields
y
−∞
f(x)dx =
+∞
y
f(x)dx = 1 −
y
−∞
f(x)dx =
1
2
0.0 0.2 0.4 0.6 0.8 1.0
1.52.02.53.03.54.0
0.0 0.2 0.4 0.6 0.8 1.0
2.02.53.03.54.0

Bayesian Statistics and Loss Functions
In statistics, consider some functional seen as a distance between θ and θ, e.g.
squared loss : 2(θ, θ) = (θ − θ)2
absolute loss : 1(θ, θ) = |θ − θ|
zero/one loss ◦|(θ, θ) = 1(|θ − θ| > ε) for some ε > 0
Deﬁne risk as the expected loss,
R(θ, θ) = (θ(y), θ) f(y|θ)
L(θ,y)
dy
(where the average is over the sample space).

Bayes’Rule: Minimize average risk
R(θ) = R(θ, θ)π(θ)dθ
and set
θ = argmin R(θ)
Hence
θ = argmin
Θ Rn
(θ(y), θ)f(y|θ)dyπ(θ)dθ
θ = argmin
Rn Θ
(θ(y), θ)π(θ|y)dθf(y)dy

Application : head/tails Bernoulli with uniform prior
f(y) = θy
(1 − θ)1−y
, where y ∈ {0, 1}
and π(θ) = 1(θ ∈ [0, 1]). Then likelihood is
L(θ, y) = f(y|θ) = θ yi
(1 − θ)n− yi
From Baye’s Theorem,
π(θ|y) ∝ π(θ) · f(y|θ) ∝ θ yi
(1 − θ)n− yi
which is a Beta distribution. Recall that B(α, β)
g(u|α, β) =
uα−1
(1 − u)β−1
B(α, β)
on [0, 1]

and E[U] =
α
α + β
, Var[U] =
αβ
(α + β)2(α + β + 1)
, while
median[U] ∼
3α − 1
3α + 3β − 2
and mode[U] =
α − 1
α + β − 2
if α, β > 1.
Here, posterior distribition is B(ny + 1, n(1 − y) + 1).
Here with 2, θ2 = E[θ|y] =
ny + 1
n + 2
while with ◦|, θ = mode[θ|y] = y (which is
also the maximum likelihood estimator).
With a Beta B(a, b) prior, posterior distribition is B(ny + a + 1, n(1 − y) + b + 1).
Here with 2, θ2 = E[θ|y] =
ny + a
n + a + b
while with ◦|,
θ = mode[θ|y] =
ny + a − 1
n + a + b − 2
(which is no longer the maximum likelihood
estimator).

Example: Gaussian case (with known variance), Yi ∼ N(µ, σ2
0). Consider
Gaussian prior, µ ∼ N(m, s2
). We are uncertain here about the value of µ. Then,
the posterior distribution for µ is a Gaussian distribution µ|y ∼ N(my, s2
y) where
my = s2
y
m
s2
+
ny
σ2
0
and s2
y =
1
s2
+
n
σ2
0
−1
Here my is Bayes estimator of µ under loss functions 1, 2 and ◦|.
When σ2
0 is no longer known, but is just a nuisance parameter, the natural
approach is to consider a joint posterior density for θ = (µ, σ2
) and then
marginalize by integrating out the nuisance parameters

OLS vs. Median Regression (Least Absolute Deviation)
Consider some linear model, yi = β0 + xT
i β + εi ,and deﬁne
(βols
0 , β
ols
) = argmin
n
i=1
yi − β0 − xT
i β
2
(βlad
0 , β
lad
) = argmin
n
i=1
yi − β0 − xT
i β
Assume that ε|X has a symmetric distribution, E[ε|X] = median[ε|X] = 0, then
(βols
0 , β
ols
) and (βlad
0 , β
lad
) are consistent estimators of (β0, β).
Assume that ε|X does not have a symmetric distribution, but E[ε|X] = 0, then
β
ols
and β
lad
are consistent estimators of the slopes β.
If median[ε|X] = γ, then βlad
0 converges to β0 + γ.

OLS vs. Median Regression
Median regression is stable by monotonic transformation. If
log[yi] = β0 + xT
i β + εi with median[ε|X] = 0,
then
median[Y |X = x] = exp median[log(Y )|X = x] = exp β0 + xT
i β
while
E[Y |X = x] = exp E[log(Y )|X = x] (= exp E[log(Y )|X = x] ·[exp(ε)|X = x]
1 > ols <- lm(y˜x, data=df)
2 > library(quantreg)
3 > lad <- rq(y˜x, data=df , tau =.5)

Notations
Cumulative distribution function FY (y) = P[Y ≤ y].
Quantile function QX(u) = inf y ∈ R : FY (y) ≥ u ,
also noted QX(u) = F−1
X u.
One can consider QX(u) = sup y ∈ R : FY (y) < u
For any increasing transformation t, Qt(Y )(τ) = t QY (τ)
F(y|x) = P[Y ≤ y|X = x]
QY |x(u) = F−1
(u|x)
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0 1 2 3 4 5
0.00.20.40.60.81.0
q
q

Empirical Quantile

Quantiles and optimization : numerical aspects
Consider the case of the median. Consider a sample {y1, · · · , yn}.
To compute the median, solve min
µ
n
i=1
|yi − µ| which can be solved using
linear programming techniques.
More precisely, this problem is equivalent to min
µ,a,b
n
i=1
ai + bi with ai, bi ≥ 0
and yi − µ = ai − bi, ∀i = 1, · · · , n.
Consider a sample obtained from a lognormal distribution
1 n = 101
2 set.seed (1)
3 y = rlnorm(n)
4 median(y)
5 [1] 1.077415

Here, one can use a standard optimization routine
1 md=Vectorize(function(m) sum(abs(y-m)))
2 optim(mean(y),md)
3 $par
4 [1] 1.077416
or a linear programing technique : use the matrix form, with 3n constraints, and
2n + 1 parameters,
1 library(lpSolve)
2 A1 = cbind(diag (2*n) ,0)
3 A2 = cbind(diag(n), -diag(n), 1)
4 r = lp("min", c(rep (1,2*n) ,0),
5 rbind(A1 , A2),c(rep(">=", 2*n), rep("=", n)), c(rep (0,2*n), y))
6 tail(r$solution ,1)
7 [1] 1.077415

More generally, consider here some quantile,
1 tau = .3
2 quantile(x,tau)
3 30%
4 0.6741586
The linear program is now min
µ,a,b
n
i=1
τai + (1 − τ)bi with ai, bi ≥ 0 and
yi − µ = ai − bi, ∀i = 1, · · · , n.
1 A1 = cbind(diag (2*n) ,0)
2 A2 = cbind(diag(n), -diag(n), 1)
3 r = lp("min", c(rep(tau ,n),rep(1-tau ,n) ,0),
4 rbind(A1 , A2),c(rep(">=", 2*n), rep("=", n)), c(rep (0,2*n), y))
6 [1] 0.6741586

Quantile regression ?
In OLS regression, we try to evaluate E[Y |X = x] =
R
ydFY |X=x(y)
In quantile regression, we try to evaluate
Qu(Y |X = x) = inf y : FY |X=x(y) ≥ u
as introduced in Newey & Powell (1987) Asymmetric Least Squares Estimation and
Testing.
Li & Racine (2007) Nonparametric Econometrics: Theory and Practice suggested
Qu(Y |X = x) = inf y : FY |X=x(y) ≥ u
where FY |X=x(y) can be some kernel-based estimator.

Quantiles and Expectiles
Consider the following risk functions
Rq
τ (u) = u · τ − 1(u < 0) , τ ∈ [0, 1]
with Rq
1/2(u) ∝ |u| = u 1
, and
Re
τ (u) = u2
· τ − 1(u < 0) , τ ∈ [0, 1]
with Re
1/2(u) ∝ u2
= u 2
2
.
QY (τ) = argmin
m
E Rq
τ (Y − m)
which is the median when τ = 1/2,
EY (τ) = argmin
m
E Re
τ (X − m) }
which is the expected value when τ = 1/2.
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
0.00.51.01.5
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−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
0.00.51.01.5

Quantiles and Expectiles
One can also write
quantile: argmin



n
i=1
ωq
τ (εi) yi − qi
εi



where ωq
τ ( ) =



1 − τ if ≤ 0
τ if > 0
expectile: argmin



n
i=1
ωe
τ (εi) yi − qi
εi
2



where ωe
τ ( ) =



1 − τ if ≤ 0
τ if > 0
Expectiles are unique, not quantiles...
Quantiles satisfy E[sign(Y − QY (τ))] = 0
Expectiles satisfy τE (Y − eY (τ))+ = (1 − τ)E (Y − eY (τ))−
(those are actually the ﬁrst order conditions of the optimization problem).

Quantiles and M-Estimators
There are connections with M-estimators, as introduced in Serﬂing (1980)
Approximation Theorems of Mathematical Statistics, chapter 7.
For any function h(·, ·), the M-functional is the solution β of
h(y, β)dFY (y) = 0
, and the M-estimator is the solution of
h(y, β)dFn(y) =
1
n
n
i=1
h(yi, β) = 0
Hence, if h(y, β) = y − β, β = E[Y ] and β = y.
And if h(y, β) = 1(y < β) − τ, with τ ∈ (0, 1), then β = F−1
Y (τ).

Quantiles, Maximal Correlation and Hardy-Littlewood-Polya
If x1 ≤ · · · ≤ xn and y1 ≤ · · · ≤ yn, then
n
i=1
xiyi ≥
n
i=1
xiyσ(i), ∀σ ∈ Sn, and x
and y are said to be comonotonic.
The continuous version is that X and Y are comonotonic if
E[XY ] ≥ E[X ˜Y ] where ˜Y
L
= Y,
One can prove that
Y = QY (FX(X)) = argmax
˜Y ∼FY
E[X ˜Y ]

Expectiles as Quantiles
For every Y ∈ L1
, τ → eY (τ) is continuous, and striclty increasing
if Y is absolutely continuous,
∂eY (τ)
∂τ
=
E[|X − eY (τ)|]
(1 − τ)FY (eY (τ)) + τ(1 − FY (eY (τ)))
if X ≤ Y , then eX(τ) ≤ eY (τ) ∀τ ∈ (0, 1)
“Expectiles have properties that are similar to quantiles” Newey & Powell (1987)
Asymmetric Least Squares Estimation and Testing. The reason is that expectiles of
a distribution F are quantiles a distribution G which is related to F, see Jones
(1994) Expectiles and M-quantiles are quantiles: let
G(t) =
P(t) − tF(t)
2[P(t) − tF(t)] + t − µ
where P(s) =
s
−∞
ydF(y).
The expectiles of F are the quantiles of G.
1 > x <- rnorm (99)
2 > library(expectreg)
3 > e <- expectile(x, probs = seq(0, 1, 0.1))

Expectiles as Quantiles
0.0 0.2 0.4 0.6 0.8 1.0
−2−1012
0.0 0.2 0.4 0.6 0.8 1.0
0246810
0.0 0.2 0.4 0.6 0.8 1.0
02468
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0

Elicitable Measures
“elicitable” means “being a minimizer of a suitable expected score”
T is an elicatable function if there exits a scoring function S : R × R → [0, ∞)
such that
T(Y ) = argmin
x∈R R
S(x, y)dF(y) = argmin
x∈R
E S(x, Y ) where Y ∼ F.
see Gneiting (2011) Making and evaluating point forecasts.
Example: mean, T(Y ) = E[Y ] is elicited by S(x, y) = x − y 2
2
Example: median, T(Y ) = median[Y ] is elicited by S(x, y) = x − y 1
Example: quantile, T(Y ) = QY (τ) is elicited by
S(x, y) = τ(y − x)+ + (1 − τ)(y − x)−
Example: expectile, T(Y ) = EY (τ) is elicited by
S(x, y) = τ(y − x)2
+ + (1 − τ)(y − x)2
−

Elicitable Measures
Remark: all functionals are not necessarily elicitable, see Osband (1985)
Providing incentives for better cost forecasting
The variance is not elicitable
The elicitability property implies a property which is known as convexity of the
level sets with respect to mixtures (also called Betweenness property) : if two
lotteries F, and G are equivalent, then any mixture of the two lotteries is also
equivalent with F and G.

Empirical Quantiles
Consider some i.id. sample {y1, · · · , yn} with distribution F. Set
Qτ = argmin E Rq
τ (Y − q) where Y ∼ F and Qτ ∈ argmin
n
i=1
Rq
τ (yi − q)
Then as n → ∞
√
n Qτ − Qτ
L
→ N 0,
τ(1 − τ)
f2(Qτ )
Sketch of the proof: yi = Qτ + εi, set hn(q) =
1
n
n
i=1
1(yi < q) − τ , which is a
non-decreasing function, with
E Qτ +
u
√
n
= FY Qτ +
u
√
n
∼ fY (Qτ )
u
√
n
Var Qτ +
u
√
n
∼
FY (Qτ )[1 − FY (Qτ )]
n
=
τ(1 − τ)
n
.

Empirical Expectiles
Consider some i.id. sample {y1, · · · , yn} with distribution F. Set
µτ = argmin E Re
τ (Y − m) where Y ∼ F and µτ = argmin
n
i=1
Re
τ (yi − m)
Then as n → ∞
√
n µτ − µτ
L
→ N 0, s2
for some s2
, if Var[Y ] < ∞. Deﬁne the identiﬁcation function
Iτ (x, y) = τ(y − x)+ + (1 − τ)(y − x)− (elicitable score for quantiles)
so that µτ is solution of E I(µτ , Y ) = 0. Then
s2
=
E[I(µτ , Y )2
]
(τ[1 − F(µτ )] + [1 − τ]F(µτ ))2
.

Quantile Regression
We want to solve, here, min
n
i=1
Rq
τ (yi − xT
i β)
yi = xT
i β + εi so that Qy|x(τ) = xT
β + F−1
ε (τ)
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5 10 15 20 25
020406080100120
speed
dist
10%
90%
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20 40 60 80
23456
probability level (%)
slopeofquantileregression

Geometric Properties of the Quantile Regression
Observe that the median regression will always have
two supporting observations.
Start with some regression line, yi = β0 + β1xi
Consider small translations yi = (β0 ± ) + β1xi
We minimize
n
i=1
yi − (β0 + β1xi)
From line blue, a shift up decrease the sum by
until we meet point on the left
an additional shift up will increase the sum
We will necessarily pass through one point
(observe that the sum is piecwise linear in )
−4 −2 0 2 4 6
51015
H
D
q
q
q
0 1 2 3 4
0246810
x
y

Geometric Properties of the Quantile Regression
Consider now rotations of the line around the support
point
If we rotate up, we increase the sum of absolute diﬀer-
ence (large impact on the point on the right)
If we rotate down, we decrease the sum, until we reach
the point on the right
Thus, the median regression will always have two sup-
portting observations.
2 > fit <- rq(dist˜speed , data=cars , tau =.5)
3 > which(predict(fit)== cars$dist)
4 1 21 46
5 1 21 46
−4 −2 0 2 4 6
5101520
H
D
q
q
q
0 1 2 3 4
0246810
x
y
q
q
q
0 1 2 3 4
0246810
x
y
q

Numerical Aspects
To illustrate numerical computations, use
1 base=read.table("http:// freakonometrics .free.fr/rent98_00. txt",header
=TRUE)
The linear program for the quantile regression is now
min
µ,a,b
n
i=1
τai + (1 − τ)bi
with ai, bi ≥ 0 and yi − [βτ
0 + βτ
1 xi] = ai − bi, ∀i = 1, · · · , n.
1 require(lpSolve)
2 tau = .3
3 n=nrow(base)
4 X = cbind( 1, base$area)
5 y = base$rent_euro
6 A1 = cbind(diag (2*n), 0,0)
7 A2 = cbind(diag(n), -diag(n), X)

Numerical Aspects
1 r = lp("min",
2 c(rep(tau ,n), rep(1-tau ,n) ,0,0), rbind(A1 , A2),
3 c(rep(">=", 2*n), rep("=", n)), c(rep(0,2*n), y))
5 [1] 148.946864 3.289674
see
1 library(quantreg)
2 rq(rent_euro˜area , tau=tau , data=base)
3 Coefficients :
4 (Intercept) area
5 148.946864 3.289674

Numerical Aspects
1 plot(base$area ,base$rent_euro ,xlab=expression(
paste("surface (",mˆ2,")")),
2 ylab="rent (euros/month)",col=rgb
(0 ,0 ,1 ,.4),cex =.5)
3 sf =0:250
4 yr=r$solution [2*n+1]+r$solution [2*n+2]*sf
5 lines(sf ,yr ,lwd=2,col="blue")
6 tau = .9
7 r = lp("min",c(rep(tau ,n), rep(1-tau ,n) ,0,0),
rbind(A1 , A2),c(rep(" >=", 2*n), rep("=", n)
), c(rep (0,2*n), y))
8 yr=r$solution [2*n+1]+r$solution [2*n+2]*sf
9 lines(sf ,yr ,lwd=2,col="blue")

Numerical Aspects
For multiple regression, we should consider some trick (R function assumes all
variables are nonnegative)
1 tau = 0.3
2 n = nrow(base)
3 X = cbind (1, base$area , base$yearc)
4 y = base$rent_euro
5 r = lp("min",
6 c(rep(tau , n), rep(1 - tau , n), rep(0, 2 * 3)),
7 cbind(diag(n), -diag(n), X, -X),
8 rep("=", n),
9 y)
10 beta = tail(r$solution , 6)
11 beta = beta [1:3] - beta [3 + 1:3]
12 beta
13 [1] -5542.503252 3.978135 2.887234

Numerical Aspects
which is consistant with
1 library(quantreg)
2 rq(rent_euro˜area+yearc , tau=tau , data=base)
3 Coefficients :
4 (Intercept) area yearc
5 -5542.503252 3.978135 2.887234

Distributional Aspects
OLS are equivalent to MLE when Y − m(x) ∼ N(0, σ2
), with density
g( ) =
1
σ
√
2π
exp −
2
2σ2
Quantile regression is equivalent to Maximum Likelihood Estimation when
Y − m(x) has an asymmetric Laplace distribution
g( ) =
√
2
σ
κ
1 + κ2
exp −
√
2κ1( >0)
σκ1( <0)
| |

Quantile Regression and Iterative Least Squares
start with some β(0)
e.g. βols
at stage k :
let ε
(k)
i = yi − xT
i β(k−1)
deﬁne weights ω
(k)
i = Rτ (ε
(k)
i )
compute weighted least square to estimate β(k)
One can also consider a smooth approximation of Rq
τ (·), and then use
Newton-Raphson.
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5 10 15 20 25
020406080100120
speed
dist

Optimization Algorithm
Primal problem is
min
β,u,v
τ1T
u + (1 − τ)1T
v s.t. y = Xβ + u − v, with u, v ∈ Rn
+
and the dual version is
max
d
yT
d s.t. XT
d = (1 − τ)XT
1 with d ∈ [0, 1]n
Koenker & D’Orey (1994) A Remark on Algorithm AS 229: Computing Dual
Regression Quantiles and Regression Rank Scores suggest to use the simplex
method (default method in R)
Portnoy & Koenker (1997) The Gaussian hare and the Laplacian tortoise suggest to
use the interior point method *

Interior Point Method
See Vanderbei et al. (1986) A modiﬁcation of Karmarkar’s linear programming
algorithm for a presentation of the algorithm, Potra & Wright (2000) Interior-point
methods for a general survey, and and Meketon (1986) Least absolute value
regression for an application of the algorithm in the context of median regression.
Running time is of order n1+δ
k3
for some δ > 0 and k = dim(β)
(it is (n + k)k2
for OLS, see wikipedia).

Quantile Regression Estimators
OLS estimator β
ols
is solution of
β
ols
= argmin E E[Y |X = x] − xT
β
2
and Angrist, Chernozhukov & Fernandez-Val (2006) Quantile Regression under
Misspeciﬁcation proved that
βτ = argmin E ωτ (β) Qτ [Y |X = x] − xT
β
2
(under weak conditions) where
ωτ (β) =
1
0
(1 − u)fy|x(uxT
β + (1 − u)Qτ [Y |X = x])du
βτ is the best weighted mean square approximation of the tru quantile function,
where the weights depend on an average of the conditional density of Y over xT
β
and the true quantile regression function.

Assumptions to get Consistency of Quantile Regression Estimators
As always, we need some assumptions to have consistency of estimators.
• observations (Yi, Xi) must (conditionnaly) i.id.
• regressors must have a bounded second moment, E Xi
2
< ∞
• error terms ε are continuously distributed given Xi, centered in the sense
that their median should be 0,
0
−∞
fε( )d =
1
2
.
• “local identiﬁcation” property : fε(0)XXT
is positive deﬁnite

Under those weak conditions, βτ is asymptotically normal:
√
n(βτ − βτ )
L
→ N(0, τ(1 − τ)D−1
τ ΩxD−1
τ ),
where
Dτ = E fε(0)XXT
and Ωx = E XT
X .
hence, the asymptotic variance of β is
Var βτ =
τ(1 − τ)
[fε(0)]2
1
n
n
i=1
xT
i xi
−1
where fε(0) is estimated using (e.g.) an histogram, as suggested in Powell (1991)
Estimation of monotonic regression models under quantile restrictions, since
Dτ = lim
h↓0
E
1(|ε| ≤ h)
2h
XXT
∼
1
2nh
n
i=1
1(|εi| ≤ h)xixT
i = Dτ .

There is no first order condition, in the sense ∂Vn(β, τ)/∂β = 0 where
Vn(β, τ) =
n
i=1
Rq
τ (yi − xT
i β)
There is an asymptotic first order condition,
1
√
n
n
i=1
xiψτ (yi − xT
i β) = O(1), as n → ∞,
where ψτ (·) = 1(· < 0) − τ, see Huber (1967) The behavior of maximum likelihood
estimates under nonstandard conditions.
One can also define a Wald test, a Likelihood Ratio test, etc.

Then the confidence interval of level 1 − α is then
βτ ± z1−α/2 Var βτ
An alternative is to use a boostrap strategy (see #2)
• generate a sample (y
(b)
i , x
(b)
i ) from (yi, xi)
• estimate β(b)
τ by
β
(b)
τ = argmin Rq
τ y
(b)
i − x
(b)T
i β
• set Var βτ =
1
B
B
b=1
β
(b)
τ − βτ
2
For confidence intervals, we can either use Gaussian-type confidence intervals, or
empirical quantiles from bootstrap estimates.

If τ = (τ1, · · · , τm), one can prove that
√
n(βτ − βτ )
L
→ N(0, Στ ),
where Στ is a block matrix, with
Στi,τj = (min{τi, τj} − τiτj)D−1
τi
ΩxD−1
τj
see Kocherginsky et al. (2005) Practical Conﬁdence Intervals for Regression
Quantiles for more details.

Quantile Regression: Transformations
Scale equivariance
For any a > 0 and τ ∈ [0, 1]
ˆβτ (aY, X) = aˆβτ (Y, X) and ˆβτ (−aY, X) = −aˆβ1−τ (Y, X)
Equivariance to reparameterization of design
Let A be any p × p nonsingular matrix and τ ∈ [0, 1]
ˆβτ (Y, XA) = A−1 ˆβτ (Y, X)

Visualization, τ → βτ
See Abreveya (2001) The eﬀects of demographics and maternal behavior...
1 > base=read.table("http:// freakonometrics .free.fr/ natality2005 .txt")
20 40 60 80
−6−4−20246
AGE
10 20 30 40 50
01000200030004000500060007000
Age (of the mother) AGE
BirthWeight(ing.) 1%
5%
10%
25%
50%
75%
90%
95%

1 > base=read.table("http:// freakonometrics .free.fr/ natality2005 .txt",
header=TRUE ,sep=";")
2 > u=seq (.05 ,.95 , by =.01)
4 > coefstd=function(u) summary(rq(WEIGHT˜SEX+SMOKER+WEIGHTGAIN+
BIRTHRECORD+AGE+ BLACKM+ BLACKF+COLLEGE ,data=sbase ,tau=u))$
coefficients [,2]
5 > coefest=function(u) summary(rq(WEIGHT˜SEX+SMOKER+WEIGHTGAIN+
BIRTHRECORD+AGE+ BLACKM+ BLACKF+COLLEGE ,data=sbase ,tau=u))$
coefficients [,1]
6 CS=Vectorize(coefstd)(u)
7 CE=Vectorize(coefest)(u)

See Abreveya (2001) The eﬀects of demographics and maternal behavior on the
distribution of birth outcomes
20 40 60 80
−6−4−20246
AGE
20 40 60 80
708090100110120130140
SEXM
20 40 60 80
−200−180−160−140−120
SMOKERTRUE
20 40 60 80
3.54.04.5
WEIGHTGAIN
20 40 60 80
20406080
COLLEGETRUE

See Abreveya (2001) The eﬀects of demographics and maternal behavior...
1 > base=read.table("http:// freakonometrics .free.fr/BWeight.csv")
20 40 60 80
−202468
mom_age
20 40 60 80
406080100120140
boy
20 40 60 80
−190−180−170−160−150−140
smoke
20 40 60 80
−350−300−250−200−150
black
20 40 60 80
−10−505
ed

Quantile Regression, with Non-Linear Eﬀects
Rents in Munich, as a function of the area, from Fahrmeir et al. (2013)
Regression: Models, Methods and Applications
1 > base=read.table("http:// freakonometrics .free.fr/rent98_00. txt")
50 100 150 200 250
050010001500
Area (m2
)
Rent(euros)
50%
10%
25%
75%
90%
50 100 150 200 250
050010001500
Area (m2
)
Rent(euros)
50%
10%
25%
75%
90%

Rents in Munich, as a function of the year of construction, from Fahrmeir et al.
(2013) Regression: Models, Methods and Applications
1920 1940 1960 1980 2000
050010001500
Year of Construction
Rent(euros)
50%
10%
25%
75%
90%
1920 1940 1960 1980 2000
050010001500
Year of Construction
Rent(euros)
50%
10%
25%
75%
90%

BMI as a function of the age, in New-Zealand, from Yee (2015) Vector Generalized
Linear and Additive Models, for Women and Men
1 > library(VGAMdata); data(xs.nz)
20 40 60 80 100
15202530354045
Age (Women, ethnicity = European)
BMI
5%
25%
50%
75%
95%
20 40 60 80 100
15202530354045
Age (Men, ethnicity = European)
BMI 5%
25%
50%
75%
95%

BMI as a function of the age, in New-Zealand, from Yee (2015) Vector Generalized
Linear and Additive Models, for Women and Men
20 40 60 80 100
15202530354045
Age (Women)
BMI
50%
95%
50%
95%
Maori
European
20 40 60 80 100
15202530354045
Age (Men)
BMI
50%
95%
Maori
European
50%
95%

One can consider some local polynomial quantile regression, e.g.
min
n
i=1
ωi(x)Rq
τ yi − β0 − (xi − x)T
β1
for some weights ωi(x) = H−1
K(H−1
(xi − x)), see Fan, Hu & Truong (1994)
Robust Non-Parametric Function Estimation.

Asymmetric Maximum Likelihood Estimation
Introduced by Efron (1991) Regression percentiles using asymmetric squared error
loss. Consider a linear model, yi = xT
i β + εi. Let
S(β) =
n
i=1
Qω(yi − xT
i β), where Qω( ) =



2
if ≤ 0
w 2
if > 0
where w =
ω
1 − ω
One might consider ωα = 1 +
zα
ϕ(zα) + (1 − α)zα
where zα = Φ−1
(α).
Efron (1992) Poisson overdispersion estimates based on the method of asymmetric
maximum likelihood introduced asymmetric maximum likelihood (AML)
estimation, considering
S(β) =
n
i=1
Qω(yi − xT
i β), where Qω( ) =



D(yi, xT
i β) if yi ≤ xT
i β
wD(yi, xT
i β) if yi > xT
i β
where D(·, ·) is the deviance. Estimation is based on Newton-Raphson (gradient
descent).

Noncrossing Solutions
See Bondell et al. (2010) Non-crossing quantile regression curve estimation.
Consider probabilities τ = (τ1, · · · , τq) with 0 < τ1 < · · · < τq < 1.
Use parallelism : add constraints in the optimization problem, such that
xT
i βτj
≥ xT
i βτj−1
∀i ∈ {1, · · · , n}, j ∈ {2, · · · , q}.

Quantile Regression on Panel Data
In the context of panel data, consider some fixed effect, αi so that
yi,t = xT
i,tβτ + αi + εi,t where Qτ (εi,t|Xi) = 0
Canay (2011) A simple approach to quantile regression for panel data suggests an
estimator in two steps,
• use a standard OLS fixed-effect model yi,t = xT
i,tβ + αi + ui,t, i.e. consider a
within transformation, and derive the fixed effect estimate β
(yi,t − yi) = xi,t − xi,t
T
β + (ui,t − ui)
• estimate fixed effects as αi =
1
T
T
t=1
yi,t − xT
i,tβ
• finally, run a standard quantile regression of yi,t − αi on xi,t’s.
See rqpd package.

Quantile Regression with Fixed Effects (QRFE)
In a panel linear regression model, yi,t = xT
i,tβ + ui + εi,t,
where u is an unobserved individual specific effect.
In a fixed effects models, u is treated as a parameter. Quantile Regression is
min
β,u



i,t
Rq
α(yi,t − [xT
i,tβ + ui])



Consider Penalized QRFE, as in Koenker & Bilias (2001) Quantile regression for
duration data,
min
β1,··· ,βκ,u



k,i,t
ωkRq
αk
(yi,t − [xT
i,tβk + ui]) + λ
i
|ui|



where ωk is a relative weight associated with quantile of level αk.

Quantile Regression with Random Eﬀects (QRRE)
Assume here that yi,t = xT
i,tβ + ui + εi,t
=ηi,t
.
Quantile Regression Random Eﬀect (QRRE) yields solving
min
β



i,t
Rq
α(yi,t − xT
i,tβ)



which is a weighted asymmetric least square deviation estimator.
Let Σ = [σs,t(α)] denote the matrix
σts(α) =



α(1 − α) if t = s
E[1{εit(α) < 0, εis(α) < 0}] − α2
if t = s
If (nT)−1
XT
{In ⊗ ΣT ×T (α)}X → D0 as n → ∞ and (nT)−1
XT
Ωf X = D1,
then √
nT β
Q
(α) − βQ
(α)
L
−→ N 0, D−1
1 D0D−1
1 .

Quantile Treatment Effects
Doksum (1974) Empirical Probability Plots and Statistical Inference for Nonlinear
Models introduced QTE - Quantile Treatement Effect - when a person might have
two Y ’s : either Y0 (without treatment, D = 0) or Y1 (with treatement, D = 1),
δτ = QY1 (τ) − QY0 (τ)
which can be studied on the context of covariates.
Run a quantile regression of y on (d, x),
y = β0 + δd + xT
i β + εi : shifting effect
y = β0 + xT
i β + δd + εi : scaling effect
−4 −2 0 2 4
0.00.20.40.60.81.0

Quantile Regression for Time Series
Consider some GARCH(1,1) ﬁnancial time series,
yt = σtεt where σt = α0 + α1 · |yt−1| + β1σt−1.
The quantile function conditional on the past - Ft−1 = Y t−1 - is
Qy|Ft−1
(τ) = α0F−1
ε (τ)
˜α0
+ α1F−1
ε (τ)
˜α1
·|yt−1| + β1Qy|Ft−2
(τ)
i.e. the conditional quantile has a GARCH(1,1) form, see Conditional
Autoregressive Value-at-Risk, see Manganelli & Engle (2004) CAViaR: Conditional
Autoregressive Value at Risk by Regression Quantiles

Quantile Regression for Spatial Data
1 > library(McSpatial)
2 > data(cookdata)
3 > fit <- qregcpar(LNFAR˜DCBD , nonpar=˜LATITUDE+LONGITUDE , taumat=c
(.10 ,.90) , kern="bisq", window =.30 , distance="LATLONG", data=
cookdata)
10% Quantiles
−2.0
−1.5
−1.0
−0.5
0.0
0.5
90% Quantiles
−2.0
−1.5
−1.0
−0.5
0.0
0.5
Difference between .10 and.90 Quantiles
0.5
0.6
0.7
0.8
0.9
1.0

Expectile Regression
Quantile regression vs. Expectile regression, on the same dataset (cars)
20 40 60 80
23456
Slope(quantileregression)
20 40 60 80
23456
Slope(expectileregression)
see Koenker (2014) Living Beyond our Means for a comparison quantiles-expectiles

Expectile Regression
Solve here min
β
n
i=1
Re
τ (yi − xT
i β) where Re
τ (u) = u2
· τ − 1(u < 0)
“this estimator can be interpreted as a maximum likelihood estimator when the
disturbances arise from a normal distribution with unequal weight placed on
positive and negative disturbances” Aigner, Amemiya & Poirier (1976)
Formulation and Estimation of Stochastic Frontier Production Function Models.
See Holzmann & Klar (2016) Expectile Asymptotics for statistical properties.
Expectiles can (also) be related to Breckling & Chambers (1988) M-Quantiles.
Comparison quantile regression and expectile regression, see Schulze-Waltrup et
al. (2014) Expectile and quantile regression - David and Goliath?

Expectile Regression, with Linear Eﬀects
Zhang (1994) Nonparametric regression expectiles
50 100 150 200 250
050010001500
Area (m2
)
Rent(euros)
50%
10%
25%
75%
90%
50 100 150 200 250
050010001500
Area (m2
)
Rent(euros)
50%
10%
25%
75%
90%
Quantile Regressions Expectile Regressions

Expectile Regression, with Non-Linear Eﬀects
See Zhang (1994) Nonparametric regression expectiles
50 100 150 200 250
050010001500
Area (m2
)
Rent(euros)
50%
10%
25%
75%
90%
50 100 150 200 250
050010001500
Area (m2
)
Rent(euros)
50%
10%
25%
75%
90%
Quantile Regressions Expectile Regressions

Expectile Regression, with Linear Eﬀects
2 > coefstd=function(u) summary(expectreg.ls(WEIGHT˜SEX+SMOKER+
WEIGHTGAIN+BIRTHRECORD+AGE+ BLACKM+ BLACKF+COLLEGE ,data=sbase ,
expectiles=u,ci = TRUE))[,2]
3 > coefest=function(u) summary(expectreg.ls(WEIGHT˜SEX+SMOKER+
WEIGHTGAIN+BIRTHRECORD+AGE+ BLACKM+ BLACKF+COLLEGE ,data=sbase ,
expectiles=u,ci = TRUE))[,1]
4 > CS=Vectorize(coefstd)(u)
5 > CE=Vectorize(coefest)(u)

Expectile Regression, with Random Eﬀects (ERRE)
Quantile Regression Random Eﬀect (QRRE) yields solving
min
β



i,t
Re
α(yi,t − xT
i,tβ)



One can prove that
β
e
(τ) =
n
i=1
T
t=1
ωi,t(τ)xitxT
it
−1 n
i=1
T
t=1
ωi,t(τ)xityit ,
where ωit(τ) = τ − 1(yit < xT
itβ
e
(τ)) .

Expectile Regression with Random Eﬀects (ERRE)
If W = diag(ω11(τ), . . . ωnT (τ)), set
W = E(W), H = XT
WX and Σ = XT
E(WεεT
W)X.
and then
√
nT β
e
(τ) − βe
(τ)
L
−→ N(0, H−1
ΣH−1
),
see Barry et al. (2016) Quantile and Expectile Regression for random eﬀects model.
See, for expectile regressions, with R,
2 > fit <- expectreg.ls(rent_euro ˜ area , data=munich , expectiles =.75)
3 > fit <- expectreg.ls(rent_euro ˜ rb(area ,"pspline"), data=munich ,
expectiles =.75)

Application to Real Data

Extensions
The mean of Y is ν(FY ) =
+∞
−∞
ydFY (y)
The quantile of level τ for Y is ντ (FY ) = F−1
Y (τ)
More generaly, consider some functional ν(F) (Gini or Theil index, entropy, etc),
see Foresi & Peracchi (1995) The Conditional Distribution of Excess Returns
Can we estimate ν(FY |x) ?
Firpo et al. (2009) Unconditional Quantile Regressions suggested to use inﬂuence
function regression
Machado & Mata (2005) Counterfactual decomposition of changes in wage
distributions and Chernozhukov et al. (2013) Inference on counterfactual
distributions suggested indirect distribution function.
Inﬂuence function of index ν(F) at y is
IF(y, ν, F) = lim
ε↓0
ν((1 − )F + δy) − ν(F)

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