Quantile and Expectile Regression

1. Arthur CHARPENTIER - Quantile and Expectile Regression Models
Quantile and Expectile Regression Models
A. Charpentier (Université de Rennes 1)
with A.D. Barry & K. Oualkacha (UQàM)
ESC Rennes, December 2016.
http://freakonometrics.hypotheses.org
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2. Arthur CHARPENTIER - Quantile and Expectile Regression Models
Mediane/Quantiles ( 1-norm)
Empirical median m(y) is solution of
m(y) = argmin
θ ∈ R
1
n
n
i=1
1
2
|yi − θ|
=rQ
1/2
(yi−θ)
.
Empirical quantile q(α, y) is solution of
q(α, y) = argmin
θ ∈ R
1
n
n
i=1
rQ
α (yi − θ) ,
with rQ
α (u) = |α − 1(u ≤ 0)| · |u|.
−2 −1 0 1 2
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3. Arthur CHARPENTIER - Quantile and Expectile Regression Models
Quantiles
Consider Y ∼ F, and a level α ∈ (0, 1), then q(α, Y ) = inf{y; FY (y) ≥ α}.
Equivalently
q(α, Y ) = argmin
θ ∈ R
E αQ
(Y − θ) , with rQ
α (u) = |α − 1(u ≤ 0)| · |u|
The empirical version, with a sample y = {y1, · · · , yn}, is
q(α, y) = argmin
θ ∈ R
1
n
n
i=1
rQ
α (yi − θ) .
The conditional α-quantile of Y |x is q(α, Y, x) = inf{y; FY |x(y) ≥ α}. Assuming
that F−1
Y |x(α) = xT
i βQ
(α), quantile regression parameters are obtained from
sample (y, X) = {(y1, x1), · · · , (yn, xn)} as
β
Q
(α, y, X) = argmin
β ∈ Rp
1
n
n
i=1
rQ
α (yi − xT
i βQ
(α)) .
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4. Arthur CHARPENTIER - Quantile and Expectile Regression Models
Expected value/Expectiles ( 2-norm)
Empirical meam y is solution of
y = argmin
θ ∈ R
1
n
n
i=1
1
2
[yi − θ]2
=rE
1/2
(yi−θ)
.
Empirical expectile µ(τ, y) is solution of
µ(τ, y) = argmin
θ ∈ R
1
n
n
i=1
rE
τ (yi − θ) ,
with rE
τ (u) = |τ − 1(u ≤ 0)| · u2
.
See −2 −1 0 1 2
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5. Arthur CHARPENTIER - Quantile and Expectile Regression Models
Expectiles
Consider Y ∼ F, and a level τ ∈ (0, 1),
µ(τ, Y ) = argmin
θ ∈ R
E{rE
τ (Y − θ)} with rE
τ (u) = |τ − 1(u ≤ 0)| · u2
.
The empirical version, with a sample y = {y1, · · · , yn} is
µ(τ, y) = argmin
θ ∈ R
1
n
n
i=1
rE
τ (yi − θ) .
The conditional τ-expectile of Y |x is
µ(τ, Y, x) = argmin
θ ∈ R
E{rE
τ (Y − θ)|x},
and assuming that µ(τ, x) = xTβE
(τ), parameters of the expectile regression are
β
E
(τ, y, X) = argmin
β ∈ Rp
1
n
n
i=1
rE
τ (yi − xiTβE
(τ)) .
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6. Arthur CHARPENTIER - Quantile and Expectile Regression Models
Quantiles and Expectiles
Observe that q(α, Y ) is solution of
α = F(q(α, Y )) = E[1(Y < q(α, Y ))]
while µ(τ, Y ) is solution of
τ =
E[|Y − µ(τ, Y )| · 1{Y < µ(τ, Y )}]
E[|Y − µ(τ, Y )|]
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7. Arthur CHARPENTIER - Quantile and Expectile Regression Models
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),
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.
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8. Arthur CHARPENTIER - Quantile and Expectile Regression Models
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 assymmetric 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 .
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9. Arthur CHARPENTIER - Quantile and Expectile Regression Models
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))|.
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10. Arthur CHARPENTIER - Quantile and Expectile Regression Models
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
).
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11. Arthur CHARPENTIER - Quantile and Expectile Regression Models
Application to Real Data
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