Quantile and Expectile Regression
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The document summarizes quantile and expectile regression models. It defines quantiles and expectiles, and how they are estimated from sample data. It also discusses quantile and expectile regression, including extensions for fixed effects and random effects panels. Key points include: - Quantiles minimize an asymmetric absolute loss function, while expectiles minimize an asymmetric squared loss function. - Quantile regression parameters are estimated by minimizing the weighted sum of losses. Expectile regression parameters are estimated similarly using weighted squared losses. - Panel data models include penalized quantile regression with fixed effects, and quantile/expectile regression with random effects and their asymptotic properties.
![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
@freakonometrics 4](https://image.slidesharecdn.com/causal-esc-161130220221/85/Quantile-and-Expectile-Regression-4-320.jpg)
![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 )|]
@freakonometrics 6](https://image.slidesharecdn.com/causal-esc-161130220221/85/Quantile-and-Expectile-Regression-6-320.jpg)
![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.
@freakonometrics 7](https://image.slidesharecdn.com/causal-esc-161130220221/85/Quantile-and-Expectile-Regression-7-320.jpg)
![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 .
@freakonometrics 8](https://image.slidesharecdn.com/causal-esc-161130220221/85/Quantile-and-Expectile-Regression-8-320.jpg)
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 @freakonometrics 1
- 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 @freakonometrics 2
- 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)) . @freakonometrics 3
- 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 @freakonometrics 4
- 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)) . @freakonometrics 5
- 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 )|] @freakonometrics 6
- 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. @freakonometrics 7
- 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 . @freakonometrics 8
- 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))|. @freakonometrics 9
- 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 ). @freakonometrics 10
- 11. Arthur CHARPENTIER - Quantile and Expectile Regression Models Application to Real Data @freakonometrics 11