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Quantile and Expectile Regression

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Quantile and Expectile Regression

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Quantile and Expectile Regression

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 11. Arthur CHARPENTIER - Quantile and Expectile Regression Models Application to Real Data @freakonometrics 11

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