Quantile regression in varying coefficient models

Transcription

1 Quantile regression in varying coefficient models Irène Gijbels KU Leuven Department of Mathematics and Leuven Statistics Research Centre Belgium July 6, 2017 Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

2 Introduction: mean and quantile regression unconditional mean and quantile Y random variable, with cumulative distribution function F Y mean or expectation of Y : E(Y ) = argmin E(Y a) 2 a quantiles of Y, with the median as a special case for 0 < τ < 1, the τ-th order quantile of Y q τ (Y ) = F 1 Y (τ) = inf{y : F Y (y) τ} Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

3 Introduction: mean and quantile regression q τ (Y ) = F 1 Y (τ) = inf{y : F Y (y) τ} F(x) 0 1/7 2/7 3/7 4/7 5/7 6/7 1 F 1 (p) /7 2/7 3/7 4/7 5/7 6/7 1 x Y Exponential(λ) (F Y strictly increasing) F Y (y) = 1 e λy (y > 0) F 1 τ) Y (τ) = ln(1 λ p (0 < τ < 1)) mean E(Y ) one characteristic of the distribution of Y quantile function F 1 Y (τ) fully characterizes Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

4 Introduction: mean and quantile regression E(Y ) = argmin a E(Y a) 2 q τ (Y ) = argmin a E [ρ τ (Y a)] q 0.5 (Y ) = argmin a E Y a with ρ τ ( ) the so-called check function, or the pinball loss function,... ρ τ (z) = { τ z if z > 0 (1 τ) z otherwise ρ τ(u) Rho_0.5 Rho_ u Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

5 Introduction: mean and quantile regression estimation of unconditional mean and quantile Y 1,..., Y n i.i.d. sample from Y E(Y a) 2 1 n n (Y i a) 2 i=1 population quantity mean E(Y ) = argmin a E(Y a) 2 argmin a 1 n estimator sample mean n (Y i a) 2 = 1 n i=1 non robust to outliers n i=1 Y i quantile / median sample quantile / median 1 n q τ (Y ) = argmin a E [ρ τ (Y a)] argmin a ρ τ (Y i a) n i=1 robust to outliers Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

6 Introduction: mean and quantile regression conditional mean and quantile / mean and quantile regression Y response variable X (1),..., X (p) covariates multiple linear regression model: Y = β 0 + β 1 X (1) β p X (p) + ε mean regression function p E(Y X (1),..., X (p) ) = β 0 + β j X (j) + E (ε X (1),..., X (p)) j=1 E ( ε X (1),..., X (p)) = 0 = E(Y X(1),..., X (p) ) = β 0 + β = (β 0, β 1,..., β p ) T argmin β estimation by least squares method E ( Y β 0 p β j X (j) j=1 p β j X (j)) 2 Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43 j=1

7 Introduction: mean and quantile regression quantile regression function Y = β 0 + β 1 X (1) β p X (p) + ε τth conditional quantile of Y given X (1),..., X (p) q τ (Y X (1),..., X (p) ) = β 0 + d j=1 β j X j +F 1 ε X (1),...,X (p) (τ X (1),..., X (p) ) if F 1 (τ X (1),..., X (p) ) = F 1 ε X (1),...,X (p) ε (τ), then p p q τ (Y X (1),..., X (p) ) = β0 + Fε 1 (τ) + β }{{} j X (j) = β0 τ + β j X (j) =β0 τ j=1 j=1 p β τ = (β0 τ, β 1,..., β p ) T argmin β τ E ρ τ Y β0 τ β j X (j) j=1 Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

8 Introduction: mean and quantile regression multiple linear regression model: Y = β 0 + β 1 X (1) β p X (p) + ε mean regression quantile regression homoscedastic regression homoscedastic setting E ( ε X (1),..., X (p)) = 0 Var ( ε X (1),..., X (p)) = σ 2 (τ X (1),..., X (p) )= F 1 (τ) ε X (1),...,X (p) F 1 ε heteroscedastic regression heteroscedastic setting ε = σ ( X (1),..., X (p)) ε ε = σ ( X (1),..., X (p)) ε σ(,..., ) 0 E (ε) = 0 Var (ε) = 1 identifiability assumption(s) Var ( ε X (1),..., X (p)) ( = σ 2 X (1),..., X (p)) } {{ } to be estimated F 1 (τ X (1),..., X (p) ) ε X (1),...,X (p) = σ (X (1),..., X (p)) Fε 1 (τ) }{{}}{{} possibly to be estimated to be estimated Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

9 Introduction: mean and quantile regression multiple linear regression model: Y = β 0 + β 1 X (1) β p X (p) + V ( X (1),..., X (p)) ε V 0 location-scale model examples heteroscedasticity structures: ) ( λ V (X (1),..., X (p) = σ 1 + γ 1X (1) γ px ) (p) (Box and Hill (1974),...) ) V (X (1),..., X (p) = σg(x (1),..., X (p), λ, γ 1,..., γ p) G known function Bianco et al. (2000), Bianco & Boente (2002),... ) ln V (X (1),..., X (p) = γ 0 + γ 1X (1) γ px (p) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

10 Introduction: mean and quantile regression example: Y = β 0 + β 1 X + V (X)ε ε independent of X V (X) 0 V (X) = γ γ > 0 q τ (Y X) = β 0 + β 1 X + γ Fε 1 (τ) = β0 τ + β 1X V (X) = γx 2 q τ (Y X) = β 0 + β 1 X + γx 2 F 1 ε (τ) = β 0 + β 1 X + γ τ X 2 V (X) = 1 + γx 2 q τ (Y X) = β 0 + β 1 X + (1 + γx 2 ) F 1 ε (τ) = β τ 0 + β 1X + γ τ X 2 Koenker & Bassett (1978),..., Koenker (2005),... Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

11 Introduction: mean and quantile regression Y = β 0 + β 1 X + V (X)ε ε N(0; 1) β 0 = 2 β 1 = 0.5 γ = 1 V (X) = γ V (X) = γ X 2 V (X) = 1 + γ X 2 q τ (Y X) = β τ 0 + β 1X q τ (Y X) = β 0 + β 1 X + γ τ X 2 q τ (Y X) = β τ 0 + β 1X + γ τ X 2 quantile function quantile function quantile function x x x Figure: Examples of regression quantiles. τ = Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

12 Introduction: mean and quantile regression example: esterase data data set involves 113 assays for the concentration of the enzyme esterase Y = the number of bindings counted by a radio-immunoassay procedure X = observed esterase concentration 1400 Quantile curves 1400 local linear Quantile curves data points global linear linear fits RIA Count RIA Count Esterase Concentration Esterase Concentration τ = Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

13 Introduction: mean and quantile regression example: triceps skinfold measurements triceps skinfold measurements of 892 girls and women up to age 50, recorded in three Gambian villages during the dry season of 1989 Y = skinfold measurement X = age of the girl/woman 40 Quantile curves Triceps Skinfold Age (yr) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

14 Introduction: mean and quantile regression Y = β 0 + β 1 X (1) β p X (p) +error term }{{} signal part several important modeling and inference questions is the model specification about the signal part correct? is the signal part a linear function of the covariates? is there a parametric function that adequately describes the signal part? is there any structure in the error term? if yes, which heteroscedastic structure? do all covariates have a significant impact on the conditional quantile function q τ (Y X (1),..., X (p) )? how to select variables that have an impact on the quantile functions and/or on the heteroscedastic error structure? robust methods? Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

15 Introduction: mean and quantile regression other issues... preventing for crossing estimated quantile functions response Y covariate X F Y X : conditional distribution function of Y X theoretically: quantile curves should not cross each other: indeed, for any given conditional distribution F Y X ( x), with x fixed, and with 0 < τ 1 τ 2 < 1, and hence {y : F Y X (y x) τ 1 } {y : F Y X (y x) τ 2 } q τ1 (x) = inf{y : F Y X (y x) τ 1 } inf{y : F Y X (y x) τ 2 } = q τ2 (x) x in practice: estimated quantile curves can cross adaptations are needed to prevent this from happening... Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

16 Introduction: mean and quantile regression methods to prevent crossing estimated quantile curves random vector (Y, X) Y = β 0 + β 1 X + ε (Y 1, X 1 ),... (Y n, X n ) i.i.d. observations from (Y, X) n quantile regression estimation: solve min (β τ 0,β 1 ) ρ τ (Y i β0 τ β 1 X i ) investigated approaches: i=1 simultaneous quantile objective function estimation consider 0 < τ 1 < τ 2 <... τ H 1 < τ H < 1; solve min (β τ 1 0,βτ 2 0,...,βτ H 0,β 1 ) H n h=1 i=1 ρ τh (Y i β τ h 0 β 1 X i ) and introduce constraints to impose non-crossing composite quantile regression; Zou & Yuan (2008), Bondell et al. (2010), Schnabel & Eilers (2013, quantile sheet), Jiang et al. (2016), Yang & Liu (2016),... Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

17 Introduction: mean and quantile regression weighted simultaneous quantile objective function estimation: solve min (β τ 1 0,βτ 2 0,...,βτ H 0,β 1 ) H n W h h=1 i=1 ρ τh (Y i β τ h 0 β 1 X i ) (how to choose the weights? preferably in a data-driven way)... Koenker (2004, 2011), Lamarche (2011), Alexander et al. (2011), Zhao & Xiao (2014), Jiang et al. (2014),... stepwise procedure (Wu & Liu (2009)) approach of He (1997), in linear quantile regression settings Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

18 Flexible quantile regression modeling varying coefficient regression model multiple linear regression model: Y = β 0 + β 1 X (1) β p X (p) + ε complex data flexible modelling varying coefficient regression model: Y (T) = β 0 (T) + β 1 (T)X (1) (T) β p (T)X (p) (T) + ε(t) (Y (T ), X (1) (T ),..., X (p) (T ), T ) random vector Hastie & Tibshirani (1993), Hoover et al. (1998),..., Honda (2004), Kim (2006),..., Wang et al. (2008),..., Antoniadis, G. & Verhasselt (2012a), Andriyana (2015), Xie et al. (2015),... other flexible models: additive models, single-index models,... Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

19 Flexible quantile regression modeling additive quantile regression: q τ (Y X (1),..., X (p) ) = β 0 + p j=1 g j ( X (j)) β 0 parameter; g 1 ( ),..., g p ( ) unknown univariate functions Yu & Lu (2004), Koenker (2011),... partially linear additive quantile regression: for 1 q < p p q τ (Y X (1),..., X (p) ) = β 0 + β 1 X (1) β q X (q) + j=q+1 β 0,..., β q parameters; g q+1 ( ),..., g p ( ) univariate functions Sherwood & Wang (2016),... single index quantile regression: q τ (Y X (1),..., X (p) ) = g 0 ( β0 + β 1 X (1) β p X (p)) β 0,..., β p parameters; g 0 ( ) univariate functions g j ( X (j)) Choudhuri et al. (1997), Wu et al. (2010), Kong & Xia (2012), Ma & He (2016), Christou & Akritad (2016), Zhao et al. (2017),... Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

20 general setting Quantile regression in varying coefficient models Y (T ) = β 0 (T ) + β 1 (T )X (1) (T ) + + β p (T )X (p) (T ) + ε = X T (T )β(t ) + V (X(T ), T ) ε where ε is independent of (X(T ), T ) X(t) = ( 1, X (1) (t),..., X (p) (t) ) T (extended) covariate vector at time t β(t) = (β 0 (t), β 1 (t),..., β p (t)) T vector of (p + 1) unknown univariate regression coefficients at time t β 0 (t) is the baseline effect special settings: V (X(T ), T ) = V (T ) V (X(T ), T ) = V a constant simple heteroscedastic setting homoscedastic setting assumptions to ensure identifiability needed in all settings Andriyana (2015), Andriyana & G. (2017), Andriyana et al. (2017),... Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

21 Quantile regression in varying coefficient models heteroscedastic varying coefficient models Y (T ) = β 0 (T ) + β 1 (T )X (1) (T ) β p (T )X (p) (T ) + V (X(T ), T ) ε V (X(T ), T ) = γ 0 (T ) + γ 1 (T )X (1) (T ) γ p (T )X (p) (T ) 0 from the model and the error structure: Y (T ) = X T (T )β(t ) + X T (T )γ(t ) ε where β(t) = (β 0 (t), β 1 (t),..., β p (t)) T and γ(t) = (γ 0 (t), γ 1 (t),..., γ p (t)) T denote: F 1 ε (τ) = inf {u : F ε (u) τ} τ-th conditional quantile can be re-written as q τ (Y (t) X(t), t) = X T (t)β(t) + X T (t)γ(t) Fε 1 (τ) = X T (t) [ β(t) + γ(t)fε 1 (τ) ] Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

22 Quantile regression in varying coefficient models general heteroscedastic varying coefficient model Y (T ) = β 0 (T ) + β 1 (T )X (1) (T ) β p (T )X (p) (T ) + V (X(T ), T ) ε V (X(T ), T ) = γ 0 (T ) + γ 1 (T )X (1) (T ) γ p (T )X (p) (T ) q τ (Y (t) X(t), t) = X T (t) [ β(t) + γ(t)fε 1 (τ) ] = X T (t)β τ (t) aim: estimate the conditional quantile estimated quantile curves should not cross each other estimate β 0 (t), β 1 (t),..., β p (t) estimate γ 0 (t), γ 1 (t),..., γ p (t), and variability function V (, ) simple heteroscedastic model: Y (T ) = β 0 (T ) + β 1 (T )X (1) (T ) β p (T )X (p) (T ) + V (T ) ε q τ (Y (t) X(t), t) = X T (t)β(t) + V (t) Fε 1 (τ) = X T (t)β τ (t) where β τ (t) = (β τ 0 (t), β 1(t),..., β p (t)) T, with now β0 τ (t) = β 0(t) + V (t)fε 1 (τ) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

23 Quantile regression in varying coefficient models observational setting: longitudinal data setup n independent subjects/individuals for each individual i: measurements repeated over a time period measurements at time points t i1,..., t ini N i different measurements for response and all explanatory variables: Y (t ij ) = Y ij X (k) (t ij ) = X (k) ij k = 1,..., p = X(t ij ) not. = X ij = (1, X (1) ij,..., X(p) ij )T total number of observations over all individuals: N = n i=1 N i Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

24 Real data example CD4 data example the data are a subset from the Multicenter AIDS Cohort Study (Kaslow et al. (1987)) 283 homosexual men who became HIV-positive between 1984 and 1991 data set contains repeated measurements of physical examinations, laboratory results, CD4 cell counts and CD4 percentages of each individual unequal numbers of repeated measurements and different measurement times for each individual number of repeated measurements ranges from 1 to 14 the number of distinct time points was 59 Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

25 Real data example response variable : Y (t) = CD4 percentage at time t after infection covariates: X (1) i the smoking status of the i-th individual (1 or 0 if the individual ever or never smoked cigarettes) X (2) i X (3) i the centered age at HIV infection for the i-th individual the centered pre-infection CD4 percentage aim: try to evaluate the median effect of cigarette smoking, pre-hiv infection CD4 cell percentage and age at HIV infection on the CD4 percentage after infection response Y (t) = β 0 (t) + β 1 (t)x (1) + β 2 (t)x (2) + β 3 (t)x (3) + V(t) ε Y (t) = V(X(t), t) ε Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

26 Real data example baseline CD Fitted coefficient function coefficient of smoking status Fitted coefficient function 95%CI Percentile method 95%CI Percentile method time since infection time since infection coefficient of age Fitted coefficient function coefficient of pre infection CD Fitted coefficient function 95%CI Percentile method 95%CI Percentile method time since infection time since infection Figure: τ = 0.5, γ = 1, estimated coefficient fcts, 95% bootstrap CI. Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

27 Real data example time since infection CD4 percentage after infection τ = 0.5 τ = 0.1 τ = time since infection CD4 percentage after infection τ = 0.5 τ = 0.1 τ = 0.9 Figure: Estimated quantile curves: τ = 0.1 (dashed curves), τ = 0.5 (solid curves) and τ = 0.9 (dotted curves) for (left) median and (right) maximum of covariate values. median covariate case: nonsmoking, 32.6 years old patient, with pre-infection CD4 of 42.3% τ = 0.5: estimated to have a CD4 percentage of 24.37% after 6 years Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

28 Real data example Example: CD4 data CD4 percentage after infection τ = 0.95 τ = 0.75 τ = 0.5 τ = 0.25 τ = 0.05 CD4 percentage after infection τ = 0.95 τ = 0.75 τ = 0.5 τ = 0.25 τ = time since infection (a) Individual time since infection (b) AHe approach. Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

29 Real data example time since infection V^(.) o V^(X(t), t) V^(t) Figure: Estimated variability functions. seems that V (T ) constant seems that V (X(T ), T ) = V (T ) suffices (simple heteroscedastic structure) R package QRegVCM (Yudhie Andriyana) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

30 Quantile estimation in varying coefficient models varying coefficient models Y (T) = β 0 (T) + β 1 (T)X (1) (T) β p (T)X (p) (T) + ε(t) ε(t ) independent of (X (1) (T ),..., X (p) (T )) aim: based on longitudinal data (Y ij, X ij ), i = 1,..., n; j = 1,..., N i, estimate τth conditional quantile function (0 < τ < 1) q τ (Y (t) X(t), t) = β τ 0 (t) {}}{ β 0 (t) +β 1 (t)x (1) (t) β p (t)x (p) (t) approximate each univariate function by normalized B-splines... use penalization to prevent from overfitting... Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

31 Quantile estimation in varying coefficient models approximate each univariate function β k ( ) by normalized B-splines... coefficient functions β k ( ): can be of different degree of smoothness; given knots; B-splines are polynomial pieces of degree ν k joined together at each knot point; for k = 0,..., p, approximate β k (t) β k (t) α k1 B k1 (t; ν k ) α kmk B kmk (t; ν k ) = α k = (α k1,..., α kmk ) T m k = u k + ν k normalized B-splines: m k α kl B kl (t; ν k ) l=1 = α T k B k(t; ν k ) B k (t; ν k ) = (B k1 (t; ν k ),..., B kmk (t; ν k )) T u k + 1 = number of equidistant knot points m k B kl (t; ν k ) = 1 l=1 Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

32 Quantile estimation in varying coefficient models estimation of global vector of all unknown coefficients α = ( α T 0,..., ) T αt p quality of the fit measured via the goodness-of-fit quantity n 1 N i N i=1 i j=1 ) p m k ρ τ (Y ij α kl B kl (t ij ; ν k )X (k) ij k=0 l=1 reducing the modelling bias: use a large number of basis functions but this leads to overfitting... prevent this to happen by adding a penalty term Eilers & Marx (1996),... Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

33 Quantile estimation in varying coefficient models... adding a penalty term: minimize ) n N 1 i p m k ρ τ (Y ij α kl B kl (t ij ; ν k )X (k) ij + N i=1 i j=1 k=0 l=1 p m k k=0 l=d k +1 λ k d k α kl γ where γ > 0 (for example γ = 1) λ k > 0, k = 0,..., p : smoothing (regularization) parameters d k = the d k th order differencing operator of the kth variable, i.e d kα kl = d k t=0 ( 1)t( d kt ) αk(l t) with d k IN d k = 1 : d k = 2 : α kl = α kl α k,l 1 2 α kl = ( α kl ) = α kl α k,l 1 = 1 α kl 2α k,l α k,l 2 denote by α k the resulting P-splines estimator for the vector α k, k = 0,..., p estimator for the τ th conditional quantile function? Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

34 Quantile estimation in varying coefficient models q τ (Y (t) X(t), t) = = β τ 0 {}} (t) { β 0 (t) +β 1 (t)x (1) (t) β p (t)x (p) (t) d m k α kl B kl (t ij ; ν k )X (k) ij k=0 l=1 p α T kb k (t; ν k )X (k) (t) = X T (t)b(t)α k=0 B 01 (t, ν 0 )... B 0m0 (t, ν 0 ) B(t) = B p1 (t, ν p)... B pmp (t, ν p) P-splines estimator for β k (t) and and the conditional regression quantile : β k (t) = m k l=1 α kl B kl (t; ν k ) q τ (Y ij X ij, t ij ) = p m k k=0 l=1 α kl B kl (t ij ; ν k )X (k) ij Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

35 Quantile estimation in varying coefficient models simple heteroscedastic varying coefficient model q τ (Y (t) X(t), t) = X T (t)β(t) + V (t) Fε 1 (τ) = X T (t) [ β(t) + V (t)fε 1 (τ) ] general heteroscedastic varying coefficient model q τ (Y (t) X(t), t) = X T (t) [ β(t) + γ(t) Fε 1 (τ) ] adapt methods to allow/estimate (general) heteroscedastic error structure... V (t) B-splines V (t) αl V BV l (t; νv ) m V l=1 } {{ } B V (t)α V γ 0 (t) approximated by B-splines γ 1 (t) approximated by B-splines. γ p (t) approximated by B-splines V (X(t), t) = X T (t)b v (t)γ Andriyana, G. & Verhasselt (2017), Andriyana & G. (2017) R package QRegVCM (Yudhie Andriyana) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

36 Real data example Example: UK employment data data from Datastream International (R-package plm under EmplUK) records of employment and remuneration (i.e. wage bill) for all United Kingdom quoted companies concerns 140 United Kingdom firms observed each year from (e.g. Arellano and Bond (1991)) data are unbalanced: some firms have more observations than others; observations correspond to different points in historical time range of observations per firm is seven to nine (i.e. min(n i ) = 7 and max(n i ) = 9) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

37 Real data example response variable Y (t): log(employment) at time t = log(number of employees) at time t 3 covariates: X (1) (t) : average annual wage per employee in the company (at year t) X (2) (t) : capital defined as the book value of gross fixed assets X (3) (t) : an index of value-added output at constant cost Y (t) = β 0 (t) + β 1 (t)x (1) (t) + β 2 (t)x (2) (t) + β 3 (t)x (3) (t) + V(t) ε we also adapt the AHe approach further to allow for Y (t) = β 0 (t) + β 1 (t)x (1) (t) + β 2 (t)x (2) (t) + β 3 (t)x (3) (t) + V(X(t), t) ε Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

38 Real data example coefficient of capital coefficient of capital year year (a) Coefficient of capital ( β 2 (t)). (b) Coefficient of capital ( γ 2 (t)). Figure: UK employment data. The estimated influence of the covariate capital on the main quantile function (a) and on the heteroscedasticity (b)), using the general AHe approach method. Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

39 Real data example year V^(.) o V^(X(t), t) V^(t) Figure: UK employment data. Variability estimators: V (X(t), t) and V (t). most appropriate: general heteroscedastic structure: V (X(t), t) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

40 Real data example baseline log employment coefficient of output year year (a) Basel. log-employment ( β 0 (t)). (b) Coef. of output ( β 3 (t)). coefficient of output x year (c) Coef. of output ( γ 3 (t)). Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

41 Shape testing in quantile regression of interest is to look into testing problems of the form H 0 : β k (t) = constant versus H 1 : β k (t) β k H 0 : β k (t) is monotone versus H 1 : β k (t) is not monotone H 0 : β k (t) is convex versus H 1 : β k (t) is not a convex function similar hypothesis for functional coefficients γ k (t) in the variability function and simultaneous testing problems (e.g. β k (t) is increasing and γ l (t) is constant) by using with B-spline approximations, tests for these problems can be derived G., Ibrahim & Verhasselt (2017) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

42 Shape testing in quantile regression Thank you! Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

43 Shape testing in quantile regression Andriyana, Y. (2016). R package QRegVCM Andriyana, Y. & Gijbels, I. (2017). Quantile regression in heteroscedastic varying coefficient models. AStA Advances in Statistical Analysis, 101, Andriyana, Y., Gijbels, I. & Verhasselt, A. (2014). P-splines quantile regression estimation in varying coefficient models. Test, 23, Andriyana, Y., Gijbels, I. & Verhasselt, A. (2017). Quantile regression in varying coefficient models: non-crossingness and heteroscedasticity. Statistical Papers, to appear. Antoniadis, A., Gijbels, I. & Verhasselt, A. (2012). Variable selection in varying coefficient models using P-splines. Journal of Computational and Graphical Statistics, 21, Gijbels, I., Ibrahim, M. & Verhasselt, A. (2017). Shape testing in quantile varying coefficient models with heteroscedastic error. Journal of Nonparametric Statistics, 29, Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

44 Shape testing in quantile regression AHe s approach (adaptation of idea of He (1987)) for identifiability reasons : (H1): q 0.5 (ε(t)) = 0 (i.e. a 0.5 (t) = 0) (H2): q 0.5 ( ε(t) ) = 1 Step 1. Under (H1), the median quantile function is q 0.5 (Y (t) X(t), t) = β 0 (t)x (0) (t) + β 1 (t)x (1) (t) + + β p (t)x (p) (t) = X T (t)β(t) estimated median regression curve: q 0.5 (Y (t) X(t), t) = β(t) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

45 Shape testing in quantile regression Step 2. from the model and since V (t) 0, we have Y (t) X T (t)β(t) = V (t) ε(t) using (H2) q 0.5 ( Y (t) X T (t)β(t) ) = V (t) estimate V ( ) using a P-splines estimation technique, and pseudo-observations Y (t ij ) X T (t ij ) β(t ij ) for Y (t) X T (t)β(t) V (t) αl V BV l (t; νv ) m V l=1 obtain ( α 1 V,..., αv ) by minimizing m V N n i 1 ρ 0.5 Y (tij) X T m V (t ij) β(t ij) αl V Bl V (t ij; ν V ) + N i i=1 j=1 l=1 estimated variability function : V mv (t) = l=1 m V l=d V +1 α V l BV l (t; νv ) λ V d Vα V l Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43

46 Shape testing in quantile regression Step 3. equipped with ( β 0 (t),..., β p (t)) from Step 1 and the estimator V (t) from Step 2, and recall that Y (t) X T (t)β(t) = V (t)ε(t) substitute β(t) by β(t) and V (t) by V (t) m h approximate : a τ h(t ij ) α q h,l Bq l (t ij; ν h ) minimize l=1 n 1 N i ρ τh Y (t ij ) X T m h m h (t ij ) β(t ij ) V (t ij ) α q h,l i=1 N Bq l (t ij ; ν h ) q + λ i j=1 l=1 l=d q h dq h α q h,l h +1 estimator of a τ h(t) : estimated quantile function â τ h(t) = m h l=1 αq h,l Bq l (t; ν h) q τh (Y (t) X(t), t) = X T (t) β(t) + V (t)â τ h (t) Irène Gijbels (KU Leuven) ICORS 2017, Wollongong, Australia July 6, / 43