Quantile Regression for Group Effect Analysis Cristina Davino 1 - PowerPoint PPT Presentation

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Quantile Regression for Group Effect Analysis Cristina Davino 1 Domenico Vistocco 2 1 Dip.to di Studi sullo Sviluppo Economico 2 Dip.to di Scienze Economiche Università di Macerata Università di Cassino cdavino@unimc.it vistocco@unicas.it 19 th International Conference on Computational Statistics Paris, 22 – 27 August 2010 all computations and graphics were done in the R language using the packages quantreg and ggplot2 Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Outline Aim of the paper 1 QR for group effect analysis 2 Basic notation The reference framework The proposed approach An empirical analysis 3 The dataset Main results Concluding remarks 4 Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Outline Aim of the paper 1 QR for group effect analysis 2 Basic notation The reference framework The proposed approach An empirical analysis 3 The dataset Main results Concluding remarks 4 Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Outline Aim of the paper 1 QR for group effect analysis 2 Basic notation The reference framework The proposed approach An empirical analysis 3 The dataset Main results Concluding remarks 4 Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Outline Aim of the paper 1 QR for group effect analysis 2 Basic notation The reference framework The proposed approach An empirical analysis 3 The dataset Main results Concluding remarks 4 Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Aim of the paper Identification of group effects in a quantile regression model C ONFIRMATIVE APPROACH 1 R OW – PARTITIONED DATA 2 Supervised approach Unsupervised approach Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Aim of the paper Identification of group effects in a quantile regression model C ONFIRMATIVE APPROACH 1 R OW – PARTITIONED DATA 2 Supervised approach Unsupervised approach Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Aim of the paper Identification of group effects in a quantile regression model C ONFIRMATIVE APPROACH 1 R OW – PARTITIONED DATA 2 Supervised approach Unsupervised approach Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Aim of the paper Identification of group effects in a quantile regression model C ONFIRMATIVE APPROACH 1 R OW – PARTITIONED DATA 2 Supervised approach Unsupervised approach Some solutions for group effect analysis Estimation of different models for each group Introduction of a dummy variable Multilevel modeling (Gelman and Hill, 2007) Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Basic notation The data structure n : number of units p : number of regressors G : number of groups or levels X [ n × p ] g x ij ( i =1,..., n ; j =1,..., p ; g =1,... G ) y [ n ] g y i ( i =1,..., n ; g =1,... G ) n g : number of units in group g Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Classical vs quantile linear regression Classical linear regression Quantile regression (Koenker and Basset, 1978) (conditional expected value) (conditional quantiles) estimation of the conditional mean of a estimation of the conditional quantiles of a response variable (y) distribution as a response variable (y) distribution as a function of a set X of predictor variables function of a set X of predictor variables E ( y | X ) = X β Q θ ( y | X ) = X β ( θ ) where: ( 0 < θ < 1 ) β i ( θ ) = ∂ Q θ ( y ) β i = ∂ E ( y ) ∂ x i ∂ x i Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Classical vs quantile linear regression Classical linear regression Quantile regression (Koenker and Basset, 1978) (conditional expected value) (conditional quantiles) estimation of the conditional mean of a estimation of the conditional quantiles of a response variable (y) distribution as a response variable (y) distribution as a function of a set X of predictor variables function of a set X of predictor variables E ( y | X ) = X β Q θ ( y | X ) = X β ( θ ) where: ( 0 < θ < 1 ) β i ( θ ) = ∂ Q θ ( y ) β i = ∂ E ( y ) ∂ x i ∂ x i Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Classical vs quantile linear regression Classical linear regression Quantile regression (Koenker and Basset, 1978) (conditional expected value) (conditional quantiles) estimation of the conditional mean of a estimation of the conditional quantiles of a response variable (y) distribution as a response variable (y) distribution as a function of a set X of predictor variables function of a set X of predictor variables E ( y | X ) = X β Q θ ( y | X ) = X β ( θ ) where: ( 0 < θ < 1 ) β i ( θ ) = ∂ Q θ ( y ) β i = ∂ E ( y ) ∂ x i ∂ x i Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Classical vs quantile linear regression Classical linear regression Quantile regression (conditional expected value) (conditional quantiles) estimation of the conditional mean of a estimation of the conditional quantiles of a response variable (y) distribution as a response variable (y) distribution as a function of a set X of predictor variables function of a set X of predictor variables E ( y | X ) = X β Q θ ( y | X ) = X β ( θ ) where: ( 0 < θ < 1 ) Q θ ( g y | g X ) = g X g β ( θ ) g y = g X g β + g e Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks Classical vs quantile linear regression Classical linear regression Quantile regression (conditional expected value) (conditional quantiles) estimation of the conditional mean of a estimation of the conditional quantiles of a response variable (y) distribution as a response variable (y) distribution as a function of a set X of predictor variables function of a set X of predictor variables E ( y | X ) = X β Q θ ( y | X ) = X β ( θ ) where: ( 0 < θ < 1 ) Q θ ( g y | g X ) = g X g β ( θ ) g y = g X g β + g e Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks The proposed approach Global estimation 1 Q θ ( y | X ) = X ˆ B ( θ ) Identification of the best model 2 for each unit density estimation 1 Y = X ˆ ˆ B ( θ ) best model identification 2 y i − ˆ θ i : argmin y i ( θ ) θ = 1 , Θ best density estimation 3 vector y best ˆ θ Identification of the best model 3 for each group g θ best , for g = 1 , G Partial estimation 4 Q θ ( y | X ) = X ˆ B ( θ ) best Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks The proposed approach Global estimation 1 Q θ ( y | X ) = X ˆ B ( θ ) Identification of the best model 2 for each unit density estimation 1 Y = X ˆ ˆ B ( θ ) best model identification 2 y i − ˆ θ i : argmin y i ( θ ) θ = 1 , Θ best density estimation 3 vector y best ˆ θ Identification of the best model 3 for each group g θ best , for g = 1 , G Partial estimation 4 Q θ ( y | X ) = X ˆ B ( θ ) best Davino, Vistocco Quantile Regression for Group Effect Analysis

Aim of the paper QR for group effect analysis An empirical analysis Concluding remarks The proposed approach Global estimation 1 Q θ ( y | X ) = X ˆ B ( θ ) Identification of the best model 2 for each unit density estimation 1 Y = X ˆ ˆ B ( θ ) best model identification 2 y i − ˆ θ i : argmin y i ( θ ) θ = 1 , Θ best density estimation 3 vector y best ˆ θ Identification of the best model 3 for each group g θ best , for g = 1 , G Partial estimation 4 Q θ ( y | X ) = X ˆ B ( θ ) best Davino, Vistocco Quantile Regression for Group Effect Analysis