Single Index Quantile Regression for Heteroscedastic Data

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Single Index Quantile Regression for Heteroscedastic Data E. Christou M. G. Akritas Department of Statistics The Pennsylvania State University JSM, 2015 E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 1 / 32

Outline 1 Introduction Motivation Possible Models 2 Single Index Quantile Regression The Proposed Estimator Main Results 3 Numerical Studies Example 1 Example 2 4 Conclusions 5 Future Work E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 2 / 32

Outline 1 Introduction Motivation Possible Models 2 Single Index Quantile Regression The Proposed Estimator Main Results 3 Numerical Studies Example 1 Example 2 4 Conclusions 5 Future Work E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 3 / 32

Introduction Motivation Least Squares Regression: Models the relationship between a d-dimensional vector of covariates X and the conditional mean of the response Y given X = x. Least Squares Estimator: 1 provides only a single summary measure for the conditional distribution of the response given the covariates. 2 sensitive to outliers and can provide a very poor estimator in many non-gaussian and especially long-tailed distributions. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 4 / 32

Introduction Motivation Quantile Regression (QR): Models the relationship between a d-dimensional vector of covariates X and the τ th conditional quantile of the response Y given X = x, Q τ (Y x). Quantile Regression: 1 provides a more complete picture for the conditional distribution of the response given the covariates. 2 useful for modeling data with heterogeneous conditional distributions, especially data where extremes are important. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 5 / 32

Outline 1 Introduction Motivation Possible Models 2 Single Index Quantile Regression The Proposed Estimator Main Results 3 Numerical Studies Example 1 Example 2 4 Conclusions 5 Future Work E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 6 / 32

Introduction Possible Models Linear QR: Q τ (Y x) = β x, for a d-dimensional vector of unknown parameters β. Disadvantage: linear assumption not always valid. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 7 / 32

Introduction Possible Models Linear QR: Q τ (Y x) = β x, for a d-dimensional vector of unknown parameters β. Disadvantage: linear assumption not always valid. Nonparametric QR: Q τ (Y x) = g(x), for an unspecified function g : R d R. Disadvantage: meets the data sparseness problem for high dimensional data. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 7 / 32

Introduction Possible Models Linear QR: Q τ (Y x) = β x, for a d-dimensional vector of unknown parameters β. Disadvantage: linear assumption not always valid. Nonparametric QR: Q τ (Y x) = g(x), for an unspecified function g : R d R. Disadvantage: meets the data sparseness problem for high dimensional data. Single-Index QR: Q τ (Y x) = g(β x β), depends on x only through the single linear combination β x. Advantage: reduces the dimension and maintain some nonparametric flexibility. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 7 / 32

Outline 1 Introduction Motivation Possible Models 2 Single Index Quantile Regression The Proposed Estimator Main Results 3 Numerical Studies Example 1 Example 2 4 Conclusions 5 Future Work E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 8 / 32

The Proposed Estimator The Model Let {Y i, X i } n i=1 be independent and identically distributed (iid) observations that satisfy Y i = Q τ (Y X i ) + ɛ i, where Q τ (Y x) = Q τ (Y β 1x) and the error term satisfies Q τ (ɛ i x) = 0. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 9 / 32

The Proposed Estimator The Model Let {Y i, X i } n i=1 be independent and identically distributed (iid) observations that satisfy Y i = Q τ (Y X i ) + ɛ i, where Q τ (Y x) = Q τ (Y β 1x) and the error term satisfies Q τ (ɛ i x) = 0. For identifiability, we impose certain conditions on β 1 : the most common one: β 1 = 1 with its first coordinate positive. the one adopted here: β 1 = (1, β ). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 9 / 32

The Proposed Estimator The Model The true parametric vector β satisfies β = arg min b E ( ρ τ (Y Q τ (Y b 1X)) ), where b 1 = (1, b ) and ρ τ (u) = (τ I (u < 0))u. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 10 / 32

The Proposed Estimator Iterative algorithms As in the single index mean regression (SIMR) problem, the unknown Q τ (Y b 1X) must be replaced with an estimator. Unlike the SIMR problem, however, there is no closed form expression for the estimator of Q τ (Y b 1 X i), and this has led to iterative algorithms for estimating β (Wu, Yu, and Yu, 2010; Kong and Xia, 2012). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 11 / 32

The Proposed Estimator The New Approach For any given b R d 1, define the function g(u b) : R R as where b 1 = (1, b ). g(u b) = E(Q τ (Y X) b 1X = u), E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 12 / 32

The Proposed Estimator The New Approach For any given b R d 1, define the function g(u b) : R R as where b 1 = (1, b ). Noting that β also satisfies where b 1 = (1, b ). g(u b) = E(Q τ (Y X) b 1X = u), g(β 1X β) = Q τ (Y X), β = arg min b E ( ρ τ (Y g(b 1X b)) ), (1) E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 12 / 32

The Proposed Estimator The New Approach The sample level version of (1) consists of minimizing S n (τ, b) = n ρ τ (Y i g(b 1X i b)). i=1 E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 13 / 32

The Proposed Estimator The New Approach The sample level version of (1) consists of minimizing S n (τ, b) = n ρ τ (Y i g(b 1X i b)). i=1 Again, g( b) is unknown but it can be estimated, in a non-iterative fashion, by first obtaining estimators Q τ (Y X i ), for i = 1,..., n, and forming the Nadaraya-Watson-type estimator ĝ NW ˆQ (t b) = n i=1 ( ) t b Q τ (Y X i )K 1 X i 2 h 2 ( ) n k=1 K t b, (2) 1 X k 2 h 2 where K 2 ( ) is a univariate kernel function and h 2 is a bandwidth. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 13 / 32

The Proposed Estimator The New Approach For the estimator Q τ (Y X i ), we use the local linear conditional quantile estimator (Guerre and Sabbah, 2012). Specifically, for a multivariate kernel function K 1 (x) = K 1 (x 1,..., x d ) and a univariate bandwidth h 1, let L n ((α 0, α 1 ); τ, h 1, x) = 1 nh d 1 n i=1 ( ) ρ τ (Y i α 0 α Xi x 1(X i x))k 1 h 1 and define Q τ (Y x) as α 0 (τ; h 1, x), where α 0 (τ; h 1, x) is defined through ( α 0 (τ; h 1, x), α 1 (τ; h 1, x)) = arg min (α 0,α 1 ) L n((α 0, α 1 ); τ, h 1, x). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 14 / 32

The Proposed Estimator The proposed estimator is obtained by β = arg min b Θ n ( ) ρ τ Y i ĝ NW (b 1X i b), (3) i=1 where Θ R d 1 is a compact set, β is in the interior of Θ. ˆQ E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 15 / 32

Outline 1 Introduction Motivation Possible Models 2 Single Index Quantile Regression The Proposed Estimator Main Results 3 Numerical Studies Example 1 Example 2 4 Conclusions 5 Future Work E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 16 / 32

Main Results Proposition 1 Let ĝ NW (t b) be as defined in (2).Under some regularity conditions, we ˆQ have sup ĝ NW Q (t b) g(t b) ( = O p a n + a n + h2) 2, b Θ,t T b where a n = (log n/n) s/(2s+d) and a n = (log n/(nh 2 )) 1/2. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 17 / 32

Main Results Proposition 1 Let ĝ NW (t b) be as defined in (2).Under some regularity conditions, we ˆQ have sup ĝ NW Q (t b) g(t b) ( = O p a n + a n + h2) 2, b Θ,t T b where a n = (log n/n) s/(2s+d) and a n = (log n/(nh 2 )) 1/2. Proposition 2 Let β be as defined in (3). Under regularity conditions, β is n-consistent estimator of β. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 17 / 32

Main Results Theorem Let β be as defined in (3). Under regularity conditions, n( β β) d N ( 0, τ(1 τ)v 1 ΣV 1), where V = E ( (g (β 1X β)) 2 (X 1 E(X 1 β 1X))(X 1 E(X 1 β 1X)) f ɛ X (0 X) ), and Σ = E ( (g (β 1X β)) 2 (X 1 E(X 1 β 1X))(X 1 E(X 1 β 1X)) ). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 18 / 32

Outline 1 Introduction Motivation Possible Models 2 Single Index Quantile Regression The Proposed Estimator Main Results 3 Numerical Studies Example 1 Example 2 4 Conclusions 5 Future Work E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 19 / 32

Numerical Studies Notation NWQR - the proposed estimator. WYY - the estimator resulting from the iterative algorithm proposed by Wu et al. (2010). WYY-1 - the estimator resulting from dividing the estimator of Wu et al. (2010) by its first component. WYY-2 - the estimator resulting from using the proposed constraint in conjunction with the iterative algorithm of Wu et al. (2010). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 20 / 32

Numerical Studies Example 1 Consider the model Y = exp(β 1X) + ɛ, (4) where X = (X 1, X 2 ), X i N(0, 1) are iid, β 1 = (1, 2) and the residual ɛ follows a standard normal distribution. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 21 / 32

Numerical Studies Example 1 Consider the model Y = exp(β 1X) + ɛ, (4) where X = (X 1, X 2 ), X i N(0, 1) are iid, β 1 = (1, 2) and the residual ɛ follows a standard normal distribution. We fit the single index quantile regression model for five different quantile levels, τ = 0.1, 0.25, 0.5, 0.75, 0.9. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 21 / 32

Numerical Studies Example 1 Consider the model Y = exp(β 1X) + ɛ, (4) where X = (X 1, X 2 ), X i N(0, 1) are iid, β 1 = (1, 2) and the residual ɛ follows a standard normal distribution. We fit the single index quantile regression model for five different quantile levels, τ = 0.1, 0.25, 0.5, 0.75, 0.9. We use sample size of n = 400 and perform N = 100 replications. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 21 / 32

Numerical Studies Example 1 For comparison of the above estimators, consider the mean square error, R( β), and the mean check based absolute residuals, R τ ( β 1 ), which are defined as R( β) = 1 N N ( β i 2) 2, R τ ( β 1 ) = 1 n i=1 n ρ τ (Y i i=1 Q LL τ (Y β 1X i )), (5) where Q LL τ (Y β 1x) is the local linear conditional quantile estimator on the univariate variable β 1X i. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 22 / 32

Table 1: mean values and standard errors (in parenthesis), R( β) and R τ ( β 1 ) for Model (4). 95% coverage probability for NWQR, WYY-2 and the second estimated coefficient of Wu et al. (2010). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 23 / 32 Numerical Studies Example 1 τ 0.1 0.25 0.5 0.75 0.9 NWQR 2.0085 2.0132 2.0149 2.0290 2.0768 (0.0603) (0.0477) (0.0481) (0.0683) (0.3905) β 1.9521 1.9890 3.4450 1.9363 1.9592 WYY-1 (0.2521) (0.2252) (3.8756) (0.3246) (0.5906) WYY-2 1.9699 1.9899 2.0158 2.0525 2.0287 (0.1593) (0.1307) (0.3937) (0.6735) (0.7240) NWQR 0.0037 0.0024 0.0025 0.0055 0.1569 R( β) WYY-1 0.0652 0.0503 192.6956 0.1084 0.3469 WYY-2 0.0260 0.0170 0.1537 0.4518 0.5197 NWQR 0.1682 0.3096 0.3909 0.3113 0.1761 R τ ( β 1 ) WYY-1 0.1783 0.3273 0.4154 0.3479 0.2265 WYY-2 0.1720 0.3160 0.4031 0.3310 0.2050 coverage NWQR 0.99 0.97 0.97 0.97 0.95 prob. WYY-2 0.93 0.90 0.92 0.80 0.70 WYY 0.55 0.76 0.86 0.66 0.28

Outline 1 Introduction Motivation Possible Models 2 Single Index Quantile Regression The Proposed Estimator Main Results 3 Numerical Studies Example 1 Example 2 4 Conclusions 5 Future Work E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 24 / 32

Numerical Studies Boston Housing Data 506 observations. 14 variables, response: medv, the median value of owner-occupied homes in $1000 s, predictors: measurements on the 506 census tracts in suburban Boston from the 1970 census. available in the MASS library in R. consider the four covariates: RM, TAX, PTRATIO, LSTAT (Opsomer and Ruppert 1998; Yu and Lu 2004; Wu et al. 2010). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 25 / 32

Numerical Studies Boston Housing Data center the dependent variable. take logarithmic transformations on TAX and LSTAT. standardize RM, log(tax), PTRATIO and log(lstat) and denote them by X 1, X 2, X 3 and X 4. consider the single index quantile regression models: Q τ (medv X) = g(x 1 + β 2 X 2 + β 3 X 3 + β 4 X 4 β), (6) for τ = 0.1, 0.25, 0.5, 0.75, 0.9. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 26 / 32

Numerical Studies Boston Housing Data τ RM log(tax) PTRATIO log(lstat) 0.1 1-1.1274-0.6098-1.1366 (0.2431) (0.3521) (0.4038) 0.25 1-0.7052-0.5828-1.2833 (0.2325) (0.4518) (0.4861) 0.5 1-0.4984-0.4497-1.0197 (0.3286) (0.6176) (0.6765) 0.75 1-0.3323-0.3961-0.8692 (0.3585) (0.6414) (0.7184) 0.9 1-0.6809-1.6961-5.2774 (0.0452) (0.0995) (0.0936) Table 2: Parametric vector estimates and standard errors (in parenthesis) for Boston housing data for Models (6) with five different quantile levels. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 27 / 32

Numerical Studies Boston Housing Data τ WYY NWQR 0.1 1.102 1.093 0.25 2.105 2.085 0.5 2.845 2.786 0.75 2.577 2.482 0.9 1.749 1.709 Table 3: Comparison of average check absolute residuals R τ ( β 1 ) defined in (5). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 28 / 32

Conclusions conditional quantiles provide a more complete picture of the conditional distribution and share nice properties such as robustness. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 29 / 32

Conclusions conditional quantiles provide a more complete picture of the conditional distribution and share nice properties such as robustness. consider a single index model to reduce the dimensionality and maintain some nonparametric flexibility. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 29 / 32

Conclusions conditional quantiles provide a more complete picture of the conditional distribution and share nice properties such as robustness. consider a single index model to reduce the dimensionality and maintain some nonparametric flexibility. iterative algorithms for estimating the parametric vector; convergence issues. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 29 / 32

Conclusions conditional quantiles provide a more complete picture of the conditional distribution and share nice properties such as robustness. consider a single index model to reduce the dimensionality and maintain some nonparametric flexibility. iterative algorithms for estimating the parametric vector; convergence issues. proposed a method that directly estimates the parametric component non-iteratively and have derived its asymptotic distribution under heteroscedastic errors. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 29 / 32

Conclusions conditional quantiles provide a more complete picture of the conditional distribution and share nice properties such as robustness. consider a single index model to reduce the dimensionality and maintain some nonparametric flexibility. iterative algorithms for estimating the parametric vector; convergence issues. proposed a method that directly estimates the parametric component non-iteratively and have derived its asymptotic distribution under heteroscedastic errors. make use of a parametrization that corresponds to the asymptotic theory derived. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 29 / 32

Future Work for high dimensional data, it is possible to improve the local linear conditional quantile estimators Q τ (Y X i ). E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 30 / 32

Future Work for high dimensional data, it is possible to improve the local linear conditional quantile estimators Q τ (Y X i ). variable selection for estimating the local linear conditional quantile estimators. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 30 / 32

Future Work for high dimensional data, it is possible to improve the local linear conditional quantile estimators Q τ (Y X i ). variable selection for estimating the local linear conditional quantile estimators. penalty term for estimating the parametric component β. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 30 / 32

References I E. Guerre and C. Sabbah. Uniform bias study and Bahadure representation for local polynomial estimators of the conditional quantile function. Econometric Theory, 28(01):87 129, 2012. E. Kong and Y. Xia. A single-index quantile regression model and its estimation. Econometric Theory, 28:730 768, 2012. J. D. Opsomer and D. Ruppert. A fully automated bandwidth selection for additive regression model. Journal of the American Statistical Association, 93:605 618, 1998. T. Z. Wu, K. Yu and Y. Yu. Single index quantile regression. Journal of Multivariate Analysis, 101(7):1607 1621, 2010. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 31 / 32

References II K. Yu and Z. Lu. Local linear additive quantile regression. Scandinavian Journal of Statistics, 31:333 346, 2004. E. Christou, M. G. Akritas (PSU) SIQR JSM, 2015 32 / 32

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