Bayesian Quantile Regression using the Skew Exponential Power Distribution Mauro Bernardia , Marco Bottoneb , Lea Petrellac,∗ a Department
of Statistical Sciences, University of Padua, Padua, Italy. of Italy, Directorate General for Economics, Statistics and Research, Italy. c Department of Methods and Models for Economics, Territory and Finance Sapienza University of Rome, Rome, Italy.
b Bank
Abstract Traditional Bayesian quantile regression relies on the Asymmetric Laplace (AL) distribution due primarily to its satisfactory empirical and theoretical performances. However, the AL displays medium tails and it is not suitable for data characterized by strong deviations from the Gaussian hypothesis. An extension of the AL Bayesian quantile regression framework is proposed to account for fat tails using the Skew Exponential Power (SEP) distribution. Linear and Additive Models (AM) with penalized splines are considered to show the flexibility of the SEP in the Bayesian quantile regression context. Lasso priors are used in both cases to account for the problem of shrinking parameters when the parameters space becomes wide while Bayesian inference is implemented using a new adaptive Metropolis within Gibbs algorithm. Empirical evidence of the statistical properties of the proposed models is provided through several examples based on both simulated and real datasets. Keywords: Bayesian quantile regression, Skew Exponential Power distribution, Asymmetric Laplace distribution, Lasso, MCMC.
1. Introduction Quantile regression has become a very popular approach to provide a wide description of the distribution of a response variable conditionally on a set of regressors. While linear regression analysis with symmetric L2 loss aims to estimate the conditional mean of a variable of interest, the quantile regression approach estimates any conditional quantile of confidence level τ ∈ (0, 1). Since the seminal works of Koenker and Basset (1978) and Koenker and Machado ∗ Corresponding
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[email protected] (Marco Bottone),
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Preprint submitted to Computational Statistics & Data Analysis
April 26, 2018
(1999), several papers have emerged in the literature considering quantile regression analysis from both a frequentist and a Bayesian point of view. As concerns the former approach, since the seminal paper of Koenker (2005), the estimation strategy relies on the minimization of an asymmetric L1 loss function. From the Bayesian perspective, instad, Yu and Moyeed (2001) introduced the Asymmetric Laplace (AL) distribution as a working likelihood to perform the inference. After the contibution of Yu and Moyeed (2001) a huge literature dealing with Bayesian quantile methods has proliferated in recent decades. Yu and Stander (2007) first proposesd the Bayesian treatment of Tobit quantile regression and Kobayashi (2017) extended it to accounts for endogeneity. Lum and Gelfand (2012) introduced the asymmetric Laplace process for quantile regression with spatially dependent errors accounting also for multiple simultaneous quantile estimation in a natural way. Reich et al. (2011) develop a Bayesian spatial quantile method for tropospheric ozone accounting for spatial variability by modeling the conditional distribution as a spatial process. Yue and Rue (2011) present a Bayesian quantile inference method based on the integrated nested Laplace approximations in additive mixed models, while Hallin et al. (2010) consider a multivariate extension of quantile based on a directional version of Koenker and Bassett’s traditional regression quantile using the asymmetric L1 optimization. Wang et al. (2016) introduce a quantile structural equation model to provide a comprehensive analysis of the interrelationships among latent variables. Kottas and Gelfand (2001) and Kottas and Krnjajic (2009) propose a Bayesian semiparametric approach for quantile regression modelling. Hu et al. (2015) introduce a Bayesian quantile regression method for partially linear additive models which explicitly models components that have linear and nonlinear effects while Chen and Yu (2009) propose a nonparametric quantile regression framework using piecewise polynomial functions with number and location of knots inferred through reversible jump Markov chain Monte Carlo. Nonparametric Bayesian quantile regression is also considered in Thompson et al. (2010) that propose to model the dependence of a quantile of one variable on the values of another using a natural cubic spline. Sriram et al. (2013) provide justification for assuming AL in the Bayesian quantile regression framework. Empirical likelihood as a working likelihood for quantile regression in Bayesian quantile inference is considered in Yang and He (2012) while an approach based on the pseudo–joint Asymmetric Laplace likelihood is implemented in Karthik et al. (2016). Novel implementation of Bayesian quantile regression in the risk measure field is considered in Bernardi et al. (2015) and Meligkotsidou et al. (2009). Count and binary Bayesian quantile regression are also considered by Lee and Neocleous (2010), Benoit and Van den Poel (2012) and Mollica and Petrella (2017), among others. Finally, the problem of variable selection in Bayesian quantile regression models based on the AL has been addressed by Yu et al. (2013), Alhamzawi and Yu (2012), Alhamzawi and Yu (2015), Alhamzawi (2016) and Ji et al. (2012). Although widely used for performing conditional quantile estimation within a Bayesian framework, the AL distribution has the main disadvantage of displaying medium tails. This may produce misleading results when extreme quan2
tiles are concerned, and, in particular, when the data are characterized by the presence of outliers and/or heavy tails. The absence of a parameter governing the tails of the AL distribution may influence the final inference. Recently, Wichitaksorn et al. (2014) tried to generalize the classical Bayesian quantile regression by using some flexible skewed distributions, obtained as scale–mixture of Normals. Despite their flexibility to model different degrees of asymmetry of the response variable, this class of distributions imposes a given structure of the tails. To overcome the aforementioned drawback of the AL distribution to adapt to fat–tailed data, we propose to extend the Bayesian quantile regression by using the Skew Exponential Power (SEP) distribution, introduced by Zhu and Zinde-Walsh (2009), as alternative working likelihood. As for the AL, the key property of the SEP distribution, when used as misspecified likelihood for quantile estimation, is that the natural location parameter coincides with the τ –th level quantile of the distribution. However, unlike the AL, the SEP distribution has a shape parameter governing the decay of the tails, providing a more reliable framework for quantile inference when the true data generating process is strongly contaminated by outliers or display fat–tails. Within the linear regression analysis several works have considered the symmetric version of the SEP, i.e., the Exponential Power (EP) distribution, for its characteristics of being robust to the presence of outliers contamination. Box and Tiao (1973) first introduced the linear regression model with EP innovations as a robust alternative to the Gaussian linear regression model, while Choy and Smith (1997), explored the robustness properties of posterior moments based on the EP distribution. More recently, Choy and Walker (2003) further extended the work of Choy and Smith (1997) by dealing with the case where the shape parameter that controls for the tail–decay assumes values greater than two. Finally, Naranjo et al. (2015) and Kobayashi (2016) suggest the use of the SEP distribution in regression and stochastic volatility models, respectively. To the best of our knowledge, this paper is the first attempt to use the SEP distribution as working likelihood in the quantile regression analysis. There are several advantages of using the SEP distribution as working likelihood to perform Bayesian quantile inference. First, the SEP distribution enconpasses the Asymmetric Laplace and the Asymmetric Gaussian related to quantiles and expectiles regression (see Aigner et al. 1976 and Newey and Powell 1987), respectively, providing a general method which has been proved to be interesting and theoretically justified in several contexts, such as risk measurement (Bellini et al. 2014) and methodological statistics (Bernardi et al. (2017), Gneiting et al. 2007 and Dawid and Musio 2015). Moreover, since the parameter that governs the tails of the SEP distribution is not fixed to a prespecified value but varies on µ, σ α 2(1−τ )σ where µ ∈ < is the location parameter, σ ∈
