Traditional Bayesian quantile regression relies on the Asymmetric Laplace distribution (ALD) due primarily to its satisfactory empirical and theoretical performances. However, the ALD displays medium tails and is not suitable for data
characterized by strong deviations from the Gaussian hypothesis. In this paper, we propose an extension of the ALD Bayesian quantile regression framework to account for fat tails using the Skew Exponential Power (SEP) distribution.
Besides having the τ-level quantile as a parameter, the SEP distribution has an additional key parameter governing the decay of the tails, making it attractive for robust modeling. Linear and Additive Models (AM) with penalized spline are used to show the exibility 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. To implement the Bayesian inference we propose a new adaptive Metropolis within Gibbs algorithm. Empirical evidence of the statistical properties of the proposed SEP Bayesian quantile regression method is provided through several examples based on both simulated and real datasets.Conditional Autoregressive Value-at-Risk (CAViaR) and Conditional Autoregressive Expectile (CARE) have become two popular approaches for direct measurement of market risk. Since their introduction in the econometric literature
by the seminal papers of Engle and Manganelli (2004) and Taylor (2008), several improvements have been proposed to the original approaches allowing for different degrees of asymmetry and local non-linearity. Furthermore, Bayesian
modeling of time-varying quantile and expectile regression relies on the Asymmetric Laplace (AL) distribution and the Asymmetric Gaussian (AG) distribution respectively, making impossible to consider the two risk measure in a
unified framework. Here we propose two extensions of the Bayesian CAViaR and CARE class of models using the Skew Exponential Power (SEP) distribution and a flexible functional form specified by P-Spline functions. The SEP distribution includes the AL and the AG as a special cases allowing for a new general class of models, that include both CAViaR and CARE models, called Bayesian Conditional Autoregressive Risk Measures (B-CARM). Further, we
consider the P-spline approximation of the models which permits to take into account for non-linearity. To estimate all the model parameters we propose a new Adaptive Independent Metropolis within Gibbs algorithm. The the effectiveness of the model is demonstrated using real data on daily return of five stock market indices
