where x$_{i}$ is a k × 1 vector of covariates, β is a k × 1 vector of unknown parameters at the p$_{0}$ -th quantile, ϵ$_{i}$ follows a GAL distribution, i.e., ϵ$_{i}$ ∼ GAL (0 , σ, γ ) and n denotes the number of observations. Note that we have suppressed the dependence of parameters on p$_{0}$ for notational simplicity. The variable z$_{i}$ is unobserved and relates to the observed discrete response y$_{i}$ , which has J categories or outcomes, via the cut-point vector ξ as follows: ξ$_{j}$$_{−}$$_{1}$ < z$_{i}$ ≤ ξ$_{j}$ ⇒ y$_{i}$ = j , ∀ i = 1 , · · · , n ; j = 1 , · · · , J, (8) where ξ$_{0}$ = −∞ and ξ$_{J}$ = ∞ . In addition, ξ$_{1}$ is typically set to 0, which anchors the location of the distribution required for parameter identification (see Jeliazkov et al., 2008). Given the data vector y = ( y$_{1}$, · · · , y$_{n}$ ) $^{′}$, the likelihood for the model expressed as a function of unknown parameters ( β, σ, γ, ξ ) can be written as, where, F$_{p}$$_{0}$ ( · ) ≡ F ( ·| 0 , 1 , γ ) denotes the cdf of the GAL distribution and I ( y$_{i}$ = j ) is an indicator function, which equals 1 if y$_{i}$ = j and 0 otherwise. Working directly with the GAL distribution is difficult, so we replace the error term with its mixture representation (1) and rewrite the FBQROR model as follows: f ( β, σ, γ, ξ ; y ) = n ∏ i =1 J ∏ j =1 P ( y$_{i}$ = j | β, σ, ξ, γ ) I ( y$_{i}$ = j ) = n ∏ i =1 J ∏ j $_{=1}$[ F$_{p}$$_{0}$ ( ξ$_{j}$ − x ′ $_{i}$β$_{p}$ σ ) − F$_{p}$$_{0}$ ( ξ$_{j}$$_{−}$$_{1}$ − x ′ $_{i}$β σ ) ] I ( y$_{i}$ = j ) (9) z$_{i}$ = x ′ $_{i}$β + σAw$_{i}$ + σC | γ | s$_{i}$ + σ √ Bw$_{i}$ u$_{i}$, ∀ i = 1 , · · · , n. (10) The above formulation (10) implies that the latent variable z$_{i}$ | β, w$_{i}$, s$_{i}$, σ, γ ∼ N ( x ′ $_{i}$β + σC | γ | s$_{i}$ + σAw$_{i}$, σ $^{2}$Bw$_{i}$ ). However, the presence of the scale parameter σ in the conditional mean is not conducive to the construction of MCMC algorithm (Kozumi and Kobayashi, 8