Taking the first, second and third derivative of equation (A.30) and evaluating at t = 0 we get, m$_{1}$ ( Y ) = E ( Y ) = ∂ ln M$_{Y}$ ( t ) ∂t ∣ t =0 = µ + √ 2 π ασ + σ (1 − 2 p ) p (1 p ) m$_{2}$ ( Y ) = V ( Y ) = ∂ $^{2}$ln M$_{Y}$ ( t ) ∂t 2 ∣ t =0 = α $^{2}$σ2 ( 1 − 2 π ) + σ 2 [ 1 − 2 p + 2 p 2 p $^{2}$(1 − p ) 2 ] m$_{3}$ ( Y ) = ∂ $^{3}$ln M$_{Y}$ ( t ) ∂t 3 ∣ t =0 = α $^{3}$σ3 √ 2 π ( 4 π − 1 ) + 2 σ 3 [ (1 p ) 3 − p 3 p $^{3}$(1 − p ) 3 ] where m$_{1}$ ( Y ) , m$_{2}$ ( Y ) and m$_{3}$ ( Y ) are the first, second and third order central moments, respectively. Hence, skewness can be obtained as m 3 ( m 3 / 2 2 $_{)}$. ■ In this appendix, we derive the conditional posteriors of the FBQROR model parameters. Specifically, the conditional posteriors of β, ν, h , and z have tractable distributions and is sampled using a Gibbs approach. The paraemters ( σ, γ ) are jointly sampled using random- walk MH algorithm (to reduce autocorrelation in MCMC draws) and δ is sampled using a random-walk MH algorithm. The derivations below follow the ordering as presented in Algorithm 1. Appendix B. Conditional Densities in FBQROR model (1) Starting with β , the conditional posterior π ( β | z, ν, h, σ, γ ) is proportional to π ( β ) × f ( z | β, ν, h, σ, γ ) and its kernel can be written as, π ( β | z, ν, h, σ, γ ) ∝ exp { − 1 2 [ n ∑ i =1 ( z$_{i}$ − x ′ $_{i}$β − Aν$_{i}$ − C | γ | h$_{i}$ ) 2 σBν$_{i}$ + ( β − β$_{0}$ ) $^{′}$B − 1 0 ( β − β$_{0}$ ) ]} ∝ exp { − 1 2 [ β ′ ( n ∑ i =1 x$_{i}$x ′ i σBν$_{i}$ + B − 1 0 ) β − β ′ ( x$_{i}$ ( z$_{i}$ − Aν$_{i}$ − C | γ | h$_{i}$ ) σBν$_{i}$ + B − 1 0 β$_{0}$ ) − ( x ′ $_{i}$( z$_{i}$ − Aν$_{i}$ − C | γ | h$_{i}$ ) σBν$_{i}$ + β ′ $_{0}$B − 1 0 )]} ∝ exp { − 1 2 [ β ′ ˜ B − $^{1}$β − β ′ ˜ B − 1 ˜ β − ˜ β ′ ˜ B − $^{1}$β + ˜ β ′ ˜ B − 1 ˜ β − ˜ β ′ ˜ B − 1 ˜ β ]} ∝ exp { − 1 2 ( β − ˜ β ) ′ ˜ B − $^{1}$( β − ˜ β ) } , 37