Comparison of Score-Driven Equity-Gold Portfolios During the COVID-19 Pandemic Using Model Confidence Sets

Astrid Ayala; Szabolcs Blazsek; Adrian Licht

doi:10.1515/snde-2022-0107

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Comparison of Score-Driven Equity-Gold Portfolios During the COVID-19 Pandemic Using Model Confidence Sets

Astrid Ayala , Szabolcs Blazsek and Adrian Licht

Published/Copyright: November 7, 2023

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 27 Issue 5

Abstract

Gold may have a hedge, safe haven, or diversifier property when added to stock portfolios. Motivated by the favorable statistical properties and out-of-sample performance of score-driven models, we investigate for equity-gold portfolios whether score-driven mean, volatility, and copula models can improve the performances of DCC (dynamic conditional correlation) portfolios, the naïve portfolio strategy, and the Standard & Poor’s 500 (S&P 500) index. We consider 2880 score-driven portfolio strategies. We use score-driven Clayton, rotated Clayton, Frank, Gaussian, Gumbel, rotated Gumbel, Plackett, and Student’s t copulas. We use several classical and score-driven models of marginal distribution. We use weekly, monthly, quarterly, semi-annual, and annual updates of portfolio weights. We use minimum-variance, maximum Sharpe ratio, and maximum utility function strategies. We use rolling data-windows for portfolio optimization for the COVID-19 investment period of February 2020 to September 2021. We classify competing portfolios by using a new robust multi-step model confidence set (MCS) test approach and provide evidence of the superiority of score-driven portfolios.

Keywords: dynamic conditional score (DCS) models; generalized autoregressive score (GAS) models; score-driven copulas; equity-gold portfolios; model confidence set (MCS)

JEL Classification: C32; C52; C58; G11

Corresponding author: Szabolcs Blazsek, Stetson-Hatcher School of Business, Mercer University, Macon, GA 31207, USA; and School of Business, Universidad Francisco Marroquín, Guatemala City, 01010, Guatemala, E-mail: blazsek_s@mercer.edu

Funding source: Universidad Francisco Marroquin

Acknowledgments

The authors are thankful for the helpful comments of Matthew Copley, and the editor and reviewer of the journal. All remaining errors are our own. Funding from Universidad Francisco Marroquín is acknowledged. Data and computer codes are available from the authors upon request. No potential conflict of interest was reported by the authors. The authors express their consent for publication.

Appendix A

Student’s t -distribution—For this distribution ϵ _k,t ∼ t[0, 1, exp(ν _k) + 2] i.i.d., where ν _k ∈ IR is a shape parameter, and exp(⋅) is the exponential function. The first two moments of r _k,t exist.

The log conditional density of r _k,t is

(A.1) ln ⁡ f ( r k , t | F t − 1 ; Θ ) = ln Γ exp ( ν k ) + 3 2 − ln Γ exp ( ν k ) + 2 2 − ln ( π ) + ln [ exp ( ν k ) + 2 ] 2 − λ k , t − exp ( ν k ) + 3 2 ln 1 + ϵ k , t 2 exp ( ν k ) + 2

where ln(⋅) is the natural logarithm function and Γ(⋅) is the gamma function.

The score function with respect to μ _k,t is (Harvey 2013):

(A.2) ∂ ⁡ ln ⁡ f ( r k , t | F t − 1 ; Θ ) ∂ μ k , t = [ exp ( ν k ) + 2 ] exp ( λ k , t ) ϵ k , t ϵ k , t 2 + exp ( ν k ) + 2 × exp ( ν k ) + 3 [ exp ( ν k ) + 2 ] exp ( 2 λ k , t ) = s μ , k , t × exp ( ν k ) + 3 [ exp ( ν k ) + 2 ] exp ( 2 λ k , t ) = s μ , k , t × K ( λ k , t )

where the scaled score function s _μ,k,t is defined in the second equality, and the scale factor K(λ _k,t) is defined in the last equality. The s _μ,k,t term trims outliers, because s _μ,k,t → _p0 when |ϵ _k,t| → ∞ (Figure A1(a)). The discounting undertaken by s _μ,k,t is identical for the positive and negative sides of the distribution. The score function with respect to λ _k,t is:

(A.3) s λ , k , t = ∂ ⁡ ln ⁡ f ( r k , t | F t − 1 ; Θ ) ∂ λ k , t = [ exp ( ν k ) + 3 ] ϵ k , t 2 exp ( ν k ) + 2 + ϵ k , t 2 − 1

The updating term s _λ,k,t Winsorizes extreme observations, because s _λ,k,t → _p c (c > 0 is a real number) when |ϵ _k,t| → ∞ (Figure A1(b)). The discounting undertaken by s _λ,k,t is identical for the positive and negative sides of the probability distribution.

The conditional mean and volatility of r _k,t, respectively, are:

(A.4) E ( r k , t | F t − 1 ; Θ ) = μ k , t

(A.5) σ ( r k , t | F t − 1 ; Θ ) = σ k , t = exp ( λ k , t ) exp ( ν k ) + 2 exp ( ν k ) 1 / 2

Figure A1:

Scaled score function s _μ,k,t and score function s _λ,k,t estimates, as functions of ϵ _t. ML estimates of the shape parameters with λ _t = 0 are used. In parentheses, we refer to the asymptotic transformation of outliers, as |ϵ _t| → ∞.

Gen- t distribution—For this distribution ϵ _t ∼ Gen-t[0, 1, exp(ν _k) + 2, exp(η _k)] i.i.d., where ν _k ∈ IR, and η _k ∈ IR are shape parameters. For exp(η _k) = 2, the Gen-t distribution is the Student’s t-distribution. The first two moments of r _k,t exist.

The log density of r _k,t is (Ayala, Blazsek, and Escribano 2022):

(A.6) ln ⁡ f ( r k , t | F t − 1 ; Θ ) = η k − λ k , t − ln ( 2 ) − ln [ exp ( ν k ) + 2 ] exp ( η k ) − ln Γ exp ( ν k ) + 2 exp ( η k ) − ln Γ [ exp ( − η k ) ] + ln Γ exp ( ν k ) + 3 exp ( η k ) − exp ( ν k ) + 3 exp ( η k ) ln 1 + | ϵ k , t | exp ( η k ) [ exp ( ν k ) + 2 ]

where sgn(⋅) is the signum function.

The score function with respect to μ _t is given by:

(A.7) ∂ ⁡ ln ⁡ f ( r k , t | F t − 1 ; Θ ) ∂ μ t = [ exp ( ν k ) + 2 ] exp ( λ k , t ) ϵ k , t | ϵ k , t | exp ( η k ) − 2 | ϵ k , t | exp ( η k ) + [ exp ( ν k ) + 2 ] × exp ( ν k ) + 3 [ exp ( ν k ) + 2 ] exp ( 2 λ k , t ) = s μ , k , t × exp ( ν k ) + 3 [ exp ( ν k ) + 2 ] exp ( 2 λ k , t ) ≡ u μ , k , t × K ( λ k , t )

where the scaled score function s _μ,k,t is defined in the second equality, and the scale factor K(λ _k,t) is defined in the last equality. The s _μ,k,t term trims extreme observations, because s _μ,k,t → _p0 when |ϵ _k,t| → ∞ (Figure A1(c)). The discounting undertaken by s _μ,k,t is not identical for the positive and negative sides of the distribution. The score function with respect to λ _k,t is:

The updating term s _λ,k,t Winsorizes outliers, because s _λ,k,t → _p c ₁ when ϵ _k,t → −∞ and s _λ,k,t → _p c ₂ when ϵ _k,t → +∞ (c ₁ > 0 and c ₂ > 0 are real numbers) (Figure A1(d)). The Winsorizing of s _λ,k,t is not identical for the positive and negative sides of the distribution.

The conditional mean and volatility of r _k,t, respectively, are (Ayala, Blazsek, and Escribano 2022):

(A.9) E ( r k , t | F t − 1 ; Θ ) = μ k , t

(A.10) σ ( r k , t | F t − 1 ; Θ ) = σ k , t = exp ( λ k , t ) [ exp ( ν k ) + 2 ] exp ( − η k ) × B 3 exp ( η k ) , exp ( ν k ) exp ( η k ) B 1 exp ( η k ) , exp ( ν k ) + 2 exp ( η k ) 1 / 2

where B(⋅, ⋅) is the beta function.

EGB2 distribution—For this distribution ϵ _k,t ∼ EGB2[0, 1, exp(ν _k), exp(η _k)] i.i.d., where ν _k ∈ IR and η _k ∈ IR are shape parameters. For the EGB2 distribution, all moments exist.

The log conditional density is (Caivano and Harvey 2014):

(A.11) ln ⁡ f ( r k , t | F t − 1 ; Θ ) = exp ( ν k ) ϵ k , t − λ k , t − ln Γ [ exp ( ν k ) ] − ln Γ [ exp ( η k ) ] + ln Γ [ exp ( ν k ) + exp ( η k ) ] − [ exp ( ν k ) + exp ( η k ) ] ln 1 + exp ( ϵ k , t )

The score function with respect to μ _k,t is given by (Caivano and Harvey 2014):

(A.12) ∂ ⁡ ln ⁡ f ( r k , t | F t − 1 ; Θ ) ∂ μ k , t = exp ( − λ k , t ) [ exp ( ν k ) + exp ( η k ) ] exp ( ϵ k , t ) exp ( ϵ k , t ) + 1 − exp ( − λ k , t ) exp ( ν k )

Moreover, the scaled score function s _μ,k,t and the scale factor K(λ _k,t), respectively, are:

(A.13) s μ , k , t = ∂ ⁡ ln ⁡ f ( r k , t | F t − 1 ; Θ ) ∂ μ k , t × Ψ ( 1 ) [ exp ( ν k ) ] + Ψ ( 1 ) [ exp ( η k ) ] exp ( 2 λ k , t )

(A.14) K ( λ k , t ) ≡ Ψ ( 1 ) [ exp ( ν k ) ] + Ψ ( 1 ) [ exp ( η k ) ] − 1 ⁡ exp ( − 2 λ k , t )

where Ψ⁽¹⁾(⋅) is the trigamma function. The updating term s _μ,k,t Winsorizes outliers, because s _μ,k,t → _p c ₁ when ϵ _k,t → −∞ and s _μ,k,t → _p c ₂ when ϵ _k,t → +∞ (c ₁ > 0 and c ₂ > 0) (Figure A1(e)). The Winsorizing undertaken by s _μ,k,t is not identical for positive and negative values. The score function with respect to λ _k,t is (Caivano and Harvey 2014):

(A.15) ∂ ⁡ ln ⁡ f ( r k , t | F t − 1 ; Θ ) ∂ λ k , t = s λ , k , t = [ exp ( ν k ) + exp ( η k ) ] ϵ k , t ⁡ exp ( ϵ k , t ) exp ( ϵ k , t ) + 1 − exp ( ν k ) ϵ k , t − 1

The updating term s _λ,k,t performs a linearly increasing and asymmetric transformation of ϵ _k,t, as |ϵ _k,t| → ∞ (Figure A1(f)).

The conditional mean and conditional volatility of r _k,t, respectively, are:

(A.16) E ( r k , t | F t − 1 ; Θ ) = μ k , t + exp ( λ k , t ) Ψ ( 0 ) [ exp ( ν k ) ] − Ψ ( 0 ) [ exp ( η k ) ]

(A.17) σ ( r k , t | F t − 1 ; Θ ) = σ k , t = exp ( λ k , t ) Ψ ( 1 ) [ exp ( ν k ) ] + Ψ ( 1 ) [ exp ( η k ) ] 1 / 2

where Ψ⁽⁰⁾(⋅) is the digamma function.

NIG distribution—For this distribution ϵ _k,t ∼ NIG[0, 1, exp(ν _k), exp(ν _k) tanh(η _k)] i.i.d., where ν _k ∈ IR and η _k ∈ IR are shape parameters, and tanh(⋅) is the hyperbolic tangent function. For the NIG distribution all moments exist.

The log conditional density is (Blazsek, Ho, and Liu 2018):

(A.18) ln ⁡ f ( r k , t | F t − 1 ; Θ ) = ν k − λ k , t − ln ( π ) + exp ( ν k ) 1 − tanh 2 ( η k ) 1 / 2 + exp ( ν k ) tanh ( η k ) ϵ k , t + ln K ( 1 ) exp ( ν k ) 1 + ϵ k , t 2 − 1 2 ln 1 + ϵ k , t 2

where K ^(j)(⋅) is the modified Bessel function of the second kind of order j.

The score function with respect to μ _k,t is given by (Blazsek, Ho, and Liu 2018):

(A.19) ∂ ⁡ ln ⁡ f ( r k , t | F t − 1 ; Θ ) ∂ μ k , t = − exp ( ν k − λ k , t ) tanh ( η k ) + ϵ k , t exp ( λ k , t ) ( 1 + ϵ k , t 2 ) + exp ( ν k − λ k , t ) ϵ k , t 1 + ϵ k , t 2 × K ( 0 ) exp ( ν k ) 1 + ϵ k , t 2 + K ( 2 ) exp ( ν k ) 1 + ϵ k , t 2 2 K ( 1 ) exp ( ν k ) 1 + ϵ k , t 2

Moreover, the scaled score function s _μ,k,t and the scale factor K(λ _k,t), respectively, are:

(A.20) s μ , k , t = ∂ ⁡ ln ⁡ f ( r k , t | F t − 1 ; Θ ) ∂ μ k , t × exp ( 2 λ k , t )

(A.21) K ( λ k , t ) ≡ exp ( − 2 λ k , t )

The updating term s _μ,k,t Winsorizes outliers, because s _μ,k,t → _p c ₁ when ϵ _k,t → −∞ and s _μ,k,t → _p c ₂ when ϵ _k,t → +∞ (c ₁ > 0 and c ₂ > 0 are real numbers) (Figure A1(g)). The Winsorizing undertaken by s _μ,k,t is not identical for the positive and negative sides of the probability distribution. The score function with respect to λ _k,t is given by (Blazsek, Ho, and Liu 2018):

(A.22) s λ , k , t = − 1 − exp ( ν k ) tanh ( η k ) ϵ k , t + ϵ k , t 2 1 + ϵ k , t 2 + exp ( ν k ) ϵ k , t 2 1 + ϵ k , t 2 × K ( 0 ) exp ( ν k ) 1 + ϵ k , t 2 + K ( 2 ) exp ( ν k ) 1 + ϵ k , t 2 2 K ( 1 ) exp ( ν k ) 1 + ϵ k , t 2

The updating term s _λ,k,t performs a linearly increasing and asymmetric transformation of ϵ _k,t, as |ϵ _k,t| → ∞ (Figure A1(h)).

The conditional mean and volatility of r _k,t, respectively, are (Blazsek, Ho, and Liu 2018):

(A.23) E ( r k , t | F t − 1 ; Θ ) = μ k , t + exp ( λ k , t ) tanh ( η k ) 1 − tanh 2 ( η k ) 1 / 2

(A.24) σ ( r k , t | F t − 1 ; Θ ) = σ k , t = exp ( 2 λ k , t − ν k ) 1 − tanh 2 ( η k ) 3 / 2 1 / 2

Appendix B

Clayton copula—The bivariate Clayton copula density function is

(B.1) c t ( u , v ) ≡ c ( u , v ; ρ t | F t − 1 ; Θ ) = ( 1 + ρ t ) ( u v ) − ( ρ t + 1 ) u − ρ t + v − ρ t − 1 − ( 1 + 2 ρ t ) / ρ t

where u and v are realizations of U[0, 1] random variables, and ρ _t ∈ [−1, ∞)\{0} (Harvey 2013; Joe 2015). The partial derivative of ln c(u, v; ρ _t) is

(B.2) s ρ , t ≡ s ρ ( u , v ; ρ t | F t − 1 ; Θ ) = ∂ ⁡ ln ⁡ c ( u , v ; ρ t ) ∂ ρ t = 1 1 + ρ t − ln ( u v ) + 1 ρ t 2 ln u − ρ t + v − ρ t − 1 + ( 1 + 2 ρ t ) u − ρ t ⁡ ln ( u ) + v − ρ t ⁡ ln ( v ) ρ t u − ρ t + v − ρ t − 1

Rotated Clayton copula—The bivariate rotated Clayton copula density function is

(B.3) c t ( u , v ) ≡ c ( u , v ; ρ t | F t − 1 ; Θ ) = ( 1 + ρ t ) [ ( 1 − u ) ( 1 − v ) ] − ( ρ t + 1 ) ( 1 − u ) − ρ t + ( 1 − v ) − ρ t − 1 − ( 1 + 2 ρ t ) / ρ t

with ρ _t ∈ [−1, ∞)\{0} (Patton 2004). The partial derivative of ln c(u, v; ρ _t) is

(B.4) s ρ , t ≡ s ρ ( u , v ; ρ t | F t − 1 ; Θ ) = ∂ ⁡ ln ⁡ c ( u , v ; ρ t ) ∂ ρ t = 1 1 + ρ t − ln [ ( 1 − u ) ( 1 − v ) ] + 1 ρ t 2 ln ( 1 − u ) − ρ t + ( 1 − v ) − ρ t − 1 + ( 1 + 2 ρ t ) ( 1 − u ) − ρ t ⁡ ln ( 1 − u ) + ( 1 − v ) − ρ t ⁡ ln ( 1 − v ) ρ t ( 1 − u ) − ρ t + ( 1 − v ) − ρ t − 1

Frank copula—The bivariate Frank copula density function is

(B.5) c t ( u , v ) ≡ c ( u , v ; ρ t | F t − 1 ; Θ ) = ρ t [ 1 − exp ( − ρ t ) ] exp [ − ρ t ( u + v ) ] { 1 − exp ( − ρ t ) − [ 1 − exp ( − ρ t u ) ] [ 1 − exp ( − ρ t v ) ] } 2

with ρ t ∈ R \ { 0 } (Joe 2015). The partial derivative of ln c(u, v; ρ _t) is

(B.6) s ρ , t ≡ s ρ ( u , v ; ρ t | F t − 1 ; Θ ) = ∂ ⁡ ln ⁡ c ( u , v ; ρ t ) ∂ ρ t = 1 ρ t + 1 exp ( ρ t ) − 1 − ( u + v ) − 2 exp ( − ρ t ) − u ⁡ exp ( − ρ t u ) − v ⁡ exp ( − ρ t v ) + ( u + v ) exp [ − ρ t ( u + v ) ] − exp ( − ρ t ) + exp ( − ρ t u ) + exp ( − ρ t v ) − exp [ − ρ t ( u + v ) ]

Gaussian copula—The bivariate Gaussian copula density function is

(B.7) c t ( u , v ) ≡ c ( u , v ; ρ t | F t − 1 ; Θ ) = 1 1 − ρ t 2 exp 2 ρ t Φ − 1 ( u ) Φ − 1 ( v ) − ρ t 2 ( Φ − 1 ( u ) ) 2 + ( Φ − 1 ( v ) ) 2 2 1 − ρ t 2

where Φ⁻¹(x) is the inverse of the distribution function of N(0, 1) and ρ _t ∈ [−1, 1] (Joe 2015; Meyer 2013). The partial derivative of ln c(u, v; ρ _t) is

(B.8) s ρ , t ≡ s ρ ( u , v ; ρ t | F t − 1 ; Θ ) = ∂ ⁡ ln ⁡ c ( u , v ; ρ t ) ∂ ρ t = ρ t 1 − ρ t 2 + Φ − 1 ( u ) Φ − 1 ( v ) ( 1 + ρ t 2 ) − ρ t ( Φ − 1 ( u ) ) 2 + ( Φ − 1 ( v ) ) 2 ρ t 2 − 1 2

Gumbel copula—The bivariate Gumbel copula density function is

(B.9) c t ( u , v ) ≡ c ( u , v ; ρ t | F t − 1 ; Θ ) = exp − ( − ln ⁡ u ) ρ t + ( − ln ⁡ v ) ρ t 1 / ρ t × ln ( u ) ln ( v ) ρ t − 1 u v ( − ln ⁡ u ) ρ t + ( − ln ⁡ v ) ρ t 2 − 1 / ρ t × ( − ln ⁡ u ) ρ t + ( − ln ⁡ v ) ρ t 1 / ρ t + ρ t − 1

with ρ _t ∈ [1, ∞) (Joe 2015; Patton 2004). The partial derivative of ln c(u, v; ρ _t) is

(B.10) s ρ , t ≡ s ρ ( u , v ; ρ t | F t − 1 ; Θ ) = ∂ ⁡ ln ⁡ c ( u , v ; ρ t ) ∂ ρ t = ξ 1 1 / ρ t − 1 ρ t 2 ξ 1 ⁡ ln ( ξ 1 ) − ρ t ξ 2 + ln ( ln ⁡ u ⁡ ln ⁡ v ) + ( 1 − 2 ρ t ) ξ 2 ρ t ξ 1 − ln ( ξ 1 ) ρ t 2 + ξ 1 1 / ρ t − 1 − ξ 1 ⁡ ln ( ξ 1 ) + ρ t ξ 2 + 1 ρ t 2 ξ 1 1 / ρ t + ρ t − 1

where ξ 1 = ( − ln ⁡ u ) ρ t + ( − ln ⁡ v ) ρ t and ξ 2 = ( − ln ⁡ u ) ρ t ⁡ ln ( − ln ⁡ u ) + ( − ln ⁡ v ) ρ t ⁡ ln ( − ln ⁡ v ) .

Figure B1:

Copula density contour plots. For all copulas, Blomqvist’s β _t = 0.70 and for Student’s t-copula ν = 5. For all copulas, the density contour plot is for the levels 0.2, 0.6, 1, 1.4, 1.8, 2.2, 2.6, and 3.

Figure B2:

Discounting property of score for score-driven copula models. s _ρ(u, v) is presented as a function of u with v fixed at the 0.1 (solid thin), 0.5 (solid thick), and 0.9 (dashed) levels. For all copulas Blomqvist’s β _t = 0.70 and for Student’s t-copula ν = 5.

Rotated Gumbel copula—The bivariate rotated Gumbel copula density function is

(B.11) c t ( u , v ) ≡ c ( u , v ; ρ t | F t − 1 ; Θ ) = exp − [ − ln ( 1 − u ) ] ρ t + [ − ln ( 1 − v ) ] ρ t 1 / ρ t × ln ( 1 − u ) ln ( 1 − v ) ρ t − 1 ( 1 − u ) ( 1 − v ) [ − ln ( 1 − u ) ] ρ t + [ − ln ( 1 − v ) ] ρ t 2 − 1 / ρ t × [ − ln ( 1 − u ) ] ρ t + [ − ln ( 1 − v ) ] ρ t 1 / ρ t + ρ t − 1

with ρ _t ∈ [1, ∞) (Patton 2004). The partial derivative of ln c(u, v; ρ _t) is

(B.12) s ρ , t ≡ s ρ ( u , v ; ρ t | F t − 1 ; Θ ) = ∂ ⁡ ln ⁡ c ( u , v ; ρ t ) ∂ ρ t = ξ 1 1 / ρ t − 1 ρ t 2 ξ 1 ⁡ ln ( ξ 1 ) − ρ t ξ 2 + ln [ ln ( 1 − u ) ln ( 1 − v ) ] + ( 1 − 2 ρ t ) ξ 2 ρ t ξ 1 − ln ( ξ 1 ) ρ t 2 + ξ 1 1 / ρ t − 1 − ξ 1 ⁡ ln ( ξ 1 ) + ρ t ξ 2 + 1 ρ t 2 ξ 1 1 / ρ t + ρ t − 1

where ξ 1 = [ − ln ( 1 − u ) ] ρ t + [ − ln ( 1 − v ) ] ρ t and

(B.13) ξ 2 = [ − ln ( 1 − u ) ] ρ t ⁡ ln [ − ln ( 1 − u ) ] + [ − ln ( 1 − v ) ] ρ t ⁡ ln [ − ln ( 1 − v ) ]

Plackett copula—The bivariate Plackett copula density function is

(B.14) c t ( u , v ) ≡ c ( u , v ; ρ t | F t − 1 ; Θ ) = ρ t 1 + ( ρ t − 1 ) ( u + v − 2 u v ) 1 + ( ρ t − 1 ) ( u + v ) 2 − 4 ρ t ( ρ t − 1 ) u v 3 / 2

with ρ _t ∈ [0, ∞)\{1} (Joe 2015). The partial derivative of ln c(u, v; ρ _t) is

(B.15) s ρ , t ≡ s ρ ( u , v ; ρ t | F t − 1 ; Θ ) = ∂ ⁡ ln ⁡ c ( u , v ; ρ t ) ∂ ρ t = − 3 ( u + v ) 2 ( ρ t − 1 ) + u + v + 2 u v ( 1 − 2 ρ t ) ( u + v ) ( ρ t − 1 ) + 1 2 − 4 ρ t ( ρ t − 1 ) u v + u + v − 2 u v ( ρ t − 1 ) ( u + v − 2 u v ) + 1 + 1 ρ t

Student’s t copula—The bivariate Student’s t-copula density function is

(B.16) c t ( u , v ) ≡ c ( u , v ; ν , ρ t | F t − 1 ; Θ ) = 1 1 − ρ t 2 Γ [ ( ν + 2 ) / 2 ] Γ ( ν / 2 ) Γ 2 [ ( ν + 1 ) / 2 ] × 1 + T ν − 1 ( u ) 2 + T ν − 1 ( v ) 2 − 2 ρ t T ν − 1 ( u ) T ν − 1 ( v ) ν 1 − ρ t 2 − ν + 2 2 1 + T ν − 1 ( u ) 2 ν − ν + 1 2 1 + T ν − 1 ( v ) 2 ν − ν + 1 2

where T ν − 1 ( x ) is the inverse of the distribution function of the Student’s t-distribution, ρ _t is the correlation coefficient, and the additional parameter ν denotes degrees of freedom (Joe 2015). The partial derivative of ln c(u, v; ν, ρ _t) is

(B.17) s ρ , t ≡ s ρ ( u , v ; ν , ρ t | F t − 1 ; Θ ) = ∂ ⁡ ln ⁡ c ( u , v ; ν , ρ t ) ∂ ρ t = ρ t 1 − ρ t 2 + ν + 2 ρ t 2 − 1 × ρ t T ν − 1 ( u ) 2 + T ν − 1 ( v ) 2 − ρ t 2 + 1 T ν − 1 ( u ) T ν − 1 ( v ) T ν − 1 ( u ) 2 + T ν − 1 ( v ) 2 − 2 ρ t T ν − 1 ( u ) T ν − 1 ( v ) − ν ρ t 2 − 1

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2022-0107).

Received: 2022-11-27

Accepted: 2023-10-16

Published Online: 2023-11-07

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Articles in the same Issue

https://doi.org/10.1515/snde-2022-0107

Keywords for this article

dynamic conditional score (DCS) models; generalized autoregressive score (GAS) models; score-driven copulas; equity-gold portfolios; model confidence set (MCS)