Score-driven multi-regime Markov-switching EGARCH: empirical evidence using the Meixner distribution

Szabolcs Blazsek; Michel Ferreira Cardia Haddad

doi:10.1515/snde-2021-0101

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Score-driven multi-regime Markov-switching EGARCH: empirical evidence using the Meixner distribution

Szabolcs Blazsek and Michel Ferreira Cardia Haddad

Published/Copyright: July 21, 2022

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 27 Issue 4

Abstract

In this paper, statistical and volatility forecasting performances of the non-path-dependent score-driven multi-regime Markov-switching (MS) exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models are explored. Three contributions to the existing literature are provided. First, we use all relevant score-driven distributions from the literature - namely, the Student’s t-distribution, general error distribution (GED), skewed generalized t-distribution (Skew-Gen-t), exponential generalized beta distribution of the second kind (EGB2), and normal-inverse Gaussian (NIG) distribution. We then introduce the score-driven Meixner (MXN) distribution-based EGARCH model to the literature on score-driven models. Second, proving the sufficient conditions of the asymptotic properties of the maximum likelihood (ML) estimator for non-path-dependent score-driven MS-EGARCH models is an unsolved problem. We provide a partial solution to that problem by proving necessary conditions for the asymptotic theory of the ML estimator. Third, to the best of our knowledge, this work includes the largest number of international stock indices from the G20 countries in the literature, covering the period of 2000–2022. We provide a discussion on the major events which caused common or non-common switching to the high-volatility regime for the G20 countries. The statistical performance and volatility forecasting results support the adoption of score-driven MS-EGARCH for the G20 countries.

Keywords: dynamic conditional score; exponential generalized autoregressive conditional heteroskedasticity (EGARCH); generalized autoregressive score; Markov regime-switching

JEL Classification: C22; C51; C52; C58

Corresponding author: Szabolcs Blazsek, School of Business, Universidad Francisco Marroquín, Guatemala City 01010, Guatemala, E-mail: sblazsek@ufm.edu

Funding source: The Cambridge Commonwealth, European & International Trust

Award Identifier / Grant number: BEX 2220/15-6

Funding source: Coordination for the Improvement of Higher Education Personnel of Brazil (CAPES)

Funding source: Universidad Francisco Marroquin

Acknowledgments

Data and computer codes are available from the authors upon request. Both authors thank Astrid Ayala, Lorenzo Cristofaro, Matthew Copley, Luis Alberiko Gil-Alana, Adrian Licht, Demian Licht, and Jacob Rasmussen. All remaining errors are the authors’ responsibility.

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: Szabolcs Blazsek acknowledges funding from the School of Business of Universidad Francisco Marroquín. Michel Haddad acknowledges funding from the Coordination for the Improvement of Higher Education Personnel of Brazil (CAPES), and from The Cambridge Commonwealth, European & International Trust, under the grant/award BEX 2220/15-6.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Expected value and volatility formulas

MXN distribution: The mean of y t | ( F t − 1 , s t , Θ ) is

(A.1) μ t ( s t ) = c ( s t ) + exp [ λ t ( s t ) ] Ψ ( 0 ) { exp [ δ 1 ( s t ) ] } − Ψ ( 0 ) { exp [ δ 2 ( s t ) ] }

where exp(⋅) is the exponential function and Ψ⁽⁰⁾(⋅) is the digamma function. The volatility of y t | ( F t − 1 , s t , Θ ) is

(A.2) σ t ( s t ) = exp [ 2 λ t ( s t ) + δ 2 ( s t ) ] cos { π ⁡ tanh [ δ 1 ( s t ) ] } + 1 1 / 2

where cos(⋅) is the cosine function and tanh(⋅) is the hyperbolic tangent function.

EGB2 distribution: The mean of y t | ( F t − 1 , s t , Θ ) is

(A.3) μ t ( s t ) = c ( s t ) + exp [ λ t ( s t ) ] tanh [ δ 2 ( s t ) ] 1 − tanh 2 [ δ 2 ( s t ) ] 1 / 2

The volatility of y t | ( F t − 1 , s t , Θ ) is

(A.4) σ t ( s t ) = exp [ λ t ( s t ) ] Ψ ( 1 ) { exp [ δ 1 ( s t ) ] } + Ψ ( 1 ) { exp [ δ 2 ( s t ) ] } 1 / 2

where Ψ⁽¹⁾(⋅) is the trigamma function.

NIG distribution: The mean of y t | ( F t − 1 , s t , Θ ) is

(A.5) μ t ( s t ) = c ( s t ) + 2 exp [ λ t ( s t ) ] tanh [ δ 1 ( s t ) ] { exp [ δ 2 ( s t ) ] + 2 } exp [ − δ 3 ( s t ) ] × B 2 exp [ δ 3 ( s t ) ] , exp [ δ 2 ( s t ) ] + 1 exp [ δ 3 ( s t ) ] B 1 exp [ δ 3 ( s t ) ] , exp [ δ 2 ( s t ) ] + 2 exp [ δ 3 ( s t ) ]

where B(⋅, ⋅) is the Beta function. The volatility of y t | ( F t − 1 , s t , Θ ) is

(A.6) σ t ( s t ) = exp [ 2 λ t ( s t ) − δ 1 ( s t ) ] 1 − tanh 2 [ δ 2 ( s t ) ] 3 / 2 1 / 2

Skew-Gen-t distribution: The mean of y t | ( F t − 1 , s t , Θ ) is

(A.7) μ t ( s t ) = c ( s t ) + 2 exp ( λ t ) tanh [ δ 1 ( s t ) ] [ exp [ δ 2 ( s t ) ] + 2 ] exp [ − δ 3 ( s t ) ] × B 2 exp [ δ 3 ( s t ) ] , exp [ δ 2 ( s t ) ] + 1 exp [ δ 3 ( s t ) ] B 1 exp [ δ 3 ( s t ) ] , exp [ δ 2 ( s t ) ] + 2 exp [ δ 3 ( s t ) ]

The volatility of y t | ( F t − 1 , s t , Θ ) is

(A.8) σ t ( s t ) = exp [ λ t ( s t ) ] { exp [ δ 2 ( s t ) ] + 2 } exp [ − δ 3 ( s t ) ] × 3 tanh 2 [ δ 1 ( s t ) ] + 1 B 3 exp [ δ 3 ( s t ) ] , exp [ δ 2 ( s t ) ] exp [ δ 3 ( s t ) ] B 1 exp [ δ 3 ( s t ) ] , exp [ δ 2 ( s t ) ] + 2 exp [ δ 3 ( s t ) ] − 4 tanh 2 [ δ 1 ( s t ) ] B 2 2 exp [ δ 3 ( s t ) ] , exp [ δ 2 ( s t ) ] + 1 exp [ δ 3 ( s t ) ] B 2 1 exp [ δ 3 ( s t ) ] , exp [ δ 2 ( s t ) ] + 2 exp [ δ 3 ( s t ) ] 1 / 2

Appendix B: Error specifications, log-densities, and score functions

EGB2 distribution: The error term follows the EGB2 distribution, as shown below:

(B.1) ϵ t ( s t ) ∼ EGB 2 { 0,1 , exp [ δ 1 ( s t ) ] , exp [ δ 2 ( s t ) ] }

where δ₁(s_t) and δ₂(s_t) are shape parameters. Parameters δ₁(s_t) and δ₂(s_t) influence both asymmetry and tail-heaviness of ϵ_t(s_t). The log conditional density of y_t is:

(B.2) ln f y | s ( y t | F t − 1 , s t , Θ ) = exp [ δ 1 ( s t ) ] ϵ t ( s t ) − λ t ( s t ) − ln Γ { exp [ δ 1 ( s t ) ] } − ln Γ { exp [ δ 2 ( s t ) ] } + ln Γ { exp [ δ 1 ( s t ) ] + exp [ δ 2 ( s t ) ] } − { exp [ δ 1 ( s t ) ] + exp [ δ 2 ( s t ) ] } ln { 1 + exp [ ϵ t ( s t ) ] }

The regime-dependent score function is given by:

(B.3) u t ( s t ) = { exp [ δ 1 ( s t ) ] + exp [ δ 2 ( s t ) ] } ϵ t ( s t ) exp [ ϵ t ( s t ) ] exp [ ϵ t ( s t ) ] + 1 − exp [ δ 1 ( s t ) ] ϵ t ( s t ) − 1

NIG distribution: The error term follows the NIG distribution, as shown below:

(B.4) ϵ t ( s t ) ∼ NIG { 0,1 , exp [ δ 1 ( s t ) ] , exp [ δ 1 ( s t ) ] tanh [ δ 2 ( s t ) ] }

where δ₁(s_t) and δ₂(s_t) are regime-dependent shape parameters. Parameters δ₁(s_t) and δ₂(s_t) influence tail-heaviness and asymmetry of ϵ_t(s_t). The log conditional density of y_t is:

(B.5) ln f y | s ( y t | F t − 1 , s t , Θ ) = δ 1 ( s t ) − λ t ( s t ) − ln ( π ) + exp [ δ 1 ( s t ) ] 1 − tanh 2 [ δ 2 ( s t ) ] 1 / 2 + exp [ δ 1 ( s t ) ] tanh [ δ 2 ( s t ) ] ϵ t ( s t ) + ln K ( 1 ) exp [ δ 1 ( s t ) ] 1 + ϵ t 2 ( s t ) − 1 2 ln 1 + ϵ t 2 ( s t )

where K^(j)(⋅) is the modified Bessel function of the second kind of order j. The regime-dependent score function is given by:

(B.6) u t ( s t ) = − 1 − exp [ δ 1 ( s t ) ] tanh [ δ 2 ( s t ) ] ϵ t ( s t ) + ϵ t 2 ( s t ) 1 + ϵ t 2 ( s t ) + exp [ δ 1 ( s t ) ] ϵ t 2 ( s t ) 1 + ϵ t 2 ( s t ) × K ( 0 ) exp [ δ 1 ( s t ) ] 1 + ϵ t 2 ( s t ) + K ( 2 ) exp [ δ 1 ( s t ) ] 1 + ϵ t 2 ( s t ) 2 K ( 1 ) exp [ δ 1 ( s t ) ] 1 + ϵ t 2 ( s t )

Skew-Gen-t distribution and its special cases: The error term is:

(B.7) ϵ t ( s t ) ∼ Skew − Gen − t { 0,1 , tanh [ δ 1 ( s t ) ] , exp [ δ 2 ( s t ) ] + 2 , exp [ δ 3 ( s t ) ] }

where δ₁(s_t), δ₂(s_t), and δ₃(s_t) are regime-dependent shape parameters that influence asymmetry, tail-heaviness, and peakedness of ϵ_t(s_t), respectively. The degrees of freedom parameter { exp[δ₁(s_t)] + 2} is greater than two, hence, the conditional variance of y_t is finite. The asymmetry parameter is given by tanh(τ_t) ∈ (−1, 1), as required for Skew-Gen-t.

The log conditional density of y_t for Skew-Gen-t is detailed below:

(B.8) ln f y | s ( y t | F t − 1 , s t , Θ ) = δ 3 ( s t ) − λ t ( s t ) − ln ( 2 ) − ln { exp [ δ 2 ( s t ) } + 2 ] exp [ δ 3 ( s t ) ] − ln Γ exp [ δ 2 ( s t ) ] + 2 exp [ δ 3 ( s t ) ] − ln Γ { exp [ − δ 3 ( s t ) ] } + ln Γ exp [ δ 2 ( s t ) ] + 3 exp [ δ 3 ( s t ) ] − exp [ δ 2 ( s t ) ] + 3 exp [ δ 3 ( s t ) ] ln × 1 + | ϵ t ( s t ) | exp [ δ 3 ( s t ) ] { 1 + tanh [ δ 1 ( s t ) ] sgn [ ϵ t ( s t ) ] } exp [ δ 3 ( s t ) ] × { exp [ δ 2 ( s t ) ] + 2 }

The regime-dependent score function is given by:

(B.9) u t ( s t ) = | ϵ t ( s t ) | exp [ δ 3 ( s t ) ] { exp [ δ 2 ( s t ) ] + 3 } | ϵ t ( s t ) | exp [ δ 3 ( s t ) ] + { 1 + tanh [ δ 1 ( s t ) ] sgn [ ϵ t ( s t ) ] } exp [ δ 3 ( s t ) ] { exp [ δ 2 ( s t ) ] + 2 } − 1

Appendix C: Proofs of propositions 1 to 3

Proof of Proposition 1

Express λ_t(s_t) as:

(C.1) λ t ( s t ) = ω ( s t ) + β ( s t ) λ t − 1 ( s t ) + α ( s t ) u t − 1 ( s t ) + α * ( s t ) sgn [ − ϵ t − 1 ( s t ) ] [ u t − 1 ( s t ) + 1 ] = ω ( s t ) + β ( s t ) E [ λ t − 1 ( s t − 1 ) | F t − 1 , s t ] + α ( s t ) E [ u t − 1 ( s t − 1 ) | F t − 1 , s t ] + α * ( s t ) sgn { − E [ ϵ t − 1 ( s t − 1 ) | F t − 1 , s t ] } { E [ u t − 1 ( s t − 1 ) | F t − 1 , s t ] + 1 } = ω ( s t ) + β ( s t ) ∑ i = 1 N Pr ( s t − 1 = i | F t − 1 , s t ) λ t − 1 ( s t − 1 = i ) + α ( s t ) ∑ i = 1 N Pr ( s t − 1 = i | F t − 1 , s t ) u t − 1 ( s t − 1 = i ) + α * ( s t ) sgn − ∑ i = 1 N Pr ( s t − 1 = i | F t − 1 , s t ) ϵ t − 1 ( s t − 1 = i ) × ∑ i = 1 N Pr ( s t − 1 = i | F t − 1 , s t ) u t − 1 ( s t − 1 = i ) + 1

from which E [ ⋅ | F t − 1 , s t ] is evaluated for both sides, and the following terms are presented:

(C.2) E [ λ t − 1 ( s t − 1 ) Pr ( s t − 1 | F t − 1 , s t ) | F t − 1 , s t ] = ∫ { y 1 , … , y t − 1 } λ t − 1 ( s t − 1 ) Pr ( s t − 1 | F t − 1 , s t ) g ( y 1 , … , y t − 1 | s t ) d ( y 1 , … , y t − 1 ) = ∫ { y 1 , … , y t − 1 } λ t − 1 ( s t − 1 ) g ̃ ( y 1 , … , y t − 1 | s t − 1 , s t ) Pr ( s t − 1 | s t ) d ( y 1 , … , y t − 1 ) = Pr ( s t − 1 | s t ) E [ λ t − 1 ( s t − 1 ) | F t − 2 , s t − 1 ]

(C.3) E [ u t − 1 ( s t − 1 ) Pr ( s t − 1 | F t − 1 , s t ) | F t − 1 , s t ] = ∫ { y 1 , … , y t − 1 } u t − 1 ( s t − 1 ) Pr ( s t − 1 | F t − 1 , s t ) g ( y 1 , … , y t − 1 | s t ) d ( y 1 , … , y t − 1 ) = ∫ { y 1 , … , y t − 1 } u t − 1 ( s t − 1 ) g ̃ ( y 1 , … , y t − 1 | s t − 1 , s t ) Pr ( s t − 1 | s t ) d ( y 1 , … , y t − 1 ) = Pr ( s t − 1 | s t ) E [ u t − 1 ( s t − 1 ) | F t − 2 , s t − 1 ]

where g and g ̃ are joint densities of {y₁, …, y_t−1} conditional on s_t and (s_t−1, s_t), respectively. Therefore, E [ λ t ( s t ) | F t − 1 , s t ] can be recursively constructed as follows:

(C.4) E [ λ t ( s t ) | y 1 , … , y t − 1 , s t ] = = β ( s t ) ∑ i = 1 N Pr ( s t − 1 = i | s t ) E [ λ t − 1 ( s t − 1 = i ) | F t − 2 , s t − 1 = i ] + α ( s t ) ∑ i = 1 N Pr ( s t − 1 = i | s t ) E [ u t − 1 ( s t − 1 = i ) | F t − 2 , s t − 1 = i ] + α * ( s t ) sgn − ∑ i = 1 N Pr ( s t − 1 = i | s t ) E [ ϵ t − 1 ( s t − 1 = i ) | F t − 2 , s t − 1 = i ] × ∑ i = 1 N Pr ( s t − 1 = i | s t ) E [ u t − 1 ( s t − 1 = i ) | F t − 2 , s t − 1 = i ] + 1 ≡ β ( s t ) ∑ i = 1 N Pr ( s t − 1 = i | s t ) E [ λ t − 1 ( s t − 1 = i ) | F t − 2 , s t − 1 = i ] + g t − 1 ( s t )

where g_t−1(s_t) is defined by the last equation and the probability Pr(s_t−1|s_t) is given by:

(C.5) Pr ( s t − 1 = j | s t = k ) = π j * π k * Pr ( s t = k | s t − 1 = j ) = π j * π k * p j , k

where k = 1, …, N and j = 1, …, N. Equation (C.4) in matrix representation is:

(C.6) E [ λ t ( s t = 1 ) | F t − 1 , s t = 1 ] ⋮ E [ λ t ( s t = N ) | F t − 1 , s t = N ] = β ( 1 ) π 1 * π 1 * p 1,1 … β ( 1 ) π N * π 1 * p N , 1 ⋮ ⋱ ⋮ β ( N ) π 1 * π N * p 1 , N … β ( N ) π N * π N * p N , N × E [ λ t − 1 ( s t − 1 = 1 ) | F t − 2 , s t − 1 = 1 ] ⋮ E [ λ t − 1 ( s t − 1 = N ) | F t − 2 , s t − 1 = N ] + g t − 1 ( s t = 1 ) ⋮ g t − 1 ( s t = N )

For ease of notation, we represent Equation (C.6) as follows:

(C.7) λ t ( 1 ) = Q ( 1 ) λ t − 1 ( 1 ) + u t − 1 ( 1 )

For the covariance stationarity of λ_t(s_t), asymptotically at the true values of parameters, it is necessary that the maximum modulus of eigenvalues of Q⁽¹⁾ is < 1 . □

Proof of Proposition 2

First, we focus on the following derivative:

(C.8) ∂ λ t ( s t ) ∂ α * ( s t ) = X t − 1 ( s t ) ∂ λ t − 1 ( s t ) ∂ α * ( s t ) + sgn [ − ϵ t − 1 ( s t ) ] [ u t − 1 ( s t ) + 1 ]

For the partial derivatives with respect to ω(s_t), β(s_t), and α(s_t), the following results are obtained:

(C.9) ∂ λ t ( s t ) ∂ ω ( s t ) = X t − 1 ( s t ) ∂ λ t − 1 ( s t ) ∂ ω ( s t ) + 1

(C.10) ∂ λ t ( s t ) ∂ β ( s t ) = X t − 1 ( s t ) ∂ λ t − 1 ( s t ) ∂ β ( s t ) + λ t − 1 ( s t )

(C.11) ∂ λ t ( s t ) ∂ α ( s t ) = X t − 1 ( s t ) ∂ λ t − 1 ( s t ) ∂ α ( s t ) + u t − 1 ( s t )

For Proposition 2, the same arguments are used as for Proposition 1. Thus, we define:

(C.12) Q ( 2 ) = E [ X t − 1 ( s t = 1 ) ] π 1 * π 1 * p 1,1 … E [ X t − 1 ( s t = 1 ) ] π N * π 1 * p N , 1 ⋮ ⋱ ⋮ E [ X t − 1 ( s t = N ) ] π 1 * π N * p 1 , N … E [ X t − 1 ( s t = N ) ] π N * π N * p N , N

For ease of notation, we represent Eqs. (C.8)–(C.11) as follows:

(C.13) λ t ( 2 , i ) = Q ( 2 ) λ t − 1 ( 2 , i ) + u t − 1 ( 2 , i )

for i = 1, …, 4. For the time-invariance of the expected value of G_t(Θ), asymptotically at the true values of parameters, it is necessary that the maximum modulus of eigenvalues of Q⁽²⁾ is < 1 . □

Proof of Proposition 3

First, we focus on the following derivative:

(C.14) ∂ λ t ( s t ) ∂ α * ( s t ) 2 = X t − 1 ( s t ) ∂ λ t − 1 ( s t ) ∂ α * ( s t ) + sgn [ − ϵ t − 1 ( s t ) ] [ u t − 1 ( s t ) + 1 ] 2 = X t − 1 2 ( s t ) ∂ λ t − 1 ( s t ) ∂ α * ( s t ) 2 + 2 X t − 1 ( s t ) ∂ λ t − 1 ( s t ) ∂ α * ( s t ) sgn [ − ϵ t − 1 ( s t ) ] [ u t − 1 ( s t ) + 1 ] + sgn 2 [ − ϵ t − 1 ( s t ) ] [ u t − 1 ( s t ) + 1 ] 2

For all combinations of the derivatives with respect to ω(s_t), β(s_t), α*(s_t), and α(s_t), i = 1, …, 16 equations similar to Eq. (C.14) are obtained with the same first-order dynamic parameter, i.e. X t − 1 2 ( s t ) . For Proposition 3, the same arguments as for Propositions 1 and 2 are used. We define:

(C.15) Q ( 3 ) = E X t − 1 2 ( s t = 1 ) π 1 * π 1 * p 1,1 … E X t − 1 2 ( s t = 1 ) π N * π 1 * p N , 1 ⋮ ⋱ ⋮ E X t − 1 2 ( s t = N ) π 1 * π N * p 1 , N … E X t − 1 2 ( s t = N ) π N * π N * p N , N

For ease of notation, those equations are represented as follows:

(C.16) λ t ( 3 , i ) = Q ( 3 ) λ t − 1 ( 3 , i ) + u t − 1 ( 3 , i )

for i = 1, …, 16. For the time-invariance of the information matrix, asymptotically at the true values of parameters, it is necessary that the maximum modulus of eigenvalues of Q⁽³⁾ is < 1 . □