The Formula of Unconditional Kurtosis of Sign-Switching GARCH(p,q,1) Processes

Vol. 12 ( 2012) 105−110

Submitted October 25, 2012 ISSN

Accepted December 20, 2012 1234-3862

Joanna Górka

*

The Formula of Unconditional Kurtosis

of Sign-Switching GARCH(p,q,1) Processes

A b s t r a c t. In the paper we argue that a general formula for the unconditional kurtosis of sign-switching GARCH(p,q,k) processes proposed by Thavaneswaran and Appadoo (2006) does not give correct results. To show that we revised the original theorem given by Thavaneswaran and Appadoo (2006) for the special case of the GARCH(p,q,k) process, i.e. GARCH(p,q,1). We show that the formula for the unconditional kurtosis basing on the original theorem and the revised version is different.

K e y w o r d s: Kurtosis, sign-switching GARCH models. J E L Classification: C22.

Introduction

In the article „Properties of a New Family of Volatility Sing Models” Thavaneswaran and Appadoo (2006) proposed a general formula for the uncon-ditional kurtosis of the sign-switching GARCH(p,q,k) process (Fornari, Mele, 1997). Unfortunately, the proposed general formula of kurtosis does not give correct results. The formula for the unconditional kurtosis of the process de-rived from the Theorem 2.1 a) in Thavaneswaran and Appadoo (2006) is not the same as the formula obtained without using this theorem (see equation 9 in For-nari and Mele (1997) or equation 27 in Górka (2008)).

1. Introductory Remarks

The general sign-switching GARCH(p,q,k) model is described by equations (Fornari, Mele, 1997):

*

Correspondence to: Department of Econometrics and Statistics, Nicolaus Copernicus Uni-versity, Gagarina 13a, Toruń, Poland E-mail: joanna.gorka@umk.pl.

t t t y =σ ε , (1) 2 2 2 1 1 1 q p k t i t i j t j l t l i j l y s σ ω α ₋ β σ _{− −} = = = = +

∑

+

∑

+

∑

Φ , (2) where ε_t~ . . .(0,1)i i d , ω> ,0 α_i≥ ,0 β_j ≥ , 0

∑

Φ ≤ ,_l ω 1 for 0 0 for 0 1 for 0 t t t t y s y y >   =_ = . − <  If 2 2 t t t

u =y −σ is the martingale difference with variance 2

var( )u_t =σ_u, the model (1)–(2) can be interpreted as ARMA(m,q) with the sign function for the

2

t

y and can be written as:

2 2 1 1 1 ( ) p m k t i i t i j t j l t l t i j l y ω α β y₋ β u₋ s₋ u = = = = +

∑

+ −

∑

+

∑

Φ + , (3) or 2 1 ( ) ( ) k t t l t l l B y B u s φ ω β ₋ = = + +

∑

Φ , (4) where 1 1 ( ) 1 ( ) 1 m m i i i i i i i B B B φ α β φ = = = −

∑

+ = −

∑

, 1 ( ) 1 p j j j B B β β = = −

∑

, max

m= {p q}, , α_i= for 0

i

>

q

and β_i = for j p0 > .

The stationarity assumptions for 2

t

y specified by (4) are the following

(Thavaneswaran, Appadoo, 2006):

(Z.1) All roots of the polynomial ( )φ B = lie outside the unit circle. 0

(Z.2) 2 0 i i ψ ∞ = < ∞

∑

, where the ψ_i are coefficients of the polynomial

1 ( ) 1 i i i B B ψ ∞ ψ =

= +

∑

satisfying the equation ( ) ( )ψ Bφ B =β( )B .

Assumptions (Z.1)–(Z.2) ensure that the variance of u is finite and that the t

2

t

y

process is weakly stationary.

Assume that k =1. Then the equation (4) has the form:

2

1 1

( )B y_t ( )B u_t s_t

φ = +ω β + Φ ₋. (5)

If the assumptions (Z.1)–(Z.2) are satisfied, then the above equation can be converted to the form:

2

1 1

( ) ( ) ( )

t t t

where 1 ( ) 1 i i i B B ψ ∞ ψ =

= +

∑

satisfies the equation ( ) ( )ψ Bφ B =β( )B , and

1 ( ) 1 _i i i B B π ∞ π =

= +

∑

satisfies the equation ( ) ( ) 1π B φ B = .

2. Author’s Results

The theorem presented below is the revised version of the part a) of the Theorem 2.1 presented in Thavaneswaran and Appadoo (2006) but for the spe-cial case of the GARCH(p,q,k) process, i.e GARCH(p,q,1).

Theorem. Suppose the y is a sign-switching GARCH(p,q,1) process specified t

by (1)–(2) and satisfying the assumptions (Z.1)–(Z.2), with a finite fourth

mo-ment and a symmetric distribution of ε_t. Then the unconditional kurtosis of the

process y is given by: t

2 2 2 2 4 1 0 2 2 _{4 4 2} 0 1 t i t i t _{t t i} i E _E K E _{E E} σ π _ε σ _{ε ε ψ} ∞     _ _     _ _ = ∞    _{   }  _ _ _{   }       = + Φ = ⋅ .   −_ − _

∑

∑

(7)

Proof. A kurtosis of the process y described by equations (1)–(2) can be writ-_t

ten as: 4 4 4 4 4 2 2 2 2 2 2 2 t t t t t t t t t E y E E K E E y E E ε σ σ ε ε σ σ                                   _ _  _ _  _ _       = = = . (8)

We note that by definition of the u (_t u_t =y_t2−σ_t2) it follows that:

( )

( )

_{2 2}

( )

2 4 4 4 4 4 4 4 4 4 4 0 var 1 t t u t t t t t t t t t t t t E u u E u E u E y E E E E E E E E σ σ ε σ σ σ ε σ σ ε       _ _                                                                     _{ }   = , = = − = − = − = −   = − .  

Let us indicate that the variance of the process 2

t

y , satisfying the assumptions of

2 2 2 2 2 1 0 0 4 4 _{2 2 2} 1 0 0 var 1 t u i i i i t t i i i i y E E σ ψ π σ ε ψ π ∞ ∞       = = ∞ ∞               _{= =} = + Φ   = − +Φ .  

∑

∑

∑

∑

(9)

On the other hand, this variance can be calculated from the equation (1). We get then 2 2 4 2 2 4 4 2 2 2 4 4 2 var _{t t t} t t t t t t t y E y E y E E E E E ε σ ε σ σ ε σ                                    _{  }                     _{  }     = − = − = − . (10)

Comparing the results of (9) and (10) we receive:

2 4 4 2 2 2 4 4 2 1 0 0 1 t t i i t t t i i E σ E ε ψ π E σ E ε E σ ∞ ∞                                    = =  ₋  _{+ Φ = − .}  

∑

∑

Hence, 2 4 4 4 2 2 2 2 1 0 0 2 2 2 2 4 ₁ 4 4 2 0 2 2 2 ₀ 2 1 1 t t t i i t i i i t t _i t t i i t t E E E E E E E E E E σ ε ε ψ π σ π σ σ ε ε ψ σ σ ∞ ∞                                     = =   ∞         _{ ∞   }    _{      }   =  _ _ _ _       _{ = }                 −_ − _ = Φ + , Φ +   −_ − _ =

∑

∑

∑

∑

_     , 2 2 2 2 4 ₁ 0 2 2 2 2 _{4 4 2} 0 1 1 t i t _i t t _{t t i} i E E E E _{E E} σ π σ σ σ _{ε ε ψ} ∞         _{ }       = ∞       _{   }  _ _  _ _ _{   }         = + Φ = ⋅ .   −_ − _

∑

∑

(11) Substituting (11) to (8) we obtain: 2 2 2 2 4 1 0 2 2 _{4 4 2} 0 1 t i t i t _{t t i} i E E K E _{E E} σ π _ε σ _{ε ε ψ} ∞     _ _     _ _ = ∞    _{   }  _ _ _{   }   _{   } = + Φ = ⋅ .   −_ − _

∑

∑



If Φ = , then the formula (7) of the unconditional kurtosis process is re-₁ 0

duced to the formula of the unconditional kurtosis processes generated by

ap-propriate GARCH models (see the Theorem 2 1. in Thavaneswaran et al.,

Example. The example concerns the sign-switching GARCH(1,1,1) model with

normal distribution, i.e.

2 2 2 1 1 1 1 1 1. t t t t t t t y y s σ ε σ ω α ₋ β σ _{− −} = , = + + + Φ (12)

If u_t =y_t2−σ_t2 is the martingale difference with variance var( )u_t =σ_u2, the

model (12) is following 2 2 1 1 1 1 1 1 1 ( ) t t t t t y = +ω α +β y₋ + −u βu₋ + Φs₋ . (13)

Then the polynomials (see the equation (5)) have the form:

1 1

( ) 1 (B )B

φ = − α +β , β( ) 1B = −β₁B. The individual weights ψ are

follow-ing: ψ₁= , α₁ ψ₂=α α₁( ₁+β₁), …, 1

1( 1 1)

i i

ψ ₌α α ₊β − _,

…. The weights π are:

1 1 1 π =α +β , π₂=(α₁+β₁)2, …, π_i=(α₁+β₁)i, …. If condition (Z.2) is satis-fied, then 2 1 1 (α +β) < and then: 1 2 1 2 1 1 2 2 2 2 2 4 1 1 1 1 1 1 1 0 i 1 ( ) ( ) 1 1 ( ) i … α α β ψ α α α β α α β ∞ = = + + + + + + = + − +

∑

, 2 1 1 2 2 4 6 1 1 1 1 1 1 1 0 i 1 ( ) ( ) ( ) 1 ( ) i π α β α β α β … α β ∞ = = + + + + + + + = − +

∑

.

Assuming that ε_t N(0 1), and substituting into (7) we obtain:

(

)

(

)

(

)

(

)

2 1 2 1 1 _{1 1} 2 1 2 1 1 1 1 2 1 1 ( ) 2 1 1 ( ) 2 2 2 2 1 1 1 1 1 2 2 2 1 1 1 3 3 2 1 1 ( ) 1 ₃ 1 ( ) 2 K ω α β α β α ω α β α β ω α β α β ω α β α Φ − − − + − − − +       + = ⋅ − + − + + Φ − − = ⋅ − + −

(

)

2 2 2 2 1 1 1 1 1 2 2 2 1 1 1 1 3 1 ( ) 1 1 2 3 ω α β α β ω α β β α        _ _            − + + Φ − − = . − − − (14)

This result is the same like the formula of the unconditional kurtosis obtained by Fornari and Mele (1997) and by Górka (2008) but it is different from the result obtained by Thavaneswaran and Appadoo (2006). Nonetheless, in each

case, if Φ = then the formula (14) reduces to a formula for the unconditional ₁ 0

kurtosis of the GARCH (1,1) process.

References

Fornari, F., Mele, A. (1997), Sign- and Volatility-switching ARCH Models: Theory and Applica-tions to International Stock Markets, Journal of Applied Econometrics, 12, 49–65. Górka, J. (2008), Description the Kurtosis of Distributions by Selected Models with Sing

Thavaneswaran, A., Appadoo, S. S. (2006), Properties of a New Family of Volatility Sing Mod-els, Computers & Mathematics with Applications, 52, 809–818.

Thavaneswaran, A., Appadoo, S. S., Samanta, M. (2005b), Random Coefficient GARCH Models,

Mathematical & Computer Modelling, 41, 723–733.

Wzór na bezwarunkową kurtozę procesu generowanego

przez model sign-switching GARCH(p,q,1)

Z a r y s t r e ś c i. W artykule zauważono, że na podstawie wzoru na bezwarunkową kurtozę procesu GARCH(p,q,k) zaproponowanego przez Thavaneswarana i Appadoo (2006) nie otrzymu-jemy poprawnych wyników. Dlatego też w niniejszej pracy przedstawiono poprawioną formułę twierdzenia Thavaneswarana i Appadoo (2006) dla szczególnego przypadku procesu GARCH(p,q,k), tzn. GARCH(p,q,1). Wykazano, że formuła na bezwarunkową kurtozę procesu generowanego przez model sign-switching GARCH(1,1,1) bazująca na oryginalnym twierdzeniu i poprawionej wersji jest inna.