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http://www.dem.umk.pl/dem 

DYNAMIC ECONOMETRIC MODELS 
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. 



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 105–110 

106 

t t ty 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 sσ ω α β σ− − −
= = =

= + + + Φ ,∑ ∑ ∑  (2) 

where ~ . . .(0,1)t i i dε ,  0 0 0i jω α β> , ≥ , ≥ ,  l ωΦ ≤ ,∑   

1 for 0
0 for 0
1 for 0

t

t t

t

y
s y

y

>
= = .
− <

 

 If 2 2
t t tu y σ= −  is the martingale difference with variance 2var( )t uu σ= , the 

model (1)–(2) can be interpreted as ARMA(m,q) with the sign function for the 
2
ty  and can be written as:  

2 2

1 1 1
( )

pm k

t i i t i j t j l t l t
i j l

y y u s uω α β β− − −
= = =

= + + − + Φ + ,∑ ∑ ∑  (3) 

or  

2

1
( ) ( )

k

t t l t l
l

B y B u sφ ω β −
=

= + + Φ ,∑  (4) 

where 
1 1

( ) 1 ( ) 1
m m

i i
i i i

i i
B B Bφ α β φ

= =

= − + = −∑ ∑ , 
1

( ) 1
p

j
j

j
B Bβ β

=

= −∑ ,  

maxm {p q}= , , 0iα =  for i q>  and 0iβ =  for j p> .  
 The stationarity assumptions for 2

ty  specified by (4) are the following 
(Thavaneswaran, Appadoo, 2006):  

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

(Z.2) 2

0
i

i
ψ

∞

=

< ∞∑ , where the iψ  are coefficients of the polynomial 

1
( ) 1 i

i
i

B Bψ ψ
∞

=

= +∑  satisfying the equation ( ) ( ) ( )B B Bψ φ β= .  

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

process is weakly stationary.  
 Assume that 1k = . Then the equation (4) has the form:  

2
1 1( ) ( )t t tB y B u sφ 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 ty B B u B sπ ω ψ π −= + + Φ ,  (6) 



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

DYNAMIC ECONOMETRIC MODELS 12 (2012) 105–110 

107 

where 
1

( ) 1 i
i

i
B Bψ ψ

∞

=

= +∑  satisfies the equation ( ) ( ) ( )B B Bψ φ β= , and 

1
( ) 1 i

i
i

B Bπ π
∞

=

= +∑  satisfies the equation ( ) ( ) 1B Bπ φ = .  

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 ty  is a sign-switching GARCH(p,q,1) process specified 
by (1)–(2) and satisfying the assumptions (Z.1)–(Z.2), with a finite fourth mo-
ment and a symmetric distribution of tε . Then the unconditional kurtosis of the 
process ty  is given by:  

22 2 2
41

0
22 4 4 2

0
1

t i
ti

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 ty  described by equations (1)–(2) can be writ-
ten as:  

4 4 4 4
4

2 2 22 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 tu  ( 2 2
t t tu y σ= − ) it follows that:  

( )
( ) ( ) 22 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
ty , satisfying the assumptions of 

the Theorem and described by the equation (6), is given by:  



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 105–110 

108 

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  

22 4 2

24 4 2 2

24 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:  
24 4 2 2 2 4 4 2

1
0 0

1t t i i t t t
i i

E E E E Eσ ε ψ π σ ε σ
∞ ∞

          
          

          
= =

 − + Φ = − . ∑ ∑  

Hence,  
24 4 4 2 2 2 2

1
0 0

22 2 2
4 1

4 4 2 0
2 22 20

1

1

t t t i i t
i i

i t
t i

t t i
i

t t

E E E E

EE
E E

E E

σ ε ε ψ π σ

π σσ
ε ε ψ

σ σ

∞ ∞ 
         

          
         

= = 

∞
  
    ∞            =

    
         =     

    

 − − = Φ + , 

Φ +
 − − = 

∑ ∑

∑
∑


 
 

,

 

22 2 2
4 1

0
2 22 2 4 4 2

0

1

1

t i
t i

t t t t i
i

EE

E E E E

σ πσ

σ σ ε ε ψ

∞
  
   

   
  =

∞                              
=

+ Φ
= ⋅ .

 − − 

∑

∑
 (11) 

Substituting (11) to (8) we obtain:  
22 2 2

41
0

22 4 4 2

0
1

t i
ti

t t t i
i

E E
K

E E E

σ π ε

σ ε ε ψ

∞
  
    

    
 =

∞                  
=

+ Φ
= ⋅ .

 − − 

∑

∑
 

  
 If 1 0Φ = , then the formula (7) of the unconditional kurtosis process is re-
duced to the formula of the unconditional kurtosis processes generated by ap-
propriate GARCH models (see the Theorem 2 1.  in Thavaneswaran et al., 
(2005)).  



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

DYNAMIC ECONOMETRIC MODELS 12 (2012) 105–110 

109 

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 2 2
t t tu y σ= −  is the martingale difference with variance 2var( )t uu σ= , the 

model (12) is following  
2 2

1 1 1 1 1 1 1( )t t t t ty y u u sω α β β− − −= + + + − +Φ .  (13) 

Then the polynomials (see the equation (5)) have the form: 
1 1( ) 1 ( )B Bφ α β= − + , 1( ) 1B Bβ β= − . The individual weights ψ  are follow-

ing: 1 1ψ α= , 2 1 1 1( )ψ α α β= + , …, 1
1 1 1( )i

iψ α α β −= + , …. The weights  π  are: 

1 1 1π α β= + , 2
2 1 1( )π α β= + , …, 1 1( )i

iπ α β= + , …. If condition (Z.2) is satis-
fied, then 2

1 1( ) 1α β+ <  and then:  
2
1

2
1 1

2 2 2 2 2 4
1 1 1 1 1 1 10 1 ( )

1 ( ) ( ) 1ii
… α

α β
ψ α α α β α α β∞

= − +
= + + + + + + = +∑ ,

2
1 1

2 2 4 6 1
1 1 1 1 1 10 1 ( )

1 ( ) ( ) ( )ii
…

α β
π α β α β α β∞

= − +
= + + + + + + + =∑ .  

Assuming that (0 1)t Nε ,  and substituting into (7) we obtain:  

( )
( ) ( )

( )

2
1

21 1 1 1
2
1

2
1 1 1 1

2

1 1 ( )
2

1 1 ( )

22 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

αω
α β α β

ω α β α β

ω α β α

Φ
− − − +

− − − +

 
  

+
= ⋅

− +

− + +Φ − −
= ⋅

− + −

 

( )22 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 1 0Φ =  then the formula (14) reduces to a formula for the unconditional 
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 Func-

tion, Dynamic Econometric Models, 8, 39–49. 



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 105–110 

110 

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. 

S ł o w a  k l u c z o w e: Kurtoza, model sign-switching GARCH.  


	Introduction
	1. Introductory Remarks
	2. Author’s Results
	References

