Covariance stationary GARCH-family models with long memory property

Journal of the Korean Statistical Society 37 (2008) 29–35 www.elsevier.com/locate/jkss

Covariance stationary GARCH-family models with long memory property O. Lee ∗ , H.M. Kim Department of Statistics, Ewha University, Seoul 120-750, Republic of Korea Received 1 November 2006; accepted 1 July 2007 Available online 30 January 2008

Abstract We propose simple models which extend GARCH model and find regions of coefficients on which the given process is nonnegative covariance stationary and has long memory property. c 2008 The Korean Statistical Society. Published by Elsevier Ltd. All rights reserved.

1. Introduction One of the most popular models of financial data is the generalized autoregressive conditional heteroscedasticity (GARCH) process. The classical GARCH ( p, q) model is given by equations t = σt z t ,

σt2 = α0 +

q X

2 αi t−i +

i=1

p X

2 β j σt− j,

(1.1)

j=1

where α0 > 0, αi ≥ 0, β j ≥ 0, p ≥ 0, q ≥ 0 are model parameters and z t are independent and identically distributed random variables with zero mean. GARCH ( p, q) model can be written as an infinite moving average of the past squared returns s2 , s < t with exponentially decaying coefficients and absolutely summable exponentially decaying autocovariance functions. Empirical studies of high frequency financial data, however, show that sample autocorrelations for squared or absolute valued observations remain fairly large for vary large lags. Long memory GARCH models are desirable in light of the observed covariance structure on many real-life time series (see, e.g. Baillie, Bollerslev, and Mikkelsen (1996), Ding and Granger (1996), and Giraitis, Leipus, and Surgailis (2007)). Consequently, many authors have discussed possible ways of constructing stationary long memory models and a number of modifications of GARCH models, for example, ARCH(∞), Integrated ARCH(∞), Linear ARCH, Fractional Integrated GARCH, Fractional Integrated Exponential GARCH, Long Memory GARCH and Martingale Difference ARCH(∞), which can produce such long memory behavior have been suggested (see, Giraitis, Robinson, and Surgailis (2000), Giraitis et al. (2007), Robinson (2001), Karanasos, Psaradakis, and Sola (2004) and Koulikov (2003a,b) etc.). ∗ Corresponding author.

E-mail address: [email protected] (O. Lee). c 2008 The Korean Statistical Society. Published by Elsevier Ltd. All rights reserved. 1226-3192/$ - see front matter doi:10.1016/j.jkss.2007.07.001

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O. Lee, H.M. Kim / Journal of the Korean Statistical Society 37 (2008) 29–35

A random sequence {t } is said to satisfy ARCH(∞) equations if for each t, t = σ t z t ,

σt2 = b0 +

∞ X

2 b j t− j,

b0 > 0, b j ≥ 0, j = 1, 2, . . . .

(1.2)

j=1

The model (1.2) was introduced by Robinson (1991) and has been extensively studied by, for example, Giraitis, Kokoszka, and Leipus (2000), Kazakeviˇcius and Leipus (2002) and Zaffaroni P(2004), which includes ARCH ( p) and GARCH ( p, q). It is proved that, by using Volterra series expansion, E(z 02 ) ∞ j=1 b j < 1 if and only if the existence P of a stationary solution with E(t2 ) < ∞. E(z 04 )1/2 ∞ b < 1 if and only if the existence of fourth-order stationary j=1 j 2 solution P of (1.2) and in this case t has short memoryPin the sense that the covariance function is absolutely summable, i.e. Cov(t2 , 02 ) < ∞. Under b0 = 0 and E(z 02 ) ∞ j=1 b j < 1, the unique Volterra series expansion-type solution P of (1.2) is given by t = σt = 0. Model (1.2) with E(z 02 ) ∞ j=1 b j = 1 is called the IGARCH(∞) model which has been suggested by Engle and Bollerslev (1986). IGARCH(∞) model may have stationary solution but the stationary solution t has a clearly infinite variance. LARCH(∞) model is defined by t = σt z t ,

σt = b0 +

∞ X

b j t− j

(1.3)

j=1

and the process allows long memory but σt may be negative or may vanish (see, Davidson (2004) and Giraitis, Leipus, Robinson, and Surgailis (2001)). FIGARCH ( p, d, q) process was introduced by Baillie et al. (1996) and Ding and Granger (1996) in order to capture long memory effects in volatility and is defined by the equation t = σ t z t ,

A(L)(1 − L)d t2 = α + B(L)vt , vt = t2 − σt2 , Pp Pq where α > 0, A(L) = 1 − j=1 α j L j , B(L) = 1 − j=1 β j L j and (1 − L)d with d ∈ (0, 1) is defined as (1 − L) = F(−d, 1; 1; L) = d

∞ X j=0

(1.4)

∞   X 0( j − d) d j L = (−1) j L j , 0(−d)0( j + 1) j j=0

where F denotes the hypergeometric function, 0(·) the Gamma function and L the lag operator. FIGARCH is an example in which the lag coefficients decline hyperbolically to zero, rather than geometrically. Our goal is to propose simple models which extend GARCH model and have nonnegativity, covariance stationarity, and long memory property. 2. Long memory solution of ARCH(∞) equations Consider the process which is generated by equations t = σ t z t ,

σt2 = a +

∞ X

θ j−1 vt− j ,

vt = t2 − σt2 .

(2.1)

j=1

For the remaining part of this paper, we assume that z t is independent and identically distributed with zero mean, unit variance and E(z t4 ) < ∞. vt in (2.1) is a sequence of martingale difference innovations and hence the model (2.1) is called MD-ARCH(∞) model. We make the following assumptions: (A1) a > 0, θ j ≥P 0. 2 (A2) E(z 02 − 1)2 ∞ j=0 θ j < 1. Theorem 2.1 (Koulikov, 2003a). Under (A1) and (A2), {(t2 , σt2 )} defined in (2.1) is covariance stationary and for each t ∈ Z , k ≥ 0 and some constant M =

a 2 E(z 02 −1)2 P 2, 1−E(z 02 −1)2 ∞ j=0 θ j

we have that

O. Lee, H.M. Kim / Journal of the Korean Statistical Society 37 (2008) 29–35

E(t2 ) = E(σt2 ) = a,

2 Cov(σt+k , σt2 ) = M

∞ X

31

θ j θ j+k ,

j=0 2 2 Cov(t+k , t2 ) = Cov(σt+k , σt2 ) + M θˆk

(θˆ0 = 1, θˆk = θk−1 , k ≥ 1).

P P∞ 2 2 Therefore in addition to (A1) and (A2), if ∞ j=0 θ j < ∞ but j=0 |θ j | = ∞, then t in (2.1) is covariance P 2 ,  2 ) = ∞. stationary and has the long memory property, that is, Cov(t+k t Now consider the model t = σt z t ,

σt2 = a +

∞ X

2 π j−1 (t− j − a),

(2.2)

j=1

P where π j ≥ 0 and ∞ j=0 π j = 1. Note that the model (2.2) can be obtained from Eqs. (1.2) with b0 = 0. To find the relations between the model (2.1) and (2.2), we need the additional assumptions: P j−1 (A3) θ0 = π0 , θ j = π j + i=0 θ j−1−i πi , ∀ j ≥ 1 P P N −1 (A4) ∞ j=N (π j + i=0 θi π j−1−i ) → 0 as N → ∞. Theorem 2.2 (Koulikov, 2003b). If (A1)–(A4) hold with E(t2 ) = a > 0, then solution {(t2 , σt2 )} of the model (2.1) satisfies the equations in (2.2) and solution {(t2 , σt2 )} of the model (2.2) satisfies the relations in (2.1). 3. Stationary long memory GARCH-family processes FIGARCH model (1.4) with zero intercept can be written as   A(L) σt2 = 1 − (1 − L)d t2 B(L)   A(L) d (1 − L) (t2 − a) = a+ 1− B(L)

(3.1)

which is a type of model (2.2). LM-GARCH ( p, d, q) model is defined by   B(L) −d 2 (1 − L) − 1 vt , (3.2) σt = a + A(L) Pp Pq where a > 0, A(L) = 1 − j=1 α j L j and B(L) = 1 − j=1 β j L j such that |A(z)| > 0 and |B(z)| > 0 for all z on the closed unit disk. Model (3.2) is a type of MD-ARCH(∞) model. In most practical applications, relatively simple models like the FIGARCH (1, d, 1)-type provide a good representation of the real data. In this section, we suggest a simple GARCH-family model and find regions of coefficients of (3.1) and (3.2) on which the process is nondegenerate, nonnegative, covariance stationary and has nonsummable autocovariance function. Define a condition R0 as R0 :

E(t2 ) = a > 0,

d ∈ (0, 1/2).

3.1. LM-GARCH (0, d, 0) model Consider the following two equations: σt2 = [1 − (1 − L)d ]t2 =

∞ X

2 π j−1 t− j,

(3.1.1)

j=1

σt2 = a + [(1 − L)−d − 1]vt = a +

∞ X j=1

θ j−1 vt− j .

(3.1.2)

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O. Lee, H.M. Kim / Journal of the Korean Statistical Society 37 (2008) 29–35

Here we have that 0(1 + j + d) . (3.3) 0( j + 2)0(d) P P P Clearly, π j = 1 and (A3) holds. It can be easily shown that θ j = ∞, and θ 2j < ∞ if d ∈ (0, 1/2). A proof of (A4) can be found in Koulikov (2003b).   2 2 0(1 − 2d) R1 : E(z 0 − 1) − 1 < 1. 0(1 − d)2 P 2 2 2 0(1−2d) Since E(z 02 −1)2 ∞ j=0 θ j = E(z 0 −1) [ 0(1−d)2 −1] < 1, under R0 and R1, the solution of (3.1.2) is nonnegative, second-order stationary with long memory property and the solution also satisfies the Eq. (3.1.1). π0 = θ0 = d,

πj =

0(1 + j − d) , 0( j + 2)0(d)

θj =

3.2. LM-GARCH (1, d, 0) model Consider the LM-GARCH (1, d, 0) model which is given by σt2 = [1 − (1 − β L)−1 (1 − L)d ]t2 =

∞ X

2 π 0j−1 t− j,

(3.2.1)

j=1

σt2 = a + [(1 − β L)(1 − L)−d − 1]vt = a +

∞ X

θ 0j−1 vt− j .

(3.2.2)

j=1

For the processes (3.2.1) and (3.2.2), the following equalities can be derived. π00 = π0 − β, θ00 = θ0 − β,

π 0j = π j + βπ 0j−1 , θ 0j = θ j − βθ j−1 ,

j ≥1

(3.4)

j ≥ 1.

(3.5)

Here θ j and π j are given in (3.3). Clearly, π 0j > 0 and θ 0j > 0 for j = 0, 1, 2, . . . provided β < d and P 02 θ j < ∞. Moreover, for j ≥ N , (A3), (3.3)–(3.5) and the relation that θ N ∼ ∞ X

π 0j

j=N

+

N −1 X

! θi0 π 0j−1−i

=

i=0

=

≤

∞ X

θ 0j

−

j−1 X

j=N

j=N

∞ X

j−1 X

θj −

j=N

j=N

∞ X

j−1 X

θj −

j=N

2

θ 0j = (θ0 − β)2 +

j=0

∞ X

→ 0 as N → ∞ yields that

! θi0 π 0j−1−i ! θi π j−1−i

+ θ N −1

j ∞ X X j=0

π j−i β i+1 − β j+2

i=0

θi π j−1−i

+ βθ N −1 → 0

j=N an bn

= 1). Now

(θ j − βθ j−1 )2

j=1

 0(1 − 2d) 0(1 − 2d)0(1 + d) 2 = (1 + β ) − 1 + β 2 − 2β 2 0(d)0(1 − d)0(2 − d) 0(1 − d)   0(1 − 2d) d + 1 − 1. = β 2 − 2β 1−d 0(1 − d)2 

!

!

and hence (A4) holds (an ∼ bn as n → ∞ signifies that limn→∞ ∞ X

N d−1 (d−1)!

P

as N → ∞,

θ 0j = ∞ and

O. Lee, H.M. Kim / Journal of the Korean Statistical Society 37 (2008) 29–35

33

Let R2 : β < d,

E(z 02 − 1)2



 0(1 − 2d) 2 d (β − 2β + 1) − 1 < 1. 1−d 0(1 − d)2

Consequently, on the region R0 and R2, (A1)–(A4) hold and the process generated by (3.2.1) or (3.2.2) is stationary and has long memory (see, Fig. 1(a)). 3.3. LM-GARCH (0, d, 1) model Define σt2 = [1 − (1 − αL)(1 − L)d ]t2 =

∞ X

2 π 00j−1 t− j,

(3.3.1)

j=1

σt2 = a + [(1 − L)−d (1 − αL)−1 − 1]vt = a +

∞ X

θ 00j−1 vt− j .

(3.3.2)

j=1

Then we have that π000 = π0 + α, θ000 = θ0 + α,

π 00j = π j − απ j−1 ,

j ≥1

θ 00j = θ j + αθ 00j−1 ,

Note that π 00j > 0 if α <

(3.6)

j ≥ 1.

(3.7)

1−d 2 .

From (3.6) and (3.7) and simple calculation, we prove that ! ! j−1 ∞ N −1 ∞ X X X X 00 00 00 πj + θj − θi π j−1−i = θi π j−1−i − αθ N00 −1 → 0 as N → ∞, j=N

j=N

i=0

j=N

since θ 00j = θ j · F(1, − j − 1; −d − j; α) ∼ Assume that R3 : α <

1−d , 2

E(z 02 − 1)2

j d−1 (1−α)(d−1)!

∞ X

→ 0 as j → ∞.

θ 2j · F(1, − j − 1; −d − j; α)2 < 1.

j=0

It follows that on R0 and R3, the process {t2 } of (3.3.1) and (3.3.2) is stationary and has nonsummable autocovariance functions (see, Fig. 1(b)). 3.4. LM-GARCH (1, d, 1) model Let σt2

  ∞ X (1 − αL) d 2 (1 − L) t2 = π ∗j−1 t− = 1− j, (1 − β L) j=1

(3.4.1)

σt2

 ∞ X (1 − β L) −d =a+ (1 − L) − 1 vt = a + θ ∗j−1 vt− j . (1 − αL) j=1

(3.4.2)



We can derive the equations π0∗ = π000 − β, θ0∗ = θ000 − β,

π ∗j = π 00j + βπ 00j−1 + β 2 π 00j−2 + · · · + β j π000 − β j+1 , θ ∗j = θ 00j − βθ 00j−1 .

(3.8) (3.9)

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O. Lee, H.M. Kim / Journal of the Korean Statistical Society 37 (2008) 29–35

Using (3.8) and (3.9), we obtain that ! ! j−1 ∞ N −1 ∞ X X X X π ∗j + θi∗ π ∗j−1−i ≤ θi∗ π ∗j−1−i + θ N00 −1 β θ ∗j − j=N

i=0

=

j=N

j=N

∞ X

j−1 X

θj −

j=N

→0

! θi π j−1−i

− (α − β)θ N00 −1

j=N

as N → ∞.

Define the region R4 by R4 : α <

1−d , 2

β < d,

E(z 02 − 1)2

∞ X

θ ∗j 2 < 1,

j=0

where j=0 θ ∗j 2 = (1 + β 2 ) j=0 θ 00j 2 − 2β j=0 θ 00j θ 00j−1 + β 2 and θ 00j = θ j · F(1, − j − 1; −d − j; α). Thus under R0 and R4 the desired result, that is the stationarity and the long memory property of the process given by (3.4.1) (or (3.4.2)), follows. (See, Fig. 1(c) and (d) for the case that d = 0.2 and d = 0.4.) P∞

P∞

P∞

(a) LM-GARCH (1, d, 0).

(b) LM-GARCH (0, d, 1).

(c) LM-GARCH (1, d, 1), d = 0.2.

(d) LM-GARCH (1, d, 1), d = 0.4.

Fig. 1. Each shaded area shows the region of coefficients on which {t2 } is covariance stationary and has long memory property.

We can derive, in the same manner as above, regions of the coefficients of the higher-order processes such as LMGARCH(2, d, 0), LM-GARCH (2, d, 1), LM-GARCH (2, d, 2) etc. on which the stationarity and the long memory property are guaranteed. Acknowledgement This research was supported by grant KRF-2005-015-C00091.

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