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. Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41
where e is a strong white noise, show poor performance in terms of nonlinear prediction criteria.
t
Following a recurrent idea see Klemes, 1974; Balke and Fomby, 1989; Lobato and Savin, 1997; Granger, 1999; Granger and Hyung, 1999; Granger and Terasvirta, 1999; Diebold and Inoue, 1999,
this paper aims to explain how infrequent stochastic breaks can create strong persistence in the estimated serial correlation. We develop two approaches. In Section 2 we consider a stationary
switching regime model with rather large spells of each regime, and directly relate the decay rate of autocorrelations to the tails of the duration distributions. In Section 3, we introduce a dynamic model
in which the switching probabilities tend to zero when the number of observations tends to infinity. The associated limiting model features a finite, but random number of regime switches. This limiting
model is used to derive the asymptotic properties of the estimated autocorrelogram. We conclude in Section 4.
2. Switching regimes with large spells
In this section we describe a regime switching model with two regimes 0 and 1. We assume that the durations of the successive spells of regimes 0 and 1 are independent and identically distributed when
they correspond to the same regime. Next, we study the dynamic properties of the binary process indicating regime 1, and especially the pattern of its autocorrelation function.
2.1. Assumption and notation We introduce a time origin 0 and assume that the system is in regime 0 at this date. The sequence
1 1
of durations spent in the two possible states is denoted by: D ,D ,D ,D , . . . ; D is the duration of
1 1
2 2
j th
1 th
the j spell of regime 0, whereas D
is the duration of the j spell of regime 1. The successive
j 1
1 p
1
regime switching dates are: t 5 D , t 5 D 1 D , t 5 D 1 D 1 D , . . . ,t 5
o D 1 D .
1 1
2 1
1 3
1 1
2 2 p
j 51 j
j 1
We assume that the various durations D ,D , j varying are independent, and that the duration D , j
j j
j 1
varying [resp. D , j varying] follow the same distribution F [resp. F ]. These distributions are
j 1
discrete with values in N . The binary process Z , t [ N is defined by:
t
Z 5
1, if there exist p such that t
, t t
t 2 p 21
2 p
0, otherwise
During a given time interval h1, . . . ,T j there is a number N of switching dates, N of spells of regime
T T
1 1
0, N of spells in regime 1, with N 1 N 5 N 1 1. The total time spent in regime 0 [resp. regime 1]
T T
T T
1 1
1
is denoted by A [resp. A ] with A 1 A 5 T. Note that A and A are not necessarily sums of the
T T
T T
T T
basic durations due to the censoring effect of the window of observations. 2.2. The autocovariance function
We now discuss the pattern of the autocovariance function of the binary process Z. For this purpose we first consider the empirical autocovariance, and next its limit obtained when the number of
observations tends to infinity.
C . Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41
31
The empirical autocovariance function is given by:
T 2
1 T 2 h
¯ ˆ
] ]]
g h 5
O
Z Z 2
Z
T t
t 2h T
T T
t 5h 11
The cross terms Z Z take value one if and only if t and t 2 h both belong to spells of regime 1. We
t t 2h
can distinguish these terms depending if they are in the same spell or in different spells of this type. We denote t, t 2 h [ I , if t and t 2 h belong to two spells of regime 0 separated by j spells of
j,T
regime 0. Then we get:
`
1 T 2 h
2
¯ ˆ
] ]]
g h 5
O O
Z Z 2
Z
T t
t 2h T
F G
T T
j 50 t,t 2h [I
j,T
`
1 1
T 2 h
2
¯ ]
] ]]
5
O
Z Z 1
O O
Z Z 2
Z
t t 2h
t t 2h
T
T T
T
t,t 2h [I j 51
t,t 2h [I
0,T j,T
When the durations spent in regime 0 and 1 are rather large, i.e. the switching dates are not very frequent, we can expect that the second term in the decomposition is small with respect to the first
one. Therefore we focus the attention on the first and the third terms, and denote by R h the residual
T 1
second term. We get:
1
N 1
2
T
A 1
T 2 h
T 1
1
S D
ˆ ]
]] ] g h 5
O
D 2 h 1 R h 2 1
T i
T
T T
T
i 51 1
where D 2 h 5 maxD 2 h,0, or equivalently:
1
N 1
1 2
T
N N A
1 T 2 h
T T
T 1
1
S D
ˆ ] ] ]
]] ] g h 5
O
D 2 h 1 R h 2 2
T 1
i T
T N T
T N
T i 51
T
Under standard ergodicity conditions, the various terms converge to constant limits:
N
T
]
lim 5
p, i.e. the limiting switching probability
T T
→ `
1
1 N
T
]
lim 5
a , i.e. the limiting proportion of spells in regime 1
N
T
T →
`
1 1
N N
T T
1 1
1 1
1 A
N N
1 1
T T
T
] ]
] ] ]
lim 5lim
O
D 5 lim
O
D 5 pa ED
1
i i
1 T
T T N
N
T
T →
` T
→ `
T
i 51 i 51
lim R h 5 R h, say
T `
T →
`
Therefore we get:
1 1
We neglect the possible censoring of the last duration D , without loss of generality.
1
N
T
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. Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41
ˆ gh
5 lim g h
T T
→ `
3
1 1
2 2
1 2
5 pa ED 2 h 1 R h 2 p a ED
1 `
1 1
1
Whenever R h is small with respect to the first term pa ED 2 h , the pattern of the
` 1
1 1
autocovariance function is the same as the pattern of ED 2 h considered as a function of h.
1 1
`
¯ ¯
It is well known that: ED 2 h 5 e F u du, where F is the survivor function associated with
h 1
1 1
F , and that this quantity measures the magnitude of the tails of D . Therefore the pattern of the
1
autocovariance function is directly related to the type of tails of the duration distribution F . For
1 2
d
¯ instance the autocovariance function has an hyperbolic decay if the survivor function F u | u
for
1
large u, i.e. if the duration distribution is of Pareto type. The above reasoning is valid if the second term R h is sufficiently small, i.e. when, intuitively, h
` 0 2
is small with respect to the values of D . For example, if the expected duration of regime 0 is close to
j
200, we get an insight into the decay rate for h 500, which corresponds to degree d. This result completes the property by Heath et al. 1997, who investigated the limiting behaviour of the
autocorrelation function when h tends to infinity. They prove that if F is of Pareto type with tail
1
parameter d, 1 , d , 2, F of Pareto type with tail parameter d . d, then the autocorrelation function
3
admits a hyperbolic decay of degree d 2 1. We infer that the estimated autocorrelogram likely
features a hyperbolic decay at a rate varying between d and d 2 1 depending on the window selected
4
for the lag. 2.3. Simulation
As an illustration we consider a simulated path of a binary series Z, of length T 5 29040. The duration distributions F and F are identical Pareto distributions with mean 200, and tail parameter
1
1.5. We provide in Fig. 1 the estimated autocorrelogram for h 200, which clearly features a slow decay typical for long memory processes see Granger and Terasvirta 1999 for a similar pattern with
a model with endogenous switching regimes. Moreover we observe smooth waves for h 200, indicating different hyperbolic rates of decay over varying ranges of lags.
Similar patterns are obtained when the regimes are perturbed by a noise. Let us consider the series defined by:
Y 5 X 1 2 Z 1 X Z 4
t 0,t
t 1,t
t
where X ,X are independent variables, independent of Z , with gaussian distributions: X
|
0,t 1,t
t 0,t
2 2
N[m ,s ], X
| N[m ,s ]. We
get: Y 5 m 1 m 2 m Z 1 s U 1 2 Z 1 s U Z ,
where
1,t 1
1 t
1 t
0,t t
1 1,t
t
2
This aspect is discussed in Appendix 1 where we compute explicitly the term corresponding to I .
1,T 3
Note that for financial tick by tick data, autocorrelations are often computed up to lag 1000 or 2000. In our case, where average intertrade durations are very short, this would span two or three trading days in calendar time.
4
Note that the above results are different from the properties derived by Cioczek-Georges and Mandelbrot 1995, Taqqu and Levy 1986, Taqqu et al. 1997 and Willinger et al. 1997, who derive the fractional Brownian motion by averaging
renewal processes. In our case, the results are derived from one single path.
C . Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41
33
Fig. 1. Autocorrelogram of the binary series.
U ,U are the associated standard gaussian variables. The quantitative series Y will likely share
0,t 1,t
some properties of temporal dependence of Z if m ± m . To illustrate this point we provide in Fig. 2
1
the autocorrelation function [acf, henceforth] of Y for m 5 0, m 5 3, s 5 s 5 1, and in Fig. 3 the
t 1
1 2
acf of Y and Y for m 5 m 5 0, s 5 1, s 5 3.
t t
1 1
We see that the long memory effect is still observed in the presence of breaks in the mean, and it disappears when breaks occur in the variance. Indeed in the latter case the Y process is a martingale
difference sequence and therefore a weak white noise. However the persistence effect is observed from the acf of its squares. As noted by Granger and Marmol 1998, ‘a frequent property of data,
particularly in the financial area, is that the correlogram is low but remains positive for many lags’. To take into account this stylized fact, Granger and Marmol 1998 proposed to add a simple noise to a
long memory process, while Ghysels et al. 1997 introduced nonlinear unobservable factors. Figs. 2 and 3 show that noisy infrequent breaks can create this specific effect. This phenomenon can easily be
explained. Indeed, we obtain:
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. Gourieroux, J. Jasiak Economics Letters 70 2001 29 –41
Fig. 2. Autocorrelogram for switching mean.
Cov Y ,Y 5
Cov E Y uZ , E Y
uZ 1 ECov Y ,Y uZ
f s d s
dg s
d
s d
t t 2h
t t 2h
t t 2h
5 Cov m 2 m Z , m 2 m Z
fs d
s d
g
1 t
1 t 2h
2
5 m 2 m
Cov Z ,Z , for h ± 0
s d
s d
1 t
t 2h
VY 5
VE Y uZ 1 EVE Y uZ
s d
s d
t t
t 2
2 2 2
5 m 2 m
1 s 2 s
VZ
f s
d g
s d
1 1
t
Therefore the autocorrelation function of Y is proportional to the autocorrelation function of Z, for
2 2
2 2
h 1, with a factor related to the ratio m 2 m s 2 s .
1 1
3. Asymptotics for small switching probabilities
