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Memory and infrequent breaks
a b ,
*
Christian Gourieroux , Joann Jasiak
a
CREST and CEPREMAP, Paris, France b
Department of Economics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3 Received 9 December 1999; accepted 15 June 2000
Abstract
We study how processes with infrequent regime switching may generate a long memory effect in the autocorrelation function. In such a case, the use of a strong fractional I(d) model for economic or financial analysis may lead to spurious results. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Long memory; Switching regime; Heavy tail
JEL classification: C22
1. Introduction
Inference on the dynamics of economic or financial time series is usually based on the autocorrelation function whose decay pattern is used to assess the persistence range of processes. Those, displaying a geometric decay rate are modelled as Autoregressive Moving Averages whereas strong fractional I(d) models are used to fit hyperbolic decay rates of so-called long memory processes. However the analysis adequate for linear dynamics may often become misleading if the true underlying dynamics is nonlinear. This point is of special importance for the ‘long memory’ property, which is often observed in macroeconomic series (see, e.g. the study by Diebold and Rudebusch (1989)), and financial series of either volatility of returns (Ding et al., 1993; Andersen and Bollerslev, 1997), or intertrade durations (Gourieroux and Jasiak, 1998; Jasiak, 1999).
Even though the hyperbolic decay of the autocorrelation function [acf henceforth] of long memory processes is a stylized fact, it is widely known that the estimated decay rate may not be accurate, and that linear formulas of the type:
d (12L ) Yt5et
*Corresponding author. Tel.: 11-416-736-5083; fax:11-416-736-5987.
E-mail address: [email protected] (J. Jasiak).
where et is a strong white noise, show poor performance in terms of nonlinear prediction criteria. 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
0 1 0 1 0
of durations spent in the two possible states is denoted by: D ,D ,D ,D , . . . ; D is the duration of1 1 2 2 j
th 1 th
the j spell of regime 0, whereas Dj is the duration of the j spell of regime 1. The successive
0 0 1 0 1 0 p 0 1
regime switching dates are: t15D ,1 t25D11D ,1 t35D11D11D , . . . ,2 t2 p5oj51(Dj 1D ).j
0 1 0
We assume that the various durations D ,D , j varying are independent, and that the duration D , jj j j 1
varying [resp. D , j varying] follow the same distribution F [resp. F ]. These distributions arej 0 1
*
discrete with values in N .
*
basic durations due to the censoring effect of the window of observations.
2.2. The autocovariance function
The empirical autocovariance function is given by:
can distinguish these terms depending if they are in the same spell or in different spells of this type. We denote (t, t2h)[I , if t and tj,T 2h belong to two spells of regime 0 separated by j spells of
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 residualT
1
where (D2h) 5max(D2h,0), or equivalently:
1
Under standard ergodicity conditions, the various terms converge to constant limits:
NT
]
lim T 5p, i.e. the limiting switching probability T→`
1 1
NT
]
lim N 5a , i.e. the limiting proportion of spells in regime 1
T
ˆ
autocovariance function is the same as the pattern of E(D 2h) considered as a function of h.
1 1 ` ¯ ¯
It is well known that: E(D 2h) 5eh F (u) du, where F is the survivor function associated with1 1
1
F , and that this quantity measures the magnitude of the tails of D . Therefore the pattern of the1
autocovariance function is directly related to the type of tails of the duration distribution F . For1
2d
¯
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 toj 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 tail1
*
parameterd, 1,d ,2, F of Pareto type with tail parameter0 d .d, then the autocorrelation function 3 admits a hyperbolic decay of degree d21. We infer that the estimated autocorrelogram likely features a hyperbolic decay at a rate varying betweend andd21 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 T529040. The duration distributions F and F are identical Pareto distributions with mean 200, and tail parameter0 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:
Yt5X (10,t 2Z )t 1X Z1,t t (4)
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 5m 1(m 2m ) Z 1sU (12Z )1s U Z , where
0 0 1,t 1 1 t 0 1 0 t 0 0,t t 1 1,t t
2
This aspect is discussed in Appendix 1 where we compute explicitly the term corresponding to I1,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
Fig. 1. Autocorrelogram of the binary series.
U ,U0,t 1,t are the associated standard gaussian variables. The quantitative series Y will likely share some properties of temporal dependence of Z if m ±m . To illustrate this point we provide in Fig. 2
0 1
the autocorrelation function [acf, henceforth] of Y for mt 050, m153, s05s151, and in Fig. 3 the 2
acf of Y and Y for mt t 05m150, s051, s153.
Fig. 2. Autocorrelogram for switching mean.
Cov Y ,Ys t t2hd 5 Cov E Y
f s
tuZ , E Yd s
t2huZdg
1ECov Y ,Ys
t t2huZd
5 Cov mfs 02m Z , m1d t s 02m Z1d t2hg 2
5 sm 2m d Cov Z ,Zs d, for h±0
0 1 t t2h
VYt 5 VE Y
s
tuZd
1EVE Ys
tuZd
2 2 2
2
5
f
sm02m1d 1s
s02s1d g
VZtTherefore 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 (m02m ) /(1 s02s1) .
3. Asymptotics for small switching probabilities
Fig. 3. Autocorrelograms for switching variance.
observations, which allows us to focus on the effect of infrequent switches on memory, and provides a tractable limit model.
3.1. The finite sample model and the limiting model
j
For a finite sample of size T, we consider a standard transition model, where p , jT 50,1 denotes the transition probability from state j to the complementary state between consecutive dates. These probabilities depend on T as well as on the corresponding distributions of durations of the two
0 1
regimes. To emphasize this dependence, we index by T the various variables, i.e.tn,T, Dj,T, Dj,T. . .
0 1
The duration variables are independent with geometric distributions of parameters pT and p ,T 5
respectively. Under the assumption:
0 1
lim TpT5l0, lim TpT5l1 (A.1.)
T→` T→`
it is known that the finite sample process tends in distribution to a limiting process after an appropriate
0 1
change of time. More precisely the normalized durations Dj,T/T and Dj,T/T tend in distribution to
0 1
limiting duration variables Dj,`, Dj,`, which are independent with exponential distributions with respective parametersl0, l1. The normalized observation period becomes [0,1], whereas the limiting
5
j j
variables lim tn,T/T5tn,`, lim NT5N , lim A /T` T 5A` are deduced from the limiting durations as
T→` T→` T→`
they were in the finite sample model.
Note that in the limiting model the number of switching dates N` is finite, but stochastic. For instance in the special case l15l0, the distribution of N is a Poisson distribution with a parameter`
l15l0.
3.2. Limiting behaviour of the autocorrelogram
We study directly the asymptotic properties of the autocorrelogram or equivalently of the empirical
ˆ
autocovariance function. In fact gT(h) tends in distribution to a limiting variable g`(h) which is not degenerate. This variable admits a mixture distribution, which takes into account the limiting number of switching dates. More precisely let us consider the first possible number of regimes and distinguish the cases of fixed and large lags h.
(i) If there is only one limiting regime N`50, with probability p0, say, we get:
ˆ
gT(h)50,;h, and alsog`(h)50,;h
(ii) If there are two limiting regimes, N`51, with probability p1, we get:
0 2
The limiting pattern corresponds to a constant function with a stochastic level.
0 1
The limiting pattern corresponds to a piecewise linear function in the lag, featuring a rather slow decay which may be confused with a hyperbolic decay in practice.
• (iii) If there are three limiting regimes, N`52, with probability p2, we get:
And so on. It is important to note that the limiting distributions depend on the initial regime fixed at 0 in our computations.
3.3. Simulation
To illustrate the limiting behaviour of the empirical autocovariance we have simulated a series of
0 1
length T53000, with switching probabilities pT5pT50.005. Despite the large number of observa-tions, the limiting distributions of the first and second order moments are not reduced to a point mass (Figs, 4 and 5).
The effect of state transitions on the pattern of the autocorrelogram and also on the estimated fractional degree can be observed by considering the joint distribution of estimated autocorrelations at two different lags (Fig. 6).
These limiting distributions may explain the large variation of the estimator of fractional coefficient for high frequency data, where the d parameter is estimated by rolling, since the estimated value is very sensitive to the subsample it is computed from.
Fig. 5. Distribution of autocorrelations.
4. Concluding remarks
To discuss the relation between infrequent breaks and long memory, we focused our study on the estimated autocorrelogram, rather than on the estimation of the fractional degree from the au-tocorrelogram. Indeed the main difficulties encountered in practice can be due to the estimated autocorrelogram itself, and not to the approach used to derive the parameter d. We first observe that long memory may be found in nonlinear time series with infrequent breaks, and that, in the limiting case of very small switching probabilities, the estimated autocorrelogram may converge to a nondegenerate distribution.
Therefore the hyperbolic decay rate of the estimated autocorrelograms may result from nonlinear dynamics with infrequent switching regimes and not from linear fractional dynamics with i.i.d. innovations.
Fig. 6. Joint distribution of autocorrelations.
Finally note that it is a common practice to round up the series to a given number of decimals either for computational ease, or due to market rules, for example, in asset prices which are multiples of a fixed tick. These rounding procedures introduce artificial switching regimes, which may create spurious long memory in the observed series.
Acknowledgements
The second author gratefully acknowledges financial support of the Natural Sciences and Engineering Research Council of Canada.
Appendix 1. Second term of the autocovariance expansion
1
]
Let us consider the term T ot,t2h[I Z Zt t2h5A1,T (say).
1,t
Z Zt t2h51
We deduce (up to the final truncation effect which is negligible):
1
If the lag h is small with respect to the values taken by the durations spent in regime 0, we get:
1
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