This paper presents a 2-regime SETAR model for the volatility with a long-memory process in the first regime and a short-memory process in the second regime. Persistence properties are studied and estimation methods are proposed. Such a process is applied to stock indices and individual asset prices.SETAR - Long-memory - FARIMA models - Stock indices

Anne Peguin-Feissolle

Dominique Guegan

Gilles Dufrénot

English

Research Papers in Economics

Modelling squared returns using a
SETAR model with long-memory
dynamics∗†
Gilles DUFRENOT‡ Dominique GUEGAN§
Anne PEGUIN-FEISSOLLE¶
July 7, 2004
Abstract
This paper presents a 2-regime SETAR model for the volatility
with a long-memory process in the ﬁr s tr e g i m ea n das h o r t - m e m o r y
process in the second regime. Persistence properties are studied and
estimation methods are proposed. Such a process is applied to stock
indices and individual asset prices.
Keywords: SETAR - Long-memory - FARIMA models - Stock in-
dices.
JEL classiﬁcation: C32, C51, G12
∗Corresponding author: Anne PEGUIN-FEISSOLLE, GREQAM, Centre de la Vieille
Charit´ e, 2 rue de la Charit´ e, 13002 Marseille, France, tel: +33.4.91.14.07.35, fax:
+33.4.91.90.02.27, Email: peguin@ehess.univ-mrs.fr
†A ﬁrst version of this paper was presented at the International Conference on Fore-
casting Financial Markets, Paris, June 2003, and at the International Workshop ”Models
with breaks in Economics and Finance: Recent Developments”, Paris, November 2003.
We gratefully acknowledge the comments from the participants to these conferences. We
also thank the Conseil R´ egional of PACA for ﬁnancial support and Aldric Pouron for pro-
viding us with the data. Helpful comments and suggestions from a referee and the editor
are also gratefully acknowledged.
‡ERUDITE, Universit´ e Paris 12 and GREQAM, France
§Ecole Normale Sup´ e r i e u r ed eC a c h a n ,M O R A - I D H E ,F r a n c e
¶GREQAM-CNRS, France
11I n t r o d u c t i o n
It is common wisdom that asymmetric volatility is a characteristic of ﬁnan-
cial markets. The asymmetric dynamics takes diﬀerent forms: regimes of
high and low volatilities vary both in size and duration according to market
microstructures, the price-volume relationships varies with economic expan-
sions and recessions variables, etc.. A variety of alternative models has been
proposed in the literature to account for this phenomena. In this paper, we
propose to describe the regimes of the volatility of returns using a SETAR
model with one regime characterized by a long-memory dynamics and the
other by weak dependence:
(
(1 − B)dXt = ε
(1)
t , if Xt−1 ≤ c :r e g i m e 1
Xt = ε
(2)
t , if Xt−1 >c:r e g i m e 2 ,
(1)
or
(
Xt = ε
(1)
t , if Xt−1 ≤ c :r e g i m e 1
(1 − B)dXt = ε
(2)
t , if Xt−1 >c:r e g i m e 2 ,
(2)
where d ∈ (0,1/2) is a fractional diﬀerence parameter, ε
(i)
t ,i=1 ,2 are strong
white noises with ﬁnite variances, B is the backward shift operator. This
model relies upon three series of arguments put forward in previous papers
of the literature.
The ﬁrst point is that SETAR-type models can be more convenient than
standard GARCH family models to describe volatility clustering in stock
returns, if the volatility series exhibit diﬀerent serial autocorrelation orders
when the volatility is large and when it is small (see, among others, Cao
and Tsay (1992), Li and Li (1996), Brooks (2001)). Usually, it is assumed
that, in all the regimes, the volatility process is driven by short-memory AR
models. In using our SETAR process, we introduce a persistence dynamics
in at least one regime.
The second point concerns the separation between the two regimes. Some
authors in the literature have argued that the persistence of volatility depends
upon the size of the shocks (see, for instance, Friedman and Laibson (1989)
and Schwert (1990). Since, in general, large shocks imply a higher risk on the
assets than ordinary shocks, our regimes in equation (1) refer to a ”high risk
regime” and ”low risk regime”. Equation (2) illustrates another situation.
Pockets of predictability may exist when high volatility is observed and the
long-memory process captures clustering dynamics in high volatilities. Our
model can be viewed as an alternative approach to the literature based on
2multiple factors models of volatility such as in Gallant, Hsu and Tauchen
(1999); these authors indeed proposed a three factor model that mimics the
long-memory behavior of ﬁnancial volatility.
Thirdly, how to model the persistence of volatility is still an open ques-
tion in the literature. Engle and Lee (1999) propose a decomposition of a
GARCH(2,2) model into permanent and transitory components. Andersen
et al. (2003) suggest an approach based on realized volatility series applied
to long-memory Gaussian processes and obtain better forecasts in compari-
son with more standard ARCH processes. In the recent years, some authors
have suggested the combination of nonlinear and long-memory models; our
paper can be seen as a further contribution in this sense (see, Franses and
Paap (2002), Franses, Van der Leij and Paap (2002), van Dijk, Franses and
Paap (2002), Gu´ egan (2000, 2003)).
T h ep a p e ri so r g a n i z e da sf o l l o w s .I ns e c t i o n2 ,w eb r i e ﬂy introduce the
properties of the model and propose simple methods to locate the parameter
c. Section 3 adapts the model to three series of returns and a comparison with
ARFIMA models is made using Diebold and Mariano (1995)’s forecastibility
criterion. Section 4 concludes.
2 The properties of the model and methods
to locate the threshold parameter
Regime 1 in model (1) is referred as the ”long-memory” regime in Granger
(1980)’s sense. We say that a stationary process (Xt)t, whose autocovariance
function is γX,i sl o n gm e m o r yi f ,∀t and ∀τ,
γX(τ) ∼ C(d)τ
2d−1,a sτ →∞ , (3)
where 0 <d<1/2a n dC(d) is a constant which depends only on d.U n d e r
the assumption that ε
(i)
t ,i =1 ,2 are strong white noises with ﬁnite vari-
ances, model (1) is both locally (in each regime) and globally stationary and
invertible. Its autocorrelation function, γX(τ),can be written as:
γX(τ)=C(τ,d)N1(c)+γε(0)N2(c), (4)
where
C(τ,d)=
Γ(1 − 2d)Γ(τ + d)
Γ(d)Γ(1 − d)Γ(τ +1− d)
. (5)
N1(c)a n dN2(c) are the percentages of observations, respectively in regime 1
a n di nr e g i m e2a n dd e p e n du p o nt h et h r e s h o l dc. γε(0) is the variance of the
3noise (ε
(2)
t )t. The theoretical autocorrelation function is thus a mixture of the
autocorrelation of the long-memory model in regime 1 and of the variance of
the white noise model in regime 2.
Even in a simple formulation such as ours, the estimation of the model
parameters (d,c) is substantially more diﬃcult than in the cases of ARFIMA
models or standard SETAR models. Ideally, one would like to apply here
methods based on maximum likelihood approach (the Whittle estimator),
but this is not feasible because the model is piecewise linear. Further, it is
not possible to apply the variety of methods suggested for standard SETAR
models because of the presence of the fractional parameter in one regime.
The approach we adopt here is as follows.
• One estimates the value of the threshold parameter c.
• The observations are separated into two sub-groups according to the
estimated value of c and one deduces N1(c), N2(c).
• Classical methods are applied to each sub-group to obtain an estima-
tion of the fractional parameter (GPH, non-parametric methods, Whit-
tle). If the model (1) is adequate to describe the data, then one must
ﬁnd a signiﬁcant parameter d for one sub-group and a non signiﬁcant
parameter for the other sub-group.
The crucial point concerns step 1 where the parameter c has to be locate
We ﬁrst construct the time series ( e Xt)t of arranged observations according
to the decreasing values of Xt−1 and then proceed as follows:
1. One considers a set of s1 initial observations of ( e Xt)t and estimate the
long-memory parameter and the corresponding t − ratio: ts1.
2. The vector ( e Xt)t is then incremented in such a way to contain s2,s 3,
...,s n observations; new long-memory parameters and their t − ratios
are computed: ts2,t s3,... , tsn.
3. Consider the set of estimated t-ratios {ts1,t s2,t s3,...,tsn}.O n et e s t sf o r
the presence of a structural break e t in the view of ﬁnding a sequence of
t-ratios such that, for instance, for tsk ≤ e t, the estimated long-memory
parameters are not statistically signiﬁc a n t ,w h i l e ,f o rtsk > e t,t h e ya r e
signiﬁcant.
A simple way to do this is to use a standard Chow test. The series of t-
ratios is regressed on a linear time trend, using incremented dummy variables:
4for k =1 ,2,...,n
tskt =( α + βDt)+( γ + δDt)t + ut, (6)
where ut is a strong white noise and
Dt =
½
1,i f t ≤ e t
0,o t h e r w i s e
We test the null hypothesis H0 : β = δ = 0 against the alternative H1 : β 6=0
or δ 6=0 . The test is implemented by considering diﬀerent values of e t and
ﬁnally one retains the value yielding the lowest p-value. Instead of using the
Chow test, one can also compute the sum of squares residuals corresponding
to the equation and select the t-ratio (and thus the threshold value) yielding
the lowest sum (see Dufr´ enot, Gu´ egan and P´ eguin-Feissolle (2003)).
3 Empirical application
In this section we apply the model to two individual assets (ST-GOBAIN
and TOTAL) and a stock index (FTSE) using daily data over the period
1998-2003. We consider the squared absolute returns
Xt = |log(pt) − log(pt−1)|
2 , (7)
where pt is the price index at time t. Table 1 shows our results. The data
successfully detect the presence of two distinct regimes in the volatility of the
three stock indices. To save place, we report the results for which the change
point in the t-ratios is obtained using the Chow-based method (the approach
based on the minimum sum of squares yields similar results). Further, we re-
port the estimations of the fractional parameters based on the GPH method:
b d corresponds to a simple ARFIMA model on the whole series, and b d1 and
b d2 to the fractional parameters of both regimes in model (1) or (2); b c is the
value of the threshold parameter, N1 and N2 are the number of points in the
regimes and the last row gives the corresponding model (1) or (2).
Several comments are in order.
• Firstly, comparing the t-ratios of the fractional parameter in the two
regimes, we ﬁnd evidence that both the models (1) and (2) are able to
capture the state-dependent dynamics of the volatility. For instance, in
the case of ST-GOBAIN, the low volatility regime is characterized by
some predictable slow dynamic adjusting behavior, whereas for TOTAL
the volatility is highly persistent in the regime of large past volatility.
5• A second point is that, for these two indices, we approximately have
the same number of observations within each regimes. The implica-
tions for market analysts is that strategies based on technical analysis
(charts, market timing, dynamic asset allocation) and those based on
investment strategies (for instance, portfolio insurance) are not mu-
tually exclusive. The former can be used when a persistent volatility
generating process is detected and the latter during the periods of un-
predictable volatilities.
• A third interesting point concerns the diﬀerence between an aggregate
stock index like FTSE and disaggregated ﬁrms’ stock indices. Although
we ﬁnd a two-regime model for the FTSE index, it is seen that the
number of points in the white noise regime is only approximately 10% of
the sample. The upshot of our result is that we might expect our model
to apply to the volatility of the ﬁrms’ assets rather than to aggregate
indices. One reason can be that our ”on/oﬀ” persistent volatility model
captures the inﬂuence of market microstructures that would not be
found when individual assets are aggregated.
Finally, it is interesting to enquire as to whether a model where the per-
sistence of the volatility is allowed in one regime only produces forecasts
that are superior to a one-regime long-memory model. Table 2 reports some
results of predictive accuracy based on Diebold and Mariano (1995)’s test
statistics. The number in bold correspond to cases where the SETAR model
yields forecasts that are statistically diﬀerent from those obtained with a
ARFIMA model. The number n in the table indicates the percentages of
points for which the residuals of our model are inferior to those of the one-
regime ARFIMA model. As is seen, the numbers are higher than 50%, which
suggests that our model oﬀers an alternative competing framework to de-
scribe the persistence of the volatility.
4C o n c l u s i o n
This paper has proposed a new model to investigate the long-memory dy-
namics of volatility that contains a mixture of long-memory and white noise
structures. Other further research topics are the following. Firstly, one can
conjecture that the model suggested here can be extended to the case where
regime 1 is described by an ARFIMA model and regime 2 by a station-
ary ARMA model or by a mixing process. Secondly, the model can be
applied to time series other than returns; for instance, it seems worthwhile
to investigate the case corresponding to |log(pt) − log(pt−1)|
4:i ti saw a yt o
6measure the kurtosis of its underlying distribution. Thirdly, the SETAR
model with long-memory behavior can be studied under the assumption of
heteroskedastic errors in order to incorporate the inﬂuence of volatility of
the long-memory structure. Finally, it would be interesting to apply our
model to realized volatility, using for instance price-range as in Alizadeh et
al. (2002), and compare its performance with other models developed in the
literature.
7References
[1] Alizadeh, S., M. Brandt and F. Diebold, 2002, Range-based estimation
of stochastic volatility models, Journal of Finance 57, 3, 1047-1091.
[2] Andersen, T.G., J. Bollerslev, F.X. Diebold and P. Labys, 2003, Mod-
elling and forecasting realized volatility, Econometrica 71, 579-625.
[3] Brooks, C., 2001, A double-threshold GARCH model for the
Franc/Deutschemark exchange rate, Journal of Forecasting 20, 135-140.
[4] Cao, C.Q. and R.S. Tsay, 1992, Nonlinear time series analysis of stock
volatilities, Journal of Applied Econometrics 7, 165-185.
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Journal of Business and Economic Statistics, 253-263.
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with long-memory property, International Conference on Forecasting Fi-
nancial Markets, Paris, June.
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stock returns volatility, in: R. Engle and H. White, eds., Cointegration,
causality an forecasting: a Festchrift in honor of Clive W.J. Granger,
475-497 (Oxford University Press).
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with an application to US unemployment, Journal of Applied Econo-
metrics 17, 347-366.
[9] Franses, P.H., M. van Der Leij and R. Paap, 2002, Modelling and fore-
casting level shifts in absolute returns, Journal of Applied Econometrics
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9Table 1. Parameter estimates for absolute squared returns
ST GOBAIN TOTAL FTSE
b d 0.0919 0.1066 0.3165
t-ratio (3.09) (2.51) (8.90)
b c 0.0001 0.0002 0.0006
N1 606 740 1236
b d1 0.0904 0.0560 0.2907
t-ratio (3.29) (1.37) (6.90)
N2 734 600 102
b d2 0.1064 0.1615 0.1835
t-ratio (1.93) (3.06) (1.49)
Model (1) (2) (1)
Note:The change point in the t-ratios was obtained using the method based on the
Chow test. b d1 and b d2 refer to the estimated fractional parameter, respectively in regime
1 and regime 2. b d is the estimated fractional parameter on the whole series. The t-ratio
must be compared 1.96 (corresponding to the critical value at the 5% level of signiﬁcance).
An o n -s i g n i ﬁcant parameter indicates that the volatility is driven by a white noise process
(and is thus unpredictable). Conversely, a signiﬁcant parameter means that the volatility
exhibit a long-memory dynamics therefore yielding to a high predictability. The last row
shows the corresponding adequate model (1) or (2)
10Table 2. Test of predictive accuracy:
our model versus an ARFIMA model (p-values)
AS SI WI NB MGN MR n
STGOBAIN 0.067 0.000 0.403 0.155 0.310 0.113 55.19%
TOTAL 0.000 0.396 0.232 0.041 0.082 0.113 51.16%
FTSE 0.488 0.000 0.009 0.199 0.391 0.250 55.57%
Note: The diﬀerent columns are: AS: Asymptotic test, SI: Sign test, WI: Wilcoxon’s
test, NB: Naive benchmark test, MGN: Morgan-Granger-Newbold’s test, MR: Meese-
Rogoﬀ’s test, n: number of times in percent where the residuals coming from the TAR
model with a long memory regime are smaller than the residuals coming from a standard
long memory model (when n>50%, it means that the TAR model seems the best). The
null hypothesis is the hypothesis of equal accuracy of diﬀerent predictive methods. The
loss function is quadratic. The test statistics follows asymptotically diﬀerent distributions:
N(0,1) for the asymptotic test, the sign test, the Wilcoxon’s test, the Meese-Rogoﬀ’s test,
F(T0,T 0) for the Naive benchmark test and a tT0−1 for the Morgan-Granger-Newbold’s
test (where T0 is the number of predicted observations, i.e. T0 =2 0 ). The Meese-Rogoﬀ
test statistic is computed with the Diebold-Rudebusch covariance matrix estimator. The
truncation lag is 10 for the asymptotic test and is given by the integer part of T
4/5
0 for
the Meese-Rogoﬀ’s test.
11

Modelling squared returns using a SETAR model with long-memory dynamics

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Modelling squared returns using a SETAR model with long-memory dynamics

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