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Preface
Long–rangedependent,orlong–memory,timeseriesarestationarytimeseries
displaying a statistically significant dependence between very distant obser-
vations. We formalize this dependence by assuming that the autocorrelation
function of these stationary series decays very slowly, hyperbolically, as a
function of the time lag.
Many economic series display these empirical features: volatility of asset
prices returns, future interest rates, etc. There is a huge statistical literature
on long–memory processes, some of this research is highly technical, so that
it is cited, but often misused in the applied econometrics and empirical eco-
nomics literature.The first purpose of this book is to presentin a formaland
pedagogicalway some statistical methods for studying long–rangedependent
processes.
Furthermore, the occurrence of long–memory in economic time series
might be a statistical artefact as the hyperbolic decay of the sample autocor-
relation function does not necessarily derive from long–range dependent pro-
cesses. Indeed, the realizations of non-homogeneousprocesses, e.g., switching
regime and change–point processes, display the same empirical features. We
thus also present in this book recent statistical methods able to discriminate
between the long–memory and change–point alternatives.
Going beyond the purely statistical analysis of economic series, it is of
interest to determine which economic mechanisms are generating the strong
dependence propertiesofeconomicseries,whether they aregenuine,orspuri-
ous. The regularities of the long–memory and change–pointproperties across
economic time series, e.g., common degree of long–range dependence and/or
common change–points, suggest the existence of a common economic cause.
A promising approach is the use of the class of micro–basedmodels in which
the set of economic agents is represented as a (self)–organizing and interact-
ing society whose composition evolves over time, i.e., something resembling
the realizationofa non–homogeneousstochastic process.Some of these mod-
els, inspired by the works of entomologists, are able to mimic some empirical
propertiesoffinancialandnon–financialmarkets.Thisimplicitlysuggeststhat
VI Preface
whatis termedas“long–memory”ineconomicsis morecomplex thanastan-
dard (nonlinear) long–range dependent process, and mastering a wide range
of statistical tools is a great asset for studying economic time series.
Thisvolume startswiththe chapter byLiudas Giraitis,Remis Leipus and
Donatas Surgailis, who have been awardedthis year the Lithuanian National
Prize for Science for their work on “Long–memory: models, limit distribu-
tions, statistical inference”. Donatas Surgailis and his numerous former stu-
dents have made, and are still making, essential contributions to this topic.
Their (encyclopedic) survey chapter reviews some recent theoretical findings
on ARCH type volatility models. They focus mainly on covariance station-
ary models which display empirically observed properties which have come
to be recognized as ”stylized facts”. One of the major issues to determine is
whetherthecorrespondingmodelforsquaresr2 ofARCHsequenceshaslong–
k
memoryorshortmemory.ItispointedoutthatforseveralARCH-typemodels
the(cid:1)behavior of Cov(rk2,r02) alone is sufficient to derive the limit distribution
of (r2 −Er2) and statistical inferences, without imposing any additional
k k k
(e.g.mixing)assumptionsonthedependencestructure.Thisfirstchapteralso
discusses ARCH(∞) processes and their modifications such as linear ARCH,
bilinearmodelsandstochasticvolatility,regimeswitchingstochasticvolatility
models, randomcoefficient ARCH and aggregation.They give an overviewof
the theoretical results on the existence of a stationary solution, dependence
structure, limit behavior of partial sums, leverage effect and long–memory
property of these models. Statistical estimation of ARCH parameters and
testing for change-points are also discussed.
Bhansali, Holland and Kokoszka consider a new and an entirely different
approachtomodelingphenomenaexhibitinglong–memory,intermittencyand
heavy-tailed marginal distributions, namely through the use of chaotic inter-
mittencymaps.Thisclassofmapshaswitnessedconsiderabledevelopmentin
recent years and it represents an important emerging branch of the subject
area of Dynamical Systems Theory. Three principal properties of these maps
are relevant and these properties qualify them as a plausible class of models
for financial returns. First, unlike some of the standard chaotic maps, the in-
termittency maps display long–memory and have correlations decaying at a
sub-exponential rate, meaning at a polynomial rate or even slower.Secondly,
the invariant density of these maps can display ’Pareto’ tails and thus go to
zero at a polynomial rate. Thirdly, as their name implies, these maps display
intermittency and generate time series, called the orbit of the map, which
display intermittent chaos, meaning the orbit of the map alternates between
laminar and chaotic regions.
Brousseauanalyzesthe time seriesofthe euro–dollarexchangerateasthe
realization of a continuous–time physical process, which implies the use of
different degrees of time resolution. The analysis takes into account various
statistical indicators, but puts special emphasis on the spectrum of the pro-
cess. Brousseaufinds that this spectrum has an identifiable pattern, which is
a core characteristic of the process. Then he simulates a process having the
Preface VII
same spectrum, and the behavior of the actual process and of the simulated
process are compared using various statistical indicators. It appears that the
simulatedprocessprovidesagood,butnotperfect,replicationofthebehavior
of the actual euro–dollar exchange rate.
The next two chapters deal with the issue of change–point detection.
Raˇckauskas and Suquet present the invariance principle by Donsker and
Prokhorov,whichcanbeusedforanalyzingstructuralchanges.Theyfocuson
invarianceprincipleswithrespecttoHo¨ldertopologies,asHo¨lderspacesbring
out well variations properties of processes.They presentsome applications of
the Ho¨lderian invariance principles to the problem of testing the stability of
a sample against epidemic change–points alternatives.
LavielleandTeyssi`ereconsiderthemultiplechange–pointproblemfortime
series, including strongly dependent processes, with an unknown number of
change–points. They propose an adaptive method for finding the sequence
of change–points τ with the optimal level of resolution. This optimal seg-
mentation is obtained by minimizing a standard penalized contrast function
J(τ,y)+βpen(τ).Theadaptiveprocedureissuchthattheoptimalsegmenta-
tiondoesnotstronglydependonthepenalizationparameterβ.Thisalgorithm
is applied to the problem of detection of change–points in the volatility of fi-
nancialtimeseries,andcomparedwiththebinarysegmentationprocedureby
Vostrikova.
Thechapterby Henryis onthe issue ofbandwidthselectionforsemipara-
metric estimatorsof the long–memoryparameter.The spectralbasedestima-
tors are derived from the shape of the spectral density at low frequencies,
where all but the lowest harmonics of the periodogram are discarded. This
allows one to ignore the specification of the short range dynamic structure of
the time series, and avoid the bias incurred when the latter is misspecified.
Such a procedureentails anorder of magnitude loss of efficiency with respect
to parametric estimation, but may be warranted when long series (earth sci-
entificorfinancial)canbeobtained.Thischapterpresentsstrategiesproposed
for the choice of bandwidth, i.e., the number of periodogramharmonics used
in estimation, with the aim of minimizing this loss of efficiency.
Teyssi`ere and Abry present and study the performance of the semipara-
metric wavelet estimator for the long–memory parameter devised by Veitch
and Abry, and compare this estimator with two semiparametric estimators
in the spectral domain: the local Whittle and the “log–periodogram” esti-
mators. The wavelet estimator performs well for a wide range of nonlinear
long–memoryprocessesinthe conditionalmeanandthe conditionalvariance,
and is reliable for discriminating between change–points and long–range de-
pendence in volatility. The authors also address the issue of the selection of
the range of octaves used as regressors by the weighted least squares esti-
mator. It appears that using the feasible optimal bandwidths for either the
spectral estimators, surveyed by Henry in the previous chapter, is a useful
ruleofthumbforselectingthelowestoctave.Thewaveletestimatorisapplied
to volatility and volume financial time series.
VIII Preface
Kateb,Seghier and Teyssi`erestudy a fast versionof the Levinson–Durbin
algorithm, derived from the asymptotic behavior of the first column of the
inverse of T (f), the (N+1)×(N+1) Toeplitz matrix with typical element
N
f,thespectraldensityofalong–memoryprocess.TheinversionofT (f)with
N
Yule–Walker type equations requires O(N3) operations, while the Levinson–
Durbin algorithm requires O(N2) elementary operations. In this chapter, an
asymptotic behavior of (T (f)), for large values of N is given so that the
N
computations of the inverseelements are performed in O(N) operations.The
numericalresultsarecomparedwiththosegivenbytheLevinson–Durbinalgo-
rithm,withparticularemphasisonproblemsofpredictingstationarystochas-
tic long–range dependent processes.
Gaunersdorfer and Hommes study a simple nonlinear structural model of
endogenousbelief heterogeneity.News about fundamentals is an independent
andidenticallydistributedrandomprocess,butneverthelessvolatilitycluster-
ing occurs as an endogenous phenomenon caused by the interaction between
different types of traders, fundamentalists and technical analysts. The belief
types are drivenby adaptive,evolutionarydynamics accordingto the success
of the predictionstrategies as measuredby accumulatedrealizedprofits,con-
ditioned upon price deviations from the rational expectations fundamental
price. Asset prices switch irregularly between two different regimes – periods
ofsmallpricefluctuationsandperiodsoflargepricechangestriggeredbyran-
dom news and reinforced by technical trading – thus, creating time varying
volatility similar to that observed in real financial data.
Cont attempts to model the volatility clustering of asset prices: large
changes in prices tend to cluster together, resulting in persistence of the
amplitudes of their changes. After recalling various methods for quantify-
ing and modeling this phenomenon, this chapter discusses several economic
mechanisms which have been proposed to explain the origin of this volatility
clustering in terms of behavior of market participants and the news arrival
process. A common feature of these models seems to be a switching between
lowandhighactivity regimeswith heavy-taileddurationsof regimes.Finally,
a simple agent-based model is presented, which links such variations in mar-
ket activity to threshold behavior of market participants and suggests a link
between volatility clustering and investor inertia.
Kirmans chapter is devoted to an account of the sort of micro-economic
founded models that can give rise to long memory and other stylised facts
about financial and economic series. The basic idea is that the market is
populated by individuals who have different views about future prices. As
time unfolds they may change their ways of forecasting the future. As they
do so they change their demands and thus prices. In many of these models
the typical rules are “chartist” or extrapolative rules and those based on the
idea that prices will revert to some “fundamental” values. In fact, any finite
number of rules can be considered. The switching between rules may result
in “herding” on some particular rule for a period of time and give rise to
longmemoryandvolatilityclustering.The firstmodelswereusedtogenerate
Preface IX
data which was then tested to see if the stylised facts were generated. More
recently theoretical results have been obtained which characterise the long
run distribution of the price process.
Alfarano and Lux present a very simple model of a financial market with
heterogeneous interacting agents capable of reproducing empirical statistical
properties of returns. In this model, the traders are divided into two groups,
fundamentalists and chartists, and their interactions are based on herding
mechanism. The statistical analysis of the simulated data shows long-term
dependence in the auto-correlations of squared and absolute returns and hy-
perbolic decay in the tail of the distribution of the raw returns, both with
estimateddecayparametersinthesamerangelikeempiricaldata.Theoretical
analysis, however, excludes the possibility of “true” scaling behavior because
oftheMarkoviannatureoftheunderlyingprocessandthefinitesetofpossible
realized returns.
ThepurposeofthechapterbydePerettiistodeterminewhetherhysteretic
series can be confused with long–memory series. The hysteretic effect is a
persistenceintheserieslikethelong–memoryeffect,althoughhystereticseries
are not mean reverting whereas long–memory series are. Hysteresis models
have in fact a short memory, since dominant shocks erase the memory of the
series,andthepersistenceisduetopermanentandnon-revertingstatechanges
atamicrostructurelevel.Inorderto checkwhetherhysteretic modelsdisplay
spuriouslong–rangedependence,amodelpossessingthehysteresispropertyis
usedforsimulatinghystereticdatatowhichstatisticaltestsforshort–memory
against long–memory alternatives are applied.
We wish to thank all contributors to this book, all the referees for their
careful and fast reports which made the selection of the chapters easier, the
Springer Verlag editor, Dr Martina Bihn, for efficient and patient manage-
ment, and Mrs Ruth Milewski for correcting the first series of manuscripts
and the final version. Teyssi`ere managed chapters 1 to 8 and 10, while Kir-
man managed chapters 9, 11, 12 and 13.
Gilles Teyssi`ere thanks Paul Doukhan for his help to return to the aca-
demic system on serious grounds, and the SAMOS (Statistique Appliqu´ee et
MOd´elisationStochastique)UniversityParis1.He wantsto expresshis grati-
tude to Rosa Maria to whom he owes bunches of flowersas the completion of
this volumespilt overinto their spare time. AlanKirmanwouldlike to thank
the other people with whom he has investigated this sort of problem, and in
particular,HansFoellmer,UlrichHorst,RichardTopolandRomainRicciotti.
Paris, Gilles Teyssi`ere
June 2006 Alan Kirman
Contents
Part I Statistical Methods
Recent Advances in ARCH Modelling
Liudas Giraitis, Remigijus Leipus, Donatas Surgailis ................. 3
Intermittency, Long–Memory and Financial Returns
Raj Bhansali, Mark P. Holland, Piotr S. Kokoszka ................... 39
The Spectrum of Euro–Dollar
Vincent Brousseau ............................................... 69
H¨olderian Invariance Principles and Some Applications for
Testing Epidemic Changes
Alfredas Raˇckauskas, Charles Suquet................................109
Adaptive Detection of Multiple Change–Points in Asset Price
Volatility
Marc Lavielle, Gilles Teyssi`ere.....................................129
Bandwidth Choice, Optimal Rates and Adaptivity in
Semiparametric Estimation of Long Memory
Marc Henry .....................................................157
Wavelet Analysis of Nonlinear Long–Range Dependent
Processes. Applications to Financial Time Series
Gilles Teyssi`ere, Patrice Abry .....................................173
Prediction, Orthogonal Polynomials and Toeplitz Matrices.
A Fast and Reliable Approximation to the Durbin–Levinson
Algorithm
Djalil Kateb, Abdellatif Seghier, Gilles Teyssi`ere .....................239
