Discussion of "Feature Matching in Time Series Modeling" by Y. Xia and H. Tong

0.12 MB·English

by  Kung-Sik Chan

StatisticalScience
2011,Vol.26,No.1,53–56
DOI:10.1214/11-STS345B
MainarticleDOI:10.1214/10-STS345
(cid:13)c InstituteofMathematicalStatistics,2011
Discussion of “Feature Matching in Time
Series Modeling” by Y. Xia and H. Tong
Kung-Sik Chan and Ruey S. Tsay
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We thank Xia and Tong for their stimulating ar- Consider the simple case that a time series of re-
1
0 ticle on time series modeling. Their emphasis on es- turns,{rt},follows ageneralized autoregressive con-
2 timation rather than model specification is interest- ditionalheteroscedastic modeloforder(1,1)orsim-

n ing. It brings new light to statistical applications in ply a GARCH(1,1) model:
a
generalandtotimeseriesanalysisinparticular.The
J r =σ ε ,
use of maximum likelihood or least squares method t t|t−1 t
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is so common, especially with the widely available σt2|t−1=ω+αrt2−1+βσt2−1|t−2,
] statistical software packages, thatonetendsto over-
E where ω > 0,α ≥ 0,β ≥ 0, 1 > α + β > 0 are pa-
look its limitations and shortcomings.
M rameters, {ε } are independent and identically dis-
There is hardly any statistical method or proce- t
tributed (i.i.d.) random variables with zero mean
t. dure that is truly “one-size-fits-all” in real applica-
a tions. We welcome Xia and Tong’s contributions as andunitvariance,andεt isindependentofpastone-
st they argue so convincingly that feature matching step-ahead conditional variances σs2|s−1,s≤t. Esti-
[
often fares better in time series modeling. On the mation of the GARCH model can be done by max-

1 imizing the Gaussian likelihood of the data, which
other hand, we’d like to point out some issues that
v
essentially matches the conditional variances with
7 deserve a careful study.
the squared returns.
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3 A natural generalization of the catch-all method
1. HIGHER ORDER PROPERTIES
1 is to estimate a GARCH model that matches the
1. The conditional mean function generally provides k-step-ahead conditional variance to the kth ahead
0 a good description of the cyclical behavior of the data, for k=1,2,...,m with a fixed m, by minimiz-
2
underlying process, and the catch-all approach can ing some weighted measure of dissimilarity of the
1
: be effectively implemented by estimating the model multi-step conditional variances to future squared
v
that matches the multi-step conditional means to returns.Variousdissimilaritymeasuresmaybeused.
i
X the data, as eminently illustrated by the authors. Here, we illustrate the usefulness of this idea by
r Here, we want to point out the natural extension adopting the negative twice Gaussian log-likelihood
a
of estimating a model by matching multi-step con- as the dissimilarity measure, that is, estimating
ditional higher moments to the data. For example, a GARCH model by minimizing
in financial time series analysis, it is pertinent to
n−m m
model the dynamics of the conditional variances. 2 2 2
S(ω,α,β)= X Xwℓ{rt+ℓ/σt+ℓ|t+log(σt+ℓ|t)},
t=1 ℓ=1
Kung-Sik Chan is Professor, Department of Statistics
where {w } is a set of fixed weights and σ2 is the
and Actuarial Science, University of Iowa, Iowa City, ℓ t+ℓ|t
Iowa 52242, USA e-mail: [email protected]. conditional variance of rt+ℓ given information avail-
Ruey S. Tsay is H. G. B. Alexander Professor, Booth ableattimet.Whenm=1,thenewmethodreduces
School of Business, University of Chicago, 5807 South to the Gaussian likelihood method. On the other
Woodlawn Avenue, Chicago, Illinois 60637, USA hand, under the assumption that the true model is
e-mail: ruey.tsay@ chicagobooth.edu. a GARCH model and for a fixed m>1, the esti-
mator is expected to be consistent and asymptoti-
This is an electronic reprint of the original article
cally normal, with details of the investigation to be
published by the Institute of Mathematical Statistics in
Statistical Science, 2011, Vol. 26, No. 1, 53–56. This reported elsewhere. However, if the GARCH model
reprint differs from the original in pagination and does notcontain thetrue model,as likely is thecase
typographic detail. in practice, the (generalized) catch-all method with
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2
K.-S. CHANAND R.S. TSAY
Fig. 1. The red line connects the one-step-ahead conditional variance, σˆ2 , from the GARCH(1,1) model fitted by the
t|t−1
catch-all method with m=30 and equal weights, whereas their counterparts from the model fitted by the Gaussian likelihood
method are connected by the blue line. The light gray bars are the squared daily returns of the CREF series.
m>1 may provide new information for estimating nicely with the increasing kurtosis of the data; see
a GARCH model that better matches the observed Tsay [(2010), Chapter 3] for a discussion on contri-
volatility clustering pattern. butionofαtokurtosisofthereturnr .Thisexample
t
We tried this approach by fitting a GARCH(1,1) illustrates the usefulness of the generalization of the
model to the daily returns of a unit of the CREF catch-all method by matching higher moments, and
stock fund over the period from August 26, 2004 to also the potential usefulness of developing a test for
August 15, 2006; this series was analyzed by Cryer model misspecification based on the divergence of
and Chan [(2008), Chapter 12], and they identified the catch-all estimates with increasing m.
the series as a GARCH(1,1) process. Gaussian like-
lihood estimation yields ωˆ =0.0164,αˆ=0.0439,βˆ= 2. INFORMATION CONTENT
0.917.Ontheotherhand,thecatch-allmethod,with As statisticians, we love data. However, data have
m=30andtheweightswℓ∝1,resultsintheestima- their limitation. Available data may not be infor-
tes: ωˆ =0.0261,αˆ =0.102 and βˆ=0.836. Figure 1 mativetoconductanymeaningfulfeaturematching.
contrasts the fitted values, σˆt2|t−1, based on the two When the information content pertaining to the se-
fitted GARCH models with the squared returns su- lected feature is low, feature matching as an estima-
perimposedaslightgraybars.Thefigureshowsthat, tion method is likely to fail. As an example, assess-
ascomparedtothemodelestimatedbytheGaussian ment of financial risk focuses on the tail behavior
likelihood method, the GARCH model fitted by the of the loss over time. The relevant feature here is
catch-all method appears to track the squared re- the upper quantiles of the loss distribution. Since
turnsmoreclosely over thevolatile periodand tran- big losses are rare, available data, no matter how
sitintotheensuingquietperiodatafasterratecom- big the sample size is, are not informative about the
mensurate with the data. It seems that by requiring extreme quantiles. Theuncertainty in any matching
the model to track multi-step squared returns, the would be high.
method chooses a GARCH model that gives more As another example, consider the monthly time
weight to the current squared return to the future series of global temperature anomalies from 1880 to
evolution of conditional variances. Indeed, as m in- 2010. The data are available from many sources on
creasesfrom1to30,theARCHcoefficientestimateαˆ the web, for example, the websites http://data.giss.
increasesfrom0.0439 to0.102, whereastheGARCH nasa.gov/gistemp/oftheGoddardInstituteforSpa-
coefficientβˆdecreasesfrom0.917to0.836.Thesesys- ce Studies (GISS), National Aeronautics and Space
tematicparametricchangessignifythatthetruemo- Administration(NASA)andhttp://www.ncdc.noaa.
delisunlikelyaGARCH(1,1)model.Italsomatches gov/cmb-faq/anomalies.html oftheNationalClima-
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DISCUSSION
tic Data Center, National Oceanic and Atmospheric
Administration (NOAA). These time series are of
interest because of the concerns about global warm-
ing.Thekey featuretomatch thenis theunderlying
trend of the global temperature. To handle the time
trend, two approaches are commonly used in the
timeseriesliterature.Thefirstapproachiscalledthe
difference-stationarity inwhichthetimeseriesisdif-
ferenced to obtain stationarity. The ARIMA models
of Box and Jenkins are examples of this approach.
Thesecondapproach iscalled thetrend-stationarity
inwhichoneimposesalineartimetrendforthedata.
Thedeviation fromthetimetrendbecomesastatio-
naryseries.Eventhoughwehave130yearsofdata,it
remainshardforthedatatodistinguishatrend-sta-
tionary model from a difference-stationary one. On
theotherhand,thelong-termforecasts ofadifferen-
ce-stationary model differ dramatically from those
Fig. 2. The directed scatterplot on the right side shows the
of a trend-stationary model. The eventual forecast- evolution of the ARMA coefficients for the linear trend plus
ing function of an ARIMA model without constant ARMA(1,1) noise model fitted to the annual global temper-
is a horizontal line with standard error approaching ature anomalies, with m in the catch-all method increasing
infinity whereas that of a trend-stationary model is from 1 to 30, while the evolution of the ARMA coefficients
for the ARMA(1,1,1) model is shown by the directed scatter-
a straight line going to positive or negative infin-
plot on the left. The two solid circles show the estimates with
ity with a finite standard error. See Tsay (2012) for
m=1.
further analysis of the data and model comparison.
Here, we explore whether feature matching may- mean, σ2 is the ℓ-step-ahead prediction variance,
cast new light on the preceding problem. For sim- that is, ℓσ2 =σ2 ℓ−1ψ2 where ψ’s are the coeffi-
plicity, we consider the annual global temperature ℓ aPj=0 j
cients in the linear MA representation of the model
anomalies. Model identification suggests two plausi-
also known as the impulse response coefficients; σ2
ble models, namely, ARIMA(1,1,1) model and lin-
is the harmonic mean of σ2,ℓ=1,2,...,m. When
ear trend plus ARMA(1,1) model. We use the plus ℓ
m=1, the catch-all method reduces to Gaussian li-
convention for the MA coefficients, that is, the
kelihood estimation. But for m > 1, the catch-all
ARIMA(1,1,1) modelspecifies (1−φB)(1−B)Y =
t
methodattempts tomatch modelpredictionwithm
(1+θB)a , where B is the backshift operator such
t
“future”values,intermsofmeansandvariances.We
that BYt = Yt−1 and at are i.i.d. with zero mean implementedthecatch-allmethodforfittingtheglo-
andvariance σ2.For simplicity, thesemodels arefit-
a baltemperatureanomalies,withm=1,2,...,30.Fi-
ted by conditional least squares with the residuals
gure 2 plots the evolution of the catch-all estimates
initialized as 0 at the first time point. We also fit-
of the ARMA coefficients for the two models. It is
ted themodelwiththegeneralized catch-all method
interesting to observe that thecatch-all estimates of
that matches the predictive distribution to the fu-
the ARIMA(1,1,1) model quickly move away from
ture m values, in terms of the k-step-ahead predic-
the Gaussian likelihood estimates and fluctuate sta-
tive means and variances, k =1,2,...,m. The loss
blyaroundanessentiallyARIMA(0,1,1)model,that
function is similar to that discussed in the previous
is, an exponential smoothing model, for a while, be-
section. After profiling out the innovation variance,
fore they drift to more extreme values. Thus, the
the generalized catch-all method fits a model to the
catch-all method suggests that for forecasting on
time series {Y |t=1,2,...,T} by minimizing
t a decadal scale, an exponential smoothing model
m may be appropriate for the temperature data. Simi-
S(θ)=XX(Yt+ℓ−Yˆt+ℓ|t)2σ2/σℓ2, larly, the catch-all estimates of the linear trend plus
t ℓ=1 ARMA(1,1) noise model quickly move away from
where θ is the vector of all parameters except the the Gaussian likelihood estimate, and fluctuate sta-
innovation variance; Yˆ is the ℓ-step predictive bly for a number of steps around an estimate with
t+ℓ|t
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K.-S. CHANAND R.S. TSAY
its MA(1) coefficient comparable to that of the ex- tionseeksparametersthatgivethehighestprobabil-
ponential smoothing model. As the corresponding ity of thedataundertheentertained model.Feature
AR(1)estimatesarequitecloseto1,thecatch-alles- matching searches for parameters that bestdescribe
timates of thetrendplus noisemodelsuggest strong thefeaturesofinterest.Theexamplesusedinthear-
similarity between the two models over a forecast ticle all have clearly defined features such as cycles
horizon on a decadal scale. When m approaches 30, and, as expected, feature matching works well. On
thecatch-allestimatesofthetrendplusARMA(1,1) theother hand,therearesituations underwhichthe
noise model drift off on a flight that suddenly ends objective of data analysis does not match well with
into a trend plus uncorrelated noise model. any specific feature. For example, with the economy
For long-term prediction, the global temperature under pressure, unemployment is of major impor-
series is of limited value. The preceding analysis tance to people of all walks. It is well known that
highlights the conundrum that using a trend-statio- unemployment rate exhibits strong business cycles,
nary model,we simply force thedata to supportthe which in sharp contrast with examples used in the
trend; for difference-stationary models, we basically article donot have a fixed (or even an approximate)
endupusingtheexponentialsmoothingmodel.This period. It is then not clear which feature to match
is a problem facing all estimation methods, not just if one is interested in understandingand forecasting
feature matching. the U.S. unemployment rate.
We consider these two examples not because we
question thevalue of featurematching in time series
modeling; rather we like to point out that care must ACKNOWLEDGMENTS
be exercised in using feature matching. In other
K.-S.ChanthankstheU.S.NationalScienceFoun-
words, feature matching may encounter the same
dation (NSF-0934617) and R. S. Tsay thanks the
problemasthetraditionalestimationmethods.They
Booth Schoolof Business,University ofChicago, for
are statistical tools. It is the statisticians who use
partial financial support.
thetools,notthetools thatprocessthedata.Which
tool to use in a given application is the choice of
astatistician. Whilewewelcome theaddition offea-
ture matching to the tool kits, we like to emphasize REFERENCES
that there are limitations in feature matching, too. Cryer,J.D.andChan,K.S.(2008).TimeSeriesAnalysis:
With Applications in R, 2nd ed.Springer, New York.
3. FEATURE VERSUS OBJECTIVE Tsay, R. S. (2010). Analysis of Financial Time Series, 3rd
ed.Wiley, Hoboken, NJ.
Modelselection indataanalysisdependscritically Tsay, R. S.(2012). An Introduction to Financial Data Ana-
on theobjective of data analysis. Likelihood estima- lysis. Wiley, Hoboken,NJ.