Front matter
Estimating Volatility for Long Holding Periods
Rudiger Kiesel William Perraudin Alex Taylor
Birkbeck College Bank of England Birkbeck College
ABSTRACT Sample volatilities calculated from shortinterval daily data do not
necessarily imply much about volatility of nancial assets over long yearly horizons
In this note we construct a modelfree volatility estimator to investigate the long
horizon volatility of various shortterm interest rate time series and study implications
for shortrate models
Introduction
The problem of estimating volatility is one of the most important
topics in modern nance Accurate specication of volatility is a
prerequisite for modelling nancial time series such as interest rates
or stocks and crucially aects the pricing of contingent claims Mod
elling volatility has therefore be widely discussed in the nancial lit
erature see eg Campell Lo and MacKinley chapter Shiryaev
chapter or Taylor chapter for overviews on the subject
The main focus in these studies has been to estimate volatility over
short time periods and deduce results for longer period volatility
from underlying models
In this note we address the problem of estimating volatility over
longer time intervals directly Recently several attempts have been
made to address this problem most notably work by Andersen
Bollerslev Diebold and Labys who use intraday observations
to estimate the distribution of daily volatility and Drost Nijman
and Werker who consider temporal aggregation of GARCH
processes In contrast to these approaches we do not assume any
underlying parametric model for the datagenerating processes Our
only assumption is that the datagenerating process is rstdierence
stationary The model free approach leads to an estimator which is
insensitive to shortperiod contamination and only reacts to eects
relevant to the time period in question Applications of the proposed
JEL No G
Key words and phrases Volatility Unit Roots Variance Ratio interestrate models
estimator can be found in Cochrane who used the estimator to ob
tain a measure of the persistence of uctuations in GNP and Kiesel
Perraudin and Taylor who estimated the long term variability
of credit spreads
Related to our estimation problem are socalled moment ratio
tests which are frequently used to investigate the weak eciency
of nancial markets see Campbell et al chapter or Pagan
for surveys and LoMacKinlay and Groenendijk et al for
applications related to this investigation
The motivation behind the estimator is as follows From the as
sumption that the datagenerating process xt is rstdierence sta
tionary ie contains a unit root we obtain from Wolds decompo
sition see eg an innite moving average representation x
xt xt xt ajtj j X
Using this representation a result by Beveridge and Nelson implies
that xt can be represented as the sum of a stationary and a random
walk component ie
xt yt zt
where
a a a t j t j t j t y j j j X X X A A A
zt zt aj t j X A with t a sequence of uncorrelated random variables
The longperiod behaviour of the variance of the process xt may
dier substantially for processes with representation This be
comes of particular importance for valuation of contingent claims
and in case of interest rate models for bond pricing since the pric
ing formulae crucially depend on the volatility Since in general
the longterm behaviour of the variance of xt is dominated by the
variance of the random walk component the use of a volatility es
timator based on daily time intervals to contingent claoimsbonds
longer time to maturity may lead to substantial pricing errors In
the next section we introduce the estimator and discuss some of its
properties We perform Monte Carlo experiments to illustrate the
properties of the estimator in section In section we apply it to
estimate long holding period variances for several interest rate se
ries By analysing the quotient of longterm to shortterm variances
variance ratio we can infer the magnitude of the random walk com
ponent in the short term interest rate process This has implications
for the appropriate modelling of the short rate and relates to recent
results on the empirical verication of various shortterm interest
rate models see Section concludes
Construction and Properties of the Estimator
We start with a general representation of a rstdierence station
ary linear process as the sum of a stationary and a random walk
component ie
xt yt zt
with
yt BLt
zt zt t
with BL a polynomial in the lag operator Lt t t uncorre lated random variables and IEtt arbitrary Such a decom
position implies that IEtxt k zt k In that sense we call zt the permanent and yt the temporary component of xt compare also
for a related model and interpretation This suggests that the long
term variability of xt is also dominated by the innovation variance
z of the random walk component Utilizing the BeveridgeNelson
decomposition of a process xt given by one can show that the
innovation variance z is invariant to the particular decomposition
of type chosen in particular only the BeveridgeNelson decom
position is guaranteed to exist see also To make the above ar
guments on the importance of the innovation variance more precise
consider the k eriod variability A standard argument compare p
shows x
k
V artxt k xt k k jj j X
with j the autocovariances of the stationary process xt xt xt Then
k k j i lim V artxt k xt lim j x S xe k k k k j X A
i where j are the autocorrelations and S xe is the spectral den
sity function at frequency of xt A further application of the
BeveridgeNelson decomposition implies
i S xe z
Therefore in order to estimate z we could use an estimator of the
spectral density at frequency zero However estimating the spectral
density function at low frequencies is extremely dicult and involves
a tradeo between bias and eciency of the estimator see eg
for such estimators and their properties So rather then relying x on estimators for the spectral density function we proceed directly
with an estimator suggested by In particular suggests
to replace the autocovariance functions with their sample estimators
and then employ wellknown limit theorems for the sample autoco
variances
Large Sample Properties
In order to use we recall that under our assumptions x is
a covariance stationary process and as such has a moving average
representation Limit theorems for the sample autocovariances of
such processes have been studied extensively see x and we intend to utilize some of these results much the same x way as did Let us start by expressing the basic estimator
T k k xj xjk xT x T k T jk X
in a dierent form Dene j xj xj T xT x then becomes
T k k xjk l xjk l xT x T k jk l T X X T k jk l T k jk l X X
Tk k k j l j lj l j j k T k j l l X X X
k k o k k
where Th h jj h T j X
and o species an error term in probability depending on the dis
tribution of the innovations Dene the vector k then we can write
l o k
k l l with the kdimensional vector k k We therefore can use limit theorems on the asymptotic distribution of to deduce
the asymptotic distribution of our estimator k These limit theorems
depend crucially on the distribution of the innovations in If
IE the limit distribution of is Gaussian see eg If IE and further regularity xx x conditions are satised the limit distribution consists of a stable
random variable multiplied by a constant vector see and
for details Hence in the rst case the asymptotic distribution x of k will be Gaussian while in the second case it will asymptotically
be distributed according to a stable law
Small Sample Adjustments
In small samples the estimator exhibits a considerable bias
To discuss possible adjustments we assume that the data generating
process is a pure unit root process ie equation becomes
xt zt t
with t uncorrelated random variables So we can write the
numerator of the estimator
T k N xj xjk xT x T jk XT k T k k j T T T jk XT n X T X k T jk nk X X X
j Dening Zjk jk and using the fact that the are uncorrelated we get P
T k k IEN IEZjk IEZjkZTT IEZTT T T jk X T k k k IE k T k T k T T T jk X
So in order to get an unbiased estimator for using the quantity
N we have to multiply it by
T kT k T k
which is just the adjustment proposed by Cochrane compare
and leads to
T T k k xj xjk xT x kT kT k T jk X
If we assume that the innovations in are uncorrelated with ex
isting fourth moment we can use the asymptotic equivalence of the
estimators and to deduce the weak convergence
pTk N k k
We denote weak convergence by
If however the last existing moment of the innovations in is of
order ie the variance exists but the fourth moment is
innite we have the weak convergence
p CT k kS
where S is a stable random variable with index and CT
a constant depending on the T and the tail behaviour of the inno
vations which is related to the index The relevant asymptotic
result for the autocovariances is Theorem in where the exact
values of the constants to be used to construct the vector l in
can be found If we drop the assumption the limit laws remain
of the same type However the variances change considerably since
they depend on the autocovariances of the process
a b c d e k se se k se se k se se
f w
m
q
y
TABLE Model with i i d Gaussian innovations
amodel xt t with t N and Z First se column are always Monte
Carlo second se column are asymptotic se assuming existence of the fourth moment
bmodel xt a xt t with t N and z a here z c Adjusted for ARcovariance structure
dmodel xt t at with t N and z a here z
e Adjusted for MAcovariance structure
f wweek mmonth qquarter yyear
For instance in case the IE we have limn Covpq
pq kkk p q k qk p and thus k P N k T V where V is the covariance matrix see x This implies
l an asymptotic standard normal distribution of k with variance V l
Monte Carlo Illustrations
In this section we illustrate our estimating procedure using simu
lated time series We consider three basic settings of rstdierence
stationary sequences with representation First as a benchmark
case we consider a pure random walk with representation as in
To study the eect of nonzero autocovariances of the series x
on the asymptotic standard error we simulate two further series
namely a sequence whose rstdierence follows an autoregressive
model of order one ARmodel implying an innite order moving
average representation and on the other hand a sequence which has
rstdierences allowing a moving average representation of order
one MA
These settings imply that the error terms in are perfectly cor
related The ARmodel corresponds to a small random walk com
ponent in our setting it accounts for roughly of variability of
xk in The MAmodel on the other hand corresponds to a
large random walk component the innovation variance of the ran
dom walk component zk in is larger due to dependence than
the innovation variance of the series xk
a b c d e k se se k se se k se se
f w
m
q
y
TABLE Model with i i d t innovations
amodel xt t with t t and Z First se column are always MonteCarlo
second se column are asymptotic se assuming existence of the fourth moment
bmodel xt a xt t with t t and z a here z c Asymptotic standard error adjusted for ARcovariance structure
dmodel xt t at with t t and z a here z
e Asymptotic standard error adjusted for MAcovariance structure
f wweek mmonth qquarter yyear
For each of these series we consider three types of innovation pro
cess As a standard model we consider iid Gaussian innovations
Then we investigate the eect of heavytailed innovations using iid
Student t innovations and nally to discuss second order de
pendence we use GARCHinnovations Each experiment con
sisted of generating a series of length with coecients in line
with coecients obtained performing the corresponding ARIMA
GARCH for the series used in and was repeated times x We report the mean of longperiod volatility estimators for periods
of length k weeks month quarters years together
with standard errors se computed from the MonteCarlo simula
tions and according to the asymptotic results for an underlying pure
unit root process with an existing fourth moment
a b c d e k se se k se se k se se
f w
m
q
y
TABLE Model with GARCH innovations
amodel xt t with t GARCH and Z First se column are
always MonteCarlo second se column are asymptotic se assuming existence of the fourth
moment
bmodel xt a xt t with t GARCH and z a here Z
c Adjusted for ARcovariance structure
dmodel xt t at with t GARCH and z a here Z
e Adjusted for MAcovariance structure
f wweek mmonth qquarter yyear
In line with the asymptotic consistency of the estimator k com
pare the estimated value converges towards the true value of the
innovation variance of the random walk component in all cases For
Gaussian and GARCH innovation cases for which the appropriate
limit theory holds the asymptotic standard errors are in line with
the observed Monte Carlo errors As expected the asymptotic stan
dard errors calculated under the assumption of an existing fourth
moment become unreliable for heavy tailed innovation ie simula
tions based on t innovations
Since for shorter series the asymptotic standard error becomes un
reliable we also tested various bootstrap based methods Motivated
by the application we have in mind we concentrated on series with
length and standard normal or GARCH innovations It turned
out that tting a loworder ARmodel to the simulated time series
and resampling from the residuals produced satisfactory bootstrap
standard errors
lag lag Model a k Bse Ase k Bse Ase
b RW
AR
MA
c RW
AR
MA
TABLE Bootstrap estimates of standard errors
a All on a time series of length with bootstrap resamples pa
rameters chosen as above
b standard Normal innovations
c GARCHInnovations values multiplied by
Applications
Empirical comparisons of continuoustime models of the shortterm
interest rate have recently been the focus of several studies see eg
In these studies the general class of singlefactor diu
sion models
dr rdt r dW
with constant coecients and W a standard Brownian motion has
been compared We will consider the subclass where we restrict the
parameter to take one of the values or so eg the Vasicek
and the CoxIngersollRoss model are included The discretetime
analog of this model class is
rt rt rt t IEt t IEt t rt jF jF
with Ft the information set at time t A model like this will generate
a time series within our framework if If we focus on the
unconditional longterm variance a standard calculation shows that
we have the following asymptotic relations under
V arrt t V arrt t ct V arrt e
c a constant Using the Cochranetype estimator we can com
pare the observed longterm variances with variances predicted from
the model setting We apply this idea to three shortterm day
maturity interest rate series The rates we use are US EURODOLLAR
with observations from UK EUROPOUND
with observations from and German EURO
MARK with observations from
rate
US EURODOLLAR
a
UK EUROPOUNDS
GER EUROMARK
TABLE Short rate volatilities
a For lags to se are based on asymptotic calculations for lags and
se are bootstrap based
To ensure the validity of the assumption we performed var
ious tests for unit roots and stationarity For all series we cant
For the unit root tests we used the augmented DickeyFuller and PhillipsPerron
procedures and for testing stationarity the KwiatkowskiPhillipsSchmidtSin test see
chapters and for a description and discussion of these tests
reject the presence of a unit root at a signicance level whereas
stationarity of the series is rejected at the signicance level Ap
plying these tests again to the rstdierence of the series indicated
no evidence of a unit root in the dierenced series The combination
of these test results allows us to conclude the ball series should be
modelled as rstdierence stationary and t into our framework
We report the results for the interest series in table From a
modelfree point of view that is within the general framework
these results indicate that using the oneday volatility estimate will
seriously overestimate longer term volatility
FIGURE VarianceRatios for shortterm interest rates
Turning to the question of modelling shortterm interest rates
within the class of onefactor diusion models we calculate and plot
the ratio of the volatility calculated over a longer holding period to
that calculated over one day multiplied by k see gure For all
rates considered the ratios are downward slopping for short holding
periods the meanreverting component dies o After a period of
stability the variance ratio begin to increase linearly showing a be
haviour roughly in line with the asymptotics of a CoxIngersollRoss
model
Conclusion
We presented a nonparametric method to estimate longterm vari
ances and the magnitude of the unit root process in various interest
rates Our results that calculating longterm variances on the basis of
shortterm variance estimates will overestimate longterm variances
Our results further indicate that within the onefactor diusion short
rate model class squareroot type processes model the behaviour of
longterm variances of short rates best Vasicektype models which
assume that the short rate follows a meanreverting process and thus
omit a unit root component in the datagenerating process will lead
to an underestimating of longterm variances since for longer time
horizons the unitroot component of the interestrate process be
comes dominant Our ndings support a model of CoxIngersoll Ross
type without a meanreverting component
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