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MODELING VOLATILITY DYNAMICS
by
Francis X. Diebold and Jose A. Lopez
Federal Rese"e Bank of New York
Research Paper No. 9522
October 1995
This paper is being circulated for purposes of discussion and COIDillent only.
The contents should be regarded as preliminary and not for citation or quotation without
permission of the author. The views expressed are those of the author and do not necessarily
reflect those of the Federal Reserve Bank of New York or the Federal Reserve System.
Single copies are available on request to:
Public Information Department
Federal Reserve Bank of New York
New York, NY 10045
MODELING VOLATILITY DYNAMICS
Francis X. Diebold
Jose A. Lopez
Department of Economics
University of Pennsylvania
3718 Locust Walle
Philadelphia, PA 19104-6297
Research and Market Analysis Group
Federal Reseive Bank of New York
33 Liberty Street
New York, NY 10045
Print date: October 23, 1995
ABSTRACT: Many economic and financial time series have been found to exhibit dynamics
in variance; that is, the second moment of the time series innovations varies over time. Many
possible model specifications are available to capture this phenomena, but to date, the class of
models most widely used are autoregressive conditional heteroskedasticity (ARCH) models.
ARCH models provide parsimonious approximations to volatility dynamics and have found
wide use in macroeconomics and finance. The family of ARCH models is the subject of this
paper. In section Il, we sketch the rudiments of a rather general univariate time-series model,
allowing for dynamics in both the conditional mean and variance. In section m, we provide
motivation for the models. In section IV, we discuss the properties of the models in depth,
and in section V, we discuss issues related to estimation and testing. In Section VI, we detail
various important extensions and applications of the model. We conclude in section VIl with
speculations on productive directions for future research.
AcJmnwJedgements· The views expressed here are those of the authors and not those of the
Federal Reseive Bank of New York or the Federal Reseive System. Helpful comments were
received from Richard Baillie, Tim Bollerslev, Pedro de Lima, Wayne Ferson, Kevin Hoover,
Peter Robinson and Til Schuermann. We gratefully acknowledge the support of the National
Science Foundation, the Sloan Foundation, and the University of Pennsylvania Research
Foundation.
I. Introduction
Good macroeconomic and financial theorists, like all
good theorists, want to get the
facts straight before theorizing; hence, the explosive
growth in the methodology and
application of time-series econometrics in the last 25
years. Man y factors fueled that growth,
Illllging from important developments in related field
s (e.g. , Box and Jenkins, 1970) to
dissatisfaction with the "incredible identifying restriction
s" associated with traditional
macroeconometric models (Sims, 1980) and the assoc
iated recognition that many tasks of
interest, such as forecasting, simply do not require a
structural model (e.g. , Granger and
Newbold, 1979). A short list of active subfields inclu
des vector autoregressions, index and
dynamic factor models, causality, integration and persi
stence, cointegration, seasonality,
unobserved-components models, state-space representa
tions and the Kalman filter, regimeswitching models, nonlinear dynamics and optimal nonl
inear filtering. Any such list must also
include models of volatility dynamics. ARCH mode
ls, in particular, provide parsimonious
approximations to volatility dynamics and have found
wide use in macroeconomics and
1
finance. The family of ARCH models is the subject
of this chapter.
Economists are typically introduced to heteroskedasti
city in cross-sectional contexts,
such as when the variance of a cross-sectional regression
disturbance depends on one or more
of the regressors. A classic example is the estimation
of Engel curves by weighted least
squares, in light of the fact that the variance of the distu
rbance in an expenditure equation may
depend on income. Heteroskedasticity is equally perv
asive in the time-series contexts
prevalent in macroeconomics and finance. For exam
ple, in Figures 1 and 2, we plot the log of
daily Deutschemark/Dollar and Swiss Franc/Dollar
spot exchange rates, as well as the daily
returns and squared returns, 1974--1991. Volatility clust
ering (that is, contiguous periods of
high or low volatility) is apparent. However, models
of cross-sectional heteroskedasticity are
not useful in such cases because they are not dynamic.
ARCH models, on the other hand,
were developed to model such time-series volatility
fluctuations. Engle (1982) used them to
model the variance of inflation, and more recently they have enjoyed widespread use in
modeling asset return volatility.
Exhaustive surveys of the ARCH literature already exist, including Engle and
Bollerslev (1986), Bollerslev, Chou and Kroner (1992), Bera and Higgins (1993) and
Bollerslev, Engle and Nelson (1994), and it is not our intention to produce another. Rather,
we shall provide a selective account of certain aspects of conditional volatility modeling that
are of particular relevance in macroeconomics and finance. In section II, we sketch the
rudiments of a rather general univariate time-series model, allowing for dynamics in both the
conditional mean and variance. We introduce the ARCH and generalized ARCH (GARCH)
models there. In section ID, we provide motivation for the models. In section IV, we discuss
the properties of the models in depth, and in section V, we discuss issues related to estimation
and testing. In Section VI, we detail various important extensions and applications of the
model. We conclude in section VII with speculations on productive directions for future
research.
II. A Time-series Model with Conditional Mean and Variance Dynamics
Wold's (1938) celebrated decomposition theorem establishes that any covariance
stationary stochastic process {xJ may be written as the sum of a linearly deterministic
component and a linearly indeterministic component with a square-summable, one-sided
moving average representation. 2 We write x,
= cl,+ y., where cl, is linearly deterministic and
y, is a linearly regular (or indeterministic) covariance stationary stochastic process (LRCSSP)
given by
y = B(L) e,,
B(L) = Lb; Li,
L- b/ <
1
i=O
i=O
E[ e e ] = {
' '
a~ <
0,
3
00
,
00 ,
b0 = 1,
if t = -r
otherwise.
The uncorrelated innovation sequence {eJ need
not be Gaussian and therefore need not be
independent. Non-independent innovations are
characteristic of nonlinear time series in
general and conditionally heteroskedastic time
series in particular.
In this section, we introduce the ARCH proc
ess within Wol d's framework by
contrasting the polar extremes of the LRCSSP
with independent and identically distributed
(i.i. d.) innovations, which allows only condition
al mean dynamics, and the pure ARC H
process, which allows only conditional variance
dynamics. We then combine these extremes
to produce a generalized model that permits vari
ation in both the first and second conditional
moments. Finally, we introduce the Generali
zed ARCH (GARCH) process, which is very
useful in practice.
A. Conditional Mean Dynamics
Suppose that y, is a LRCSSP with i.i.d ., as oppo
sed to merely white noise,
3
innovations. The ability of the LRCSSP to
capture conditional mean dynamics is the sour
ce
of its power. The unconditional mean and vari
2
ance are ElY,l = o and E[ y,2 ] =
bj , which
t=O
are both time-invariant. However, the condition
al
mea
n
is
time
-var
ying
and
is given by
•
m
y1 j Qt-I ] = ~ b; e _;, where the information
set is Q1_ 1 = { e,-1> e,_ , ... }1
o; 't
E[
2
1=1
Because the volatility of many economic time
series seems to vary, one would hope
that the LRCSSP could capture conditional vari
ance dynamics as well, but such is not the case
for the model as presently specified. The cond
itional variance of y, is constant at
E [ ( y1 - E [ y IQ _ ] )2 jQt-1 ] = o! . This pote
1
ntially unfortunate restriction manifests itself
1 1
in
the properties of the k-step-ahead conditional
prediction erro r variance. The least squares
forecast is the conditional expectation,
m
E[Y,.k
I Q,1 = ~ bk•i~-j,
1•0
4
and the associated prediction error is
k-1
Yt+k - E [ Yt+k
I n, l = ~
••0
bi e,.k-i•
which has a conditional prediction error variance of
As k -
a; L- bi
00
the conditional prediction error variance converges to the unconditional variance
,
2
•
i=O
Note that for any k, the conditional prediction error variance depends only on k
and not on 0 1_1; thus, readily available and potentially useful infonnation is discarded.
B. Conditional Variance Dynamics
By way of contrast, we now introduce a pure ARCH process, which displays only
conditional variance dynamics. We write
e,
I n,-1 -
N(O,
h,}
h, = w + y(L)t:;,
w>O,
y(L) =
L- yiLi,
Yi;, 0 'ef i,
y(I) <I.
i=l
The process is parameterized in tenns of the conditional density of e,
I n,-1>
which is assumed
to be nonnal with a zero conditional mean and a conditional variance that depends linearly on
past squared innovations..Note that even though the e,'s are serially uncorrelated, they are not
independent. The stated conditions are sufficient to ensure that the conditional and
unconditional variances are positive and finite as well as that y, is covariance stationary.
The unconditional moments are constant and are given by E[ y ] = O and
1
E[( y1
-
E[y,
])2] =
w
I - y(I)
. As for the conditional moments, by construction, the
5
conditional mean of the process is zero, and the conditional variance
is potentially timevarying. That is, E[y,
I
Qt-1]
=
O and E[(Y, - E[y,
I
Qt-1 ]}2 1 Q1_1]
= w + y(L) e;.
C. Conditional Mean and Variance Dynamics
We can incozporate both conditional mean and conditional variance
dynamics by
introducing ARCH innovations into the standard LR.CSSP. We write
Y, = B(L)e,,
e, I Q1_1
-
N(o, Ji.}
Ji.= w +y(L )e;,
subject to the conditions discussed earlier. Both the unconditional mean
and variance are
constant; i.e., E[ y,] = O and
However, the conditional mean and variance are time-varying; i.e.,
-
E[y, I Q,_i] = ~ biet-i,
1=1
E[(Y, - E[y, I Q,-iJ)2
I Qt-1]
= w + y(L)e ;.
Thus, this model treats the co.nditional mean and variance dynamics
in a symmetric fashion by
allowing for movement in each, a common characteristic of economic
time series.
D. The Generalized ARCH Process
In the previous subsections, we used an infinite-ordered ARCH
process to model
conditional variance dynamics. We now introduce the GARCH proce
ss, which we shall
subsequently focus on almost exclusively. The finite-ordered GARC
H model approximates
6
infinite-ordered conditional variance dynamics in the same way that finite-ordered
ARMA
models approximate infinite-ordered conditional mean dynamics.4
The GARCH(p,q) process, introduced by Bollerslev (1986), is given by
e1
h,
I OH
- N{O, h,}
)e; + p(L)h,,
= w + o:(L
The stated conditions ensure that the conditional variance is positive and that y,
is covaria
nce
5
stationary. The ARCH model of Engle (1982) emerges when p(L) = 0. If both
o:(L) and
P(L) are zero, then the model is simply i.i.d. noise with variance w. The
GARCH(p,q) model
can be represented as a restricted infinite-ordered ARCH model:
h
"t
w
= I - P(l)
+
o:(L)
I - P(L)
e2 =
I
w
I - P(l) +
~
2
6' 0;f!t-i.
The first two unconditional moments of the pure GARCH model are constant
and given
by
E[y j = O and
1
The conditional moments are E [ Y1 I 0 _ 1 ] = 0 and
1
7
m.
Motivating GARCH Processes
GARCH models have been used extensively in macroeconomics and finance becaus
e of
their attractive approximation-theoretic properties. However, these models do
not arise
directly from economic theory, and various efforts have been made to imbue them
with
economic rationale. Here, we discuss both approximation-theoretic and econom
ic motivations
for the GARCH framework.
A. Approximation-Theoretic Considerations
The primary and most powerful justification for the GARCH model is approximation
theoretic. That is, the GARCH model provides a flexible and parsimonious approx
imation to
conditional variance dynamics, in exactly the same way that ARMA models provid
e a flexible
and parsimonious approximation to conditional mean dynamics. In each case,
an infiniteordered distributed lag is approximated as the ratio of two finite, low-ordered
lag operator
polynomials. The power and usefulness of ARMA and GARCH models come
entirely from
the fact that ratios of such lag operator polynomials can accurately approximate
a variety of
infinite-ordered lag operator polynomials. 6 In short, ARMA models with GARC
H innovations
offer a natural, parsimonious, and flexible way to capture the conditional mean
and variance
dynamics observed in a time series.
B. Economic Considerations
Economic considerations may also lead to GARCH effects, although the precise
links
have proved difficult to establish. Any of the myriad economic forces that produc
e persistence
in economic dynamics may be responsible for the appearance of GARCH effects
in
volatility.
In such cases, the persistence happens to be in the conditional second momen
t, rather than the
first.
8
To take one example, conditional heteroskedasticity may arise in situations in which
"economic time" and "calendar time" fail to move together. A well-known example from
financial economics is the subordinated stochastic process model of Clark (1973). In this
model and its subsequent extensions, the number of trades occurring per unit of calendar
time
(I,) is a random variable, and the price change per unit of calendar time (eJ is the sum
of the I.
intra-period price changes (o~, which are assumed to be normally distributed:
i.i.d.
oi Using a simple transformation,
N( 0, TJ ).
e. can be written more directly as a function of I.,
i.i.d.
N( 0, I).
Thus, e, is characterized by conditional heteroskedasticity linked to trading volume. If the
number of trades per unit of calendar time displays serial correlation, as in Gallant, Hsieh
and
Tauchen (1991), the serial correlation induced in the conditional variance of returns (measur
ed
in calendar time) results in GARCH-like behavior. Similar ideas arise in macroeconomics
.
The divergence between economic time and calendar time accords with the tradition of "phaseaveraging" (e.g., Friedman and Schwartz, 1963) and is captured by the time-deformatio
n
models of Stock (1987, 1988).
Several other explanations for the existence of GARCH effects have been advanced,
including parameter variation (Tsay, 1987), differences in the interpretability of informa
tion
(Diebold and Nerlove, 1989), market microstructure (Bollerslev and Domowitz, 1991),
and
agents' "slow" adaptation to news (Brock and LeBaron, 1994). Currently, a consensus
economic model producing persistence in conditional volatility does not exist, but it would
be
foolish to deny the existence of such persistence; measurement is simply ahead of theory.
9
IV. Properties of GARCH Processes
Here we highlight some important properties of GARCH
processes. To facilitate the
discussion, we generate a realization of a pure GAR CH(l
, I) process of length 500 that we will
use repeatedly for illustration. 7 The parameter values are
w = l, rx = .2 and p = .7, and the
underlying shocks are N(0, I). 8 This parameterization deliv
ers a persistent conditional
variance and has finite unconditional variance and kurtosis. 9
We plot the realization and its
first 25 sample autocorrelations in Figure 3. The sample
autocorrelations are indicative of
white noise, as expected.
A. The Conditional Variance is a Serially Correlated Rand
om
Variable
The conditional variance associated with the GARCH mode
l is
h1 = w + rx(L )t; + P(L) Ji..
Recall that the unconditional variance of the process is given
by
2
0
y
W
=--1-rx (l)P(l) .
Replacing w with o;(I - rx(I) - P(I)) yields
h1 = o~(l - rx(I) - PO) )+ rx(L )t; + P(L) I\,
so that
I\ - o~ = rx(L )t; - o~rx (l) + P(L) I\ - ~P( l)
= rx(L)(e~ - o;) + P(L) (I\ - o;).
Thus, the conditional variance is itself a serially correlated
random variable.
We plot the conditional variance of the simulated GAR CH(l
, I) process and its sample
autocorrelation function in Figure 4. The high persistenc
e of the conditional variance is due to
the large sum of the coefficients, rx+P = 0.90.
10
B.
e! Has an ARMA Representation
If e, is a GARCH(p,q) process, ~ has the ARMA representation
t;
=
=
[a(L) + j3(L)]~ j3(L) v, + v ,
1
where v1 = ~ - 11t is the difference between the squared innovation and the conditional
(,I) +
variance at time t. To see this, note that, by supposition, h, =
(,I)+
a(L)~
+
f3(L)11t-
Adding and subtracting j3(L)~ from the right side gives
h1
= (,I)+
= (,I) +
a(L)~
[a(L)
Adding ~ to each side gives
+ j3(L)e:
+
- j3(L)e:
+
j3(L)11t
j3(L)]~ - j3(L)[e: -11t].
so that
t;
= (,I)+
[a(L)
+
j3(L)]~ - j3(L)[e: -11t]
+
[t;
-11t],
[a(L) + P(L)]e ; - P(L)v, + v,.
Thus, e~ is an ARMA([max(p,q)], p) process with innovation v,, where v E[-h , oo), and it is
1
1
covariance stationary if the roots of a(L)+P (L)=l are outside the unit circle.
=(,I)+
The square of our GARCH(l,l) realization is presented in Figure 5; the persistence in
~, which is essentially a proxy for the unobservable 11t, is apparent. Differences in the
behavior of~ and h1 are also apparent, however. In particular, ~ appears "noisy." To see
why, use the multiplicative form of the GARCH model, e, = h.112 z with z, ~ N ( 0, I). It is
1
easy to see that
t;
is an unbiased estimator of Ii.,
E[ ~
2
because z.
P(~ <
I OH ~
.! 11t) >
2
I n,_ 1 ]
= E[11t
I n,_i]E[z.2 I n,_ 1 ]
= EJ11t
I n,_i],
x:i>· However, because the median of a Jec 1>is .455,
1/2. Thus, the
e; proxy introduces a potentially significant error into the
analysis of small samples of Ii., t = 1, ... , T, altliough the error diminishes as T increases.
11
C. The Conditional Prediction Erro r Variance Dep
ends
on the Conditioning
Information Set
Because the conditional variance of a GARCH proc
ess is
a serially correlated random
variable, it is of interest to examine the optimal k-ste
p-ahead prediction, prediction error and
conditional prediction error variance. Immediately,
the k-step-ahead prediction is
E [ Y,.k I n,] = 0, and the prediction error is
l
Yt•k - E [Y,.k I n, = etok·
This implies that the conditional variance of the pred
iction error,
E[(Y,.k - E[Y,.k I n,])2 In, ]= E[e;.k In,}
depends on both k and n, because of the dynamics
in the conditional variance. Simple
calculations reveal that the-expression for the GAR
CH(p, q) process is given by
In the limit, this conditional variance reduces to
the unconditional variance of the process,
lim E [F.2
k-~
,.k
IQ
'
l
=
<,)
I - a(l) - P(l)
•
For finite k, the dependence of the prediction error
variance on
the current information
set n, can be exploited to produce better interval forec
asts, as illustrated in Figure 6 for k = l.
We plot the one-step-ahead 90% conditional and unco
nditional interval forecasts of our
simulated GARCH(l,l) process along with the actu
al realization. We construct the conditional
prediction intervals using the conditional variance
E[t:;. 1 In_]= 1\.1 = w + at:; +Pl\= 1 + .2t:; + .7J\;
[ii;}:~;.
thus, the conditional prediction intervals are ~l.6 4
The 90% unconditional interval,
on the other hand, is simply [f.os, f. .J, where f. deno
tes the a percentile of the unconditional
9
distribution of the GARCH process. The ability of
the conditional prediction intervals to adapt
to changes in volatility is clear.
12
D. The Implied Unconditional Distrib ution Is Symme tric and Leptok urtic
The moment structure of GARCH processes is a complicated affair. In addition to the
earlier-referenced surveys, Milhoj (1985) and Bollerslev (1988) are good sources. Howeve
r,
straightforward calculation reveals that the unconditional distribution of a GARCH process
is
symmetric and leptokurtic, a characteristic that agrees nicely with a variety of financial market
data. The unconditional leptokurtosis of GARCH processes follows from the persistence
in
conditional variance, which produces the clusters of "low volatility" and "high volatility"
episodes associated with observations in the center and in the tails of the unconditional
distribution.
GARCH processes are not constrained to have finite unconditional moments, as shown
in Bollerslev (1986). In fact, the only conditionally Gaussian GARCH process with
unconditional moments of all orders occurs when a(L)
= ~(L) = 0, which is the degenerate
case of i.i.d. innovations. Otherwise, depending on the precise parameterization,
unconditional moments will cease to exist beyond some point. For example, most parame
ter
estimates for financial data indicate an infinite fourth moment, and some even indicate an
infinite second moment. Our illustrative process has population mean 0, variance I 0,
skewness 0, and kurtosis 5.2.
E. Temp(lral Aggregation Produces Convergence to Normality
Convergence to nonnality under temporal aggregation is a key feature of much
economic data and is also a property of covariance stationary GARCH processes. The key
insight is that a low-frequency change is simply the sum of the corresponding high-frequency
changes; for example, an annual change is the sum of the internal quarterly changes, each
of
which is the sum of its internal monthly changes, and so forth. Thus, if a Gaussian central
limit theorem can be invoked for sums of GARCH processes, convergence to nonnality under
temporal aggregation is assured. Such theorems can be invoked so long as the process is
13
covariance stationary, as shown by Dieb old (198
8) using a central limi t argument from Whi te
(1984) that requires only the existence of an unco
nditional second moment. Dros t and Nijman
(1993) extend Dieb old's result by showing that
a parti cula r generalization of the GAR CH class
is closed unde r temporal aggregation, and by char
acterizing the prec ise way in which temporal
aggregation leads to reduced GAR CH effects. 10
V. Estimation and Testing of GARCH Models
Foll owin g the majority of the literature, we focu
s primarily on maximum-likelihood
estimation (ML E) and associated testing procedur 11
es.
A. Approximate Maximum Likelihood Estimat
ion
As always, the likelihood function is simply the
join t density of the observations,
L(6 ; Y1, ... ,YT )= f(Y1•···,YT; 6).
This join t density is non-Gaussian and does not
have a known closed-form expression, but it
can be factored into the prod uct of conditional
densities,
L(6; Yi, ... , YT) = f(yT I QT-1; 6) ~YT-1 I QT-2; 6)
... ~Yp+1 IQP; 6) l{Yp• ...• Y1; 6),
where, if the conditional densities are Gaussian
,
1-l\( 6r 112
exp (-.! _i_ ) .
.fEi
f(y, I Qt-I; 6} = -
2 h1(6)
The f( Yp• ... ,y ; 6 )term is often igno red because
1
a closed-form expression for it does not exist
and beca use its deletion is asymptotically inconseq
uential. Thus, the approximate log
likelihood is
T
T
2
lnL (6; Yp+l' ... , YT ) = -T-p
Y, ) ·
- ln(21t) - -l "~ In 1\(6) - -1 "~ h(
2
2t=p+ l
2 t=p+l •'t 6
It may be maximized numerically using iterative
procedures and is easily generalized to models
riche r than the pure univariate GAR CH process,
such as regression models with GAR CH
14
disturbances. In that case, the likelihood is ihe same with et
of y,. The unobserved conditional variances {h,(8)} T
t=p+l
calculated at iteration j using
eG- 1>,
= Yt
- E [ Yt
I Qt-1; 8] in place
that enter the likelihood function are
the estimated parameter vector at iteration j-1. The
necessary initial values of the conditional variance are set at the first iteration to the
sample
variance of the observed data and at all subsequent iterations to the sample varianc
e of a
simulated realization with parameters eG- 1>.
The assumption of conditional normality is not always appropriate. Nevertheless,
Weiss (1986) and Bollerslev and Wooldridge (1992) show that even when normality
is
inappropriately assumed, the resulting quasi-MLE estimates are asymptotically normal
ly
distributed and consistent if the conditional mean and variance functions are specifi
ed
correctly. Bollerslev and Wooldridge (1992), moreover, derive asymptotic standar
d errors for
the quasi-MLE estimates that are robust to conditional non-normality and are easily
calculated
as functions of the estimated parameters and the first derivatives of the conditional
mean and
variance functions.
B. Exact Maximum Likelihood Estimation
Diebold and Schuermann (1993) propose a numerical procedure for constructing the
exact likelihood function of an ARCH process using simulation techniques in conjun
ction with
nonparametric density estimation, thereby retaining the information contained in {yp,
... ,y 12
1}.
Consider the ARCH(p) process, Yt = t\, where et
I Qt-1
-
N( 0, h,}
t
ht = w + a 1e~_ 1 + ... + «lf-p, w > 0, «; ;;,; 0, v' i = I, ... , p, and
«; < 1. The
••I
conditional normality assumption is adopted only because it is the most common; alterna
tive
distributions can be used with no change in the procedure. Let 8 = ( <a>, « , ••• , aP }
1
The initial likelihood term f( Yp, ... ,y ; 8) for any given parameter configuration 8
is
1
simply the unconditional density of the first p observations evaluated at {Yp, ... ,y },
which can
1
be estimated to any desired degree of accuracy using well-known techniques of simula
tion and
15
consistent nonparametric density estimation. At any iteration j, a curren
t "best guess" of the
0
parameter vector e >exists. Therefore, a very long realization of
the process with parameter
0
vector e > can be simulated and the value of the joint unconditional
density evaluated at
{yP, ... ,y1 } canbeconsistentlyestimatedanddenotedas f(yP, ... , y
0
1; e >) This estimated
unconditional density can then be substituted into the likelihood where
the true unconditional
density appears. By simulating a large sample, the difference betwe
en ~Yp, ... , y ; e<i l) and
1
f(Yp• ..., y1;
e0 >) is made arbitrarily small, given the consistency of the density estimation
technique. The full conditionally Gaussian likelihood, evaluated at
em, is then
L(B<il;y1>···,Y1)" r(Yp•·--,Y1;0<il)
II [/iii1i,(emi-112
t=p+l
21t
exp[-21
(y~">)
ht 0
J
which may be maximized with respect to 0 using standard numerical
techniques.
ll
C. Testing
Standard likelihood-ratio procedures may be used to test the hypothesis
that
no ARCH
effects are present in a time series, but the numerical estimation requir
ed under the ARCH
alternative makes that a rather tedious approach. Instead, the Lagra
nge-multiplier (LM)
approach, which requires estimation only under the null, is preferable.
Engle (1982) proposes
a simple LM test for ARCH under the assumption of conditional norm
ality that
involves only a
least-squares regression of squared residuals on an intercept and lagge
d squared residuals.
Under the null of no ARCH, TR2 from that regression is asymptotica
lly distributed as x\>•
where q is the number of lagged squared residuals included in the regres
sion.
A minor limitation of the LM test for ARCH is the underlying assum
ption of
conditional normality, which is sometimes restrictive. 13 A more impor
tant limitation is that it
is difficult to generalize to the GARCH case. Lee (1991) and Lee
and King (1993) present
such a generalization, but as discussed in Bollerslev, Engle and Nelso
n (1994), the GARCH
16
parameters cannot be separately identified in models close to the null -- the 1M test
for
GARC H(l,l) is the same as that for ARCH(!).
Thus, less formal diagnostics are often used, such as the sample autocorrelation
function of squared residuals. McLeod and Li (1983) show that under the null hypoth
esis of
no non-linear dependence among the residuals from an ARMA model, the vector of
normalized sample autocorrelations of the squared residuals,
where 6 2 is the estimated residual variance and t
= 1, ... , m, is asymptotically distributed as
a multivariate normal with a zero mean and a unit covariance matrix. Moreover,
the
associated Ljung-Box statistic,
Q,, (m)
= T(T
+2>:E f>,,(t}2,
•=I
T-t
is asymptotically X~mJ under the null. If the null is rejected, then non-linear depend
ence, such
as GARCH, may be present. 14
After fitting a GARCH model, it is often of interest to test the null hypothesis that
the
standardized residuals are conditionally homoskedastic. Bollerslev and Mikkelsen
(1993)
argue that one may use the Ljung-Box statistic on the squared standardized residua
l
autocorrelations, but that the significance of the statistic should be tested using a ~m-k)
distribution, where k is the number of estimated GARCH parameters. This adjustm
ent is
necessary due to the deflation associated with fitting the conditional variance model.
A related testing issue concerns the effect of GARCH innovations on tests for other
deviations from classical behavior. Diebold (1987, 1988) examines the impact of
GARCH
17
effects on two standard serial correlation diagnostics, the Bartlett standard errors and
the
Ljung-Box statistic. As is well-known, in the large-sample Gaussian white-noise case,
p(-r) i.i.d.
- N ( 0, ~·) , -r-= 1, 2, ...
and
m
Q(m) = T(T+2) L
t=l
1
.(f-'t)
p(-r)
a
2
-
X~m)•
where p(-r) denotes the sample autocorrelation at lag -r. In the GARCH case, howev
er, an
adjustment must be made,
p(-r)
i.i.d.
N( 0, ~(I+ Yy;~-r)) ), -r = I, 2, . . ,
where y y,(-r) denotes the autocovariance function of y,2 at lag -r and a4 is the squared
unconditional variance of y,. The adjustment is largest for small -r and decreases
monotonically as -r- if the process is covariance stationary. Similarly, the robust
Ljung-Box
statistic is
00
Q(m) = T(T+2)
t-1
-(
t=l (T--r)
o4
) p(-r)2
o4 + Yy2('t)
~ X~mJ·
The formulae are made operational by replacing the unknown population parameters
with the
usual consistent estimators.
It is important to note that the standard error adjustment serves to increase the standar
d
errors; failure to perform the adjustment results in standard error bands that are "too
tight."
Similarly, failure to adjust the Ljung-Box statistic c:auses empirical test size to be larger
than
nominal size -- often much larger, due to the cumulation of distortions through summa
tion.
Thus, failure to use robust serial correlation diagnostics for GARCH effects may produc
ea
spurious impression of serial correlation.
18
A more general approach that yields robust sample autocovariances and related
statistics
is obtained by adopting a generalized method of moments (GMM) perspective,
as proposed by
West and Cho (1994). 15 Define "1 =
6
(e;, e,e1-1, ..., e.e.-m)',
= (E[e;J Eft;e,_iJ ... , Eft;e,-m])' and g.(6) = X. - 6 as.((m +l)xl)
vector sand 60MM
as the value of 6 that satisfies the condition
Note that, because there are as many parameters being estimated as there are orthog
onality
conditions, GMM simply yields the standard point estimates of the autocovarianc
es. Their
standard errors and related test statistics are asymptotically robust, because as
shown by
Hansen (1982) under general conditions allowing for heteroskedasticity and serial
correlation
of unknown form,
~ N(O,V) where
/f(eoMM - e)
&(:~MM) ] S -I E [ ag,( !:MM}
V = { E [a
rr
1
and S is the spectral density matrix of g.(6) at frequency zero. This expression
for V is made
operational by replacing all population objects with consistent estimates. The
GMM-estimated
autocovariances of y, and their standard errors will be robust to possible conditi
onal
heteroskedasticity in e,, as will the Ljung-Box statistic computed using the GMMestimated
autocovariances.
VI. Applications and Extensions
There are numerous applications and extensions of the basic GARCH model.
In this
section, we highlight those that we judge most important in macroeconomic and
financial
contexts. It is natural to discuss applications and extensions simultaneously becaus
e many of
the extensions are motivated by applications.
19
A. Functional Form and Density Form
Numerous alternative functional forms for the conditional variance have
been suggested
16
in the literature. One of the most interesting is Nelson's (1991) expon
ential GARCH(p,q) or
EGARCH(p,q) model,
Y1 = et =
1,
''t
112
Z,,
i.i.d.
z, - N( 0, I),
In(!\}
=w +
ta; g( z,_;)
+
t P; In(!\-;}
1= 1
,,...
g(z,} = 8z1 + Y(/z,/ -E[/z,/]).
The log specification ensures that the conditional variance is positive,
and the model allows for
an asymmetric response to the z. innovations depending on their sign.
Thus, the effect of a
negative innovation on volatility may differ from that of a positive innova
tion. This allowance
for asymmetric response has proved useful for modeling the "leverage
effect" in the stock
market described by Black (1976). 17
With respect to density form, non-Gaussian conditional distributions are
easily
incorporated into the GARCH model. This is important, because it is
commonly found that
the Gaussian GARCH model does not explain all of the leptokurtosis
in asset returns. With
this in mind, Bollerslev (1987) proposes a conditionally Student-t GARC
H model, in which
the degrees-of-freedom is treated as another parameter to be estimated.
Alternatively, Engle
and Gonzalez-Rivera (1991) propose a semiparametric methodology in
which the conditional
variance function is parametrically specified in the usual fashion, but
the conditional density is
estimated nonparametrically.
B. GARC H-M: Time-Varying Risk Premi a
20
Consider a regression model with GARCH disturbances of the usual sort, with one
additional twist: the conditional variance enters as a regressor, thereby affecting the
conditional mean. Write the model as
Y1 =
x.'P
e, I n,_ 1
+ Yh. +t,,
-
N ( o,
Ii.).
This GARCH-in-Mean (GARCH-M) model is useful in modeling the relationship between
risk
and return when risk (as measured by the conditional variance) varies. Engle, Lillien and
Robins (1987) introduce the model and use it to examine time-varying risk premia in the tenn
structure of interest rates.
C. !GAR.CH: Persistence in Variance
A special case of the GARCH model is the integrated GARCH (IGARCH) model,
introduced by Engle and Bollerslev (1986). A GARCH(p,q) process is integrated of order
one
in variance if I - a(L) - P(L) = 0 has a root on the unit circle. The IGARCH process
is
potentially important because, as an empirical matter, GARCH roots near unity are commo
n in
high-frequency financial data.
The earlier ARMA result for the squared GARCH process now becomes an ARIMA
result for the squared IGARCH process. As before,
thus, [I -a(L) - P(L)JC : =
<il -
e";
= <il +[a(L) +p(L)J t;-p(L) v +v ;
1
1
P{L) v, + v,. When the autoregressive polynomial
contains a unit root, it can be rewritten as
e;
[ 1 - a(L) - p (L)]
= q>(L)(l -L) t; = <il - p (L) v, + v,.
Thus, the differenced squared process is of stationary ARMA fonn.
Unlike· the conditional prediction error variance for the covariance stationary GARCH
process, the IGARCH conditional prediction error variance does not converge as the forecast
horizon lengthens; instead, it grows linearly with the length of the forecast horizon. Fonnall
y,
21
E[
e;.k I n,] = (k-l)w + 11..1so that Jim E [ e;.k I n,] =
an infinite unconditional variance.
k-•
00 •
Thus, the IGARCH process has
Clearly, a parallel exists between the IGARCH process and the vast literatu
re on unit
roots in conditional mean dynamics (see Stock, 1994). This parallel, howev
er, is partly
superficial. In particular, Nelson (1990b) shows that the IGARCH(l,1) proces
s (with w ., 0)
is nevertheless strictly stationary and ergodic, which leads one to suspect that
likelihood-based
inference may proceed in the standard fashion. This conjecture is verified
in the theoretical
and Monte Carlo work of Lee and Hansen (1994) and Lumsdaine (1992, 1995).
Although conditional variance dynamics are often empirically found to be
highly
persistent, it is difficult to ascertain whether they are actually integrated. (Again
, this
difficulty parallels the unit root literature.) Circumstantial evidence agains
t IGARCH arises
from several sources, such as temporal aggregation. Little is known about
the temporal
aggregation of IGARCH processes, but due to the infinite unconditional second
moment, we
conjecture that a Gaussian central limit theorem is unattainable. (To the best
of our
knowledge, no existing Gaussian central limit theorems are applicable.) If
so, this bodes
poorly for the IGARCH model, because actual series displaying GARCH effects
seem to
approach normality when temporally aggregated. It would then appear likely
that highly
persistent covariance-stationary GARCH models, not IGARCH models, provid
e a better
approximation to conditional variance dynamics.
The possibility also arises that some findings of IGARCH may be due to
misspecification of the conditional variance function. In particular, Diebo
ld (1986) suggests
that the appearance of IGARCH could be an artifact resulting from failure
to allow for
structural breaks in the unconditional variance, if in fact such breaks exist.
This is borne out
in various contexts by Lastrapes (1989), Lamoureux and Lastrapes (1990),
and Hamilton and
Susmel (1994). Accordingly, Chu (1993) suggests procedures for testing param
eter instability
in GARCH models.
22
D. Stochastic Volatility Models
A simple first-order stochastic volatility model is given by
e1 =
01
Zi = exp( ; ) Zi,
Zi - N( 0, 1 ),
h,
=
(A)
+
Pl\-1
+ ,,,,
111 - N ( 0, o~ ).
Thus, as opposed to standard GARCII models, h, is not deterministic conditional on QH; the
conditional variance evolves as a first-order autoregressive process driven by a separate
innovation. Moreover, the exponential specification ensures that the conditional variance
remains positive. It is clear that the stochastic volatility model is intimately related to Clark's
(I 973) subordinated stochastic process model -- in fact, for all practical purposes, it is Clark's
model. For further details, see Harvey, Ruiz and Shephard (1994), and for alternative
approaches to estimation, which can be challenging, see Jacquier, Polson and Rossi (1994) and
Kim and Shephard (1994). Although there has been substantial recent interest in stochastic
volatility models, their empirical success relative to GARCII models has yet to be established.
E. Multivariate GARCH Models
Cross-variable interactions are key in macroeconomics and finance. Multivariate
GARCII models are used to capture cross-variable conditional volatility interactions. The first
multivariate GARCII model, developed by Kraft and Engle (1982), is a multivariate
generalization of the pure ARCII model. The multivariate GARCII (p,q) model is proposed in
Bollerslev, Engle and Wooldridge (1988). The N-dimensional Gaussian GARCII(p,q) process
is e,
I n,_ 1
-
N (0, H,), ·where H. is the (NxN) conditional covariance matrix given by
vech(If.)
=
W
+
t
i=l
A; vech{ eH t;_;}
+
t
j=l
23
Bi vech(lf. ~
vech(.) is the vector-half operator that converts {NxN
) matrices into (N(N + 1)/2 xl) vectors of
their lowe r triangular elements, W is an (N(N + 1)/2
xl) parameter vector, and A; and B; are
((N{N + 1)/2) x {N(N + 1)/2)) parameter matrices. Like
lihood-based estimation and inference
are conceptually straightforward and parallel the univ
ariate case. The approximate log
likelihood function for the conditionally-Gaussian mult
ivariate GARCH(p,q) process, aside
from a constant, is
In practice, however, two complications arise. First
, the conditions needed to ensure that H. is
positive definite are complex and difficult to verify.
Second, the model lacks parsimony; an
unrestricted parameterization of H. is too profligate to
be of much empirical use. As written
above, the model has (N(N + 1)/2)[1 +(p+ q)N( N + 1)/2]
= O(N4) parameters, which makes
numerical maximization of the likelihood function extre
mely difficult, even for low values of
N, pan dq.
Various strategies have been proposed to deal with the
positive definiteness and
parsimony complications. Engle and Kroner (1993)
propose restrictions that guarantee
positive definiteness without entirely ignoring these
cross-variable interactions. Bollerslev,
Engle and Wooldridge (1988) enforce further parsimon
y by requiring that the A; and B;
matrices be diagonal, reducing the number of param
eters to (N(N + 1)/2)[1 +p+ q] = O{N2).
However, the parsimony of this "diagonal" model come
s at potentially high cost, because
much of the potential cross-variable volatility interactio
n, a key point of multivariate analysis,
is assumed away.
F. Common Volatility Patterns: Multivariate Mod
els With Factor Structure
Multivariate models with factor structure, such as the
latent-factor GARCH model
(Diebold and Nerlove, 1989) and the factor GARCH
model (Engle, 1987 and Bollerslev and
24
Engle, 1993), capture the idea of commonality of volatility shocks,
which appears empirically
relevant in systems of asset returns in the stock, foreign exchange, and
bond markets. 18
Models with factor structure are also parsimonious and are easily constr
ained to maintain
positive definiteness of the conditional covariance matrix.
In the latent-factor model, movements in each of the N time series are
driven by an
idiosyncratic shock and a set of k < N common latent shocks or "facto
rs". 19 The latent factors
display GARC H effects, whereas the idiosyncratic shocks are i.i.d.
and orthogonal at all leads
and lags. The one-factor model is important in practice, and we descri
be it in some detail.
The model is written as e, = .i..F + v,, where e., A and u, are (Nxl)
vectors and F, is a scalar.
1
F, and u, have zero conditional means and are orthogonal at all leads
and lags. The factor F,
follows a GARCH(p,q) process,
F,
1
n,_ 1 -
N(o,
I\)
h1 = c.> + a(L)F,2 + PCL)h.,
so that the conditional distribution of the obseived vector is
e,I0,_ 1 -
N ( 0, ff.}
.... = .i...i..'h' + r,
T-f
where r = cov(v,) = diag(y ,
1
-
H..JJ,t
= )...J2h-~
••• ,
yN). Thus, the j'11 time-t conditional variance is
+ y.J
= )...J2( c.>
+
t
i=I
2
q
a.F,
. + L..J
't"'
1 -1
i=I
ll._h .)
t'1•'t-1
+
Y·J'
and the j ,1<'1' time-t conditional covariance is
Note that the latent factor F, is unobseivable and not directly included
in '21-1 = {e1_1,
Effectively, the latent-factor model is a stochastic volatility model.
In general, the numbe r of parameters in the k-factor model is
N(k + 1) + k 2(1 +p +q) = O(N), so the number of parameters in the one-fa
ctor case is
25
••• ,
e1 }
2N +(I +p+ q), a drasti~ reduction relative to the general multi
variate case. Moreover, the
conditional covariance matrix is guaranteed to be positive defin
ite, so long as the conditional
variances of the common and idiosyncratic factors are_constraine
d to be positive.
A simulated realization from a bivariate model with one comm
on GARCH(l, 1) factor
is sho.wn in Figures 7-9. The model is parameterized as
h, = I
+
(v 1,, v2.)'
.2F/. 1
+
.71\_ 1,
i.i.d.
-
N(O, I).
The realization of the common factor underlying the system
is precisely the one presented in
our earlier discussion of univariate GAR.CH models. The laten
t-factor GAR.CH series exhibit
the volatility clustering present in the common factor. As befor
e, the squared realizations of
the two series indicate a degree of persistence in volatility.
Furthermore, as expected, the
conditional second moments of the two series are similar to that
of F, because, as shown
above, they are simply multiples of Ii.Diebold and Nerlove (1989) suggest a two-step estimation proce
dure. The first step
entails performing a standard factor analysis; i.e., factoring
the unconditional covariance
matrix as H = ;\.;\. 1o2
+
r, where o2 is the unconditional variance ofF,, and extracting an
.estimate of the time series of factor values
{F,}:1.
The second step entails estimating the
latent-factor GAR.CH model treating the extracted factor series
F, as if it were the actual series
F,.
The Diebold-Nerlove procedure is clearly suboptimal relative
to fully simultaneous
maximum likelihood estimation, because the F, series is not
equal to the F, series, even
asymptotically. Harvey, Ruiz and Sentana (1992) provide a
better approximation to the exact
26
likelihood function that involves a correction factor to account for the fact that the F, series is
unobservable. 20 For example, using an ARCH(l) specification, the conditional variance of the
latent factor F, in the Diebold-Nerlove model is
Ii. = var(F,IC:J.-i) = w
+ aF,: 1 = w. + aE [F1:t1Q._ 1}
Using the identity F1-1 = F1-1 + (F1-1 - Ft-1}
r
Er F,:1101-1 J = E [ Ft-I + (Ft-1-F,-1) 10,-1 J = Er F,~110,-1 J + P(-1 = F,~1 + P,-1,
where p,.1 is the correction factor. Thus, h, is expressed as Ii. = w + a F ~ + p _ }. The
1 1
1 1
correction factor can be constructed using the appropriate elements in the conditional
covariance matrix of the state vector estimated by the Kalman filter.
Finally, we note that recently-developed Markov-chain Monte Carlo techniques
facilitate exact maximum-likelihood estimation of the latent-factor model (or, more precisely,
approximate maximum-likelihood estimation with the crucial distinction that the approximation
error is under the user's control and can be made as small as possible). For details see Kim
and Shephard (1994).
G. Optimal Prediction Under Asymmetric Loss
Volatility forecasts are readily generated from GARCH models and used for a variety
of purposes, such as producing improved interval forecasts, as discussed previously. Less
obvious but equally true is the fact that, under asymmetric loss, volatility dynamics can be
exploited to produce improved point forecasts, as shown by Christoffersen and Diebold
(1994). If, for example, Y,+k is normally distributed with conditional mean µ,.klO, and
conditional variance 11..klO. and L(et+k) is any loss function defined on the k-step-ahead
prediction error e,.k = Y,.k - Y,.k• then the optimal predictor is Y,.k = µ,.klO, + a,, where a,
depends only on the loss function and the conditional prediction error variance
var( e,.klO,}
= var(y,.klO,} = 11..klO,.
The optimal predictor under asymmetric loss is not the
conditional mean, but rather the conditional mean shifted by a time-varying adjustment that
27
depends on the conditional variance. The intuition
for this is simple - when, for example,
positive prediction errors are more costly than nega
tive errors, a negative conditionally
expected error is desirable and is induced by setting
the bias 0:1 > 0. The optimal amount of
bias depends on the conditional prediction error varia
nce of the process. As the conditional
variation around µ,.d 0 1 grows, so too does the optim
al amount of bias needed to avoid large
positive prediction errors.
To illustrate this idea, consider the linlin loss function,
so-named for its linearity on
each side of the origin (albeit with possibly different
slopes):
L(Y,.k -:r,.k)
ajy,.k -y,.kl,
=
l
•
if Y,.k -y,.k > 0
.f
•
b/y,.k-yt•k'• 1 Y,.k-Yt+k
s;
O.
Christoffersen and Diebold (1994) show that the optim
al predictor of Yi+k under this loss
function is
where cl> is the Gaussian cumulative density function.
In contrast, a pseudo-optimal predictor,
which accounts for loss asymmetry but not condition
al variance dynamics, is
Yt+k
where
CJ~
=
µ,.k/0,
+ 0
k q,-1( a:b),
is the unconditional variance of Y,+t·
In Figure 10, we show our GARCH(l,l) realizatio
n together with the one-step-ahead
linlin-optimal, pseudo-optimal and conditional mean
predictors for the loss parameters a = .95
and b = .05. Note that the optimal predictor injec
ts more bias. when conditional volatility is
high, reflecting the fact that it accounts for both loss
asymmetry and conditional
heteroskedasticity. This conditionally optimal amo
unt of bias may be more or less than the
constant bias associated with the pseudo-optimal pred
ictor. Of course, the conditional-mean
28
predict or injects no bias, as it accounts for neither loss asymmetry nor conditi
onal
heteroskedasticity.
H. Evaluating Volatility Forecasts
Although volatility forecast accuracy comparisons are often conducted using
meansquared error, loss functions that explicitly incorporate the forecast user's econom
ic loss
function are more relevant and may lead to different rankings of models. West
et al. (1993)
and Engle et al. (1993) make important contributions along those lines, propos
ing economic
loss functions based on utility maximization and profit maximization, respect
ively.
Lopez (1994) proposes a volatility forecast evaluation framework that subsum
es a
variety of economic loss functions. The framework is based on transforming
a model' s
volatility forecasts into probability forecasts by integrating over the distribution
of t\. By
selecting the range of integration corresponding to an event of interest, a forecas
t user can
incorporate elements of her loss function into the probability forecasts. For
example, given
e, I0 1_1 - D( 0, Ii.) and a volatility forecast
an options trader interested. in the event
h.,
e, E [ L•. ,. Uc. 1 ] would generate the probability forecast
p :
t
Pr(L
<e <
c.t
t
--, Jr.,l
U )·: Pr[ L•. , <7 < u•.
c,t
/hi
where z. is the standardized innovation,
: ·~,
f( z,) is the functional form of the distribution
D ( o, I ) , and [1•. ,, u•. 1 ] is the standardized range of integration. In contras
t, a. forecast user
such as a portfolio manag er or a central bank interested in the behavior of y,
= µ1 + e,, where
µ1 = E [ y1
P
I
I 0 1_1 ], would generate the probability forecast
=
Pr(L
y.t
<y <
t
U ) = Pr[ Ly., - lli
y.t
A
29
< z, <
A µ,l
uy.t -
=
where
11,
is the forecasted conditional mean and [ ly. T•t' uy, T•t]
is the standardized range of
integration.
The probability forecasts so-generated can be evaluated
using statistical tools tailored to
the user 's loss function. In particular, probability scori
ng rules can be used to assess the
accuracy of the probability forecasts, and the significan
ce of differences across models can be
tested using a generalization of the Diebold-Mariano
(1995) procedure. Moreover, the
calibration tests of Seillier-Moiseiwitsch and Dawid
(1993) can be used to examine the degree
of equivalence between an even t's predicted and obse
rved frequencies of occurrence within
subsets of the probability forecasts specified by the user.
vn.
Directions for Future Research
Fifteen years ago, little attention was paid to condition
al volatility dynamics in
modeling macroeconomic and financial time series;
the situation has since changed
dramatically. GARCH and related models have prov
ed tremendously useful in modeling such
dynamics. However, perhaps in contrast to the impr
ession we may have created, we believe
that the literature on modeling conditional volatility
dynamics is far from settled, and that
complacency with the ubiquitous GAR CH( l, 1) mode
l is not justified.
Almost without exception, low-ordered (and hence poten
tially restrictive) GARCH
models are used in applied work. For example, amon
g hundreds of empirical applications of
the GARCH model, almost all casually and uncriticall
y adopt the GAR CH( l,1) specification.
EGARCH applications have followed suit with the vast
majority adopting the EGA RCH (l, I)
specification. Similarly, applications of the stochastic
volatility model typically use an AR( l)
specification. However, recent findings suggest that
such specifications - as well as the
models themselves, regardless of the particular speci
fication -- are often too restrictive to
maintain fidelity to the data.
30
It appears, for example, that the conditional volatility dynamics of stock market returns
(as well as certain other asset returns) contain long memory. Ding, Engle and Granger (1993)
find positive and significant sample autocorrelations for daily S&P 500 returns at up to 2500
lags and that their rate of decay is slower than exponential. A model consistent with such
long-memory volatility findings is the fractionally-integrated GARCH (FIGARCH) model
developed by Baillie, Bollerslev and Mikkelsen (1993), building on earlier work by Robinson
(1991). FIGARCH is a model of fractionally-integrated conditional variance dynamics, in
parallel to the well-known fractionally-integrated ARMA models of conditional mean dynamic
s
(e.g., Granger and Joyeux, 1980). The FIGARCH model implies a hyperbolic rate of decay
for the autocorrelations of the squared process that is slower than exponential.
To motivate the FIGARCH process, begin with the GARCH(l,l) process,
e,
'1t
I 0 1_1
-
N(O,
h,}
= w + a(L)t:; + P(L)h.-
Rearranging the conditional variance into ARMA form, the FIGARCH (p,d,q) equation is
[! - a(L) - P(L)]
e; = <!>(L) (I -Lt e; = w
(1-P(L)) u,.
That is, the [ I - a(L) - P(L)] polynomial can be factored into a stationary ARMA
+
component and a long-memory difference operator. If O< d < l, the process is
FIGARCH(p,d,q). If d=O, then the standard GARCH(p,q) model obtains; if d = 1, then the
IGARCH(p,q) model obtains. Bollerslev and Mikkelsen (1993) conjecture that the coefficients
in the ARCH representation of a FIGARCH process (d < 1) are dominated by those of an
IGARCH process. If so, then FIGARCH (d < 1) would be strictly stationary (though not
covariance stationary), because IGARCH is strictly stationary.
Long memory is only one of many previously unnoticed features of volatility.
Interestingly, as we study volatility more carefully, more and more anomalies emerge.
Volatility patterns tum out to differ across assets, time periods, and transformations of the
31
data. The complacency with the "standard" GARCH mode
l is being shattered, and we think it
unlikely that any one consensus model will take its place.
The implications of this
development are twofold. First, real care must be taken
in tailoring volatility models to the
relevant data, as in Engle and Ng (1993). Second, becau
se all volatility models are likely to
be misspecified, care should be taken in assessing models'
robustness to misspecification.
To illustrate the deviations from classical GARCH models
that turn out to be routinely
present in real data, we present in Figure 11 the sample autoc
orrelation functions of the
absolute and squared change in the Jog daily closing value
of the S&P
1990. The autocorrelation functions are shown to displaceme
nt -c
500 stock index, 1928-
= 200 in order to assess the
evidence for long memory, and dashed lines indicate the
Bartlett 95 % confidence interval for
white noise. Note that substantially more persistence is found
in absolute returns than in
squared returns, in keeping with Ding, Engle, and Granger
(1993), and that both absolute and
squared returns appear too persistent to accord with any of
the "standard" volatility models. In
addition, these patterns are different over time. In Figur
e 12, we show squared returns over
various subperiods: 1928-1940, 1941-1970, 1971-1980
and 1981-1990. It seems clear that
most of the Jong memory is driven by the 1928-1940 perio
d. To the extent that there is any
long memory in the post-1940 period, it seems to be comi
ng from the 1970's. Interestingly,
there seems to be no GARCH effects in the 1980's as show
n by the negligible autocorrelations
2
for e,.
Other assets, including interest rates, foreign exchange rates,
and other stock indexes,
display a bewildering variety of volatility patterns, as discu
ssed in Mor (1994). Sometimes
there seems to be long memory; sometimes not. Sometimes
the autocorrelation patterns of
match those of je,I, and sometimes the autocorrelation patte
rns of le,I appear much more
persistent. The patterns differ across assets and often seem
to indicate structural change. For
example, the long memory seemingly present in exchange
rate volatility seems concentrated in
the 1970's, while long memory in interest rate volatility
is typically concentrated in the
e;
32
1980's. These observed phenomena, as well as occasional long-horizon spikes in
autocorrelations and the appearance of oscillatory autocorrelation behavior, are again
inconsistent with standard specifications.
An additional illustration of the inadequacies of GARCH models is provided by West
and Cho (1994). Using weekly exchange rates, they show that for horizons longer than one
week, out-of-sample GARCH volatility forecasts loose their value, even though volatility
seems highly persistent. The good in-sample perfonnance of GARCH models breaks down
rapidly out-of-sample. 21 In addition, standard tests of forecast optimality, such as regressi
ons
of realized squared returns on an intercept and the GARCH forecast, strongly reject the null
of
the optimality of the GARCH forecast with respect to available infonnation. West and Cho
suggest time-varying parameters and discrete shifts in the mean level of volatility as possible
explanations.
In light of the emerging evidence that GARCH models are likely misspecified and the
unlikely occurrence of happening upon a "correct" specification, it is of interest to conside
r
whether GARCH models might still perfonn adequately in tracking and forecasting volatilit
y -that is, whether their good properties are robust to misspecification. In a series of papers
{Nelson, 1990a, 1992, 1993; Nelson and Foster, 1991, 1994), Nelson and Foster find that
the
usefulness of GARCH models in volatility tracking and short-tenn volatility forecasting is
robust to a variety of types of misspecification; thus, in spite of misspecification, GARCH
models can consistently extract conditional variances from high-frequency time series. More
specifically, if a process is well approximated by a continuous-time diffusion, then broad
classes of GARCH models provide consistent estimates of the instantaneous conditional
variance as the sampling frequency increases. This occurs because the sequence of
GARCH {l,I) models used to fonn estimates of next period's conditional variance average
increasing numbers of squared residuals from the increasingly recent past. In this way, a
33
sequence of GARCH(l,l) models can consistently estimate
next period's conditional variance
despite potentially severe misspecification.
34
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40
Figure 1
Figure2
Daily Spot SF/$ (1974-1991)
Daily Spot DM/$ (1974-1991)
·~ --- --- --- ~
•.- --- --- --- --- -,
·-
1.8
1.8
1.,
1.6
1.4
1.4
u
0.2
Timo
Daily DM/$ Returns (1974-1991)
Daily SF/$ Returns (1974-1991)
0.00
0.06
0.04
.....
.....
Timo
Squared DM/$ Returns (1974-1991)
.,
Squared SF/$ Returns (1974-1991)
•
'10
3
3
25
2
2
,..
Time
41
Figu re3
GAR CH( l,1) Realization
Sample Autocorrelation Function
10
0
-10
Tme
•
Figu re4
Conditional Variance
Sample Autocorrelation Function
.,.
30
•••
25
20
~~
0
------- ------
-o.•1,----.------...----.,,,r---,=----"""·,
Tme
•
Figu res
Squared GAR CH( l,l) Realization
120
80
50
••
42
Figure6
GARC H(l,l) Realization with One-Step-Ahead
90% Conditional and Unconditional Confidence Intervals
10
8
6
-10
0
100
20
300
Time
43
400
500
Figure 7
Factor-GARCH Series 1
Factor-GARCH Series 2
8
8
•
4
j. ij
.
Figur es
Factor-GARCH Series 1 Squared
Factor-GARCH Series 2 Squared
80
.
70
70
80
60
'°
40
'°
30
30
40
44
Figure 9
Conditional Variance of Series 1
Sample Autocorrelation Function
.
30
,.
•••
20
15
0
k
Conditional Variance of Series 2
Sample Autocorrelation Function
.
30
,.
20
Time
k
Conditional Covariance
Sample Autocorrelation Function
.
30
,.
20
,.
-0.501,- --..-,---,u ---,.--m, ...----.J,
k
45
Figure 10
GARC H(l,l) Realization with
Linlin Optimal, Pseudo-Optimal, and Conditional Mean Predictors
15, ---- ---, ---- --,- ---- ---r ---- -,-- --~
10
-15- ---~ ----- -=b. ----- ,d-,. ----- ----. -,..,. .---- 0
100
200
300
400
500
Notes to Figure: The linlin loss parameters are set to a = .95 and b = .05, so
a/(a+b ) =
.95. The GARC H(l,1) parameters are set to a=.2 .and P=.75. The dotted linethat
is
the
GARC H(l,l) realization. The horizontal line at zero is the conditional mean
predict
horizontal line at 1.65 is the pseudo-optimal predictor, and the time-varying solid or, the
line is the
optimal predictor.
46
Figure 11
Autocorrelation Function-S&P
Ie I - Jan 28 to May 90
Autocorrelation Function-S&P
2
~
- Jan 28 to May 90
;~- ---- -~- -~- ---,
•
•
0
0
.
0
'=!---
1--
.
;-_- ----- - -- -- -_._- •
--
--------------- --
•'------~--------__,
l 0
20
•0
80
80
100
120
140
160
180
200
47
··~
Figure 12.
Autocorrelation Function-S&P
e 2 - Jan 28 to Dec 40
Autocorrelation Function-S&P
e 2 - Jan 41 to Dec 70
••
••
.
•
~---1--- ----
--
__,_
;;.___,,.
10
Autocorrelation Function-S&P
e 2 - Jan 71 to Dec 80
20
<10
___________
60
BO
100
120
1•0
160
180
_,
200
Autocorrelation Function-S&P
e2 - Jan 81 to May 90
.•
••
---•-~·--- ----!--- ---- r--- ---- -
---- ---- ---- ----
•'- --- --- --- --- --- -'
,o
20
"0
&0
ao
100
120
uo
tac
1ao
200
•
48
Endnotes
1. ARCH is short for AutoRegressive Conditional Heterosked
asticity. ·
2. A process is linearly detenninistic if it can be predicted to any desire
d degree of accuracy
by linear projection on sufficiently many past observations.
3. Recall that the defining characteristic of white noise-is a lack of
serial correlation, which is
a weaker condition than serial independence.
4. The obvious empirically useful approximation to an LRCSSP (whic
h is an infinite-ordered
moving average) with infinite-ordered ARCH errors is an ARMA proce
ss with GARCH
errors. See Weiss (1984), who studies ARMA processes with finiteordered ARCH errors.
(The GARCH process had not yet been invented.)
5. Nelson and Cao (1992) show that, for higher order GARCH proce
sses, the nonnegativity
constraints are sufficient, but not necessary, for the conditional varian
ce to be positive.
6. See, for example, Jorgenson (1966).
7. Setting Yo= 0 and ho= E(y,2), we generate 1500 observations,
and we discard the first
1000 to eliminate the effects of the start-up values.
8. The parameter values for a and
empirical literature.
Pare typical of the parameter estimates reported in the
9. For a precise statement of the necessary and sufficient condition
for finite kurtosis, see
Bollerslev (1986).
10. Their results, however, require a finite fourth unconditional mome
nt, a condition likely to
be violated in financial contexts.
11. Alternative approaches may of course be taken. Geweke (1989
), for example, discusses
Bayesian procedures.
12. Generalization to the GARCH case has not yet been done.
13. However, Bollerslev and Wooldridge (1992) introduce a modif
ied LM test robust to nonnonnal conditional distributions.
14. As always, rejection of the null does not imply acceptance of the
alternative. Tests for
conditional heteroskedasticity, for example, often have power again
st alternatives of serial
correlation as well; see Engle, Hendry and Trumble (1985).
15. Robinson (1991) also treats the issue of robustness by proposing
general classes of
heteroskedasticity-robust serial correlation tests and serial correlationrobust heteroskedasticity
tests.
16. In fact, Robinson (1987) goes so far as to propose nonparametric
estimation of the
conditional variance function, thereby eliminating the need for param
etric specification of
functional fonn.
49
17. Negative shocks appear to contribute more to stock marlcet volatility than do positive
shocks. This phenomena is called the leverage effect, because a negative shock to the market
value of equity increases the aggregate debt/equity ratio (other things the same), thereby
increasing leverage.
18. Models of "copersistence" in variance and cointegration in variance are based on similar
ideas; see Bollerslev and Engle (1993).
19. Despite the similarity in their names, the latent-factor GARCH model discussed here is
different from the factor GARCH model. In the latent-factor GARCH case, the observed
variables are linear combinations of latent GARCH processes, whereas in the factor GARCH
case, linear combinations of the observed variables follow univariate GARCH processes. As
pointed out by Sentana (1992), the difference between the two models is similar to the
difference between standard factor analysis and principal components analysis.
20. See also King, Sentana and Wadhwani (1994), Demos and Sentana (1991), and Sentana
(1992).
21. Note, however, that West and Cho (1994) evaluate volatility forecasts using the meansquared error criterion, which may not be the most appropriate. For further discussion, see
Bollerslev, Engle and Nelson (1994) and Lopez (1995).
50
@FedFRASER