JOURNAL OF APPLIED ECONOMETRICS

J. Appl. Econ. 17: 425– 446 (2002)

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jae.683

NEW FRONTIERS FOR ARCH MODELS

ROBERT ENGLE*
Department of Finance, Stern School of Business, New York University, New York, NY 10012, USA

SUMMARY
In the 20 years following the publication of the ARCH model, there has been a vast quantity of research
uncovering the properties of competing volatility models. Wide-ranging applications to financial data have
discovered important stylized facts and illustrated both the strengths and weaknesses of the models. There are
now many surveys of this literature. This paper looks forward to identify promising areas of new research.
The paper lists five new frontiers. It briefly discusses three—high-frequency volatility models, large-scale
multivariate ARCH models, and derivatives pricing models. Two further frontiers are examined in more
detail—application of ARCH models to the broad class of non-negative processes, and use of Least Squares
Monte Carlo to examine non-linear properties of any model that can be simulated. Using this methodology, the
paper analyses more general types of ARCH models, stochastic volatility models, long-memory models and
breaking volatility models. The volatility of volatility is defined, estimated and compared with option-implied
volatilities. Copyright  2002 John Wiley & Sons, Ltd.

1. INTRODUCTION
Who could imagine 20 years ago, the flowering of research and applications that would develop
around the ARCH model? It certainly was not an instant success. Years went by before anyone
except my students and I wrote a paper on ARCH. But as applications shifted to financial markets,
and as richer classes of models were developed, researchers saw how volatility models could be
used to investigate the fundamental questions in finance. How are assets priced and what is the
tradeoff between risk and return? ARCH models offered new tools for measuring risk, and its
impact on return. They also provided new tools for pricing and hedging non-linear assets such as
options.
This conference and this paper are designed to reflect on these developments and look forward to
the next important areas of research. In this paper I will give a rather brief idiosyncratic assessment
of the important accomplishments of the last 20 years in volatility modelling. Then I will point to
five frontiers on which I think new developments can be expected in the next few years. For two
of these, I will give some new results to show the directions I see developing.

2. WHAT WE HAVE LEARNED IN 20 YEARS

The research that made as of this possible is the finding that almost every asset price series
exhibits volatility clustering that can be modeled by ARCH/GARCH. This was pointed out early
by French, Schwert and Stambaugh (1987) and Bollerslev (1987), and is especially clear in some
of the surveys of empirical work. It inspired all the rest including a quest for better models.
The number of new models proposed, estimated and analysed has been dramatic. The alphabet

Ł Correspondence to: Robert Engle, Department of Finance, New York University, 44 West 4H, NY, NY 10012, USA.
E-mail: rengle@stern.nyu.edu

Copyright  2002 John Wiley & Sons, Ltd. Received 13 December 2001
Revised 7 June 2002
426 R. ENGLE

soup of volatility models continually amazes. The most influential models were the first: the
GARCH model of Bollerslev (1986), and the EGARCH of Nelson (1991). Asymmetric models
of Glosten, Jaganathan and Runkle (1993) Rabemananjara and Zakoian (1993), Engle and Ng
(1993) and power models such as Higgins and Bera (1992), Engle and Bollerslev (1986), and
Ding, Granger and Engle (1993) joined models such as SWARCH, STARCH, QARCH and many
more. The linguistic culmination might be that of Figlewski (conference presentation, UCSD,
1995), the YAARCH model—an acronym for Yet Another ARCH model. Coupled with these
models was a sophisticated analysis of the stochastic process of data generated by such models as
well as estimators of the unknown parameters. Theorems for the autocorrelations, moments and
stationarity and ergodicity of these processes have been developed for many of the important cases;
see, for example, Nelson (1990), He and Terasvirta (1999a,b) and Ling and McAleer (2002a,b).
Work continues and new models are continually under development, but this is a well-studied
frontier. The limiting distribution of the MLE for GARCH models waited for Lumsdaine (1996)
and Lee and Hansen (1994) for rigorous treatments. There are now a collection of survey articles
that give a good appreciation of the scope of the research. See, for example, Bollerslev, Chou
and Kroner (1992), Bollerslev, Engle and Nelson (1994), Bera and Higgins (1993), and recent
pedagogical articles by Engle (2001) and Engle and Patton (2001). A very recent survey is Li,
Ling and McAleer (2002).
Another topic for ARCH models is their usefulness in trading options. It was initially supposed
that volatility models could give indications of mispricing in options markets leading to trading
opportunities. Early studies such as Engle, Kane and Noh (1994) suggested the profitability of
such strategies. More recent data fails to find evidence of significant trading opportunities, at
least in the US index options market. This is not surprising since GARCH models have a limited
information set and are available to all traders today. The same question is often asked in terms
of forecast accuracy. Do GARCH models out-forecast implied volatility models? The answer is
complex depending upon the statistical approach to forecast evaluation, but generally it is found
that implied volatilities are more accurate forecasts of future volatility than are GARCH models.
See, for example, Poon and Granger (2002).
The theory of asset pricing is based upon the reward for bearing risk. ARCH models have been
developed to measure the price of risk. The first such model was the univariate ARCH-M model
of Engle, Lilien and Robins (1987). Estimation of the CAPM began with Bollerslev, Engle and
Wooldridge (1988) and has been extended and improved by a series of interesting papers including
McCurdy and Stengos (1992), Engel et al. (1995), and de Santis, Gerard and Hillion (1997).
With the introduction of Value at Risk, a new role for ARCH models emerged. A variety
of studies examined the usefulness of volatility models in computing VaR and comparing these
methods with the exponential smoothing approach favoured by Riskmetrics. See, for example,
Christoffersen and Diebold (2000), Christoffersen, Hahn and Inoue (2001) and Alexander (1998).
GARCH methods proved successful but suffered if errors were assumed to be Gaussian.
These chapters of research on ARCH models are full and may have reached the point of dimin-
ishing returns. However, new directions are always available and these are the main focus of
this paper.

3. FIVE NEW FRONTIERS

Five new frontiers are identified below. These are areas where substantial research can be expected
over the next few years. The problems are important, soluble, and already have some important
Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
 NEW FRONTIERS FOR ARCH MODELS 427

new papers. For two of the areas, I will give some new results suggesting a possible direction for
future research.

3.1. High-Frequency Volatility Models

The study of volatility models within the day is in its infancy yet is a natural extension of the
daily models examined so widely. Several alternative formulations have been introduced including
Andersen and Bollerslev (1997) and Bollerslev, Cai and Song (2000). Such models focus on the
time of day or ‘diurnal’ effect and have the requirement that they be useful forecasting many days
in the future. These models have regularly spaced observations in calendar time but ultimately
it will be desirable to find models based on irregularly spaced data as this is the inherent limit
of high frequency data. Engle (2000) calls such tick data ‘ultra high frequency’ data and gives
some models which indicate that the arrival rate of trades, spreads and other economic variables
may be important variables for forecasting volatility at this frequency. Such a model could give a
continuous record of instantaneous volatility where events such as trades and quote revisions as
well as time itself, modify the volatility estimate.
Continuous time models are ubiquitous in financial theory and derivative pricing. However most
estimation of these models begins with equally observed prices and focuses on the mean process
possibly with jumps. Continuous time stochastic volatility models, possibly with volatility jumps
are a new class of models with interesting derivative implications.
In addition to these models, there is now increasing interest in using intra-daily data to estimate
better daily models. Andersen et al. (2001), for example, build models based upon ‘realized
volatility’ and Andersen and Bollerslev (1998) use this measure to evaluate traditional GARCH
specifications.

3.2. Multivariate Models

Although the research on multivariate GARCH models has produced a wide variety of models and
specifications, these have not yet been successful in financial applications as they have not been
capable of generalization to large covariance matrices. As computation becomes cheaper, and new
parsimonious models are formulated, the potential for building ever larger time varying conditional
covariance and correlation matrices increases. Models such as the vec and BEKK model of Engle
and Kroner (1995) have attractive properties as linear systems. The constant conditional correlation
(CCC) model of Bollerslev (1990) has the attraction of computational simplicity. A new model
called Dynamic Conditional Correlation (DCC) by Engle (2002) combines some of these features
to introduce a parsimonious correlation model to go with a conventional volatility model. Engle
and Sheppard (2001) estimate and test models of up to 100 assets. Ledoit and Santa-Clara (1998)
combine bivariate models to form multivariate models in a way which can be greatly expanded.
Correlation models can be estimated directly on intraday data. However as the frequency
increases, the asynchronicity of trades and returns leads to a serious underestimate of comovements.
This has been observed since Epps (1979) and the solutions of Scholes and Williams (1977) are
widely employed in spite of both theoretical and empirical difficulties. These are not appropriate
for ultra high frequency data and new solutions must be found.
Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
428 R. ENGLE

3.3. Options Pricing and Hedging

The pricing of options when the underlying asset follows a GARCH model is a topic of
future research. Most approaches are based on simulation but the appropriate approach to risk
neutralization must be investigated. This was barely addressed in Engle and Mustafa (1992) when
they simply simulated GARCH returns with the riskless rate as mean, calculating an ‘implied
GARCH’ model. Duan (1995, 1997) in a series of papers has proposed a local risk neutralization
which is based on the assumption that quadratic utility is a good local approximation to the
representative agent’s risk preferences. Engle and Rosenberg (2000) develop hedging parameters
for GARCH models and Rosenberg and Engle (2001) jointly estimate the pricing kernel and
the empirical GARCH density. This paper is in the line of several which use both options and
underlying data to estimate both the risk neutral and objective densities with ever more complex
time series properties in an attempt to understand the enormous skew in index options volatilities.
An alternative strategy is the GARCH tree proposed by Richken and Trevor (1999) which
adapts binomial tree methods for the path-dependence of GARCH models. These methods involve
a variety of assumptions that must be examined empirically. They are computationally much faster
than the simulation estimators discussed above.

4. MULTIPLICATIVE ERROR MODELS FOR MODELLING NON-NEGATIVE PROCESSES

4.1. Introduction
GARCH type models have been introduced for other variables. Most notable is the Autoregressive
Conditional Duration model ACD of Engle and Russell (1998) that surprisingly turns out to be
isomorphic to the GARCH model. In this section, I explore a much wider range of potential
applications of GARCH type models for any non-negative time series process.
Frequently we seek to build models of time series that have non-negative elements. Such series
are generally common and are particularly common in finance. For example, one could model
the volume of shares traded over a 10-minute period. Or one might want to model the high price
minus the low price over a time period, or the ask price minus the bid price, or the time between
trades, or the number of trades in a period, or many other series. There are two conventional
approaches to this problem: the first is to ignore the non-negativity, and the second is to take logs.
We discuss the disadvantages of these approaches.
Consider a time series fxt g, t D 1, . . . , T, where xt ½ 0 for all t. Suppose in addition that
P
xt < jxt
1 , . . . , x1  > 0, for all  > 0, and for all t
1
which says that the probability of observing zeros or near zeros in x is greater than zero. Let the
conditional mean and variance of the process be defined as:
t  E
xt jxt
1 , . . . , x1 , t2  V
xt jxt
1 , . . . , x1 
2
A linear model is given by
x t D  t C εt , εt j=t
1 ¾ D
0, t2 
3
It is clear that the distribution of the disturbances must be carefully specified. Since the mean is
positive and x is non-negative, the disturbances cannot be more negative than the mean. Thus the
Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
 NEW FRONTIERS FOR ARCH MODELS 429

range of the disturbance will be different for every observation. The variance and other higher
moments are unlikely to be constant. Efficient estimation via Maximum Likelihood is going to
be very difficult, although least squares will remain consistent. The probability of a near zero is
given by
Pt
1
xt <  D Pt
1
εt < 
t 
hence the error distribution must be discontinuous at
t in order to satisfy (1).
The second conventional solution is to take logs. The model might then be written as
log
xt  D mt C ut
4
where
t D emt E
eut , t2 D e2mt V
eut 
5
This solution will not work if there are any exact zeros in fxt g. Sometimes a small constant is
added to eliminate the zeroes. However, this is more of a theoretical solution than a practical
one since the finite sample estimates are typically heavily influenced by the size of this constant.
Furthermore, the assumption in (1) that observations very near zero are possible, requires that
P
u <
A > 0, for all A > 0. This is only true of very peculiar distributions.
Estimation of (4) requires the specification of both m and u. Clearly the relation between
mt and t depends upon the distribution of u. Thus even one-step forecasts require knowing the
distribution of u.

4.2. Theoretical Model

The proposed model, which solves all these problems, is the multiplicative error model that could
be abbreviated as MEM. This model specifies an error that is multiplied times the mean. The
specification is
xt D t εt , εt j=t
1 ¾ D
1, t2 
6
thereby automatically satisfying (2). The range of the disturbance would naturally be from
zero to infinity thereby satisfying (1). If the disturbance is i.i.d., then the variance of x is
proportional to the square of its mean. This is a strong restriction but is not in conflict with
other parts of the model. If it is not i.i.d. then a non-negative distribution with a unit mean
and time varying variance can be specified. There are many candidates. The residual in x is
naturally measured as the proportional deviation from the estimated mean as the standardized
residual, xt /
O t . This would be homoscedastic although the additive residual xt
 O t
εt
1
Ot D 
would not.
Vector models can be formulated in just the same way. Let an arrow represent a column
vector and let þ represent the Hadamard product of two matrices, which is element-by-element
multiplication. See Styan (1973).
xEt D 
E t þ Eεt , and V
Ext  D  E t ‘ þ V
Eεt  D diag

E t E t V
Eεt  ð diag

E t
7
Thus the positive definiteness of the covariance matrix is automatically guaranteed by the
MEM structure.
Estimation of the MEM can typically proceed by maximum likelihood once the distribution
of the disturbance has been specified. A natural choice of distribution is the exponential as
Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
430 R. ENGLE

it has non-negative support. An exponential random variable with mean one is called a unit
exponential. Assuming that the disturbance is a unit exponential, then the univariate log likelihood
is simply

T  
xt
L
 D
log
t


8
tD1
t


where theta represents the vector of parameters to be estimated. The first-order conditions for a
maximum of this likelihood function are
  
∂L

T
xt
t ∂t
D
9
∂ tD1
2t ∂

By the law of iterated expectations, the expected value of the first-order condition when evaluated
at the true parameter value will be zero regardless of whether the density of x is truly unit
exponential. This implies that the log likelihood in (8) can be interpreted as a Quasi Likelihood
function and that parameters that maximize this are QMLE. Application of the theorem originally
given in White (1980) requires regularity conditions on the mean function and its determinants,
and gives general expressions for the covariance matrix.
A fairly general class of mean functions can be entertained for this problem. Suppose the mean
is linear in lagged x and in a k ð 1 vector of predetermined or weakly exogenous variables zt .
Then a (p, q) mean specification would be


p

q
0
t D ω C ˛j xt
j C ˇj t
j C zt
10
jD1 jD1

The parameters of this model may be restricted to ensure positive means for all possible
realizations, and to ensure stationary distributions for x. If z are positive variables, then sufficient
conditions for non-negativity are clearly that all parameters are positive. However, these are not
necessary. See Nelson and Cao (1992) for an examination of sufficient conditions. Sufficient
conditions for the covariance stationarity of x from Bollerslev, Engle and Nelson (1994) are that
z is covariance stationary and

p

q
˛j C ˇj < 1
11
jD1 jD1

This result can be formalized from Engle and Russell (1998) based upon a theorem in Lee and
Hansen (1994) for GARCH models. In their case, x is the duration between successive events, but
the theorem applies to any non-negative process. In this theorem, the process is assumed to be a
first-order GARCH type model possibly with unit or explosive roots.

Corollary to Lee and Hansen (1994)

If

(1) Et
1
xt   0,t D ω0 C ˛0 xt
1 C ˇ0 0,t
1

(2) εt  xt /0,t is
Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
 NEW FRONTIERS FOR ARCH MODELS 431

(i) strictly stationary and ergodic

(ii) non-degenerate
(iii) has bounded conditional second moments
(iv) supt E[ln
ˇ0 C ˛0 εt j=t
1 ] < 0 a.s.

(3) 0 
ω0 , ˛0 , ˇ0  is in the interior of 

T 
xt  D ω C ˛xt C ˇt
1 for t > 1
(4) L
 D
log
t  C where t
 t t D ω/
1
ˇ for t D 1
tD1

Then:

(a) The maximizer of L will be consistent and asymptotically normal with a covariance matrix
given by the familiar robust standard errors as in Lee and Hansen.
p
(b) The model can be estimated with GARCH software by taking x as the dependent variable
and setting the mean to zero.
(c) The robust standard errors of Bollerslev and Wooldridge (1992) coincide with those in Lee
and Hansen.

From this corollary it is apparent that even mildly explosive models may be estimated consistently
by QMLE. From an examination of the mean specification in (10) it is apparent that the (p,q)
version of this MEM model with exogenous variables can also be estimated using GARCH software
p
by making xi the dependent variable, specifying it to have zero mean and an error process
assumed normal GARCH(p, q) with exogenous variables z. The estimated ‘conditional variance’
is then the conditional mean of x. Multi-step forecasts of x are computed simply by multi-step
forecasts of the conditional variance.
The clear advantage of the exponential error assumption is that estimation is consistent regardless
of the correctness of this distribution. The disadvantage is that it is not fully efficient. However,
it is perfectly straightforward to formulate more general likelihood functions which allow more
flexible shapes of density function or time variation in higher moments of this density function.

4.3. Empirical Models

Several papers have already developed models based on the multiplicative error structure. Of
course, the first to mention is the family of ARCH and GARCH models themselves. The basic
model common to all these processes1 and its square are

rt D ht εt
rt2 D ht ε2t
12

In the squared version, the dependent variable is non-negative with mean h and a non-negative
multiplicative i.i.d. error with unit mean. This can be estimated directly by taking the absolute
value of returns as the dependent variable of a GARCH model.

1 Although the dependent variable does not need to be signed, the lagged variables in the conditional variance can still
include sign information if asymmetric models are sought.

Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
432 R. ENGLE

The second paper in this direction is the ACD model of Engle and Russell (1998) where the
dependent variable is modelled as the time between events. The model proposed is

xi D i εi


p

q
0
i DωC ˛j xi
j C ˇj i
j C zi
13
jD1 jD1

essentially a GARCH(p, q) with exogenous variables for the square root of durations.
Manganelli (2000) has used multiplicative error models for volume in a transactions model of
market microstructure. He estimated models of returns, duration and volume as a trivariate system
of equations and then examined impulse responses through this system.
Engle and Gallo (2002) and Chou (2001) estimated models on realized volatility and high low
ranges to obtain new more efficient volatility estimators. These models all had this form.
I will now present some illustrative results using realized volatilities of dollar/DM exchange
rates from Andersen et al. (2001). They construct a series of daily variances from squaring and
averaging 5-minute returns obtained from Olsen and associates for a period from 12/1986 to
4/1996. For almost 10 years of daily data, we have a return and a ‘realized variance’ and we want
to use these to model volatilities.
The data in Table I show that the average of the squared returns is close to the average realized
variance. The squared return, however, has a much larger range and a larger standard deviation.
Estimation of a GARCH(1,1) model with robust standard errors gives the rather conventional
results in Table II.
Introducing lagged realized variance, v(
1) into this equation produces the results in Table III.
Realized variance does have explanatory power beyond past squared returns to predict squared
returns. In fact, lagged squared returns are no longer significant in this model.
Now I will estimate a model for realized volatility which is the square root of realized variance
(Table IV). Again, a GARCH(1,1) will be specified but in this case it should be recognized that
this is a multiplicative error model with an exponential error assumption. Standard errors are

Table I. Sample: 12/02/1986 4/18/1996

R2 V

Mean 0.4904 0.5287

Median 0.1501 0.3984
Maximum 12.441 5.2454
Minimum 0 0.0518
Std. dev. 1.0140 0.4837

Table II. Dependent variable: R

Coefficient Std. error z-Statistic Prob.

C 0.007 0.003 2.386 0.017

ARCH(1) 0.046 0.010 4.676 0.000
GARCH(1) 0.940 0.012 75.55 0.000

Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
 NEW FRONTIERS FOR ARCH MODELS 433

Table III. Dependent variable: R

Coefficient Std. error z-Statistic Prob.

C 0.0122 0.0069 1.7525 0.0797

ARCH(1) 0.0077 0.0144 0.5322 0.5946
GARCH(1) 0.8698 0.0324 26.8602 0.0000
V(
1) 0.0908 0.0293 3.1006 0.0019

computed using the Bollerslev–Wooldridge (1992) formulation. Two models are estimated; the
second also includes past squared returns.
As can be seen, the first equation is similar to a conventional GARCH model except that the
coefficient on the ARCH term is much larger and the coefficient on the GARCH term much smaller
than usual. This process is still highly persistent but is more volatile; it has a higher volatility of
volatility. This will be discussed in much detail later in this paper. In the second set of results, the
previous squared return is introduced. It has a small coefficient but a very small standard error2
and is quite significant in relation to the asymptotic normal distribution. In fact, the coefficient is
quite similar to that in Table II. There is apparently information in the squared return that helps
to predict realized volatility.
This estimator is potentially inefficient as it assumes an exponential density when another
density could be better. In fact the squared standardized residuals which in this case are
estimates of the disturbances, have a mean of one but a mode which is close to one as
well and a standard deviation which is 0.75, revealing underdispersion. A plot is given in
Figure 1. A more flexible set of density functions for non-negative disturbances is the gamma
density, which is a continuous version of a Chi Square. Setting the mean to unity leaves
one shape parameter in the density, which is the degrees of freedom/2 in a Chi Square.
The results are in Table V for both specifications. These densities are strongly preferred over
the exponential, which achieved a log likelihood of only
619. A chi square of 7 degrees
of freedom divided by 14, which is its mean, has a plot similar to that in Figure 1. The
significance of the squared returns is supported again with very similar coefficients and standard
errors.

Table IV. Dependent variable: SQR(V)

Coefficient Std. error z-Statistic Prob.

C 0.0325 0.0075 4.3470 0.0000

ARCH(1) 0.2860 0.0248 11.5519 0.0000
GARCH(1) 0.6538 0.0307 21.2816 0.0000
C 0.0304 0.0066 4.5854 0.0000
ARCH(1) 0.2014 0.0228 8.8291 0.0000
GARCH(1) 0.6912 0.0276 25.0348 0.0000
R(
1)2 0.0546 0.0081 6.7344 0.0000

2 Interestingly, when conventional standard errors are used, the t-statistic is only 1. It is not clear why the robust standard
errors are so much smaller.

Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
434 R. ENGLE

Kernel Density (Epanechnikov, h = 0.1536)

1.2

1.0

0.8

0.6

0.4

0.2

0.0
0 1 2 3 4 5 6 7 8 9 10 11
RESID01^2
Figure 1. Estimated residual density function

Many further extensions are possible including more flexible densities and time varying
densities. Preliminary efforts to estimate time varying gamma densities for this data set indicated
that the assumption of i.i.d. gamma was quite acceptable. For other data, this may not be
the case.
The potential applications of this set of statistical methods are enormous. As experience grows
with various types of data it will become clearer which densities will be most useful and how best
to parameterize any changing density. The forecast performance of such models can be investigated
with both one-step and many-step forecasts (Table V).

Table V. Dependent variable: SQR(V)

Coefficient Std. error z-Statistic Prob.

C 0.0325 0.0025 12.7904 0.0000

ARCH(1) 0.2856 0.0129 22.0913 0.0000
GARCH(1) 0.6541 0.0146 44.7447 0.0000
DF/2 3.4768 0.0824 42.2075 0.0000
Log likelihood 132.1059 Akaike info criterion
0.1046
Avg. log likelihood 0.0539 Schwarz criterion
0.0951
Number of coefs. 4 Hannan–Quinn criterion
0.1012
C 0.0303 0.0022 13.6604 0.0000
ARCH(1) 0.2010 0.0140 14.3499 0.0000
GARCH(1) 0.6918 0.0135 51.3626 0.0000
R(
1)2 0.0546 0.0071 7.7330 0.0000
DF/2 3.5527 0.0828 42.8839 0.0000
Log likelihood 160.9815 Akaike info criterion
0.1274
Avg. log likelihood 0.0658 Schwarz criterion
0.1156
Number of coefs. 5.0000 Hannan–Quinn criterion.
0.1231

Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
 NEW FRONTIERS FOR ARCH MODELS 435

5. SIMULATION METHODS FOR CONDITIONAL EXPECTATIONS

5.1. Least Squares Monte Carlo
A common problem is calculating the expected value of some future outcome given state variables
that we know today. This expectation depends upon the stochastic process generating the data but
only in the simplest cases are these expectations available analytically. Examples of this problem
in time series analysis are the expectation of a random variable many steps in the future when the
process is non-linear or the expectation of a non-linear function of a random variable even when
the process is linear. In finance, an example might be the payoff of an option as a function of
today’s underlying price or in a more complex setting, the payoff of an option strategy that could
include path dependent decisions, as a function of today’s state variables. In the ARCH context,
we would like to know the expectation of future volatility as a function of today’s information or
perhaps the expectation of the log of future volatility. For risk management, we may want to know
the expected value of a density function below a threshold or the probability of a joint tail event.
All these problems can be formulated as the following problem.
For a stochastic process
fr1 , . . . , rTCM g ¾ F

14

find
E
yT jz1,t , . . . , zk,t , where yT D u
rtC1 , rtC2 , . . . , rtCT 

15
zj,t D vj
rt , rt
1 , . . . , rt
M , j D 1, . . . , k
A potential solution to this problem is a simulation of simulations, which is, unfortunately,
unworkable. This consists of simulating the process from time 1 to t and computing the z’s. For
each of these simulations, a set of future r’s is simulated and the resulting yT ’s are averaged to
estimate the expectation given the z’s. Then the expected values of y given z are tabulated, or
approximated by a generous parameterization or a non-parametric regression. If N replications are
needed, this requires N2 simulations which is typically too time consuming.
The alternative strategy is Least Squares Monte Carlo as proposed by Longstaff and Schwartz
(2001). Their approach requires only N simulations and is thus computationally far more efficient.
They used the method to price American-style options by simulation. This has been thought to be
impossible because it requires knowing the value of the option before the expiration point—hence
the need for simulations of simulations. Their method can be used to find the early exercise frontier
for the options and then price these options.
The Least Squares Monte Carlo method estimates a cross-sectional regression with each
simulation being one observation with the dependent variable yT . The independent variables are
a non-parametric or generously parameterized function of the zj,t ’s. The result is a consistent
estimate of (15) by the standard properties of ordinary least squares. Since each observation is
independent, then the efficiency of this estimation will depend upon the distribution of the errors.
If in addition
V
yT jz1,t , . . . , zk,t  D s2
z1,t , . . . , zk,t 
16

where s is a known function, then weighted least squares will be the best linear unbiased estimator
of this model.
This solution is closely related to the method of ‘reprojection’ introduced by Gallant and Tauchen
(1998) to find properties of simulated series. Their approach, however, initially constructs a very
long time series and thus does not deal easily with overlapping observations.
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436 R. ENGLE

5.2. Estimating the Volatility of Volatility

To illustrate the problem we will define and estimate measures of the volatility of volatility that
can be estimated for a wide range of conditional volatility models as well as stochastic volatility
models and compared with implied volatilities. Consider the volatility of a price process that
follows geometric Brownian motion:
dp D pdt C pdw
d log p D

 2 /2dt C dw
17
The volatility is generally measured by the standard deviation of d log p. Even in more general
diffusion processes, it is natural to measure the volatility as the unconditional standard deviation
of d log p. Now replace p with an instantaneous volatility called v, then the standard deviation
of d log v would be called the volatility of volatility or VoV. Similarly, v could be the realized
2
volatility over a fixed time period, given in discrete time by v2T D
rT
k C . . . C rT2 . Again VoV
would be the standard deviation of d log v.
The same definition can be used for expected volatility. If for example v2T/t is the forecast of the
variance from time T
k to T, based on information at time t, then the VoV can be constructed
for this volatility letting t increase while holding T fixed. Again the definition gives VoV as the
standard deviation of d log vT/t , or in discrete time:3

VoV D V
log vT/t
log vT/t
1  ³ V[Et
log
vT 
Et
1
log
vT ]
18
This measure depends on the conditional variances but is calculated unconditionally over all
realizations. A conditional version could also be examined.
A particular application of this formula is to implied volatilities, which, under some conditions,
can be interpreted as the square root of a forecast of variance from today to the expiration of the
option at T. This measure can be constructed for any time series on implied volatilities; there may
be minor issues associated with the rollover from one contract to another, as maturity will not be
exactly constant. Such a number can be compared with forecasts from GARCH or other volatility
models. However, it is not easy to calculate this value for most volatility models. For example,
the VIX implied volatility series is approximately the 22-day Black–Scholes at-the-money implied
volatility of the S&P100 options. The estimated VoV for the sample period of the 1990s, as used
below, is
VoVVIX D stdev
log
VIX/VIX
t
1 D 0.060
The standard error of an autoregression in log(VIX ) is almost the same number.
To apply the least squares Monte Carlo method to estimation of the VoV for a process, estimate
a cross-sectional regression:
yT,i D f
rt,i , zt
1,i  C εi
19
where  1/2

k
yT,i D log  2
rn,tCj 
20
jD1

3 This approximation is exact if the Jensen inequality terms cancel. This will be true in a second-order Taylor expansion
if the coefficient of variation is the same at t and t
1.

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 NEW FRONTIERS FOR ARCH MODELS 437

is the log realized standard deviation of returns in the ith simulation. This is the log volatility
regression. An alternative is the variance regression when logs are not taken. The regressors are
returns at an intermediate time t and a set of variables based on information prior to time t. The
fitted value is an estimate of the expectation of y made at time t. A similar regression can be
estimated with information only to time t
1. Let this be
yT,i D g
zt
1,i  C εi
21
The difference between f and g is the improvement in forecast attributed to information on rt . It
is the vector of innovations and its standard deviation is the estimated VoV.
VoV D f
O
O g
22
The plot of f against r is interpreted as the News Impact Curve of Engle and Ng (1993) for
the statistic yT . Engle and Ng considered only the one-step-ahead conditional variance but this
can be extended to many statistics. Clearly the NIC will depend in general on the values of z,
and therefore it is common to evaluate this plot at an average value of z although any values can
be used.
The VoV in (22) is most easily calculated as the standard deviation of the difference between
the residuals of (19) and (21). To form an asymptotic confidence interval, let u be the squared
difference between the two residuals. Since each residual has mean zero and observations are
independent, applying the central limit theorem and the delta method,
N1/2
u1/2
VoV


! N
0, u2 /VoV2 
23
N!1

where u2 is the sample variance of u.

The regressors assume that the model and its parameters are known but only information up to
time t or t
1 is observed. The choice of regressors for (19) and (21) is simple in only a few
special cases. For all types of GARCH models, the innovation in the variance equation is the
difference between the squared return and its conditional variance. The expectation at time t
1
is based just upon the conditional variance at t. For GARCH, all forecasts of variances are linear
combinations of these two variables. In the log volatility equation, it might be expected that logs
of conditional variances and innovations will occur. In general, this should be a non-parametric
specification. The parametric approximation follows some experimentation. The results are only
very slightly altered by a wide range of specifications. The following model was used for all
conditional volatility models:

log
O T  D c C ˛1 rt2 C ˛2 jrt j C ˛3 jrt j/ ht C ˇ1 ht C ˇ2 log
ht  C ˇ3 ht C εi
24

2 2
where O T2 D
rtC1 p which will be positive with probability 1. Typically the most
C . . . C rtCk
important variables are
jrt j/ ht , log
ht , but for some models, others are also significant.

5.3. Application to GARCH(1,1)

To examine the performance of this methodology as well as to estimate the previously incomputable
VoV, a series of GARCH models was simulated. The model is defined to be
ht D ω C ˛ht
1 ε2t
1 C ˇht
1
25
Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
438 R. ENGLE

The design parameters are given in Table VI. There are some models with unusually high alphas
and others with very low alpha but high persistence as measured by the sum of alpha and beta.
Horizons of one month (22 days) and one year (252 days) are considered with 100,000
simulations. The log volatility equation is estimated for each of the 11 models to compute the
volatility of volatility. The volatility of volatility is given in Table VI along with the standard
errors of the estimates from (23).
It is interesting to note that at the monthly horizon, the values of volatility of volatility are
largest for the models with larger ˛. However, at the annual horizon, the most persistent models
generally are the most volatile. The standard errors in all cases are quite small. The daily VoV’s
of the monthly forecasts from the models we are accustomed to seeing, are well below the 0.06
for the option data.

5.4. Application to Other GARCH Models

The same approach can be applied to a wide range of GARCH models for which analytic results are
not available. In each case, the models have been estimated by maximum likelihood for a decade
of S&P500 daily returns ending on 2 August 2001. The GARCH(1,1), Component GARCH and
symmetric EGARCH(1,1) are the first three models considered. For these models, most analytical
results are known.4 However for the remaining models, this is not true.
The first new model is the SQGARCH or square root garch model of Engle and Ishida (2001).
This model parameterizes the volatility of volatility so that the variance of the variance is linear in
the variance. It is a discrete time analogue of square root diffusion models used by Cox, Ingersoll
and Ross (1985), Heston (1993) and many others. The conditional variance equation is given by
 
1/2 rt2
htC1 D ω C ˇht C ˛ht
1
26
ht

It is easy to see that if the exponent on the innovation term is 1, it is simply a GARCH(1,1). The
estimated parameters are in Table VII.

Table VI. GARCH Volatility of volatility

Alpha Beta Monthly Std err. Annual Std err.

VoV VoV

0.400 0.400 0.04977 0.00094 0.00812 0.00054

0.300 0.600 0.06176 0.00073 0.01085 0.00113
0.200 0.700 0.04584 0.00066 0.00553 0.00016
0.100 0.800 0.02619 0.00029 0.00317 0.00004
0.100 0.850 0.03674 0.00046 0.00531 0.00013
0.100 0.870 0.04398 0.00044 0.00990 0.00017
0.050 0.900 0.02010 0.00022 0.00253 0.00004
0.050 0.940 0.02842 0.00030 0.01162 0.00014
0.030 0.960 0.01849 0.00020 0.00791 0.00010
0.020 0.970 0.01219 0.00013 0.00514 0.00007
0.040 0.955 0.02545 0.00025 0.01395 0.00021

4 The log likelihood for these three models is:
3741.492,
3730.631, and
3737.322 respectively. The symmetric
p
EGARCH is the same as the conventional EGARCH but omits the term rt / ht .

Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
 NEW FRONTIERS FOR ARCH MODELS 439

Table VII. SQGARCH Parameters

Coefficient Std. error z-Statistic

Omega 0.008874 0.001596 5.560236

Alpha 0.041878 0.003685 11.36383
Beta 0.990080 0.001990 497.5850
Log likelihood
3747.891 Akaike info criterion 2.562960

A second model is a non-linear GARCH model that is a hybrid between an integrated model
and a mean reverting model. This integrated model has no intercept and therefore will implode
eventually as shown by Nelson (1990). However, if the variance gets too small, it converts to
a mean reverting GARCH in a piecewise linear fashion. Thus it behaves like an integrated
GARCH (or Riskmetrics style Exponentially weighted moving average model) for large conditional
variances and like a conventional GARCH for small series. The formula is:

htC1 D ht C ˛
rt2
ht  C
υ
ht Ih<υ
27

where I is an indicator function. Notice that it is an IGARCH for variances larger than delta, and
mean reverting below delta. The estimated values are given in Table VIII.
The daily variance of the S&P is about 1%. The estimated value of delta is not very significant
but is considerably above the mean variance indicating that the model is primarily mean reverting
except for large volatilities.
Several existing models were also considered such as the power model of Engle and Bollerslev
(1986):
p
htC1 D ω C ˛rt C ˇht
28

which has parameter estimates shown in Table IX.

Notice that p is significantly below 2 by conventional criteria. Consequently, no analytical
formula for a multi-step forecast can be found, even if a particular distributional assumption is

Table VIII. Non-linear GARCH Parameters

Coefficient Std. error z-Statistic

Alpha 0.054196 0.004749 11.41158

Gamma 0.001208 0.001935 0.624299
Delta 3.194072 4.471226 0.714362
Log likelihood
3741.520 Akaike info criterion 2.558606

Table IX. Engle–Bollerslev Power model

Coefficient Std. err z-Statistic

Omega 0.000587 0.002340 0.251006

Alpha 0.067071 0.009086 7.381861
p 1.712818 0.117212 14.61294
Beta 0.941132 0.005546 169.7078
Log likelihood
3739.091

Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
440 R. ENGLE

made on the innovations. This model might appear explosive since alpha and beta sum to more
than unity, however that configuration implies explosive behaviour only when p D 2.
Ding, Granger and Engle (1993) proposed the Power ARCH or PARCH model. In its symmetric
form it is given by
2
htC1 D ω C ˛rt C ˇht
29

and the estimated parameters are in Table X. In this case the exponent applies to all the variables
and is estimated at close to 1/2, suggesting a standard deviation style model.
A model written entirely in square root form was proposed by Taylor (1986) and used by
Schwert (1989). It is defined by

htC1 D ω C ˛jrt j C ˇ ht
30

and has parameter estimates given in Table XI.

Notice that this fits slightly better than the conventional GARCH(1,1) for this data set.
These models can all be compared from the perspective of volatility of volatility since they can
all be simulated. Using the same strategy as employed for the GARCH(1,1) models, the results
shown in Tables XII and XIII are found.

Table X. PARCH model

Coefficient Std. err z-Statistic

Omega 0.006680 0.001653 4.041563

Alpha 0.064930 0.005608 11.57887
Gamma 0.665636 0.082814 8.037719
Beta 0.941625 0.005211 180.704
Log likelihood
3738.040

Table XI. Taylor-Schwert Model

Coefficient Std. err z-Statistic

Omega 0.007678 0.001667 4.605529

Alpha 0.065232 0.005212 12.51587
Beta 0.942517 0.005104 184.6524
Log likelihood
3739.032

Table XII. Volatility of volatility with Gaussian shocks

Monthly Standard Annual Standard

VoV error VoV error

GARCH(1,1) 0.01683 0.00058 0.00244 0.00017

COMPONENT 0.02791 0.00056 0.01578 0.00056
NL-GARCH 0.02570 0.00042 0.02102 0.00187
SQ-GARCH 0.02448 0.00064 0.00944 0.00059
Taylor/Schwert 0.03318 0.00047 0.01790 0.00055
EB-POWER 0.02765 0.00057 0.01209 0.00067
PARCH 0.03199 0.00051 0.01831 0.00051
EGARCH 0.03191 0.00046 0.01264 0.00054

Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
 NEW FRONTIERS FOR ARCH MODELS 441

Table XIII. Volatility of volatility with non-Gaussian shocks

Monthly Annual Monthly Annual Monthly Annual

t(6) t(6) t(4) t(4) crashes crashes

GARCH(1,1) 0.0299 0.0033 0.0316 0.0092 0.0364 0.0068

COMPONENT 0.0427 0.0199 0.0437 0.0239 0.0513 0.0388
NL-GARCH 0.0395 0.0213 0.0413 0.0275 0.0491 0.0447
SQ-GARCH 0.0409 0.0140 0.0477 0.0194 0.0393 0.0176
Taylor/Schwert 0.0409 0.0169 0.0374 0.0169 0.0472 0.0272
EB-POWER 0.0422 0.0169 0.0451 0.0210 0.0440 0.0192
PARCH 0.0423 0.0180 0.0403 0.0198 0.0494 0.0331
EGARCH 0.0418 0.0163 0.0407 0.0187 0.0465 0.0183

This wide range of GARCH type models have monthly VoV that range from 0.016 to 0.033 per
day. These would correspond to annualized volatilities from 25% to 50%, although one might not
want to annualize these in the conventional way since volatilities are mean reverting. The most
volatile models are the Taylor/Schwert, the PARCH and the EGARCH. The annual volatilities are
considerably lower ranging from 0.002 to 0.02. The largest VoV are the NLGARCH, the PARCH
and Taylor/Schwert.
Table XIII calculates the same measures when the simulated data are generated with non-
Gaussian innovations. Shocks are generated as t(6), t(4) and crash, where a crash innovation
is standard Gaussian with probability 0.995 and Gaussian (
5,2) otherwise. The fatter-tailed
innovations increase the VoV but not dramatically. If the implied volatility series VIX is a standard
for the monthly volatility forecast, then all of these models are too smooth. The VoV of the VIX
is 0.06, which exceeds all of these cases. Of course, the VIX refers to the S&P100 which is more
volatile than the S&P500, and it has variability attributable to time variation in risk premia and
mispricing. Either the multi-step forecasts of these models are insufficiently responsive to news,
or options prices incorporate important additional sources of volatility.

5.5. Stochastic Volatility Models

The techniques developed for the conditional volatility models can be extended to any volatility
model that can be simulated. In this section, we develop this approach and apply it to several
types of stochastic volatility model. Several recent surveys of this material should be examined
by the interested reader: The main difference, is that the past information set can no longer be
summarized by a conditional variance, since this is no longer defined by the model. Instead, a
more general conditioning data set is introduced.
The natural generalization of equation (24) considers simply long lag distributions of absolute
and squared returns. Thus the equations that are estimated are:

50
log
O T  D c C ˛1 jrt j C ˛2 rt2 C ˛3 jrt j C 2

ˇ1j jrt
j j C ˇ2j rt
j 
31
jD1

and the parameters labeled alpha measure the impact of news.

Three stochastic processes are simulated. The first is a conventional stochastic volatility process,
the second is a ‘breaking volatility’ model that has changes in volatilities at random times, and the
Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
442 R. ENGLE

third is a long-memory stochastic volatility model. The generating equation for the SV model is

rt D t εt
log
t  D  C  log
t
1  C t

εt , vt  ¾ IN
0, I
32

as originally proposed by Taylor (1982) and later Nelson (1988). When this model is estimated
by Bayesian methods using the approach of Kim, Shephard and Chib (1998), the parameters are
given by the middle column of Table XIV.
When the same model was estimated using Shephard’s Maximum Likelihood routine, almost
identical parameters were obtained. For a different sample period of the S&P, Jaquier, Poulson
and Rossi (1994) (JPR) obtained similar parameter values given in the last column (Sandmann
and Koopman, 1998; Shephard and Pitt, 1997).
The ‘breaking volatility’ model is also a type of stochastic volatility model since it models the
volatility as a latent variable. In this model, volatility is constant until a random event occurs. At
this time a new volatility is drawn from a fixed distribution. The process can be defined as:

rt D t εt
 2
2 t
1 with probability p
t D
exp
t C  otherwise

εt , t  ¾ IN
0, I
33

Two sets of parameters were used for this process based on a very casual matching of moments
(Table XV).
The third process is the Long Memory Stochastic Volatility model as proposed by Breidt, Crato
and de Lima (1998) and studied by Deo and Hurvich (2000). It is an alternative to the FIGARCH
Model of Baillie and Mikkelson (1996).

Table XIV. Stochastic volatility

parameters

Parameter KSC JPR

Rho 0.987 0.97

Kappa 0.133 0.15
Mu
0.0024 0.0

Table XV. Breaking volatility parameters

Parameter Breaking Breaking

vol I vol II

p 0.99 0.999
Kappa 1 1
Mu
0.5
0.5

Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
 NEW FRONTIERS FOR ARCH MODELS 443

Table XVI. Long memory stoch-

astic volatility parameters

Parameter Value

D 0.47
Kappa 0.6
Beta 0.4

Table XVII. VoV For stochastic volatility models

Monthly VoV Std Err Annual VoV Std Err

Breaking Vol I 0.05300 0.00091 0.02054 0.00053

Breaking Vol II 0.02559 0.00093 0.02356 0.00035
Stochastic Vol KSC 0.04315 0.00067 0.01535 0.00045
Stochastic Vol JPR 0.04322 0.00082 0.00818 0.00037
Long Memory SV 0.04797 0.00085 0.01743 0.00033

p
rt D ˇ exp
ut εt

1
Bd ut D t

εt , t  ¾ IN
0, I
34

Parameter estimates used for this model are shown in Table XVI.
The results from estimating the VoV for these five models are given in Table XVII.
From these results it appears that all these models have moderately high volatility of volatility
for long forecasts. Presumably this is because the persistence is governed by a different parameter
from the volatility of volatility. A related point has been made by Carnero, Pena and Ruiz (2001) in
the context of the stochastic volatility model. The difficulty of estimation and forecasting for these
models has made it impossible to observe this property before now. The long memory models
have the property that the long maturity VoV is high and rather closer to that of the shorter
horizon. The breaking volatility models are not systematically calibrated to the data and therefore
it is not clear which is to be considered most representative of this class of models. However,
the more frequently breaking model does have the highest VoV of all the Gaussian models at the
monthly horizon.
The use of the least squares Monte Carlo method can illuminate these and many other features of
time series models. In fact the applications of this general structure seem unlimited. The forecast
equation for these stochastic volatility models can be estimated, the expected payoff or utility
from path dependent hedging strategies can be assessed, the structure of derivative prices can
be analysed, and many other tasks can be rather simply evaluated, reducing the need to have
analytically simple models in order to obtain closed form forecasting results.

6. CONCLUSIONS
ARCH models have come a long way from 1982 to the present. In some respects their very success
has made it less interesting to continue research on volatility models. There is a sense in which this
Copyright  2002 John Wiley & Sons, Ltd. J. Appl. Econ. 17: 425–446 (2002)
444 R. ENGLE

is a known body of econometrics. In other ways, new horizons are opening all the time. The five
frontiers discussed in this paper are all fresh new areas of research with important applications and
lots of room for both theoretical and empirical developments. These five areas—high frequency
volatility, high dimension correlation, derivative pricing, modeling non-negative processes, and
analysing conditional simulations by Least Squares Monte Carlo—could well occupy our research
efforts for another decade, and would pay handsome dividends in terms of useful and reliable
research and practitioner tools. But probably, even brighter frontiers lie ahead and we can only
await them with anticipation.

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