Journal of Econometrics 52 (1992) 5-59. North-Holland
ARCH modeling in finance*
A review of the theory and empirical evidence
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Northwestern Unirwsity, Er,anston, IL 60208, USA
Ray Y. Institute of Technoloa, Atlanta, GA 30332, USA
Kenneth F. Kroner
University of Arizona, Tucson, AZ 85721, USA
Although volatility clustering has a long history as a salient empirical regularity characterizing high-frequency speculative prices, it was not until recently that applied researchers in finance have recognized the importance of explicitly modeling time-varying second-order moments. Instrumental in most of these empirical studies has been the Autoregressive Conditional Heteroskedasticity (ARCH) model introduced by Engle (1982). This paper contains an overview of some of the developments in the formulation of ARCH models and a survey of the numerous empirical applications using financial data. Several suggestions for future research, including the implementation and tests of competing asset pricing theories, market microstructure models, information transmission mechanisms, dynamic hedging strategies, and the pricing of derivative assets, are also discussed.
*An earlier version of this paper by T. Bollerslev, R. Chou, N. Jayaraman and K. Kroner was entitled: ¡®ARCH Modeling in Finance: A Selective Review of the Theory and Empirical Evidence, with Suggestions for Future Research¡¯. We would like to thank our colleagues who helped supply the references cited in this survey. Among many others, we would especially like to thank , , Gennaro, , , , , , , J. Culloch, Curdy, , . , , , partici- pants at the Conference on Statistical Models of Financial Volatility at UCSD on April 6-7, 1990, and an anonymous referee for very helpful and detailed comments on an earlier draft. , , and would like to acknowledge financial support from NSF #SES90-22807, the Georgia Tech Foundation, and the Center at the University of Arizona, respectively.
0304-4076/92/$05.000 1992-Elsevier Science Publishers B.V. All rights reserved
6 T. Bollerslec et al., ARCH modeling in finance
1. Introduction
Uncertainty is central to much of modern finance theory. According to most asset pricing theories the risk premium is determined by the covariance between the future return on the asset and one or more benchmark portfo- lios; e.g., the market portfolio or the growth rate in consumption. In option pricing the uncertainty associated with the future price of the underlying asset is the most important determinant in the pricing function. The con- struction of hedge portfolios is another example where the conditional future variances and covariances among the different assets involved play an impor- tant role.
While it has been recognized for quite some time that the uncertainty of speculative prices, as measured by the variances and covariances, are chang- ing through time [see, e.g., Mandelbrot (1963) and Fama (1965)1, it was not until recently that applied researchers in financial and monetary economics have started explicitly modeling time variation in second- or higher-order moments. One of the most prominent tools that has emerged for characteriz- ing such changing variances is the Autoregressive Conditional Heteroskedas- ticity (ARCH) model of Engle (1982) and its various extensions. Since the introduction of the ARCH model several hundred research papers applying this modeling strategy to financial time series data have already appeared. In this paper we survey those contributions that we consider to be the most important and promising in the formulation of ARCH-type models and their applications in the modeling of speculative prices. Several interesting topics in empirical finance awaiting future research are also discussed.
The plan of this paper is as follows. We begin in section 2 with a brief overview of some of the important theoretical developments in the parame- terization and implementation of ARCH-type models, and continue in sec- tion 3 with applications of the ARCH methodology to stock return data. Sections 4 and 5 cover the modeling of interest rates and foreign exchange rates, respectively. A detailed bibliography is given at the end of the paper.
Following the seminal paper by Engle (1982) we shall refer to all discrete time stochastic processes (~~1 of the form
E, = ZtO; 7
z,i.i.d., E(zl)=0,var(zr)=1,
with a, a time-varying, positive, and measurable function of the time t – 1 information set, as an ARCH model. For now E, is assumed to be a
T. Bollersle~ et al., ARCH modeling in finance 7
univariate process, but extensions to multivariate settings are straightforward as discussed below. By definition E, is serially uncorrelated with mean zero, but the conditional variance of F~ equals ut2, which may be changing through time. In most applications E, will correspond to the innovation in the mean for some other stochastic process, say {y,}, where
y,=g(x,_,;b) +sr,
and g(x,_ ,; b) denotes a function of x,_ 1 and the parameter vector b, where x,_ 1 is in the time t – 1 information set. To simplify the exposition, in most of the discussion below we shall assume that &t is itself observable.
Let f(z,) denote the density function for z,, and let 0 be the vector of all the unknown parameters in the model. By the prediction error decomposi- tion, the log-likelihood function for the sample Ed, e7._,, . . . , F, becomes, apart from initial conditions,¡¯
L(6) = i: [log+&) – logUt,]. t=l
The second term in the summation is a Jacobian term arising from the transformation from z, to F,. Note that (4) also defines the sample log-likeli- hood for y7, y,_ , , . . . , y, as given by (3). Given a parametric representation
for f(z,), maximum likelihood estimates for the parameters of interest can be computed directly from (4) by a number of different numerical optimization techniques.
The setup in eqs. (1) and (2) is extremely general and allows for a wide variety of models. At the same time, the economic theory explaining tempo- ral variation in conditional variances is very limited. Consequently, in the remainder of this section we shall concentrate on some of the more success- ful time series techniques that have been developed for modeling at2. These models for the temporal dependence in conditional seconds moments bear much resemblance to the time series techniques for conditional first moments popularized in the early seventies. Just as the integration of time series techniques for the conditional mean into structural econometric model building has led to a much deeper and richer understanding of the underlying dynamics, similar results have already started to emerge in the modeling of conditional variances and covariances.
¡®Throughout this paper, the dependence of E( and a, on the parameter vector 6 are suppressed for notational convenience.
8 T. Bollerslec et al., ARCH modeling in finance 2.1. The linear ARCH(q) model
As Engle (1982) suggests in his seminal paper, one possible parameteriza- tion for at2 is to express ut2 as a linear function of past squared values of the process,
at2=w+ ~a,&:_i=w+a(L)E:,
where w > 0 and (Y~2 0, and L denotes the lag operator. This model is known as the linear ARCH(q) model. With financial data it captures the tendency for volatility clustering, i.e., for large (small) price changes to be followed by other large (small) price changes, but of unpredictable sign. In order to reduce the number of parameters and ensure a monotonic declining effect of more distant shocks, an ad hoc linearly declining lag structure was often imposed in many of the earlier applications of the model; i.e., (Y~= a(q + 1 – i)/(q(q + 1)) as in Engle (1982, 1983).
For z, normally distributed, the conditional density entering the likelihood function in (4) takes the form
log f(8tVf1) = – 0.5 log2rr – 0.5&,2U,?
Maximum likelihood (ML) based inference procedures for the ARCH class of models under this distributional assumption are discussed in Engle (1982) and Pantula (1985). Although the likelihood function is highly nonlinear in the parameters, a simple scoring algorithm is available for the linear ARCH(q) model defined in (5). Furthermore, a (LM) test for (Y,= .. . =a4 = 0 can be calculated as TR* from the regression of &f on F:-,, . . , E:-~, where T denotes the sample size. This same test is generally valid using consistently estimated residuals from the model given in (3). An alternative but asymptotically equivalent testing procedure is to subject &: to standard tests for serial correlation based on the autocorrela- tion structure, including conventional portmanteau tests as in Ljung and Box (1978). In addition, Gregory (1989) suggests a nonparametric test for ARCH(q) derived from a finite state Markov chain approximation, while Robinson (1991) presents an LM test for very general serially dependent heteroskedasticity. The small sample performance of some of these estima- tors and test statistics have been analyzed by Engle, Hendry, and Trumble (1985), Diebold and Pauly (1989), Bollerslev and Wooldridge (19911, and Gregory (1989). Interestingly, the well-known small sample downward bias for the parameter estimates in autoregressive models for the mean carries over to the estimates for (Y,, . . . , aq also.*
¡®It is also worth noting that in the presence of ARCH(q) effects, standard tests for serial correlation in the mean will lead to overrejections; see Weiss (1984), Taylor (1984), Milhoj (1985), Diebold (1987), and Domowitz and Hakkio (1987) for further discussion.
T. Bollerslec et al., ARCH modeling in finance 9
As an alternative to ML estimation, ARCH-type models can also be estimated directly with Generalized Method of Moments (GMM). This was suggested and implemented by Mark (1988) Bodurtha and Mark (1991) Glosten, Jagannathan, and Runkle (1991), Simon (1989) and Rich, Raymond, and Butler (1990a, b), and in a closely related context by Harvey (1989) and Ferson (1989). A comparison of the efficiency of exact ML, Quasi Maximum Likelihood (QML), and GMM estimates using different instrument sets would be interesting. Bayesian inference procedures within the ARCH class of models are developed in a series of papers by Geweke (1988, 1989a, b), who uses Monte Carlo methods to determine the exact posterior distribu- tions.
An observationally equivalent representation for the model in cl), (2) and (5) is given by the time-varying parameter MA(q) model,
&,=W,+ &z,;q-i,
where ~,,a,,,. . .,atq are i.i.d. with mean zero and variance w, (Y,, . . , arq, respectively. This relationship between the time-varying parameter class of models and the linear ARCH(q) model has been further studied by Tsay (1987) Bera and Lee (1989,1991), Kim and Nelson (1989) Wolff (1989) Cheung and Pauly (1990) and Bera, Higgins, and Lee (1991). Similarly, in Weiss (1986b) and Higgins and Bera (1989a) comparisons to the bilinear time series class of models are considered.
2.2. The linear GARCH(p, q) model
In many of the applications with the linear ARCH(q) model a long lag length q is called for. An alternative and more flexible lag structure is often provided by the Generalized ARCH, or GARCH(p, q), model in Bollerslev (1986)¡±
2=w+ ;a;e:~_B,;+u~¡®,=w+a(L)a:+B(L)u~2.
To ensure a well-defined process all the parameters in the infinite-order AR representation a,* = ME: = (1 – P(L))-¡®~(L)E: must be nonnegative, where it is assumed that the roots of the polynomial P(h) = 1 lie outside the unit circle; see Nelson and Cao (1991) and Drost and Nijman (1991). For a GARCH(l,l) process this amounts to ensuring that both ~yi and p, are nonnegative. It follows also that E, is covariance stationary if and only if
3The simple GARCH(1, 1) model was independently suggested by Taylor (1986).
10 T. Bollersleuet al., ARCH modelingin finance
a(l) + p(l) < 1.4 Of course, in that situation the GARCH(p, q) model corre- sponds exactly to an infinite-order linear ARCH model with geometrically declining parameters.
An appealing feature of the GARCH(p, q) model concerns the time series dependence in E:. Rearranging terms, (7) is readily interpreted as an ARMA model for E,¡± with autoregressive parameters c4L) + p(L), moving average parameters --p(L), and serially uncorrelated innovation sequence {E,¡±- a:}. Following Bollerslev (1988), this idea can be used in the identification of the orders p and q, although in most applications p = q = 1 is found to suffice.5
Much of modern finance theory is cast in terms of continuous time stochastic differential equations, while virtually all financial time series are available at discrete time intervals only. This apparent gap between the empirically motivated ARCH models and the underlying economic theory is the focus of Nelson (1990b), who shows that the discrete time GARCH(1, 1) model converges to a continuous time diffusion model as the sampling interval gets arbitrarily smalL6 Along similar lines, Nelson (1992) shows that if the true model is a diffusion model with no jumps, then the discrete time
variances are consistently estimated by a weighted average of past residuals as in the GARCH(1, 1) formulation. Another possible reason for the success of the GARCH(p, q) models in estimating conditional variances is discussed in Brock, Hsieh, and LeBaron (1991). They show that if E: is linear in the sense of Priestley (1981), the GARCH(p, q) representation may be seen as a parsimonious approximation to the possibly infinite Wold representation
While aggregation in conventional ARMA models for the conditional
mean is straightforward, temporal aggregation within the ARCH class of models is not obvious. However, in an insightful recent paper, Drost and Nijman (1991) show that the class of GARCH(p, q) models is closed under temporal aggregation, appropriately defined in terms of best linear projec- tions. Also, Diebold (1986b, 198S>, using a standard central limit theorem type argument, shows convergence towards normality of a martingale process with ARCH errors under temporal aggregation.
2.3. Nonnormal conditional densities
At the same time that high-frequency financial data exhibit volatility
clustering, it is also widely recognized that the unconditional price or return 41n an interesting recent paper, Hansen (1990) derives sufficient conditions for near epoch
dependence and the application of standard asymptotic theory in a GARCH(1, 1) model.
5As pointed out in Milhoj (1990) within the context of the ARCH(l) model, the asymptotic
standard error for the autocorrelations and the partial autocorrelations for e: exceeds l/ fi in the presence of ARCH, thus leading to potentially lower power of such tests.
%ee also the comparison in Taylor (1990a) of the statistical properties of the GARCH(1, 1) and autoregressive random variable (ARV) models motivated by diffusion formulations.
T. BollersleLlet al., ARCH modeling in finance 11
distributions tend to have fatter tails than the normal distribution; for some of the earliest evidence see Mandelbrot (1963) and Fama (1965). Although the unconditional distribution for E, in the GARCH(p, q> model with condi- tional normal errors as given by cl), (2), (6), and (7) have fatter tails than the normal distribution [see Milhclj (198.5) and Bollerslev (1986)1, for many financial time series it does not adequately account for the leptokurtosis. That is, the standardized residuals from the estimated models, i, = ElfGt-¡®, often appear to be leptokurtic.¡¯
Following White (1982), asymptotic standard errors for the parameters in the conditional mean and variance functions that are robust to departures from normality have been derived by Weiss (1984, 1986a). Bollerslev and Wooldridge (1991) present a consistent estimator for this robust variance-co- variance matrix in an ARCH framework that requires only first derivatives, together with an illustration of the small sample performance of the estima- tor and the properties of the robust TR’ tests in Wooldridge (1988,199O). It is found that the conventional standard errors based on the outer product of the quasi-gradient obtained under the assump- tion of conditional normality tend to understate the true standard errors for the parameters in the conditional variance equation when conditional lep- tokurtosis is present. These ideas are also illustrated empirically in Baillie and Bollerslev (1991).¡±
While the QML based inference procedures are straightforward to imple- ment, fully efficient maximum likelihood estimates may be preferred in some situations. In addition to the potential gains in efficiency, the exact form of the error distribution also plays an important role in several important applications of the ARCH model, such as option pricing and the construction
of optimal forecast error intervals; see Engle and Mustafa (1992) and Baillie and Bollerslev (1992). Bollerslev (1987) suggests using the standardized Student-t distribution with the degrees of freedom being estimated.¡± Other parametric densities that have been considered in the estimation of ARCH models include the normal-Poisson mixture distribution in Jorion (1988), the power exponential distribution in Baillie and Bollerslev (1989), the normal-lognormal mixture distribution in Hsieh (1989a), and the generalized exponential distribution in Nelson (1990~). In a related context, McCulloch (1985) suggests the use of an infinite variance leptokurtic stable Paretian
¡®It follows from Jensen¡¯s inequality that with a correctly specified conditional variance. the excess kurtosis in E,u,~¡¯ cannot exceed the excess kurtosis in E,; see Hsieh (1989a).
¡®At the same time, abstracting from any inference, Nelson (1990d) has shown that the normal quasi-likelihood increases with more precise volatility estimates (appropriately defined), while this is not generally true for nonnormal likelihood functions.
¡°In the continuous time conditionally normal GARCHCI, 1) diffusion approximation discussed in Nelson (1990b), the innovations observed over short time intervals are approximately t-distrib- uted.
12 i? Bollerslec et al., ARCH modeling in finance
distribution in the maximum likelihood estimation of the so-called Adaptive Conditional Heteroskedasticity, or ACH, model.
As an alternative to maximum likelihood, a semiparametric density estima- tion technique could be used in approximating f(t,). Following Gallant and Nychka (19871, in Gallant and Tauchen (1989), Gallant, Hsieh, and Tauchen (19891, and Gallant, Rossi, and Tauchen (19901, f(z,) is replaced by a polynomial expansion, whereas Engle and Gonzalez-Rivera (1991) suggest a density estimator originally developed by Tapia and Thompson (19781.¡± By avoiding any specific distributional assumption, semiparametric density esti- mation gives an added flexibility in the specification. Of course, compared to full information maximum likelihood with a correctly specified density, the semiparametric approach invariably involves a loss in asymptotic efficiency. However, with markedly skewed distributions the efficiency of the semipara- metric estimator compares favorably with the QML estimates obtained under the assumption of conditional normality; see Engle and Gonzalez-Rivera (1991).
2.4. Nonlinear and nonparametric ARCH
In the GARCH(p, q) model (7) the variance only depends on the magni- tude and not the sign of E*. As discussed in section 3.3 below, this is somewhat at odds with the empirical behavior of stock market prices where leverage effects may be present. In the Exponential GARCH(p,q), or
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