(εt2 ) is not Gaussian, theKalman ﬁlter can be used to provide the best linear unbiased estimator of ht given Y t .Moreover, if (8.6) were a Gaussian state space model, the Kalman ﬁlter would providethe exact likelihood function. In fact, a by-product of the ﬁlter are the innovations et/t−1 ,
12
See also Carraro and Sartore (1987).
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which are the one-step-ahead forecast errors and their corresponding mean square errors,Ft/t−1 . Together they deliver the likelihood (ignoring constants):
T (θ ) = −
TT211  et/t−1log Ft/t−1 −2 t=12 t=1 Ft/t−1
As the state space is linear but not Gaussian, the ﬁlter gives a quasi-likelihood function which can be used to obtain a consistent estimator θ̂ and asymptotically normalinference (see Ruiz (1994)).This way of estimating ht is used by Melino and Turnbull (1990), after estimating θ bythe generalised method of moments (see Section 8.4.3.1). Harvey et al. (1994) examinethe QML estimator.
8.4.1.2 The Hamilton ﬁlter for maximum likelihood estimation of discrete SV modelsThe discrete SV model (8.3) is a non-linear and non-Gaussian state space model. Inthe two-regimes case, the transition equation can be written in a linear form (see (8.5))and the measurement equation can be linearised by the log transformation, but boththe equations are non-Gaussian. However, the joint process (yt , ht ) is Markovian andthus the general ﬁlter presented in Section 8.4.1 gives an analytic recursion, since theintegrals become simple sums over the possible values of ht . The input is the ﬁlteredprobability13 P (ht−1 |Y t−1 ) and the algorithm gives the prediction probability, the onestep-ahead forecast density and the subsequent ﬁltered probability:P (ht |Y t−1 ) =

P (ht |ht−1 )P (ht−1 |Y t−1 )
ht−1
f (yt |Y t−1 ) =

f (yt |ht )P (ht |Y t−1 )
ht
f (yt |ht )P (ht |Y t−1 )P (ht |Y t ) = f (yt |ht )P (ht |Y t−1 )ht
The combination of the one-step-ahead forecast densities:T
T (θ ) =
f (yt |Y t−1 )t=1
provides the likelihood function, the maximisation of which gives the maximum likelihoodestimators of the parameters.13
The initial probability P (h0 |Y 0 ) can be taken equal to the unconditional (ergodic) probability P (h0 ) = π .
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8.4.2 A general smoother for non-Gaussian parameter-driven modelsWe might also want to obtain the estimation of ht given all the information available, thatis conditional on Y T . Such a procedure is called smoothing and as before it is possibleto derive a formal backward algorithm which delivers the smoothed densities f (ht |Y T ).Let f (ht+1 |Y T ) be the input of the iteration,14 t = T − 1, T − 2, . . . , 2, 1. We candecompose the joint density of ht+1 , ht , conditional on the information set Y t , in theproduct of the transition density by the ﬁltered density (available from the ﬁlter):step 1
f (ht+1 , ht |Y t ) = f (ht+1 |ht )f (ht |Y t )
By conditioning with the prediction density obtained from the ﬁlter, we obtain the following conditional density:step 2
f (ht |ht+1 , Y t ) =
f (ht+1 , ht |Y t )f (ht+1 |Y t )
The joint density of ht+1 , ht , conditional on the information set Y T , is given by the productof the conditional density f (ht |ht+1 , Y T ) by the input of the algorithm f (ht+1 |Y T ). TheTTinformation set ht+1 , Y T is included in the information set ht+1 , Y t , εt+1, ηt+2, whereTTTTεt+1 = (εt+1 , . . . , εT ) and ηt+2 = (ηt+2 , . . . , ηT ) . Given that εt+1 , ηt+2 is independent ofht , ht+1 , Y t , we can conclude that f (ht |ht+1 , Y T ) = f (ht |ht+1 , Y t ) (computed at step 2)and then:step 3
f (ht+1 , ht |Y T ) = f (ht |ht+1 , Y T )f (ht+1 |Y T ) = f (ht |ht+1 , Y t )f (ht+1 |Y T )
Finally, by marginalisation we obtain the smoothed density of ht (output):step 4f (ht |Y T ) = f (ht+1 , ht |Y T ) dht+1 = f (ht |ht+1 , Y t )f (ht+1 |Y T ) dht+1Again, only in the linear and Gaussian case, and in the Markovian and discrete case isit possible to obtain an analytic backward recursion: the Kalman smoother and the Kimsmoother (Kim, 1994).8.4.2.1 The Kalman smoother for continuous SV modelsLet ĥt+1/T = E(ht+1 |LY T ) = E(ht+1 |Y T ) and Qt+1/T = MSE(ht+1 |LY T ) = MSE(ht+1 |Y T ). The Kalman smoother15 computes these quantities recursively for t =T − 1, T − 2, . . . , 2, 1:ĥt/T = ĥt/t + βQt/t Q−1t+1/t (ĥt+1/T − ĥt+1/t )Qt/T = Qt/t + β 2 Q2t/t Q−2t+1/t (Qt+1/T − Qt+1/t )where ĥt/t , Qt/t , ĥt+1/t , Qt+1/t are stored from the Kalman ﬁlter.1415
For the ﬁrst iteration (t = T − 1), the input is simply the ﬁnal output of the ﬁlter f (hT |Y T ).See also de Jong (1989).
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For the log transformation of the continuous SV model (8.6), the Kalman smootheris useful in estimating the unobserved log-volatility, in fact it provides the best linearunbiased estimator of ht given (y1 , y2 , . . . , yT ) .8.4.2.2 The Kim smoother for discrete SV modelsThe input is the smoothed probability P (ht+1 |Y T ) and the recursion is simply:P (ht |Y T ) =
 P (ht+1 |ht )P (ht |Y t )P (ht+1 |Y T )st+1
P (ht+1 |Y t )
where P (ht |Y t ) and P (ht+1 |Y t ) are stored from the Hamilton ﬁlter.8.4.3 Other estimation methods for continuous SV modelsFor the discrete SV model the Hamilton ﬁlter allows us to obtain the maximum likelihoodestimator of the parameters. On the contrary, for the continuous SV models, the Kalmanﬁlter provides only an approximation of the likelihood function. Let us review some otherpossible estimation methods useful for the continuous SV model.Like most non-Gaussian parameter-driven models, there are many different waysto perform estimation. Some involve estimating the likelihood; others use method ofmoments procedures.8.4.3.1 Method of momentsThe simplest approach is the method of moments, based on matching empirical andtheoretical moments. In the SV case there are many possible moments to use in estimatingthe parameters of the model. This is because yt2 behaves like an ARMA(1,1) model andmoving average models do not allow sufﬁcient statistics which are of a smaller dimensionthan T . This suggests that the use of a ﬁnite number of moment restrictions is likely to2lead to loss of information. Examples include those based on yt2 , yt4 , yt2 yt−r, although thereare many other possibilities. As a result, we may well want to use more moments thanthere are parameters to estimate, implying that they will have to be pooled. A reasonablysensible way of doing this is via the Generalised Method of Moments (GMM).We can consider, for example, the vector gT of the ﬁrst r autocovariances of yt2 or oflog(yt2 ) as moment constraints. There are more moments than parameters and the issueis how to weight all the available information. The GMM approach of Hansen (1992)suggests minimising the quadratic form gT WT gT by varying the parameters θ and theweighting matrix WT should reﬂect the relative importance given to matching each ofthe chosen moments. Applications of this method to SV models are the seminal work ofMelino and Turnbull (1990) and the extensive study of Andersen and Sørensen (1996).The main advantage of the GMM approach comes from the fact that it does not requiredistributional assumptions. However, this is not useful for the SV model since it is a fullyspeciﬁed parametric model. On the contrary, as argued by Shephard (1996), there are anumber of drawbacks to the GMM estimation of the SV model. First of all, GMM can onlybe used if ht is stationary; if β is close to one (as we will ﬁnd for many high-frequencyﬁnancial data sets), we can expect GMM to work poorly. Second, parameter estimates
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are not invariant to the parameterisation and the model (8.2) is not fundamentally moreinteresting thanyt = εt γ exp(ht /2)ht = βht−1 + ηtThird, as already observed, the squares yt2 behave like an ARMA(1,1) model; if ση2 issmall (as we will ﬁnd in practice), ρyt2 (r) will be small but positive for many r. Thisimplies that for many series the number of moments to be considered will have to bevery high to capture the low correlation in the volatility process. Finally, GMM does notdeliver an estimate (ﬁltered or smoothed) of ht , consequently a second form of estimationwill be required.The GMM and QML approaches are the simplest way of estimating the SV modelsand they are about equally efﬁcient, with the relative performance being dependent onthe speciﬁc parameter values (see Andersen and Sørensen (1997)).8.4.3.2 Simulation-based methodsAll the other estimation approaches are based on simulation techniques. In the last 10years there has been a growing interest in simulation-based16 methods which proposeseveral ways of resolving the inference problem for this class of models (see Billio (1999,2002b)). In fact, it is clear that one can easily recursively simulate (path simulations) fromthe system (8.2) for any given value of parameters, θ .A ﬁrst approach relies on simulation-based methods which are relatively simple toimplement, but which are less efﬁcient than the maximum likelihood approach: see, forexample, the simulated method of moments (Dufﬁe and Singleton, 1993), the indirectinference method (Gouriéroux et al., 1993) or the efﬁcient method of moments (Gallantand Tauchen, 1996; Gallant et al., 1997). A second approach considers the problemof the computation (or of the approximation) of the likelihood and then of the maximum likelihood estimator through importance sampling methods (Danielsson and Richard,1993; Danielsson, 1994; Durbin and Koopman, 1997). In a Bayesian framework, a thirdapproach considers Markov Chain Monte Carlo (MCMC) techniques based on the dataaugmentation principle, which yields samples out of the joint posterior distribution of thelatent volatility and all model parameters, and allows the parameter estimates and the latentvolatility dynamics to be obtained (Jacquier et al., 1994; Kim et al., 1998; Chib et al.,2002). Finally, a fourth approach utilises MCMC methods to compute (or approximate) themaximum likelihood estimator (see the simulated expectation maximisation (Shephard,1993; Geyer, 1994, 1996; Billio et al., 1998).In practice, the choice between these different simulation-based approaches depends onseveral criteria, such as efﬁciency and computing time. Unfortunately, in general there is atrade-off between these criteria. Methods like the simulated maximum likelihood and thesimulated likelihood ratio have several advantages in the estimation of SV models. Sincethey are likelihood methods, the classical theory of maximum likelihood carries over tothe simulated case and standard likelihood ratio tests can be constructed. MCMC-basedapproaches are certainly more time-consuming, but also allow estimation of the latentvolatility dynamics by simulating from the smoothing/posterior distribution of ht .16Simulation techniques make use of sequences of pseudo-random numbers which are generated by a computerprocedure.
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Let us brieﬂy introduce part of these methods and their application to SV models.8.4.3.2.1 Indirect inference approachThe so-called indirect inference methodology was recently introduced in the literatureby Smith (1993), Gouriéroux et al. (1993), Gallant and Tauchen (1996) for a simulationbased inference on generally intractable structural models through an auxiliary model,conceived as easier to handle. This methodology allows the use of somewhat misspeciﬁedauxiliary models, since the simulation process in the well-speciﬁed structural model andthe calibration of the simulated paths against the observed one through the same auxiliarymodel will provide an automatic misspeciﬁcation bias correction. There are several waysof implementing this idea.17The original approach is the indirect inference method of Gouriéroux et al. (1993).Consider an auxiliary model fa (yt |Y t−1 ; π ) for the observed data (for example18 thegeneral linear state space model obtained by the log transformation (8.6)). Let π̂T =T (Y T ) denote the QML estimator of π based on fa as a function T (·) of the observeddata set Y T . The indirect inference estimator of structural parameters θ is given by:
θ̂II = arg min[π̂T − π̃NT (θ )] WT [π̂T − π̃NT (θ )]θ
where WT is a weighting matrix and π̃NT (θ ) is the π estimator obtained on a simulatedpath of Ỹ NT for a given value of θ (i.e. that is given by the binding function π̃NT (θ ) =lim NT (Ỹ NT ), which is approximated by NT (Ỹ NT ) for large N ). This approach may
N→∞
be very computationally demanding as one needs to evaluate the binding function π̃NT (θ )for each value of θ appearing in the numerical optimisation algorithm.The estimator of Gallant and Tauchen (1996) circumvents the need to evaluate thebinding function by using the score vector ∂fa (yt |Y t−1 ; π)/∂π (score generator) to deﬁnethe matching conditions. If the auxiliary model fa (yt |Y t−1 ; π) is chosen ﬂexibly witha suitable non-parametric interpretation, then the estimator achieves the asymptotic efﬁciency of maximum likelihood and has good power properties for detecting misspeciﬁcation (Gallant and Long, 1997; Tauchen, 1997), hence the term Efﬁcient Method ofMoments (EMM). EMM delivers consistent estimates of the structural parameter vectorunder weak conditions on the choice of the auxiliary model. However, extrapolating fromthe generalised method of moments evidence, it is natural to conjecture that the quality ofinference may hinge on how well the auxiliary model approximates the salient features ofthe observed data. This intuition is formalised by Gallant and Long (1997), who show thata judicious selection of the auxiliary model, ensuring that the quasi-scores asymptoticallyspan the true score vector, will result in full asymptotic efﬁciency.1917For all these methods, it is necessary to recycle the random numbers used in the calculation when θ changes,in order to have good numerical and statistical properties of the estimators based on these simulations.18Another possible auxiliary model is an ARMA(p, q) on the logarithms of the squared data (see Monfardini(1998)).19In fact, as the score generator approaches the true conditional density, the estimated covariance matrixfor the structural parameter approaches that of maximum likelihood. This result embodies one of the mainadvantages of EMM, since it prescribes a systematic approach to the derivation of efﬁcient moment conditionsfor estimation in a general parametric setting.
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Andersen et al. (1999) perform an extensive Monte Carlo study of EMM estimation of astochastic volatility model. They examine the sensitivity to the choice of auxiliary modelusing ARCH, GARCH and EGARCH models for the score as well as non-parametricextensions. EMM efﬁciency approaches that of maximum likelihood for larger samplesizes, while inference is sensitive to the choice of auxiliary model in small samples, butrobust in larger samples.20The indirect inference theory, however, crucially depends on the correct speciﬁcationassumption concerning the structural model. There is now an emerging literature (see,for example, Dridi and Renault (2000) and Dridi (2000)) which focuses on proceduresmore robust to the structural model speciﬁcation. In particular, Dridi and Renault (2000)propose an extension to the indirect inference methodology to semiparametric settingsand show how the semiparametric indirect inference works on basic examples usingSV models.8.4.3.2.2 Importance samplingA more direct way of performing inference is to compute the likelihood by integratingout the latent ht process. As previously seen, the integral (8.8) has no closed form andit has to be computed numerically. However, the likelihood function naturally appears asthe expectation of the function Tt=1 f (yt |Y t−1 , H t ; θ ) with respect to the p.d.f. P deﬁnedby21 Tt=1 f (ht |Y t−1 , H t−1 ; θ ), from which it is easy to recursively draw. Therefore, anunbiased simulator of the whole likelihood function T (θ ) is Tt=1 f (yt |Y t−1 , n H̃ t ; θ )where n H̃ t are recursively drawn from the auxiliary p.d.f. P . The likelihood is thenapproximated by the empirical mean:1 f (yt |Y t−1 , n H̃ t ; θ )N n=1 t=1N
T
and this simulated likelihood can be numerically maximised. However, this basic simulatormay be very slow, in the sense that the simulator may have a very large variance and thensome accelerating technique is needed. One solution is to consider the general methodof importance sampling based on a sequence of conditional p.d.f.s q(ht |Y T , H t−1 1). Letus denote this probability distribution by Q and the corresponding expectation by EQ .We have:TT (θ ) = EP
f (yt |Y t−1 , H t ; θ )t=1
T= EQ
f (yt |Y t−1 , H t ; θ )f (ht |Y t−1 , H t−1 ; θ )q(ht |Y T , H t−1 ; θ )t=1

20Care must be taken, however, to avoid over-parameterisation of the auxiliary model, as convergence problemsmay arise if the quasi-score is extended to the point where it begins to ﬁt the purely idiosyncratic noise inthe data.21It is important to note that this p.d.f. is neither f (H T ; θ ), except when yt does not cause ht , nor f (H T |Y T ; θ ).
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Therefore, an unbiased simulator of T (θ ) is:TEQ
f (yt |Y t−1 , n H̃ t ; θ )f (n h̃t |Y t−1 , n H̃ t−1 ; θ )
t=1

q(n h̃t |Y T , n H̃ t−1 ; θ )
where n H̃ T is drawn in Q. The problem is then how to choose the importance function: the natural answer is by reducing the Monte Carlo variance. It is easy to calculatethe theoretical optimal choice f (H T |Y T ; θ ) = Tt=1 f (ht |Y T , H t−1 ) (i.e. the smoothingdensity of ht ), for which one simulation is sufﬁcient, but it is clearly not computable.Then it is possible to consider the smoothing density of an approximating model, or toﬁx a parametric family of importance functions, choosing the member that minimisesthe Monte Carlo variance (which is eventually computed in an approximated way). Forthe SV model (8.2), the ﬁrst solution is proposed by Sandmann and Koopman (1998)by using as approximating model the linearised version (8.6). In the aim of the secondsolution, Danielsson and Richard (1993) propose a sequentially optimised importancesampling, which Danielsson (1994) applies to the SV model.22 In both cases, the simulated maximum likelihood estimates of model parameters are obtained by numericaloptimisation of the logarithm of the simulated likelihood.238.4.3.2.3 Bayesian approachIn the Bayesian setting, there are also serious difﬁculties in estimating the SV model. Ingeneral, the posterior density f (θ |Y T ) and the posterior expectation of θ cannot be computed in a closed form. Again, this complex setting requires a simulation-based approach.The data augmentation principle, which considers the latent variable ht as nuisance parameters, and the utilisation of Gibbs sampling (Gelfand and Smith, 1990), by iteratingsimulations from f (H T |Y T , θ ) (data augmentation step) and f (θ |Y T , H T ) (parametersimulation step), allow simulation from the joint posterior distribution f (H T , θ |Y T ),derivation of the distribution of interest as the marginal distribution of θ and approximation of the posterior expectation by a sample average. When conditional distributionscannot be directly simulated, the corresponding steps in the Gibbs algorithm are replacedby Metropolis–Hastings steps.24 Moreover, the prior modelling on the parameters is usually quasi-non-informative.One way of considering this approach is to regard it as an empirical Bayes procedure,reporting the mean of the posterior distributions as an estimator of θ . This is the approachfollowed by Jacquier et al. (1994) who show that empirical Bayes outperforms QML andGMM in the SV case.In Jacquier et al. (1994) the posterior distribution of the parameters was sampled byMCMC methods using a one-move approach (i.e. the latent variables ht were sampled eachat time from f (ht |Y T , H −t , α, β, ση2 ), where H −t denotes all the elements of H T excluding ht ). Although this algorithm is conceptually simple, it is not particularly efﬁcient froma simulation perspective, as is shown by Kim et al. (1998), who develop an alternative,22
The details will not be dealt with here as they are quite involved, even for the simplest model.As for non-efﬁcient methods, numerical and statistical accuracy is obtained by recycling the random numbersused in the calculation for each parameter value.24Such hybrid algorithms are validated in Tierney (1994).23
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more efﬁcient, multi-move MCMC algorithm. The efﬁciency gain in the Kim et al. (1998)algorithm arises from the joint sampling of H T in one block conditioned on everythingelse in the model. Finally, Chib et al. (2002) develop efﬁcient Markov Chain Monte Carloalgorithms for estimating generalised models of SV deﬁned by heavy-tailed Student-t distributions, exogenous variables in the observation and volatility equations, and a jumpcomponent in the observation equation (see Section 8.5.1).8.4.3.2.4 An MCMC approach to maximum likelihood estimationAlthough the Bayesian approach is straightforward to state and computationally attractive,it requires the elicitation of a prior, which is often regarded by some econometriciansas being difﬁcult in dynamic models. Even if this is not an insurmountable problem,alternatives are available which allow us to perform maximum likelihood estimation usingMCMC methods.The ﬁrst possibility is the Simulated Expectation Maximisation (SEM) algorithm proposed by Shephard (1993). The EM algorithm exploits the following decomposition ofthe log-likelihood function:log f (Y T ; θ ) = log f (Y T , H T ; θ ) − log f (H T |Y T ; θ )= E[log f (Y T , H T ; θ)|Y T ] − E[log f (H T |Y T ; θ )|Y T ]and iterates:θ i+1 = arg max Eθ i [log f (Y T , H T ; θ)|Y T ]θ
This is an increasing algorithm such that the sequence θ i converges to the ML estimator.The problem is that, although log f (Y T , H T ; θ ) has in general a closed form, the same isnot true for its conditional expectation. In the SEM algorithm this expectation is replacedby an approximation based on simulations. Thus, the problem is now to be able to drawin the conditional distribution of H T given Y T and θ . Shephard (1993), in the contextof a non-linear state space model, uses the Hastings–Metropolis algorithm to solve thisproblem, and applies it to the SV model.Another possible approach is the Simulated Likelihood Ratio (SLR) method proposedby Billio et al. (1998). The general principle is:f (Y T ; θ )= Eθf (Y T ; θ )

 f (Y T , H T ; θ )  TYf (Y T , H T ; θ ) 
(8.9)
where θ is an arbitrary ﬁxed value of the parameters. Obviously:arg max f (Y T ; θ ) = arg maxθ
θ
f (Y T ; θ)f (Y T ; θ)
and with n H̃ T , n = 1, 2, . . . , N , simulated paths in the conditional distributionf (H T |Y T ; θ), the SLR method amounts to maximising:
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1  f (n H̃ T , Y T ; θ )N n=1 f (n H̃ T , Y T ; θ )N
with respect to θ . The method can be implemented by simulating the conditional distribution25 f (H T |Y T ; θ ). As already noted, it is impossible to simulate directly this distribution,thus a Hastings–Metropolis approach is suggested.Contrary to the SEM approach, the SLR method allows for the computation of thelikelihood surface and then of likelihood ratio test statistics. It needs only one optimisationrun and not a sequence of optimisations; it is possible to store the simulated paths, andthen only one simulation run is required. Moreover, as the simulation is made for onlyone value of the parameter, the objective function will be smooth with respect to θ , evenif simulations involve rejection methods.Billio et al. (1998) apply the SLR method also to the SV model (8.2).
8.5 EXTENSIONS OF SV MODELSThe basic SV models can be generalised in a number of directions. Straightforwardgeneralisations might allow εt to have heavy-tailed Student-t distributions and exogenousvariables in the observation and volatility equations.Moreover, the ARCH in mean model of Engle et al. (1987) can be extended to theSV framework, by specifying yt = µ0 + µ1 exp(ht ) + εt exp(ht /2). This model allows ytto be moderately serially correlated, but in the discrete SV model the Hamilton ﬁlter nolonger works, because yt , ht are not jointly Markovian.
8.5.1 Extensions of continuous SV modelsIn the context of continuous SV models, Harvey et al. (1994) concentrated their attentionon models based on Student-t error; Mahieu and Schotman (1998) analysed the possibilityof using a mixture distribution. Jacquier et al. (1995) have computed the posterior densityof the parameters of a Student-t-based SV model. This particular type of model in factcan be viewed as an Euler discretisation of a Student-t-based Levy process but withadditional stochastic volatility effects; further articles are available in (continuous-time)mathematical options and risk assessment literature.26 By building on the work of Kimet al. (1998), Chib et al. (2002) develop efﬁcient Markov Chain Monte Carlo algorithmsfor estimating these models. They also consider a second type of model which containsa jump component27 in the observation equation to allow for large, transient movements.25The resulting Monte Carlo approximation of (8.9) could be only locally good around θ , and so Geyer (1996)suggests updating θ to the maximiser of the Monte Carlo likelihood and repeating the Monte Carlo procedureusing the new θ . By updating θ a few times, one should obtain better approximations of the relative likelihoodfunction near the true maximum likelihood estimate.26Leading references include Eberlein (2001) and Eberlein and Prause (2001). The extension to allow forstochastic volatility effects is discussed in Eberlein and Prause (2001) and Eberlein et al. (2001).27Jump models are quite popular in continuous-time models of ﬁnancial asset pricing. See, for example, Merton(1976), Ball and Torous (1985), Bates (1996), Dufﬁe et al. (2000).
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Moreover, a natural framework for extension of continuous SV models might be basedon adapting the Gaussian state space so that:yt = εt exp(zt ht /2)ht = Tt ht−1 + ηtηt ∼ N (0, Ht )and then on allowing ht to follow a more complicated ARMA process. Another simpleexample would be:   2σ η1 01β 0ht =ht−1 + ηt , ηt ∼ N 0,zt =10 10 ση22Now, the second component of ht is a random walk, allowing the permanent level of thevolatility to slowly change. This is analogous to the Engle and Lee (1992) decompositionof shocks into permanent and transitory. A model along the same lines has been suggestedby Harvey and Shephard (1993), who allow (ignoring the cyclical AR(1) component): 0 011 1Tt =Ht =zt =00 10 ση22This uses the Kitagawa and Gersch (1984) smooth trend model in the SV context, whichin turn is close to putting a cubic spline through the data. This may provide a goodsummary of historical levels of volatility, but it could be poor as a vehicle for forecastingas conﬁdence intervals for forecasted volatilities ht+r may grow very quickly with r.Another suggestion is to allow ht to be a