Regime-Switching and the Estimation of Multifractal Processes · Calvet, Fisher, and Mandelbrot...

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Regime-Switching and the Estimation of Multifractal Processes Laurent Calvet Department of Economics, Harvard University Adlai Fisher Faculty of Commerce, University of British Columbia This Draft: September 25, 2002 First Draft: March 2, 2002 Preliminary and Incomplete. Please do not cite without the authors’ permission Littauer Center, Cambridge, MA 02138-3001, [email protected]; Finance Division, 2053 Main Hall, Vancouver, BC, Canada V6T 1Z2, adlai.[email protected]. We thank John Campbell and Guido Kuersteiner for helpful comments. Excellent research assistance was provided by Xifeng Diao. We are very appreciative of nancial support provided for this project by the Social Sciences and Humanities Research Council of Canada under grant 410-2002-0641.

Transcript of Regime-Switching and the Estimation of Multifractal Processes · Calvet, Fisher, and Mandelbrot...

Page 1: Regime-Switching and the Estimation of Multifractal Processes · Calvet, Fisher, and Mandelbrot (1997 a,b,c) introduce the multifractal model of asset returns (MMAR), a di ffusion

Regime-Switching and the Estimation ofMultifractal Processes∗

Laurent CalvetDepartment of Economics, Harvard University

Adlai FisherFaculty of Commerce, University of British Columbia

This Draft: September 25, 2002First Draft: March 2, 2002

Preliminary and Incomplete.Please do not cite without the authors’ permission

∗Littauer Center, Cambridge, MA 02138-3001, [email protected]; Finance Division,2053 Main Hall, Vancouver, BC, Canada V6T 1Z2, [email protected]. We thankJohn Campbell and Guido Kuersteiner for helpful comments. Excellent research assistance wasprovided by Xifeng Diao. We are very appreciative of financial support provided for this projectby the Social Sciences and Humanities Research Council of Canada under grant 410-2002-0641.

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Abstract

We propose a discrete-time stochastic volatility model in which regime-switching serves three purposes. First, it captures low frequency vari-ations, which is its traditional role. Second, it specifies intermediatefrequency dynamics that are usually assigned to smooth autoregressiveprocesses. Finally, it generates a fat-tailed conditional distribution ofreturns. A single mechanism thus captures three important featuresof the data that are typically addressed as distinct phenomena in theliterature. Maximum likelihood estimation is developed and shownto perform well in typical sample sizes. We also construct a simu-lated method of moments (SMM) estimator in which scaling statis-tics, log-covariagrams, log-periodograms, and tail statistics are usedas potential identifying restrictions. We estimate on exchange ratedata a process with four parameters and more than a thousand states.The estimated process performs well both in-sample and out-of-samplewhen compared with previous models. The paper thus contributes tothe regime-switching literature by offering an effective technique forparsimoniously specifying high-dimensional state spaces. We also ex-tend the emerging multifractal literature by proposing a convenienttime series construction and by developing an econometric toolkit ofestimation and testing methods.

JEL Classification: G0, C5.

Keywords: Long memory, Markov regime-switching, maximum likeli-hood estimation, Paretian tail, scaling, simulated method of moments,stochastic volatility, time deformation, volatility component.

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1. Introduction

Since being introduced by Hamilton (1989, 1990), regime-switching models haveproven to be extremely useful to economists and econometricians, and the basicformulation has been extended in many interesting directions.1 A common threadin the literature is to use regime-switching mainly for the specification of low-frequency dynamics.2 For financial volatility modeling, which is the subject ofour paper, this tendency is particularly apparent in the class of regime-switchingARCH and GARCH processes (Cai, 1994; Hamilton and Susmel, 1994; Gray,1996; Klaassen, 2002). This paradigm then typically separates volatility dynam-ics into roughly three categories. A thick-tailed conditional distribution of returnscaptures high-frequency variations, smooth ARCH or GARCH transitions modelmid-range frequencies, and regime-switching is only applied to very low frequen-cies. In this paper, we propose an alternative paradigm based on regime-switchingvolatility at all frequencies.In the previous literature, it is rare to find an application that makes use

of more than a few regimes. One explanation is that because regime-switchingis typically used only for low-frequency dynamics, a small number of states cansufficiently describe most data sets. A more practical limitation is that for ageneral formulation, the number of parameters grows quadratically in the numberof states. A natural solution is to place some form of restrictions on regimeparameters and switching probabilities. For example, Bollen, Gray, and Whaley(2000) use a set of parameter restrictions to tightly specify a four regime model.Since we endeavor to parsimoniously model volatility dynamics across a wide rangeof frequencies, we seek a correspondingly tighter set of restrictions on regimeparameters and switching probabilities. The approach we take permits us toroutinely estimate good-performing models with over a thousand states and onlyfour parameters using standard maximum likelihood methods.Our solution builds on Calvet and Fisher (2001), which develops a time-

stationary formulation of multifractal processes as well as a weakly convergentsequence of discretizations. Now beginning our exposition directly with a discrete-time regime-switching process, we specify total volatility as the multiplicative

1In addition to the switching ARCH and GARCH models, another notable extension usestime-varying transition probabilities (Diebold, Lee, and Weinbach, 1994; Filardo, 1994). SeeHamilton and Raj (2002) for an excellent survey of regime-switching methods and applications.

2One exception is Dueker (1997), who investigates regime switches in the tail-thickness ofthe conditional density of stock returns and finds a highly leptokurtotic regime with duration ofabout one week.

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product of a large but finite number of distinct volatility components. The dy-namics of each volatility component are mutually independent and first orderMarkov, and their laws of motion are identical except for time scale. Specifically,the only difference between components is that each has a different expectedtime to arrival of innovations, where arrival times are driven by discretized Pois-son processes of differing intensities. The specification is completed by assumingthat the logarithm of Poisson intensities progresses geometrically. This parsimo-nious specification delivers long-memory in volatility, thick tails, a decompositioninto volatility components with heterogenous decay rates, and is consistent withmoment-scaling properties observed in many financial time series (Calvet andFisher, 2002).The present paper also contributes to the literature by developing a toolkit

of econometric methods that can be used to estimate and test multifractal pro-cesses. Calvet, Fisher, and Mandelbrot (1997a,b,c) introduce the multifractalmodel of asset returns (MMAR), a diffusion that provides the foundation for sub-sequent multifractal processes. The combinatorial construction used in this earlywork is however not well-suited to standard econometric techniques. In partic-ular, regime-changes take place at predetermined dates making the model non-stationary. Early efforts to test the model thus focus on unconditional moments,and their scaling across different observation horizons has received particular at-tention.Financial economists are already familiar with a scaling property in the second

moment through the variance ratio statistic.3 Consistent with the multifractalmodel, similar scaling relationships have also been observed in other momentsof financial time series.4 LeBaron (1999, 2001) investigates the robustness andinformativeness of scaling properties, and suggests the need for more formal tests.We are aware of only two prior papers that attempt to fill this gap. Calvetand Fisher (2002) use Monte Carlo simulations to generate a statistic that testsa model’s ability to replicate scaling features of the data. Lux (2001) developsa GMM estimator based on analytically calculated high-frequency covariancesof the multifractal model and examines its finite sample performance. While

3See Campbell, Lo, and MacKinlay (1997) for a review.4Dacorogna et al. (1993) find evidence of scaling in absolute returns. Calvet, Fisher, and

Mandelbrot (1997c) and Calvet and Fisher (2002) conduct a more detailed investigation ofmoment-scaling across a range of moments. Further evidence of moment scaling is provided byAndersen, Bollerslev, Diebold and Labys (2001) and others. Several contributions also documentsimilarities between financial data and physical data, including Ghashgaie et al. (1996), Gallucioet al. (1997), and Pasquini and Serva (2000).

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offering improvements over previous methods, both of these papers focus on thenon-stationary original version of the MMAR.Our model and estimation techniques are easy to use and directly extend from

the original Hamilton regime-switching formulation. Maximum likelihood (ML)estimation is thus standard, and only requires special knowledge of the set ofrestrictions imposed on regimes and transition probabilities. Monte Carlo simu-lations demonstrate the effectiveness of our ML estimator in typical sample sizes.We also find it useful to develop a simulated method of moments (SMM) method-ology. We investigate SMM moment conditions covering the main features offinancial data that the multifractal model is designed to capture. First, we usethe scaling of sample moments at different observation frequencies, which hasbeen explored extensively in previous work. Moment-scaling is closely tied to ahyperbolic decay rate in autocovariances for the multifractal model. This providesour second and third sets of moments which relate respectively to log-log regres-sions on the sample autocovariogram, and to estimators from log-periodogramregressions (Geweke and Porter-Hudak, 1983; Robinson, 1995; Hurvich, Deo, andBrodsky, 1998; Andrews and Guggenberger, 2000; Shimotsu and Phillips, 2001).Our fourth and fifth sets of moment conditions are motivated by Lux (2001), whouses moments related to high frequency autocovariances to estimate the MMARin a GMM framework. The final moment condition is a tail statistic estimator,as developed in Hill (1975), Hall (1982), and Phillips, McFarland, and McMahon(1996). Despite the multifractal literature’s early emphasis on using moment-scaling for estimation, we find that combinations of log-covariogram regressions,the log-periodogram regressions and the tail-statistics produce the most preciseSMM estimators.In an application to exchange rate data, we provide a comparison of our pro-

cess with other models. The multifractal model performs well both in-sampleand out of sample in comparison with GARCH and Markov-switching GARCH.In particular, out-of-sample forecasting tests show that the model is comparableto GARCH(1,1) by a root mean square error criterion, and delivers uniformlybetter results than MS-GARCH. Thus, we find that a pure Markov switchingmodel can capture the same dynamics that in previous literature have requirednot only regime-switching but also linear GARCH transitions and a thick-tailedconditional distribution of returns. It is fascinating that the single mechanism ofregime-switching can play all three of these roles, and moreover perform them allremarkably well. The innovation that achieves this surprising economy of mod-eling technique is based upon scale-invariance, an approach that has long been

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championed by Mandelbrot (1963, 1997). In our regime-switching formulation,scale invariance equates with a very specific set of restrictions on the regime pa-rameters and transition probabilities of a very high dimensional state space.Section 2 presents the discrete-time model. Section 3 defines the MLE estima-

tor and provides empirical results. Section 4 addresses SMM estimation. Section5 discusses alternative processes and provides in and out of sample comparisons.Section 6 concludes.

2. The Multifrequency Regime-Switching Model

This section develops a discrete-time GaussianMarkov process withmulti-frequencystochastic volatility. The process has a finite number k of latent volatility statevariables, each of which corresponds to a different frequency.

2.1. Stochastic Volatility

We consider an economic or financial time series Xt defined in discrete time on theregular grid t = 0, 1, 2, ...,∞. In applications, Xt will often be the log-price of afinancial asset or exchange rate. Define the innovations xt ≡ Xt−Xt−1. A commonmodeling methodology assumes that every period, the system is hit by a shock,which progressively decays over time (e.g. Engle, 1982). We consider insteadan economy with k components M1,t,M2,t, ...,Mk,t, which decay at heterogeneousfrequencies γ1, .., γk. Such a model could be very unwieldy as the number kbecomes very large. We will see, however, that the model can be parsimoniouslydescribed by a small set of parameters.We model the innovations xt ≡ Xt −Xt−1 by

xt = σ(M1,tM2,t...Mk,t)1/2εt, (2.1)

where each random multiplier Mk,t satisfies E(Mk,t) = 1. The innovations εtare assumed to be iid standard Gaussians N (0, 1). The parameter σ is a positiveconstant, which, by construction, is equal to the unconditional standard deviationof the innovation (Ex2t )1/2. Equation (2.1) has the natural interpretation of astochastic volatility model with σt = σ(M1,tM2,t...Mk,t)

1/2.We conveniently stackthe multipliers into a vector

Mt =¡M1,t,M2,t, ...,Mk,t

¢.

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Given m = (m1, ..,mk̄) ∈ Rk̄, denote by g(m) the productQk̄

i=1mi. We can nowwrite the time t volatility as σt = σ[g (Mt)]

1/2.The properties of volatility are driven by the stochastic dynamics of the vector

Mt. We assume for simplicity that Mt is a first-order Markov process: for alla ∈ R, P (Mt ≤ a |Mt−1,Mt−2,Mt−3, ...) = P (Mt ≤ a |Mt−1 ) . This design willfacilitate the iterative construction of the innovation process {xt} through time,and will permit maximum likelihood estimation of the parameters.5 It is naturalto think ofMt as a volatility state vector. We note that this vector is latent, as theeconometrician usually observes the innovation σ[g (Mt)]

1/2εt, but not the vectorMt itself. Each element of Mt is called a volatility component or volatility statevariable.We now specify the dynamics of these state variables. Each of the volatility

componentsMk,t, k ∈©1, ..., k

ª, follows a process that is identical except for time

scale. We first define a vector of indicator functions χt =¡χ1,t, χ2,t, ..., χk,t

¢.For

each k ∈ ©1, ..., kª, the transitions of the volatility state variable Mk,t are givenby

Mk,t =Mk,t−1 if χk,t = 0Mk,t ∼ iid M if χk,t = 1.

Thus, the value of a volatility state variableMk,t remains constant except when theassociated indicator variable χk,t = 1. When the indicator does take a value of one,a new draw is taken for the volatility state variable. This draw is iid across all kand t, always taken from the same distributionM . Each of the indicator variablesis drawn independently each period t, but the indicator variables associated withdifferent volatility components have different probabilities of taking the value one.For each k ∈ ©1, ..., kª, let

γk ≡ P¡χk,t = 1

¢for all t.

The probabilities γ ≡ (γ1, γ2, ..., γk) satisfy

γk = 1− (1− γ1)(bk−1) . (2.2)

Since 1− γk = (1− γ1)(bk−1), we observe that the logarithm of the staying prob-

ability is decreasing exponentially in base b powers as k increases.

5This innovation, introduced in Calvet and Fisher (2001), distinguishes our construction fromprevious multifractal processes that are generated by recursive operations on the entire samplepath.

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In financial time series, volatility is both a highly variable process exhibitingmany outliers, and a highly persistent process often characterized by long mem-ory. Economic intuition and earlier work suggest that the multiplicative structure(2.1) is appealing to model these properties. When a low level multiplier changes,volatility varies very discontinuously and has strong degree of persistence. Thiscontrasts with stationary ARMA models, in which a shock slowly decays overtime and thus leads to relatively smooth volatility processes. Additional shocksare required to capture the outliers, but these are often disconnected from thelong-memory aspect of the model. Furthermore, such models presuppose a spe-cific frequency, and thus typically fail to explain the time-invariance propertiesexhibited by financial time series. In contrast, the multifractal stochastic volatil-ity model jointly specifies persistence, outliers and scaling.

2.2. Parameters

The parameter k determines the number of volatility frequencies in the model.We will take this number as fixed, and view the choice of k as a model selec-tion problem. Note that this is consistent with the methodology employed forARMA(p, q) or GARCH (p, q) models, where estimation is developed for a fixednumber of lags and the determination of p and q is viewed as model selection. Thetwo parameters γk and b are sufficient to describe the frequencies of the volatilitycomponents. We also require a scaling parameter σ to give the unconditionalexpected volatility, and a random variable M to describe the distribution of thevolatility state variables. For convenience, we assume that the distribution of Mcan be specified by a finite parameter vector ϕ. The full parameter vector is

ψ ≡ (ϕ, σ, b, γ∗).We now explore different methods of estimating the parameters of the model. Weimplement these methods on exchange rate data and test the properties of theestimators using Monte Carlo methods.

3. Maximum Likelihood Estimation

Volatility in this model follows a pure Markov switching process. When the distri-bution ofM is discrete, the state space is finite. It is thus practical to use standardfiltering methods to obtain maximum likelihood estimates. After showing how toobtain the maximum likelihood estimator, we demonstrate its performance in

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Monte Carlo experiments. Using a binomial specification for the multiplier M ,we apply the MLE estimator to four exchange rate series and find that all preferspecifications with a large number k of volatility component frequencies.

3.1. Updating the State Vector

Assume that the multiplier M takes a finite number of values bm. The vectorMt =

¡M1,t,M2,t, ...,Mk,t

¢can therefore take d = bkm values m1, ...,md ∈ Rk.

The dynamics of the Markov chain Mt are characterized by the transition matrixA = (ai,j)1≤i,j≤d with components aij = P(Mt+1 = mj|Mt = mi).6

Let n (.) denote the density of a standard normal. Since xt = σg (Mt) εt, thedensity of xt conditional on Mt satisfies

fxt¡x¯̄Mt = mi

¢=

1

σg (mi)n

·x

σg (mi)

¸. (3.1)

Denote the conditional probabilities

Πjt ≡ P

¡Mt = mj|x1, .., xt

¢over the unobserved statesm1, ...,md. We can stack these conditional probabilitiesin the row vector Πt =

¡Π1t , ..,Π

dt

¢ ∈ Rd+. We know that Πtι

0 = 1, where ι =(1, ., 1) ∈ Rd.Using Bayes’ rule, we easily update the new belief Πt+1 from the old belief Πt

and the new innovation xt+1. Consider the function

ω(xt) =

"n (xt/σg (m

1))

σg (m1), ...,

n¡xt/σg

¡md¢¢

σg (md)

#.

and let x ∗ y denote the Hadamard product (x1y1, .., xdyd) of any x, y ∈ Rd. Weeasily show in the Appendix that

Πt+1 =ω(xt+1) ∗ (ΠtA)

[ω(xt+1) ∗ (ΠtA)] ι0 . (3.2)

6We note that aij =Qk̄

k=1

h(1− γk) 1{mi

k=mjk} + γkP(M = mj

k)i, wheremi

k denotes themth

component of vector mi, and 1{mik=m

jk} is the dummy variable equal to 1 if m

ik = mj

k, and 0

otherwise. In Calvet and Fisher (2001), the transition matrix is different because an innovationto a lower frequency multiplier is assumed to cause innovations in all higher frequency multipliers.This paper assumes instead that the innovation indicators χk,t, k ∈

©0, 1, .., k

ªare independent,

which delivers slightly superior empirical results.

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This formula expresses the conditional probability Πt+1 as a function of the ob-servation xt+1 and the probability Πt calculated in period t. These results implythat we can recursively calculate Πt. Since the multipliers (M1,1, ..,Mk,1) are in-

dependent, the initial vector Π0 is uniquely determined by Πj0 =

Qkl=1 P(M = mj

l )for all j.

3.1.1. The Likelihood Function

Having solved the conditioning problem, it is straightforward to show in the Ap-pendix that the log likelihood function is

lnL (x1, ..., xT ;ψ) =TXt=1

ln[ω(xt) · (Πt−1A)].

This defines the maximum likelihood estimator (MLE). This model can be viewedas a special case of the latent factor Markov switching models introduced byHamilton (1989). For fixed k̄, we know that the maximum likelihood estimatoris consistent and asymptotically efficient as T → ∞. An important differencebetween our model and standard Markov switching models is that we have par-simoniously parameterized the transition matrix A. This allows us to estimatethe model with reasonable precision even in the presence of a very large statespace. While the Expectation Maximization (EM) algorithm proposed in Hamil-ton (1990) is not directly applicable with constrained transition probabilities, wefind that numerical optimization of the likelihood function produces good results.

3.2. Monte Carlo Performance of the ML Estimator

We now assess the validity of the MLE estimator. For simplicity, we examine aMarkov process in which the multiplier M can take the values m0 and 2 − m0

with equal probability. The other parameters are the growth rate b = 2.5 and thestandard deviation σ = 1.Table 1 presents Monte Carlo results. The structural parameters of the model

are estimated on samples of length T1 = 2, 500, T2 = 5, 000, and T3 = 10, 000.Welet k = 8, and choose the parameter γk so that the lowest frequency componentM1,t changes on average twice in a sample of length T1. Three processes, definedby three choices of m0, are considered. In Panel A, the parameters m0 and σ areestimated, while the other parameters are assumed known to the econometrician.We see that the MLE estimates ofm0 and σ are unbiased and have small standard

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deviations (FSSE) and root mean squared error (RMSE). Standard asymptotictheory suggests to use the information matrix as an approximation of the standarddeviations of the ML estimates. We see in Panel A that the AASE and the FSSEare numerically very close.These results are slightly modified in Panel B, where the MLE procedure is

also applied to the frequency parameter γk. The ML estimator of m0 is unbiased,for all three processes. For m0 = 1.2 and m0 = 1.4, the estimator γ̂k exhibits asubstantial upward bias, while σ̂ remains unbiased. We obtain opposite resultswhen m0 = 1.6. The parameter estimate γ̂k is unbiased and σ̂ exhibits a strongupward bias. We also observe that the AASE exhibits a downward bias and is apoor approximation of the true standard deviation of the ML estimator. In bothpanels, the precision of the estimator improves when the sample length increasesfrom T1 = 2, 500 to 10, 000. Overall the ML estimator provides an efficient andgood estimator of the structural parameters of the multifractal model.

3.3. Exchange Rate Data

Our empirical analysis uses daily exchange rate data for the Deutsche Mark(DEM), Japanese Yen (JPY), British Pound (GBP), and Canadian Dollar (CAD)versus the U.S. Dollar. To correspond with the ending of fixed exchange rates inMarch of 1973, all series except the Canadian Dollar begin on June 1, 1973. TheCanadian Dollar was essentially held at parity with the U.S. Dollar through theend of 1973, hence we begin this series one year later than the others on June 1,1974. All series end on June 30, 2002, except the Deutsche Mark, which ends onDecember 31, 1998 because of its replacement with the Euro.Table 2 shows the square root of the average squared return over the entire

sample period for each series, and gives the same statistic after breaking eachsample into quarters. We observe that the sample standard deviation of returnscan be very different across subperiods, which is consistent with the long-memoryvolatility property of the multifractal model. Figure 1 plots the log returns ofeach series. We observe apparent volatility clustering at a range of frequencies ashas been noted in previous studies of exchange rate data.

3.4. ML Estimation Results

We report in Table 3 the results of the ML estimation for the four currencies anda frequency number k̄ varying from 1 to 10. The estimated multiplier m̂ and scaleparameter σ̂ tend to decline with k̄. As the number of multipliers increases, less

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volatility inM is required to match the same data set. When k̄ = 1, the parameterγk̄ is between 0.064 and 0.199 across currencies, corresponding to durations of oneto three weeks. The parameter γk̄ is substantially higher for larger values of k̄.Thus for k̄ = 10, the estimates of γk̄ are close to 1 and correspond to shocksof duration 1 to 1.5 day. The corresponding estimate of γ1 is obtained from(2.2). In unreported calculations, the duration of low frequency shocks is found toincrease with k̄, and become as large as few decades for k̄ = 10. Furthermore, weobserve for each currency that the log-likelihood is substantially higher for largervalues of k̄. The increase is monotonic for all currencies except the DEM. Thissuggests to examine higher values of k̄ in later research. These results confirmthe intuition that financial data contains shocks of very heterogeneous durations.Our specification parsimoniously captures this property, and the empirical resultssuggest a very large increase in the frequency range as k̄ increases.

4. Simulated Moment Estimation

This section investigates simulated moment estimation of the multifractal model.Our simulated method of moments (SMM) estimator follows in the spirit of Ingramand Lee (1991) and Duffie and Singleton (1993). Simulated moment estimatorsare interesting to consider for several reasons. We previously noted that whenthe distribution of M is discrete, the state space of our model is finite and themaximum likelihood estimator can be written analytically. When the distributionof M is continuous, however, maximum likelihood estimation is not practical. Bycontrast, we expect that the implementation of simulated moments estimatorsshould not be substantially different for discrete or continuous distributions ofM .Further, simulation based estimators provide diagnostics that may help to deter-mine the features of the multifractal model that distinguish it from alternativeprocesses.In the previous literature, Calvet and Fisher (2002) use simulations to evaluate

fit of multifractal processes estimated from an analytical method of momentsapproach, but do not use simulations in estimation. Lux (2001) uses simulationsto evaluate the performance of a GMM estimator of a multifractal process, butthe moment conditions in his study are based on analytical expected values ratherthan simulated moments. We might expect the use of simulated moments toreduce the biases reported in these two papers. By permitting the use of momentconditions that have no simple analytical expression, we may also conjecture thatusing simulations will increase estimation efficiency relative to previous moment-

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based estimators.

4.1. SMM Estimation

Following Ingram and Lee (1991) and Duffie and Singleton (1993), we assume adata vector x = {xt}Tt=1 and estimate the true parameter vector ψ0 that generatedthe data. In the time-series literature it is common to simulate a single path ofconsiderable length for a given parameter vector. One then takes time averagesof sample moments, and computes a HAC weighting matrix to adjust for possibleautocorrelation in the time-series of moment conditions.We slightly depart from this methodology for two reasons. First, the moment

conditions may have correlations at long lags that would be difficult to accuratelycorrect with a HAC weighting matrix. Second, we would like to use momentconditions — such as the tail statistic — that require an entire time-series samplefor their calculation.Specifically, for any candidate parameter vector ψ we simulate J paths of

length T using the model (2.1). Denote each path by Yj (ψ) = {Yj,t (ψ)}Tt=1,(1 ≤ j ≤ J). For each path, calculate a vector of sample moments h[Yj (ψ)]. Ourspecific choice of sample moments is discussed in the following section. We arrangethe simulated paths in a J × T matrix Y (ψ) = [Y1 (ψ) , ..., YJ (ψ)]

0, and define

H [Y (ψ) , x] = h (x)− 1J

JXj=1

h[Yj (ψ)].

The function H provides a measure of how well the model fits sample momentsof the data. In particular, we can define an objective function

G [Y (ψ) , x,W ] = H 0WH

for any positive definite weighting matrix W . Maximizing the objective functionG with respect to the parameter vector ψ provides a simulated method of momentsestimator bψSMM (W ) for the process.Note that our approach uses simulated sample paths that are known to be in-

dependent. We can thus be sure that repeated sampling will cause our estimatedcovariance matrix to converge to the true covariance matrix of the moment con-ditions. Moreover, the covariance matrix will already be sized correctly for oursample length, since all simulated paths have the same length as the data.

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In practice, we start with an initial weighting matrixW0 of an identity matrix.We then invert the simulated matrix of moment conditions at the first stageestimator to obtain a weighting matrix

Wbψ =(

JXj=1

HhY (bψ), Yj(bψ)iH hY (bψ), Yj(bψ)i0 /J)−1 .

We iterate on the weighting matrix and estimated parameter vector until conver-gence of the parameter vector.

4.2. SMM Moment Conditions

The set of moment conditions we use in estimation is motivated by known prop-erties of multifractal processes. In particular, we choose moment conditions thatcapture the moment-scaling properties and hyperbolically declining autocovari-ances.

Scaling Properties Following Calvet and Fisher (2002), multifractal processescan be defined by the moment-scaling property

E (|Xt+∆t −Xt|q) = cq (∆t)τ(q)+1 . (4.1)

which holds for all t and ∆t within the defined domain of the process, and all qthat keep the expectation finite. It is then natural to define the sample sum

S(4t, q) ≡int[T/∆t]X

i=0

¯̄X(i+1)4t −Xi4t

¯̄q. (4.2)

WhenX(t) is multifractal, the addends are identically distributed, and the scalinglaw (4.1) yields E[S (4t, q)] =int[T/∆t] c(q)(4t)τ(q)+1 when the qth moment exists.This implies

lnE[S(4t, q)] = τ(q) ln(4t) + c∗(q) (4.3)

where c∗(q) = ln c(q)+lnT . For each admissible q, equation (4.3) provides testablemoment conditions describing how the partition function varies with incrementsize ∆t.To put equation (4.3) to empirical use, we choose sets of exponents Q and

intervals Υ. For each q ∈ Q and each ∆t ∈ Υ, we calculate the sample sum

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S(4t, q). We then use OLS regression to obtain an estimate bτ(q). For each valueof q, the estimated value bτ(q) will be used as one of the elements of the momentvector h (.). We also use the sum of squared errors (SSE) from the linear regressionthat provided bτ(q) as an additional element of the moment vector h (.).The slopes and SSE from linear regressions of lnS(4t, q) on ln4t were used

in global test statistics and found to be able to discriminate well between models(Calvet and Fisher, 2002). By the methods above, we now use these simulatedmoments in estimation. We also specify Q = {0.5, 1, 2, 3, 4}, to capture a rangeof relevant moments. The set Υ is defined by taking unique elements of the setcreated by rounding the sequence a0, a1, a2, ..., T/50 to the closest integer, wherea = 1.3. Using the estimated slope and SSE for each of the elements of Q gives aset of ten moment conditions.

Autocovariances Calvet and Fisher (2002) also show that multifractal pro-cesses have hyperbolically declining autocovariances, and similar properties forautocorrelations of other moments. We similarly show in the Appendix that

Cov(|xt|q , |xt+n|q) ∝ nτθ(q)−2τθ(q/2)−1. (4.4)

Consider the sample averages C(∆t, q) =PT−∆t

t=1 |xt+4txt|q /T. We regress thelogarithm of C(∆t, q) on the length∆t for each q ∈ Q and all∆t ∈ Υ∪{0}. Denotethe estimated slopes by bφ (q). These slopes and the SSE from the correspondingregressions provide an additional ten moment conditions.

High Frequency Autocovariances (HFAC) We use six moment conditionssuggested by Lux (2001), which are informative on the high-frequency autocorre-lation structure of volatility. These are also closely related to moments used byAndersen and Sorensen (1996).

Log-Periodogram Regressions For a given q > 0, we calculate the log-

periodogram Ij =¯̄̄PT

t=1 |xt|qeiωjt¯̄̄2/ (2πT ) on the grid ωj = 2πj/T, j = 1, ...,m.

We then regress ln Ij on 1 and −2 lnω. The regression coefficient corresponding to−2 lnωj is the estimator d̂.We also experimented with the bias reduction methodssuggested by Andrews and Guggenberger (2000), but in Monte Carlo tests thisproduced higher root mean square errors for the SMM estimator. We use thevalue m = T/2 in all reported results.

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4.3. Monte Carlo Simulations

We report in Table 4 the performance of the SMM procedure based on seven setsof moments. Estimations i to vi separately examine the following conditions.

i. Moment-scaling (4.1), which has been used most extensively in the literature.

ii. Covariograms. We calculate the rate of decline of the covariograms, as in (4.4).

iii. Log-periodograms. We use simple log-periodogram regression to obtain esti-mators bdq of the qth moment |xt|q of the absolute value of returns.iv. HFAC1 considers the moments of x2t−|xtxt−1|, x2t−|xtxt−2|, and x4t−

¯̄x2tx

2t−1¯̄.

v. HFAC2 considers x2t − |xtxt−4|, x2t − |xtxt−8|, and x4t −¯̄x2tx

2t−4¯̄.

vi. Tail Indices obtained by linearly fitting the empirical c.d.f.s of the 100 largestabsolute returns.

The covariogram ii and the tail index vi the display the lowest RMSE. Theyprovide information on both the fat tails and long memory of volatility. Wechecked that our estimator of the tail index vi is slightly more stable than thestandard Hill (1975) and Hall (1982) statistic. Combining ii and vi defines theSMM estimator vii. This estimator is unbiased. The RMSE for m is 0.021, whichis about 50% larger than the RMSE obtained with the ML estimator. The RMSEfor σ appears to be slightly smaller with SMM. These results suggest that theSMM methodology works reasonably well in the two parameter case.

4.4. Model Selection with Simulated Moment Conditions

We conduct in Table 5 specification tests for the number of volatility components.We use a set of moment conditions to test the model fit at the ML parameter esti-mates obtained in Table 3. The nine moment conditions are motivated by Table 4:sample variance, tails statistics, and the slopes and SSE’s of the log-covariogramregressions (for q = 1, 2, 4). In contrast with estimator vii, we consider two tailstatistics, which are respectively based on the most extreme 100 and 2,400 ab-solute returns. This is motivated by the following observation. Using a smallnumber of outliers correctly focuses the estimation on the most extreme events,but at the cost of relatively large sample variability. On the other hand, usinga larger number of less extreme returns alleviates noisiness, but at the cost ofincluding more events from the bell. We find that the inclusion of both statisticsslightly increases the discriminating power of the specification test. The weight-ing matrices W1, ..,W10 correspond to the inverse of the covariance matrix of the

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moment conditions under each of the 10 models. Expectations and covariancesare obtained by simulating 10,000 paths with the same sample length as the data.The p-values show the probability of observing a larger statistic conditional onthe assumed process. Asymptotic theory suggests that the diagonal entries havemost power against each model.The results show that specifications with larger number of volatility compo-

nents tend to have better fit, consistent with the increased likelihoods observedin Table 3. We view this as a confirmation that the data display heterogeneity infrequency. The multifractal model is accepted at the 5% confidence level for threeexchange rates. For the Japanese Yen, the p-values are monotonically increas-ing with k̄ on the diagonal of Table 5B but fall slightly short of 1% for k̄ = 10.Further work will thus extend the analysis by including additional frequencies.For now since the ML and SM analyses both confirm the validity of including alarge number of frequencies, we will henceforth focus on the case k̄ = 10 for allcurrencies.

5. Comparison with Alternative Models

We compare the multifractal model with some alternative models: GARCH(1,1)and Markov-switching GARCH (MS-GARCH). Future work will also considerstochastic volatility models.We consider a GARCH(1,1) with Student distribution: xt = h

1/2t et, where

the random variables et are IID Student with unit variance and ν degrees offreedom (d.f.’s). Conditional variance follows the recursion ht+1 = ω+αε2t + βht.Markov-switching GARCH combines the short-run dynamics of GARCH with lowfrequency regime shifts (Gray, 1996). Klaassen (2002) refines Gray’s specificationand obtains better forecasting performance on three of the four foreign exchangerates considered in this paper (Deutsche Mark, Japanese Yen and British Pound).We thus choose this later model. Volatility is determined by a latent state st ∈{1, 2}, which follows a first-order Markov process with transition probabilitiespij = P(st+1 = j|st = i). The econometrician observes the return xt in each periodbut not the latent state. We denote by Et and V art the expectation and varianceconditional on past returns {xt}ts=1. For i = {1, 2}, let ht+1(i) = V art(xt+1|st+1 =i) denote the variance of xt+1 conditional on past returns and st+1 = i. Klaassen(2002) assumes

ht+1(i) = ωi + αiε2t + βiEt [ht(st) |st+1 = i ] . (5.1)

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Note that (5.1) conditions volatility on a larger information set than Gray’ spec-ification: ht+1(i) = ωi + αiε

2t + βiEt−1 ht(st).

We report in Table 6 the ML estimates of GARCH(1,1) and MS-GARCH. Thecoefficient 1/ν reports the inverse of the d.f.’s estimated in the Student distribu-tion. This is a convenient renormalization, that has been frequently used in theliterature (e.g. Bollerslev, 1986). The coefficients σ1 and σ2 represent the stan-dard deviation of returns conditional on the volatility state: σ2i = ωi/(1−αi−βi),i = 1, 2. They are found to be easier to interpret across models than the coef-ficients ωi. The log-likelihood is substantially higher with the multifractal thanwith GARCH(1,1) for all exchange rates. Our process thus provides a better over-all fit of the data than GARCH(1,1). We observe that the likelihood reported forGARCH(1,1) match the numbers obtained in our setup with 3 or 4 frequencies.Note that this occurs even though GARCH(1,1) and the multifractal process havethe same number of parameters. We find that the multifractal is the preferredspecification for all currencies.We next examine the performance of MS-GARCH(1,1). This process provides

better log-likelihoods than our model for all currencies. The difference is slight forthe Deutsche Mark, British Pound and the Japanese Yen, and is more pronouncedfor the Canadian Dollar. MS-GARCH(1,1) has 8 parameters as compared to 4for GARCH(1,1) and the multifractal. This leads us to compute the AIC and theSchwarz BIC criteria. The multifractal model is then the preferred specificationin almost all cases. The exception is the Canadian Dollar where the multifractalis only preferred by BIC.We finally investigate in Table 8 the out-of-sample forecasting properties of

the three models. The first two columns correspond to parameter estimates fromthe OLS regression e2t+1 = γ0+ γ1Ete2t+1+ut. For an unbiased forecast, we expectγ0 = 0 and γ1 = 1. This methodology is for instance used by West and Cho (1995)and West and McCracken (1998). Asymptotic standard errors in parenthesis havenot been corrected for autocorrelation or parameter uncertainty in the forecastingmodel. RMSE is the mean square forecast error, and R2 is RMSE divided bythe sum of demeaned squared returns in the out-of-sample periods. We note thatthe multifractal implies lower RMSE and higher R2 than MS-GARCH(1,1) forall currencies. Our model also performs well in forecast regressions, and yieldsfor three currencies both lower values of γ0 and higher values of γ1. We alsoobserve that Markov-switching GARCH is also dominated by GARCH(1,1), andis thus the worst performer out-of-sample. It is interesting that MS-GARCH(1,1)provides the best fit in-sample and the worst forecast out-of-simple. This suggests

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that this process may overfit the data in-sample but does not represent a goodstructural model of volatility.Table 8 also helps compare the multifractal with GARCH(1,1). The forecast

regressions, RMSE and R2 seem to favor the multifractal process for the Markand the Canadian Dollar, and GARCH(1,1) for the Yen and the Pound. It isinteresting that this first attempt to use the multifractal model for forecasting,we are matching the performance of GARCH(1,1), which has been shown to bean excellent process for forecasting purposes in the literature (e.g. Andersen andBollerslev, 1998).

6. Conclusion

This paper proposes an expanded role for regime-switching in modeling volatil-ity. Traditional approaches, such as SWARCH (Cai, 1994; Hamilton and Sus-mel, 1994) and MS-GARCH (Gray, 1996; Klaassen, 2000), consider separatelythree categories of volatility dynamics. High-frequency variation is captured by athick-tailed conditional distribution of returns, mid-range frequencies by smoothARCH or GARCH components, and only very low frequencies are modeled withregime-switching. We suggest an alternative approach based on regime-switchingdynamics at all frequencies. The model is very tightly parameterized in spite ofa high dimensional state space. Using four long series of daily exchange rates,we find that the process matches or dominates the performance of traditionalmodels across a range of in-sample and out-of-sample measures of fit. Thus, theprimary contribution of the paper is to show that regime-switching can be usedin a much wider variety of settings than previously envisioned. In particular,our pure regime switching model provides a viable alternative to traditional ap-proaches that combine regime-switching, linear volatility dynamics, and flexibletail distributions.Researchers often focus on applications of immediate practical value when

evaluating statistical models. We have similarly shown that our process does wellby several standard measures of performance. Another motivation for statisticalmodels is that good description is the first step in explanation. If we can find astatistical model that represents data well among many different dimensions, wecan then seek to identify economic mechanisms that might replicate the observedfeatures. Whereas standard models would require us to explain a triumvirate ofstatistical phenomena — regime-switching, linear dynamics, and thick tails — ourmodel offers the possibility of much more concise explanation. Specifically, our

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results invite the theorist to explain the economic mechanism that might causea self-similar form of regime-switching to provide such a useful description offinancial data.Returning to the more immediate contributions of our work, this is the first

paper to develop and use a comprehensive set of standard econometric tools toestimate and test multifractal processes. We develop the first maximum likeli-hood estimator for multifractal processes, based on a straightforward applicationof the filter developed by Hamilton (1989). We show that this estimator workswell in Monte Carlo simulations. The application to exchange rates leads toseveral observations. First, the likelihood function of the multifractal model de-creases monotonically as we add volatility components to the model. This findingis important because it confirms substantial heterogeneity in the persistence ofvolatility shocks, which is one of the primary motivations of the multifractal ap-proach. Consistent with our intuition, we also observe that the spacing of volatilitycomponents across frequencies becomes tighter and the contribution of individualvolatility components becomes smaller as we increase the number of components.We develop a simulated method of moments estimator for our process using a

variety of conditions that have been used previously in the literature. Specifically,we consider SMM estimators based on combinations of sample variance, moment-scaling statistics, log-covariogram regressions, log-periodogram regressions, high-frequency autocorrelations and tail statistics. In a Monte Carlo study, the mostefficient SMM estimator is found to combine log-covariogram regressions withthe sample variance and two tail statistics. It is still, however, substantiallyoutperformed by the MLE estimator. We then use the set of simulated momentconditions identified by our Monte Carlo study as overidentifying restrictions ofthe previously estimated MLE specifications. These again show that a largernumbers of volatility components is strongly preferred. In three of the four datasets, we also fail to reject the multifractal model at the five percent confidencelevel.The multifractal model is then compared with with two previous processes: the

Student GARCH(1,1) of Bollerslev (1987), and theMarkov-Switching GARCH(1,1;1,1) of Klaassen (2000). Among the many possibilities, these are chosen becauseof their demonstrated good performance with exchange rate data under a varietyof metrics. Like our multifractal model, these models conveniently permit max-imum likelihood estimation and analytical forecasting. The in-sample likelihoodis considerably higher for the multifractal than for GARCH, even though theseprocesses have the same number of parameters. The likelihood functions of the

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multifractal model and MS-GARCH are comparable. However, the multifractalmodel is again favored in-sample by model selection criteria such as AIC and BIC,which penalize for the larger number of parameters in MS-GARCH.Out-of-sample evidence further validates the multifractal model. It uniformly

gives lower mean square forecast errors than the MS-GARCH model, which ap-pears to suffer from overfitting the in-sample data. The multifractal and GARCHappear comparable from an out of sample forecasting perspective - each has thelowest mean square forecast error on two data sets. Summarizing the in- and out-of-sample evidence, the multifractal model fares very well, outperforming each ofthe competitors in at least one area while never being dominated itself. Theseresults are especially encouraging given that this is the first comprehensive eval-uation of the multifractal model, and that refinements of the specification mayvery well lead to further improvements.

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7. Appendix

7.1. Bayesian Updating

Denote the set of past observations It ≡ {xs}ts=1. We note that the conditionalprobability Πj

t+1 = P (Mt+1 = mj |It, xt+1 ) satisfiesΠjt+1 = fxt+1

¡xt+1|Mt+1 = mj

¢P¡Mt+1 = mj |It

¢/fxt+1 (xt+1 |It ) ,

which, by (3.1), can be rewritten as

Πjt+1 =

n[xt+1/σg (mj)]³Pd

i=1 aijΠit

´σg (mj) fxt (xt+1 |It )

. (7.1)

This implies (3.2) sinceP

j Πjt+1 = 1.

7.2. Likelihood Function

We know that L (x1, ..., xT ;ψ) =PT

t=1 ln f(xt |x1, ..., xt−1 ). Bayes’ rule impliesthat

f(xt |x1, ..., xt−1 ) =dX

i=1

P(Mt = mi|x1, ..., xt−1)f(xt|Mt = mi)

=dX

i=1

P(Mt = mi|x1, ..., xt−1) 1

σg (mi)n

µx

σg (mi)

¶and thus f(xt |x1, ..., xt−1 ) = ω(xt) · (Πt−1A).

7.3. Cumulants and Covariogram

ConsiderK2q(n) = E(|xt|2q |xt+n|2q). Since xt = σ(M1,tM2,t...Mk,t)1/2εt and xt+n =

σ(M1,t+nM2,t+n...Mk,t+n)1/2εt+n, we infer that

|xt|2q |xt+n|2q = σ4q |εt|2q |εt+n|2q (M1,tM1,t+nM2,tM2,t+n...Mk,tMk,t+n)q

Multipliers in different stages of the cascade are statistically independent, andtherefore

K2q(n) = σ4q[E( |εt|2q)]2kY

k=1

E(Mqk,tM

qk,t+n).

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By Bayes’ rule, E(Mqk,tM

qk,t+n) = E(M2q)(1− γk)

n + [E(M q)]2[1− (1− γk)n] and

thusE(M q

k,tMqk,t+n)/[E(M

q)]2 = 1 + aq(1− γk)n,

where aq = V ar(Mq)/[E(Mq)]2. Let n = bk̄−l and g = 1− γk̄. Since (1 − γk)n =

gbk−l

, we obtain

logbK2q(n) = logb cq,k +k−lX

i=−l+1logb

³1 + aqg

bi´,

where cq,k = σ4q[E( |εt|2q)]2[E(Mq)]2k . The quantity gbiis close to 1 when i ≤ 0,

and is negligible for larger values of i. Hence logbK2q(n) ∼ l logb (1 + aq). Sinceτ θ(q) = − logb E(M q) − q − 1, we infer that logb (1 + aq) = 2τ θ(q) + 1 − τ θ(2q),and thus

K2q(n) ∝ nτθ(2q)−2τθ(q)−1.

This is a hyperbolic function with coefficient d = τ θ(2q)/2 − τ θ(q). We observethat d = [1− logb (1 + aq)]/2 < 1/2. Furthermore, d > 0 if and only if 1 + aq < b,or equivalently E(M2q) < b[E(M q)]2.We can now derive the covariogram γ2q(n) = Cov(|xt|2q , |xt+n|2q) = K2q(n)−

[E(|xt|2q)]2. Since [E(|xt|2q)]2 = σ4q[E( |εt|2q)]2[E(Mq)]2k, we infer that

γ2q(n) = cq,k

kY

k=1

[1 + aq(1− γ1)nbk−1]− 1

.

When 1 << l << k̄, we know that γ2q(n) ∼ cq,k(bk̄/n)logb(1+aq) >> 1 and thus

γ2q(n) ∼ K2q(n) ∝ nτθ(2q)−2τθ(q)−1.7

7.4. Spectral Density and Log-Periodogram

Consider a covariance-stationary process {Xt}∞t=1 with mean zero. Let γn =Cov(Xt, Xt+n). The spectral density is defined as

s(ω) =1

∞Xn=−∞

γne−inω =

1

"γ0 + 2

∞Xn=1

γn cos(nω)

#.

7These results are consistent with the combinatorial binomial construction. In Calvet andFisher, 2002), we thus show that Cov(|X(t,∆t)|q , |X(t+ s,∆t)|q) ∼ s2d−1, where d = τθ(q)/2−τθ(q/2).

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When γn = n2d−1 (0 < d < 1/2), simple calculus implies that the function s(ω) ∼ω−2d when ω → 0.The periodogram of a sample {Xt}Tt=1 is defined by

I(ω) =1

2πT

¯̄̄̄¯

TXt=1

Xteiωt

¯̄̄̄¯2

.

A log-periodogram regression is based on I(ω) ∼ ω−2d as ω → 0. We consider aset of frequencies {ω1, ..., ωJ} and calculate the corresponding {I(ω1), .., I(ωJ)}.We then regress ln I(ωj) on ωj, and the corresponding slope is an estimate of−2d. This idea was first proposed by Geweke and Porter-Hudak (1983). Theconsistency of the procedure was shown by Robinson (1995) and Hurvich, Deo andBrodsky (1998). Recent refinements include Andrews and Guggenberger (2000)and Shimotsu and Phillips (2001).

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