Time-Varying Long-Memory in Volatility: Detection and ... - CiteSeerX

Time-Varying Long-Memory in Volatility: Detection and Estimation with Wavelets Mark J. Jensen Brandon Whitcher Department of Economics EURANDOM University of Missouri P.O. Box 513 118 Professional Building 5600 MB Eindhoven Columbia, MO 65211 The Netherlands [email protected] [email protected] January 6, 2000

Abstract

Previous analysis of high frequency nancial time series data has found volatility to follow a long-memory process and to display an intradaily U-shape pattern. These ndings implicitly assume that a stable environment exists in the nancial world. To better capture the nonstationary behavior associated with market collapses, political upheavals and news annoucements, we propose a nonstationary class of stochastic volatility models that features time-varying parameters. The generality of our nonstationary stochastic volatility model better accommodates several empirical features of volatility and nests stationary stochastic volatility models within it. To estimate the time-varying longmemory parameter, we use the log linear relationship between the local variance of the maximum overlap discrete wavelet transform's coecients and their scaling parameter to produce a semiparameteric, OLS estimator. Because wavelets are a set of well localized basis functions in time and scale, they are an ideal tool for analyzing nonstationary behavior. We apply our estimator to a years worth of ve-minute Deutsche mark-U.S dollar return data. JEL Classi cation: C13; C14; C22; Keywords: continuous record asymptotics, discrete wavelet transform, foreign exchange rates, locally stationary process, long-memory, micro marketstructure, semiparametric estimator, stochastic volatility, tick-by-tick data, wavelet variance

1 Introduction Previous analysis of the volatility in nancial series, as measured by the log-squared, squared or absolute return of a nancial instrument, have proved fruitful in identifying economic series which exhibit long-memory [Breidt et. al. (1998), Andersen and Bollerslev (1997a), Bollerslev and Mikkelsen (1996), Baillie, et. al. (1996), Ding, et. al. (1993), and Dacorogna et. al. (1993)]. In the past the volatility of returns has been modeled as short-memory ARCH or stochastic volatility models, whose correlation decays exponentially fast to zero as the time interval increases, and as a near integrated IGARCH model [Engle and Bollerslev (1986)]. In the presence of long-memory these other models of volatility provide incorrect inferences about the return's long-run volatility; predicting that an exogenous shock to volatility will either quickly dissipate or persist forever. Because the correlation between observations far apart in time is so strong for long-memory, the impact of a shock to a longmemory volatility model decay to zero at a slow hyperbolic rate. The accuracy of measuring and forecasting volatility plays an important role in the pricing of nancial securities and derivatives and, in light of the incorrect inference that would be made with either a shortmemory or a integrated model of volatility when long-memory is present, an estimator is needed that is robust to both short and long-memory. Recent interdaily analysis of high frequency measures of volatility in nancial instruments has again found the existence of long-memory. Anderson and Bollerslev (1997a) nd that regardless of the level of time aggregation constructed from the tick-by-tick return data, be it semi-daily, daily or weekly, volatility still exhibits long-memory behavior, and although Bollerslev and Wright (1998) nd through Monte Carlo studies a downward bias in the long-memory estimate as the size of temporal aggregation increases, Andersen and Bollerslev (1997a) long-memory estimates are invariant to sampling frequency. High frequency nancial data have also allowed researchers to pursue market microstructure studies of volatility. Mueller et. al. (1990), Ballie and Bollerslev (1990), Zhou (1996) and Anderson and Bollerslev (1997b, 1998) each identify a U-shape pattern to the volatility of foreign exchange rates over each regional market's operating hours, whereas Wood et. al. (1985), Harris (1986), and Lockwood and Linn (1989) nd a similiar U-shape pattern in the 1

variability of U.S. stock returns during the operating hours of the New York market. This intradaily pattern in volatility is consistent with the two agent trading model developed by Admanti and P eiderer (1988), where the entry of informed traders generally leads to a thicker market which attracks discretionary liquidity traders. However, market microstructure studies have not in general incorporated the interdaily, long-memory behavior of volatility into their analysis, arguing that studying long-run properties requires long time-spans of uninterrupted data that is free from structural breaks, policy changes, and other intradaily behavior. Nor have interdaily studies of volatility generally used the time-of-the-day eects of intradaily volatility in modeling long-memory behavior. Andersen and Bollerslev (1997a) ndings of long-memory with a years worth of ve-minute foreign exchange rate returns bridges this dichotomy. They accomplish this ggregating the returns, essentially ignoring the energy associated with the returns intradaily frequencies and leaving only the low frequency, long-term dynamics of returns. These previous ndings of long-memory in the volatility of high-frequency nancial time series, model conditional variance as a stationary process. This assumes that a stationary environment exists in the nancial world, ignoring the known intradaily patterns and the unknown irregular occurrences of market crashes, mergers, and political coups. In this paper we merge the ndings of intradaily time-of-the-day eects with the interdaily results of long-memory by modeling volatility as a nonstationary, time-varying, long-memory process. We generalize the stochastic volatility model of Breidt et. al. (1998) by allowing the autoregressive, moving average, and fractional dierencing parameter to vary over time and show that this model of stochastic volatility is a member of the nonstationary class of models de ned by Dahlhaus (1996,1997) as locally stationary processes. Using this model of stochastic volatility we are capable of producing responses to shocks that are not only persistent in the long-memory hyperbolic sense, but depending on when the shock took place vary in their length of persistence. While wavelets have made a substantial impact in a broad array of disciplines, currently economics has not been one of them.1 In this study we nd that wavelet analysis based on 1 See Jensen (1999, 2000a, 2000b), Ramsey (1999), and Ramsey and Lampart (1998a, 1998b) for examples of how wavelets have been used in economics.

2

the maximum overlap discrete wavelet transform (MODWT) is an eective technique for quantifying the time-varying long-memory behavior of volatility.2 While Fourier analysis is well equipped with its localized frequency basis functions and spectral representation to analyze stationary processes, wavelet analysis is an advancement in harmonic analysis that is better suited to handle the time-varying behavior of a nonstationary time series. Wavelets accomplish this by being both well localized in time and frequency. This means that at high frequencies, the wavelet has small time support, while at low frequencies, the time support of the wavelet function is large. Unlike the standard wavelet transform, the MODWT is not an orthogonal basis, rather it produces an over-determined, or redundant representation of the series, which has advantages in regards to statistical inference. The MODWT decomposes a series into its \details" and a single \smooth". The MOWDT details essentially describe the information that is lost at a particular point in time as the original series is temporally aggregated over longer and longer time intervals; i.e., a measure of the variance at a particular time scale, and the smooth represents the output from the longest time interval of aggregation. Applying an ordinary temporal aggregate to a nite data set results in a decline of observations which can not make up for the lost in the high frequency behavior of the data. However, because the MOWDT is a redundant, zero-phase lter, the details at each time scale and the smooth contain the same number of observations as the orginal series and line up in time with the events of the original series. Thus, the MODWT decomposition of volatility across time and scale is an ideal tool for intra and interdaily analysis. In Section 2 we de ne the nonstationary class of locally stationary processes, and in Section 3 present a stochastic volatility model where the latent variable is a locally stationary process. We show in Section 4 how the MODWT produces a log-linear relationship between the wavelet coecients local variance and the scaling parameter that is equal to the longmemory parameter. We then apply in Section 5 the estimator to a years worth of ve-minute Deutsche-mark, US-dollar foreign exchange rate data. 2 See Percival and Walden (2000) for an extensive introduction to wavelets as applied to time series analysis.

3

2 Locally Stationary Processes Dahlhaus (1996,1997) de nes a locally stationary process as the triangular array, Xt;T (t = 0; 1; : : :; T ? 1), with transfer function A0 , drift , and spectral representation  Z t Xt;T =  T + ei!tA0t;T (!) dZ (!) 

(1)

where A1. the stochastic process Z (! ) has bounded spectral densities hk , k  2 (h1  0, h2  1), de ned by the cumulants of the random measure dZ (! ), cumfdZ (!1; : : :; dZ (!k )g  

k X

=1

l

!

!l hk (!1; : : :; !k?1) d!1


d!k

P where  (! ) = 1 j =?1  (! + 2j ) is the period 2 extension of the Dirac function.

A2. there exists a continuous 2 -periodic function, A(u; ! ) : [0; 1]  [?;  ] ! C , with A(u; !) = A(u; ?!), and a constant K such that



sup A0t;T (! ) ? A(t=T; ! )  KT ?1 t;!

(2)

for all T . A3. the drift (u) is continuous in u 2 [0; 1]. Our notation is such that u will always represent a time point in the rescaled time domain, [0; 1]; i.e., u = t=T . A helpful example of a locally stationary process can be constructed from a stationary, invertible, moving average process, Yt , with spectral representation Z Yt = 21 ei!t(1 + e?i!t ) dZ (!) where the transfer function, A(! ) = (1 + e?i!t )=2 , and jj < 1. Now de ne the process Xt;T to be    t Xt;T =  T +  Tt Yt 

4

where ;  : [0; 1] ! < are continuous functions. Xt;T is a locally stationary process with the time-varying transfer function   A(u; !) = A0t;T (!) = 2(u) 1 + e?i!t : where u = t=T . The time-path of Xt;T will exhibit the periodic behavior of a stationary moving average process but with time-varying amplitude equal to  (u). As T ! 1, more and more of the cycles of Xt;T with amplitudes close to  (u0), where u 2 [u0 ? ; uo + ], will be observed. This de nition of asymptotic theory is similiar to those found in nonparametric regression estimation. Since future observations of Xt;T tell us nothing about the nonstationary processes's behavior at earlier t, in our setting T ! 1 has the interpretation of measuring the series over the same time period but at a more frequent sampling rate. Phillips (1987) referred to this form of asymptotics as continuous record asymptotics since in the limit a continuous record of observations is obtained. In the context of Phillips (1987), the locallystationary process Xt;T can be regarded as a triangular array of a dually index random n g1 variable, ffXnt gTt=1 n=1 where as n ! 1, Tn ! 1 and the length between observations, kn, approaches zero such that kn Tn = N so that the time interval [0; N ] may be considered xed. Given that the existing asymptotic results for locally stationary processes are due mostly to Dahlhaus (1997), we choose to follow his notation and use the dual indexed random variable, Xt;T , to describe a locally stationary series.

3 Time-Varying Stochastic Volatility In this section we introduce a time-varying version of the stochastic volatility model developed by Wiggins (1987), Melino and Turnbull (1990), Harvey et. al. (1994) and Taylor (1994), keeping it general enough so that it also extends the long-memory stochastic volatility model proposed by Breidt et. al. (1998). De ne yt;T to be the time-varying stochastic volatility model

yt;T = expfHt;T =2gt (t=T; B)(1 ? B)d(t=T )Ht;T = (t=T; B)t 5

(3) (4)

where B is the lag operator, Ht?1;T = BHt;T , jd(u)j < 1=2, t iid  N (0; 1), t iid  N (0; 2) and the two Gaussian white noise processes are independent of each other. The functions (u; B) and (u; B) are, respectively, order p and q polynomials whose roots lie outside the unit circle uniformly in u and whose coecient functions, j (u), for j = 1; : : :; p, and k (u), for k = 1; : : :; q, are continuous on 1, and are dierentiable with bounded derivatives for u 2 [0; 1]. In the above time-varying stochastic volatility model, the logarithmic transform of the squared volatility is not only a random process, it is also a locally stationary process that has a time-varying spectral representation. This means that the conditional variance of yt;T is both time-varying in the commonly held view of being a random process and time-varying in the sense that the level of persistence associated with a shock to conditional variance is dependent on when the shock takes place. The locally stationary behavior of Ht;T is exible enough to capture both the dynamics of Harvey et al. (1994) original stochastic volatility model by setting (u; B ) = (B ), (u; B) = (B) and d(u) = 0 for all u 2 [0; 1], and also the long-memory stochastic volatility model of Breidt et. al. (1998) when d(u) = d for all u 2 [0; 1]. We now prove that log-squared volatility is a locally stationary process in the sense of Dahlhaus (1996,1997). For clarity and guidance in understanding the class of locally stationary models we rst show that Ht;T is locally stationary when the order of the AR polynomial is one (p = 1) and the order of the MA polynomial equals zero (q = 0). By backward substitution and use of the binomial series representation of the fractional P dierencing operator, (1 ? B )d(u) = 1 l=0 ? (l ? d(u)) =[?(l +1)?(d(u))], Ht;T can be written as

Ht;T

1

8 < lY

?1

9

= l + d(t=T ))  : = (?1)l :  t ?T j ; ?(?( l + 1)?(d(t=T )) t?l j =0 l=0 X



Thus, Ht;T can be represented in the spectral form as Z

Ht;T = ei!tA0t;T (!) dZ (!) 6

(5)

with

8 < lY

1

?1

A0t;T (!) = p (?1)l :  t ?T j 2 l=0 j =0 X



9 = ;

?(l + d(t=T )) ?i!l ?(l + 1)?(d(t=T )) e :

Now de ne

1 X l + d(u)) e?i!l: l(u) ?(?( A(u; !) = p l + 1)?(d(u)) 2 l=0 Under the conditions d(u) = 0 and j(u)j < 1, for all u 2 [0; 1], Dahlhaus (1996) shows that A(u; !) satis es A2 of Section 2. Since the spectral representation of Ht;T does not involve evaluating the dierencing coecients function, d(
), at arguments other than t=T , the binomial representations of (1 ? B )?d(u) in A(u; ! ) and A0t;T (! ) cancel, giving us the same convergence as found by Dahlhaus (1996). Hence, Ht;T is a locally stationary process when p = 1 and q = 0. When p > 1, q = 0 and d(u) = 0 for all u 2 [0; 1], Kunsch (1995) shows that Ht;T has

the solution

Ht;T =

1

X

=0

t;T ;l

l

t?l

(6)

P with 1 l=0 j t;T ;lj < 1 uniformly in t and T . Dahlhaus (1996) in Theorem 2.3 proves that Ht;T has the spectral representation Z

Ht;T = ei!tA0t;T (!) dZ (!) with

1 X A0t;T (!) p 2 l=0

t;T ;l

e?i!t

and is locally stationary with

A(u; !) = p (u; e?i!)?1: 2

By the same argument used above to prove that a volatility process with p = 1 and q = 0 and jd(u)j < 0:5 is locally stationary, Ht;T is a locally stationary process with 1 X ?(l + d(t=T )) e?i!l ; (7) A0t;T (!) = p1 t;T ;l ?(l + 1)?(d(t=T )) 2 l=0 A(u; !) = p1 (u; e?i! )(1 ? e?i! )?d(u) ; (8) 2 7

and supt;! jA0t;T (! ) ? A(u; ! )j < K=T . Since the spectral representation of the MA(q ) process, Xt;T , equals

Xt;T = with

Z

ei!tA0t;T (!) dZ (!)

A0t;T (!) = p

q X

 (t=T )e?i!l 2 l=0 l which only involves evaluating the moving average coecient functions, l (
), at t=T , it follows that A0t;T (!) = A(u; !) = p

 (u; e? ): 2 i!

(9)

From (7)-(9), the process, Ht;T , as de ned in Eq. (4) with p > 0 and q > 0 is a locally stationary process with (u; e?i! ) (1 ? e?i! )?d(u) : A(u; !) = p  (10) 2 (u; e?i! )

3.1 Statistical properties The following properties hold for the time-varying stochastic volatility model de ned in Eq. (3) - (4). Since Ht;T is a locally stationary process for xed T [Martin and Flandrin (1985)] the Wigner-Ville spectrum is 1 X Cov(H[uT ?s=2];T ; H[uT +s=2];T ) e?i!s fT (u; !) = 21 s=?1 where [
] is the integer portion of the argument. Dahlhaus (1996) in Theorem 2.2 shows that fT (u; !) tends in squared mean to the time-varying spectral density, f (u; !)  jA(u; !)j2, for all u 2 [0; 1]. By synthesis of f (u; ! ) we obtain the local covariance function of Ht;T

(u;  ) =

Z 

?

ei! f (u; !) d!:

Given that t and t are assumed to be Gaussian and since it is possible for a martingale increment to be uncorrelated and time-dependent, yt;T is a martingale dierence with timevarying variance,Var(yt;T ) = exp(H2 (t=T )=2), where

H2 (u) = (u; 0) =

Z 

?

f (u; !) d!; 8

and Cov(yt;T ; yt+;T ) = 0, for  6= 0. Even though Ht;T is an unobservable latent variable, taking the logarithm of the squared values in Eq. (3) we obtain 2 = E [log 2 ] + H + (log 2 ? E [log 2 ]) xt;T = log yt;T t;T t t t =  + Ht;T + t

(11)

which has similar statistical properties to those of the latent variable, Ht;T . The process xt;T is the sum of a locally stationary ARFIMA process with white noise, t, whose mean equals, E [xt;T ] = 0, and whose local autocovariance function is

x(u;  ) = (u;  ) + 2( ); where  ( ) is the Dirac function. By applying the spectral representation theorem [Brockwell and Davis (1993) Theorem 4.8.2] to t it follows that xt;T , like Ht;T , is a locally stationary long-memory process but with spectral representation Z h p i xt;T =  + ei!t Aot;T (!) + = 2 dZ (!);

and time-varying spectral density 2 2 (u; e?i! )j2 ?i! j?2d(u) +  fx(u; !) = 2 jj j 1 ? e (u; e?i!)j2 2 2 (u; 1)j ! ?2d(u)  jj (u; 1)j2 as ! ! 0.

(12) (13)

4 Estimating Time-Varying Long-Memory in Volatility Since the tell-tale sign of long-memory is the presence of energy near the origin of the spectral density function, a semiparametric estimator of d(u) that does not require identifying the order of (u; B ) and (u; B ) is initially more desirable than an ecient parameterized estimator. In this paper we utilize the wavelet based semiparametric estimator of a locally stationary I (d) process developed by Jensen and Whitcher (1999) to estimate the timevarying stochastic volatility model's long-memory parameter, d(u). 9

4.1 Maximum Overlap Discrete Wavelet Transforms Using a modi ed version of the discrete wavelet transform called the maximum overlap discrete wavelet transform (MODWT) [Percival and Guttorp (1994) and Percival and Mofjeld (1997)] (also know by the name \stationary DWT" [Nason and Silverman (1995)] and \translation-invariant DWT" [Coifman and Donoho (1995)]) Jensen and Whitcher (1999) show that MODWT coecients from a locally stationary I (d) process are themselves a locally stationary processes in the sense of Dahlhaus (1996,1997) with time-varying variances that are log-linear in the dierencing parameter d(u). They then prove that the OLS estimator of d(u) calculated by regressing the sample wavelet variance for wavelet coecients whose support contains the point [uT ] across scales is a consistent estimator of the time-varying dierencing parameter. The MODWT essentially decomposes a time series into a number of dierent scales, or levels of resolution.3 When a multiresolution analysis is performed, the wavelet details and smooth form an additive decomposition of the original series; i.e., when summed over at a point in time the wavelet's details and smooth equals the original series. At each scale the MODWT coecients constitute a time series describing the original series at coarser and coarser levels of resolution, not in a time aggregate manner, but in a manner where information that is being lost when the original series is aggregated over longer and longer time intervals is being conveyed. The MODWT coecients are calculated from the Daubechies families of compactly supported wavelet lters, which are well localized in time [Daubechies (1992)]. Using Daubechies least asymmetric family of wavelet lters, the MODWT is constructed via approximate linear ltering operations thus allowing wavelet coecients at various scales to be aligned in time with the events of the original series. This property makes the MODWT a particularly useful tool in the analysis of time-dependent processes. To have a clear understanding of the MODWT and to facilitate the introduction of the semiparametric wavelet estimator of d(u) consider applying the MODWT (of order J , where J represents the longest time interval over which the original time series is aggregated) to the 3

See Mallat (1989) for the seminal article on wavelets as a multiresolution analysis.

10

locally stationary, long-memory process Xt;T , for t = 0; : : :; T ? 1. For scales j = 1; : : :; J , let fehj;l j l = 0; : : :; Lj g, where L1 = L < T and Lj = (2j ? 1)(L ? 1) + 1, be the level-j P wavelet lters such that l he j;l = 0, and let fgeJ;l j l = 0; : : :; LJ g be the level-J scaling lters P such that l geJ;l = 1. The level-j wavelet coecients of Xt;T are obtained through ltering via f W j;t;T =

?1

Lj X

hj;lXt?l mod T ;T t = 0; : : :; T ? 1; e

=0

l

(14)

and the level-J scaling coecients

VeJ;t;T =

?1

LJ X

=0

l

geJ;lXt?l mod T ;T t = 0; : : :; T ? 1:

(15)

f Notice from the wavelet coecients, W j;t;T , t  Lj , whose computation do not involve the circularity assumption, the MODWT wavelet lter is compactly supported on the time f interval [t ? Lj + 1; t]. The level-j wavelet coecients, W j;t;T , are associated with changes of Xt;T on the scale j  2j ?1 and the level-J scaling coecients VeJ;t;T are associated with changes of Xt;T on the scale 2J . For computing a multiresolution analysis, the level-j wavelet details are de ned by

D e

j;t;T =

?1

Lj X

=0

f hej;l W t = 0; : : :; T ? 1; j;t+l mod T ;T

(16)

l

and the level-J wavelet smooths

S

e J;t;T

=

?1

LJ X

=0

geJ;lVeJ;t+l mod T ;T t = 0; : : :; T ? 1:

(17)

l

As with the wavelet and scaling coecients, the level-j wavelet details De J;t;T are associated with changes of Xt;T on the scale j and level-J wavelet smooths SeJ;t;T are associated with changes of Xt;T on the scale 2J . Together the details and smooths form the additive decomposition

Xt;T =

J X j

=1

D e

j;t;T

+ SeJ;t;T :

11

4.2 OLS Estimator of d(u) f Using the transfer function of ehj;l , Jensen and Whitcher (1999) show that W j;t;T is a locally stationary process with mean zero and time-varying variance

2 2 j (2d(u)?1) f Var(W j;t;T ) = X (u; j ) !  (u) 2

where

1?d(u)  2 ?2d(u) (21?d(u) ? 1)

(u)2 = 2



as j ! 1;

(18)

:

1 ? 2d(u) Taking the logrythmic transformation of Eq. (18), Jensen and Whitcher (1999) obtain the linear relationship log X2 (u; j ) = (u) + D(u) log 2j

(19)

f where D(u) = 2d(u) ? 1. Using the time-scale nature of W j;t;T , Jensen and Whitcher (1999) propose estimating the time-varying variance, X2 (u; j ), with the sample variance calculated with wavelet coecients whose support contain the point t. The support of the level-j wavelet lter was found to include several coecients whose value was close to zero, hence a `central portion' of the wavelet lter j was de ned and utilized instead. In other words, X2 (u; j ) was estimated with the time-varying sample variance of wavelet coecients 1 XW f2 (20) ~X2 (u; j ) = # j;t+l;T : j l2 j

Jensen and Whitcher (1999) proved that by replacing log X2 in Eq. (19) with log ~X2 , the OLS estimator of H (u) is consistent. Hence, the OLS estimator d^(u) = (H^ (u) + 1)=2 is a consistent estimator of the time-varying dierencing parameter. 2 is a time-varying long-memory process plus white noise and does Although xt;T = log yt;T not formally equal Xt;T , the local second order properties of xt;T and Xt;T are asymptotically equivalent. The ratio of the time-varying spectrum of xt;T and Xt;T fx (u; !) ! K; fX (u; !) for some K < 1. Thus, the OLS wavelet estimator of the time-varying stochastic volatility model's dierencing parameter should be consistent. 12

5 Intraday Long-Memory Behavior of the Deutsche MarkU.S. Dollar Exchange Rate Recently, the Deutsche- mark U.S.-dollar exchange rate has been of great interest in investigating the intradaily and long-run behavior of the foreign exchange market [Andersen and Bollerslev (1997a,1997b,1998), Muller et. al. (1997), Baillie and Bollerslev (1990)]. The spot DM-$ market has the largest turnover of any market, is highly liquid, and has low transaction costs. Furthermore, the spot market for DM-$ is a 24-hour market comprised of sequential but not mutually exclusive regional trading centers, so except for those endogenous slow downs in the level of trading associated with weekends and regional holidays, the spot DM-$ market is essentially always open. Such a market is ideal for analyzing the time-varying long-memory behavior of volatility.

5.1 Data source Using the Olsen and Associates collected tick-by-tick DM-$ spot exchange rate of the bid and ask price quotes that appeared on the interbank Reuters network from October 1, 1992, through September 30, 1993, we constructed 74,880 ve-minute returns using the linear interpolated average method described in Andersen and Bollerslev (1997b). Because of market inactivity all returns from Friday 21.00 Greenwich mean time (GMT) through Sunday 21.00 GMT were excluded. This left us with 260 daily trading cycles each consisting of 288 ve-minute trades. In order to determine if the intradaily behavior of the time-varying long-memory parameter follows the typical pattern found in the volatility of the DM-$ return, we plot in Fig. 1 the average sample mean of each of the 288 ve-minute absolute returns over all 260 trading days. It shows as found by Andersen and Bollerslev (1997b,1998) the time-of-day eects caused by the openings and closings of the Far Eastern, European and U.S. markets, the drop in volatility during the lunch hours of the Hong Kong and Tokoyo markets (between intervals 40 and 60), and the sharp increase in volatility during the afternoon trading sessions in Europe and the opening of the U.S. market (interval 156).

13

Level j 1 2 3 4 5 6 7 8 9 10 11 12

Scale j 5 min. 10 min. 20 min. 40 min. 80 min. = 1 hr. 20 min. 160 min. = 2 hr. 40 min. 320 min. = 5 hr. 20 min. 640 min. = 10 hr. 40 min.  1/2 day 1280 min. = 21 hr. 20 min.  1 day 2560 min. = 42 hr. 40 min.  2 days 5120 min. = 85 hr. 20 min.  3.5 days 10,240 min. = 170 hr. 40 min.  7 days

Table 1: Translation of levels in a MODWT to appropriate time scales for the DM-$ return series (
T = 5 minutes).

5.2 Wavelet coecients We estimate the time-varying dierencing parameters of the DM-$ log-squared return using the MOWDT wavelet coecients computed from the Daubechies (1992) \least asymmetric" class of wavelet lters with 8 nonzero coecients; i.e., L = L1 = 8 in Eq (14). In Fig. 2 we plot a multiresolution analysis of the DM-$ log-squared returns, consisting of the wavelet details Dej;t;T , for levels j = 1; : : :; 12, and wavelet smooth Se12;t;T , for the entire data set. Table 1 provides conversions from levels of the MODWT to time scales for the DM$ time series. For example, the wavelet detail De 1 is associated with changes (dierences of weighted averages) of 5 minutes while De 10 is associated with changes of approximately 2 days. Notice that at j = 9, the time scale, 9
t = 512
5 min. = 1280 min.  1 day. Hence, the wavelet details for j = 1; : : :; 8 are associated with intraday activity and for levels j  9 are associated with interday activity.

5.3 Time-varying dierencing parameters Using the above MODWT coecients, we calculate the time-varying sample wavelet variances ~2 (t=T; j ) for scales j = 5; : : :; 12. To reduce the boundary eects caused by the circularity assumption of the MODWT, the wavelet coecients were computed by rst 14

(R)  fX ; X ; : : :; X ; X ; X re ecting the series to create Xt;T 1;T 2;T T ;T T ;T T ?1;T ; : : :; X1;T g and then assuming periodic boundary conditions for the series of length 2T . Regressing flog ~2 (u; j )gj =5;:::;12 on a constant and flog 2j gj =5;:::;12 for a given u produces the estimates of the time-varying dierencing parameters plotted in Fig. 3. In all but a few countable number of instances, the time-varying dierencing parameter estimates are postive. Since a global estimate of the dierencing parameter is essentially the average of the time-varying dierencing parameter estimates over the entire time period, our ndings of predominately non-negative dierencing parameters supports the ndings of others that the volatility of intradaily foreign exchange series exhibit long-memory behavior. The two largest estimated values of d(t) found in Fig. 3 are both greater than one. The rst occurrence corresponds to Christmas Day and the second to New Years. As noted by Andersen and Bollerslev (1998) these two day are eectively \weekends" since the low quote activity causes the information contained in these two trading days to be meaningless. This nding is encouraging in that it suggests that the semiparametric wavelet estimator of the time-varying dierencing parameter is able to detect short-lived temporal occurrences. The three most negative values of d(t) in Fig. 3 correspond to the rst (October 2, interval 188), second (June 4, interval 188) and eighth (September 21, interval 243) largest absolute return values. As mentioned earlier, the rst and second largest log-squared returns occurred on days in which scheduled economic information was announced. The other event corresponds with the Russian crisis on September 21, 1993. The estimated value of the time-varying dierencing parameter during these events suggests that volatility becomes anti-persistent in reponse to both scheduled releases of economic news and unexpected political upheavals. Although the information is new and does temporarily aect the market, market participants quickly trade the asset to its new price and then rely on their longterm understanding of the markets behavior and the return-generating process's inherent property of long-memory when carrying out future trades.

15

5.4 Intraday periodicity In Fig. 4 we plot the average value of d(t) at each of the daily 288 ve-minute intervals. Unlike the regional U-shape behavior of volatility found in the average absolute returns of Fig. 1, where volatility increases and the market thickens due to the openening and closing of a regional trading center [ Mueller et. al. (1990), Ballie and Bollerslev (1990) and Anderson and Bollerslev (1997b, 1998)], Fig. 4 reveals that the intraday pattern of long-memory in the DM-$ exchange market is highest when the two most active trading centers (London and New York) are closed. Because a larger long-memory parameter leads to a smoother or less volatile process, the period of the day when both the London and New York markets are closed is a tranquil time with small and infrequent changes in the DM-$ exchange rate. Whereas when the London and New York markets are both open (14:30 GMT to 16:30 GMT), the exchange rate is on average the most turbulent. Since the volume on the London market is the largest of either three markets and is nearly twice that of the New York market, the small longmemory value associated with the closing of the London market (16:30 GMT) suggests that heavier market activity leads to lower degrees of long-memory with its accompaning large and frequent changes in volatility. It is dicult to determine if the decrease in the degree of long-memory that occurs when London and New York are simulataneously trading is the result of public or private information. In the equity markets, French and Roll (1986) argue that private price information held by informed traders must be exploited prior to market closing, and thus, higher levels of volatility are to be expected immediately before the market closes. On the other hand, Harvey and Huang (1991) and Edrington and Lee (1993) argue that, unlike the equity markets, the foreign exchange market is a continuous market, giving informed traders a liquid market in which to capitalize on their private infomation almost 24 hours out of every day. As a result, the increase in volatility during trading versus non-trading hours is due to the concentration of public information that is released during the trading hours. Both hypothesis are plausible explaination for the small long-memory parameter associated with the London and New York markets concurrently trading. The private information 16

hypothesis holds since the level of long-memory in Fig. 4 continually decreases up to the closing of the London market, before beginning to increase. However, the decline in longmemory parameter may also be the result of the large number of U.S. macroeconomic annoucements that take place during the opening hours of trading. To distinguish between the two hypothesis further research is required. The intradaily pattern of d(t) in Fig. 4 also suggest that the three major nancial markets, Tokyo, London, and New York, may be fully integrated. Baillie and Bollerslev (1990) surmise that the regional U-shape of volatility is caused by local market makers holding open position during the day but few overnight. Such positions could cause the increase in the level of trading that occurs during the opening and closing hours of a regional market and be the reason for the dierent markets not being fully integrated. From Fig. 4, the average level of persistence in volatility as measured by d(t) is at its highest point during the opening of the East Asian markets and declines monotonically through the Tokyo and Hong Kong market's lunch hour, the opening of the European and New York markets, and reaches its low as the European market becomes inactive. It is at this point during the day that Andersen and Bollerslev (1998) nd the volatility in the DM-$ exchange rate to be at its daily high. The level of persistence in volatility then monotonically increases through the closing of New York and the opening of the Paci c markets. Hence, through the entire 24 hour trading day, the only opening and closing aects on the intradaily average of d(t) are the opening of the Tokoyo market and the closing of the London market. Our nding of the smallest average value of the long-memory parameter occuring as the London market closes adds strength to Andersen and Bollerslev (1997a) argument that long-memory is a fundamental component of volatility and is not an artifact of the external shocks and regime shift posited by Lamoureux and Lastrapes (1990). Most information shocks come during Europe's and New York's market hours [Harvey and Huang (1991)]. However, it is during these hours that the degree of long-memory and long-term persistence in volatility is at at its lowest point.

17

References Admati, A.T. and P. P eiderer (1988) \A Theory of Intraday Patterns: Volume and Price Variability," The Review of Financial Studies, 1, 3-40. Andersen T.G. and T. Bollerslev (1997a), \Heterogeneous Information Arrivals and Return Volatility Dynamics: Uncovering the Long-run in High Frequency Returns," Journal of Finance 52, 975-1005. Andersen, T.G. and T. Bollerslev (1997b), \Intraday Periodicity and Volatility Persistence in Financial Markets," Journal of Empirical Finance 4, 115-158. Andersen, T.G. and T. Bollerslev (1998), \Deutsche Mark-Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer Run Dependencies," Journal of Finance, 53, 219-265. Baillie, R.T., and T. Bollerslev (1990) \Intra-day and Inter-Market Volatility in Foreign Exchange Rates," Review of Economic Studies, 58, 565-585. Baillie, R.T., T. Bollerslev, and H.O.A. Mikkelsen (1996) \Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity," Journal of Econometrics 74, 3-30. Bollerslev, T. and H.O.A. Mikkelsen, (1996) \Modeling and Pricing Long-Memory in Stock Market Volatility," Journal of Econometrics 73, 151-184. Bollerslev, T. and J.H. Wright (1998) \Semiparametric Estimation of Long-Memory Volatility Dependencies: The Role of High Frequency Data," unpublished manuscript, Department of Economics, Duke University. Breidt, F.J., N. Crato, and P. de Lima (1998) \The Detection and Estimation of Long Memory in Stochastic Volatility," Journal of Econometrics, 83, 325-348. Brockwell, P.J., and R.A. Davis (1991) Time Series: Theorys and Methods, 2nd ed. (New York: Springer). Coifman, R.R. and D. Donoho (1995) \Translation-Invariant De-Noising," in Wavelets and Statistics, (Lecture Notes in Statistics 103) eds. A. Antoniadis and G. Oppenheim (New York: Springer-Verlag) 125-150. Daubechies, I. (1992), Ten Lectures on Wavelets, (Philadelphia: SIAM). Dahlhaus, R., (1996) \On the Kullback-Leibler Information Divergence of Locally Stationary Processes," Stochastic Processes and their Applications 62, 139-168. Dahlhaus, R. (1997) \Fitting Time Series Models to Nonstationary Processes," Annals of Statistics, 25, 1-37. Darorogna, M.M., U.A. Muller, R.J. Nagler, R.B. Olsen, and O.V. Pictet (1993) \A Geographical Model for the Daily and Weekly Seasonal Volatility in the Foreign Exchange Market," Journal of Internationial Money and Finance 12, 413-438. 18

Ding, Z., C. Granger, and R.F. Engle, (1993) \A Long Memory Property of Stock Market Returns and a New Model," Journal of Empirical Finance 1, 83-106. Ederington, L.H, and J.H. Lee (1993) \How Markets Process Information: News Releases and Volatility," Journal of Finance, 48, 1161-1191. Engle, R.F. and T. Bollerslev (1986) \Modelling the Persistence of Conditional Variances," Econometric Reviews, 5, 1-50. French, K.R. and R. Roll (1986) \Stock Return Variances: The Arrival of Information and the Reaction to Traders," Journal of Financial Economics, 17, 5-26. Harris, L. (1986) \A Transaction Data Study of Weekly and Intradaily Patterns in Stock Returns," Journal of Finanancial Economics, 16, 99-117. Harvey, A.C., E. Ruiz, and N. Shephard (1994) \Multivariate Stochastic Variance Models," Review of Economic Studies 61, 247-264. Harvey, C.B., and R.D. Huang (1991) \Volatility in the Foreign Currency Future Market," Review of Financial Studies, 4, 543-569. Jensen, M.J. (1999a) \Using Wavelets to Obtain a Consistent Ordinary Least Squares Estimator of the Fractional Dierencing Parameter," Journal of Forecasting, 18, 17-32. Jensen, M.J. (1999b) \An Approximate Wavelet MLE of Short and Long Memory Parameters." Studies in Nonlinear Dynamics and Econometrics, 3, 239-253. Jensen, M.J. (2000) \An Alternative Maximum Likelihood Estimator of Long-Memory Processes Using Compactly Supported Wavelets," Journal of Economic Dynami and Control, 24, 361-387. Jensen, M.J. and B. Whitcher (1999) \A Semiparametric Wavelet-Based Estimator of a Locally Stationary Long-Memory Model," Preprint, Department of Economics, University of Missouri. Kunsch, H.R. (1995) \A Note on Causal Solutions for Locally Stationary AR Processes," Preprint, ETH Zurich. Lamoureux, C.G. and W.D. Lastrapes, (1990) \Persistence in Variance, Structural Change and the GARCH Model," Journal of Buisiness and Economic Statistics 8, 225-234. Lockwood, L.J. and S.C. Linn (1990), \An Examination of Stock Market Return Volatility During Overnight and Intraday Periods, 1964-1984," Journal of Finance, 45, 591-601. Mallat, S. (1989) \A Theory of Multiresolution Signal Decomposition: The Wavelet Representation," IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 674-693. Martin, W. and P. Flandrin (1985) \Wigner-Ville Spectral Analysis of Nonstationary Processes," IEEE Transactions of Acoustic Speech Signal Process 33, 1461-1470. Melino, A. and M. Turnbull (1990) \Pricing Foreign Currency Options with Stochastic 19

Volatility," Journal of Econometrics 45, 239-265. Muller, U.R., M.M. Dacorogna, R.B. Olsen, O.V. Pictet, M. Schwarz, and C. Morgenegg (1990), \Statistical Study of Foreign Exchange Rates, Empirical Evidence of a Price Change Scaling Law, and Intraday Analysis," Journal of Banking and Finance 14, 11891208. Nason, G.P. and B.W. Silverman (1995) \The Stationary Wavelet Transform and Some Statistical Applications," in Wavelets and Statistics, (Lecture Notes in Statistics 103) eds. A. Antoniadis and G. Oppenheim (New York: Springer-Verlag) 281-299. Percival, D.B. and P. Guttorp, (1994) \Long-Memory Processes, the Allan Variance and Wavelets," in Wavelets in Geophysics, eds. E.Foufoula-Georgiou and P. Kumar (San Diego: Academic Press) 325-344. Percival, D.B. and H.O. Mofjeld (1997) \Analysis of Subtidal Coastal Sea Level Fluctuations Using Wavelets," Journal of the American Statistical Association 92, 868-880. Percival, D.B. and A.T. Walden (2000), Wavelet Methods for Time Series Analysis, (Cambridge, U.K.: Cambridge University Press) forthcoming. Phillips, P.C.P (1987) \Time Series Regression with a Unit Root," Econometrica, 55, 277301. Ramsey, J.B. (1999) \The Contribution of Wavelets to the Analysis of Economic and Financial Data," Phil. Trans. R. Soc. Lond. A, 357, 2593-2606. Ramsey, J.B., and C. Lampart (1998a) \Decomposition of Economic Relationships by Time Scale Using Wavelets: Money and Income," Macroeconomic Dynamics, 2, 49-71. Ramsey, J.B., and C. Lampart (1998b) \The Decompositon of Economic Relationshps by Time Scale Using Wavelets: Expenditure and Income," Studies in Nonlinear Dynamics and Econometrics, 3, 23-42. Taylor, S. (1994) \Modeling Stochastic Volatility: A Review an Comparative Study," Mathematical Finance 4, 183-204. Wiggins, J.B. (1987) \Option Values Under Stochastic Volatility: Theory and Empirical Estimates," Journal of Financial Economics 19, 351-372. Wood, R.A., T.H. McInish, and J.K. Ord (1985) \An Investigation of Transaction Data for NYSE Stocks," Journal of Finance, 40, 723-741. Zhou, B. (1996) \High-Frequency Data and Volatility in Foreign-Exchange Rates," Journal of Economic Business and Statistics, 14, 45-52.

20

0.08

0.07

0.06

Avg. Absolute Return

0.05

0.04

0.03

0.02

0.01

0.0

1

24

47

70

93

116

139

162

185

Five Minute Interval

208

231

254

277

300

Figure 1: Intraday average of the absolute value for the 288 daily ve-minute returns..

21

Log Squared DEM-USD D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 S12

0

20000

40000

60000

80000

Figure 2: Multiresolution analysis for the DM-$ log-squared return series MOWDT wavelet coecients and scaling coecents over the time period October 1, 1992 through September 29, 1993.

22

1

d(t)

0.5

0

−0.5 10

11

12

1

2

3

4 Month

5

6

7

8

9

Figure 3: Semiparametric wavelet estimates of the time-varying dierencing parameter, d(t), f for the DM-$ log-squared returns using MOWDT wavelet coecients (W j;t;T , j = 5; : : :; 12, t = 1; : : :; 74880) calculated from a LA(8) wavelet lter.

23

0.38

0.36

0.34 FTSE

0.32

0.3

d(t)

Nikkei

NYSE

0.28

0.26

0.24

0.22

0.2

0

2

4

6

8

10

12 14 Hourly GMT

16

18

20

22

24

Figure 4: Intraday average over all 260 trading days for each of the 288 daily ve-minute intervals of d(t). The arrows show when the particular market is open, the Nikkei, 1:00 GMT - 7:00 GMT, the FTSE, 10:00 GMT - 16:30 GMT, and the NYSE, 14:30 GMT - 22:00 GMT.

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