estimating high-frequency foreign exchange rate volatility with nonparametric arch models

journal of statistical planning

Journal of Statistical Planning and and inference Inference 68 (1998) 247-269 ELSEVIER

Estimating high-frequency foreign exchange rate volatility with nonparametric ARCH models

C h r i s t i a n M . H a f n e r * Institute for Statistik & Okonometrie, Humboldt-Universit~it zu Berlin, Spandauer Strasse l,

D-10178 Berlin, Germany

Received 8 May 1996; received in revised form 13 November 1996; accepted 5 December 1996

Abstract

High-frequency foreign exchange rate (HFFX) series are analyzed on an operational time scale using models of the ARCH class. Comparison of the estimated conditional variances focuses on the asymmetry and persistence issue. Estimation results for parametric models confirm standard results for HFFX series, namely high persistence and no significance of the asymmetry coefficient in an EGARCH model. To find out whether these results are robust against alternative specifi- cations, nonparametric models are estimated. Local linear estimation techniques are applied to a nonparametric ARCH model of order one (CHARN). The results show significant asymmetry of the volatility function. To allow for both flexibility and persistence, a higher-order multiplicative model is fitted. The results show important asymmetries in volatility. In contrast to the EGARCH specification, the news impact curves have different shapes for different lags and tend to increase slower at the boundaries. (~) 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

High-frequency financial data analysis has been subject of many recent investiga-

tions. It is important for the econometric analysis of financial markets mainly for two

reasons. First, after having found that GARCH processes fit daily and weekly foreign

exchange (FX) rates well in most cases, the topic o f temporal aggregation (Drost and

Nijman, 1993) arose. It was shown that under suitable conditions the parameters of a

model estimated at a given frequency can be uniquely transformed to the parameters

o f the same model at a different frequency. Recent investigations, e.g. of Andersen and

Bollerslev (1995) revealed that the temporal aggregation o f GARCH processes does not hold empirically for high-frequency data. Second, by using tick-by-tick data one

is able to observe microeconomic features o f the market that disappear when the data are aggregated. Two examples are the negative autocorrelation of the returns and the

* Correspondence address: SFB 373, Humboldt-Universit~it zu Berlin, Germany, and CORE, Universit6 Catholique de Louvain, Louvain-la-Neuve, Belgium. E-mail: [email protected].

0378-3758/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S0378-3758(97)00 144-4

248 C.M. HaJher l Journal of Statistical Plannin9 and Inference 68 (1998) 247-269

risk-dependent bid-ask spreads. Analyzing these features yields information about the efficiency of the market.

The literature on high-frequency data is still very scarce. An early paper by Wasserfallen and Zimmermann (1985) already reported typical statistical properties, namely negative autocorrelation of the returns at the first lags. Goodhart and Figliuoli (1991) extended the analysis. The problems of both papers are the not representative data sets. Along with the first conference on high-frequency data in finance in Zfirich 1995, however, the distribution of data sets have much improved. A review about high-frequency financial data is given by Guillaume et al. (1994).

Modelling volatility of FX returns, the predominant model of the past was the GARCH model with its various parametric extensions. It was soon recognized, however, that a standard GARCH model has three major drawbacks, as summarized by Nelson (1991): (i) it imposes a symmetrical influence of lagged residuals on the volatility, (ii) the necessary parameter constraints can cause optimization problems, and (iii) similar to the distinction between ARMA and ARIMA models it is in most cases difficult to decide whether the estimated conditional variance is weakly stationary or not, i.e. the distinction between GARCH and IGARCH. Furthermore, written explicitly, GARCH is an additive model in terms of lagged residuals. Unlike the mean, however, the variance is not linear, and a multiplicative form seems more natural.

A way to overcome the misspecification of standard GARCH models is to mod- ify the functional form of the conditional volatility equation, allowing for asymmetry, for instance. In most practical situations, however, it is not known beforehand which functional form is appropriate, and therefore extensive use of specification tests has to be made. An alternative, more flexible and somewhat easier approach is to use recent results of nonparametric autoregression models. The main purpose of this paper is to formulate some types of nonparametric ARCH models and to compare the model fits to the results obtained from parametric ARCH models. It is focused on the well-known HFDF93 data set of Olsen & Associates, containing the most liquid rates DMark/US Dollar (DEM/USD), Yen/Dollar (JPY/USD), and Yen/DMark (JPY/DEM). Through- out the paper the analysis is restricted to the univariate case to find out the one- dimensional data generating processes. Any crosscorrelations or arbitrage relationships between the FX rates are not considered.

A general problem of the econometric analysis of high-frequency financial time series is the interaction between complex seasonality and conditional heteroskedasticity, both crucially depending on the choice of the time scale. There is still no generally accepted methodology that copes with all features in a rigorous and simultaneous way. In this paper, the analysis is performed sequentially, i.e. first the seasonality is removed, then the conditional heteroskedasticity is modelled. The statistical inference of the second stage does not take into account any estimation error of the first stage, so the results will be interpreted with care.

The remainder is organized as follows. After a description of the series and the time scale, the results of parametric ARCH models are reported. Section 4 deals

C M. Ha]her~Journal of Statistical Plannin O and Inference 68 (1998) 247-269 249

with the local linear estimation of a nonparametric ARCH model of order one, called CHARN. Finally, in Section 5 higher-order multiplicative models are estimated by

spline methods.

2. The foreign exchange market and the time scale

The foreign exchange market is by far the largest financial market. According to the Wall Street Journal of 1 March 1990, the average daily FX trading volume is $ 650 billion. Compared to this, the NYSEs largest volume day, 19 October 1987, only saw $ 21 billion of volume. The market is decentralized with the main trading locations in Tokyo, Hong Kong, Singapore, London, Frankfurt, and New York. It is an electronic market, active 24 h a day. Banks act as market makers and place bid- and ask-quotes on the screen. Central information collectors such as Reuters provide the quotes for the market makers. Actual trade takes place over the phone. Therefore, there is no information about actual prices and trading volume. So far, the largest part of trading occurs in US Dollars, which assumes the role of a num6raire for the minor rates. Although there is some important central-bank intervention money, the largest part of the FX market volume is speculated by the market makers.

The data set HFDF93 was acquired from Olsen & Associates, Zfirich. It contains 1472241 quotes for DEM/USD, 570840 for JPY/USD and 158979 for JPY/DEM during the time 1 October 1992, 0 : 00 : 00 and 30 September 1993, 23 : 59 : 59 GMT. For each pair of bid- and ask-quotes, the time in GMT, the quoting bank and the location of the bank are recorded.

Before looking at the statistical properties of the FX rates it is necessary to introduce some notation. Let {At, t>~0} and {Bt, t>~0} denote the ask- and bid- quote processes in continuous time for some choice of the time unity (20 min, say). The logarithmic price process {qt, t >10} is defined as qt =-(logAt + log Bt)/2. This transformation has become standard, since it has the appealing property that the logarithmic price process of the inverted exchange rate, given by {At l} and {Btl}, is just {-qt} . The returns in continuous time would be measured by the process {dqt}, but we can only observe returns in discrete time, which are defined by yt = q t - qt-At, t = At, 2At, . . . . To keep the notation simple, the dependence of Yt on At is not written explicitly, but it is important to keep this in mind. Because the time series properties of bid- and ask-quotes are very similar, the loss of information due to the aggregation is small. Attempts to exploit the information contained in the bid-ask spreads for volatility models have not

been very successful. Section 4 refers to an example. For some 6 > 0 let V~(y t )=- (Ely t - E(yt)lr) 1/~ be the volatility of the returns

depending on At. Two special cases of the volatility definition are the mean absolute deviation for 6 = 1 and the standard deviation for 6 = 2. For a given sample period S and time interval At, the number of observations is given by n = SlAt. Mean and volatility of Yt are estimated by E(Y t )= n-1 ~ i n l YiAt and Vr(yt): (n -1 ~in_-i lyi:~,- E(yt )lr) I/6, respectively.

250 C.M. Hafner/ Journal of Statistical Plannino and Inference 68 (1998) 247-269

A

I

0

o.

o

o

o

l Ol•O

DEM/USD 20min calendar I I I I I

,:l

o12 01, 01, o18 110 Lag ('10 3)

Fig. 1. Autocorrelation Function of DEM/USD squared (dashed) and absolute (solid) returns for 20 min intervals in calendar time up to lag 1000.

To begin with, it is necessary to deal with the subject of seasonal volatility. Con- sider the following procedure: The quotes are observed every 20 min, i.e. At = 20 min. The returns from this series are calculated according to the definition given above. Fig. 1 depicts the autocorrelation function of the absolute and squared returns of the

DEM/USD rate. Daily and weakly patterns are clearly visible and much more distinct for the former. Not all autocorrelation plots are shown, but it is worth noting that the

Yen rates have less pronounced seasonality. In order to take the seasonality into account, we adopt the O-time scale of

Dacorogna et al. (1993) which is more flexible than the dummy-variable method of

Baillie and Bollerslev (1990). The basic idea is to define the time intervals to be longer in low business periods and shorter in busy ones. Without information about volume in FX markets other 'activity' measures have to be used. The empirical 'scaling law'

first noted by Miiller et al. (1990) states

V (y, ) = c , ( xt) (1)

with a constant C1 depending on the FX rate. The scaling exponent ~6 is obtained by the slope coefficient of the best linear projection of log Vr(yt) o n log(At). For the estimation, a medium range of time intervals is recommended, because (i) for large time intervals n is small such that the variance of the volatility estimate increases, and (ii) for small time intervals the variance of the volatility estimate is small but the returns are of the order of the spreads, which induces a 'microeconomic' variance

C.M. Hafner / Journal of Statistical Planning and Inference 68 (1998) 247-269 251

due to the uncertainty about the true returns. Details on this point are given by Miiller et al. (1995). For calculation of the volatilities, the mean was neglected here as in

Miiller et al. (1990) due to the small-value compared to the standard deviation, i.e. l~ , (yt) /~,=3.4E-04 and E(yt)/l?l of the same order for DEM/USD. According to Miiller et al. (1990), the estimates of 71 and ~2 are, respectively, 0.58 and 0.52 for most foreign exchange rate series, indicating a substantial deviation from a Gaussian random walk.

Recall from Fig. 1 that there is both a daily and weekly seasonal pattern of" the FX rates. Therefore, the period P is in this case one week, and for a time unity of 20 min we have P = 504. Based on a regular time scale let one period consist of J time intervals, i.e. P = J A t . In analogy to the whole sample we define volatility for each season separately as

V&j(yt)-- V~(yt I t m o d e = j - 1), j = 1, . . . ,J , (2)

which is estimated straightforwardly by calculating the volatility conditional on the seasons.

Rearranging the scaling law (1) and substituting the volatility of the whole sample by the volatility of season j now gives the definition of activity in season j:

1 (V~,j(yl)~ 1/~ a j = - ~ CI J , j = 1, . . . ,J . (3)

Due to the pronounced seasonal pattems of absolute returns and to the findings of Ding et al. (1993), 6 = 1 will be used in the following.

If the market activity at a certain point t in calendar time is of interest, the estimate aj is rather crude if large time intervals were used. Because the sample period is fixed, however, it is no solution to use very small time intervals due to the larger variance. To get a continuous activity function nevertheless, At = 20 min was used and the resulting function was smoothed with a Nadaraya-Watson kernel smoother, quartic kemel, and bandwidth h = 5. This was seen as a compromise between too small and too large bandwidths. Too small bandwidths yield too wiggly curves with large variances and thus highly unstable activity functions. Too large bandwidths, on the other hand, do not capture all detailed patterns of activity. The importance of balancing these two effects is obvious for our in-sample analysis, and even more so for out-of-sample prediction.

Fig. 2 shows the activity function for DEM/USD. Activity clusters are clearly visible. Typical is a trimodal intraday pattern, corresponding to the openings of the major markets in Far East, Europe and America. Moreover, the activity at the weekend is negligibly small. Similar seasonal patterns are obtained for the Yen rates (not shown).

The new time scale ~ based on the smoothed, continuous activity function a(t) is defined such that time intervals with high activity are shortened and time intervals with small activity are enlarged. More precisely, after transformation of the time scale the same activity is to be expected in each time interval. There is a one-to-one deterministic

252 CM. Hafnerl Journal of Statistical Planning and Inference 68 (1998) 247-269

A

i

o

0

o

4 °

"3 o

i ° / o o

010 110

Smoothed Activity I I I I I

,io 3'.0 ,'.o 51o index of 20min interval/week 1"10 21

Fig. 2. Smoothed activity as a function of 20 min intervals in calendar time during a week for the DEMAJSD rate. A Nadaraya-Watson smoother with quartic kernel and bandwidth h = 5 was used.

relationship between calendar time t and operational time 0, and the mapping function

O(t) is

v~(t) = ~-2 a(t ')dt ' ,

with constant

.•'o P

C2 = a(t') dt',

(4)

so that one period P in 0-time corresponds exactly to one period in calendar time. Note that this is a deterministic mapping function based on the seasonality in the volatility of returns. It can be regarded as a deseasonalization technique.

Because the activity function is merely an average measure, there is sometimes no quote in the new time interval. In these cases the next occurring quote is taken and empty intervals are skipped. The number of records are thus reduced from 26280 20 min intervals per year to 25 476 for DEM/USD, 24 838 for JPY/USD, and 23 265 for JPY/DEM. All subsequent analyses are based on the 0-scale. Despite the risk of confusion, however, the time indices 0 are renamed t and numbered as an equispaced scale, i.e. t = 1,2, . . . ,n.

In Table 1, summary statistics about the FX returns in 0-time are given. The coefficient of kurtosis reveals substantial differences from normality. Even more striking are the values of the skewness. Considering standard errors of X//-~ under the null

C. M. Hafner l Journal of Statistical Plannin# and Inference 68 (1998) 247-269

Table 1 Summary statistics of the FX returns in O time

253

FX rate n Min Max Mean Variance Skewness Kurtosis

DEM/USD 25476 -0.0069 0.0101 5.73E--06 6.5E--07 0.240 9.40 JPY/USD 24 838 -0.0087 0.0129 -4 .90E-06 6.7E-07 0.182 17.30 JPY/DEM 23265 -0.0101 0.0068 - l . 1 0 E - 0 5 7.0E-07 -0.167 8.71

Table 2 Autocorrelations of FX returns, squared and absolute returns in 0 time for selected lags k

k DEM/USD JPY/USD JPY/DEM

Pk(Yt) Pk(Y2t) Pk(lYt[) Pk(Yt) Pk(Y 2) Pk(lY, I) Pk(Yt) Pk(Y 2) Pk(lYtl)

1 -0.0789* 0.2041" 0.2067* -0.1008" 0.2563* 0.2624* -0.0079 0.1084" 0.1687" 2 --0.0152" 0.0981" 0.1412" --0.0115 0.0836* 0.1644" -0.0100 0.0761" 0.1278" 3 -0.0235* 0.0749* 0.1336 0.0034 0.0415" 0.1344" -0.0012 0.0702* 0.1120" 4 -0.0027 0.0889* 0.1281" --0.0199" 0.0429* 0.1255" 0.0159" 0.0509* 0.0969* 5 0.0055 0.0763* 0.1193" 0.0056 0.0308* 0.1109" -0.0014 0.0490* 0.0942* 6 -0.0049 0.0682* 0.1030" 0.0085 0.0338* 0.1069" -0.0097 0.0511" 0.0954* 7 0.0145" 0.0494* 0.0953* 0.0055 0.0545* 0.1201" 0.0059 0.0738* 0.1000" 8 0.0111 0.0431" 0.0872* --0.0102 0.0510" 0.1170" 0.0156" 0.0751" 0.1007" 9 0.0059 0.0580* 0.0947* 0.0042 0.0493* 0.1103" -0.0021 0.0632* 0.0913"

10 0.0134" 0.0562* 0.0909* -0.0070 0.0365* 0.0998* --0.0025 0.0306* 0.0687* 25 0.0129" 0.0303* 0.0572* -0.0008 0.0264* 0.0853* --0.0042 0.0277* 0.0497* 50 0.0097 0.0234* 0.0484* --0.0061 0.0253* 0.0755* -0.0097 0.0291" 0.0446*

100 --0.0092 0.0261" 0.0670* 0.0084 0.0208* 0.0795* -0.0047 0.0231" 0.0595* 250 0.0037 0.0105 0.0249* -0.0074 0.0138" 0.0547* -0.0067 0.0166" 0.0335* 500 --0.0053 0.0178" 0.0360* 0.0003 0.0059 0.0358* 0.0053 0.0129 0.0211"

1000 -0.0018 0.0068 0.0230* 0.0010 0.0062 0.0273* -0.0029 0.0071 0.0198"

Note: The asterisk marks asymptotic 5% significance.

hypothesis of normality, these are highly significantly different from zero. Thus, the distributions of the FX returns appear to be significantly asymmetric in O-time.

Consider the autocorrelations of returns, squared and absolute returns in Table 2. The dollar rates have significant negatively autocorrelated returns at the first lags. Goodhart and Figliuoli (1991) and Guillaume et al. (1994) report similar results for ultra-high frequencies. The economic explanation is that banks have to perform inventory rebalancing if they hold open positions longer than a few minutes. This is confirmed by the fact that negative autocorrelation disappears when the data are aggregated.

Note that the autocorrelations of the much less liquid JPY/DEM returns are not significant at the first lags. This indicates the commodity status of the dollar rates which banks use for inventory rebalancing in favor of rates with less volume and thus larger bid-ask spreads and transaction costs.

Consider Fig. 3, which shows the autocorrelations of DEM/USD squared and absolute returns in 0-time up to lag 1000. Absolute returns are clearly higher autocorrelated than squared retums. Ding et al. (1993) showed that models of the ARCH type are

254 CM. Hafner/ Journal of Statistical Planning and InJerence 68 (1998) 247-269

o

,-4

0

,4 t

t ~-

o

0

0

0

DEM/USD 20min theta I I I t I

I ' ' I I ' 0 0 2.0 4.0 6 0 S 0 i0.0 Lag 1"i0 2 }

Fig. 3. Autocorrelation Function of DEM/USD squared (dashed) and absolute (solid) returns in ~ time up to lag 1000.

able to produce such behavior. The ACF of absolute returns declines very slowly, so that a hyperbolical rather than an exponential decline seems plausible. This gives some motivation for the fractionally integrated GARCH (FIGARCH) model of Baillie et al. (1996). Indeed, the long memory in volatility is striking: There is still significant positive autocorrelation around lag 2000, approximately four weeks of time lag.

3. Parametric GARCH models

In this section, estimation results of parametric GARCH models of the FX rate series are presented as a benchmark for nonparametric models. It is focused on two main issues: persistence and asymmetry of the estimated volatility equation.

Recall that the dollar rates are negatively autocorrelated at the first lags. To overcome this, an AR(3) and AR(2) model was fitted to the returns of the DEM/USD and JPY/USD returns, respectively. The orders were chosen by the Schwarz information criterion. Maximum likelihood estimation results are given in Table 3. l

I As two referees correctly remarked, the reported standard errors would only be correct i f there was no estimation error in the time change. If one would also take this error into account, the resulting standard errors would, of course, be affected. However, as the investigations of Mfiller et al. (1995) show, the standard errors for the estimation of the scaling exponent are unusually small. Also, the observed seasonality patterns are very stable, so that the error induced by the time change should be small compared with the statistical inference error.

C.M. HafnerlJournal of Statistical Plannin 9 and Inference 68 (1998) 247-269 255

Table 3 MLE estimation results for the AR(2) and AR(3) model fitted to the non-standardized returns of DEM/USD and JPY/USD, respectively

FX rate DEM/USD JPY/USD

~1 -0.0811 -0.1031 (0.006) (0.0063)

~2 --0.0236 -0.0219 (0.007) (0.0063)

~3 -0.0265 (0.006)

logL 145 403.6 141346.8

Note: qSi denotes the autoregressive parameter of the ith lag. Heteroskedasticity-consistent standard errors are given in parentheses.

Table 4 QMLE Estimation results for the GARCH(I,1) and IGARCH(1,1) models

FX rate GARCH IGARCH

DEM/USD JPY/USD JPY/DEM DEM/USD JPY/USD JPY/DEM

~o 0.0626 0.0345 0.0324 0.0201 0.0258 0.0097 (0.0060) (0.0030) (0.0035) (0.0018) (0.0020) (0.0011 )

0.1018 0.1405 0.0845 0.1071 0.1519 0.0757 (0.0071) (0.0079) (0.0056) (0.0053) (0.0066) (0.0041)

fl 0.8382 0.8401 0.8862 (0.0106) (0.0076) (0.0073)

logL -34638.6 -30549.7 -31672.3 -34818.9 -32649.6 -31743.8 Mean(~t) 0.0053 -0.0086 -0.0168 0.0050 -0.0084 -0.0174 Var(~ t) 0.999 1.0 1.0 0.93 0.97 0.954 Skew(~ t ) 0.2164 0.0369 -0.1497 0.24 0.04 -0.17 Kurt( ~t ) 9.73 12.04 6.71 11.43 12.04 6.70

Note: Heteroskedasticity-consistent standard errors are given in parentheses, logL denotes the log likelihood value.

The residuals e, of the estimated AR(3) model were now used for the estimation of ARCH models. Of course estimating variance parameters affects the estimation of mean parameters, so the estimation should be performed simultaneously. This was also

done, but no remarkable differences were detected. Therefore, the mean-zero residual series et served as the basic series for volatility models.

To start with, the results of a GARCH(1,1 ) model

e, = o ' , ¢,, (5)

2 O~ E'2-- 1 2 a t = 09 + + f l a t _ l , (6)

with it ~ N.i.d.(0, 1) are given in Table 4. The estimation was performed by QMLE using the optimization algorithm of Berndt et al. (1974, BHHH). The standard errors

256 c. M. Hafner / Journal of Statistical Plannin9 and InJbrence 68 (1998) 247 269

are heteroskedasticity consistent, see White (1980), and conditional on the results of the conditional mean estimation. Because the original series has a very small unconditional variance, numerical problems arose with the use of the BHHH algorithm. Therefore, the

residuals were divided by their standard deviation, so that the unconditional variance

of the transformed series is equal to one. This only affects the value of ~o but not the

parameters of interest, namely ~ and ft. If the value of ~o of the original series is also of interest, this can be obtained by using cb -- ( 1 - ~ - 1~)(~2. For DEM/USD, for e.g., o5 = 3.88E-08 and the value of the log likelihood function becomes 146 893.4.

Note that ~ +/~--0.94, 0.98 and 0.97, respectively, so the parameter estimates imply a conditional variance close to the unit root. However, the corresponding likelihood-

ratio test statistics are 360.6, 4199.8 and 143.0, so that the null hypothesis of a unit

root is rejected in all cases. The distribution of the estimated residuals ~t is still highly leptokurtic for all rates.

The asymptotic standard deviation of X/~-4/n of the estimated kurtosis indicates sig-

nificant discrepancy of the estimated residuals from normality, i.e. misspecification of the conditional distribution for all rates. The leptokurtosis implied by conditional het-

eroskedasticity is not strong enough to account for the leptokurtosis observed for the unconditional return distribution if ~t is assumed to be normally distributed. Plausible alternative assumptions will not be pursued in this context, so for the following models

conditionally Gaussian residuals will be assumed, too. Turning to the persistence issue, the IGARCH(1,1) model

O'~ =O3 -]- ~'~--I -+- (1 -- ~)O'LI (7)

is estimated and the results are also reported in Table 4. Obviously, the fit is outper-

formed by the GARCH model. Note that the kurtosis of the estimated residuals has even increased for DEM/USD. Also, the variance of the estimated residuals is 0.93

for DEM/USD, i.e. the fit overestimates the unconditional variance. These facts indi-

cate misspecification of the IGARCH model. The degree of persistence in the volatility does not seem to be large enough to justify a permanent impact of shocks on volatility

forecasts. The next topic is asymmetry. For this, the Exponential GARCH (EGARCH) of

Nelson (1991) is estimated as

log a t2 ----e9 + 0~,_1 + ~(1~,-, I - 2 x / ~ ) + fl log err2-1, (8)

with ~t ~ N.i.d.(0, 1). Results are reported in Table 5. The likelihood value is slightly smaller than the GARCH likelihood. Most striking is the nonsignificance of the asymmetry parameter 0. Based on this result one would argue that shocks to the DEM/USD rate to either side have the same impact on volatility. However, we will see in the next section that this conclusion may be misleading.

The first 500 estimated conditional variances (this covers approximately one week) for the different models are depicted in Fig. 4. The top panel shows the residuals

C. M. Hafner l Journal of Statistical Plannino and Inference 68 (1998) 247-269

Table 5 QMLE results for the EGARCH(I,1) model

257

FX rate DEM/USD JPY/USD JPY/DEM

~o 0.0055 0.0127 0.0047 (0.0012) (0.0038) (0.0032)

0 -0.0022 -0.0118 -0.0160 (0.0048) (0.0119) (0.0102)

7 0.1426 0.2454 0.2140 (0.0096) (0.0210) (0.0157)

[~ 0.9426 0.8754 0.8440 (0.0044) (0.0115) (0.0157)

log L - 3 4 652.3 - 3 2 864.8 -31 877.4 Mean(~t) 0.0062 -0.0079 -0.0149 Var(~t) 1.0144 0.9416 0.9791 Skew(~t) 0.25 0.0588 -0.1159 Kurt(~ t) 9.51 12.01 7.30

Note: As in Table 4.

Fig. 4. The first 500 residuals of the fitted AR(3) model to DEM/USD together with the corresponding estimated conditional variances of the GARCH, IGARCH and EGARCH (from top to bottom) models. The scale is (E+01).

258 C M. Hafner l Journal of Statistical Planning and Inference 68 (1998) 247-269

Table 6 Summary statistics of the estimated FX volatilities and residual diagnostics

Model GARCH IGARCH EGARCH

DEM/USD Min(cr 2 ) 0.42 0.27 0.35 Max(a 2) 20.81 23.15 20.51 Mean(at 2 ) 1.02 1.19 1.03 Var(at 2 ) 0.56 1.06 0.51

Q2(10) 10.84 20.47 63.94 Q2(100) 97.32 94.34 121.88 Q2(1000) 1197.43 1258.26 1166.17

JPY/USD Min(at 2 ) 0.23 0.19 0.33 Max( a 2 ) 41.48 45.14 25.95 Mean(at 2 ) 1.09 1.17 0.97 Var(trt 2 ) 1.96 2.44 0.40

Q2(10) 39.67 43.87 90.94 Q2(100) 218.75 204.23 529.59 Q2(1000) 1307.03 1182.87 2770.36

JPY/DEM Min(a 2 ) 0.32 0.20 0.43 Max(at 2 ) 14.21 13.34 12.61 Mean(at 2 ) 1.03 1.13 0.98 Var(a 2 ) 0.49 0.69 0.14

Q2 ( 10 ) 32.43 46.29 103.95 Q2(100) 197.96 215.11 770.61 Q2 ( 1000 ) 1359.60 1152.20 3189.90

Note: Q2(m) denotes the Box-Ljung statistic of squared residuals.

of the fitted AR(3) model to the DEM/USD rate. After the crisis of the European Monetary System (EMS) in September 1992, this subsample is still dominated by large volatilities. Two major jumps occur during this first week. Although the GARCH and IGARCH estimated volatilities appear to react similarly to news, it can be seen that IGARCH overweights outliers relative to GARCH (note that the same scales are used). This holds even stronger for EGARCH if the shock is very large, as for the second jump in the subsample. Thus, the conclusion of Engle and Ng (1993) that EGARCH tends to overweight outliers is confirmed.

Summary statistics of the estimated volatilities are given in Table 6 together with residual diagnostics. Qz(m) denotes the Box-Ljung statistic of the squared residuals, i.e.

l m Q 2 ( m ) = n ( n + )~-]~ (n - i)-lpZi, i=1

where 2 2 p i = f O 1 T ( ~ t , ~ t _ i ) , and Q2(m) is asymptotically Z 2 distributed with degrees of freedom equal to m minus the number of estimated parameters, cf. McLeod and Li (1983).

C.M. Hafner / Journal of Statistical Planning and InJerence 68 (1998) 247-269 259

4. A simple nonparametric model

Consider the general nonlinear autoregressive model of order one

Yt = f (Y t -1 ) + 0"(Yt-1 )~t, (9)

where it ~ i.i.d.(0, 1), and the conditional mean f ( - ) and conditional variance 0"2(") are

smooth functions satisfying ergodicity conditions given by Ango Nze (1992). Accord- ing to Bossaerts et al. (1996) (9) is called a conditionally heteroskedastic autoregressive nonlinear (CHARN) model. The use of nonparametric methods in time series analysis has been extensive since Robinson (1983) provided consistency results for a-mixing processes. It is known that stationary Markov chain processes have the a-mixing property, so for the model in (9), where {yt} is a Markov chain, it is sufficient to assume that it is also stationary.

For estimation of the functions, kernel weighted local linear estimation was used. For a recent treatise on local polynomial estimation see the monograph of Fan and Gijbels (1996). The Nadaraya-Watson estimate is equivalent to local constant estimation. Local linear estimation (LLE) was chosen in favor of the Nadaraya-Watson (NWE) or Gasser-Mfiller (GME) estimator. Under fixed design, the Gasser-Mfiller estimator is preferrable to NWE because of its better bias behavior. Under random design, however, the variance of GME is larger by the factor 1.5. Asymptotically, local linear estimation combines the advantages of GME and NWE, having the same bias as GME and the same variance as NWE. The LLE performs better than NWE and GME especially at the boundaries. A more practical reason is that LLE corresponds to a local least-squares problem for which easy and fast efficient algorithms are available. Not only the regression function, but also its derivatives up to the order of the polynomial are estimated simultaneously.

Consider again the model (9). The task is to estimate the mean function f ( x ) = E(yt [Yt-I = x ) and the variance function 0"2(X)= E(y2tlYt_l = x ) - { E ( y t l Yt-, = X ) } 2.

Assume that the unknown functions are continuously differentiable up to order l. Then, local polynomials of degree l yield estimates of f ( x ) and 0"2(x) that are the solution of the following weighted least-squares problems:

( , )2 /~(x) = arg min ~ y, - ~ flj(Yt-i -- X ) j Kh(yt-, - x), (10)

fl C~I+I t=2 j=0

£(x) = arg min y~ - ~-~sj(yt-i - x) j Kh(yt-i - x), (11) sE~/+l t=2 j=O

where Kh(u )=h- lK(u /h ) denotes a bounded, symmetric kernel function. Here, the quartic kernel was used. The estimators ,f(x) and d2(x) are given by

/ ( x ) -- and

260 C. M. Hafner / Journal of Statistical Plannin9 and Inference 68 (1998) 247-269

i

o

L

o

o

o

d"

o

o

t-

o %D

DEM/USD Volatility Function I I I I I I I

i

....... / • .' , ' /

3.o 3.o -;.o olo ~Io ~'.o ~Io lagged return 1"10 )

Fig. 5. The estimated volatility function for DEM/USD with uniform confidence bands. Shown is the truncated range (-0.003,0.003). The bandwidths used were hi = 0.0092 and h2 = 0.0076.

H/irdle and Tsybakov (1997) proved asymptotic normality o f these estimators under

conditions given by Tweedie (1975), Ango Nze (1992), and Diebolt and Gu6gan

(1993).

Bandwidths were chosen by cross-validation criteria as in H~irdle and Vieu (1992).

Analytical expressions for uniform confidence bands in the current setting are not avail-

able. To give some evidence for the reliability of the estimates nevertheless, asymptotic

95% uniform confidence bands are plotted for the i.i.d, case, see e.g. Hfirdle (1990,

Ch. 4). 2 To save space only the volatility estimates are shown. Corresponding to the

results reported in Table 2, the mean function estimate declines linearly for the dol-

lar rates, and is not significant for JPY/DEM. In the volatility pictures, boundaries

are skipped, because for all rates more than 99% of the returns are in the interval

[ - 0.003; 0.003 ].

The DEM/USD volatility estimate is shown in Fig. 5 using the cross-validation optimal bandwidth h2 =0.0076 for the estimation o f E ( y 2 l y t _ l ) , and hi = 0.0091 for

the estimation o f E(yt ] Y t - i ). The variance function is skewed and thus reveals asymmetry. Volatility increases

more for a large increase of the DEM/USD rate than for a large decrease o f the same size. If the true volatility was additively separable, each function being symmetric and

depending on one lag, then the variance conditioned only on the first lag would also be symmetric. Thus, under the hypothesis o f additivity a general class of parametric

2 Obvious errors in the formulae of Algorithm 4.3.2. in Hfirdle (1990) were corrected.

C. M. Hafner / Journal of Statistical Plannin# and Inference 68 (1998) 247-269

JPY/USD Vol Fct I I I I I I I

261

o

0

o .

-3'.0 -Lo -;.o olo 11° 21o 31o l a g g e d r e t u r n ( ' 1 0 - 3 )

Fig. 6. The estimated volatility function for JPY/USD with uniform confidence bands. Shown is the truncated range ( -0 .003, 0.003). The bandwidths used were hi = 0.0032 and h2 = 0.0032.

A o

o

JPY/DEM Vol Fct I t I I I I I

C4

0

,-4

0

,-4

U~

0

i

-3'o -2'.0 -1'.o olo 11o 21o ~io l a g g e d r e t u r n ( ' 1 0 - 3 )

Fig. 7. The estimated volatility function for JPY/DEM with uniform confidence bands. Shown is the truncated range (-0.003,0.003). The bandwidths used were h] = 0.0041 and h2 = 0.0021.

262 C.M. Hafner l Journal of Statistical Plannin9 and Inference 68 (1998) 247-269

Table 7 Residual statistics for the local linear estimation of a CHARN model: mean, variance, skewness and kurtosis

FX rate DEM/USD JPY/USD JPY/DEM

Mean(t) -0.0098 -0.0012 0.0003 Var(~) 0.9522 0.8943 0.9325 Skew(t) 0.2516 0.3079 -0.1853 Kurt(~t) 8.8450 16.4050 8.4518

Ql(10) 42.48* 21.58" 16.11 QI(100) 161.61" 130.51" 110.14 QI(1000) 1114.05 1244.63" 1098.62"

Q2(10) 932.84* 368.14" 483.81" Q2(100) 2882.81" 1225.34" 1734.27" Q2(1000) 11 551.10" 5032.68* 5044.25*

Note: Q1 (') and Q2() denote the Portmanteau-statistics for the residuals and squared residuals, respectively. An asterisk marks significance at the 95% level.

Table 8 First 10 autocorrelations of the CHARN residuals, squared and absolute residuals

k DEM/USD JPY/USD JPY/DEM

pk(~,) 2̂ .2 . pk(g,) pk(l~,l) pk(~,) pk(~) pa(l~,l) pk(~,) pk(¢,) Pk(l~,l)

1 0 . 0 0 0 9 0 .0578 0 .1197 -0.0038 0.01454 0.0702 0.0009 0 .0020 0.0258 2 -0.0203 0.0735 0 .1194 -0.0184 0 .0637 0 . 1 2 5 7 -0.0103 0.0561 0.0971 3 -0.0208 0 .0667 0 .1185 -0.0039 0 . 0 3 4 6 0.1117 0.0020 0.0611 0.0955 4 -0.0046 0 .0816 0 .1141 -0.0185 0 .0322 0.1035 0.0141 0 .0410 0.0777 5 0 .0021 0 .0647 0 .1084 -0.0037 0 . 0 2 5 4 0 . 0 9 4 3 -0.0020 0 .0335 0.0741 6 -0.0051 0 .0625 0.0922 0.0054 0 .0251 0 . 0 8 7 2 -0.0113 0 .0445 0.0787 7 0 . 0 1 7 9 0 .0434 0 .0856 -0.0028 0 .0406 0.1012 0.0023 0.0541 0.0767 8 0 . 0 1 2 2 0 .0384 0 .0783 -0.0101 0 . 0 4 9 8 0.0985 0.0152 0.0603 0.0809 9 0 .0113 0.0585 0 .0878 -0.0010 0 .0416 0 . 0 9 3 5 -0.0032 0 .0464 0.0723

10 0 .0131 0 .0435 0.0804 0.0021 0 .0343 0 . 0 7 9 1 -0.0020 0 .0199 0.0508

models , including the standard G A R C H model , is re jected by this result. Basical ly, the

same holds for the yen rates. I f the D M and the dollar drop against the yen, volat i l i ty

increases less than i f they rise by the same amount.

Residual statistics are g iven in Table 7, and the first ten autocorrelat ions o f the

residuals, squared and absolute residuals in Table 8. Consider, first, the autocorrelat ions

o f the residuals and the corresponding Q~ statistics. Al though the Q~ statistics o f the

dollar rates are significant, the linear structure in the residuals is much weaker than in

the returns. N o w consider the autocorrelat ions o f squared and absolute residuals and the

Q2-statistics. By compar ing these values with the corresponding values for the returns

in Table 2 it becomes obvious that the C H A R N mode l o f order one is not able to

cope with the long-range dependence o f squared and absolute returns. This holds for

all rates. As could be expected, only at the first lag the large posi t ive autocorrelat ions

were substantially reduced.

C.M. Hafner / Journal o f Statistical Planning and Inference 68 (1998) 247-269 263

In Bossaerts et al. (1996), the bid-ask spread was used as a second argument in

the volatility function. This was motivated by the fact that spread and volatility are highly correlated. Economically speaking, this is due to the nature of bid-ask spreads. The spread can be considered as a compensation for the market maker, having two components: the transaction costs and the risk component. Risk is higher in less active markets and thus the bid-ask spread widens, because the bank takes the risk of having an open position for a longer time interval than in busy hours. However, although the residual diagnostics in Bossaerts et al. (1996) were improved, the long-memory problem remained. Thus, in the next section the CHARN model is generalized by allowing for higher-order autoregressions.

5. Multiplicative nonparametric ARCH models

To capture the long memory in volatility it is necessary to generalize the nonparametric model described in the preceding section. A direct nonparametric analogue to a GARCH model such as

C t = ~Tt~t,

2 ¢72 a, = g ( t - l , e , - l ) .

with some unknown function g(.) is only possible to estimate iteratively, and conver- gence is highly sensitive to the initialization of g(.) and to the starting value of the conditional variance.

However, a GARCH model can be approximated by an ARCH model of sufficiently high order, so it is natural to think of a purely nonparametric ARCH(q) model. Un- fortunately, estimating a nonrestricted model of high-order raises the 'curse of dimensionality'. Recently, several authors have estimated additive models for the conditional mean, see e.g. Chen and Tsay (1993). This restriction conveys to the variance function, where it is more natural to specify a multiplicative model. By taking logarithms this is transformed to an additive model. It is a suitable and well-known way to ensure that the variance function is always positive. For instance, the EGARCH and stochastic volatility models have this specification.

As noted above, the conditional mean E(yt I ) of DEM/USD is specified as an AR(3) model. The linear mean function is kept, and the mean-zero residual series et : Y t - E(yt[~t-l) is used for volatility models. Now, we write a multiplicative nonparametric ARCH model of order q as

~t ~- tTt~t,

q at = I ' - [ gj (Ct - - j ) , (12)

j=l

with it ~ i.i.d.(0, 1 ) and continuous, bounded and positive functions gj(.). For estimation of at one could use marginal integration as in Linton and Nielsen (1995), but

264 C.M. Ha/her~Journal of Statistical Planning and Injerence 68 (1998) 247-269

for high orders this also runs into the 'curse of dimensionality'. Therefore, (12) is transformed to

q log let[ = ~ log gj(et-/)+ log I~tl, (13)

j = l

and standard estimation techniques for additive models are applied. In general, the error term log ICtl does not have mean zero. If ~t is normally distributed, then E(log ICt[)~ -0.635. The normality assumption is used to simplify this subject, knowing that in most financial applications the conditional distribution is leptokurtic. As experiments showed, however, our results were very robust with respect to changes to various fat-tailed distributions.

Hence, the model can be estimated as

e* = log at + ~*,t (14)

with e* -= log le,[+ 0.635, ~* -z log 13,1+ 0.635 and E ( ~ * ) = 0. For the nonparametric estimation of the functions 9j('), basically two methods are

available: the backfitting procedure of Hastie and Tibshirani (1990), and the estimation based on marginal integration, as in Linton and Nielsen (1995). The integration estimator has the advantage of a closed form expression, which implies that it is noniterative. Also, confidence intervals can be constructed. On the other hand, for large dimensions q and large data sets, the estimator is computationally expensive and the choice of the bandwidths is a delicate problem. We used the backfitting procedure of Hastie and Tibshirani (1990), because efficient implementations in standard packages exist. 3 The backfitting procedure is iterative with the following steps: 1. Initialization: log 9j(et-j) = 0, j = 1 . . . . . q 2. Repeat for j = 1, . . . ,q, 1 . . . . . q . . . .

loggj(et_j)=Sj ( e * - k#j~-~'logyk(et-k)let-J)'

3. until the residual sum of squares { }2 q

- l o g t=q+l

converges. In step 2, Sj(YtIXt) denotes a scatterplot smoother of the response variable Y given X.

As Chen and Tsay (1993) note, the backfitting procedure works in the time-series context if the series is geometrically ergodic and e-mixing, i.e. the dependence structure is not too strong. For the scatterplot smoother in step 2, a standard nonparametric estimator such as kernel or spline estimator can be applied under these conditions. Because no kernel weighted local linear estimate was available for the gain macro in

3 The backfitting macro of S-plus was used, see Venables and Ripley (1994).

C M. Hafner l Journal of Statistical Plannin9 and Inference 68 (1998) 247-269 265

S-plus, we chose the spline method. The simplest case is the following: A response variable Y and a one-dimensional explanatory variable X are observed at i = 1 . . . . . n. Then a spline smoother minimizes

~-~{Y i - -m(x i ) }2+2 {mtt(x)}Zdx, (15) i=1

where m(x) denotes the regression function in the class of cubic polynomials, and 2 a positive smoothing parameter that determines the degree of penalty for a too wiggly function re(x).

As kernel estimates, spline smoothers also can be written as linear functions of the response variable, i.e.

m~ = W,~y,

where tV is a [n×n] weighting or smoother matrix depending on 2, and Y = ( y l . . . . , yn),. Under technical conditions, Silverman (1984) derived the asymptotic equivalent kernel estimate, where the kemel is of higher-order with adaptive bandwidth.

For the choice of the smoothing parameter, Chen and Tsay (1993) suggest a generalized cross-validation criterion. For the model (14), this takes the form

[ 12 GCV(2) EtLq+l e* -- ~"~f=l loggj (e t - j )

= (16) / ; ( n - q ) { 1 - [ l + ~ - ~ f = l ( t r W ; , j - 1 ) ] n

where tr W;,j is the trace of the j th smoother matrix. Eq. (16) is minimized with respect to 2. More generally, a different smoothing parameter 2 can be used for each lag.

An alternative way to determine the smoothing parameter is to fix the degrees of freedom, 1 + ~jq--1 [tr WZj - 1]. This is the case in the S-plus macrogam, where the default number of degrees of freedom is four. This was accepted here. Experiments with smaller or larger degrees of freedom did not improve the results reported in the following.

Models up to order q = 15 were estimated. As an illustration, the resulting plots for MNARCH(5) are shown in Fig. 8. Interestingly, the corresponding estimated functions of higher-order models (not shown) are very similar. Recall that GARCH volatility depends additively on functions of the residuals ct, whereas EGARCH models are multiplicative in the standardized residuals it = ct/~Tt. T h u s , these models cannot be compared directly with the model (12). However, if in the EGARCH model (8) the residuals were standardized by the unconditional standard deviation, it is obvious that the functions log g j(.) would be piecewise linear and downscaled for higher lags. For the estimated MNARCH(5) model (see Fig. 8) this seems to hold quite well for the first lag, whereas for lags two to five the impact of outliers on volatility increases slower than linearly.

266 C.M. Hafner / Journal of Statistical Plannin9 and Inference 68 (1998) 247-269

. , , ,, i

• s I

o • 1 I

~ • ~ ~ s ~ 1

-0,0C6 O0 0.006 0.010 - 0 . 0 ~ 0 0 00~ 0.010 wl w2

w

i s "

"Q,006 0.0 0 . ~ 0.010 "0.006 w3

0.0 0 . 0 ~ 0.010 w4

S e r i e s : r e s 5

s p

I i i i i r l I I r l i b I ~ I o l

-*.0on 0.0 o,oo6 OOlO o ~o ,,oo 6oo ~ o v,S Lag

Fig. 8. Estimated nonparametric functions log 0j ( ' ) ,J = 1 . . . . . 5, for the MNARCH(5) model. The functions are plotted together with confidence intervals given by the S-Plus output, see Venables and Ripley (1994). The bottom right plot shows the ACF of absolute residuals up to lag 1000.

Table 9 Residual diagnostics for selected models

Model ~0 ~1 R 2 (%) Q2(10) Q2(lOO)

GARCH(1,1) 0.18 (0.03) 0.81 (0.02) 4.19 10.84 97.32 IGARCH(1,1) 0.32 (0.03) 0.57 (0.02) 4.02 20.47* 94.34 EGARCH(1,1) 0.21 (0.03) 0.76 (0.02) 3.41 63.94* 121.88

CHARN 0 (0) 2.38 (0.02) 1.66 932.84* 2882.81" MNARCH(5) 0 (0) 0.58 (0.02) 3.87 37.38* 327.36* MNARCH(10) 0 (0) 0.60 (0.02) 4.01 6.33 146.28" MNARCH(12) 0 (0) 0.59 (0.02) 4.03 6.07 128.34 MNARCH(15) 0 (0) 0.58 (0.02) 3.88 5.57 113.35

Note: R2: coefficient of determination of the regression ~t 2 = c~o + ~1 ~2 _~_ OI; Q2(m): Box-Ljung statistic of squared residuals with m lags. An asterisk marks 95% significance.

Res idua l d iagnos t i cs for q = 5 , 10, 12, 15 are repor ted in Tab le 9. As in Pagan and

Schwer t (1990) , p a r a m e t e r es t imates and coefficient o f de t e rmina t i on R 2 are g iven for

the l inear regress ion

g,2 =~0 +~la7 + v,,

C.M. Hafner/Journal of Statistical Planning and Inference 68 (1998) 247 269 267

0.0 1.0 2.0 3.0 4.0 S.O

o

zt 0.0 1.0 2.0 3.0 4.0 5.0

i | i |

Fig. 9. The first 500 estimated DEMAJSD volatilities of CHARN, MNARCH(5) and MNARCH(12) (from top to bottom). The scale of the top panel is (E-01).

with parameters 70 and ~l, and error term vt. OLS estimation is applied and reported standard errors are heteroskedasticity consistent, see White (1980). Ideally, ~0 should be zero and cq one. Finally, the Box-Ljung statistics of the squared residuals are reported.

The results show that MNARCH(q) models with q sufficiently large yield serially uncorrelated squared residuals, even though the memory appeared to be much longer. Considering the R 2 and the Q2(m) statistics, the MNARCH(q) fits are similar to the parametric GARCH fits. The first 500 DEM/USD volatilities estimated by CHARN, MNARCH(5) and MNARCH(12) are plotted in Fig. 9, where again the scales are adjusted for comparison. It becomes obvious that MNARCH models of higher-order can reproduce some of the GARCH volatility behavior with large ft. Low-order models tend to estimate volatility with sudden bursts rather than a smooth dynamic behavior. The additional flexibility, however, allows to detect asymmetries at certain lags. In this case the second lag has an impact on volatility that would not be captured by standard

GARCH models.

6. Summary and conclusion

The demand for more flexible ways of modeling volatility has become important in current research since the well-known leverage effect in stock returns revealed standard

268 C M. Hafner / Journal of Statistical Planning and Inference 68 (1998) 247-269

GARCH models to be misspecified. In this paper, some new nonparametric approaches

are discussed. A nonparametric ARCH model o f order one was fitted to high-frequency

FX rate series via local linear estimation on the basis of a redefined time scale. The

results show negative autocorrelation, represented by a linear mean function, and con-

ditional heteroskedasticity. The volatility function is skewed, so asymmetry is apparent.

By adding more lags one is more able to cope with the long memory in volatility, as

it was shown using multiplicative nonparametric models. These models are relatively

parsimonious and reveal individual nonlinearities of the lagged variables. This stands in

contradiction to standard GARCH models. Particularly, the EGARCH model as one of

the standard parametric models allowing for asymmetry is not favored by the results.

It appears to be a strong assumption that the news impact curve as defined by Engle

and Ng (1993) increases exponentially, because the impact of outliers on volatility is

overweighted. The news impact curves appear to have different shapes for different

lags. This result can be used for subsequent selection and refinement of parametric

models.

Acknowledgements

The author wishes to thank P. Bossaerts, M. Dacorogna, C. Gourirroux, W. H~irdle

and an anonymous referee for helpful comments. Financial support by the Deutsche

Forschungsgemeinschaft is greatfully acknowledged.

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estimating high-frequency foreign exchange rate volatility with nonparametric arch models

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