Journal of Forecasting J. Forecast. 17, 497±514 (1998)

Non-linearity and Exchange Rates MARCELO FERNANDES* Universite Libre de Bruxelles, Belgium

ABSTRACT The conditional heteroscedastic models (CHM) are commonly used to describe the dynamics of nominal exchange rates. However, some investigations have already pointed out that the CHMs are not able to fully explain all non-linearities exhibited by the exchange rate series. This paper analyses the performance of univariate CHMs in modelling the nonlinearities of nominal exchange rate series vis-aÁ-vis the US dollar. Twelve currencies are examined on a weekly basis: The Belgian, Swiss and French francs; the Canadian dollar; the German mark; the Danish and Norwegian kroners; the British and Irish pounds; the Italian lira; the Japanese yen and the Dutch guilder. The CHMs captured in a satisfactory way the volatility clustering of the series and show volatility peaks in historically nervous periods of the international market. Moreover, the results of the BDS tests for whiteness applied to the standardized residuals show the good speci®cation of the models. # 1998 John Wiley & Sons, Ltd. KEY WORDS

conditional heteroscedastic models; BDS test; exchange rates

In the absence of structural models, the conditional heteroscedastic models (CHMs) introduced by Engle (1982) are natural candidates for the econometric analysis of nominal exchange rates since they are able to capture properties such as volatility clustering and leptokurtosis. There are numerous investigations of exchange rates based on CHMs, generally using a GARCH (Generalized AutoRegressive Conditional HeteroscedasticÐBollerslev, 1986) speci®cation. Engle, Ito and Lin (1990) tried to identify the source of volatility clustering by analysing the di€usion of intermarket information. The `meteor shower' hypothesis was shown to be more consistent than the `heat wave' assumption with the actual di€usion process of information, since there was more evidence of intra-daily volatility spill-overs among markets than idiosyncratic volatility shocks. Similar results were also found by Baillie, Bollerslev and Redfearn (1993) when examining the US market of foreign exchange rate in the 1920s. Baillie and Bollerslev (1990) proposed a seasonal GARCH model to describe intra-day exchange rates. They could not reject both `meteor shower' and `heat wave' hypotheses, such that both common and idiosyncratic volatility shocks may exist among di€erent markets. Bollerslev (1990) estimated a multivariate GARCH for some European currencies aiming to evaluate the impact of the European Monetary System (EMS) on the behaviour of exchange * Correspondence to: Marcelo Fernandes, ULB, C.E.M.E. CP 139 Avenue F.D. Roosevelt 50, B-1050 Bruxelles.

CCC 0277±6693/98/070497±18$17.50 # 1998 John Wiley & Sons, Ltd.

Received November 1996 Accepted October 1997

498

Marcelo Fernandes

rates. Surprisingly, the results suggested a more volatile behaviour after the implementation of the EMS. Bollerslev and Engle (1993) found some evidence of co-persistence in the conditional variance between the British pound and the German mark vis-aÁ-vis the US dollar. Hsieh (1989a,b) used a variety of CHMs associated to di€erent non-normal distributions to describe the daily volatility of ®ve major currencies. The CHMs were able to capture all conditional heteroscedasticity of the series, but could not fully explain all singularities in the data. Similar outcomes were also found by Peel and Spreight (1994) when estimating GARCH models to interwar exchange rates. Following Hsieh's (1989a) framework, this paper analyses the performance of univariate CHMs in modelling the non-linearities of nominal exchange rate series vis-aÁ-vis the US dollar. Twelve currencies are examined on a weekly basis: the Belgian (BEF), Swiss (CHF) and French francs (FRF); the Canadian dollar (CAD); the German mark (DEM); the Danish (DKK) and Norwegian kroners (NOK); the British (GBP) and Irish pounds (IEP); the Italian lira (ITL); the Japanese yen (JPY) and the Dutch guilder (NLG). The estimated CHMs capture in a satisfactory way the volatility clustering of the series and show volatility peaks in historically nervous periods of the international market. As expected, quite similar models were selected for the `German mark bloc', composed of currencies more under German in¯uence and the German mark itself. Moreover, the results of the BDS test (Brock et al., 1996) applied to the standardized residuals show that the CHMs were able to fully describe the non-linear structure in most currencies. The remainder of this paper is organized as follows. The next section contains a description of the data and reports some preliminary results as well as the methodology for model estimation, selection and testing. The third section discusses the estimation results of the selected models and the behaviour of the volatility. The fourth section describes the BDS test and its properties when applied to rescaled residuals of CHMs. A bootstrap-based methodology to ®nd more accurate critical values is also conducted. The ®nal section o€ers some conclusions. DATA DESCRIPTION, PRELIMINARY ANALYSIS, AND METHODOLOGY The data series analysed were extracted from the Citibase data bank and correspond to the weekly log-return of the Wednesday closure spot price against the US dollar in the New York Exchange market. The sample ranges from the ®rst week of April 1980 to the last week of May 1992 (634 observations), such that structural breaks due to the EMS crises are avoided. Summary statistics are reported in Table I, where the most surprising feature is clearly the high ®rst autocorrelation coecient. The log-returns of all exchange rates have sample mean around zero as well as almost symmetric, but leptokurtotic, distributions. Given the high ®rst autocorrelation, the ®rst step consists in identifying the best ARIMA model to approximate the conditional mean dynamics. The Augmented Dickey±Fuller (ADFÐ Dickey and Fuller, 1979) test including a constant was used to determine the integration order of each series. The test was computed varying its lag structure from zero to twelve lags. In all cases, the null hypothesis of unit root was easily rejected. In Table II the ADF statistics corresponding to the minimum lag structure necessary to achieve uncorrelated residuals are displayed. Both Akaike and Schwarz's information criteria selected an autoregressive model of order one without constant for all currencies. The estimated coecients are also presented in Table II along their p-values corrected for heteroscedasticity of unknown form (Newey and West, 1987). The Ljung± Box statistic Q2(k) was computed for the squared residuals to test for heteroscedasticity varying k # 1998 John Wiley & Sons, Ltd.

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Table I. Descriptive statistics BEF

CAD

CHF

DEM

DKK

FRF

GBP

IEP

ITL

JPY

Mean Median (104) Std dev. (102) Skewness Kurtosis

3.01 8.08 1.35 ÿ0.16 3.28

0.20 0.00 0.52 1.08 11.35

ÿ3.82 ÿ8.52 1.38 0.27 4.30

ÿ0.68 6.72 1.37 ÿ0.22 3.25

2.81 3.33 1.35 ÿ0.16 3.71

5.25 5.39 1.35 ÿ0.03 3.76

ÿ0.96 4.55 1.51 ÿ0.48 3.59

ÿ4.42 ÿ5.70 1.32 0.20 3.41

6.78 1.07 1.24 ÿ0.16 3.36

ÿ9.77 2.59 1.29 ÿ0.67 4.78

ÿ0.38 4.26 4.3 5.52 1.37 1.18 ÿ0.17 ÿ0.13 3.25 6.23

Autocorrelation function lags 1 2 3 4 8 16

0.30 0.08 0.07 0.03 0.05 0.03

0.17 0.01 ÿ0.06 ÿ0.05 ÿ0.03 0.03

0.22 ÿ0.02 0.04 0.07 0.08 ÿ0.02

0.29 0.06 0.05 0.01 0.06 0.04

0.26 0.03 0.07 0.05 0.09 ÿ0.03

0.28 0.05 0.05 0.01 0.05 ÿ0.02

0.30 0.06 0.06 ÿ0.01 0.07 0.01

0.29 0.12 0.07 0.01 ÿ0.01 0.02

0.27 0.20 0.07 0.05 0.07 0.06 0.02 ÿ0.02 0.07 ÿ0.01 0.03 ÿ0.01

(104)

0.26 0.06 0.07 0.002 0.06 0.02

0.30 0.06 0.05 0.02 0.06 0.002

NLG

NOK

Table II. Preliminary results Currency BEF CAD CHF DEM DKK FRF GBP IEP ITL JPY NLG NOK

ADF ÿ18.41b ÿ21.18b ÿ8.20b ÿ18.71b ÿ19.28b ÿ6.44b ÿ7.61b ÿ6.42b ÿ6.42b ÿ6.48b ÿ19.15b ÿ6.06b

Q2(8)

AR(1) coecient 0.3019 0.1726 0.2562 0.2913 0.2629 0.2981 0.2195 0.2815 0.2985 0.2907 0.2680 0.2009

(0.001) (0.021) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.005)

42.5810 35.4629 18.4606 59.5419 73.5550 53.7352 67.1201 52.8687 49.9497 14.9830 65.9795 96.9928

(0.001) (0.001) (0.002) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.010) (0.001) (0.001)

a Rejection

of the null hypothesis considering the 5% critical value. Rejection of the null hypothesis considering the 1% critical value. P-values adjusted to heteroscedasticity of unknown form in parentheses.

b

from 4 to 20. Although in all cases the null hypothesis of uncorrelated squared residuals was easily rejected, only the results of Q2(8) are displayed in Table II. The last step in this preliminary analysis is to identify the nature of possible non-linearities in each series. I perform the third moment test proposed by Hsieh (1989a), which was designed to determine whether the non-linearity is multiplicative (non-linear in variance) or additive (nonlinear in mean) with respect to the noise. The test is based on the property that, in general, additive and hybrid non-linear processes present non-zero conditional expectation in opposition to the multiplicative processes. Assuming that the conditional expectation F(It71) is twice di€erentiable, a second-order Taylor expansion around zero can be carried out, and it is possible to show that F…Itÿ1 † ˆ 0 ) rxxx …i; j† ˆ # 1998 John Wiley & Sons, Ltd.

E‰xt xtÿi xtÿj Š sx3

ˆ0

8 i; j 4 0 J. Forecast. 17, 497±514 (1998)

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Marcelo Fernandes

Table III. Results of the third moment test Currency

BEF CAD CHF DEM DKK FRF GBP IEP ITL JPY NLG NOK

Raw data

Filtered data

W (1,1)

W (1,2)

W (2,2)

W (1,1)

W (1,2)

W (2,2)

ÿ0.8088 (0.2876) 1.2609 (0.1802) ÿ0.8233 (0.2843) ÿ0.9607 (0.2515) ÿ1.1922 (0.1960) 0.0140 (0.3989) 0.8554 (0.2767) 0.8842 (0.2699) ÿ0.7942 (0.2910) ÿ2.1013 (0.0439) ÿ1.2825 (0.1753) 0.7432 (0.3027)

0.6534 (0.3223) ÿ0.4918 (0.3535) ÿ0.0138 (0.3989) 1.0094 (0.2397) 0.5016 (0.3518) 1.0137 (0.2387) 0.4891 (0.3540) ÿ0.6399 (0.3251) 0.9343 (0.2578) ÿ1.6088 (0.1094) 0.9498 (0.2541) 2.3418 (0.0257)

ÿ1.2004 (0.1941) ÿ0.6794 (0.3167) ÿ0.7697 (0.2967) ÿ1.0426 (0.2317) ÿ0.1505 (0.3944) ÿ0.9311 (0.2586) 1.0139 (0.2386) 1.0837 (0.2218) ÿ0.7026 (0.3117) ÿ1.3201 (0.1669) ÿ0.9586 (0.2520) ÿ0.2557 (0.3861)

ÿ0.8400 (0.2803) 0.6070 (0.3318) 0.4694 (0.3573) ÿ0.8795 (0.2710) ÿ0.7835 (0.2935) ÿ0.2899 (0.3825) 0.5257 (0.3475) 0.8340 (0.2818) ÿ0.7727 (0.2960) ÿ0.4409 (0.3620) ÿ1.2142 (0.1909) 0.4486 (0.3608)

0.6685 (0.3223) ÿ0.3131 (0.3799) 0.3402 (0.3765) 0.9974 (0.2426) ÿ0.5217 (0.3482) 0.6826 (0.3160) ÿ0.2895 (0.3826) ÿ0.7188 (0.3081) 0.5901 (0.3352) ÿ0.3294 (0.3779) 0.7627 (0.2983) 1.7978 (0.0793)

ÿ1.0814 (0.1941) ÿ1.3014 (0.1711) ÿ0.1435 (0.3949) ÿ0.7887 (0.2923) 0.5451 (0.3439) ÿ0.8035 (0.2889) 0.6357 (0.3260) 0.7069 (0.3107) ÿ0.4577 (0.3593) 0.2334 (0.3882) ÿ0.7598 (0.2989) ÿ0.4050 (0.3675)

P-values of the test statistic are in parentheses.

Under some mild regularity conditions, the third sample moment rxxx is asymptotically normal with zero mean and variance consistently estimated by V…r† ˆ T2

Sx2t x2tÿi x2tÿj ‰Sx2t Š3

p Therefore, the statistic W(i, j) ˆ TrxxxV71(r) is asymptotically standard normal under the null hypothesis of multiplicative non-linearity. According to the Monte Carlo simulations carried out by Hsieh (1989a), the third moment test has higher power against non-linear moving average, threshold autoregressive models and the tent map for i, j ˆ 1, 2. However, the test does not seem to have power against hybrid models in small samples. Table III exposes the results obtained by the third moment test. After linear ®ltering, the Norwegian kroner is the only currency for which there is some weak evidence of additive non-linearity. The di€erences between the results in the raw and ®ltered data indicate that the third moment test is quite sensitive to linear dependence. For instance, after performing the linear ®lter, the Japanese yen no longer appears to be characterized by additive non-linearity. Although the theoretical invariance of the asymptotic distribution of bicovariance-based statistics when applied to estimated residuals has not been established, the simulations carried out by Hsieh # 1998 John Wiley & Sons, Ltd.

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Non-linearity and Exchange Rates

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Table IV. Typology of conditional heteroscedastic models

AR(1)-GARCH(1,1) AR(1)-IGARCH(1,1)

Model

Reference

yt ˆ ryt71 ‡ et ht ˆ o ‡ ae2tÿ1 ‡ bht71 yt ˆ ryt71 ‡ et

Bollerslev (1986) Engle and Bollerslev (1986)

ae2tÿ1

AR(1)-EGARCH(1,1) GARCH-EAR(1,1) AR(1)-GARCH(1,1)-M

‡ (1 7 a)ht71 ht ˆ o ‡ yt ˆ ryt71 ‡ et ÿ1=2 ÿ1=2 j ‡ g2et71htÿ1 ln ht ˆ g0 ‡ g1 j et71htÿ1 ÿ2 yt ˆ r* exp(ÿhtsy )yt71 ‡ et ht ˆ o ‡ ae2tÿ1 ‡ bht71 p

Nelson (1991) ‡ g3 ln ht71

yt ˆ ryt71 ‡ f htÿ1 ‡ et

Engle, Lilien and Robins (1987)

ae2tÿ1 ‡ bht71 p

AR(1)-IGARCH(1,1)-M

ht ˆ o ‡ yt ˆ ryt71 ‡ f htÿ1 ‡ et

AR(1)-EGARCH(1,1)-M

ht ˆ o ‡ ae2tÿ1 ‡ (1 7 a)ht71 p yt ˆ ryt71 ‡ f htÿ1 ‡ et

AR(1)-MACH(1)-L

ÿ1=2 ÿ1=2 ln ht ˆ g0 ‡ g1 j et71htÿ1 j ‡ g2et71htÿ1 ‡ g3 ln ht71 yt ˆ ryt71 ‡ et

ht ˆ o ‡

ÿ1=2 let71htÿ1

LeBaron (1992)

Yang and Bewley (1992)

(1989a) have indicated quite conservative nominal sizes for autoregressive and moving average processes. Table IV presents the group of eight CHMs that were estimated for each currency. The IGARCH (Integrated GARCH) is characterized by the persistence of volatility shocks even in the in®nite horizon. The EGARCH (Exponential GARCH) de®nes an asymmetric response of the volatility with respect to the sign of the past errors (leverage e€ect). The GARCH-M (GARCH in Mean) and the GARCH-EAR (GARCH Exponential Autoregressive) specify the conditional mean as a function of the volatility. As GARCH-M models are more suitable for a risk premium analysis relating spot and forward exchange rates, the GARCH-EAR is a more interesting candidate for currencies characterized by non-linearities both in mean and variance. Finally, the MACH-L (Linear Moving Average Conditional Heteroscedastic) assumes transitory volatility shocks, such that there is no di€erence between the unconditional and the conditional expectation of the one-step-ahead volatility. In all models, only the ®rst two conditional moments of the error term were speci®ed, zero and ht , respectively, such that quasi-maximum likelihood methods are invoked for estimation purpose. The model selection was based on the signi®cance of the interesting parameters in each model and the Schwarz's criteria. After choosing the appropriate model, the Iterated Cumulative Sums of Square (ICSSÐIÂnlan and Tiao, 1993) algorithm was applied to the standardized residuals ethtÿ1=2 to detect possible sudden changes in the unconditional variance. When necessary, the previously selected model was re-estimated introducing dummy variables in the conditional variance equation to take the detected instabilities into account. This procedure was originally proposed by Fernandes and Monteiro (1997) to analyse the persistence in volatility, since the latter can be # 1998 John Wiley & Sons, Ltd.

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Marcelo Fernandes

overestimated if instabilities in the unconditional variance exist and are not explicitly considered. Finally, the Ljung±Box test for both level and squares of the ®nal standardized residuals was conducted to detect additional serial correlation and heteroscedasticity, respectively. CONDITIONAL HETEROSCEDASTIC MODELS FOR EXCHANGE RATES In this section I discuss the results obtained by the estimation of the CHMs and link them to historical events to shed some light on certain topics of interest. More speci®cally, three issues are covered: the `German mark bloc', the estimated volatility peaks, and the ecient market hypothesis. The selected models for each currency are ®rst described and then the events behind the volatility peaks are presented. Since the ®rst-order autoregressive coecient is still signi®cant in almost all cases, the possibility of ineciencies in the foreign exchange market is examined. Con®rming the results of the third moment test, the Norwegian kroner is best approximated by the hybrid GARCH-EAR(1,1) model. Figure 1 illustrates the negative correlation between the volatility and the autoregressive component of the Norwegian kroner. A quick glance at the plot also shows that the average autoregressive term is about twice the previous estimate obtained by the linear autoregressive model. A dummy variable, assuming one when 07/11/90 4 t 4 27/05/ 92, was introduced to consider an instability in the unconditional variance detected by the ICSS algorithm.

Figure 1. The Norwegian kroner: autoregressive component and estimated volatility # 1998 John Wiley & Sons, Ltd.

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Non-linearity and Exchange Rates

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The German mark, the French franc and the Dutch guilder are modelled by an AR(1)EGARCH(1,1) without the asymmetric e€ect of the innovations. One explanation for this symmetry is the double nature of the exchange market. A nominal depreciation of a currency higher than expected does not clearly indicate a favourable or unfavourable situation in the market. The fact that the coecients of these currencies are quite similar indicate once again the strong relationship among them. The model selection procedure resulted in an AR(1)-GARCH(1,1) model for the Danish kroner and the Belgian franc, such that the volatility dynamics are not so di€erent from the one estimated for the German mark. Again, the coecients of these two currencies are very close. The half-life of volatility shocks in both currencies is around six weeks, hence the persistence in volatility is not so high. The similar structure for these ®ve currencies was expected since the BEF, the DKK, the FRF and the NLG are the currencies historically more under the in¯uence of the German monetary policyÐforming the `German mark bloc'. The graphs presented in the Appendix con®rm the close relationship among these currencies. The Irish pound is also best described by an AR(1)-EGARCH(1,1) model without the leverage e€ect. The Swiss franc is the only major currency not belonging to the `German mark bloc' to have low persistence in volatility (half-life around 6 weeks). The Japanese yen, the British pound and the Italian lira are characterized by complete persistence in volatility since they are better described by an AR(1)-IGARCH(1,1) model. Despite the same speci®cation, the estimated parameters are very di€erent, pointing out singularities in each currency. Finally, the Canadian dollar is the only currency where the log-returns seem uncorrelated. As the ICSS algorithm suggested, two dummies were incorporated into the GARCH(1,1) model selected for the Canadian dollar (D1t ˆ 1 for 10/11/82 4 t 4 01/05/85 and D2t ˆ 1 for 08/05/85 4 t 4 27/05/92) (see Table V). The plots of the estimated volatilities in the Appendix clearly deserve a more detailed analysis. Note that, until the end of 1983, the EMS members were not coordinating their monetary policies, therefore realignments were occurring approximately every eight months. Thus, the volatility peaks of the European currencies tend to be a consequence of country-speci®c events. After 1984, the environment became more favourable to spill-over e€ects among the European currencies with the introduction of the homogenous band system. Therefore, the volatility peaks began to coincide more often. In June 1985, the bankruptcy of two banks in Alberta jeopardized the credibility and stability of the Canadian bank sector, clearly a€ecting the Canadian dollar. The extremely high volatilities of almost all currencies in the beginning of the second quarter of 1985 re¯ect the great tension of the international market. The US dollar was highly overvalued as a consequence not only of the substantial ®scal expansion and rigorous monetary policy adopted by the US authorities but also of a possible `bubble' (Krugman, 1986). Since both Germany and Japan enjoyed high gains in productivity at this time, the USA was facing serious trade de®cits. As a depreciation of the US dollar was feared to slow down the world growth rate, the Group of Seven tried to establish the necessary coordination for its realignment. Even after the collapse of the `bubble', the stress was not over, since the major central banks were selling about $4 billion in this period. In the United Kingdom and Ireland, the in¯ation rates were well above those expected, complicating the situation even more. The highest volatility levels were reached in January 1986, re¯ecting the problems faced by the US authorities to adjust the balance of payments. The US dollar depreciation was not e€ective as the foreign ®rms preferred to reduce their mark-ups to maintain competitiveness with US products. By the same token, country-speci®c events also help to explain these peaks in # 1998 John Wiley & Sons, Ltd.

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Table V. Estimation results BEF r

CAD

0.3126 (0.01)

r* 5

o(10 ) a b g0

2.0372 (0.35) 0.1381 (0.25) 0.7445 (0.00)

8.4120 (0.00) 0.3449 (0.00) 0.3435 (0.00)

4.8440 (0.44) 3.0501 (0.69)

ÿ2.531 (0.00) 2.3001 (0.00) 8.7710 (0.12) 4.3745 (0.50)

CHF

DEM

DKK

FRF

GBP

IEP

ITL

0.2595 (0.04)

0.3054 (0.01)

0.3282 (0.01)

0.3110 (0.01)

0.2727 (0.04)

0.2937 (0.01)

0.3162 (0.02)

0.2499 0.3041 (0.02) (0.01)

0.6176 (0.28) 0.2126 (0.06)

1.9923 (0.00) 0.3399 (0.00)

3.0874 (0.52) 0.1055 (0.07) 0.7527 (0.01)

g1 g3 d1(105) d2(105) Q(8) Q2(8)

4.9029 (0.43) 3.1310 (0.68)

ÿ3.127 (0.09) 0.3425 (0.00) 0.6726 (0.00)

4.2348 (0.52) 8.8471 (0.12)

2.4793 (0.07) 0.1856 (0.03) 0.6462 (0.01)

4.8976 (0.00) 3.4584 (0.63) 2.2941 (0.81)

3.7232 (0.04) 0.4930 (0.00) ÿ3.111 (0.10) 0.3765 (0.09) 0.6783 (0.00)

3.9299 (0.56) 8.5136 (0.13)

ÿ3.017 (0.11) 0.3017 (0.04) 0.6831 (0.00)

8.4073 (0.14) 4.9226 (0.43)

5.0730 (0.41) 6.5314 (0.26)

JPY

NLG

ÿ4.128 (0.07) 0.3891 (0.04) 0.5615 (0.03)

6.3060 (0.28) 4.3892 (0.49)

7.3050 4.9170 (0.20) (0.43) 10.417 9.9410 (0.07) (0.08)

NOK

0.8363 (0.00) 1.2783 (0.04) 0.2090 (0.02) 0.6694 (0.00)

5.4813 (0.00) 4.7498 (0.45) 5.5504 (0.35)

The parameters d1 and d2 correspond to the dummy variables suggested by the ICSS algorithm. The ®rst CAD dummy assumes a value of one when 10/11/82 4 t 4 01/05/85 and zero otherwise. The second CAD dummy has unity values when 08/05/85 4 t 4 27/05/92. The DKK dummy is equal to one when 26/12/90 4 t 4 27/05/92. The NOK dummy is very similar, being one when 07/11/90 4 t 4 27/05/92 and zero otherwise. P-values corrected for heteroscedasticity of unknown form are in parentheses.

volatility. The Bank of Canada intervened several times in the exchange market believing that speculative attacks were keeping the CAD undervalued after the depreciation of the US dollar. In the United Kingdom, the situation was aggravated by the OPEC decision to abolish the constraint on oil production, which compelled the British authorities to devalue the sterling to maintain the competitiveness of their products. In October 1989 there was another volatility peak in the yen possibly due to reverting the expectations with respect to US policy to strengthen the dollar. Interestingly enough, the European currencies exhibited a very stable behaviour during the beginning of 1987 until the end of 1990, such that the EMS did not need to promote any realignment. This stability was very important for EMS credibility since this period was quite turbulent, comprising the crash of the international stock market in October 1987 and the Gulf War in the ®rst quarter of 1990. The credibility of the EMS also helped to inhibit a more volatile behaviour in 1991, when Germany was facing huge expenditures due to the reuni®cation process. The `bubble' surrounding the US dollar in 1985 is also causing the ®rst-order autoregressive coecient to be signi®cant in all currencies except the Canadian dollar. A formal bubble test (West, 1987; Flood and Garber, 1994) requires a well-speci®ed theoretical model of expectations, which is clearly outside the scope of this article. Nevertheless, the signi®cance of the autoregressive parameter can also be understood as a consequence of the expected depreciation rate of the US dollar in 1985, which is not incompatible with the ecient market hypothesis (Hodrick, 1987). # 1998 John Wiley & Sons, Ltd.

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ADDITIONAL NON-LINEARITIES AND HIDDEN STRUCTURES The CHMs are able to capture volatility clustering as well as most leptokurtosis even when a conditional normal distribution for the errors is considered. However, a more careful analysis is necessary to test whether the CHMs were able to remove all non-linearities in the series. If the model is well speci®ed, the rescaled disturbances should be independent and identically distributed (IID). The BDS test developed by Brock et al. (1996) is appropriate to test deviations from the IID property as it has sucient power against a wide class of linear, non-linear, and non-stationary models. The BDS test statistic is based on the correlation integral, which establishes a measure of spatial correlation between two vectors. The correlation integral is de®ned for xm t ˆ (xt , xt‡1 , . . . , xt‡m71) as X 1 m m I…jxi ÿ xj j 5 e† T!1 …T ÿ m†…T ÿ m ‡ 1† i6ˆj

Cm …e† ˆ lim

i 6ˆ j

where T stands for the number of observations and I(
) is the indicator function. Brock et al. (1996) showed that Cm(e) ˆ C1(e)m under the null hypothesis of the i.i.d. process. Strong consistency and asymptotically standard normality are also proven for the test statistic T

BDSm …e† ˆ

p CTm …e† ÿ CT1 …e†m T sTm …e†

where sTm (e) is a non-trivial function of the correlation integral. The theoretical aspects of the choice of the dimension m and the smoothing parameter e are not well established. However, it is clear that e must be large in high dimensions, otherwise there will be too few observations for computing the statistic with a minimum level of precision. Moreover, some smooth ®lters can be applied to the p data without altering the distribution of the BDS statistic as long as the parameters can be T-consistently estimated (de Lima, 1996). However, this nuisance parameter-free property does not hold in the speci®c case of standardized residuals of CHMs. The non-zero variance of the coecient estimators a€ects the sampling distribution of the BDS statistic. More formally, using a ®rst-order Edgeworth expansion, it is possible to decompose the statistic in the following terms: T

BDSm …e† ˆ BDSm …e† ‡ yT mT The ®rst term on the right-hand side is just the statistic evaluated in the true parameter vector, which converges in distribution to a standard normal. The second term is related to the nuisance parameter bias, where yT is the vector of estimated parameters and mT standards for the departure of certain moment conditions. In the case of rescaled residuals of CHMs, yT converges in distribution to a normal with zero mean and positive ®nite variance, while mT converges in probability to a constant vector. Hence, the asymptotic distribution of BDSTm (e) should be normal with zero mean and variance greater than one. However, Monte Carlo simulations presented in Brock, Hsieh and LeBaron (1991) indicates that the distribution of the BDS statistic when applied to standardized residuals of GARCH models depends signi®cantly on the sample size and the dimension m. In small samples, the distribution of the BDS statistic has a variance well above unity and is quite leptokurtotic. Moreover, the distribution seems to deteriorate considerably whenever higher dimensions are # 1998 John Wiley & Sons, Ltd.

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considered. In large samples, increasing m does not seem to a€ect the distribution signi®cantly if an appropriate smoothing parameter is used. On the other hand, the variance is surprisingly well below unity, which can be attributed to the in¯uence of higher-order terms in the Edgeworth expansion. For instance, in a sample size of 500 observations the distribution seems to be fairly normal with a variance slightly below unity for low dimensions. When the dimension is high (m ˆ 10), the variance is close to unity, but excess kurtosis becomes a problem. Since the ®nite sample distribution of the BDS statistic when applied to CHM's rescaled innovations is not clear, I performed a non-parametric bootstrap method to determine the critical values for the test. The motivation was to capture the e€ects of higher-order terms in the Edgeworth expansion that might be playing a role in the ®nite sample distribution of the test (Efron and Tibshirani, 1986). One thousand samples were randomly drawn, with replacements, from the empirical distribution of the standardized residuals corrected by the degrees of freedom. Therefore, it was possible to construct a distribution for the BDS statistic under the null hypothesis of i.i.d. series. This simple, but computer-intensive, exercise is not exactly appropriate to investigate the e€ects of nuisance parameters on the test distribution, but suces in the task of providing more precise critical values. The asymptotic approximation for the distribution of the BDS statistic was shown to be not a€ected by skewness and excess kurtosis, but rather by bimodality and platykurtosis (Brock, Hsieh and LeBaron, 1991). Table VI shows that an extreme leptokurtosis and a very small variance characterize the rescaled innovations of the Canadian dollar. The distribution of the JPY disturbances seems to be heavy tailed (the same applies to the NOK residuals) with some Table VI. Distributions of the standardized residuals

Mean Median Variance Skewness Excess kurtosis Interquartile interval Interdecile interval Upper tail index Lower tail index

BEF

CAD

CHF

0.0329 0.0363 1.0117 ÿ0.0446 0.1239 1.3275 2.5505 1.8395 2.0076

0.0037 0.0000 0.1662 0.7154 4.2817 1.1085 2.1504 2.0839 1.8050

ÿ0.0188 ÿ0.0390 0.8681 0.1469 0.5668 1.2514 2.4631 2.1597 1.8065

GBP Mean Median Variance Skewness Excess kurtosis Interquartile interval Interdecile interval Upper tail index Lower tail index

0.0197 0.0412 1.0509 ÿ0.2763 0.4065 1.2556 2.4985 1.8425 2.1640

IEP ÿ0.0331 ÿ0.0469 1.0078 0.0721 0.3275 1.3498 2.4275 1.9097 1.7054

ITL 0.0642 0.0814 0.9345 0.0180 0.0571 1.2658 2.6450 1.8857 2.3515

DEM

DKK

FRF

0.0061 0.0382 1.0085 ÿ0.0924 0.1259 1.3821 2.5195 1.7762 1.8695

0.0325 0.0224 1.0182 ÿ0.0162 0.0293 1.3597 2.6098 1.9129 1.9262

0.0368 0.0321 1.0070 0.0722 0.3915 1.2951 2.5173 1.8651 2.0281

NLG

NOK

0.0106 0.0360 1.0084 ÿ0.0328 0.2524 1.3438 2.5110 1.8440 1.8931

0.0545 0.0544 1.1558 0.0105 1.2545 1.2513 2.4194 1.8992 1.9738

JPY ÿ0.0382 0.0680 0.8721 ÿ0.3374 1.5637 1.1481 2.3804 1.8773 2.2610

The interquartile and interdecile intervals were normalized by the variance of the distribution. These statistics should take the values 1.35 and 2.56, respectively, in the normal case. The upper (lower) tail index was computed considering the ratio of the ®rst (last) decile over the ®rst (third) quartile. Both indexes should be 1.90 in the normal distribution.

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kind of distortion in the lower tail (as the ITL residuals) and slightly asymmetric. The other standardized residuals are fairly normal. In the absence of bimodality and thin tails, the distribution of the BDS statistic resulting from the bootstrap procedure should not diverge too much from the asymptotic one. For space reasons, it is not feasible to show the bootstrapped distribution of the BDS statistic for each currency, dimension, and smoothing parameter. Table VII reports the empirical size of the asymptotic 5% critical values in order to give an idea of the size distortions. In what follows, I summarize the other results. First, the bootstrapped distributions were not always close to the one reported in Brock, Hsieh and LeBaron (1991). These di€erences are natural since their Monte Carlo simulations were based on the standardized residuals of a speci®c GARCH process (a ˆ 0.1 and b ˆ 0.8). Skewness, leptokurtosis, and tail distortion increase monotonically as m grows, but much more faster than expected considering their results. These deviations from the Table VII. Empirical sizes of the BDS test with 5.0% nominal size e

m

BEF

CAD CHF

1.5s

2 3 4 5 6 7 8 9 10 11 12

4.2 4.9 5.5 5.1 6.1 6.6 7.5 7.9 7.4 7.8 8.5

5.0 5.3 5.5 5.8 6.1 6.5 5.3 5.2 5.3 5.1 5.1

5.4 4.9 5.1 5.4 5.6 5.2 5.4 5.5 6.0 6.2 5.7

6.3 6.3 6.3 6.0 6.0 5.5 5.8 5.5 5.0 5.2 5.8

s

2 3 4 5 6 7 8 9 10 11 12

4.9 5.6 6.1 6.1 6.8 7.3 8.8 10.7 13.7 17.9 25.2

5.3 5.9 5.5 6.1 6.2 6.1 4.9 3.8 3.6 3.4 3.6

5.0 5.7 5.9 6.2 6.3 6.7 6.6 7.0 7.7 8.9 10.7

0.5s

2 3 4 5 6 7 8 9 10 11 12

6.3 9.6 12.7 19.2 21.3 46.8 61.2 78.5 99.4 100.0 100.0

5.9 6.5 7.5 8.2 10.4 15.2 23.3 33.3 49.4 67.3 81.2

5.4 6.8 8.0 12.4 20.2 32.5 47.1 64.2 85.9 100.0 99.9

# 1998 John Wiley & Sons, Ltd.

DEM DKK

FRF

GBP

6.8 5.1 5.2 4.9 5.0 5.2 4.9 5.0 5.0 4.6 4.2

5.7 5.4 5.0 5.5 6.1 6.5 6.3 6.3 6.4 5.8 5.6

6.6 6.7 6.3 5.4 5.4 5.4 5.8 5.8 5.9 6.4 5.9

6.8 6.6 7.5 7.2 6.1 6.8 8.0 8.3 8.6 10.1 12.7

6.7 5.9 5.8 6.2 6.5 7.0 6.9 8.1 10.2 13.1 16.8

4.9 5.4 4.6 5.7 5.8 6.6 7.0 7.1 7.7 8.2 11.1

7.6 11.8 14.3 20.4 30.3 44.5 59.5 72.9 95.9 100.0 100.0

8.0 10.0 13.1 19.9 31.0 47.0 62.9 78.6 87.2 100.0 100.0

6.8 6.7 8.9 14.1 18.7 30.3 48.4 62.0 81.5 99.7 100.0

IEP

ITL

JPY

NLG

NOK

6.6 5.2 5.8 5.8 5.8 5.7 5.6 5.2 5.8 6.3 6.7

6.1 5.5 5.4 5.3 6.2 6.3 5.3 5.2 5.2 6.3 6.7

4.2 4.2 3.9 4.6 5.2 4.9 5.1 5.5 5.1 4.5 4.2

4.4 4.9 4.9 5.6 4.9 5.4 5.6 5.9 5.7 5.6 5.3

5.1 6.3 6.1 6.8 6.2 5.6 6.3 6.3 6.2 6.3 6.0

6.3 7.7 7.4 5.8 5.0 5.9 6.2 7.1 8.0 8.1 8.5

6.4 6.0 5.7 5.5 4.6 5.0 5.9 6.3 8.6 11.0 13.3

7.0 6.3 6.2 6.3 6.3 6.3 7.1 7.3 8.4 10.4 14.7

4.2 4.6 4.6 4.3 4.9 5.4 5.6 6.5 6.7 7.2 8.6

4.2 5.6 5.3 4.9 4.8 4.8 4.6 6.7 8.0 8.8 11.9

5.3 6.5 5.8 6.7 6.3 6.4 6.6 6.5 7.1 8.2 9.9

8.0 9.0 9.9 12.4 19.5 33.0 48.3 64.7 91.9 100.0 99.3

8.5 9.1 12.2 19.0 29.8 46.1 60.1 72.4 98.2 100.0 100.0

7.4 8.8 12.0 17.2 26.3 39.3 36.6 71.6 98.0 100.0 99.6

4.8 5.8 7.7 10.0 15.9 24.5 38.2 57.7 86.9 98.2 68.2

6.2 9.0 12.2 17.8 26.4 40.5 56.2 71.1 93.0 100.0 100.0

7.4 8.2 9.6 13.7 19.8 32.5 45.9 62.8 84.2 99.7 99.6

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normal distribution are especially signi®cant when the smoothing parameter is small (e ˆ 0.5s), allowing reliable distributions for testing purposes only in low dimensions (m 5 4). When the smoothing parameter is set to the standard deviation, the distribution of the BDS statistic deteriorates only in high dimensions (m 4 8). Therefore serious consideration should be given exclusively to the results obtained for the following sets of parameters (e ˆ 1.5s, 8m), (e ˆ s, m 4 8) and (e ˆ 0.5s, m 4 4). In Table VIII the p-values of the BDS test indicate that, in general, the CHM models are a good approximation for the actual process. Striking evidence of hidden structures was found only in the Canadian dollar, the British pound and the Japanese yen. It is interesting to note that both GBP and JPY were modelled by IGARCH processes, suggesting that some non-linear structures may be generating spurious persistence in volatility. The models for the currencies of Table VIII. Results of the BDS test: bootstrapped p-values e

m

BEF

CAD

CHF

DEM DKK

FRF

GBP

IEP

ITL

JPY

NLG

NOK

1.5s

2 3 4 5 6 7 8 9 10 11 12

0.812 0.912 0.872 0.662 0.434 0.440 0.458 0.500 0.194 0.576 0.580

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.214 0.274 0.596 0.752 0.762 0.836 0.766 0.702 0.706 0.708 0.740

0.372 0.428 0.912 0.626 0.332 0.220 0.186 0.170 0.168 0.148 0.136

0.512 0.730 0.580 0.566 0.488 0.494 0.592 0.630 0.732 0.802 0.850

0.618 0.494 0.870 0.748 0.398 0.270 0.216 0.186 0.190 0.172 0.170

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.396 0.426 0.856 0.764 0.400 0.336 0.292 0.200 0.192 0.164 0.122

0.944 0.656 0.634 0.712 0.898 0.810 0.790 0.742 0.666 0.632 0.574

0.470 0.118 0.062 0.046 0.018 0.004 0.000 0.002 0.004 0.004 0.008

0.668 0.744 0.634 0.310 0.124 0.086 0.072 0.070 0.078 0.080 0.072

0.816 0.886 1.000 0.766 0.664 0.738 0.978 0.796 0.714 0.616 0.542

s

2 3 4 5 6 7 8 9 10 11 12

0.906 0.646 0.856 0.718 0.722 0.946 0.960 0.656 0.560 0.420 0.504

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.070 0.128 0.372 0.464 0.424 0.536 0.598 0.492 0.532 0.630 0.742

0.306 0.362 0.980 0.622 0.380 0.408 0.450 0.550 0.662 0.726 0.700

0.756 0.996 0.784 0.792 0.826 0.866 0.612 0.442 0.478 0.492 0.594

0.610 0.560 0.874 0.502 0.282 0.288 0.280 0.408 0.330 0.446 0.666

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.008

0.298 0.406 0.942 0.672 0.340 0.342 0.336 0.358 0.214 0.142 0.142

0.768 0.590 0.582 0.570 0.756 0.572 0.546 0.524 0.540 0.526 0.470

0.408 0.176 0.098 0.078 0.028 0.006 0.006 0.008 0.016 0.016 0.020

0.330 0.538 0.672 0.356 0.174 0.178 0.098 0.094 0.062 0.080 0.096

0.698 0.938 1.000 0.850 0.778 0.974 0.714 0.532 0.480 0.536 0.748

0.5s

2 3 4 5 6 7 8 9 10 11 12

0.352 0.460 0.166 0.090 0.030 0.048 0.044 0.086 0.148 0.074 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.012 0.012 0.000

0.262 0.426 0.534 0.588 0.170 0.286 0.476 0.634 0.330 0.524 0.578

0.264 0.630 0.624 0.274 0.092 0.052 0.038 0.180 0.158 0.140 0.000

0.876 0.810 0.498 0.290 0.094 0.166 0.512 0.512 0.372 0.918 0.962

0.480 0.554 0.934 0.696 0.116 0.010 0.016 0.078 0.062 0.112 0.000

0.000 0.000 0.000 0.060 0.182 0.162 0.774 0.696 0.778 0.964 0.876

0.712 0.692 0.118 0.054 0.028 0.096 0.068 0.086 0.144 0.078 0.000

0.974 0.788 0.984 0.998 0.912 0.828 0.436 0.562 0.458 0.854 0.908

0.198 0.230 0.522 0.414 0.580 0.398 0.520 0.792 0.524 0.746 0.828

0.394 0.684 0.432 0.176 0.088 0.110 0.088 0.224 0.468 0.846 0.908

0.852 0.740 0.890 0.780 0.544 0.612 0.580 0.950 0.966 0.514 0.314

Italics indicate that the p-value was drawn from a distribution considered to be unreliable.

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the `German mark bloc' seem to be well speci®ed. Remarkably, the results do not change whether one considers the p-values from the BDS statistic when e ˆ s and m 4 8. On the other hand, the other region where the test distribution was considered unreliable (e ˆ 0.5s, m 5 4) rejects more often the null hypothesis of IID series. CONCLUSIONS The very extensive literature on conditional heteroscedastic models is clearly a consequence of their success in capturing important features in ®nancial data. The CHMs are designed to handle the leptokurtosis and the volatility clustering common in these series. Moreover, Nelson (1990) has shown that the CHMs can be interpreted as discrete approximations of a di€usion process. However, the large amount of evidence of non-linearities in the ®nancial data raises the possibility that the CHM standardized residuals may still exhibit hidden structures. This paper is speci®cally concerned with the adequacy of CHMs to approximate the behaviour of nominal exchange rates. In the following, I summarize the main results and present my conclusions. The Norwegian kroner is characterized by non-linearities in both conditional mean and variance, such that the GARCH-EAR model (LeBaron, 1992) was selected. The currencies of the `German mark bloc' are described by a similar model as expected. Volatility shocks seem to be asymptotically persistent in the major currencies not belonging to the `German mark bloc', with the exception of the Swiss franc. The volatility peaks identi®ed by the CHMs correspond to periods of actual instability in the exchange rate market. The beginning of the depreciation of the US dollar in 1985 and the diculties in January 1986 faced by the US authorities in undertaking de®cit cuts are found in these periods. Interestingly, spill-over e€ects in European currencies are more substantial after the monetary policy coordination among EMS members in 1984. Country-speci®c events a€ecting the exchange rates, such as the in¯ation faced by the United Kingdom and Ireland at the end of 1985, also help to explain the volatility peaks. However, these interesting insights provided by the estimation of CHMs are not enough to prove that they are appropriate to model exchange rates in the absence of a theoretical model. Therefore, I carried out the BDS test to determine whether all non-linearities were removed from the series. Since the distribution of the BDS statistic is not exactly known when applied to the standardized residuals of CHMs, a non-parametric bootstrap procedure was performed to generate the test distribution under the null of i.i.d. series for each case. The bootstrapped distributions con®rm, to some extent, the results of the Monte Carlo simulation reported in Brock, Hsieh and LeBaron (1991). Finally, the test results indicate that the estimated CHMs are good approximations in nine of the twelve currencies. The test rejected only the models for the Canadian dollar, the British pound and the Japanese yen. This explains the conclusion of Hsieh (1989a) that conditional heteroscedasticity captures substantially, but not completely, the non-linearities in the data. His article analysed ®ve daily exchange rates; more precisely, the three currencies above plus the Swiss franc and the German mark. Strong evidence of additional non-linearities was encountered only in the sterling and the yen. Furthermore, Hsieh has used Monte Carlo simulations to settle the e€ects of nuisance parameters on the BDS test applied to the rescaled residuals of a speci®c GARCH process (a ˆ 0.25 and b ˆ 0.70). Although the results indicated a fairly normal distribution with variance considerably below one, Hsieh continued the analysis using the asymptotic critical values. As seen above, this procedure can be risky especially in high dimensions. Finally, # 1998 John Wiley & Sons, Ltd.

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the di€erent data frequency also helps in clarifying the divergent results, since temporal aggregation reduces non-linearities in general (Granger and Lee, 1989). APPENDIX: PLOTS OF ESTIMATED VOLATILITIES Appendix graphs A, B and C appear on pages 512±514.

ACKNOWLEDGEMENTS I am indebted to Renato FloÃres, Guido Friebel, Jo~ao Victor Issler, Marcos B. Monteiro, PierreYves Preumont, Ilton dos Santos, Luiz Schymura and Reinaldo C. Souza for valuable comments. I acknowledge with gratitude the ®nancial support from Conselho Nacional de Desenvolvimento CientõÂ ®co e TecnoloÂgico, CNPq-Brazil (grant 200608/95-9). The usual disclaimer applies. REFERENCES Baillie, R. T. and Bollerslev, T., `Intra-day and inter-market volatility in foreign exchange rates', The Review of Economic Studies, 58 (1990), 565±85. Baillie, R. T. and Bollerslev, T., `The long memory of the forward premium', Journal of International Money and Finance, 13 (1994), 565±71. Baillie, R. T., Bollerslev, T. and Redfearn, M. R., `Bear squeezes, volatility spillovers and speculative attacks in the hyperin¯ation 1920s foreign exchange', Journal of International Money and Finance, 12 (1990), 511±21. Bollerslev, T., `Generalized autoregressive conditional heteroskedasticity', Journal of Econometrics, 31 (1986), 307±27. Bollerslev, T., `Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH', The Review of Economics and Statistics, 72 (1990), 498±505. Bollerslev, T. and Engle, R. F., `Common persistence in conditional variances', Econometrica, 61 (1993), 167±186. Brock, W. A., Dechert, W., Scheinkman, J. A. and LeBaron, B., `A test for independence based on the correlation dimension', Econometric Reviews, 15 (1996), 197±235. Brock, W. A., Hsieh, D. A. and LeBaron, B., Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence, Cambridge, MA: The MIT Press, 1991. de Lima, P., `Nuisance parameter free properties of correlation integral based statistics', Econometric Reviews, 15 (1996), 237±59. Dickey, D. N. and Fuller, W. A., `Distribution of the estimators for autoregressive time series with a unit root', Journal of the American Statistical Association, 74 (1979), 427±31. Efron, B. and Tibshirani, R., `Bootstrap methods for standard errors, con®dence intervals and other measures of statistical accuracy', Statistical Science, 1 (1986), 54±77. Engle, R. F., `Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom in¯ation', Econometrica, 50 (1982), 987±1007. Engle, R. F. and Bollerslev, T., `Modelling the persistence of conditional variances', Econometric Reviews, 5 (1986), 1±50. Engle, R. F., Ito, T. and Lin, W. L., `Meteor showers or heat waves? Heteroskedastic intra-daily volatility in the foreign exchange market', Econometrica, 58 (1990), 525±42. Engle, R. F., Lilien, D. M. and Robins, R. P., `Estimating time-varying risk premia in term structure: the ARCH-M model', Econometrica, 55 (1987), 391±408. # 1998 John Wiley & Sons, Ltd.

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Fernandes, M. and Monteiro, M. B., `Um procedimento para anaÂlise da persisteÃncia na volatilidade', The Brazilian Review of Econometrics, 17(1), 1997, 15±43. Flood, R. P. and Garber, P. M., Speculative Bubbles, Speculative Attacks and Policy Switching, Cambridge, MA: The MIT Press, 1994. Granger, C. W. J. and Lee, T. H., `The e€ect of aggregation on nonlinearity', Department of Economics Working Paper 89-43, The University of California at San Diego, 1989. Hodrick, R. J., The Empirical Evidence on the Eciency of Forward Exchange Markets, London: Harwood Academic Publishers, 1987. Hsieh, D. A., `Testing nonlinear dependence in daily foreign exchange rates', Journal of Business, 62 (1989a), 339±68. Hsieh, D. A., `Modeling heteroscedasticity in daily foreign-exchange rates', Journal of Business and Economic Statistics, 7 (1989b), 307±17. IÂnclan, C. and Tiao, G. C., `Use of cumulative sums of squares for retrospective detection of changes of variance', Journal of the American Statistical Association, 89 (1993), 913±23. Krugman, P., `Is the strong dollar sustainable?', in The U.S. DollarÐRecent Developments, Outlook, and Policy Options, Federal Reserve Bank of Kansas, 1986, 103±32. LeBaron, B., `Some relations between volatility and serial correlations in stock market returns', Journal of Business, 65 (1992), 297±303. Nelson, D. B., `ARCH models are di€usion approximations', Journal of Econometrics, 45 (1990), 1±17. Nelson, D. B., `Conditional heteroskedasticity in asset returns: a new approach', Econometrica, 59 (1991), 347±71. Newey, W. K. and West, K. D., `A simple, positive semi-de®nite, heteroskedasticity and autocorrelation consistent covariance matrix', Econometrica, 59 (1987), 819±47. Peel, D. A. and Spreight, A. E. H., `Testing for non-linear dependence in inter-war exchange rates', Weltwirtschaftliches-Archiv, 130 (1994), 391±417. West, K., `A speci®cation test for speculative bubbles', Financial Research Memorandum 58, Princeton University, 1985. Yang, M. and Bewley, R., `Moving average conditional heteroskedasticity processes', School of Economics Discussion Paper 23, The University of New South Wales, 1992. Author's biography: Marcelo Fernandes is a PhD student at the EÂcole de Commerce Solvay, Universite Libre de Bruxelles. His research interests include non-linear econometrics and non-parametric techniques applied to ®nance. Author's address : Marcelo Fernandes, Ecole de Commerce Solvay, Universite Libre de Bruxelles, Belgium.

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