Forecasting exchange rate volatility

Economics Letters 76 (2002) 59–64 www.elsevier.com / locate / econbase

Forecasting exchange rate volatility Jon Vilasuso* Department of Economics, School of Business & Economics, West Virginia University, P.O. Box 6025, Morgantown, WV 26506 -6025, USA Received 20 May 2001; received in revised form 30 November 2001; accepted 11 December 2001

Abstract Exchange rate volatility forecasts are obtained using a fractionally integrated GARCH model. Gains in forecast accuracy associated with a fractionally integrated model compared to a GARCH or IGARCH model are shown to be substantial in many cases.  2002 Elsevier Science B.V. All rights reserved. Keywords: Long memory; Fractional integration JEL classification: C22; F31

1. Introduction A number of exchange rate volatility ‘stylized facts’ have been documented since the abandonment of the Bretton Woods system of fixed parities more than 25 years ago. First is the phenomenon of volatility clustering where large exchange rate changes are typically followed by other large changes, eventually giving way to more tranquil periods (Baillie and Bollerslev, 1991). And second, periods of high exchange rate volatility have displayed remarkable persistence, in some cases lasting years (Engel and Bollerslev, 1986). The aim of this paper is to estimate conditional volatility models in an effort to capture the salient features of exchange rate volatility, and evaluate the models in terms of out-of-sample forecast accuracy.

2. The empirical model Following Engle (1982), consider the time series y t and the associated prediction error e t ; y t 2 Et21 y t where Et21 is the expectations operator conditioned on time t 2 1 information. A generalized * Tel.: 11-304-293-7871. E-mail address: [email protected] (J. Vilasuso). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00036-8

 2002 Elsevier Science B.V. All rights reserved.

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J. Vilasuso / Economics Letters 76 (2002) 59 – 64

autoregressive conditional heteroskedasticity (GARCH) model is defined such that e t 5 z t st , z t is i.i.d. with zero mean and unit variance, and

s 2t 5 w 1 a (L)e 2t 1 b (L)s 2t

(1)

where w . 0, and a (L) and b (L) are polynomials in the lag operator L(L i x t 5 x t2i ) of order q and p, respectively. Assume that ai , bi $ 0 for all i. Expression (1) can be written in the form of an ARMA(maxh p,qj,q) model:

F (L)e 2t 5 w 1 [1 2 b (L)]vt

(2)

where vt ; e 2t 2 s 2t and F (L) 5 [1 2 a (L) 2 b (L)]. One limitation of the covariance stationary GARCH model for exchange rate studies is that the GARCH model is a short-memory model as a volatility shock decays at a fast geometric rate. The only way the model is able to represent the observed persistence of exchange rate volatility is ‘by approximating a unit root’ (Breidt et al., 1998), which supports the integrated GARCH (IGARCH) specification of Engle and Bollerslev (1986):

F (L)(1 2 L)e 2t 5 w 1 [1 2 b (L)]vt

(3)

The IGARCH model, however, is not an entirely satisfactory description of exchange rate volatility either as one property of the model is infinite memory. That is, in forecasting applications a volatility shock is permanent, affecting forecasts at all horizons. Baillie et al. (1996) have recently introduced the fractionally integrated GARCH (FIGARCH) model

F (L)(1 2 L)d e 2t 5 w 1 [1 2 b (L)]vt

(4)

where 0 # d # 1 is the fractional difference parameter. The FIGARCH model admits greater flexibility in modeling the conditional variance as it nests the covariance stationary GARCH model (d 5 0) and the IGARCH model (d 5 1) as special cases. For intermediate values of d, the FIGARCH model holds that a volatility shock is persistent, but eventually the process is mean reverting as the effects of a shock die out at a slow hyperbolic rate.

3. The persistence of exchange rate volatility Conditional volatility models are fit to daily spot exchange rates vis-a-vis the US dollar for the period 13 March 1979 to 31 December 1997 (this leaves 2 years of data in which to evaluate out-of-sample volatility forecasts). The currencies examined are the Canadian dollar, the French franc, the German mark, the Italian lira, the Japanese yen, and the British pound. All data was obtained from the Federal Reserve Bank of Chicago data files. Unit root tests (not shown) indicate that each nominal exchange rate is I(1), and in the analysis to follow, we work with exchange rate returns. For comparison, we fit alternative conditional heteroskedasticity models to each exchange rate return series. The models include a GARCH(1,1) model, an IGARCH(1,1) model with F1 5 1, and a FIGARCH(1,d,1) model given by

s 2t 5 w 1 b1 s 2t21 1 [1 2 b1 L 2 (1 2 F1 )(1 2 L)d ]e 2t

(5)

Canada

France

Germany

Italy

Japan

UK

FIGARCH GARCH IGARCH FIGARCH GARCH IGARCH FIGARCH GARCH IGARCH FIGARCH GARCH IGARCH FIGARCH GARCH IGARCH FIGARCH GARCH IGARCH

m w

b1 f1 d

20.002

20.002

20.003

20.002

20.005

20.002

20.006

20.005

20.005

20.005

20.006

20.007

0.003

0.003

0.003

0.001

0.002

0.002

(0.003)

(0.007)

(0.007)

(0.006)

(0.006)

(0.006)

(0.007)

(0.025)

(0.010)

(0.004)

(0.003)

(0.025)

(0.006)

(0.005)

(0.007)

(0.007)

(0.008)

0.005

0.002

0.001

0.014

0.005

0.005

0.018

0.007

0.006

0.004

0.001

0.003

0.014

0.003

0.003

0.004

0.006

0.005

(0.001)

(0.000)

(0.000)

(0.002)

(0.001)

(0.001)

(0.002)

(0.002)

(0.002)

(0.000)

(0.000)

(0.001)

(0.001)

(0.001)

(0.001)

(0.001)

(0.001)

(0.001)

(0.009)

0.576

0.856

0.863

0.679

0.877

0.877

0.674

0.889

0.888

0.776

0.830

0.850

0.725

0.923

0.922

0.354

0.916

0.914

(0.039)

(0.031)

(0.033)

(0.029)

(0.029)

(0.027)

(0.028)

(0.024)

(0.022)

(0.019)

(0.060)

(0.065)

(0.018)

(0.028)

(0.022)

(0.008)

(0.070)

(0.081)

0.277

0.976

1.000

0.165

0.993

1.000

0.216

0.988

1.000

0.196

0.995

1.000

0.352

0.990

1.000

0.244

0.989

1.000

(0.031)

(0.006)

(0.018)

(0.010)

(0.021)

(0.010)

(0.024)

(0.025)

(0.021)

(0.013)

(0.024)

(0.010)

0.454

0.632

0.547

0.781

0.502

0.485

(0.037)

(0.037)

(0.035)

(0.039)

(0.029)

(0.029)

Q 2 (20) 10.85

18.45

23.32

10.75

13.39

13.03

12.70

13.12

15.29

12.61

23.51

19.58

21.82

23.88

22.64

12.50

10.23

9.74

Notes: column headings refer to the estimated FIGARCH(1,d,1) model, GARCH(1,1) model, and IGARCH(1,1) model with f1 ;1. Robust standard errors are 2 shown in parentheses. Q (20) refers to a Ljung–Box portmanteau statistic for up to 20th order serial correlation in the squared standardized residuals.

J. Vilasuso / Economics Letters 76 (2002) 59 – 64

Table 1 Maximum likelihood parameter estimates of conditional heteroskedasticity models

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J. Vilasuso / Economics Letters 76 (2002) 59 – 64

Parameter estimates, collected in Table 1, are obtained using (quasi) maximum likelihood under the maintained assumption of conditional normality. Robust standard errors due to Bollerslev and Wooldridge (1992) are shown in parentheses. One complication that arises in estimating the FIGARCH model is that the infinite lag associated with the fractional difference operator shown in Eq. (5) must be truncated. FIGARCH parameter estimates are based on truncating a power series expansion at 250 terms. The conditional mean of each exchange rate return series is modeled as a constant, m. Beginning with the GARCH parameter estimates, Ljung–Box portmanteau statistics for up to 20th order serial correlation in the squared standardized residuals suggest that the GARCH model adequately captures time-varying volatility. For each currency, estimates of F1 are near one. In fact, the null hypothesis F1 5 1 cannot be rejected at conventional significance levels for each currency with the exception of the Canadian dollar. Consequently, there is little discernable difference between the GARCH and IGARCH models for these currencies, consistent with previous research. The dynamics of the conditional variance of currency returns, however, are best represented by the FIGARCH model. The parameter restrictions d 5 0 and d 5 1 corresponding to the stable GARCH and IGARCH models, respectively, are easily rejected. Then tendency for the GARCH model to approximate a unit root is expected if exchange rate volatility is persistent. Using Monte Carlo simulations, Bollerslev and Mikkelsen (1996) show that fitting a mis-specified GARCH model to a true FIGARCH process tends to produce IGARCH parameter estimates. Estimates of the fractional difference parameter range from 0.454 for the Canadian dollar to 0.781 for the Italian lira. Estimated FIGARCH parameters are consistent with a long-memory process, which more realistically describes the dynamic properties of exchange rate volatility.

4. Exchange rate volatility forecasts In this section, we evaluate out-of-sample volatility forecasts for the period 1 January 1998 to 31 December 1999. Out-of-sample volatility forecasts are calculated using the parameter estimates of the conditional heteroskedasticity models shown in Table 1. Volatility forecasts are then compared to the square of the daily exchange rate return, and accuracy is judged based on a mean square error (MSE) and a mean absolute error (MAE) criterion. A 1-, 5-, and 10-day forecast horizon, k, are considered. We also evaluate the null hypothesis of no difference in forecast accuracy associated with the alternative heteroskedasticity models. Specifically, the null maintains that the predictive performance of the FIGARCH model relative to either the GARCH or IGARCH specification is the same. This ]]]]] hypothesis is evaluated using the statistic S1 5 d¯ /œ2pf(v 5 0) /N developed by Diebold and Mariano (1995) where d¯ is the sample mean of the difference in forecast errors, f(v 5 0) is the spectral density function of the forecast error differences evaluated at the zero frequency, and N is the number of forecasts. Under the null of no difference in forecast accuracy, S1 | N(0,1). To compute f(v 5 0), we use the Newey and West (1987) estimator of the covariance matrix with bandwidth parameter k 2 1 for horizons k . 1. Out-of-sample exchange rate volatility forecast errors are summarized in Table 2. For both the MSE and MAE criteria, out-of-sample forecasts generated by the FIGARCH model are superior for each currency at every forecast horizon. Gains in forecast accuracy attributed to the FIGARCH model are substantial. At a 10-day horizon, for example, the reduction in MSE ranges from 8% (512[11.15 /

Table 2 Out-of-sample exchange rate volatility forecasts Mean square error (MSE)

France

Germany

Italy

Japan

UK

1-Day horizon

5-Day horizon

10-Day horizon

1-Day horizon

5-Day horizon

10-Day horizon

MSE

MSE

MSE

S1

MAE

S1

MAE

S1

MAE

S1

S1

S1

FIGARCH GARCH IGARCH

0.15 0.20 0.20

22.64 22.64

0.16 0.25 0.28

22.05 22.01

0.17 0.27 0.27

22.10 22.10

0.16 0.21 0.22

24.61 24.63

0.17 0.20 0.21

22.76 22.70

0.18 0.22 0.22

21.98 21.98

FIGARCH GARCH IGARCH

0.42 0.52 0.52

23.32 23.32

0.53 0.75 0.77

23.82 24.06

0.57 0.77 0.77

22.92 22.92

0.37 0.41 0.42

24.46 24.53

0.36 0.47 0.48

24.31 24.52

0.39 0.48 0.48

23.13 23.10

FIGARCH GARCH IGARCH

0.43 0.54 0.52

23.27 23.18

0.51 0.70 0.70

23.35 23.33

0.52 0.71 0.71

22.32 22.32

0.36 0.41 0.41

25.13 25.14

0.36 0.44 0.44

23.46 23.43

0.37 0.46 0.47

22.29 22.22

FIGARCH GARCH IGARCH

0.44 0.48 0.52

21.87 22.49

0.45 0.71 0.71

23.41 23.41

0.47 0.67 0.67

23.02 23.01

0.41 0.41 0.42

20.23 20.22

0.32 0.45 0.44

23.89 23.81

0.38 0.44 0.44

22.22 22.23

FIGARCH GARCH IGARCH

9.63 13.08 13.21

21.86 21.90

10.35 10.56 10.57

21.29 21.30

11.15 12.17 12.16

23.88 23.88

1.63 1.73 1.75

22.40 22.53

1.42 1.48 1.48

21.67 21.67

1.48 1.63 1.63

23.68 23.69

FIGARCH GARCH IGARCH

0.20 0.26 0.26

22.24 22.24

0.23 0.38 0.39

22.58 22.60

0.24 0.37 0.37

22.85 22.86

0.21 0.25 0.25

22.41 22.50

0.20 0.27 0.28

22.54 22.59

0.21 0.26 0.26

22.94 22.98

J. Vilasuso / Economics Letters 76 (2002) 59 – 64

Canada

Mean absolute error (MAE)

Notes: out-of-sample forecasts cover the period 1 January 1998 to 31 December 1998. Volatility forecasts are based on the estimated parameters of the conditional heteroskedasticity models shown in Table 1. The statistic S1 is used to evaluate the null hypothesis that the forecasting accuracy of the FIGARCH model is the same as either the GARCH or IGARCH model, and is distributed N(0,1).

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12.16]) for the Japanese yen to 37% (512[0.17 / 0.27]) for the Canadian dollar. Reductions in the MAE are comparable. Reductions in the MSE at a 1-day horizon are also substantial even though a short forecast horizon does not necessarily favor the FIGARCH model. The reduction in MSE at a 1-day horizon ranges from 8% (512[0.44 / 0.48]) for the Italian lira to 26% (512[9.63 / 13.08]) for the Japanese yen. Overall, the FIGARCH model produces superior out-of-sample volatility forecasts. For each currency, the relative gains in volatility forecast accuracy associated with the FIGARCH model are statistically significant at the 5-percent level with the exception of the Japanese yen and the Italian lira for a few cases.

5. Conclusion With the move to a flexible exchange rate system in 1973, nominal exchange rate volatility has exhibited remarkable persistence. In this paper, we adopt the FIGARCH model introduced by Baillie et al. (1996) to describe exchange rate volatility dynamics. Overall, the FIGARCH model is better equipped to capture the salient features of exchange rate volatility than are the commonly used GARCH and IGARCH models. And perhaps more important, the FIGARCH model generates superior out-of-sample volatility forecasts, and the gains in forecast accuracy are substantial.

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