Structural Breaks and GARCH Models of Exchange Rate Volatility

Transcription

1 Structural Breaks and GARCH Models of Exchange Rate Volatility David E. Rapach Department of Economics Saint Louis University 3674 Lindell Boulevard Saint Louis, MO Phone: Fax: Jack K. Strauss Department of Economics Saint Louis University 3674 Lindell Boulevard Saint Louis, MO Phone: Fax: November 10, 2005 (Revised) Abstract We investigate the empirical relevance of structural breaks for GARCH models of exchange rate volatility using both in-sample and out-of-sample tests. Employing eight daily U.S. dollar exchange rate return series for , we find significant evidence of structural breaks in the unconditional variance of all eight exchange rate return series using a modified version of the Inclán and Tiao (1994) iterated cumulative sum of squares algorithm that allows for dependent processes. This implies unstable GARCH processes for these exchange rates, and we show that GARCH(1,1) parameter estimates often exhibit substantial variation across the sub-samples defined by the structural breaks. In the out-of-sample analysis, we examine whether we can improve real-time exchange rate volatility forecasts by adjusting the estimation window for GARCH(1,1) forecasting models to accommodate potential structural breaks. The adjustments we consider include rolling windows of various sizes, as well as a method where the estimation window for the GARCH(1,1) forecasting model is determined by applying the modified iterated cumulative sum of squares algorithm to the observations available at the time of forecast formation. Considering two benchmark forecasting models that assume a stable GARCH process a GARCH(1,1) model estimated using an expanding window and the RiskMetrics model we find that it almost always pays to use an estimation window that accommodates potential structural breaks when generating real-time exchange rate volatility forecasts. Overall, our results indicate that structural breaks are an empirically relevant feature of GARCH models of exchange rate volatility. JEL classifications: C22; C53; G10; G12 Key words: Exchange rate volatility; GARCH(1,1) model; Structural breaks; Iterated cumulative sum of squares; Out-of-sample forecasts; Estimation window Corresponding author. We thank Mike McCracken, Chris Neely, Mark Wohar, and seminar participants at the 2005 Midwest Econometrics Group Meetings for very helpful comments. The usual disclaimer applies. The results reported in this paper were generated using GAUSS 6.0.

2 1 1. Introduction Spurred by the seminal contributions of Engle (1982) and Bollerslev (1986), an extensive literature models time-varying volatility in high-frequency asset returns using generalized autoregressive conditional heteroskedastic (GARCH) processes. A number of studies use GARCH models especially GARCH(1,1) models to characterize and forecast time-varying volatility in exchange rate returns; see, for example, Engle and Bollerslev (1986), Baillie and Bollerslev (1989, 1991), Bollerslev and Engle (1993), West and Cho (1995), Neely (1999), and Hansen and Lunde (2004b). Accurately modeling and forecasting time-varying volatility in exchange rates have important implications for financial decisionmaking, including the pricing of derivatives and portfolio risk management in an international setting. In addition, a large body of theoretical research ties exchange rate volatility to trade and welfare. 1 Researchers often assume (explicitly or implicitly) that a stable GARCH process governs conditional exchange rate volatility, so that the unconditional variance of exchange rate returns is constant. However, international financial markets are periodically subject to sudden large shocks, such as the Exchange Rate Mechanism crisis in Europe in the early 1990s and the East Asian crisis of the late 1990s. These types of shocks can cause abrupt breaks in the unconditional variance of exchange rate returns and are equivalent to structural breaks in the parameters of the GARCH processes governing the conditional volatility of exchange rate returns. Recent theoretical studies show that structural breaks have potentially important implications for estimated GARCH models of exchange rate volatility in the extant literature, which are typically highly persistent (almost-integrated). Building on insights from Diebold (1986), Hendry (1986), and Lamoureux and Lastrapes (1990), recent research by Mikosch and Stărică (2004) and Hillebrand (2004) demonstrates that neglected structural breaks in the parameters of GARCH processes induce upward biases in estimates of the persistence of GARCH processes. 2 By failing to 1 See Clark et al. (2004) for a recent survey. 2 Mikosch and Stărică (2004) show this for Whittle estimators, while Hillebrand (2004) shows that this holds for a wide class of estimators, including maximum likelihood and quasi maximum likelihood estimators, the most common estimators of GARCH models in the literature. Mikosch and Stărică (2003, 2004) and Perron and Qu (2004) also show that structural breaks can give rise to spurious evidence of long-range dependence or long memory in financial volatility data.

3 2 account for structural breaks, estimated GARCH models of exchange rate volatility in the extant literature can overstate the degree of persistence in exchange rate volatility. Structural breaks also have potentially important implications for forecasts of exchange rate volatility. In out-of-sample volatility forecasting exercises, the use of an expanding data window (or a fixed data window to reduce computational costs) to estimate GARCH forecasting models is common and appropriate under the assumption of a stable GARCH process. However, this approach is unlikely to perform well in the presence of sudden structural breaks in volatility. Along this line, West and Cho (1995) posit that the forecasting performance of GARCH(1,1) models of exchange rate volatility could be improved by allowing for structural breaks in the unconditional variance of exchange rate returns. In addition, recent research by Stărică et al. (2005) shows that long-horizon forecasts of stock return volatility generated by GARCH(1,1) models assuming parameter stability are often inferior to forecasts that allow for frequent changes in the unconditional variance of stock returns. In light of the above considerations, the present paper investigates the empirical relevance of structural breaks for GARCH(1,1) models of exchange rate volatility using both in-sample and out-ofsample tests. In our in-sample tests, we employ a modified version of the Inclán and Tiao (1994) iterated cumulative sum of squares (ICSS) algorithm that allows for dependent processes. We use the algorithm to test for (potentially multiple) structural breaks in the unconditional variance of daily returns for U.S. dollar exchange rates vis-à-vis seven OECD countries and a trade-weighted U.S. dollar exchange rate. We find significant evidence of structural breaks in the unconditional variance for all eight exchange rate return series. Inspection of the estimated GARCH(1,1) processes across the sub-samples defined by the structural breaks often reveals sharp differences in parameter estimates, and the GARCH(1,1) models fitted to the different sub-samples are sometimes considerably less persistent than models fitted to the entire sample. In our out-of-sample analysis, we consider different methods of accommodating potential structural breaks when forming exchange rate volatility forecasts in real time. We compare these methods against two natural benchmarks that assume a stable GARCH process: (1) a GARCH(1,1) model, where

4 3 an expanding estimation window is used to generate real-time volatility forecasts; (2) the popular RiskMetrics model (which is based on an integrated GARCH(1,1) process), where the RiskMetrics forecasts are generated using an expanding window. We compare forecasts of daily exchange rate return volatility generated by the two benchmark models to forecasts generated by four competing models that make adjustments in the estimation window in order to accommodate potential structural breaks. The first two competing models use rolling windows with sizes equal to equal to one-half and one-quarter of the length of the in-sample period when estimating the GARCH(1,1) forecasting model. In applied work, researchers sometimes use a rolling estimation window to allow the parameters of the GARCH process to evolve over time. The third competing model uses a method where the estimation window for the GARCH(1,1) forecasting model is determined by applying the modified ICSS algorithm to the observations available at the time of forecast formation. The final competing model simply uses the average of the daily squared returns over the previous 250 days. This moving average forecasting model clearly assumes away any GARCH dynamics, but it allows for a frequently changing unconditional variance. Stărică et al. (2005) find that this model consistently outperforms a GARCH(1,1) model estimated using an expanding window with respect to forecasting daily stock return volatility in a large number of industrialized countries. Considering loss functions based on mean square forecast error (MSFE) and Value-at-Risk (VaR), as well as forecast horizons of 1, 20, 60, and 120 days, we find that it almost always pays to use an estimation window that accommodates potential structural breaks when forming out-of-sample forecasts of exchange rate volatility. Overall, our results indicate that structural breaks are an empirically relevant feature of GARCH models of exchange rate volatility. The rest of the paper is organized as follows. Section 2 outlines our econometric methodology. Section 3 presents the empirical results for the in-sample and out-of-sample tests. Section 4 concludes.

5 4 2. Econometric Methodology 2.1. In-Sample Tests Let et = 100log( Et / Et 1), where E t is the nominal exchange rate at the end of period t, so that e is the percent return for the exchange rate from period t 1 to period t. Following West and Cho t (1995), we treat the unconditional and conditional mean of as zero. We consider eight exchange rate e t return series that satisfy this assumption in our applications in Section 3 below. Suppose we observe e t for t = 1,, T and are interested in testing whether the unconditional variance of e is constant over the available sample. A constant unconditional variance implies a stable GARCH process governing conditional volatility, while a structural break in the unconditional variance implies a structural break in the GARCH process as well. Inclán and Tiao (1994) develop a cumulative sum of squares statistic to test the null hypothesis of a constant unconditional variance against the alternative hypothesis of a break in the unconditional variance. The Inclán and Tiao (1994) statistic is given by t IT = T D, (1) 0.5 sup ( / 2) k k 2 where D = ( C / C ) ( k/ T) and C = e for k = 1,, T. The value of k that maximizes k k T k k t= 1 t 0.5 ( T /2) D k is the estimate of the break date. When e t is distributed iid N σ, Inclán and Tiao 2 (0, e ) (1994) show that the asymptotic distribution of the IT statistic is given by sup W ( ) r r, where W () r = W() r rw(1) is a Brownian bridge and W( r) is standard Brownian motion. As demonstrated in Monte Carlo simulations in de Pooter and van Dijk (2004) and Sansó et al. (2004), the IT statistic can be plagued by substantial size distortions when e is not distributed iid t N σ. This will be the case when follows a GARCH process. Kokoszka and Leipus (2000), Kim et 2 (0, e ) e t al. (2000), and Sansó et al. (2004) suggest applying a nonparametric adjustment to the IT statistic that allows e t to obey a wide class of dependent processes, including GARCH processes. Following de Pooter

6 5 and van Dijk (2004) and Sansó et al. (2004), we use a nonparametric adjustment based on the Bartlett kernel. The adjusted IT statistic can be expressed as 0.5 AIT sup T Gk k =, (2) 0.5 where G ˆ k λ m 1 = [ Ck ( k/ T) C ], ˆ ˆ 2 [1 lm ( 1) ] ˆ T λ = γ + + γl, 0 l= 1 1 T ˆ ˆ l = T ( e )( ) t l 1 t e =+ t l, ˆ γ σ σ 2 1 ˆ σ = T C, and the lag truncation parameter m is selected using the procedure in Newey and West T (1994). 3 Under general conditions, the asymptotic distribution of AIT is also given by sup ( ) r W r. Critical values for the AIT statistic can be generated via simulation and are provided in, for example, Sansó et al. (2004, Table 1). In Monte Carlo simulations, de Pooter and van Dijk (2004) and Sansó et al. (2004) find that the AIT statistic has good size properties for a variety of dependent processes, including GARCH processes. Inclán and Tiao (1994) develop an iterated cumulative sum of squares (ICSS) algorithm based on the IT statistic to test for multiple breaks in the unconditional variance; see Steps 0-3 in Inclán and Tiao (1994, p. 916). Alternatively, the ICSS algorithm can be based on the AIT statistic in order to avoid the size distortions that plague the IT statistic when e follows a GARCH process. The ICSS algorithm t based on the AIT statistic begins by testing for a structural break over the entire sample, t = 1,, T, using the AIT statistic. If the AIT statistic is not significant, the data do not support a structural break in the variance of et. If the AIT statistic detects a significant break at, say, t = T1, then the algorithm applies the AIT statistic to test for a break over each of the two sub-samples defined by the break at t = T1 ( t = 1,, T ; t = T1 + 1,, T ). If neither of the AIT statistics is significant for the sub-samples, the data 1 support a single break in the variance over the entire sample. If either of the AIT statistics is significant over the two sub-samples, then the algorithm tests for breaks in the new sub-samples defined by any 3 Andreou and Ghysels (2002) adjust the IT statistic using the VARHAC estimator of den Haan and Levin (1997). We obtain similar results in our empirical applications using this adjustment.

7 6 significant AIT statistic. The algorithm proceeds in this manner until the AIT statistic is insignificant for all of the sub-samples defined by any significant breaks. Andreou and Ghysels (2002), de Pooter and van Dijk (2004), and Sansó et al. (2004) find that the ICSS algorithm based on the AIT statistic generally performs well in extensive Monte Carlo simulations with respect to detecting the correct number of unconditional variance breaks for a variety of GARCH processes. In empirical applications using stock returns in emerging markets, de Pooter and van Dijk (2004) and Sansó et al. (2004) show that the standard ICSS algorithm based on the IT statistic detects an implausibly large number of variance breaks, while the ICSS algorithm based on the AIT statistic selects more reasonable estimates of the number of breaks. In our applications in Section 3 below, we use the ICSS algorithm based on the AIT statistic (the modified ICSS algorithm ) and the 5% significance level to test for multiple breaks in the unconditional variance of eight daily U.S. dollar exchange rate return series. If we detect significant evidence of at least one structural break, then we estimate GARCH(1,1) models over the different regimes defined by the significant structural breaks and compare these models to a GARCH(1,1) model estimated over the full sample. 4 the form, The canonical GARCH(1,1) model for e t with mean zero (conditional and unconditional) takes 0.5 et ht t = ε, (3) h t = + +, (4) 2 ω αet 1 βh t 1 where ε t is iid with mean zero and unit variance. In order to ensure that the conditional variance, positive, we require ω > 0 and α, β 0. The GARCH(1,1) process specified in equations (3) and (4) is h t, is 4 Instead of using a nonparametric adjustment to the IT statistic as in equation (2), we could use a parametric adjustment that assumes e t follows a GARCH(1,1) process, as in Kim et al. (2000) and de Pooter and van Dijk (2004). However, this considerably increases computational costs when testing for multiple structural breaks using the modified ICSS algorithm, as GARCH(1,1) models have to be estimated for the entire sample and any subsamples defined by significant breaks. This especially increases computational costs when, as described in Section 2.2 below, at each forecast date, we use the modified ICSS algorithm to determine the estimation window for a GARCH(1,1) forecasting model. As noted in the text, the nonparametric adjustment used for the AIT statistic performs well in Monte Carlo simulations involving GARCH processes.

8 7 stationary if α + β < 1; when α + β = 1, we have the integrated GARCH(1,1) (IGARCH(1,1)) model of Engle and Bollerslev (1986). For a stationary GARCH(1,1) process, the unconditional variance for e t is given by ω /(1 α β). Note that when α = 0 in equation (4), β is unidentified (and set to zero), so that ht = ω and is characterized by conditional homoskedasticity. The GARCH(1,1) process is typically e t estimated using quasi maximum likelihood estimation (QMLE), where the likelihood function corresponding to ε ~ N (0,1) is used and the restrictions ω > 0 and α, β 0 are imposed. The QMLE t parameter estimates are consistent and asymptotically normal; see, for example, Jensen and Rahbek (2004) Out-of-Sample Tests We compare out-of-sample forecasts of volatility generated by two benchmark forecasting models and four competing forecasting models. The first benchmark model is a GARCH(1,1) model estimated using an expanding window ( GARCH(1,1) expanding window model). More specifically, we divide the sample for a given exchange rate return series into in-sample and out-of-sample portions, where the in-sample portion spans the first R observations and the out-of-sample portion the last P observations. In order to generate the first out-of-sample forecast at the one-period horizon, we estimate the GARCH(1,1) model given by equations (3) and (4) using QMLE and data from the first observation through observation R. The initial forecast is given by 2 hˆ ˆ ˆ ˆ ˆ R+ 1 R, EXP = ωr, EXP + αr, EXPeR + βr, EXPh R, EXP, where ˆ R, EXP ω, ˆR α, EXP, and ˆR β, EXP are the estimates of ω, α, and β, respectively, in equation (4) and h ˆR, EXP is the estimate of h R obtained using data from the first observation through observation R. We then expand the estimation window by one observation in order to form a forecast for period R + 2, h ˆR + 2 R+ 1, EXP. We proceed in this manner through the end of the available out-of-sample period, leaving us with a series of P out-of-sample forecasts, { hˆ 1, } T t t EXP t = R + 1. The GARCH(1,1) expanding window model is a natural

9 8 benchmark model that is appropriate for forecasting when the data are generated by a stable GARCH(1,1) process. The second benchmark model we consider is the RiskMetrics model based on an expanding window, a popular model often included in studies of out-of-sample volatility forecasting performance. The RiskMetrics model is a restricted version of the GARCH(1,1) model in equation (4), with ω = 0, β = 0.94, and α + β = 1, so that the conditional volatility process is assumed to be an IGARCH(1,1) process. 5 Note that the RiskMetrics model does not involve the estimation of any parameters, making it easy to implement. We denote the forecasts for the RiskMetrics model by { hˆ 1, } T tt RM t = R + 1. In addition to its popularity, recent results in Hillebrand (2005) make the RiskMetrics model a relevant benchmark. Hillebrand (2005) shows that under fairly general conditions, if structural breaks in a GARCH(1,1) process are neglected, the estimates of ω and α + β in equation (4) go to zero and one, respectively. Thus, the popular RiskMetrics model specification is what we would expect when structural breaks in volatility are important but neglected. The four competing forecasting models all make adjustments to the estimation window in order to account for potential changes in the unconditional variance of exchange rate returns. The first competing model is a GARCH(1,1) model estimated using a rolling window with size equal to one-half of the length of the in-sample period ( GARCH(1,1) 0.50 rolling window model). The forecasts are formed as described above, with the exception that the GARCH(1,1) forecasting model is estimated using a rolling window with size equal to one-half of the length of the in-sample period; that is, the first forecast uses estimates of equation (4) based on observations 0.5R through R, the second forecast uses estimates based on observations 0.5R + 1 through R + 1, and so on. We denote the forecasts for the GARCH(1,1) 0.50 rolling window model by { ˆ } T. A rolling window with size equal to one-half of the htt 1, ROLL(0.5) t = R + 1 length of the in-sample period is a longer rolling window that represents a compromise between having a relatively long estimation window to accurately estimate the parameters of the GARCH(1,1) process and 5 RiskMetrics Group (1996) recommends using β = 0.94 for daily data.

10 9 not relying too extensively on data from separate regimes. We also consider a GARCH(1,1) model estimated using a shorter rolling window with size equal to one-quarter of the length of the in-sample period (the second competing model; GARCH(1,1) 0.25 rolling window model), so that the first forecast uses estimates based on observations 0. 75R through observation R. By using a shorter estimation window, this forecasting model has fewer observations available for estimating the parameters of the GARCH(1,1) process, 6 but it runs a lower risk of using data from different regimes. We denote the forecasts generated by the GARCH(1,1) 0.25 rolling window model by { hˆ 1, (0.25)} T tt ROLL t = R + 1. The third competing model is a GARCH(1,1) model estimated using a window whose size is determined by applying the modified ICSS algorithm to an expanding window ( GARCH(1,1) with breaks model). We first apply the modified ICSS algorithm to observations one through R. Suppose we find significant evidence of one or more structural breaks according to the ICSS algorithm and that the final break is estimated to occur at time T B. We then estimate a GARCH(1,1) model using observations T B + 1 through R to form an estimate of h R + 1. If there is no significant evidence of a structural break according to the ICSS algorithm, we estimate a GARCH(1,1) model using observations one through R to form an estimate of h R + 1. To compute the second out-of-sample forecast, we apply the modified ICSS algorithm to observations one through R + 1 and proceed as described above. Continuing in this manner through the end of the available out-of-sample period, we generate a series of forecasts corresponding to the GARCH(1,1) with breaks model, { hˆ 1, } T t t BREAKS t = R + 1. A potential drawback to this forecasting model is that a relatively short sample will be available for estimating the GARCH(1,1) parameters when a break is detected relatively close to the forecast date. 7 6 In our applications in Section 3 below, this leaves us with approximately 1,500 observations for estimating the GARCH(1,1) 0.25 rolling window model. This should be enough observations to obtain accurate estimates of the GARCH parameters; see, for example, Hwang and Valls Pereira (2004) and Straumann (2005). 7 Our strategy of comparing volatility forecasts generated by GARCH(1,1) models that use an expanding window, long and short rolling windows, and a window selected by applying a structural break test to the data available at the time of forecast formation is similar to an out-of-sample forecasting exercise in Pesaran and Timmermann (2004). They consider point forecasts of the inflation rate, industrial production growth, real GDP growth, and the interest rate in the G7 countries generated by linear autoregressive models. The autoregressive forecasting models are

11 10 The final forecasting model is the simple moving average model used in Stărică et al. (2005). It uses the average of the squared returns over the previous 250 days to form the volatility forecast for day 250 t : ˆ = 2 (1/ 250) e tt MA t i. This model assumes there are no GARCH dynamics present in the volatility h 1, i= 1 process and allows the unconditional variance to change steadily over time. Stărică et al. (2005) find that this model often outperforms a GARCH(1,1) model at longer horizons when forecasting daily stock return volatility in industrialized countries. In our applications in Section 3 below, we consider forecast horizons of 1, 20, 60, and 120 days. In general, we denote the series of s -step-ahead out-of-sample forecasts generated by model i as { hˆ } T = + for i EXP, RM, ROLL(0.5), ROLL(0.25), BREAKS, MA. We generate tt si, t R s = ˆtt si, h for i = EXP, ROLL(0.5), ROLL(0.25), BREAKS and s > 1 using equation (4.116) or (4.117) in Franses and van Dijk (2000) to iterate forward using the fitted GARCH(1,1) processes. Following convention, the RiskMetrics forecast ( h ) for s > 1 is equal to the one-day-ahead forecast multiplied by the forecast ˆtt s, RM ˆt t s, MA horizon, s. The moving average model forecast ( h ) for s > 1 is also formed by multiplying the oneday-ahead forecast by s. In order to compare forecasts across models, we consider two loss functions. The first is an aggregated version of the familiar MSFE metric. The conventional MSFE at horizon s for model given by: i is T 1 2 ( ˆ ) 2 t t t s, i MSFE = [ P ( s 1)] e h. (5) A difficulty in assessing the predictive accuracy of models of conditional volatility using equation (5) is t= R+ s that h t is not directly observed, and so we must use a proxy. In equation (5), following much of the literature, squared returns serve as a proxy for the latent volatility, h t. Awartani and Corradi (2004) and estimated using an expanding window, short and long rolling windows, and a post-break window. The post-break window is selected by using the Bai and Perron (1998, 2003) method and the Schwarz information criterion to determine the location of the latest break in the data available at the time of forecast formation.

12 11 Hansen and Lunde (2004a) show that MSFE produces a consistent empirical ranking of forecasting models when squared returns serve as a proxy for the latent volatility. Patton (2005) also shows that MSFE is an appropriate loss function when using squared returns as a volatility proxy, while a number of other popular loss functions, such as absolute error, are inappropriate. Even though MSFE produces a consistent ranking of models when using squared returns as a proxy for h t, as emphasized by Andersen and Bollerslev (1998), squared returns still tend to be a very noisy proxy for latent volatility. In order to reduce some of the idiosyncratic noise in the day-to-day movements in squared returns, we follow Granger and Stărică (2005) and Stărică et al. (2005) and use an aggregate MSFE criterion: 2 s 2 s e t = j = e 1 t ( j 1) htt si, = j = h 1 t ( j 1) t si, MSFE [ P ( s 1)] ( e h ) = T 1 2 ˆ t t t s, i 2 t= R+ s, (6) where and ˆ ˆ. Aggregating helps to reduce the idiosyncratic noise in squared returns at horizons beyond one period and provides a more informative metric for comparing volatility forecast accuracy. The second loss function we consider is the González-Rivera et al. (2004) VaR loss function. Let 0.05 VaR ti, be the forecast of the 0.05 quantile of the cumulative distribution function for the cumulative return, s e = t e ( j 1), generated by model i and formed at time j = 1 t t s. We follow González-Rivera et al. (2004) and evaluate the forecasting models with respect to VaR using the following mean loss function: ( ) ti, t ti, T ( ti, )( t ti, ) MVaR = [ P ( s 1)] 0.05 d e VaR t= R+ s, (7) where d = 1 e < VaR and 1 ( ) is the indicator function that takes on a value of unity when the argument is satisfied. This asymmetric loss function penalizes more severely observations for which 0.05 e VaR, < 0. The t t i MVaR criterion has the advantage that it does not require observations of the latent volatility, h t. It is also well-motivated, as VaR is an important risk management tool. We use the

13 12 following simulation procedure to generate VaR 0.05 ti, at horizon s. A given model i produces the sequence of point forecasts for the latent volatility, hˆt ( j 1) t s, i, for j = 1,, s at time t s. Assuming ε ~ N (0,1), t we simulate a sequence of returns, { e } s =, using equations (3) and (4) and compute the simulated t ( j 1) j 1 cumulative return, s t = j = 1 t ( j 1) e e. We repeat this process 5,000 times, leaving us with an empirical 0.05 distribution of simulated cumulative returns. VaR ti, is the 250 th element of the ordered simulated cumulative returns. 8 In addition to using the MSFE and MVaR loss functions to rank the forecasting models, we employ the White (2000) reality check to test whether the expected loss associated with the forecasts generated by at least one of the four competing models is significantly less than the expected loss of the forecasts generated by one of the benchmark models. Define the loss at time t for forecasting model j relative to benchmark model i as f = L L, where Lt, is given for each loss function by the ti,, j ti, t, j expression after the summation operator in equation (6) or (7), and let [ ( 1) ] 1 The White (2000) statistic is given by ( ) 0.5 l k= 1,, l i,1 i, l f = P s f. i, j t= R+ s t, i, j V = max [ P ( s 1)] f,, f, (8) where l is the number of competing models. ( l = 4 in our applications.) The null hypothesis is that none of the competing models has superior predictive ability in terms of expected loss over the benchmark model, whereas the one-sided (upper-tail) alternative hypothesis is that at least one of the competing models has superior predictive ability over the benchmark model. Following White (2000), a p-value corresponding to V l is generated using the stationary bootstrap of Politis and Romano (1994). We perform the White (2000) reality check with the GARCH(1,1) expanding window and RiskMetrics models serving in turn as the benchmark model. The reality check allows us to test whether any of our T 8 We also experimented with simulations where ε t follows a student s t-distribution in equation (1) in order to allow for excess kurtosis in ε t. The results are qualitatively similar to those reported in Section 3 below.

14 13 four methods of accommodating structural breaks in the unconditional variance of exchange rate returns improves real-time volatility forecasting performance relative to a benchmark model based on the assumption of a stable GARCH(1,1) process. The reality check helps to control for data mining when considering a multiple number of competing models. This is important in our applications, as we consider a variety of ways to accommodate potential structural breaks when forming out-of-sample forecasts. We also compute the Hansen (2005) studentized version of the V l statistic,, where we again generate the corresponding p-value using the stationary bootstrap of Politis and Romano (1994). 9 The Hansen (2005) version of the White (2000) reality check is designed to be a more powerful test of superior predictive ability. A word of caution is in order with respect to the use of the White (2000) and Hansen (2005) statistics and the stationary bootstrap. Recent research shows that making inferences concerning relative predictive accuracy across forecasting models can be tricky and depends on a number of factors, such as the size of the in-sample period relative to the out-of-sample period ( P/ R), type of estimation window used (expanding, rolling, or fixed), and whether the models being compared are nested or non-nested. We recognize that it is not necessarily the case that all of the required technical conditions for the strict validity of the stationary bootstrap are satisfied in our applications, and we report bootstrapped p-values SPA T n 10 for the White (2000) V l and Hansen (2005) SPA T n statistics as a rough guide to assessing statistical significance. 3. Estimation Results 3.1. Data We use daily nominal exchange rate data from Global Financial Data to compute the daily return of the U.S. dollar against the currencies of Canada, Denmark, Germany, Japan, Norway, Switzerland, and 9 Hansen (2005) discusses the generation of three bootstrapped p-values (consistent, lower bound, and upper bound) SPA corresponding to the T n statistic. We report the consistent p-value in Section 3 below. 10 A partial list of relevant studies includes Clark and McCracken (2001, 2004a), Corradi and Swanson (2005a), Giacomini and White (2005), McCracken (2004), West (1996), and West and McCracken (1998). See Corradi and Swanson (2005b) for an informative review of these issues.

15 14 the U.K. for 1/2/1980-8/13/2005. We also consider the daily return corresponding to the U.S. tradeweighted exchange rate for the same period. Table 1 reports summary statistics for the eight exchange rate return series. Heteroskedastic and autocorrelation consistent standard errors for the mean, standard deviation, skewness, and excess kurtosis are computed as in West and Cho (1995). Results in Panel A of Table 1 indicate that none of the means is significantly different from zero and that Japan and Switzerland are the only currencies that exhibit significant skewness. All of the currencies display significant excess kurtosis, a well-known stylized fact of dollar exchange rate returns. The West and Cho (1995) modified Ljung-Box statistics reported in Table 1 are robust to conditional heteroskedasticity, and they give no significant evidence of autocorrelation in the any of the exchange rate return series. With respect to the squared returns in Panel B of Table 1, the Ljung-Box statistics give clear indication of serial correlation, and the Engle (1982) Lagrange-multiplier statistics evince significant evidence of ARCH effects. Overall, the results in Table 1 provide support for modeling U.S. dollar exchange rates as GARCH processes, helping to explain the popularity of GARCH models in the literature In-Sample Test Results We apply the modified ICSS algorithm to the eight exchange rate return series for 1/2/1980-8/31/2005. Figure 1 portrays the eight return series, together with +/- three-standard-deviation bands, where the standard deviations are computed for each of the sub-samples defined by the structural breaks in variance identified by the modified ICSS algorithm. The exact dates of the structural breaks are reported in Table 2. From Figure 1, we see that the modified ICSS algorithm selects a single structural break in the unconditional variance of returns for Germany; two structural breaks for Japan, Norway, and Switzerland; three structural breaks for Canada and the U.S.; and four structural breaks for the U.K. The modified ICSS algorithm thus identifies one or more variance breaks for each of the eight exchange rate return series. A number of the variance breaks detected by the modified ICSS algorithm appear to be associated with significant economic events. The first break for Denmark occurs in September of 1993 and results in

16 15 a decrease in volatility. This break closely corresponds to a change in Danish monetary policy that tolerated greater interest rate volatility (Fernández de Lis, 2002). The single break for Germany occurs at the end of 1996 and signals a decrease in volatility. This break likely represents a decrease in uncertainty resulting from final discussions concerning the debut of the Euro and the modeling of the proposed European Central Bank after the German Bundesbank. For Japan, the first break is in May of 1997 and leads to an increase in volatility, while the second break is in March of 2000 and represents a decrease in volatility. These breaks roughly match the beginning and end of the East Asian crisis and delineate a period of increased volatility during the crisis. The first break for Switzerland in October of 1995 signifies a decrease in volatility and corresponds to an abandonment of monetary targeting by the Swiss bank (Rich, 2003). The second break for Switzerland in August of 1998, associated with an increase in volatility, occurs near the Euro s successful debut, which challenged the leading role played by the Swiss franc as a safe haven currency (Rich, 1998). The first two breaks for the U.K. (in February of 1984 and September of 1985) define a regime of increased volatility encompassing the substantial Exchange Rate Mechanism realignment in 1985, while the third break for the U.K. (in March of 1991) represents an increase in volatility that roughly corresponds to the beginning of the Exchange Rate Mechanism crisis in Europe. The final volatility break for the U.S. in April of 2000 brings about an increase in volatility and occurs very near the bursting of the tech bubble that signaled the end of the long bear market in the U.S. 11 Given that many of the variance breaks appear to correspond to significant economic events, we can be more confident that the modified ICSS algorithm identifies important breaks in the unconditional variance of exchange rate returns. In Table 2, we report full-sample QMLE GARCH(1,1) parameter estimates for the eight exchange rate return series, as well as QMLE GARCH(1,1) parameter estimates for each of the sub- 11 The first two structural breaks for the U.S. occur in September of 1995 and December of 1996 and delineate a regime of just over one year marked by a notable reduction in U.S. trade-weighted exchange rate volatility. It is difficult to point to specific economic events to account for the reduction in U.S. dollar volatility around this time; see, for example, Bank for International Settlements (1997).

17 16 samples defined by the structural breaks identified by the modified ICSS algorithm. 12 Inspection of the parameter estimates reveals that the GARCH(1,1) processes are quite persistent when estimated over the full sample, with ˆ α + ˆ β ranging from to 0.998, in line with the extant literature. However, the persistence in the full-sample estimates sometimes masks important differences in persistence across subsamples. Most markedly, there are sub-samples in Canada, the U.K., and the U.S. where ˆ α = 0, so that the sub-sample is characterized by conditional homoskedasticity. There are also sub-samples where ˆ α > 0 but ˆ α + ˆ β is well below unity: for example, the second sub-sample in Japan ( ˆ α + ˆ β = ); the second sub-sample in Switzerland ( ˆ α + ˆ β = ); and the third sub-sample in the U.S. ( ˆ α + ˆ β = ). We see from Table 2 that all of the structural breaks bring about sizable shifts in the intercept term, ˆω, of the GARCH(1,1) model and that these shifts often lead to substantial changes in the unconditional variance across regimes. Overall, the significant in-sample evidence of structural breaks in all eight exchange rates and the variation in the GARCH(1,1) parameter estimates across the sub-samples defined by the structural breaks suggest that variance breaks are an empirically relevant feature of U.S. dollar exchange rates Out-of-Sample Test Results Tables 3-6 report out-of-sample volatility forecasting test results for horizons of 1, 20, 60, and 120 days. The out-of-sample period is comprised of the last 500 observations of the 1/2/1980-8/13/2005 full-sample period and covers the 10/1/2003-8/31/2005 period for each country (with the exception of the U.S., where the out-of-sample period begins in 9/9/2003). The first row in each panel reports the mean loss for the GARCH(1,1) expanding window model, while the remaining rows present the ratio of the mean loss for the other five models to the mean loss for the GARCH(1,1) expanding window model. The tables also report p-values corresponding to the White (2000) V l and Hansen (2005) SPA T n statistics with 12 We used the GAUSS module Constrained Maximum Likelihood 2.0 to obtain the QMLE parameter estimates of the GARCH(1,1) models. Within the module, we used the Newton-Raphson algorithm and analytical derivatives, and we imposed the restrictions, ω > 0, α, β 0. We experimented with different starting values for the parameters, and the reported parameter estimates in Table 2 correspond to the global maximum.

18 17 the GARCH(1,1) expanding window and RiskMetrics models serving in turn as the benchmark model and the two GARCH(1,1) rolling window, moving average, and GARCH(1,1) with breaks models serving as the competing models. At the 1-day horizon (see Table 3), neither the GARCH(1,1) expanding window nor RiskMetrics benchmark model delivers the lowest mean loss for any country and either loss function. Among the competing models, the GARCH(1,1) 0.50 (0.25) rolling window model has a lower MSFE than the GARCH(1,1) expanding window model for all eight (seven) countries. The two GARCH(1,1) rolling window models also have a lower MSFE than the RiskMetrics model in almost every case in Table 3. This is the first evidence that allowing for instabilities in the GARCH(1,1) models leads to out-of-sample forecasting gains. The GARCH(1,1) with breaks model has a lower MSFE than the GARCH(1,1) expanding window (RiskMetrics) model for six (seven) countries, and the moving average model outperforms the GARCH(1,1) expanding window (RiskMetrics) model according to the MSFE metric for five (six) countries. The four competing models, all of which adjust the estimation window to accommodate potential structural breaks, usually lead to reductions in MSFE of 1-4% relative to the two benchmark models, and there is often significant evidence of superior predictive ability relative to either of the benchmark models according to the p-values corresponding to the Hansen (2005) SPA T n statistics. 13 With respect to the MVaR loss function, the competing models again often outperform either of the benchmarks. The GARCH(1,1) 0.50 (0.25) rolling window model has a lower MSFE than either benchmark model for six (seven) countries. The GARCH(1,1) with breaks model has a lower MSFE than the GARCH(1,1) expanding window (RiskMetrics) model for five (six) countries, while the moving average model has a lower MSFE than either benchmark for four countries. There are a number of instances where the SPA T n either benchmark for the statistic evinces significant evidence of superior predictive ability relative to MVaR loss function. 13 SPA There is more evidence of superior predictive ability for the Hansen (2005) T n statistic than the White (2000) V l statistic. This is not surprising, as the SPA T n statistic is designed to be more powerful.

19 18 We report results for the 20-day horizon in Table 4. As in Table 3, there are no cases where either the GARCH(1,1) expanding window or RiskMetrics benchmark model delivers the lowest mean loss. For either the MSFE or MVaR loss function, the two GARCH(1,1) rolling window models almost always outperform either of the benchmark models. The GARCH(1,1) with breaks model outperforms the GARCH(1,1) expanding window (RiskMetrics) model for three (five) countries for the MSFE loss function, and it outperforms the GARCH(1,1) expanding window (RiskMetrics) model for seven (eight) countries for the MVaR loss function. The moving average model performs similarly to the GARCH(1,1) with breaks model. Compared to the results for s = 1 in Table 3, we see more sizable reductions in mean loss relative to the benchmark models for s = 20 in Table 4. For the MSFE loss function, the bestperforming competing models reduce mean loss by approximately 5-15% relative to the GARCH(1,1) expanding window benchmark model. For the MVaR loss function, the reductions in mean loss range from approximately 1-10% for the best-performing competing models relative to the GARCH(1,1) expanding window model. While there is no evidence of superior predictive ability relative to either benchmark using the V l statistic, there is significant evidence of superior predictive ability in a number of cases using the SPA T n statistics for both loss functions. Results for s = 60 are reported in Table 5. There is only one case where either of the benchmark models has the lowest mean loss (the GARCH(1,1) expanding window model for the U.S. and the MVaR loss function). The GARCH(1,1) 0.50 rolling window model has a lower MSFE than the GARCH(1,1) expanding window (RiskMetrics) benchmark for six (seven) countries, while the GARCH(1,1) 0.25 rolling window model has a lower MSFE relative to the benchmark for five (seven) countries. For the MSFE loss function, the GARCH(1,1) with breaks model outperforms the GARCH(1,1) expanding window (RiskMetrics) model for four (seven) countries, while the moving average model outperforms the GARCH(1,1) expanding window (RiskMetrics) model for three (five) countries. The SPA T n statistics for the MSFE loss function provide significant evidence of superior predictive ability relative to the

20 19 benchmark models (especially the RiskMetrics model) in a number of cases. With respect to the MVaR loss function, the GARCH(1,1) 0.50 rolling window, GARCH(1,1) 0.25 rolling window, GARCH(1,1) with breaks, and moving average models generate a lower mean loss than the GARCH(1,1) expanding window (RiskMetrics) benchmark model for three, six, five, and seven (five, seven, seven, and eight) countries, and the SPA T n statistics supply evidence of superior predictive ability in some cases. When s = 60, the best-performing competing models attain mean loss reductions of around 8-25% for the MSFE loss function and up to nearly 14% for the MVaR loss function relative to the GARCH(1,1) expanding window model. At the 120-day horizon (see Table 6), we continue to find evidence that allowing for instabilities in the GARCH(1,1) processes yields forecasting gains. There is only a single case where either the GARCH(1,1) expanding window or RiskMetrics benchmark model has the lowest MSFE (the GARCH(1,1) expanding window model for the U.S.). For the MVaR loss function, the GARCH(1,1) expanding window (RiskMetrics) benchmark model has the lowest mean loss for zero (two) countries. For the MSFE loss function, the two GARCH(1,1) rolling window and GARCH(1,1) with breaks models consistently outperform the two benchmark models, while the moving average model has a lower MSFE than the GARCH(1,1) expanding window (RiskMetrics) benchmark model for two (five) countries. The best-performing competing models reduce the mean loss up to a very sizable 45% relative to the GARCH(1,1) expanding window benchmark model for the MSFE loss function. This continues the trend from Tables 3-5, where the reductions in mean loss associated with the best-performing competing models tend to increase as the horizon increases. For the MVaR loss function, the GARCH(1,1) 0.50 (0.25) rolling window outperforms the GARCH(1,1) expanding window benchmark model for seven (four) countries and the RiskMetrics benchmark model for eight (four) countries. The GARCH(1,1) with breaks model has a lower MVaR for seven (five) countries, while the moving average model achieves a lower MVaR than the GARCH(1,1) expanding window (RiskMetrics) model for seven (three) countries. The best-performing competing models provide reductions in MVaR of up to almost

21 20 18% relative to the GARCH(1,1) expanding window model. There are numerous instances where the V l and/or SPA T n statistics provide significant evidence that the competing models offer superior predictive ability relative to either benchmark for both loss functions. Summarizing the results in Tables 3-6, we find that accommodating structural breaks in the unconditional variance of exchange rate returns often improves real-time forecasts of exchange rate return volatility. GARCH(1,1) expanding window and RiskMetrics benchmark models that assume stable GARCH processes are almost always outperformed by one or more competing models that adjust the estimation window to account for potential breaks in the unconditional variance of exchange rate returns. The improvements in forecasting associated with accommodating breaks are manifested using two different loss functions, the first based on MSFE and the other on VaR, and they become more sizable as the forecast horizon increases. Complementing the in-sample results discussed in Section 3.1, the results reported in Tables 3-6 can be interpreted as out-of-sample evidence that structural breaks are an empirically relevant characteristic of exchange rate volatility Optimal Window Size It is interesting to examine the relative forecasting performance of the different methods of adjusting the estimation window for the GARCH(1,1) forecasting models. Of the three methods we consider, the GARCH(1,1) 0.50 rolling window model often displays the best forecasting performance. For the MSFE loss function, the GARCH(1,1) 0.50 rolling window model generates the lowest mean loss for three, four, four, and five countries at horizons of s = 1,20,60,120, respectively. The GARCH(1,1) 0.25 rolling window produces the lowest mean loss for three, two, two, and zero countries, 14 In an earlier version of this paper, we used weekly exchange rate data and considered Markov switching GARCH(1,1) models of exchange rate volatility (Klaasenn, 2002; Haas et al., 2004). In out-of-sample exchange rate volatility forecasting exercises, Markov switching GARCH(1,1) forecasting models estimated using an expanding window were consistently outperformed by GARCH(1,1) forecasting models estimated using windows of various sizes. When using daily data, we encountered convergence problems for some countries in estimating the Markov switching GARCH(1,1) forecasting model using an expanding window, which entails estimating the Markov switching GARCH(1,1) model 500 times to generate the out-of-sample forecasts.

22 21 while the GARCH(1,1) with breaks model produces the lowest mean loss for only one, one, one, and zero countries for the MSFE loss function. We next discuss recent research that sheds light on these results. Suppose we wish to forecast a variable, and we know that a structural break has occurred at a specific date in the recent past. At first blush, it may seem optimal to use only post-break data when estimating the forecasting model in order to avoid biased model parameter estimates and forecasts. However, as noted by Clark and McCracken (2004b, p.1), there is a balance between using too much or too little data to estimate model parameters. While discarding data from the pre-break period helps to reduce biases, it also leads to greater variances in the estimates of the forecasting model s parameters, thereby creating a potential tradeoff between using pre- and post-break data. In recent research, Clark and McCracken (2004b) and Pesaran and Timmermann (2005) derive the optimal least squares estimation window size for a linear regression forecasting model that minimizes MSFE. Importantly, they show that it can be optimal to include pre-break data in the optimal estimation window and that the optimal amount of pre-break data to include depends on a number of factors, such as the sizes and directions of the changes in the parameters of the model and the exact timing of the structural breaks. Unfortunately, given that research on the optimal estimation window size for forecasting models is in its early stages, results are not available for GARCH processes and loss functions other than MSFE, so we are limited in what we can conclude with respect to the optimal window size for GARCH(1,1) forecasting models of exchange rate volatility. Nevertheless, the results in Clark and McCracken (2004b) and Pesaran and Timmermann (2005) help to explain why the GARCH(1,1) 0.50 rolling window model which can frequently include data prior to the most recent break when estimating the GARCH(1,1) forecasting model often displays the best performance for the MSFE loss function in our out-of-sample forecasting exercises. As emphasized by Pesaran and Timmermann (2005), identifying the optimal estimation window size is made considerably more complicated when the exact timing of the break dates is unknown and must be estimated. Uncertainties surrounding the timing of breaks limit the usefulness of analytical results concerning the optimal window size. Given these uncertainties, Pesaran and Timmermann (2005)