The predictability of exchange rate volatility

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Economics Letters 98 (2008) 220 – 228 www.elsevier.com/locate/econbase

The predictability of exchange rate volatility ☆ Burkhard Raunig ⁎ Oesterreichische Nationalbank Economic Studies Division, Otto-Wagner-Platz 3, POB 61, A-1011 Vienna, Austria Received 30 May 2006; received in revised form 19 April 2007; accepted 27 April 2007 Available online 10 May 2007

Abstract The model-free test procedure used in this paper suggests that exchange rate volatility is hard to predict more than 1 month ahead with time series methods. Moreover, predictability declines rather quickly with horizon. © 2007 Elsevier B.V. All rights reserved. Keywords: Exchange rates; Volatility; Predictability JEL classification: C 120; C 530; G 100

1. Introduction Asset prices, risk calculations, trading and hedging strategies as well as monetary policy decisions depend to some extent on the volatility (i.e. variance or standard deviation) of financial returns. But is volatility predictable over a certain forecasting horizon and if yes to what degree? This paper investigates these questions for a set of exchange rate returns. Forecasts from more sophisticated volatility models cannot beat a simple estimate of the unconditional volatility if volatility is unpredictable over a given horizon. It is therefore natural to compare forecasts

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The opinions expressed do not necessarily reflect those of the Oesterreichische Nationalbank. ⁎ Tel.: +43 1 404 20 7219; fax: +43 1 404 20 7299. E-mail address: [email protected].

0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.04.035

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from competing models with statistical loss functions or some economic metric. See Poon and Granger (2003) for a survey. Such comparisons reveal the relative performance of forecasting models but give no clear answer about predictability per se because the results are model-dependent. For example, assume we evaluate forecasts from two volatility models X and Y over a given forecasting horizon. These forecasts may be no better than an estimate of the unconditional volatility but there could be a model Z we did not consider which outperforms both X and Y. Christoffersen and Diebold (2000) address this problem and test for volatility predictability with a model-free procedure. They find predictability of 10–15 days for stock index volatility and about 10 days for exchange rate volatility. Raunig (2006) uses an alternative model-free test which is somewhat more powerful in a Monte Carlo experiment to examine the predictability of DAX stock index volatility. In this paper we use a simulation based version of this test to assess the predictability of the volatility of returns on British Pound (GBP), Japanese Yen (YEN), Swiss Francs (CHF), Swedish Krona (SEK), Danish

Fig. 1. Shows P value discrepancy plots from a Monte Carlo experiment (10,000 replications, B = 9999) in which et is randomly drawn from a standard normal distribution N(0,1), a fat-tailed t distribution with 5 degrees of freedom t(5) and a skewed χ2 distribution with 4 degrees of freedom chi(4).

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Table 1 Skewness and kurtosis of nonoverlapping exchange rate returns Currency

AUD

SK K SK K SK K SK K SK K SK K SK K

CHF DKK GBP JPY NOK SEK

1 day

10 days

20 days

40 days

j = 8292

j = 827

j = 412

j = 205

−4.66⁎⁎⁎ 145.21⁎⁎⁎ −0.00 6.00⁎⁎⁎ 0.15⁎⁎⁎ 12.92⁎⁎⁎ −0.12⁎⁎⁎ 6.62⁎⁎⁎ −0.71⁎⁎⁎ 13.10⁎⁎⁎ 0.36⁎⁎⁎ 11.30⁎⁎⁎ 2.49⁎⁎⁎ 55.11⁎⁎⁎

− 1.72⁎⁎⁎ 17.73⁎⁎⁎ − 0.27⁎⁎⁎ 4.05⁎⁎⁎ 0.03 3.66⁎⁎⁎ − 0.33⁎⁎⁎ 7.20⁎⁎⁎ − 0.83⁎⁎⁎ 6.84⁎⁎⁎ 0.04 4.55⁎⁎⁎ 0.88⁎⁎⁎ 8.13⁎⁎⁎

−1.46⁎⁎⁎ 9.85⁎⁎ −0.09 3.38 −0.13 3.11 −0−21⁎ 4.85⁎⁎⁎ −0.46⁎⁎⁎ 4.04⁎⁎⁎ 0.06 3.77⁎⁎⁎ 0.51⁎⁎⁎ 5.05⁎⁎⁎

−0.78⁎⁎⁎ 4.84⁎⁎⁎ −0.24 2.87 −0.15 2.80 0.09 3.87⁎⁎ −0.52⁎⁎⁎ 4.09⁎⁎⁎ −0.24 3.12 0.56⁎⁎⁎ 4.66⁎⁎⁎

SK and K denote skewness and kurtosis coefficients, j denotes the number of observations. ⁎⁎⁎ and ⁎⁎ indicate statistical significance at the 0.01 and 0.05 level, respectively.

Krone (DKK), Norwegian Krone (NOK) and Australian Dollar (AUD) spot exchange rates against the US Dollar (USD) over horizons ranging from 1–45 trading days. 2. Methodology Consider a sample of exchange rate returns et, t = 1,…, j where any conditional mean dynamics has been removed and let Dt(et|Ωt−1) and D(et) denote the conditional and the unconditional distribution of these returns. Unpredictability with respect to the information set Ωt implies coincidence of these distributions, i.e. Dt(et|Ωt−1) = D(et) for all t (Clements and Hendry, 1998, Ch. 2). For Ωt = {et, et−1,…} this means that past returns or functions of them do not help to forecast any aspect of the distribution of current returns. Past squared returns should then not help to forecast the conditional variance of current returns (i.e. Var(et|Ωt−1) = Var(et) = E(et2 ) = const.). Factoring the joint distribution of the et into the product of their conditional distributions and exploiting repeatedly the definition of unpredictability yields j

Dðej ; ej1 ; N ; e1 Þ ¼ j Dðet Þ: t¼1

ð1Þ

Thus, returns should be independently and identically distributed (iid) if volatility is unpredictable from past returns.

Fig. 2. Simulation based P values (B = 9999) from predictability tests with 3 lags in test regression for the full sample period and two equally long sub-samples S1 and S2.

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Raunig (2006) transforms the et in such a way that tests on the transformed data do not depend on the distribution of the original data. The transformation (Berkowitz, 2001; Fruehwirth-Schnatter, 1996) is nt ¼ U1 ½Dðet Þ; t ¼ 1; N ; j

ð2Þ

and maps returns first via their unconditional distribution D into the unit interval and then via the inverse of the standard normal distribution Φ− 1 into the corresponding quantiles of the standard normal distribution N(0,1). By independence of the et the nt are iid N(0,1) if volatility is unpredictable and if D is continuous. Squared transformed returns nt2 are then of course also iid. On the other hand, the et2 are correlated if volatility is predictable and this causes correlation in the nt2 . Following the idea of the classical ARCH (autoregressive conditional heteroskedasticity) test in Engle (1982) we may regress current nt2 on their first d lags. An F test should reject the restriction α1 = α2 = … = αd = 0 in the regression n2t ¼ a0 þ a1 n2t1 þ N þ ad n2td þ ut

ð3Þ

if volatility is predictable. The F statistic in Eq. (3) is only asymptotically valid but the test does not depend on the distribution of the et because the nt are iid N(0,1) under the null for any continuous distribution D. D is usually unknown in empirical work and must be estimated. We estimate it with the empirical distribution function (EDF) ˆ DðeÞ ¼ j1

j X

1fet Veg;

ð4Þ

t¼1

where 1 is an indicator variable that takes the value 1 if et ≤e and zero otherwise, substitute it into Eq. (2) to get ˆ t Þ; t ¼ 1; N ; j nˆt ¼ U1 ½ Dðe

ð5Þ

the empirical counterpart of Eq. (2) and test for zero slope coefficients in nˆ2t ¼ a0 þ a1nˆ2t1 þ N þ ad nˆ2td þ ut :

ð6Þ

A simulation based F test can easily be computed since the values of the nˆ t depend only the sample size j and not on the distribution of the et. We can therefore randomly draw j numbers from any continuous distribution, transform them via Eq. (5) and compute the F statistic for zero slope coefficients τ⁎ in Eq. (6) for B permutations of the squared data. We can use permutations and need not draw each time a new random sample because in a sample of size j the values of nˆ t cannot change. What can change is just the ordering of the nˆ t in the random sample. The simulation based P value (Dufour and Khalaf, 2001) for an observed test statistic τ given τ1⁎, τ2⁎,…, τB⁎ simulated statistics is ˆ ¼ pðsÞ

B 1 X 1ðs⁎i zsÞ þ 1: ðB þ 1Þ i¼1

ð7Þ

The distribution of τ⁎ approximates the finite sample distribution of the F statistic and accounts for the fact that the data have first been transformed via their EDF. If B is chosen large enough one would expect that Fig. 3. R2 from predictability tests with 3 lags in test regression together with simulated 90% and 95% confidence intervals for the full sample period and two equally long sub-samples S1 and S2.

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these P values are more accurate in small samples than P values from the usual F distribution. P value discrepancy plots (Davidson and MacKinnon, 1998) from a Monte Carlo experiment (10,000 replications, B = 9999) in which the et are randomly drawn from a normal distribution, a fat-tailed t distribution with 5 degrees of freedom (df ) and a skewed χ2 distribution with 4 df suggest that the simulation based P values are indeed more accurate than the standard P values when the sample size is small. Fig. 1 further suggests that both kinds of P values are fairly accurate when the sample size is large. 3. Empirical study The data (US Federal Reserve Board data base) cover the period 02/01/1973 — 13/01/2006 and consist of 8294 daily AUD, CHF, DKK, GBP, JPY, NOK and SEK spot exchange rates against the USD. As in Christoffersen and Diebold (2000) we use nonoverlapping h-day returns (h = 1,…, 45) calculated from daily exchange rates St, t = 1,…, T as rh, j = ln(ST / ST−h), rh, j−1 = ln(ST−h / ST−2h),…, to avoid dependencies induced by overlapping data. From these series we remove possible conditional mean dynamics with an AR(1) model. The residuals are the “centered” returns eh,t, t = 1,…, j to which we apply the test. Table 1 reports skewness (SK) and kurtosis (K) coefficients of the distribution of eh,t for h = 1, 10, 20 and 40 trading days (results for the remaining values of h are available upon request). All daily return distributions have fatter tails than the normal distribution (i.e. K is significantly larger than 3). Most of the daily return distributions are also slightly skewed (SK = 0 for symmetric distributions). However, both coefficients tend to the values implied by a normal distribution as h rises and for h = 40 the distributions are approximately normal. This tendency to normality is consistent with declining conditional heteroskedasticity and hence declining volatility predictability. The first column in Fig. 2 shows simulation based P values (B = 9999) from the predictability test computed with three lags in the test regression. They suggest predictability of exchange rate volatility about 20–25 trading days into the future. Beyond this horizon the P values frequently exceed conventional significance levels. Volatility is therefore likely to be unpredictable more than 1 month ahead. To see whether predictability has changed over time the full sample period was split up into two equally long subsamples. The second and third columns of Fig. 2 show the P values for the sub-samples. The results are essentially in line with the full sample findings. Only for the AUD and the CHF volatility predictability appears to be stronger over the 1973–1989 period. Clustering effects in the original data indicate volatility predictability and these effects are reflected in the transformed data. The R 2 from Eq. (6) should therefore provide information about the strength of volatility predictability because it measures the strength of the clustering effects in the nˆ t2 series. Moreover, confidence intervals for the R2 in Eq. (6) do not depend on the distribution of the original data and can easily be generated as a byproduct in the simulation of the test statistic τ⁎. The R2 from a regression of et2 on lagged et2 would be another measure of predictability. However, in this case a bootstrap procedure would be needed to calculate accurate confidence intervals. Both measures usually differ from R 2 s obtained with Mincer–Zarnowitz type regressions where a proxy for the unobservable volatility is regressed on forecasts from a specific volatility model. Such R 2 s depend on the forecasting performance of the model as well as on the proxy for the true volatility. Andersen et al. (2005) show that this measure of forecasting performance tends to be downward biased. Fig. 3 shows the R2 from regression (6) using 3 lags together with simulation based 90% and 95% confidence intervals for the full sample and the two sub-samples. Over the full sample the R2 is about 0.12 at the daily horizon and about 0.06 at the 10 day horizon for AUD/USD exchange rate returns. The R2 is here

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even higher (about 0.24) in the first sub-sample reflecting stronger predictability during the first part of the sample period. In all other cases the R2s are smaller and around 0.08–0.05 at the daily horizon and around 0.03–0.02 at the 10 day horizon. Thus the degree of volatility predictability does not seem to be particularly high and after about 20 trading days the R2 is often inside the confidence intervals. To see whether the results depend on the number of lags in the test regression the tests where repeated with 1, 5 and 7 lags. The results are similar. Some authors (e.g. Ding et al., 1993) find that volatility predictability is more strongly reflected in absolute returns than in squared returns. To assess whether the empirical findings change when the test is based on absolute transformed returns the tests where repeated for 1, 3, 5 and 7 lags in Eq. (6) with the nˆ t2 replaced by |nˆ t|. The results are again similar (the empirical results referred to in this paragraph are available upon request). 4. Conclusions The empirical results from the model-free test indicate that exchange rate volatility is predictable at shorter horizons such as 1 or 2 weeks but difficult to forecast more than 1 month ahead with time series models. This finding is broadly consistent with the model-free results in Christoffersen and Diebold (2000) as well as with model based results in West and Cho (1995), Galbraith and Kisinbay (2005) and Taylor et al. (2004). The empirical findings do not of course rule out that a simple estimate of the unconditional exchange rate volatility could not be beaten by longer term volatility forecasts based on a larger or different information set. For example, Taylor et al. (2004) find that option implied volatility forecasts better than time series based volatility over the one and three month horizon. Combining time series based forecasts with option implied volatility may also improve forecasting accuracy as Dunis et al. (2003) demonstrate. References Andersen, T.G., Bollerslev, T., Meddahi, N., 2005. Correcting the errors: volatility forecast evaluation using high-frequency data and realized volatilities. Econometrica 73, 279–296. Berkowitz, J., 2001. Testing density forecasts, with applications to risk management. Journal of Business and Economic Statistics 19, 465–475. Christoffersen, P.F., Diebold, F.X., 2000. How relevant is volatility forecasting for financial risk management? Review of Economics and Statistics 82, 12–22. Clements, M., Hendry, D.F., 1998. Forecasting Economic Time Series. Cambridge University Press. Davidson, R., MacKinnon, J.G., 1998. Graphical methods for investigating the size and power of test statistics. Manchester School 66, 1–26. Ding, Z., Granger, C.W.J., Engle, R.F., 1993. A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106. Dufour, J.M., Khalaf, L., 2001. Monte Carlo test methods in econometrics. In: Baltagi, B. (Ed.), A Companion to Econometric Theory. Blackwell Publishers, Oxford, pp. 494–519. Dunis, C., Laws, J., Chauvin, S., 2003. FX volatility forecasts and the informational content of market data for volatility. European Journal of Finance 9, 242–272. Engle, R.F., 1982. Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007. Fruehwirth-Schnatter, S., 1996. Recursive residuals and model diagnostics for normal and non-normal state space models. Environmental and Ecological Statistics 3, 291–309. Galbraith, J.W., Kisinbay, T., 2005. Content horizons for conditional variance forecasts. International Journal of Forecasting 21, 249–260. Poon, S., Granger, C.W.J., 2003. Forecasting volatility in financial markets. Journal of Economic Literature 41, 478–539.

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