volatility forecasting of exchange rate by quantile regression

International Review of Economics and Finance 20 (2011) 591–606

Contents lists available at ScienceDirect

International Review of Economics and Finance

j ourna l homepage: www.e lsev ie r.com/ locate / i re f

Volatility forecasting of exchange rate by quantile regression☆

Alex YiHou Huang a,⁎, Sheng-Pen Peng b, Fangjhy Li c, Ching-Jie Ke a

a College of Management, Yuan Ze University, Taiwanb Department of Real Estate Management, Hsing Kuo University of Management, Taiwanc Department of Finance, Hsing Kuo University of Management, Taiwan

a r t i c l e i n f o

☆ The authors thank two anonymous referees andbenefited from valuable comments of participants inFinance and Banking Conference, Sydney, Australia. THuang thanks the support from the National Science C⁎ Corresponding author and assistant professor of

fax: +886 3 4354624.E-mail address: [email protected] (A.Y. Hu

1059-0560/$ – see front matter © 2011 Elsevier Inc.doi:10.1016/j.iref.2011.01.005

a b s t r a c t

Article history:Received 11 October 2009Received in revised form 7 July 2010Accepted 18 October 2010Available online 31 January 2011

Exchange rates are known to have irregular return patterns; not only their return volatilitiesbut the distribution functions themselves vary with time. Quantile regression allows one topredict the volatility of time series without assuming an explicit form for the underlyingdistribution. This study presents an approach to exchange rate volatility forecasting by quantileregression utilizing a uniformly spaced series of estimated quantiles. Based on empiricalevidence of nine exchange rate series, using 19 years of daily data, the adopted approachgenerally produces more reliable volatility forecasts than other key methods.

© 2011 Elsevier Inc. All rights reserved.
JEL classification:C53E27G17
Keywords:Exchange rateVolatilityQuantile regression

1. Introduction

Volatility forecasting has been a key research subject in financial economics for the past few decades. Volatility can beinterpreted as the level of uncertainty in a financial asset, and is applicable to many risk management processes. It is also a criticalinput variable in pricing financial derivatives and plays a central role in investment decisions. Malik and Hammoudeh (2007) andGutierrez, Martinez, and Tse (2009) documented that financial volatility can be transmitted across assets and global markets.Consequently, better understanding and measurement of volatility for key macroeconomic and international financial variablescan significantly benefit management of financial risks.

Exchange rate volatility, in particular, has also been a major research subject. Prior studies show a significant negativerelationship between exchange rate volatility and international trade as trading firms are risk averse (Arize, Osang, & Slottje, 2000;Arize, Osang, & Slottje, 2008; Choudhry, 2005; De Vita & Abbott, 2004; Fang, Lai, & Miller, 2009; Hooper & Kohlhagen, 1978).Government and central bank interventions have been documented as a key factor affecting exchange rate volatility (Beine,Benassy-Quere, & Lecourt, 2002; Frenkel, Pierdzioch, & Stadtmann, 2005; Sideris, 2008). Exchange rate volatility is proved to haveimpacts on macroeconomic conditions such as aggregate supply shocks (Hau, 2002), inflation volatility (Gonzaga & Terra, 1997),and distribution costs for consumer goods (Burstein, Neves, & Rebelo, 2003). Significant interdependences are also documentedbetween exchange rate volatility and economic performances including firm's profitability (Baum, Caglayan, & Barkoulas 2001),

editor Carl Chen for insightful suggestions which constructively improve the article. This paper is alsothe 5th International Conference on Asian Financial Markets, Nagasaki, Japan and the 22nd Australasianhe authors thank Wen-Cheng Hu, Chih-Chun Chen, and Guo-An Li for their excellent research assistanceouncil of Taiwan (NSC96-2415-H-155-001).finance at: 135 Yuan-Tung Road, Chung-Li, Taoyuan 32002, Taiwan, ROC. Tel.: +886 3 4638800x2668

ang).

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592 A.Y. Huang et al. / International Review of Economics and Finance 20 (2011) 591–606

private investments (Serven, 2003), and carry trade profits (Menkhoff et al., 2010). Thus, the uncertainty of exchange rate attractsgreat deal of attentions due to its effectiveness on macroeconomic behaviors and international economic dynamics, and accuratevolatility forecasts of exchange rates are therefore in high demand.

One way to estimate exchange rate volatility is to determine potential factors of exchange rate and generate the forecastaccording to their relationships. However, Engel and West (2005) found that it is difficult to link the floating exchange ratessignificantly to macroeconomic fundamentals such as real money balances, outputs, interest rates and so on, and fundamentalvariables do not help predict changes of exchange rates in the future. A more common approach is to generate exchange ratevolatility forecast based on time series autoregressive modeling, and examples can be seen in Engle and Bollerslev (1986), Scottand Tucker (1989), Kroner and Lastrapes (1993), Heynen and Kat (1994), West and Cho (1995), Neely (1999), Beine et al. (2002),Vilasuso (2002), and Hansen and Lunde (2005) among others. Under the autoregressive framework, conditional variance isproduced based on specifications of time-dependence, moving average, and heteroscedasticity.

The popular conditional variance models require certain error distributional assumptions in estimation. Due to externalinterventions, Rapach and Strauss (2008) showed that series of exchange rate volatility often contain structural breaks, andHolmes (2008) documented non-stationarity of real exchange rate under Markov regime switching framework. Patton (2006)argued that there are different degrees of correlations in regimes of economic cycles and of time dependences for exchange rateseries. Therefore, the distributions of exchange rate returns are likely time-varying and occasionally jump. The traditionalconditional variance model based on specific distributional assumption may fail to produce volatility forecasts that are consistentwith current structure of return pattern. This paper is motivated by the argument and extends the application of quantileregression in exchange rate volatility forecasting.

Engle and Manganelli (2004) presented the CAViaR model, a time-dependence conditional quantile modeling, to generateestimates of Value-at-Risk (VaR). The author documented the successful in-sample modeling of quantiles by the method for stockprices and index. Taylor (2005) further applied the CAViaR model to produce single pair of symmetric quantiles and generate thevolatility forecast from the quantile estimates by eithermethods of approximation or regression. The author showed that themethodsoutperform the conditional varianceGARCHandmoving averagemodels in volatility forecasting for stock indices and individual stocks.

This paper extends the application of CAViaR for volatility forecasting by constructing a systematic regression relationshipbetween volatility and a uniformly spaced series of quantiles. Specifically, instead of pair of tail quantiles, series of percentiles areproduced by CAViaR model and used to explain volatility dynamics. This research differs from prior studies in several ways. First,the current return distribution is completely captured by the spaced series of quantiles without the need for specific distributionalassumptions. Second, a significant relationship between quantiles and volatility is established through regression framework.Third, the dynamics of exchange rate returns, possibly vary between regimes, are explained through the quantile predictionswithout explorations of regime changes or price breaks. The empirical outcomes of this research provide evidence of reliableperformances of the adopted method.

The next section will conduct brief reviews of existing volatility models and behaviors of exchange rate volatility. The paperwill then go on to present the methodology and discuss empirical findings. The final section concludes this research.

2. Volatility modeling and characteristics of exchange rate volatility

2.1. Financial volatility modeling

By definition, the volatility of a financial asset is the standard deviation, σ, of its return distribution. Given a sample of mreturns, its natural estimate σ̂ is given by

1 Forsee e.g.

2 Plea

σ̂2 =1

m−1∑m

t=1rt−rð Þ2; where rt = ln

StSt−1

� �: ð1Þ

Here rt is the continuous return of a financial asset or portfolio during day t, r is the mean of m sample returns, and St is themarket value of the asset on day t.

There are a number of models that forecast volatility based on its past values: historical average, moving average, exponentiallyweighted moving average (EWMA), autoregressive moving average (ARMA), and so on.1 The simplest specification is thehistorical volatility, which uses Eq. (1) to calculate σ̂ t−1and assumes that it is equal to σ̂t . The issues with this method are highnoise in daily data, dependence on the sample size, and ignorance of time varying return structure.2

Time series models improve on the historical volatility method by continuously updating parameters to take into account thelatest data. The most popular time series model in volatility forecasting is the generalized autoregressive conditionalheteroscedasticity (GARCH) method pioneered by Bollerslev (1986). The model treats volatility as a time-varying and time-dependent quantity, and the following GARCH(1,1) functional form is most commonly applied.

σ2t = λ1 + λ2σ

2t−1 + λ3ε

2t−1; ð2Þ

measurement of exchange rate volatility, the moving average approach is commonly referred as the moving sample standard deviation (MSSD) methodArize et al. (2000) and Peridy (2003).se see Poon and Granger (2003) for detail arguments.

;

whereTaylormodel

3 A d4 Plea5 Plea

explore

593A.Y. Huang et al. / International Review of Economics and Finance 20 (2011) 591–606

�t−1 is the lagged error term, and the λ′s are parameters for estimation. Various modifications to the GARCH model have
wherebeen proposed to incorporate dynamics of financial asset return; examples include EGARCH (Nelson 1991), RS-GARCH (Gray1996), and FIGARCH (Baillie et al., 1996). Regardless of model specifications, the parameters are often estimated by the maximumlikelihood approach assuming an i.i.d. error term with a particular known distribution. Examples using the GARCH method toforecast exchange rate volatility can be seen in Scott and Tucker (1989), West and Cho (1995), Beine et al. (2002), and Hansen andLunde (2005) among others.
However, it is important to note that the actual volatility, σ, is not associated with any standardized distributional function.Many prior studies have found that the return distributions of financial assets are rarely Gaussian but some other forms (Hull andWhite 1998), and common examples include Student's t-distribution, the stable Paretian distribution, a mixture of normaldistributions, and the generalized error distribution (Mittnik et al., 2002; Venkataraman, 1997). Others have also found that returndistributions are time-varying, fat-tailed, and asymmetrical (Leeves, 2007; Neftci, 2000). Therefore, estimating σ based on aconstant distributional assumption of returns can easily lead one to misconstrue the real level of uncertainty.

Another approach to volatility forecasting is the stochastic volatility (SV) modeling in studying the structure of return behaviorsuch as drift mechanisms, diffusion functions, and jumping patterns.3 Recently, SV modeling has gone through several significantrefinements such as applications of the Lévy process in derivative pricing.4 The main drawbacks of the method are first, thedifficulty in empirical application due to lack of closed form solution in estimation process, and second, correct dynamics of returnsare hard to be specified. For example, jumping processes in exchange rate due to arbitrary government interventions are difficultto be fully modeled and estimated. Examples using the SVmodels for exchange rate volatility measurement can be seen in Heynenand Kat (1994), Andersen et al. (2001), and Beine et al. (2007) among others.

An alternative approach is options-based volatility forecasting. The method requires the prices of financial assets and theiraccording options, along with a specific option pricing model, to generate the volatility forecast implied by the arbitrage-freecondition. Multiple forecasts are often derived for a single financial asset by choosing different contracts of options, and suchphenomenon is called the volatility smile. Galati et al. (2007) demonstrated that a risk-neutral probability density function can bederived from the exchange rate options prices and therefore served as information content for the ex-post volatility forecasts.5

Option series must be available for application of the method, however, and many exchange rates do not have public tradedoptions derivatives.

Taylor (2005) utilized Value at Risk estimates from quantile regression for volatility forecasting. First, the author applied theCAViaR quantile regression models proposed by Engle and Manganelli (2004) to produce pair of symmetric conditional tailquantiles. The volatility forecasts can then be produced using the interval approximation approach shown in Pearson and Tukey(1965), which proposed the following simple approximations to the standard deviation based on symmetric quantiles of asample.

σ̂ =Q̂ 0:99ð Þ−Q̂ 0:01ð Þ

4:65; σ̂ =

Q̂ 0:975ð Þ−Q̂ 0:025ð Þ3:92

;σ̂ =Q̂ 0:95ð Þ−Q̂ 0:05ð Þ

3:25ð3Þ

Q̂ 1−θð Þ and Q̂ θð Þ are the estimated quantiles for a cumulative probability θ. The denominators of Eq. (3) are based on the
wherecentral distances between estimated quantiles under Pearson curves and are slightly different from the denominators of 4.653(=2.326×2), 3.92 (=1.96×2), and 3.29 (=1.645×2) respectively under Gaussian distribution. In addition, Taylor (2005)extended this idea by proposing a regression model as follow to establish relationship between quantile intervals and squaredvolatility.
σ̂2t + 1 = α1 + β1 Q̂ t + 1 1−θð Þ−Q̂ t + 1 θð Þ

� �2; ð4Þ

σ̂2t + 1 is the squared volatility forecast for time t+1, and α1 and β1 are parameters to be estimated from the sample data.

(2005) concluded that the quantile regression approach to volatility forecasting outperformed GARCH and moving averages.

2.2. Exchange rate volatility

The nominal exchange rate is defined as the relative price of the currencies between two countries, and the real exchange ratemeasures the relative price of the goods of two countries. The relation between the two exchange rates can be specified as

� = e × P d= P f

� �ð5Þ

� is the real exchange rate, e is the nominal exchange rate, and Pd and Pf are respectively the domestic and foreign price
wherelevels. Thus, real exchange rate is a better representation for relative value of goods net of inflation difference between economies.Tenreyro (2007) provided evidence of systematic bias for regression analyses on the relationship between nominal exchange rate
etailed survey can be found in Andersen et al. (2002).se see Wu (2006) for an example.se also see Bonser-Neal and Tanner (1996), Galati et al. (2005), and Hammoudeh et al. (2010) as examples using information of exchange rate markets tovolatility dynamics.


volatility and trade. The author showed no significant effects from the nominal exchange rate volatility to international tradewhenthe biases are corrected.

The fluctuation of exchange rates can influence the prices of commodity across countries and the domestic consumption levels.As the exchange rate depreciates, commodities imported from foreign countries become relatively expensive and less competitivein domestic market, and consequently, profits of import firms drop. The degree of stability of the exchange rate has high impact onfirms' expected revenues and hedging costs. When exchange rate volatility increases, higher potential loss the firms have to sufferand less activity of trade would be conducted.

Hooper and Kohlhagen (1978) showed that exchange rate uncertainty has significant effects on the prices of internationaltrades. Using MSSD measurement, Arize et al. (2000) found that the increases in real effective exchange rate volatility exert asignificant negative impact on export for 13 less developed countries, and Cho et al. (2002) documented significant effects fromexchange rate volatility on growth of agricultural trade for ten developed countries. Peridy (2003) showed that impacts of theexchange rate volatility, measured by both MSSD and GARCH(1,1), on exports vary across country-specific and industry-specificsectors. De Vita and Abbott (2004) found significant negative impact from real exchange rate volatility to US exports for variousmarkets of destination. By GARCH(1,1), Choudhry (2005) provided evidence of significant negative impacts from both real andnominal exchange rate volatility to US exports to Canada and Japan.

Lahiri and Mesa (2006) found an inverse relation between exchange rate volatility and optimal local content requirement onexport-oriented foreign direct investment. Baak et al. (2007) investigated the negative correlations between exchange rate volatilityand export sectors in four East Asian countries, and Arize et al. (2008) documented negative relationship between real exchange ratevolatility, measured by ARCH(1), and export for Latin American countries. Lai et al. (2008) showed that when capital mobility is low,exchange rate volatility is positively related to interest rate differential. Al-Abri and Goodwin (2009) explored nonlinear impacts ofexchange rate shocks to import prices, and a significant cointegrating relationship between the two was documented.

Other empirical characteristics of exchange rate volatility have also been studied. Gonzaga and Terra (1997) documentedsignificant effects on real exchange rate volatility from inflation volatility, and Baum et al. (2001) showed significant impacts onfirms' profitability from exchange rate volatility. Hau (2002) showed that the relationship between real exchange rate volatilityand monetary and aggregate supply shocks depends on level of openness to trade. Burstein et al. (2003) found large distributioncosts for consumer good caused by stabilization policy for real exchange rate. Exchange rate volatility has been shown to varyamong countries and is more volatile for emerging markets than developed countries (Hausmann et al., 2006). By 5-minuterealized volatility, Frömmel et al. (2008) showed that order flows from financial customers and banks contribute to explainingexchange rate volatility but not the flows by commercial customers. Menkhoff et al. (2010) explored the relations betweenexchange rate volatility to carry trade profits where a significant risk-return relation in exchange rate is documented. Hence,forecasting volatility should improve the profitability of exchange rate forecasts.

It is worth tomention an important difference between trading of currency and trading of other financial assets. Many financialtime series are public traded solely with double auction, where the buyers and sellers usually have no interest to stabilize the pricemovements. Currency trading, on the other hand, is highly involved with governments whose trading purpose is sometimes thestabilization of the exchange rate. Beine et al. (2002) provided evidence of the effects on exchange rate volatility from central bankinterventions. Frenkel et al. (2005) documented significant relationship between central bank interventions and yen/dollarexchange rate volatility. Sideris (2008) found interventions in currencymarkets for six central and east European economies in thehope to smooth exchange rate volatility caused by speculative attacks.

Interventions of currency trading make modeling of exchange rate volatility a more challenging task. Kroner and Lastrapes(1993) used GARCH-in-mean for exchange rate volatility modeling, and Beine et al. (2002) found FIGARCH provides betterexchange rate volatility forecast than GARCH. Serven (2003) documented the threshold effect, a larger negative relationship, fromreal exchange volatility to private investment in developing countries, which are highly open and have less financial controlsystems. Holmes (2008) provided evidence of non-stationarity of real exchange rate under Markov regime switching framework,and Han (2008) characterized the asymmetric effects on exchange rates from macroeconomic shocks based on evidence ofintraday data. Fang et al. (2009) documented asymmetrical impacts of exchange rate volatility to exports between time periods ofappreciation and depreciation where GARCH(1,1)-in-mean model is used.

In sum, exchange rate volatility is one key factor in trading activity and other macroeconomic conditions such as monetary andaggregate supply shocks. It can be modeled with conditional variance GARCH methods where characteristics of long-termpersistence, time-dependence, asymmetric, and regime switching are specified. However, significant evidence shows thatdynamics of exchange rate returns can be affected by arbitrary government interventions, and consequently, the returndistributional structures of exchange rate vary through time. Nikolaou (2008) applied quantile regression to analyze behaviors ofquantiles for real exchange rate returns and found impacts of shocks vary between their magnitudes and lead to differentdynamics such as mean-reverting tendencies. An alternative approach to model exchange rate volatility in incorporating suchcharacteristic rather than conditional variance modeling, therefore, is designed with quantile regressions by previous researchesand further developed in this study.

3. Methodology

This research extends the work of Taylor (2005), summarized in the previous section, presenting an alternative model for therelationship between volatility forecast and estimated quantiles. The method requires one to generate estimates of the returndistributional quantiles without any specific assumptions concerning the distribution function. The estimated quantiles can


therefore be applied directly to volatility forecasts, either by approximating the relationship between quantiles and the variance orby regression models.

Engle and Manganelli (2004) presented following four different autoregressive schemes in generating the conditionalquantiles for VaR estimates, known as the CAViaR model.

6 One(2006)

7 ChuHuang

Adaptive : Qt θð Þ = Qt−1 θð Þ + β 1 + exp G rt−1−Qt−1 θð Þ½ �ð Þ½ �−1−θ� �

ð6Þ

Symmetric absolute value : Qt θð Þ = β1 + β2Qt−1 θð Þ + β3jrt−1j ð7Þ

Asymmetric slope : Qt θð Þ = β1 + β2Qt−1 θð Þ + β3 max rt−1;0ð Þ−β4 min rt−1;0ð Þ ð8Þ

Indirect GARCH 1;1ð Þ : Qt θð Þ = β1 + β2Q2t−1 θð Þ + β3r

2t−1

� �1=2 ð9Þ

In all these examples, G is a positive finite number, rt is the portfolio return, and the β′s are parameters to be estimated based onthe following minimization condition.

minβ

∑rt≥Q t θð Þ

θjrt−Qt θð Þj + ∑rtbQ t θð Þ

1−θð Þjrt−Qt θð Þj( )

: ð10Þ

The concept of least absolute error estimator in regression quantiles is originated by Koenker and Basset (1978), and theCAViaR model in VaR estimation extends the approach to forecast tail quantiles. Without any explicit distributional assumptions,the VaR estimates are generated by analyzing the serial dependence of quantiles along with return behaviors to reflect latestmarket conditions. The key feature of the adaptive scheme of Eq. (6) is to assign a large drift of current VaR estimates when thepast quantiles were exceeded by the extreme return shocks and a mild pull-back when the returns were stable and below the pastquantiles. The symmetric scheme of Eq. (7) is able to capture the positive risk-return relation, and the asymmetric scheme ofEq. (8) allows the VaR estimates response asymmetrically to positive and negative returns. When the error term of Eq. (9) is i.i.d,the Indirect GARCH(1,1) is the same as the GARCH(1,1) model.6

Taylor (2005) applied Eqs. (3) and (4) to utilize the VaR estimates by CAViaR model into volatility forecast by using 98, 95, and90% quantile intervals. In particular, the distance between symmetric quantiles of a distribution can be used as the key explanatoryvariable to explain the second moment behavior. While this method has been found to produce robust empirical results, it isnatural to ask whether the choice of any single quantile pair can be justified and optimized. For example, one might wonder if justone pair of quantiles is enough to accurately explain volatility behavior, or which pair provides best results. In this study, severalalternative versions of Taylor's (2005) methods are provided in order to address this issue.

Instead of using one pair of quantiles, this study uses a uniformly spaced series of quantiles.7 Themovements of these quantilesnot only reflect the tail behaviors, but thewhole distributional pattern. In thismanner one can incorporate distributional behaviorsinto volatility forecasting and avoid overestimation from extreme asymmetrical tails or underestimation from highly symmetricaltails. Specifically, following model is employed:

σ̂t + 1 = α1 + β1F Q̂ t + 1 θð Þ� �

; ð11Þ

F(∙) represents an unspecified function and Q̂ t + 1 θð Þ is a vector of quantiles {θ, 2θ,…,mθ} to be estimated at time t+1 (θN0,
wheremθb1). Ideally, one should use as many quantiles as possible to fully explain the pattern of distribution. In this study, θ is set to be0.01, covering the percentiles {1, 2,…, 99}. This paper also considers three different functions of F(∙), which are shown in thefollowing equations. In each case, Q is the conditional mean of all quantile estimates.
Standard deviation SDð Þ : F ⋅ð Þ = 1m−1

∑99

m=1Q 0:01mð Þ−Q� �2 !1=2

: ð12Þ

Weighted SD : F ⋅ð Þ = ∑99

m=1W Q 0:01mð Þ−Q� �2 !1=2

: ð13Þ

Median SD : F ⋅ð Þ = 1m−2

∑99

m=1Q 0:01mð Þ−Q 0:5ð Þð Þ2

!1=2

: ð14Þ

can apply the backtesting procedure with likelihood ratio tests to evaluate the quantile estimates between these different schemes; see Kuester et al.for an example.ng et al. (2010) applied quantile regression to reveal entire distribution of the differences in mispricing between E-mini and floor-traded index futures.(forthcoming) employed the method of uniformly spaced series of quantiles to forecast the return volatility of stock index.

Table 1Descriptive statistics of exchange rates and consumer price indices.

Panel A: Nominal exchange rates

AUD CAD EUR GBP HKD JPY KRW SGD TWD

Mean 1.44 1.33 0.93 0.61 7.77 120.01 956.94 1.70 30.00Median 1.36 1.34 0.91 0.61 7.77 118.96 878.28 1.69 30.16S.D. 0.20 0.14 0.14 0.05 0.03 14.48 224.32 0.18 3.30Kurtosis 0.64 −1.07 −1.27 −0.78 −1.52 −0.13 −0.48 −0.08 −1.56Skewness 1.24 0.21 0.37 −0.11 −0.09 0.29 0.64 0.43 0.05Minimum 1.12 1.07 0.73 0.50 7.70 80.63 658.32 1.39 24.52Maximum 2.09 1.61 1.21 0.73 7.82 159.65 1962.50 2.17 35.46

Panel B: Consumer price indices

Aust. Canada Europe U.K. H.K. Japan Korea Sing. Taiwan U.S.

Mean 93.79 94.83 106.18 90.95 91.09 96.72 87.76 94.61 90.95 92.22Median 93.50 94.67 106.50 91.33 99.90 98.21 89.16 98.00 96.83 92.90S.D. 15.22 12.09 5.46 16.29 20.49 4.15 22.19 8.11 11.58 14.93Kurtosis −0.79 −0.77 −1.14 −0.81 −0.67 0.67 −1.18 −0.79 −0.83 −0.97Skewness 0.01 −0.09 0.02 −0.23 −0.83 −1.37 −0.16 −0.72 −0.76 −0.02Minimum 63.40 70.25 96.80 58.76 47.10 85.88 48.11 77.70 67.25 64.60Maximum 121.30 117.50 116.10 121.10 117.40 101.83 123.25 105.30 105.01 120.80

Panel C: Real exchange rates

AUD CAD EUR GBP HKD JPY KRW SGD TWD

Mean 1.42 1.29 0.93 0.62 8.09 113.28 1013.20 1.64 30.43Median 1.35 1.30 0.90 0.62 8.07 112.95 934.93 1.69 29.26S.D. 0.21 0.16 0.13 0.06 1.22 15.54 164.02 0.18 4.54Kurtosis 0.20 −1.07 −1.20 −0.76 −0.81 −0.48 0.84 −1.03 −1.48Skewness 1.08 0.22 0.44 −0.24 0.31 −0.03 1.00 −0.51 0.27Minimum 1.11 1.03 0.74 0.50 6.23 72.13 812.03 1.28 23.99Maximum 2.08 1.62 1.21 0.75 10.68 149.81 1967.96 1.92 38.74

This table presents summary statistics of nominal exchange rates, consumer price indices, and real exchange rates for nine currencies against US dollar. All dataseries are extracted from Thomson Datastream database, and sample period is between January 1, 1987 and May 31, 2007, except Euro with sample periodbetween January 1, 1999 andMay 31, 2007. The nine currencies are Australian dollar (AUD), Canadian dollar (CAD), Euro (EUR), pound sterling (GBP), Hong Kongdollar (HKD), Japanese yen (JPY), Korea won (KRW), Singapore dollar (SGD), and Taiwan dollar (TWD). Abbreviations in panel B are Aust. as Australia, U.K. asUnited Kingdom, H.K. as Hong Kong, Sing. as Singapore, and U.S. as United States.

8 All series except Euro/Dollar have 5090 sample points while the Euro/Dollar series begins at January 1, 1999 with 2095 data points.


Instead of a single symmetric pair, this approach incorporates all the estimated quantiles into the volatility forecast. All thestandard deviation (SD) functions given above represent the width of the distribution of estimated quantiles around theirconditional mean, albeit in slightly different ways. The second SD function applies a weight parameter, W, to each squareddeviation in the sum.W is set to be θ/25 when θ≤0.5, and (1−θ)/25 otherwise. This ensures that centralized quantiles will have agreater impact on the volatility forecast. The third median SD function recognizes that many return distributions are somewhatirregular, the median is more representative of their center than the mean.

The adopted approach improves on previous quantile-basedmodels by using the evenly spaced series of estimated quantiles toexplain volatility. Not only does this remove the arbitrary choice of quantile pair in representing the spread of the distribution, theseries of quantiles providesmore complete information on the returns. If the right and left tails of the return distribution, as well assections around the mean or median, are driven by different forces, this approach can provide better understanding regarding thereturn distribution dynamics. One drawback of the approach is the requirement of significant computing resources; however,recent improvements in technology have made this shortcoming less critical.

4. Empirical findings

4.1. Data and model

The empirical application uses daily exchange rate series from nine currencies, all against US dollar, and they are Australiandollar (AUD), Canadian dollar (CAD), Euro (EUR), pound sterling (GBP), Hong Kong dollar (HKD), Japanese yen (JPY), Korea won(KRW), Singapore dollar (SGD), and Taiwan dollar (TWD). The sample period is from January 1, 1987 to May 31, 2007.8 Nominalseries and the consumer price indices (CPI) for the ten economies are all extracted from ThomsonDatastream database to compose

9 60 data points are used to generate the threshold condition under adaptive quantile regression. 500 in-sample is used for CAViaR where one extra data poinis scarified in composing the continuous return.10 Please see Taylor (2005) and Kuester et al. (2006) in details.

Table 2Descriptive statistics of real exchange rate volatility forecasts for AUD/USD.

Models Mean Median S.D. Skewness Kurtosis Minimum Maximum

MA 10.01 9.44 3.22 0.85 0.53 4.09 21.64MC 10.18 9.60 3.28 0.85 0.53 4.16 22.01GARCH 10.01 9.44 3.18 0.86 0.53 4.14 21.41IGARCH 10.00 9.43 3.19 0.86 0.58 4.09 21.67GJRGARCH 10.01 9.44 3.19 0.85 0.54 4.11 21.53

Interval approximation modelsIA99 AD 11.61 10.52 4.40 1.25 1.47 2.77 27.87IA95 AD 10.05 9.55 3.38 0.99 0.74 2.15 23.74IA99 SYM 11.34 10.83 2.63 0.67 1.48 6.09 22.49IA95 SYM 10.25 10.05 2.45 1.05 2.43 5.41 29.95IA99 ASYM 11.20 10.71 2.75 0.94 2.01 4.77 32.61IA95 ASYM 10.24 9.91 2.58 0.95 1.61 4.91 27.45

Interval regression modelsIR98 AD 10.04 9.51 2.81 0.71 0.16 4.09 18.39IR90 AD 10.06 9.57 2.73 0.68 0.12 3.78 20.33IR98 SYM 9.96 9.59 2.58 0.50 0.03 4.32 27.69IR90 SYM 10.02 9.65 2.85 0.64 0.02 4.36 20.35IR98 ASYM 10.01 9.55 2.47 0.47 0.07 4.58 25.37IR90 ASYM 10.02 9.64 2.62 0.55 0.32 2.53 20.32

Regression by quantiles modelsSD AD 10.07 9.53 2.84 0.71 0.14 4.51 18.81WSD AD 10.11 9.63 2.81 0.69 0.14 4.90 19.23MSD AD 10.07 9.54 2.84 0.71 0.15 4.53 18.78SD SYM 9.95 9.50 2.89 0.65 0.08 4.06 22.57WSD SYM 10.05 9.60 2.87 0.71 0.01 4.62 20.78MSD SYM 9.94 9.50 2.88 0.65 0.09 3.98 22.53SD ASYM 9.99 9.54 2.68 0.53 0.34 4.12 21.41WSD ASYM 10.03 9.62 2.67 0.59 0.27 3.87 19.69MSD ASYM 9.98 9.53 2.68 0.54 0.35 4.23 21.37

This table presents summary statistics of real exchange rate volatility forecasts for 26 models of AUD/USD with sample period of January 1, 1987 to May 31, 2007Total number of forecasts is 4029 and the forecast is volatility in annual percentage format. Detail descriptions of volatility models please refer to Sections 3 and 4and abbreviation of S.D. stands for standard deviation.


.,

the real exchange rates according to Eq. (5). The earliest 1061 data points are required for model estimationwhere the first 561 forquantile estimation9 and the next 500 for volatility forecasting. Volatility forecasts are generated for following models:

1. Moving Average (MA): the historical volatility method based on Eq. (1). A simple 30-day average is used in this method.2. Monte Carlo (MC) simulation: a process that simulates random price movements of financial assets, and volatility forecast can

be generated from the simulated returns. Following one-factor Brownian motion simulation function is used where S0 and STare prices at the start and end of period, respectively with μ as the return mean.

ST = S0exp μ−σ2

2

!+ σ�

" #: ð15Þ

3. GARCHmethod: three GARCHmodels are applied, first, the GARCH(1,1) model described in Eq. (2), and second, the integratedGARCH (IGARCH) as proposed by Nelson (1991), where the coefficients in Eq. (2) integrate to one. Third, the GJRGARCHmodel,proposed by Glosten et al. (1993) is also applied where an indicator function is imposed on the coefficient of lag return with thefunctional form as

σ2t = λ1 + λ2σ

2t−1 + 1−I εt−1 N 0½ �ð Þλ3ε

2t−1 + I εt−1 N 0½ �ð Þλ4ε

2t−1: ð16Þ

4. Interval approximation (IA) method: the approach of approximating second moments from quantiles originally proposed byPearson and Tukey (1965) and applied by Taylor (2005). As shown in Eq. (3), three different quantile pairs were empiricallyadopted, and based on previous applications and reviews of this methodology,10 two intervals, 99% and 95%, and three quantileregression functions: adaptive (AD), symmetric (SYM), and asymmetric (ASYM) are applied. In sum, a total of six models areused in this category: IA99-AD, IA95-AD, IA99-SYM, IA95-SYM, IA99-ASYM, and IA95-ASYM.

t

Fig. 1. Volatility forecasts and post estimated historical volatility of AUD/USD. This figure presents the daily post estimated historical volatility and volatilityforecasts of three key models of real AUD/USD exchange rate between October 1991 and May 2007. Detail descriptions of exchange rate series and volatilitymodels please refer to Sections 3 and 4.


5. Interval regression (IR) method: the VaR-based approach proposed by Taylor (2005), described in Eq. (4). Three differentintervals were examined as the explanatory variable, and cases of 98% and 90% performed better than an interval of 95%.Therefore, both quantile pairs, 98% and 90%, are used with the three quantile regression functions mentioned above, composingthe following six models: IR98-AD, IR90-AD, IR98-SYM, IR90-SYM, IR98-ASYM, and IR90-ASYM.

6. Regression-by-quantiles method: approach adopted by this study as presented in Section 3. Three quantile regression functionsare used and three approaches to combine the series of quantiles into an explanatory variable: standard deviation (SD),weighted SD (WSD), and median SD (MSD) are applied. Following nine models are considered in this category: (1) SD-AD, (2)WSD-AD, (3) MSD-AD, (4) SD-SYM, (5) WSD-SYM, (6) MSD-SYM, (7) SD-ASYM, (8) WSD-ASYM, and (9) MSD-ASYM.

In sum, total of 26 models are applied for nine series of real exchange rate over 19 years data for volatility forecasts. Table 1provides summary statistics of nominal exchange rates, consumer price indices, and real exchange rates for the sample data. Dueto the long sample period, the ranges between maximum and minimum exchange rates are generally large and the exchange rateof Korean won (KRW) against USD is most volatile with highest coefficient of variance.

Table 2 shows the summary statistics of AUD/USD exchange rate volatility forecasts for all 26 models. In this series, forecasts bytraditional methods such as moving average and GARCH models generally have higher standard deviations than intervalregression models and regression-by-quantiles method. The interval approximation models, however, generate forecasts withlargest ranges in average comparing to other approaches. Statistics of volatility forecasts for other exchange rate series areexcluded in the paper due to space limitation but available upon request. Fig. 1 illustrates the post estimated 30-day volatility andvolatility forecasts of three key models of AUD/USD between January 1991 and May 2007. For model of IA99-AD, we can observe avolatilemovement of its forecasts, which fluctuatemuchmore than forecasts of other twomodels. The trend of forecasts by SD-AD,on the other hand, is rather stable and closely correlates with the post estimated volatility.

4.2. Forecast evaluation by R2 coefficient

In order to evaluate the forecasts, following Taylor (2005), below regression model is applied.

σPHVt = a + b⋅σ̂ Forecast

t + εt : ð17Þ

Here, σtPHV is the post estimated historical volatility and σ̂Forecast

t is the volatility forecast. This evaluation is to determine theout-of-sample performances of the models by comparing how well the forecasts explain the actual post estimated daily volatility.The R2 coefficients of the evaluation regressions can then be used as information content in the comparison. Raunig (2008)

11 Both realized volatility and implied volatility can be used as the benchmarks of volatility predictions when the representative intraday data are available andrisk premium of implied volatility can be controlled. Due to data limitations, only post estimated daily volatility is used in this study.12 Adaptive function has been documented superior than the symmetric and asymmetric functions in empirical works of Engle and Manganelli (2004), Taylo(2005), and Kuester et al. (2006).

Table 3R2 measure of the information content for volatility forecasts by evaluation of 20-day post estimated historical volatility.

Models AUD CAD EUR GBP HKD JPY KRW SGD TWD Mean

MA 27.92 45.16 21.89 27.48 0.35 19.78 38.98 41.42 7.33 25.59MC 27.92 45.16 21.88 27.48 0.35 19.78 38.97 41.42 7.33 25.59GARCH 27.09 43.98 20.66 26.25 0.15 18.44 35.12 39.88 6.82 24.27IGARCH 27.17 43.92 20.47 26.21 0.16 18.42 38.15 40.13 7.41 24.67GJRGARCH 27.11 44.08 20.67 26.22 0.15 18.49 36.46 40.01 7.17 24.49

Interval approximation modelsIA99 AD 26.23 31.20 16.55 26.55 0.07 20.48 33.76 43.11 6.97 22.77IA95 AD 26.75 39.87 7.78 26.07 0.34 18.44 31.59 44.36 5.52 22.30IA99 SYM 15.68 38.66 14.25 28.18 0.38 14.35 44.38 38.99 0.60 21.72IA95 SYM 20.92 42.31 3.94 29.75 0.03 16.51 42.40 44.85 6.98 23.08IA99 ASYM 13.12 36.31 13.60 25.07 0.26 11.41 38.29 38.43 0.47 19.66IA95 ASYM 18.52 40.75 2.83 27.96 0.00 13.05 42.57 44.38 5.49 21.73

Interval regression modelsIR98 AD 31.25 43.46 16.96 28.95 0.01 18.00 34.16 43.00 6.22 24.67IR90 AD 29.72 42.46 16.62 29.25 0.15 18.28 40.42 43.40 7.95 25.36IR98 SYM 20.48 42.67 13.72 29.45 0.01 13.89 28.62 44.48 2.27 21.73IR90 SYM 25.35 45.50 10.09 28.59 0.64 17.24 34.48 42.06 5.90 23.32IR98 ASYM 19.82 42.40 14.64 29.07 0.04 12.18 28.86 42.45 1.34 21.20IR90 ASYM 22.35 41.51 9.60 27.97 0.10 15.41 32.46 41.39 6.04 21.87

Regression by quantiles modelsSD AD 31.02 43.38 18.02 29.34 0.01 18.64 38.36 43.50 7.31 25.51WSD AD 30.46 43.90 16.81 29.36 0.01 18.78 42.03 43.66 8.12 25.90MSD AD 31.16 43.51 18.21 29.41 0.01 18.63 38.61 43.61 7.40 25.62SD SYM 25.73 44.07 12.23 30.96 0.00 15.90 33.70 43.01 3.57 23.24WSD SYM 26.87 44.81 11.01 29.99 0.00 16.72 34.84 40.15 5.05 23.27MSD SYM 26.02 44.22 11.99 31.00 0.00 15.98 34.04 42.87 3.54 23.30SD ASYM 22.50 43.37 12.04 29.41 0.05 14.80 31.49 42.74 3.46 22.21WSD ASYM 23.68 43.39 11.55 29.15 0.01 15.68 32.03 40.80 4.71 22.33MSD ASYM 22.81 43.50 11.81 29.41 0.04 14.74 31.81 42.64 3.37 22.24

This table presents the R2 information content of post forecast evaluation with 20-day historical volatility for nine real exchange rates and 26 volatility models.Higher the R2, closer to the post estimated historical volatility the forecast is. Detail descriptions of exchange rate series and volatility models please refer toSections 3 and 4.


showed that exchange rate volatility is hard to predict more than 1 month ahead with time series methods. Therefore, this studymainly uses post estimated historical volatilities in 20- and 30-day for evaluations as well as 60-day for robust check.11

Table 3 presents the R2 coefficients of the evaluation regressions by 20-day post estimated historical volatility for all ninereal exchange rate series and for all 26 models. For the AUD/USD series, most of volatility forecasts are able to explain the 20-dayhistorical volatility more than 20% where the IR98-AD model has the highest R2 as 0.3125 and IA99-ASYM has the lowest R2 as0.1312. The three regression-by-quantiles models based on adaptive function, SD-AD, WSD-AD, and MSD-AD, have the next bestR2 measures, all above 30%. For the CAD/USD series, R2 measures ofmostmodels are above 40%with IR90-SYMperforming the bestwith 45.5% explaining power.

Through the rest of exchange rate series, MA performs best for EUR/USD, MSD-SYM performs best for GBP/USD, three intervalapproximation models performs best for JPY/USD, KRW/USD, and SGD/USD, and WSD-AD performs the best for TWD/USD. Theexplaining powers for HKD/USD series are significantly lower than others with R2 coefficients smaller than 1% for all models. NoGARCH models perform best for any of series, and apparently, no single family method overwhelmingly outperforms others.However, when averaging the measures across all series, the adopted regression-by-quantiles WSD-AD model performs the bestwith highest mean R2 of 0.2590 and IA99-ASYM model performs the worst with lowest mean R2 of 0.1966.

Table 4 presents the R2 coefficients of the evaluation regressions by 30-day post estimated historical volatility for all series andall models. Overall, MA, interval approximation method, and regression-by-quantile method all have best R2 measures for 2exchange rate series where interval regression method performs best for 3 series. Again, GARCH models do not provide bestforecasts for any series and no method significantly outperforms others. In averaging the R2 measures, the three regression-by-quantiles models with adaptive function, SD-AD, WSD-AD, and MSD-AD, have the best results of 0.2622, 0.2651, and 0.2634,respectively. Within the same regression-by-quantiles category, the adaptive function provides better results than other two, andthis outcome is consistent with prior studies.12 The interval regression method and traditional MA, MC, and GARCH models havesimilar level of mean R2 where interval approximation method has the worst mean R2 outcome in average.

r







This table presents the R2 information content of post forecast evaluation with 30-day historical volatility for nine real exchange rates and 26 volatility modelsHigher the R2, closer to the post estimated historical volatility the forecast is. Detail descriptions of exchange rate series and volatility models please refer toSections 3 and 4.


.

Table 5 presents the R2 coefficients of the evaluation regressions by 60-day post estimated historical volatility and providessimilar observations as Tables 3 and 4. The interval approximation method and regression-by-quantile method each have best R2

measures for 3 exchange rate series and interval regression method performs best for 2 series. The three regression-by-quantilesmodels with adaptive function, again, have the best mean R2 coefficients of 0.2642, 0.2667, and 0.2656, respectively. Between thethreemodels, weight SD (WSD) always outperforms other two slightly and against the rest of models, and the outcomes show thatmethod by quantiles with heavier weights around the mean explain volatility better than models with simple average.

In sum, post estimation evaluations by all 3 historical volatilities reveal consistent outcomes. No singlemethodoutperformsothers formajority sample exchange rate series, and GARCH models do not perform best in any case of evaluations. When averaging the R2

measures across series, the adopted regression-by-quantilesmethodwith adaptive function andweight composition always has the bestoutcomes with highest explaining power for all 3 historical volatilities. It implies that even if there is no single method overwhelminglybetter than others in this empirical exercise, the proposed method consistently provides most reliable volatility forecasts in average.

4.3. Alternative evaluation statistics and sub-periods

In order to confirm the findings by R2 coefficients, alternative forecast evaluation statistics are applied with all data series andmodels. In particular, the root mean square error (RMSE) and Theil's U (Thiel, 1966) statistics are applied where the Theil's U iscomputed based on following formula.

Theil� U =∑T

t=1 σ̂Forecastt −σPHV

t

� �2∑T

t=1 σPHVt

� �20B@

1CA

1=2

: ð18Þ

Table 6, which shares the same structure as Table 4, presents the outcomes of RMSE by the 30-day post estimated historicalvolatility. Themodel performs better are the ones with smaller RMSE. For AUD/USD as example, the RMSE for first group of modelsincluding MA, MC, and GARCH methods have the RMSE between 0.1222 and 0.1235, where the second group of intervalapproximation model generally has higher RMSE between 0.1232 and 0.1635. The adopted models of regression-by-quantiles

13 Model performs better than (equal to; worse than) the naïve model has the Theil's U smaller than (equal to; larger than) 1. Model with the statistic furthebelow 1 improves more from the naïve model.







This table presents the R2 information content of post forecast evaluation with 60-day historical volatility for nine real exchange rates and 26 volatility modelsHigher the R2, closer to the post estimated historical volatility the forecast is. Detail descriptions of exchange rate series and volatility models please refer toSections 3 and 4.


.

method have all of their RMSE below 0.12, where the three models with adaptive autoregressive function (i.e. SD-AD, WSD-AD,andMSD-AD) have their RMSE below 0.1, the smallest among all. This is also the case for four other exchange series including CAD/USD, JPY/USD, KRW/USD, and TWD/USD, where the regression-by-quantiles models with adaptive function perform the best. ForGBP/USD and SGD/USD series, it is regression-by-quantiles models with the symmetric function performing the best with smallestRMSE. For EUR/USD, the interval regression models, particularly IR98-SYM and IR98-ASYM, generate two smallest RMSE with0.0461 and 0.0465, respectively. However, the regression-by-quantiles models with adaptive function have next best results ofRMSE between 0.0474 and 0.0479.

Table 7 presents the outcomes by Theil's U statistic, also with the 30-day post estimated historical volatility. Analogous to theRMSE, themodel performs better are the oneswith smaller Theil's U.13 For AUD/USD as example, threemodels have smallest Theil's Uof 0.263 including IR98-AD, SD-AD, and MSD-AD, andmodels have two largest Theil's U are IA99-AD and IA99-ASYMwith 0.330 and0.307, respectively. Thus, for AUD/USD series, the regression-by-quantiles and interval regression models generally perform betterthanothers. For other series, the regression-by-quantilesmodelswith adaptive functionhave smallest Theil's U for four exchange ratesincluding CAD/USD, GBP/USD, HKD/USD, and JPY/USD. The regression-by-quantiles models with symmetric function have smallestTheil's U for three exchange rates including EUR/USD, SGD/USD, and TW/USD. The only series that the regression-by-quantilesmodelsdo not perform best is the KRW/USD where IA95-SYM and IA95-ASYM both have the best result with the Theil's U.

During the sample period of this study, some exchange rate series may suffer from structural breaks, for both level price andvolatility, due to changes of exogenous conditions. For instance, Frenkel et al. (2005) documented significant interventions of Bankof Japan (BoJ) on the JPY/USD exchange rate until March 2004, and Hillebrand and Schnabl (2008) demonstrated significantstructural break of JPY/USD volatility at the time of early 2004. Therefore, the forecast evaluation of the yen/dollar series in thisstudy should be conducted separately for sub-periods before and after March 2004 to see if the performances of volatilityforecasting vary among various structures. For another example, the Euro dollar had significantly depreciated since itsintroduction from 0.83 (EUR/USD in real terms) in 1999 to 1.2 in late 2000 but then appreciated to 0.75 in 2004. Fig. 2 presents thetrend of real EUR/USD series between 1999 and May 2007, and in addition to above observations, we can see that after 2004, theseries fluctuated around 0.8 without significant upward or downward trends. Since in-sample size of 1061 is required for

r

Table 6Root mean square error measure of the information content for volatility forecasts by evaluation of 30-day post estimated historical volatility.






This table presents the root mean square error (RMSE) information content of post forecast evaluation with 30-day historical volatility for nine real exchange ratesand 26 volatility models. Lower the RMSE, closer to the post estimated historical volatility the forecast is. Detail descriptions of exchange rate series and volatilitymodels please refer to Sections 3 and 4.


estimation in this study, the earliest forecast available of the EUR/USD series is for September 2002 (so no forecast available for thedepreciation period), the forecast evaluation is applied for two sub-periods prior and after year 2004 for Euro–US dollar series.

Table 8 presents the R2 coefficients of the evaluation regressions by 30-day post estimated historical volatility of JPY/USD andEUR/USD series separately for the sub-periods. The first and second columns show the R2 coefficients for all models of JPY/USDbefore and after March 2004, respectively. During the sub-period prior to March 2004, where the BoJ intervention of the yenexchange rate is significant, the best three models of largest R2 coefficients are IA99-AD, WSD-AD, and SD-AD, and the IA99-ASYMmodel performs significantly worse than others with R2 of 0.0870. During the sub-period after March 2004, with no BoJintervention in the JPY/USD exchange rate, the R2 coefficients are generally low where MA, MC, and SD-AD are models with threebest results of 0.0479, 0.0466, and 0.0417, respectively. The outcome shows that, first, the performances of volatility forecasting forJPY/USD exchange rate without BOJ intervention are generally worse than ones in period with the central bank intervention.Second, although the adopted regression-by-quantiles models do not dominate other models in both sub-periods, they stillgenerate reliable forecasts with relatively higher explanatory powers.

The third and fourth columns show the R2 coefficients for all models of EUR/USD before and after 2004, respectively. In the thirdcolumn, during the time where Euro significantly appreciated against the Dollar, the best three models are MC, MA, and SD-ADwith three largest R2 coefficients of 0.1255, 0.1045, and 0.0997, respectively. In average, the regression-by-quantiles modelsperform better than the interval models. During the sub-period after 2004 where the average EUR/USD exchange rate was around0.8, the volatility models generally perform better than they did in sub-period before 2004. In this sub-period, the best threemodels are MSD-AD, MA, and WSD-AD with R2 coefficients of 0.3375, 0.3348, and 0.3320, respectively. Similarly to results before2004, the regression-by-quantiles models may not overwhelmingly dominate the traditional models such as MA, MC, and GARCHmodels but perform generally well with relatively high R2 coefficients and significantly better than the interval models.

In sum, the evaluations of volatility forecast by both statistics of RMSE and Theil's U basically confirm the outperformance ofadoptedmodels. The regression-by-quantilesmodels with adaptive and symmetric functions respectively generate smallest RMSEfor 5 and 2 exchange rate series (out of total 9 series). These regression-by-quantiles models together have smallest Theil's U for8 exchange rate series, with KRW/USD as the only exception. In addition, with consideration of structural changes in exchange ratevolatility, the forecast evaluations are applied for both series of JPY/USD and EUR/USD for different sub-periods. The outcomes

Table 7Theil's U measure of the information content for volatility forecasts by evaluation of 30-day post estimated historical volatility.






This table presents the Theil's U information content of post forecast evaluation with 30-day historical volatility for nine real exchange rates and 26 volatilitymodels. Lower the Theil's U, more improvement than the naïvemodel the forecast has. Detail descriptions of exchange rate series and volatility models please refeto Sections 3 and 4.

Fig. 2. Real exchange rate of Euro/US dollar. This figure presents the real exchange rate of EUR/USD between January 1999 and May 2007. Data including nominaexchange rate and consumer price indices are extracted from Datastream database.


r

l

image of Fig.�2

Table 8R2 measure of volatility forecasts evaluation with 30-day historical volatility for Japanese yen and Euro dollar by sub-periods.

Models Japanese yen Euro dollar

October 1990–February 2004 April 2004–May 2007 September 2002–December 2003 January 2004–May 2007

MA 17.25 4.79 10.45 33.48MC 16.35 4.66 12.55 28.56GARCH 16.04 3.04 9.20 32.38IGARCH 16.02 2.97 9.01 32.14GJRGARCH 16.11 3.06 5.21 32.32

Interval approximation modelsIA99 AD 20.44 0.14 5.21 29.72IA95 AD 18.94 0.19 1.68 13.61IA99 SYM 12.10 0.14 8.68 26.99IA95 SYM 15.53 0.09 1.77 11.42IA99 ASYM 8.70 0.14 6.26 22.24IA95 ASYM 11.87 0.16 1.53 10.11

Interval regression modelsIR98 AD 15.86 0.09 8.74 29.79IR90 AD 16.62 0.28 6.35 30.64IR98 SYM 11.59 1.46 2.87 28.46IR90 SYM 16.13 1.21 7.20 23.31IR98 ASYM 10.07 2.91 8.55 35.61IR90 ASYM 13.37 1.67 3.18 23.43

Regression by quantiles modelsSD AD 18.96 4.17 9.97 32.44WSD AD 19.16 3.33 9.41 33.20MSD AD 18.93 4.16 9.06 33.75SD SYM 14.26 2.82 6.16 28.65WSD SYM 15.34 2.65 6.95 27.30MSD SYM 14.34 2.88 6.98 28.21SD ASYM 12.84 2.22 7.40 28.57WSD ASYM 13.66 2.39 8.03 26.75MSD ASYM 12.77 2.08 6.79 28.23

This table presents the R2 of forecast evaluation with 30-day historical volatility for exchange rates of JPY/USD and EUR/USD for 26 volatility models. Outcomes arereported for two sub-periods of each exchange rate. Detail descriptions of volatility models and determinations of sub-periods please refer to Sections 3 and 4


.

show that the adopted regression-by-quantiles models consistently perform well in different sub-periods with relatively reliablevolatility forecasts.

5. Concluding remarks

Volatility is one key indicator of financial asset uncertainty, and exchange rate volatility, in particular, has been evidencedhaving significant impacts to many macroeconomic conditions such as international trade and monetary shocks. With reliableexchange rate volatility forecast, operations of governments and central banks in exchange markets can be more efficient andhedging costs of firms can be reduced. In this paper, a quantile regression approach to volatility forecasting is presented. Quantileapproaches assume no explicit distribution and do not require derivative data series, large amount of intraday data, and complexprocesses in estimation. The new method is compared to other key volatility forecasting techniques including moving average,Monte Carlo simulation, GARCH models, and existing quantile-based methods. The adopted approach improves on previous byusing an evenly spaced series of estimated quantiles to forecast volatility.

An empirical study using the daily returns of nine real exchange rates shows that the adopted models outperform traditionalmethods in average. The method provides closest forecasts of post estimated historical volatilities in 20-, 30-, and 60-days for 2, 2,and 3 individual exchange rate series, respectively. In particular, the regression-by-quantiles method with adaptive function andweighted composition performs best with highest average explanatory power in all three evaluations. It shows although no singlemodel overwhelmingly outperforms others for individual series but the adopted method consistently provides reliable volatilityforecasts. In addition, when different evaluation statistics are used, the regression-by-quantiles models outperform others in mostsample series. At last, the outperformance of the adoptedmodels is robust to different sub-periods for Japanese yen and Euro dollarwhere their exchange rate volatilities structurally change during the entire sample period.

One possible extension from this study would be to estimate higher moments of the return distribution from the quantileseries, thereby gaining a deeper understanding of the volatility behavior and return dynamics. Weighted schemes of the quantileexplanatory function might be optimized through simulation study, and the threshold structures under adaptive autoregressivequantile regression could be adjusted according to sample characteristics. These are certainly areas deserving further exploration.


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