Conditional Volatility and Distribution of Exchange

Studies in Nonlinear Dynamics & Econometrics Volume 11, Issue 3

2007

Article 1

Conditional Volatility and Distribution of Exchange Rates: GARCH and FIGARCH Models with NIG Distribution Rehim Kiliç∗

∗

Georgia Institute of Technology, [email protected]

c Copyright 2007 The Berkeley Electronic Press. All rights reserved.

Conditional Volatility and Distribution of Exchange Rates: GARCH and FIGARCH Models with NIG Distribution∗ Rehim Kiliç

Abstract This paper extends the Fractionally integrated GARCH (FIGARCH) model by incorporating Normal Inverse Gaussian Distribution (NIG). The proposed model is flexible and allows one to model time-variation, long memory, fat tails as well as asymmetry and skewness in the distribution of financial returns. GARCH and FIGARCH models for daily log exchange rate returns with Normal, Student’s t and NIG error distributions as well as GARCH/FIGARCH-in-mean models with t errors are estimated and compared both in terms of sample fit as well as out-of-the-sample predictive ability in several dimensions. The FIGARCH model with symmetric and asymmetric NIG errors outperform alternatives both in-sample fit and 1-day and 5-day ahead predictions of the quartiles of the exchange rate return distributions.

∗

I would like to thank to two anonymous referees and the editor of the Journal. Their comments have improved the paper substantially. I would also like to thank to seminar participants at Georgia Institute of Technology, School of Mathematics and School of Economics, participants of 2006 Midwest Econometrics Group Meeting and North American Summer Meeting of Econometric Society, 2007. Lastly, I would like to thank to Richard T. Baillie for his useful comments on an earlier version of the paper and Lars Forsberg for sharing his Gauss code with me. The usual disclaimer applies.

Kiliç: Volatility and FIGARCH-NIG Model

1

1

Introduction

Empirical research has documented three stylized facts on daily and weekly floating exchange rates: they are approximately martingales, conditionally heteroscedastic (volatility clustering) with long term dependence in the conditional second moment, and they exhibit excess kurtosis, see for instance, Baillie (1989), Hsieh (1989), and Baillie et al. (1996). Generalized Autoregressive Conditionally Heteroscedastic (GARCH) (Engle 1982, and Bollerslev 1986) is used to model the time varying volatility in financial markets. However, as shown in Baillie et al. (1996) GARCH models are not capable of capturing the “hyperbolic memory” observed in the conditional volatility process.1 Fractionally Integrated GARCH (FIGARCH) model of Baillie et al. (1996) has proved to be useful in capturing both the volatility clustering and the long term persistence parsimoniously. The FIGARCH model has been used extensively in modeling volatility dynamics and long memory in commodities, equities and exchange rate returns in a number of recent papers. In addition to Baillie et al. (1996), examples include Beltratti and Morana (1999), Baillie, C ¸ e¸cen and Han (2001), Baillie and Osterberg (2000), Brunetti and Gilbert (2000) and Kılı¸c (2004) among others. A related line of research on occasional structural breaks and long memory discusses the relevance of long memory in volatility. Some skeptical views about long memory have been offered by Mikosch and Starica (1998), Baine and Laurent (2001), Bredit and Hsu (2002) and Granger and Hyung (2004). These studies show that presence of occasional structural breaks in the data can generate slowly decaying autocorrelations and hence may lead to findings of long memory in the conditional volatility of exchange rate and stock returns. Therefore part of the long memory may be caused by the presence of neglected breaks in the series and findings of long memory in volatility measures might be spurious. On the other hand, Diebold and Inoue (2001) argue that long memory may be a useful description, particularly for forecasting purposes, even if the data generating process shows structural breaks and weak dependence. Morana and Belteratti (2004) provide supporting evidence on the existence of long memory in the variance process and argue that the presence of long memory in the volatility cannot be fully explained by unaccounted structural breaks. Several alternative distributions have been proposed in the literature to 1 For a technical definition of long memory and hyperbolic memory in the context of volatility, see Baillie et al. (1996) and Davidson (2004).

Published by The Berkeley Electronic Press, 2007

Studies in Nonlinear Dynamics & Econometrics

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Vol. 11 [2007], No. 3, Article 1

better account for the deviations from normality in the conditional distributions of returns in financial markets including the Students’ t distribution (due to Bollerslev 1987) and the General Error Distribution (due to Nelson 1991). More recently, Barndorff-Nielsen (1997), Andersson (2001) and Jensen and Lunde (2001) introduced Normal Inverse Gaussian (NIG) distribution with conditional volatility following a GARCH or a stochastic volatility process. Andersson (2001) and Forsberg and Bollerslev (2002) show that volatility models with NIG distributed errors fit better in the tails of the distribution than the t distribution. Jensen and Lunde (2001) show that the NIG distribution fits better not only at the tails of daily stock index returns but also at the center of the distribution. As shown in Barndorff-Nelson (1997), NIG distribution has semi-heavy tails and hence it is a natural candidate to consider in modeling the distribution of daily foreign exchange rate and equities returns. Moreover, NIG distribution is closed under temporal aggregation. For example if daily returns are NIG distributed, then weekly returns, defined by the summation of the daily returns, will also be NIG.2 It should also be emphasized that the NIG distribution is obtained as a special case of the General Hyperbolic (GH) distribution which is a normal variancemean mixture with the Inverse Gaussian (IG) being the mixing distribution (for details see the discussions in Barndorff-Nielsen 1978 and Jørgensen 1982). More recently, Forsberg and Bollerslev (2002) provide empirical justification for the relevance of NIG distribution for the conditional returns by drawing on the mixture of distributions hypothesis of Clarke (1973) and the notion of realized volatility as discussed in Andersen et al. (2001). In other words, Forsberg and Bollerslev (2002) links the conditional variance to the conditional realized volatility constructed from the sum of high-frequency intraday returns and thereby provide empirical foundation for the NIG distribution in exchange rate and stock returns. Building upon the recent literature, in this paper, we extend the GARCHNIG to FIGARCH model with NIG errors. This addition to the literature allows us to study jointly volatility clustering and long range persistence in volatility process as well as some important distributional characteristics of exchange rate returns (including asymmetries and fat tails) in a parsimonious parametric way. This paper performs a careful empirical assessment of the performance of proposed model against several alternatives, including 2

That P is, if P xi ∼ i.i.d.N IG(a, b, µi , δi ) for i = 1, · · · , m m 2 N IG(a, b, m µ , i i=1 i=1 δi ).

http://www.bepress.com/snde/vol11/iss3/art1

then

Pm

i=1

xi

∼


3

GARCH and FIGARCH models with normal, t and NIG errors as well as GARCH and FIGARCH-in-mean models with student’s t errors by utilizing several in and out of sample tools. Our analysis in the paper demonstrates that the proposed model outperforms the alternatives in several dimensions including in terms of extensive in-sample diagnostic statistics as well as in terms of 1- and 5-day ahead predictive ability of the distribution of daily log exchange rate returns. Findings of the paper show that while a FIGARCH model with a symmetric NIG distribution fits better for Dollar-Euro returns, a FIGARCH model with an asymmetric NIG error distribution characterizes the Canadian Dollar-Dollar, Dollar-British Pound, German Mark-Dollar, Swiss FrancDollar and Yen-Dollar returns. Our results also reveal presence of a statistically significant hyperbolic decay memory in conditional volatility of daily exchange rate returns across different distributions. The estimated long memory parameters range between about 0.22 to 0.52 across currencies. When evaluated in the light of the findings of Davidson (2004), these estimates reveal quite persistent conditional volatility dynamics for the DollarEuro and Yen-Dollar returns, relatively less persistent conditional volatility dynamics for Canadian Dollar, German Mark, and British Pound and moderate persistence in the Swiss Franc returns. Our findings also show that there exists a statistically significant asymmetric component in the distribution of five out of six daily exchange rate returns and modeling the asymmetry of returns parametricaly as suggested by the NIG distribution generally improves the performance of the estimated FIGARCH models. The rest of the paper is organized as follows. Section 2 discusses GARCH and FIGARCH models with NIG errors. Estimation results and within and out-of-sample evaluation of alternative models are presented and discussed in section 3, and the final section concludes the paper.

2

FIGARCH Model with NIG Errors

Following Barndorff-Nielsen (1997) and Jensen and Lunde (2001), we use the following parametrization of the density function of an NIG distributed



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Vol. 11 [2007], No. 3, Article 1

random variable x, √ a x−µ 2 2 NIG(x; a, b, µ, δ) = exp ( a − b + b (1) πδ δ −1 x−µ x−µ ×q K1 aq δ δ p where q(y) = 1 + y 2 with 0 ≤ |b| ≤ a, µ ∈ R, and δ > 0, and K1 (.) denotes the modified Bessel function of the third order and index one. The parameters of an NIG distribution can be interpreted as follows: a and b are shape parameters with a determining the steepness and b determining the asymmetry of the distribution. The larger the value of a the steeper the density is. Moreover, for b = 0 the density is symmetric. The parameter δ is a scale parameter, and µ is the location parameter as such when b = 0, µ denotes the mean of the distribution.3 Andersson (2001) and Forsberg and Bollerlev (2002) set b = 0, while Jensen and Lunde (2001) consider a 6= 0 and b 6= 0 which allow one to characterize both the peakedness and skewness of the distribution. It should also be noted that as a → ∞, the normal distribution is the limiting case. For details of statistical properties of NIG distribution see Barndorff-Nielsen (1978) and Jørgensen (1982). To present the F IGARCH model with NIG error distribution, suppose that a discretely sampled return series can be written as √ b γ σt + zt σt , for t = 1, · · · , T. (2) rt = µ + a where zt is a zero-mean and unit variance process. Following Andersson (2001) and Jensen and Lunde (2001), let the √density of zt to be given by an √ 3/2 b γ NIG distribution as such, zt ∼ NIG(a, b, − a , γ a ), where γ = a2 − b2 . This way of specifying the NIG distribution implies that the conditional distribution of returns will be NIG as well. That is rt |Ωt−1 ∼ NIG(a, b, µ,

γ 3/2 σt ), a

3

The first four central moments of the NIG distribution as parameterized here are 2 (b/a) 5 , and µ4 = 3/2 , µ3 = 3δ 3 √ given by,µ1 = µ + √ δ(b/a) 2 , µ2 = √ δ 2 2 2 1−(b/a)

4

4(b/a)2 +1

3 aδ 3 √

1−(b/a)2

7 .

a

1−(b/a)

a

For details, see Barndorff-Nielsen (1997).


1−(b/a)


5

where Ωt−1 is the information set generated by the past daily returns, Ωt−1 = σ(rt−1 , rt−1 , rt−2 , · · · ). The conditional mean and variance of returns are given by √ b γ E[rt | Ωt−1 ] = µ + σt , for t = 1, · · · , T a and V ar(rt |Ωt−1 ) = σt2 for t = 1, · · · , T. This parametrization of the NIG distribution allows to model temporal dependence in the conditional variance of the random variable to be given solely √ 2 2 by σt2 . Let ut = rt − E(rt |Ωt−1 ) = rt − σt b aa−b − µ be the innovation of the return process. The GARCH(p, q) − NIG model is then obtained by assuming a GARCH(p, q) specification for the conditional variance process, σt2

=ω+

q X j=1

αj u2t−j

+

p X

2 βj σt−j .

(3)

j=1

The F IGARCH(p, d, q) model of Baillie et al. (1996) which captures the hyperbolic decay in the volatility process is given by φ(L)(1 − L)d u2t = ω + [1 − β(L)]vt ,

(4)

where φ(L) = [1 − β(L) − α(L)], all the roots of φ(L) and [1 − β(L)] lie outside the unit circle, vt = u2t − σt2 , and 0 < d < 1. For 0 < d < 1, φ(L) is a finite order polynomial. As it is evident from (4), FIGARCH model nests GARCH and integrated GARCH (IGARCH) models in the sense that when d = 0 FIGARCH model reduces to GARCH model while for d = 1 it becomes an IGARCH model. The conditional variance of ut , or infinite ARCH representation of FIGARCH process, is given by σt2 = ω(1 − β(1)) + [1 −

φ(L) (1 − L)d ]u2t = ω/(1 − β(1)) + λ(L)u2t , (5) 1 − β(L)

where λ(L) = λ1 L + λ2 L2 + · · · . For the F IGARCH(p, d, q) process to be well defined and the conditional variance to be positive for all t, all the coefficients in the infinite ARCH representation in (5) need to be nonnegative, i.e. λj ≥ 0 for j = 1, 2, · · · . The general conditions for nonnegativity of lag coefficients in λ(L) are not easy to establish, but as illustrated in Baillie et al. (1996), it is possible to show sufficient conditions in a case by case



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Vol. 11 [2007], No. 3, Article 1

basis. In a recent paper, Conrad and Haag (2006) derive necessary and sufficient conditions for the nonnegativity of the conditional variance in the F IGARCH(p, d, q) model of the order p ≤ 2 and sufficient conditions for the general model with p > 2. For 0 < d ≤ 1, λ(1) = 0, the second moment of the unconditional distribution of ut is infinite, and hence, similar to an integrated GARCH (IGARCH) process, FIGARCH process is not covariance stationary. The conditions for the strict stationarity of the FIGARCH models are not well known (see Zaffaroni 2004). Estimation of F IGARCH(p, d, q) model with a given error distribution is carried out through Quasi Maximum Likelihood (QMLE) as discussed in Baillie et al. (1996). The log-likelihood function for the FIGARCH model with NIG errors is given by √ 3 √ 2 2 2 2 logL(ζ, ut ) = T 2 ln(a) − ln(π) − ln( a − b ) + a − b (6) 2 ! T T T X X X u 1 t ln(σt2 ) + b √ 2 2 − ln(qt ) + ln(K1 (aqt )) − 2 σt ( a −b ) t=1 t=1 t=1 a

where T is the sample size (adjusted for the initial values), ζ = is the relevant parameter vector (for example in F IGARCH(1, d, 1) model with a moving averagerterm added to the conditional mean process ζ ′ = (µ, θ, ω, d, β, φ, a, b)), qt =

1+

u2t 2 2 )(3/2) (a −b σt2 a2

, K1 is the modified Bessel function of the third

order and index one. Although, there is no formal results that show the asymptotic consistency and normality of QMLE in the context of general F IGARCH(p, d, q) model, extensive simulation experiments reported in Baillie et al. (1996) indicate that the QMLE performs very well for the sample sizes typically encountered with high-frequency financial data. As discussed in Baillie et al. (1996) the implementation of QMLE necessitates conditioning on pre-sample values and a truncation of the infinite lag polynomial in equation (5). Given the findings on the performance of QMLE in Baillie et al. (1996), we set the truncation lag to 1000 for each series.4 4

Recently Davidson (2004) suggested a new long memory model, called “hyperbolic GARCH” (HYGARCH) model which allows one to directly estimate what is known as the amplitude (i.e. size of the variations in the conditional volatility process) along with the long memory parameter d. In the FIGARCH model the amplitude is restricted to be unity. It is beyond the scope of this paper to estimate HYGARCH model. It may worth to mention however that Davidson (2004) fails to reject a unit amplitude parameter in



7

A feature of ARCH models is the memory of the conditional volatility process (see Baillie et al. 1996, Zaffaroni 2004 and Davidson 2004). The memory of ARCH models characterizes how long shocks to the volatility take to dissipate. There are usually two cases discussed in the literature relating to the memory of ARCH processes, namely the geometric (short) memory and hyperbolic (long) memory. The stationary GARCH models are usually considered to have short memory as such shocks have relatively “less” persistent effects or “short-lived effects” on the conditional volatility. The FIGARCH process implies a slow hyperbolic rate of decay for the autocorrelations of u2t , which is a characteristic of long memory processes. Davidson (2004) shows that the length of the memory of the conditional volatility process is a function of the parameters of the underlying GARCH and FIGARCH models. For example, in the short memory GARCH(1, 1) model, higher the parameter β1 the longer the memory is. The memory of F IGARCH(p, d, q) process behaves in a different way as such the length of the memory of the process increases as d approaches to zero. Interestingly enough, as d increases and becomes unity, (when FIGARCH model reduces to an IGARCH model) the memory of the process jumps to short memory and when d is exactly zero, then the process becomes short memory GARCH model. Therefore, characterization of FIGARCH model as an intermediate case in terms of its memory between GARCH and IGARCH models might be misleading. Indeed, FIGARCH process possesses more memory than GARCH and IGARCH models and hence the magnitude of the estimated long memory parameter in a FIGARCH model should provide useful information on the length of the memory in conditional volatility process. It should also be emphasized that if the true data generating process is FIGARCH and a GARCH model is incorrectly estimated, the coefficient estimates will suggest IGARCH or near IGARCH behavior which has the counter intuitive implication that shocks to the conditional variance process should have indefinitely lasting effects (Baillie et al. 1996). As the discussion so far indicates, the proposed model can be quite useful in modelling both hyperbolic memory and other salient features of exchange rate return the HYGARCH model for several major daily exchange rate returns (including British Pound, German Mark, Japanese Yen, Swiss franc among others). Therefore, Davidson (2004) argues that FIGARCH models would “explain these data pretty well”. Davidson (2004) reports statistically significant amplitude parameter estimates that are different from unity for some Asian currencies which have experienced currency crises during late 1990s.



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Vol. 11 [2007], No. 3, Article 1

distributions which will be explored empirically in the following section.

3 3.1

Long memory, Conditional Volatility and Distribution of Exchange Rates Data and Initial Analysis

The data for the analysis in this paper contains daily nominal spot exchange rates between the US Dollar and British Pound, Canadian Dollar, Euro, German Mark, Japanese Yen, and Swiss Francs. The data is obtained from Board of Governors of the Federal Reserve System. Exchange rates are the noon buying rates in New York City for cable transfers payable in foreign currencies. The Pound and Euro rates are US Dollar equivalents while others are national currencies per US Dollar. The sample period for the US DollarEuro rate is January 04, 1999 to December 26, 2006, and for the German Mark-Dollar rate is January 2, 1979 to February 11, 2002. For all others the sample period is January 2, 1979 to December 26, 2006. Therefore, we have a sample of 2007 return series for Dollar-Euro, 5870 return series for German Mark-Dollar and 7029 observations for the remaining returns in our data. Following the standard daily exchange rate returns are practice, St constructed as rt = 100 × log St−1 , where St denotes the spot exchange rate at day t. For Euro returns, we use data for 1999-2004 for the estimation and use observations for 2005 and 2006 for the forecasting. For the German Mark-US Dollar returns, we use observations for 1979-1999 for estimation and 2000-2002 for forecasting. And finally for the remaining four series, we use observations for 1979 to 2002 for estimation and leave observations for 2003-2006 for the forecasting. Table 1 reports the summary statistics together with the Ljung-Box portmanteau statistics for up to 20th-order serial correlation in the returns and squared returns during estimation period for each series. The reported results indicate that for all series, daily returns are quite small with considerable amount of variation. The sample variance of the Canadian Dollar return is the smallest among all. Except for Canadian Dollar-Dollar and DollarEuro rates all other series are negatively skewed. Kurtosis statistics indicate that all returns have excess kurtosis with the lowest kurtosis value for Euro returns. Reported Ljung-Box statistics show serial correlation in all series



Dollar-Euro

9

Figure 1: Daily spot exchange rates and returns J. Yen-Dollar

C. Dollar-Dollar

Dollar-B. Pound

G. Mark-Dollar

S. Franc-Dollar



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Table 1: Summary statistics for daily exchange rate returns T mean med min max var skew kurt Q(20) Q2 (20) C. Dollar-Dollar 6027 0.005 0.000 -1.864 1.728 0.083 0.069 5.892 37.716 1053.699 J. Yen-Dollar 6027 -0.003 0.022 -4.408 3.300 0.546 -0.186 4.591 39.431 815.671 G. Mark-Dollar 5332 0.001 0.015 -6.315 3.907 0.520 -0.315 6.782 51.725 476.051 S. Francs-Dollar 6027 -0.008 0.014 -5.630 3.366 0.476 -0.507 6.586 33.217 609.977 US Dollar-Pound 6027 -0.004 0.000 -3.843 4.589 0.401 -0.067 6.055 56.115 1397.558 US Dollar-Euro 1509 0.009 -0.016 -2.473 2.709 0.413 0.009 3.648 20.761 50.943 Key: Exchange rates are the noon buying rates in New York City for cable transfers payable in foreign currencies. Q(20) and Q2 (20) denote the Ljung-Box (1978) tests for up to 20th-order serial correlation in daily returns and squared returns. 5% and 1% critical values for the test are 31.41 and 37.566 respectively.

except for Dollar-Euro returns at conventional significance levels. The DollarPound and Mark-Dollar returns are the most persistent series among all. The reported Panels of Figure 1 displays daily returns and squared returns over the sample period. The plots in the Figure clearly indicate the occurrence of tranquil and volatile periods. This is also supported by the very high LjungBox statistics reported for the daily squared returns in the last column of Table 1.

3.2

Estimated GARCH and FIGARCH Models with Normal, Student’s t and NIG errors

Parameter estimates, standard errors, several diagnostic statistics for GARCH and FIGARCH models with Normal, students’ t and NIG errors both with b = 0 and b 6= 0 as well as GARCH and FIGARCH-in-mean models with students’ t distribution are presented in Tables 2 through 7. Note that G − N, G − t, G − NIG and G − NIGb stand for the GARCH(1, 1) model with normal, t, NIG with b = 0 and NIG with b 6= 0 respectively while the abbreviation F G − N, F G − t, F G − NIG and F G − NIGb are used to denote the F IGARCH(1, d, 1) model with the normal, t, NIG errors with b = 0 and b 6= 0 respectively. Similarly, Gm − t and F Gm − t denote the GARCH and FIGARCH -in-mean models with t errors.5 Each table gives results for a series. In each table first panel reports parameter estimates together with Quasi-Likelihood standard errors. In the second panels in each table, 5

The likelihood functions were maximized using the BHHH algorithm using numerical derivatives in Gauss. In each case different starting values for the parameters are used to check the global maximum. The results are found to be robust to different initial values. Following Conrad and Haag (2006), we have also checked the necessary and sufficient conditions for the non-negativity of conditional variance process for the FIGARCH models. All estimated models passed the conditions stated in Conrad and Haag (2006). These results can be obtained upon request.



11

we report summary diagnostic statistics for the estimated models (including the Lijung-Box statistics and excess skewness and kurtosis values from the standardized residuals) as well as log-likelihood values and Akaike and Schwartz Information criteria. Following Jensen and Lunde (2001), the residuals used in diagnostic tests and plots are all obtained by subtracting from the series the estimated location and by dividing the estimated scale. Therefore for the GARCH and FIGARCH models with Normal errors and NIG errors with b = 0, residuals are computed as ut = rtσ−µ . Residuals for the t rt −µ−

√

a2 −b2

bσ

t a models with NIG errors where b 6= 0 is given by ut = , and for σt r√ t −µ the models with t distribution is given by ut = . Finally the residuals ν−2

σt

ν

for the GARCH/FIGARCH-in-mean models with t distribution is given by r −µ−δσ2 ut = t √ ν−2t . In cases where there were an MA term for the conditional σt

ν

mean process, we also filtered the MA term out. The third panel in each table presents Ljung-Box statistic for the presence of serial correlation up to orders one and five in the probability integral transforms (PIT) of standardized residuals from each estimated model.6 Finally the last panel (row) in each table gives the robust Wald test in the FIGARCH models for testing the null that d = 1 against the alternative that d < 1. In the following, we discuss briefly estimation results for each currency and then provide an overview of general findings. Parameter estimates indicate that except for Dollar-Euro returns, for all daily exchange rates an MA component characterizes the conditional mean of daily returns.7 For each of the exchange rate returns, estimates of hyperbolic decay parameter, d, are significant and greater than 0 but less than unity (as the reported robust Wald tests in the last panel of each table also suggest) and are in the range of between 0.22-0.52 across different currencies. It should be noted that the magnitude of the estimated long memory parameters are quite similar across different distributions for each currency. In the light of Davidson’s (2004) findings, our estimates of hyperbolic decay parameters R xt The PIT is defined as zt = −∞ f (u)du, where f (u) is the probability density function for the random variable x. As shown in Diebold, Gunther and Tay (1998), if f (.) is the correct distribution, then zt ∼ U (0, 1) and hence zt and its power transformations should be i.i.d. as well. See Forsberg and Bollerslev (2002) for an application of this approach in checking for the adequacy of estimated GARCH models with different error distributions. 7 In most cases, the MA term is significant or marginally significant. In cases it is not significant, dropping the MA term increases the Ljung-Box values for serial correlation in the standardized residuals. 6



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in FIGARCH models indicate longer memory and hence more persistent volatility dynamics in Dollar-Euro and Yen-Dollar rates relative to other currencies. Estimated GARCH and FIGARCH-in-mean parameters δ in the GARCH and FIGARCH models with student’s t errors are not significantly different from zero.8 Estimated volatility dynamics parameters α ˆ and βˆ in the GARCH models and βˆ and φˆ in the FIGARCH models are all very significant. Sum of the parameter estimates for α and β are quite close to unity and although not reported in several cases 95% confidence intervals include unity implying very high persistence in GARCH models. It should be noted that estimated volatility dynamics parameters are similar across different error distributions over the same series. Estimated βs in GARCH models are usually high and in the order of 0.9s and αs are quite low but statistically highly significant. ˆ are lower In the FIGARCH models, autoregressive parameter estimates (β) (usually in the order of 0.5-0.6) and estimates for φ are usually in the order of 0.2-0.3. Estimated degree of freedom parameter in the GARCH and FIGARCH models with the t distribution, νˆ, are all significant. The estimated value for the Dollar-Euro log returns is the highest (in the order of 11.0 across different models) and ranges between 5.0-7.0 for all other currencies. Estimates of the steepness parameters, a ˆs, in the GARCH and FIGARCH models with NIG errors are all significantly greater than zero and range between 1.0-2.3 for most of the series, except for Dollar-Euro returns, for which the estimated value is higher and in the order of 3.8.9 These results indicate a leptokurtic distribution for the daily log exchange rate returns. In relative terms, Yen-Dollar and Dollar-Pound log daily returns have the least peaked distribution than the others. It should also be noted that estimated values for 8

In only two cases, namely in the GARCH/FIGARCH-in-mean models for Japanese Yen and Swiss Franc, estimates for δ are significant at 10% significance level. GARCH and FIGARCH-in-mean models with Normal errors are also estimated but results are not reported to conserve space. The parameter estimates are quite similar across currencies. Those results can be obtained upon request. 9 Note that standard errors for both estimated ν in the student’s t distribution and steepness parameter in the NIG distribution are higher in the case of Dollar-Euro returns. This might be partly due to relatively small sample size we have for this series. To our best knowledge, Anderson (2001) and Forsberg and Bollerslev (2002) are the only studies that report parameter estimates for the stochastic volatility models and GARCH models with symmetric NIG errors (i.e. b = 0) in the context of exchange rate returns. Overall estimated a ˆs are qualitatively quite similar to the reported values in these papers.



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Table 2: Estimated GARCH and FIGARCH Models for daily Canadian Dollar-US Dollar exchange rate returns q

b a2 −b2 2 rt = µ + θut−1 + σt + ut when a 2 ut ∼ NIGb6=0 ; and rt = µ + θut−1 + δσt + ut when ut ∼ tν with GARCH/FIGARCH-in-mean; 2 2 2 d 2 GARCH(1, 1) : σt2 = ω + βσt−1 + αu2 t−1 , F IGARCH(1, d, 1) : σt = ω + βσt−1 + (1 − βL − (1 − φL)(1 − L) )ut .

rt

=

µ + θut−1 + ut

µ θ δ d ω β α φ ν a b ll AIC SIC m3 m4 Q Q2 Qpit (1) Qpit (5) Q2 pit (1) Q2 pit (5) Wd=1

G-N 0.003 (0.003) 0.055 (0.013) . . . . 0.001 (0.000) 0.911 (0.005) 0.081 (0.005) . . . . . . . . -590.6 1191.3 1224.8 0.265 5.361 22.68 20.80 0.06 3.94 0.39 8.03 .

when

G-t 0.000 (0.000) 0.046 (0.013) . . . . 0.001 (0.000) 0.908 (0.006) 0.086 (0.008) . . 6.375 (0.498) . . . . -435.8 883.6 923.8 0.295 5.389 24.27 20.47 0.18 4.05 0.97 4.73 .

ut

G-NIG 0.001 (0.003) 0.045 (0.013) . . . . 0.001 (0.000) 0.918 (0.007) 0.079 (0.007) . . . . 1.910 (0.186) . . -437.8 887.6 927.8 0.306 5.467 24.61 23.19 0.07 4.32 3.18 7.00 .

∼

{N, tν , NIGb=0 };

Canadian Dollar-US Dollar Gm-t G-NIGb FG-N FG-t -0.003 -0.016 0.003 0.000 (0.005) (0.006) (0.003) (0.003) 0.046 0.043 0.054 0.047 (0.013) (0.013) (0.003) (0.013) 0.051 . . . (0.073) . . . . . 0.489 0.528 . . (0.036) (0.061) 0.001 0.001 0.002 0.001 (0.000) (0.000) (0.000) (0.000) 0.917 0.918 0.623 0.636 (0.006) (0.007) (0.019) (0.013) 0.086 0.078 . . (0.008) (0.007) . . . . 0.288 0.264 . . (0.019) (0.031) 6.384 . . 6.806 (0.498) . . (0.508) . 2.041 . . . (0.210) . . . 0.115 . . . (0.040) . . -435.5 -461.7 -578.6 -425.9 885.0 937.4 1169.2 865.9 931.9 984.3 1209.4 912.8 0.287 0.118 0.270 0.308 5.403 4.491 5.367 5.435 24.19 24.62 22.78 24.18 20.12 20.99 11.57 13.37 0.22 1.81 0.04 0.13 3.97 5.63 3.89 4.01 0.80 3.58 0.50 0.70 4.02 6.28 8.10 4.15 . 200.30 60.62

FGm-t -0.003 (0.005) 0.047 (0.013) 0.051 (0.074) 0.528 (0.061) 0.001 (0.000) 0.636 (0.031) . . 0.260 (0.032) 6.811 (0.509) . . . . -425.7 867.4 921.0 0.301 5.448 24.12 13.18 0.10 3.99 0.29 3.36 60.9

FG-NIG 0.001 (0.003) 0.046 (0.013) . . 0.544 (0.065) 0.001 (0.000) 0.615 (0.032) . . 0.285 (0.030) . . 2.035 (0.203) . . -428.7 871.4 918.3 -0.310 5.476 24.26 12.61 0.06 4.30 0.00 2.99 49.61

FG-NIGb -0.017 (0.006) 0.044 (0.013 . . 0.516 (0.058) 0.001 (0.000) 0.608 (0.030) . . 0.270 (0.029) . . 2.078 (0.209) 0.121 (0.041) -424.5 865.1 918.7 0.121 4.453 24.33 12.94 1.69 2.92 2.63 2.74 69.81

Key:The numbers in parenthesis are Quasi-Likelihood Standard Errors. Q and Q2 are defined as in Table 1. m3 and m4 are the estimated skewness and kurtosis of residuals. Wd=1 is the robust Wald test for testing d = 1 versus d < 1 in the FIGARCH models. The test is asymptotically χ2 1 distributed. G-N, G-t, G-NIG and G-NIGb stand GARCH model with Normal, student’s t and NIG errors with b = 0 and b 6= 0 respectively. FG-N, FG-t, FG-NIG and FG-NIGb refer to the FIGARCH model with Normal, student’s t, NIG errors with b = 0 and without b 6= 0 respectively. Gm-t and FGm-t stands for the GARCH and FIGARCH in the mean models with student’s t errors. AIC and SIC are the Akike and 2 the Schwartz Information Criteria. Values corresponding to Qpit (1), Qpit (5) and Q2 pit (1), Qpit (5) give the Lijung-Box tests up to 1 and 5th order serial correlation in PITs and squared PITs for the standardized returns from each model assumption. The 5% critical value from the χ2 (1) and χ2 (1) are 3.841 and 11.070 respectively. The 1% critical value from the χ2 (1) and χ2 (1) are 6.635 and 15.086 respectively.



14

Vol. 11 [2007], No. 3, Article 1

Table 3: Estimated GARCH and FIGARCH models for daily German MarkUS Dollar exchange rate returns q rt

=

ut

∼

b

a2 −b2

µ + θut−1 + ut when ut ∼ {N, tν , NIGb=0 }; rt = µ + θut−1 + σt2 + ut when a NIGb6=0 ; and rt = µ + θut−1 + δσt2 + ut when ut ∼ tν with GARCH/FIGARCH-in-mean;

2 2 2 d 2 GARCH(1, 1) : σt2 = ω + βσt−1 + αu2 t−1 , F IGARCH(1, d, 1) : σt = ω + βσt−1 + (1 − βL − (1 − φL)(1 − L) )ut .

G-N G-t µ 0.006 0.014 (0.009) (0.009) θ 0.029 0.026 (0.014) (0.014) δ . . . . d . . . . ω 0.015 0.012 (0.002) (0.002) β 0.878 0.896 (0.008) (0.011) α 0.096 0.084 (0.006) (0.009) φ . . . . ν . 7.285 . (0.645) a . . . . b . . . . ll -5497.4 -5388.8 AIC 11004.7 10789.7 SIC 11037.6 10829.1 m3 -0.243 -0.283 m4 4.874 4.951 Q 39.44 40.16 2 Q 33.01 30.43 Qpit (1) 8.65 8.91 Qpit (5) 12.69 13.26 2 Qpit (1) 7.86 8.64 Q2 9.03 16.72 pit (5) Wd=1 . . Key: Look at Table 2.

G-NIG 0.014 (0.009) 0.026 (0.014) . . . . 0.012 (0.002) 0.894 (0.011) 0.085 (0.009) . . . . 2.326 (0.268) . . -5392.7 10797.4 10836.8 -0.283 4.939 40.11 30.64 10.08 14.55 8.71 17.71 .

Gm-t 0.032 (0.017) 0.025 (0.014) -0.042 (0.036) . . 0.012 (0.002) 0.896 (0.011) 0.084 (0.009) . . 7.272 (0.643) . . . . -5388.1 10790.2 10836.3 -0.269 4.957 40.01 30.37 9.71 13.80 9.29 16.35 .

German Mark-US Dollar G-NIGb FG-N FG-t 0.053 0.005 0.014 (0.020) (0.009) (0.009) 0.026 0.029 0.025 (0.014) (0.015) (0.014) . . . . . . . 0.509 0.504 . (0.039) (0.066) 0.012 0.018 0.017 (0.002) (0.003) (0.005) 0.894 0.648 0.651 (0.011) (0.035) (0.056) 0.085 . . (0.009) . . . 0.227 0.206 . (0.027) (0.039) . . 7.116 . . (0.612) 2.307 . . (0.267) . . -0.107 . . (0.050) . . -5390.2 -5492.5 -5381.8 10794.4 10997.0 10777.7 10840.5 11036.5 10823.8 -0.092 -0.242 -0.279 4.313 4.907 4.993 39.75 40.11 40.74 31.39 24.87 23.75 3.66 8.44 8.65 4.91 12.64 13.19 4.97 7.73 7.94 10.08 8.84 16.37 . 160.45 57.2


FGm-t 0.0029 (0.017) 0.025 (0.014) -0.037 (0.074) 0.505 (0.065) 0.014 (0.004) 0.652 (0.056) . . 0.207 (0.039) 7.100 (0.611) . . . . -5381.3 10778.7 10831.3 -0.266 4.993 40.38 23.70 10.88 14.63 10.84 15.98 57.37

FG-NIG 0.014 (0.009) 0.025 (0.014) . . 0.508 (0.064) 0.013 (0.004) 0.652 (0.056) . . 0.207 (0.039) . . 2.267 (0.257) . . -5386.2 10786.4 10832.4 -0.279 4.979 40.48 23.63 10.05 14.52 8.51 17.33 58.38

FG-NIGb 0.047 (0.019) 0.025 (0.014 . . 0.502 (0.064) 0.013 (0.004) 0.649 (0.056) . . 0.211 (0.039) . . 2.241 (0.256) -0.091 (0.048) -5384.3 10784.6 10837.3 -0.081 4.349 40.04 24.24 3.62 4.29 4.83 6.28 60.8


15

Table 4: Estimated GARCH and FIGARCH Models for daily Japanese YenUS Dollar exchange rate returns q rt

=

ut

∼

b

a2 −b2



G-N G-t µ -0.001 0.018 (0.009) (0.008) θ 0.030 0.034 (0.014) (0.012) δ . . . . d . . . . ω 0.013 0.008 (0.001) (0.002) β 0.919 0.933 (0.006) (0.008) α 0.054 0.053 (0.004) (0.007) φ . . . . ν . 4.567 . (0.305) a . . . . b . . . . ll -6030.1 -5775.7 AIC 12070.2 11563.5 SIC 12103.8 11603.7 m3 -0.484 -0.592 m4 5.559 5.782 Q 30.87 30.39 2 Q 29.90 32.00 Qpit (1) 0.71 0.79 Qpit (5) 8.90 2.24 2 Qpit (1) 0.04 0.83 2 Qpit (5) 11.70 10.97 Wd=1 . . Key: Look at Table 2.

G-NIG 0.018 (0.007) 0.034 (0.012) . . . . 0.009 (0.002) 0.931 (0.009) 0.053 (0.007) . . . . 1.100 (0.102) . . -5770.7 11553.4 11593.6 -0.588 5.762 30.21 31.59 0.79 2.24 0.83 10.50 .

Gm-t 0.043 (0.017) 0.033 (0.012) -0.059 (0.037) . . 0.009 (0.002) 0.931 (0.008) 0.054 (0.007) . . 4.570 (0.306) . . . . -5774.5 11563.0 11609.9 -0.576 5.775 27.86 31.30 1.72 8.40 0.49 8.02 .

Japanese Yen-US Dollar G-NIGb FG-N FG-t 0.078 0.003 0.019 (0.015) (0.009) (0.008) 0.033 0.029 0.034 (0.012) (0.015) (0.012) . . . . . . . 0.255 0.339 . (0.019) (0.049) 0.009 0.042 0.031 (0.002) (0.007) (0.008) 0.931 0.516 0.567 (0.009) (0.054) (0.064) 0.051 . . (0.006) . . . 0.362 0.309 . (0.050) (0.052) . . 4.641 . . (0.305) 1.154 . . (0.110) . . -0.134 . . (0.031) . . -5758.1 -6023.7 -5772.2 11530.3 12059.4 11558.4 11577.2 12099.6 11605.3 -0.325 -0.506 -0.598 5.759 5.552 5.771 26.95 30.40 30.54 30.44 31.71 30.89 5.27 0.56 1.88 14.10 8.69 9.67 2.07 0.014 0.22 9.53 11.12 8.02 . 1469.4 183.0

FGm-t 0.048 (0.017) 0.032 (0.012) -0.070 (0.037) 0.329 (0.048) 0.032 (0.009) 0.550 (0.067) . . 0.303 (0.055) 4.651 (0.306) . . . . -5770.5 11557.0 11610.6 -0.578 5.749 27.78 30.12 1.42 8.06 0.70 7.99 198.8

FG-NIG 0.019 (0.007) 0.033 (0.012) . . 0.315 (0.044) 0.030 (0.009) 0.548 (0.069) . . 0.314 (0.058) . . 1.115 (0.102) . . -5766.8 11547.5 11594.5 -0.598 5.763 30.51 30.46 0.97 9.45 0.00 6.89 242.1

FG-NIGb 0.083 (0.015) 0.032 (0.012 . . 0.299 (0.041) 0.030 (0.009) 0.539 (0.072) . . 0.324 (0.063) . . 1.189 (0.114) -0.143 (0.032) -5752.8 11521.7 11575.3 -0.334 4.633 27.17 29.42 4.06 7.67 2.71 6.75 287.6



16

Vol. 11 [2007], No. 3, Article 1

Table 5: Estimated GARCH and FIGARCH Models for daily Swiss Franc-US Dollar exchange rate returns q

b a2 −b2 2 rt = µ + θut−1 + σt + ut when a 2 ut ∼ NIGb6=0 ; and rt = µ + θut−1 + δσt + ut when ut ∼ tν with GARCH/FIGARCH-in-mean; 2 2 2 d 2 GARCH(1, 1) : σt2 = ω + βσt−1 + αu2 t−1 , F IGARCH(1, d, 1) : σt = ω + βσt−1 + (1 − βL − (1 − φL)(1 − L) )ut .

rt

=

µ + θut−1 + ut

when

G-N G-t 0.005 0.013 (0.009) (0.009) θ 0.033 0.019 (0.014) (0.013) δ . . . . d . . . . ω 0.014 0.009 (0.002) (0.002) β 0.921 0.933 (0.008) (0.009) α 0.053 0.052 (0.005) (0.007) φ . . . . ν . 6.944 . (0.677) a . . . . b . . . . ll -6525.5 -6429.5 AIC 13061.1 12871.0 SIC 13094.6 12911.2 m3 -0.209 -0.252 m4 4.198 4.289 Q 30.16 33.61 Q2 9.60 9.98 Qpit (1) 1.02 0.01 Qpit (5) 4.53 3.73 Q2 0.07 0.62 pit (1) Q2 4.17 5.00 pit (5) Wd=1 . . Key: Look at Table 2. µ

ut

∼

G-NIG 0.014 (0.009) 0.018 (0.013) . . . . 0.010 (0.002) 0.931 (0.009) 0.052 (0.007) . . . . 2.070 (0.255) . . -6425.6 12863.2 12903.4 -0.253 4.279 33.87 9.92 0.01 4.10 3.04 5.21 .

{N, tν , NIGb=0 };

Gm-t 0.049 (0.022) 0.018 (0.013) -0.059 (0.037) . . 0.010 (0.002) 0.931 (0.009) 0.052 (0.007) . . 6.949 (0.680) . . . . -6427.9 12869.8 12916.7 -0.235 4.273 32.49 9.99 0.02 3.98 0.18 5.23 .

Swiss Franc-US Dollar G-NIGb FG-N FG-t 0.084 0.005 0.013 (0.022) (0.009) (0.009) 0.016 0.034 0.020 (0.013) (0.014) (0.013) . . . . . . . 0.347 0.422 . (0.027) (0.050) 0.010 0.026 0.020 (0.003) (0.004) (0.004) 0.931 0.623 0.664 (0.009) (0.015) (0.027) 0.051 . . (0.007) . . . 0.328 0.276 . (0.015) (0.026) . . 6.912 . . (0.656) 2.106 . . (0.263) . . -0.170 . . (0.053) . . -6418.9 -6526.6 -6430.2 12851.8 13065.2 12874.4 12898.7 13105.4 12921.4 -0.033 -0.215 -0.252 4.201 4.222 4.286 32.70 30.69 34.06 9.65 9.47 11.22 3.98 0.93 0.02 6.85 4.53 3.72 1.43 0.06 0.48 3.21 3.98 4.78 . 603.4 131.5

FGm-t 0.046 (0.021) 0.018 (0.013) -0.070 (0.040) 0.410 (0.049) 0.020 (0.005) 0.656 (0.026) . . 0.294 (0.026) 6.908 (0.658) . . . . -6428.8 12873.6 12927.2 -0.237 4.272 32.92 10.62 0.00 3.90 0.13 4.5 144.6

FG-NIG 0.013 (0.009) 0.018 (0.013) . . 0.410 (0.049) 0.019 (0.004) 0.657 (0.026) . . 0.252 (0.025) . . 2.059 (0.249) . . -6426.5 12867.0 12913.9 -0.255 4.284 34.38 10.88 0.01 4.07 1.96 3.16 145.6

FG-NIGb 0.080 (0.022) 0.016 (0.013 . . 0.393 (0.047) 0.020 (0.005) 0.607 (0.024) . . 0.300 (0.025) . . 2.083 (0.255) -0.160 (0.052) -6420.4 12856.8 12910.4 -0.047 4.204 30.21 9.21 4.23 7.59 1.39 3.34 169.9

a are uniform across GARCH and FIGARCH specifications for each currency and do not change much weather we have b = 0 or not in the NIG distribution. Estimated as are similar to the estimated values reported in the literature on stock and exchange rate markets (see reported values in Jensen and Lunde 2001 for daily stock index and individual equity returns as well as the estimates reported in Forsberg and Bollerslev (2002) for the daily ECU-US Dollar returns. Estimates for the asymmetry parameter, b, are significant for all series except for the Dollar-Euro log returns. These estimates are similar to the estimates reported in the literature on stock index and equities returns as in Jensen and Lunde (2001) in terms of magnitude. Note that the estimated values are negative for all series except for the Canadian Dollar-Dollar returns. One possible interpretation of a non-zero b in models with NIG errors is as the



17

impact of volatility on the log-daily returns and as such is comparable to the GARCH/FIGARCH-in-mean effect. Although one can possibly argue about presence of volatility effect in the foreign exchange rate markets by using an International Capital Asset Pricing model as in Engle (1996), given the statistical insignificance of GARCH/FIGARCH-in-mean parameters (δ), following Jensen and Lunde (2001), we explain the non-zero parameter estimates for b in the NIG distribution by the fact that they mainly account for the skewness of the return distribution that dominates the GARCH/FIGARCH-in-mean term.10 It should also be noted that the estimated bs have the same signs as with the skewness statistics reported in Table 1 for all series except for the US Dollar-Euro returns for which ˆb is not significantly different from zero. A comparison of estimated GARCH and FIGARCH models with any given distribution assumption, in terms of the diagnostic statistics and information criteria reported in the second and third panels of each table, reveals that FIGARCH models with NIG errors perform better than the alternatives in several dimensions. First of all, although not directly comparable, log-likelihood values are the highest for FIGARCH models with NIG errors for almost all series except for the German Mark. Both AIC and SIC values rank FIGARCH models with NIG errors (either with b = 0 or b 6= 0) number one for all series except for the German Mark-US Dollar log daily returns for which FIGARCH model with student’s t distribution performs better than the alternatives. It should also be noted that GARCH specification with NIG errors attains higher log-likelihood value as well as lower AIC and SIC values for the Swiss Francs than the FIGARCH specification with NIG errors. In terms of lowering the excess skewness and kurtosis values for the standardized residuals, GARCH and FIGARCH models with NIG errors (especially the models where b 6= 0) perform relatively better than the alternatives. The reported Ljung-Box statistics for the presence of serial correla10

As noted by Engel (1996) and Baillie and Bollerslev (2000) among others, the Euler equation for risk averse investor equalizing real returns in the spot and forward foreign exchange rate markets can be approximated as ∆st = (ft−1,1 − st−1 ) − 0.5V art (st ) + Cov(pt , st )+Cov(qt , st )+ut where st is the log exchange rate at date t, ft−1,1 s the 1-period ahead forward rate, pt is the relative price ratio between home and foreign countries, qt is the logarithm of the inter temporal marginal rate of substitution in consumption between two time periods. This line of reasoning may suggest that log exchange rate returns might be effected by the variation in the degree of risk in the exchange rate markets. One can also think that the composite term might be reflecting a time-varying risk premium in the foreign exchange rate markets. This issue although interesting is beyond the scope of this paper and is not investigated here.



18

Vol. 11 [2007], No. 3, Article 1

Table 6: Estimated GARCH and FIGARCH Models for daily US Dollar-Euro exchange rate returns q rt

=

µ + ut

when

ut

∼

{N, tν , NIGb=0 };

rt

=

µ +

b

a2 −b2 2 σt + ut a

when

ut

∼

NIGb6=0 ;

µ + δσt2 + ut when ut ∼ tν with GARCH/FIGARCH-in-mean; GARCH(1, 1) : σt2 = 2 2 ω + βσt−1 + αu2 F IGARCH(1, d, 1) : σt2 = ω + βσt−1 + (1 − βL − (1 − φL)(1 − L)d )u2 t−1 , t. US Dollar-Euro G-N G-t G-NIG Gm-t G-NIGb FG-N FG-t FGm-t FG-NIG FG-NIGb µ 0.014 0.012 0.011 0.134 0.020 0.013 0.011 0.075 0.011 0.025 (0.016) (0.016) (0.016) (0.076) (0.071) (0.016) (0.016) (0.064) (0.016) (0.064) δ . . . -0.304 . . . -0.158 . . . . . (0.189) . . . (0.160) . . d . . . . . 0.220 0.255 0.251 0.254 0.255 . . . . . (0.051) (0.073) (0.072) (0.073) (0.073) ω 0.006 0.005 0.005 0.005 0.005 0.057 0.046 0.046 0.039 0.046 (0.004) (0.004) (0.004) (0.004) (0.004) (0.017) (0.020) (0.020) (0.018) (0.020) β 0.968 0.968 0.968 0.967 0.968 0.595 0.612 0.609 0.612 0.612 (0.014) (0.016) (0.016) (0.015) (0.017) (0.026) (0.037) (0.037) (0.037) (0.037) α 0.018 0.020 0.020 0.020 0.020 . . . . . (0.007) (0.009) (0.009) (0.008) (0.009) . . . . . φ . . . . . 0.374 0.368 0.371 0.364 0.358 . . . . . (0.026) (0.036) (0.037) (0.036) (0.037) ν . 11.023 . 11.274 . . 11.068 11.151 . . . (3.110) . (3.276) . . (3.123) (3.226) . . a . . 3.783 . 3.821 . . . 3.832 3.864 . . (1.257) . (1.279) . . . (1.276) (1.302) b . . . . -0.027 . . . . -0.046 . . . . (0.226) . . . . (0.206) ll -1464.1 -1455.1 -1454.6 -1454.2 -1454.6 -1462.6 -1453.5 -1453.2 -1452.8 -1453.0 AIC 2936.2 2920.1 2919.2 2920.3 2921.2 2935.1 2914.0 2920.4 2917.7 2920.0 SIC 2957.4 2946.7 2945.8 2952.3 2953.1 2961.7 2950.9 2957.6 2949.6 2957.2 m3 0.016 0.029 0.030 0.024 0.072 0.000 0.007 0.007 0.008 -0.080 m4 3.642 3.651 3.651 3.621 3.650 3.644 3.659 3.637 3.661 3.654 Q 20.34 20.34 20.34 20.01 20.34 19.80 19.82 19.50 19.87 19.81 Q2 32.74 32.39 32.40 31.73 32.50 28.41 28.06 27.68 28.16 28.04 Qpit (1) 0.09 0.08 0.11 0.21 0.48 0.08 0.06 0.13 0.08 1.20 Qpit (5) 4.72 6.29 5.64 6.46 11.73 4.80 6.43 6.31 5.78 13.51 2 Qpit (1) 0.00 0.74 6.74 0.67 0.00 0.01 0.55 0.39 1.96 0.087 Q2 6.12 5.37 11.12 5.80 8.69 6.98 5.23 5.85 3.76 9.55 pit (5) Wd=1 . . . . . 233.4 105.5 107.9 105.4 104.0 Key: Look at Table 2. and

rt

=



19

Table 7: Estimated GARCH and FIGARCH Models for daily US Dollar-UK Pound exchange rate returns q rt

=

ut

∼

b

a2 −b2



G-N G-t µ 0.001 0.007 (0.007) (0.007) θ 0.049 0.029 (0.014) (0.013) δ . . . . d . . . . ω 0.004 0.003 (0.001) (0.000) β 0.944 0.941 (0.004) (0.006) α 0.048 0.054 (0.003) (0.006) φ . . . . ν . 5.908 . (0.496) a . . . . b . . . . ll -5352.4 -5213.9 AIC 10714.8 10439.8 SIC 10748.3 10479.0 m3 -0.173 -0.202 m4 4.643 4.695 Q 26.47 31.70 2 Q 19.83 18.29 Qpit (1) 1.01 0.38 Qpit (5) 6.68 6.12 2 Qpit (1) 0.19 0.05 2 Qpit (5) 7.34 4.86 Wd=1 . . Key: Look at Table 2.

G-NIG 0.007 (0.007) 0.026 (0.013) . . . . 0.003 (0.001) 0.940 (0.007) 0.054 (0.006) . . . . 1.640 (0.179) . . -5209.5 10431.0 10471.2 -0.203 4.693 32.91 18.35 0.18 5.65 0.43 8.29 .

Gm-t 0.022 (0.013) 0.029 (0.013) -0.049 (0.035) . . 0.003 (0.001) 0.941 (0.006) 0.054 (0.006) . . 5.898 (0.495) . . . . -5212.9 10439.7 10486.7 -0.188 4.697 31.07 17.70 0.42 6.02 0.13 5.23 .

US Dollar-UK Pound G-NIGb FG-N FG-t 0.038 0.001 0.007 (0.015) (0.007) (0.007) 0.025 0.049 0.029 (0.013) (0.014) (0.013) . . . . . . . 0.397 0.458 . (0.028) (0.053) 0.003 0.012 0.010 (0.001) (0.002) (0.002) 0.940 0.649 0.636 (0.007) (0.015) (0.028) 0.054 . . (0.006) . . . 0.301 0.274 . (0.015) (0.028) . . 6.034 . . (0.478) 1.659 . . (0.182) . . -0.087 . . (0.037) . . -5206.3 -5351.0 -5210.4 10426.6 10713.9 10434.9 10473.5 10754.1 10481.8 -0.096 -0.170 -0.198 4.676 4.720 4.760 32.15 26.34 31.30 17.47 17.13 17.87 5.47 1.01 0.36 10.12 6.62 5.99 2.58 0.16 0.03 8.98 7.02 4.41 . 478.1 103.6

FGm-t 0.021 (0.013) 0.028 (0.013) -0.046 (0.036) 0.458 (0.053) (0.010 (0.002) 0.635 (0.028) . . 0.275 (0.027) 6.027 (0.478) . . . . -5209.6 10435.2 10488.8 -0.184 4.761 30.68 17.49 0.25 5.51 0.10 4.87 104.5

FG-NIG 0.007 (0.007) 0.026 (0.013) . . 0.449 (0.052) 0.010 (0.002) 0.630 (0.027) . . 0.279 (0.022) . . 1.664 (0.175) . . -5206.2 10426.5 10473.4 -0.199 4.763 32.49 17.85 0.19 5.56 0.20 6.52 113.5

FG-NIGb 0.037 (0.014) 0.024 (0.013 . . 0.446 (0.051) 0.010 (0.002) 0.621 (0.026) . . 0.280 (0.027) . . 1.679 (0.177) -0.087 (0.037) -5203.1 10422.3 10475.9 -0.091 4.745 31.76 17.26 0.45 10.28 2.31 8.60 117.8



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tion in standardized residuals with 20 lags are insignificant at conventional significance levels for majority of the estimated models except for the the models for German Mark-US Dollar log daily returns. Despite the reduction in the Ljung-Box value, estimated models for Mark-Dollar returns are significant.11 Reported Ljung-Box statistics for the squared standardized residuals reveal that all estimated models for each log-return series adequately captures the serial correlation in the squared residuals. It should be emphasized that FIGARCH models achieve much lower Ljung-Box statistics compared to GARCH models for the squared standardized residuals under each of the alternative distributions. Following Diebold, Gunther and Tay (1998) and Forsberg and Bollerslev (2002), we utilize PITs to further evaluate in-sample fit of the alternative models. For this purpose in Figure 2, we report empirical quantile (QQplot) plots for the PITs from the FIGARCH models with normal, student’s t and NIG (with b = 0) error distributions. Careful inspection of the plots in Figure 2 illustrates that FIGARCH-NIG models performs much better than FIGARCH-N and FIGARCH-t models as the quantile plots for the PITs for the residuals of FIGARCH models with NIG error are almost indistinguishable from the quantiles of a Uniform distribution for all but Dollar-Euro returns.12 Since for the estimated models to be correctly conditionally calibrated, corresponding sequence of PITs should also be i.i.d. through time as discussed in Diebold, Gunther and Tay (1998) and Forserg and Bollerslev (2002), in the third panels of Tables 2-7, we report the Ljung-Box statistics for first and fifth order serial correlation in PITs and squared PITs. Reported Ljung-Box statistics show no significant temporal dependencies in PITs and squared PITs from all estimated models for all currencies except for MarkDollar returns. Ljung-Box statistics for the fifth order serial correlation for 11

Caution must be observed in interpreting the Ljung-Box statistics as asymptotic distribution of these statistics for GARCH and FIGARCH models has not yet been studied. Note also that the log likelihood values are comparable for each currency under the same distribution assumption only. Higher order MA terms and AR terms also included for the Mark-Dollar conditional mean series and none of the estimated models were better than the reported ones. These results can be obtained upon request. 12 QQ plots from F IGARCHm − t and F IGARCH − N IGb are qualitatively similar those reported for F IGARCH − t and F IGARCH − N IG models respectively and hence not reported. Similarly, QQ plots for GARCH models with NIG errors usually performs better than the GARCH models with normal and t errors and qualitatively similar to those reported for the FIGARCH models in Figure 2. For space considerations these results are not displayed but can be obtained upon request.



21

Figure 2: QQ plots for the PITs from FIGARCH models with Normal, student’s t and NIG errors Canadian Dollar-US Dollar

German Mark-US Dollar

Japanese Yen-US Dollar

Swiss Franc-US Dollar

US Dollar-Euro

US Dollar-UK Pound



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the PITs and squared PITs indicate some temporal dependency in the PITs from GARCH and FIGARCH models with Normal errors for some currencies and occasionally models with student’s t and NIG errors with b 6= 0. For German Mark returns, reported Ljung-Box statistics show that temporal dependencies disappear when NIG errors (with b 6= 0) are used. On the other hand, for Dollar-Pound and Euro-Dollar rates, Ljung-Box statistic for PITs up to order 5, indicate that GARCH −NIGb and F IGARCH −NIGb models performs worse than the alternatives. However, ˆbs are not significant in the case of Dollar-Euro and only marginally significant in the case of Dollar-Pound. Notice that GARCH − NIG and F IGARCH − NIG models perform quite well in these cases.

3.3

Comparison of alternative models via simulation

In order to further investigate in-sample performance of FIGARCH model with NIG errors, in this part, we compare the fit of FIGARCH models with NIG and t distributions. Following Kim et al. (1998) and Jensen and Lunde (2001), we simulate the sampling distribution of the Likelihood Ratio (LR) statistics. To illustrate the approach, let M1 denote the F IGARCH model with NIG (with b = 0 or b 6= 0) errors and let M0 denote the F IGARCH or F IGARCH-in-mean model with student’s t distribution. Then the loglikelihood ratio statistic that can be used to evaluate the fit of alternative models is given by LR = 2(log f (y|M1; ζˆ1 ) − log f (y|M0; ζˆ0 ))

where log f (y|M1; ζˆ1 ) and log f (y|M0 ; ζˆ0) are the log likelihoods, with ζˆ1 and ζˆ0 denoting the vector of parameter estimates from the models M1 and M0 respectively. Assuming that M1 is the true model with parameters given ζ10 , we replace ζ10 by its estimate ζˆ1 , and generate 499 simulations, y i , from the null model. In generating the data errors are drawn randomly from the distribution of the residuals assumed under the null model and data is calibrated on the parameter estimates as reported in Tables 2-7. For each simulated series both M1 and M0 models are estimated and LRi , i = 1 · · · , 499 are recorded. The simulated LR values form the sampling distribution of LR statistic under the assumed null model. For a detailed discussion of this approach in the context of GARCH and Stochastic Volatility models with different error distributions see Kim et al. (1998) and Jensen and Lunde (2001).



23

Table 8: P-values for non-nested model comparison: rank of observed LR statistics out of 500 LR tests conducted under the F IGARCH − NIG, F IGARCHm − t, or F IGARCH − t models F IGARCH − NIGb vs. F IGARCHm − t F IGARCH − NIG vs. F IGARCH − t Observed Rank Rank Observed Rank Rank Series LR value null:FG-NIGb null:FGm-t LR value null:FG-NIG null:FG-t Canadian Dollar 2.292 80 473 -5.638 7 268 German Mark -5.925 272 96 -8.803 245 24 Japanese Yen 35.319 356 448 10.802 218 497 Swiss Franc 16.786 287 498 7.472 172 493 Euro 0.417 264 258 1.388 326 156 UK Pound 12.904 323 489 8.406 303 409 Key: Observed LR values and their ranks in the simulated LR statistics as described in the text.

Figure 3 displays approximate sampling distributions of LR statistic for F IGARCH − NIGb vis − a ` − vis F IGARCHm − t and F IGARCH − NIG vis − a ` −vis F IGARCH −t, respectively for each null hypothesis. Displayed histograms are based on 499 simulations plus the observed LR. In addition to histograms, in Table 8, we report the rankings for the observed LR statistics. In comparing F IGARCH − NIGb model versus F IGARCHm − t, if the null of the F IGARCH − NIGb model is true, then LR can be expected to be positive on average for Canadian Dollar, Japanese Yen, Swiss Franc and UK Pound series, negative for German Mark and around zero for US DollarEuro returns. On the other hand, if the null F IGARCHm − t is the true true model, we should except on average more realizations of LR statistic to be negative for Canadian Dollar and German Mark, positive for Japanese Yen, Swiss Franc and UK Pound and in the neighborhood zero for Euro. Histogram plots displayed on the third and forth columns of Figure 3 reveal that in comparing F IGARCH − NIG vis − a` − vis F IGARCH − t, we can expect that LR to be on average more positive for Canadian Dollar, Japanese Yen, Swiss Franc, and UK Pound, negative for German Mark and Euro under the null F IGARCH − NIG model. If the null of F IGARCH − t is true then we should expect LR statistic to be positive for all return series except for the Canadian Dollar. The recorded rankings of observed LR statistic for all series except for US Dollar-Euro returns choose F IGARCH − NIGb over the F IGARCHm − t. For these returns, ranking of all observed LR are not extreme when F IGARCH − NIGb model is the null while the rankings of LR are more in the tails of the simulated distributions and hence are more extreme under the null of F IGARCHm − t, thereby failing to reject the null model of F IGARCH − NIGb against the alternative of F IGARCHm − t. This is consistent with the estimation results and diagnostic statistics reported in Tables 2-7. For US Dollar-Euro return series, rankings of both



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F IGARCH −NIGb and F IGARCHm−t are about the same and in neither case are extreme failing to reject both null models. When the null model is FIGARCH-NIG against the alternative FIGARCHt, the ranking of observed LR is in the tail area only for the Canadian Dollar, hence failing to reject the null of F IGARCH − NIG. Moreover, German Mark, Japanese Yen, Swiss Franc and UK Pound observed values of LR statistic are in the tail area under the null of F IGARCH − t hence rejecting that null in favor of F IGARCH − NIG model. For US Dollar-Euro returns ranking of LR statistic are not extreme under either null models suggesting the appropriateness of FIGARCH model both with symmetric NIG distribution and t distribution. Overall findings suggest that FIGARCH model with NIG errors with b 6= 0 is chosen over the FIGARCH-in-mean model with t errors for all series except for US Dollar-Euro. Only for the Canadian Dollar returns, FIGARCH model with t distribution is selected over FIGARCH with a symmetric NIG distribution. Strikingly simulation results are consistent with the comparison and ranking of FIGARCH models with alternative error distributions in terms of reported log-likelihood, AIC and SIC values in Tables 2-7. Overall, within-sample evaluation of GARCH and FIGARCH models via extensive statistical tools under different distribution assumptions reveals that a parsimonious F IGARCH(1, d, 1) specification with asymmetric NIG distribution can characterize the dynamics of conditional volatility process as well as the distribution of Canadian Dollar, German Mark, Japanese Yen, Swiss Franc, and UK Pound exchange rate returns considerably well compared to alternative models investigated. On the other hand, for the US Dollar-Euro returns the F IGARCH(1, d, 1) model with symmetric NIG distribution fits comparably well.

3.4

Out-of-sample Predictions

In this section, we discus out-of-sample performance of NIG error distribution and compare its performance with that of Normal and student’s t error distribution in the FIGARCH model. Following Forsberg and Bollerslev (2002), Table 9 reports the unconditional coverage probabilities for one-dayahead and five-day-ahead VaRs, or quantile predictions, obtained by applying estimated FIGARCH models with Normal, student’s t and NIG error distributions as reported in Tables 2-7 to the out-of-sample data for each daily log-return series. The results strikingly support FIGARCH model with NIG



25

Figure 3: Non-Nested Testing: Histograms for the simulated LR statistic

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Table 9: Quantile Predictions-Value-at-Risk for FIGARCH models with Normal, student’s t and NIG errors Canadian Dollar-US Dollar 5-day ahead 95 99 1 5 10 90 95 99 97.66 99.74 0.55 1.06 5.68 92.54 96.49 97.87 97.73 99.38 0.78 2.81 6.76 94.04 98.20 99.35 97.71 99.81 0.79 3.17 7.28 93.84 98.08 99.33 95.03 99.01 1.06 5.10 10.21 90.10 94.09 99.09 95.11 99.03 1.01 5.11 10.15 90.10 94.29 99.06 German Mark-US Dollar FG − N 0.10 1.01 2.91 96.91 99.13 99.99 0.02 1.03 3.17 91.62 97.47 99.91 FG − t 0.15 1.27 2.91 96.76 98.37 99.56 0.07 2.14 5.06 96.33 98.41 99.38 F Gm − t 0.44 2.06 4.67 95.88 98.77 99.60 0.12 2.10 5.87 96.31 98.34 99.37 F G − NIG 0.99 4.78 9.62 90.17 95.12 99.13 0.93 4.80 9.02 90.03 95.05 99.11 F G − NIGb 1.02 4.90 9.89 90.07 95.04 99.11 0.98 4.92 9.19 90.23 95.05 99.10 Japanese Yen-US Dollar FG − N 1.42 8.2 12.83 86.50 92.23 98.88 0.59 7.48 11.33 85.46 88.97 97.87 FG − t 1.52 5.87 9.85 88.93 93.13 98.02 1.92 7.63 10.62 90.51 93.60 98.18 F Gm − t 1.80 5.68 9.72 88.34 93.09 98.21 1.98 7.37 10.44 90.42 93.64 98.21 F G − NIG 1.20 4.98 9.99 90.31 95.20 99.04 1.22 5.55 10.18 90.33 95.48 99.67 F G − NIGb 1.14 4.98 10.03 90.14 95.04 99.02 1.14 5.25 10.07 90.20 95.35 99.41 Swiss Franc-US Dollar FG − N 0.31 3.75 9.20 93.89 96.96 99.61 0.38 5.04 8.26 86.74 93.10 97.17 FG − t 0.71 2.63 5.12 92.02 96.43 99.27 2.61 5.60 10.63 92.70 95.25 98.94 F Gm − t 0.72 2.64 5.11 91.98 96.35 99.20 2.60 5.43 11.07 92.84 95.24 98.95 F G − NIG 0.98 4.96 9.92 90.07 95.04 99.01 0.99 4.95 9.90 90.13 95.05 99.14 F G − NIGb 1.01 5.02 10.07 90.06 95.04 99.01 0.99 5.04 9.91 90.03 95.05 99.10 US Dollar-Euro FG − N 1.95 6.60 12.92 87.18 92.60 97.99 . 11.36 19.30 88.44 92.50 . FG − t 2.18 6.75 11.60 88.23 93.75 97.77 . 6.63 9.28 81.39 89.17 . F Gm − t 1.91 6.05 13.14 87.22 92.58 96.49 . 7.46 8.88 82.03 87.98 . F G − NIG 0.96 4.89 9.90 90.68 95.50 99.62 . 4.47 9.08 91.02 95.89 . F G − NIGb 0.96 4.87 9.93 90.55 95.40 99.60 . 4.59 9.04 91.03 95.69 . US Dollar-UK Pound FG − N 0.48 2.33 7.55 91.50 96.95 99.71 0.61 4.88 8.53 94.60 98.26 99.88 FG − t 0.67 2.88 6.86 93.52 97.47 99.03 0.44 1.33 4.24 90.86 94.92 98.26 F Gm − t 0.71 2.42 5.94 93.64 98.32 99.01 0.67 1.48 3.37 90.42 95.98 98.55 F G − NIG 1.11 4.97 10.29 90.12 95.10 99.00 1.19 4.98 10.14 90.21 95.02 99.16 F G − NIGb 1.10 4.97 10.21 90.10 95.09 99.01 1.18 4.99 10.12 90.16 95.00 99.06 Key: Table reports 1-day ahead and 5-day ahead empirical quantile predictions for the daily out-of-sample returns that are standardized by the estimated FIGARCH models with Normal, student’s t and Normal Inverse Gaussian errors as reported in Tables 2-7. Nominal FG − N FG − t F Gm − t F G − NIG F G − NIGb

1 0.14 0.51 0.58 0.90 0.96

5 1.81 2.06 3.43 4.97 4.98

1-day 10 6.25 4.43 6.01 9.94 9.90

ahead 90 94.00 94.24 93.12 90.07 90.04

error distribution. Quantile predictions from FIGARCH models with NIG errors (both when b = 0 and b 6= 0) are much closer to their respective nominal sizes than the alternative error distributions and in all cases they are easily within the 95% confidence bands.13 In Table 10, we report the Ljung-Box test for serial correlation of order 1 and 5 in the corresponding sequence of PITs and squared PITs. As the reported Ljung-Box statistics indicate Ljung-Box values are usually much 13 Based on a binomial approximation, p a 95% confidence interval for the quantile predictions can be computed as ±1.96 κ ˆ (1 − κ ˆ )/n where κ ˆ refers to the empirical size and n is the number of 1-day ahead and 5-day ahead quantile predictions. Note that n = 1003 and 200 for 1-day and 5-day ahead VaRs respectively for Canadian DollarDollar, Japanese Yen-Dollar, Swiss Franc-Dollar and Dollar-Pound returns. For German Mark-Dollar returns n = 538 and 107 for 1-day and 5-day ahead quantile predictions and for Dollar-Euro returns, these are 498 and 99 respectively.



27

Table 10: Lijung-Box Test for serial correlation in PITs and squared PITs for the out-of sample predictions 1-day ahead 5-day ahead Q(5) Q2 (1) Q2 (5) Q(1) Q(5) Q2 (1) Q2 (5) Canadian Dollar-US Dollar FG − N 1.91 3.67 2.93 5.80 0.23 10.41 0.10 8.41 FG − t 1.96 3.85 1.14 1.80 0.23 10.56 0.25 9.93 F Gm − t 1.98 3.94 1.86 2.14 0.46 11.62 0.29 10.32 F G − NIG 0.20 4.44 0.09 2.64 2.17 3.22 1.40 2.16 F G − NIGb 0.21 3.80 0.08 2.60 2.17 3.21 1.42 2.14 German Mark-US Dollar FG − N 0.02 2.70 0.93 3.12 0.01 4.06 0.07 4.11 FG − t 0.01 2.47 0.55 3.77 0.01 4.02 0.21 3.72 F Gm − t 0.01 2.48 0.54 3.71 0.01 4.41 0.26 4.30 F G − NIG 0.01 1.83 0.38 3.27 0.64 2.22 0.24 2.20 F G − NIGb 0.01 1.83 0.38 3.22 0.60 2.11 0.25 2.12 Japanese Yen-US Dollar FG − N 1.83 5.99 1.15 7.35 0.06 4.81 0.05 3.08 F Gm − t 2.31 6.64 2.43 5.14 0.03 4.81 0.42 6.76 FG − t 2.12 5.63 2.21 5.38 0.01 4.11 0.40 6.04 F G − NIG 2.52 5.56 2.29 4.43 0.54 3.38 0.02 2.69 F G − NIGb 1.90 4.63 2.20 3.69 0.47 3.30 0.04 2.61 Swiss Franc-US Dollar FG − N 0.01 3.57 0.01 3.52 2.10 2.65 1.74 3.09 FG − t 0.04 3.58 0.10 3.45 2.13 3.11 2.23 3.19 F Gm − t 0.02 3.13 0.08 3.21 2.08 3.10 2.12 3.01 F G − NIG 0.69 0.70 0.57 1.63 0.63 1.67 1.75 2.61 F G − NIGb 0.69 0.70 0.50 1.62 0.63 1.67 1.75 2.60 US Dollar-Euro FG − N 2.94 12.34 1.67 11.23 2.18 4.84 1.95 4.35 FG − t 2.98 12.31 4.12 12.29 2.18 4.81 2.23 6.03 F Gm − t 3.27 12.97 5.05 13.90 3.02 5.16 2.73 6.44 F G − NIG 0.93 2.82 1.27 4.02 0.09 3.51 0.32 4.31 F G − NIGb 0.90 2.48 1.29 3.52 0.09 3.15 1.02 3.81 US Dollar-UK Pound FG − N 0.45 2.97 0.27 4.30 0.25 2.64 0.47 2.32 FG − t 0.38 2.83 0.48 2.52 0.19 2.44 0.06 0.98 F G − NIG 0.12 2.33 0.15 3.12 0.08 2.36 0.05 2.25 F G − NIGb 0.12 2.14 0.18 2.67 0.07 2.34 0.01 0.89 Key: Table reports Lijung-Box statistics of order 1 and 5 for the PITs and squared PITs corresponding to 1-day ahead and 5-day ahead empirical quantile predictions for the daily out-of-sample returns that are standardized by the estimated FIGARCH models with Normal, student’s t and Normal Inverse Gaussian errors as reported in Tables 2-7. Q(1)

lower for the FIGARCH-NIG models than the FIGARCH-N, FIGARCHt and FIGARCHm-t models and in some cases, PITs and squared PITs for the FIGARCH models with Normal and student’s t errors display some serial correlation. Overall, the out-of-sample quantile predictions reported in Table 9 and the Ljung-Box statistics for the corresponding PITs and squared PITs reported in Table 10 provide strong support for the NIG distribution.

4

Conclusion

In this paper, we proposed the FIGARCH-NIG model to study the hyperbolic decay, time varying dynamics in conditional volatility and peakedness, asymmetry and fat tailed distribution of daily foreign exchange rate returns. The estimated GARCH and FIGARCH models with normal, t and the NIG distributions as well as GARCH/FIGARCH-in-mean models with t distri-



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bution compared in terms of several criteria. Models are evaluated both in terms of extensive in-sample fit as well as out-of-sample predictive ability of the quantiles of the distributions of daily log exchange rate returns. The analysis in the paper showed that a parsimonious F IGARCH(1, d, 1) model with NIG errors can characterize several salient features of the distribution, long memory and time-varying volatility dynamics in daily exchange rate returns considerably well. Our findings indicate that there are pronounced long memory component in the conditional volatility process as well as significant steepness and asymmetry in the distribution of several major exchange rate returns studied. Findings in the paper show that there is some variation across currencies in terms of the degree of asymmetry and steepness as well as long-range dependence. Extensive analysis reported in the paper reveal that the proposed model outperforms alternatives not only within sample fit but also in predictive ability of out-of-sample quantiles of the empirical distribution of daily exchange rate returns in 1-day and 5-day ahead forecasts and hence supports the usefulness of the model. An extension of empirical results to other financial and commodities returns may prove useful.

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Conditional Volatility and Distribution of Exchange

Conditional Volatility and Distribution of Exchange

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