- Email: [email protected]
Assessing inefficiency in euro bilateral exchange rates
ARTICLE IN PRESS
Physica A 367 (2006) 319–327 www.elsevier.com/locate/physa
Assessing inefficiency in euro bilateral exchange rates Benjamin M. Tabaka,, Daniel O. Cajueirob a
Banco Central do Brasil, SBS Quadra 3, Bloco B, 9 andar, DF 70074-900, Brazil Universidade Cato´lica de Brası´lia—Mestrado em Economia de Empresas, SGAN 916, Mo´dulo B—Asa Norte, DF 70790-160, Brazil
b
Received 28 October 2005; received in revised form 8 December 2005 Available online 5 January 2006
Abstract This paper assesses inefficiency for 10 euro bilateral exchange rates. We study the dynamics of these time series by estimating Tsallis q entropic index and Hurst exponents using the local Whittle estimator. Empirical results suggest that US, Canadian and Singapore dollar are amongst the most efficient currencies, while Japanese yen and Swedish krona are amongst the most inefficient. r 2006 Elsevier B.V. All rights reserved. Keywords: Euro bilateral exchange rates; Hurst exponents; Tsallis q entropic index; Long-range dependence
1. Introduction The analysis of the dynamics of exchange rates is crucial for the design of policy modeling, portfolio and risk management, and is an essential question in international finance as both exchange rates and their volatility are important determinants of international capital flows, foreign direct investment and macroeconomic performance. Recent developments in statistical physics may shed some light in the dynamics of exchange rates and may help in the design of exchange rate policies. An important question is how shocks to exchange rates propagate. If exchange rates or their volatility display long-range dependence then shocks tend to decay at a much slower rate than with traditional ARMA models. An important implication of long memory is that economic and financial models have to take such features into consideration. For example, if exchange rates have long memory then shocks may last for a long time and policy makers may be inclined to intervene in the foreign exchange market as the speed of adjustment to stability (or a new equilibrium level) may be slow. Generally, central bank interventions are motivated by the desire to check short-run trends or to correct longer-term misalignments. Such intervention has implication on the dynamics of exchange rates. For example, Kearns and Rigobon [1] test for the efficacy of central bank interventions in Australia and Japan and find that a US$ 100 million purchase appreciates the Australian dollar by 1.3–1.8% but the yen by just 0.2%. These results suggest that interventions may change the dynamics of exchange rates and that their market Corresponding author. Tel.: +55 61 4143092; fax: +55 61 4143045.
E-mail address: [email protected] (B.M. Tabak). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.12.007
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microstructure effects could be important. Therefore, long memory may cause and be caused by foreign exchange rate interventions. In this paper we will not address testing for causality between intervention and the degree of long-range dependence, although it is an important issue. We will focus on testing the degree of long-range dependence using the local Whittle estimator due to Robinson [2] and assessing relative inefficiencies in different currencies using as our measure of efficiency the Tsallis entropic index. Despite extensive empirical work using US dollar exchange rates, empirical tests using euro exchange rates are still lacking. In this paper we try to fill this gap by focusing on euro bilateral exchange rates and in particular on the euro-US dollar, euro-Canadian dollar, euro-Japanese yen, euro-Australian dollar, euro-New Zealand dollar, euro-Singapore dollar, euro-Norwegian kroner, euro-Swedish krona, euro-Swiss franc and euro-British pound exchange rates. The paper is organized along the following lines. Section 2 provides a brief literature review. While Section 3 discusses the methodology employed in the paper, Section 4 describes the data. Section 5 presents empirical results. Finally Section 6 concludes the paper.
2. Brief literature review Cheung [3] suggests that major exchange rate series displayed evidence of long-range dependence.1 Jin et al. [4] find that 14 out of 19 exchange rate series display evidence of long memory dynamics, using a wavelet OLS estimator.2 The authors find evidence that most exchange rates present antipersistence. These results provide motivation for studying long-range dependence for exchange rates. Han [5] study the Korean won–US dollar exchange rate using a high frequency sample and finds evidence of long memory in volatility, but that the estimated long memory parameter changes and empirical results may be subject to exogenous shifts and multiple breaks in the time series. During crisis period, November 1997, the long memory parameter is the greatest. This is in line with recent work from Grech and Mazur [6], that have tested whether Hurst exponents may be used to predict financial crashes and find positive evidence. Recent empirical work has found evidence of long-range dependence in volatility of asset prices. For example, Bollerslev and Wright [7] have shown that the temporal dependencies in the volatility process of foreign exchange rates can be best characterized by long-range dependence models. Additionally, Baillie et al. [8] study high frequency European exchange rates and suggest that currency volatility is well represented by a FIGARCH model.3 Very little research has been undertaken for euro bilateral exchange rates. Belaire-Franch and Opong [9] study the behavior of euro exchange rates testing for random walk behavior using a variance ratio methodology. Using Chow and Denning’s [10] version of the multiple variance ratio test, and adjusted critical values to prevent size distortions, the authors find that the random walk hypothesis can be rejected only for the Canadian dollar. Additionally, using signs and rank based tests and adjustments for multiplicity suggest that only for the Canadian and Singapore dollar we can reject the random walk hypothesis. It is important to notice that this tests were performed using a maximum of 30 days as the investment horizon. Therefore, are focusing on very short-term forecasting power. Empirical evidence presented so far suggests that long-range dependence may be a characteristic of both exchange rates and its volatility. In this line, part of the remainder of this paper will focus on testing whether this assertion holds true for a variety of euro bilateral exchange rates. On the other hand, Cajueiro and Tabak [11] have provided statistical evidence that, beside the long-range dependence phenomenon, the Tsallis q entropic index can be assess other types of inefficiency. Therefore, this measure will also be used to assess inefficiency in these markets. 1
Except for the British pound. The authors result’s contrasts with empirical results using the GPH estimator. 3 Their study focuses on British pound, Swiss franc and Deutsche mark vis-a-vis the US dollar. 2
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3. Methodology 3.1. The local Whittle estimator In this paper, since it is essential to know the degree of long memory which a given market presents, the Hurst’s exponent is estimated using the local Whittle estimator due to Robinson [2].4 As it is known, a given market is said to have long-range dependence with persistent behavior if the Hurst’s exponent H40:5, with antipersistent behavior is if Ho0:5 and random walk behavior if H ¼ 0:5.5 The local Whittle estimator is a semi-parametric estimator, which only requires specifying the parametric form of the spectral density when the frequency l is close to zero, f ðlÞGjlj12H
as l ! 0
(1)
when G is a constant. The computation involves an additional parameter m, an integer less than N=2, where N is the size of the time series, and such that, as N ! 1, 1 m þ ! 0. m N
(2)
This means that as N gets large, m gets large as well, although slower. For a spectral density of form (1), the Whittle approximation of the Gaussian likehood function is obtained by minimizing ! m Iðlj Þ 1X QðG; HÞ ¼ þ logðGl12H Þ , (3) j m j¼1 Glj12H where lj ¼ 2pj=N and Iðlj Þ is periodogram. So this estimator sums the frequencies only up to 2pm=N. ^ Replacing above G by its estimate G, m Iðlj Þ 1X G^ ¼ . m j¼1 l12H j
(4)
One may define ! m m 1X IðlÞ 2H 1 X ^ RðHÞ ¼ QðG; HÞ 1 ¼ log logðlj Þ . m j¼1 lj12H m j¼1
(5)
Robinson [2] showed that under certain technical assumptions, H^ ¼ arg min RðHÞ
(6)
converges in probability to actual value H, i.e., m1=2 ðH^ HÞ!d Normalð0; 1=4Þ.
(7)
Therefore, the choice of m is quite important. The larger the value of m, the faster H^ converges to H. On the other hand, if the series also presents short-range behavior, then m should be small. In this paper the values of m ¼ N=8 and m ¼ N=16 are used.6 4
The local Whittle is particularly interesting since Robinson [2] showed that under mild conditions its estimation of the long-range dependence parameter converges asymptotically to the actual value. Moreover, it presents robustness to short-range dependence that may be actually present in the data and also periodicity [12]. 5 See Hurst [13] for the original work on testing for long-range dependence. 6 One should note that using this limiting value of m, larger frequencies are not considered. Therefore, the short-range dependence phenomena are not affecting our conclusions.
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3.2. The Tsallis distribution There is no doubt that one of the most important contributions brought by non-extensive thermodynamics was the so-called non-extensive distributions.7 The most common distribution results from the maximization of the non-extensive entropy (see [15]). R 1 pðxÞq dx Sq ¼ k , (8) q1 where k is a constant, subjected to the usual constraints Z pðxÞ dx ¼ 1, Z hx xi hðx xÞ2 i
ðx xÞpðxÞq dx ¼ 0, Z
ðx xÞ2 pðxÞq dx ¼ s2
(9)
(10)
(11)
i.e., pðxÞ ¼
1 ð1 þ bðq 1Þðx xÞÞ1=ðq1Þ , zq
(12)
where zq ¼
p bðq 1Þ
1=2
Gðð3 qÞ=2ðq 1ÞÞ Gð1=ðq 1ÞÞ
(13)
is the appropriate normalization factor, GðÞ is the gamma function and b is the so-called Lagrange parameter associated to with constraint (11) and satisfies b ¼ 1=ð2s2 Þ. If q ¼ 1, it is obvious that Eq. (12) recoveries the Gaussian distribution. Furthermore, the ordinary variance is given by s2 ¼
1 bð5 3qÞ
(14)
for qo 53 and it diverges for values qX 53. On the other hand, it is easy to show that the kurtosis coefficient K depends only on q and is given by K¼
3ð5 3qÞ . 7 5q
(15)
Eq. (15) is particularly useful to calculate q from real data. A theoretical relationship has been established between q and H. If one considers that the variance is finite, i.e., qo 53 and t ! 1, then Borland [16] showed that hxðbtÞ2 i ¼ b2=ð3qÞ hxðtÞ2 i.
(16)
On the other hand, it is known that (see [17]) hxðbtÞ2 i ¼ b2H hxðtÞ2 i.
(17)
Thus, one gets H¼ 7
1 . 3q
See Borges [14] for a comprehensive presentation of these distributions.
(18)
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However, Cajueiro and Tabak [11] have presented statistical evidence showing that Eq. (18) is in general only valid for fractional Brownian motion returns. While the Hurst exponent is defined as a measure of long-range dependence presented in data, the Tsallis q parameter also measures other types of phenomena that the Hurst exponent H does not take into account. Therefore, in this paper, these two measures of efficiency are used separately to take into account the degree of development of the given markets—while we use the local Whittle method to estimate the long-range dependence phenomenon, we use Eq. (15) to estimate the Tsallis q entropic index. One may notice, as remarked above, that usually these measures do not provide the same rank of efficiency for these countries. 4. Data The data employed in this study consists of daily nominal exchange fates for Australian dollar, Canadian dollar, New Zealand dollar, Japanese yen, British pound, Norwegian kroner, Singapore dollar, Swedish krona, Swiss franc and US dollar, all relative to the euro from 5th January 1999 to 22 April 2005 (more than 6 years of data). The total number of observations is 1644. The exchange rate data were obtained from Bloomberg database. This short historical data availability may impact the results. However, Cajueiro and Tabak [18] suggest that 4 years of data are a sufficiently long time series to perform long-range dependence tests. The authors employ 1008 observations to study the dynamic behavior of emerging and developed countries. We study changes in the natural log of exchange rates and its volatility by evaluating the absolute value of such changes.8 5. Empirical results Table 1 presents Tsallis q entropic index and Hurst exponents for 10 euro bilateral exchange rates. If exchange rates were normally distributed the Tsallis q entropic index would be close to 1. The closer to 1 the more efficient is the time series. The Swiss franc is the most inefficient with a coefficient above 1.3. Hurst exponents are presented with their associated standard errors and a Wald statistic to test for the restriction that these Hurst exponents are close to 0.5. The Wald statistic has been constructed by taking the square difference of the Hurst exponent to 0.5 and dividing it by the square standard error. Such statistic has a w2 distribution with one degree of freedom. Table 2 presents a ranking for euro exchange rates using the Tsallis q entropic index. As we can see the US, Canadian and Singapore dollar rank amongst the most efficient currencies. This result is expected as the euro is traded predominantly against the US dollar. For April 2004 this currency pair represented 76% of foreign exchange turnover involving the euro. Furthermore, the euro/US dollar is the most actively traded currency pair, accounting for approximately 30% of global turnover (see [22]). The Swiss franc has an average daily turnover of 28 billion US dollars (in the euro area) and ranks fifth in terms of turnover, after the US dollar (247 US billion dollars), euro (177 US billion dollars), British pound (45 US billion dollars) and Japanese yen (39 US billion dollars). Therefore, we would not expect to see it as the most weak form inefficient currency, as it has the highest q. In Table 3 we rank exchange rates according to Wald statistics and present their respective p-values. A pvalue under 0.05 suggests that the null of absence of long-range dependence can be rejected at the 5% significance level. We find weak evidence of long-range dependence only for the New Zealand dollar and the Swedish krona. These results do not seem to be driven by liquidity and currency turnover. The BIS [22] report suggests that the average daily turnover for the New Zealand and Swedish krona are around 1 and 7 US billion dollars (daily average for April 2004). However, Canadian and Australian dollar have an average daily turnover of 5 and 6 US billions dollars, and seem to be more efficient than the New Zealand dollar and Swedish krona. 8
The reader is also referred to Cajueiro and Tabak [19–21], in which the authors test for time-varying degree of long-range dependence using approximately 1000 observations.
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Table 1 The first column presents Tsallis q entropic index Currencies
Australian dollar Canadian dollar Swiss franc British pound Japanese yen Norwegian krone New Zealand dollar Swedish krona Singapore dollar US dollar
q
m=8
1.249733 1.149488 1.326573 1.185974 1.257248 1.266709 1.262765 1.25212 1.153443 1.124752
m=16
H
s
W
H
s
W
0.452 0.498 0.454 0.474 0.480 0.487 0.437 0.401 0.499 0.539
0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035
1.88932 0.003758 1.718131 0.537481 0.329005 0.145246 3.226133 8.107876 0.00058 1.259306
0.461 0.455 0.437 0.441 0.451 0.521 0.377 0.419 0.492 0.515
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
0.628453 0.825858 1.644282 1.423184 0.962931 0.185625 6.136701 2.66144 0.023868 0.097138
Next we present Hurst exponents (H) for m=8 and m=16, followed by its standard error (s) and the associated Wald statistic (W), respectively. The Wald statistic is given by W ¼ ðH 0:5Þ2 =ðs2 Þ and has a w12 distribution. It tests the null that the estimated Hurst exponent H is equal to 0.5. The s stands for the standard error associated to the H.
Table 2 Ranking euro exchange rates according to the Tsallis q entropic index Currencies
q
US dollar Canadian dollar Singapore dollar British pound Australian dollar Swedish krona Japanese yen New Zealand dollar Norwegian krone Swiss franc
1.124752 1.149488 1.153443 1.185974 1.249733 1.25212 1.257248 1.262765 1.266709 1.326573
Table 3 Ranking euro exchange rates according to the Wald statistics for changes in exchange rates m=8
Singapore dollar Canadian dollar Norwegian krone Japanese yen British pound US dollar Swiss franc Australian dollar New Zealand dollar Swedish krona
m=16
W
p-value
0.00058 0.003758 0.145246 0.329005 0.537481 1.259306 1.718131 1.88932 3.226133 8.107876
0.980784 0.951119 0.703121 0.566246 0.463478 0.261783 0.189934 0.169279 0.072472 0.004407
Singapore dollar US dollar Norwegian krone Australian dollar Canadian dollar Japanese yen British pound Swiss franc Swedish krona New Zealand dollar
W
p-value
0.023868 0.097138 0.185625 0.628453 0.825858 0.962931 1.423184 1.644282 2.66144 6.136701
0.877221 0.755291 0.666584 0.427923 0.363473 0.32645 0.23288 0.199739 0.102808 0.01324
The Wald statistic is given by W ¼ ðH 0:5Þ2 =s 2 and has a w12 distribution. It tests the null that the estimated Hurst exponent H is equal to 0.5. The s stands for the standard error associated to the H.
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Table 4 We present Hurst exponents (H), estimated for absolute changes of exchange rates (volatility), for m=8 and m=16, followed by its standard error (s) and the associated Wald statistic (W), respectively m=8
Australian dollar Canadian dollar Swiss franc British pound Japanese yen Norwegian krone New Zealand dollar Swedish krona Singapore dollar US dollar
m=16
H
s
W
H
s
W
0.674 0.659 0.747 0.757 0.715 0.670 0.695 0.773 0.686 0.679
0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035
24.954 20.738 50.031 54.300 37.773 23.744 31.280 61.298 28.343 26.171
0.698 0.676 0.829 0.760 0.766 0.717 0.743 0.836 0.749 0.737
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
15.988 12.570 44.204 27.502 28.942 19.202 24.082 46.086 25.243 22.846
The Wald statistic is given by W ¼ ðH 0:5Þ2 =s 2 and has a w12 distribution. It tests the null that the estimated Hurst exponent H is equal to 0.5. The s stands for the standard error associated to the H.
Table 5 Ranking euro exchange rates according to the Wald statistics for volatility m=8
Canadian dollar Norwegian krone Australian dollar US dollar Singapore dollar New Zealand dollar Japanese yen Swiss franc British pound Swedish krona
m=16
W
p-value
20.738 23.744 24.954 26.171 28.343 31.280 37.773 50.031 54.300 61.298
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Canadian dollar Australian dollar Norwegian krone US dollar New Zealand dollar Singapore dollar British pound Japanese yen Swiss franc Swedish krona
W
p-value
12.570 15.988 19.202 22.846 24.082 25.243 27.502 28.942 44.204 46.086
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
The Wald statistic is given by W ¼ ðH 0:5Þ2 =s 2 and has a w12 distribution. It tests the null that the estimated Hurst exponent H is equal to 0.5. The s stands for the standard error associated to the H.
Belaire-Franch and Opong [9] find evidence of weak form efficiency for most euro exchange rates. The authors employ variance ratio statistics, with a sidack and bootstrap correction for multiplicity, and reject the null of absence of short-term predictability only in two cases: the Canadian and Singapore dollar. Our results are in line with their findings as we find weak evidence of market inefficiency using Hurst exponents for changes in exchange rates. Table 4 present Hurst exponents evaluated for absolute changes of exchange rates, which are used as a proxy for volatility. In this case Hurst exponents are high (all above 0.65), regardless of the choice of m, suggesting a strong degree of long-range dependence. This result is in line with most of the financial literature, which finds strong evidence of long-range dependence in volatility for a variety of markets (see for example [21]). Table 5 presents a ranking for efficiency using Hurst exponents evaluated for the volatility of exchange rates. These results do not depend on liquidity or market turnover, suggesting that persistence in volatility stems from other sources rather than from liquidity. Market microstructure differences may play a role in explaining these results, such as for example, differences in traders that operate in each currency. Understanding this phenomenon in depth is beyond the scope of this paper, but is an important issue to be
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addressed by the literature, although there is very little research focusing on studying for possible causes of long-range dependence. 6. Conclusions This paper examines the dynamics properties of euro bilateral exchange rates. The empirical results suggest that deviations from normality, measured by the q entropic index of the Tsallis distribution, are more pronounced for the Australian dollar, Swedish krona, Japanese yen, New Zealand dollar, Norwegian krone and the Swiss franc, with q above 1.2, while q are all below 1.2 for the US, Canadian and Singapore dollar and the British pound. We develop a Wald statistic to test the null that a specific exchange rate has a Hurst exponent close to 0.5 (random walk behavior). We can reject the null at 1% significance level only for New Zealand (for m=16). Rejection of the null of absence of long memory is strong only for volatility, as we reject the null at the 1% significance level for all exchange rates, regardless of the choice of m. Empirical results suggest that all exchange rates have a strong degree of persistence. The British pound, Swiss franc, Japanese yen and Swedish krona are amongst the most persistent. Pan and Liu [23] have shown that fractional cointegration may be a feature of exchange rate dynamics. However, it could be changing over time, a feature that has been presented before in Refs. [18–21]. Further research could follow along these lines and test for time-varying degrees of long-range dependence. Such tests may be useful in enhancing our knowledge of the dynamics of exchange rates. The role of intervention in exchange rate markets and market microstructure on exchange rate dynamics is an important issue that should be addressed in the literature. Its impact on long-range dependence parameters could provide useful insights to evaluate the impacts of these interventions in the long run. The main question that emerges is whether central banks are able to correct long-term misalignments. References [1] J. Kearns, R. Rigobon, Indentifying the efficacy of central bank interventions: evidence from Australia and Japan, J. Int. Econ. 66 (2005) 31–48. [2] P. Robinson, Gaussian semiparametric estimation of long-range dependence, Ann. Stat. 23 (1995) 1630–1661. [3] Y. Cheung, Long memory in foreign-exchange rates, J. Bus. Econ. Stat. 11 (1993) 93–101. [4] H.J. Jin, J. Elder, W.W. Koo, A reexamination of fractional integrating dynamics in foreign currency markets, Int. Rev. Econ. Finance, forthcoming. [5] Y.W. Han, Long memory volatility dependency temporal aggregation and the Korean currency crisis: the role of a high frequency Korean won (KRW)-US dollar ($) exchange rate, Jpn. World Econ. 17 (2005) 97–109. [6] D. Grech, Z. Mazur, Can one make any crash prediction in finance using the local Hurst exponent idea?, Physica A 336 (2004) 133–145. [7] T. Bollerslev, J.H. Wright, Semiparametric estimation of long memory volatility dependencies: the role of high frequency data, J. Econometrics 98 (2000) 81–106. [8] R.T. Baillie, A.A. Cecen, C. Erkal, Y.-W. Han, Measuring non-linearity, long memory and self-similarity in high-frequency European exchange rates, J. Int. Financial Markets, Inst. Money 14 (2004) 401–418. [9] J. Belaire-Franch, K.K. Opong, Some evidence of random walk behavior of euro exchange rates using rank and signs, J. Banking Finance 29 (2005) 1631–1643. [10] K.V. Chow, K.C. Denning, A simple multiple variance ratio test, J. Econometrics 58 (1993) 385–401. [11] D.O. Cajueiro, B.M. Tabak, Is the expression H ¼ 1=ð3 qÞ valid for real financial data? Working Paper of Chatolic University of Brasilia, 2005. [12] A. Montanary, M.S. Taqqu, V. Teverovsky, Estimating long-range dependence in the presence of periodicity: an empirical study, Math. Comput. Modelling 29 (1999) 217–228. [13] E. Hurst, Long term storage capacity of reservoirs, Trans. Am. Soc. Civil Eng. 116 (1951) 770–799. [14] E.P. Borges, Manifestac- o˜es dinaˆmicas e termodinaˆmicas de sistemas na˜o-extensivos, Tese de Doutorado, Centro Brasileiro de Pesquisas Fı´ sicas, 2004. [15] C. Tsallis, D.J. Bukman, Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E 54 (1996) 2197–2200. [16] L. Borland, Microscopic dynamics of the nonlinear Fokker–Planck equation: a phenomenological model, Phys. Rev. E 57 (1998) 6634–6642. [17] J. Feder, Fractals, Plenum Press, New York, 1988.
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[18] D.O. Cajueiro, B.M. Tabak, Ranking efficiency for emerging markets II, Chaos Solitons Fractals 23 (2005) 671–675. [19] D.O. Cajueiro, B.M. Tabak, The Hurst’s exponent over time: testing the assertion that emerging markets are becoming more efficient, Physica A 336 (2004) 521–537. [20] D.O. Cajueiro, B.M. Tabak, Ranking efficiency for emerging markets, Chaos Solitons Fractals 22 (2004) 349–352. [21] D.O. Cajueiro, B.M. Tabak, Testing for time-varying long-range dependence volatility for emerging markets, Physica A 346 (2004) 577–588. [22] BIS, BIS Triennial Survey, Basel, 2004. [23] M.-P. Pan, Y.A. Liu, Fractional cointegration, long memory, and exchange rate dynamics, Int. Rev. Econ. Finance 8 (1999) 305–316.
Physica A 367 (2006) 319–327 www.elsevier.com/locate/physa
Assessing inefficiency in euro bilateral exchange rates Benjamin M. Tabaka,, Daniel O. Cajueirob a
Banco Central do Brasil, SBS Quadra 3, Bloco B, 9 andar, DF 70074-900, Brazil Universidade Cato´lica de Brası´lia—Mestrado em Economia de Empresas, SGAN 916, Mo´dulo B—Asa Norte, DF 70790-160, Brazil
b
Received 28 October 2005; received in revised form 8 December 2005 Available online 5 January 2006
Abstract This paper assesses inefficiency for 10 euro bilateral exchange rates. We study the dynamics of these time series by estimating Tsallis q entropic index and Hurst exponents using the local Whittle estimator. Empirical results suggest that US, Canadian and Singapore dollar are amongst the most efficient currencies, while Japanese yen and Swedish krona are amongst the most inefficient. r 2006 Elsevier B.V. All rights reserved. Keywords: Euro bilateral exchange rates; Hurst exponents; Tsallis q entropic index; Long-range dependence
1. Introduction The analysis of the dynamics of exchange rates is crucial for the design of policy modeling, portfolio and risk management, and is an essential question in international finance as both exchange rates and their volatility are important determinants of international capital flows, foreign direct investment and macroeconomic performance. Recent developments in statistical physics may shed some light in the dynamics of exchange rates and may help in the design of exchange rate policies. An important question is how shocks to exchange rates propagate. If exchange rates or their volatility display long-range dependence then shocks tend to decay at a much slower rate than with traditional ARMA models. An important implication of long memory is that economic and financial models have to take such features into consideration. For example, if exchange rates have long memory then shocks may last for a long time and policy makers may be inclined to intervene in the foreign exchange market as the speed of adjustment to stability (or a new equilibrium level) may be slow. Generally, central bank interventions are motivated by the desire to check short-run trends or to correct longer-term misalignments. Such intervention has implication on the dynamics of exchange rates. For example, Kearns and Rigobon [1] test for the efficacy of central bank interventions in Australia and Japan and find that a US$ 100 million purchase appreciates the Australian dollar by 1.3–1.8% but the yen by just 0.2%. These results suggest that interventions may change the dynamics of exchange rates and that their market Corresponding author. Tel.: +55 61 4143092; fax: +55 61 4143045.
E-mail address: [email protected] (B.M. Tabak). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.12.007
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microstructure effects could be important. Therefore, long memory may cause and be caused by foreign exchange rate interventions. In this paper we will not address testing for causality between intervention and the degree of long-range dependence, although it is an important issue. We will focus on testing the degree of long-range dependence using the local Whittle estimator due to Robinson [2] and assessing relative inefficiencies in different currencies using as our measure of efficiency the Tsallis entropic index. Despite extensive empirical work using US dollar exchange rates, empirical tests using euro exchange rates are still lacking. In this paper we try to fill this gap by focusing on euro bilateral exchange rates and in particular on the euro-US dollar, euro-Canadian dollar, euro-Japanese yen, euro-Australian dollar, euro-New Zealand dollar, euro-Singapore dollar, euro-Norwegian kroner, euro-Swedish krona, euro-Swiss franc and euro-British pound exchange rates. The paper is organized along the following lines. Section 2 provides a brief literature review. While Section 3 discusses the methodology employed in the paper, Section 4 describes the data. Section 5 presents empirical results. Finally Section 6 concludes the paper.
2. Brief literature review Cheung [3] suggests that major exchange rate series displayed evidence of long-range dependence.1 Jin et al. [4] find that 14 out of 19 exchange rate series display evidence of long memory dynamics, using a wavelet OLS estimator.2 The authors find evidence that most exchange rates present antipersistence. These results provide motivation for studying long-range dependence for exchange rates. Han [5] study the Korean won–US dollar exchange rate using a high frequency sample and finds evidence of long memory in volatility, but that the estimated long memory parameter changes and empirical results may be subject to exogenous shifts and multiple breaks in the time series. During crisis period, November 1997, the long memory parameter is the greatest. This is in line with recent work from Grech and Mazur [6], that have tested whether Hurst exponents may be used to predict financial crashes and find positive evidence. Recent empirical work has found evidence of long-range dependence in volatility of asset prices. For example, Bollerslev and Wright [7] have shown that the temporal dependencies in the volatility process of foreign exchange rates can be best characterized by long-range dependence models. Additionally, Baillie et al. [8] study high frequency European exchange rates and suggest that currency volatility is well represented by a FIGARCH model.3 Very little research has been undertaken for euro bilateral exchange rates. Belaire-Franch and Opong [9] study the behavior of euro exchange rates testing for random walk behavior using a variance ratio methodology. Using Chow and Denning’s [10] version of the multiple variance ratio test, and adjusted critical values to prevent size distortions, the authors find that the random walk hypothesis can be rejected only for the Canadian dollar. Additionally, using signs and rank based tests and adjustments for multiplicity suggest that only for the Canadian and Singapore dollar we can reject the random walk hypothesis. It is important to notice that this tests were performed using a maximum of 30 days as the investment horizon. Therefore, are focusing on very short-term forecasting power. Empirical evidence presented so far suggests that long-range dependence may be a characteristic of both exchange rates and its volatility. In this line, part of the remainder of this paper will focus on testing whether this assertion holds true for a variety of euro bilateral exchange rates. On the other hand, Cajueiro and Tabak [11] have provided statistical evidence that, beside the long-range dependence phenomenon, the Tsallis q entropic index can be assess other types of inefficiency. Therefore, this measure will also be used to assess inefficiency in these markets. 1
Except for the British pound. The authors result’s contrasts with empirical results using the GPH estimator. 3 Their study focuses on British pound, Swiss franc and Deutsche mark vis-a-vis the US dollar. 2
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3. Methodology 3.1. The local Whittle estimator In this paper, since it is essential to know the degree of long memory which a given market presents, the Hurst’s exponent is estimated using the local Whittle estimator due to Robinson [2].4 As it is known, a given market is said to have long-range dependence with persistent behavior if the Hurst’s exponent H40:5, with antipersistent behavior is if Ho0:5 and random walk behavior if H ¼ 0:5.5 The local Whittle estimator is a semi-parametric estimator, which only requires specifying the parametric form of the spectral density when the frequency l is close to zero, f ðlÞGjlj12H
as l ! 0
(1)
when G is a constant. The computation involves an additional parameter m, an integer less than N=2, where N is the size of the time series, and such that, as N ! 1, 1 m þ ! 0. m N
(2)
This means that as N gets large, m gets large as well, although slower. For a spectral density of form (1), the Whittle approximation of the Gaussian likehood function is obtained by minimizing ! m Iðlj Þ 1X QðG; HÞ ¼ þ logðGl12H Þ , (3) j m j¼1 Glj12H where lj ¼ 2pj=N and Iðlj Þ is periodogram. So this estimator sums the frequencies only up to 2pm=N. ^ Replacing above G by its estimate G, m Iðlj Þ 1X G^ ¼ . m j¼1 l12H j
(4)
One may define ! m m 1X IðlÞ 2H 1 X ^ RðHÞ ¼ QðG; HÞ 1 ¼ log logðlj Þ . m j¼1 lj12H m j¼1
(5)
Robinson [2] showed that under certain technical assumptions, H^ ¼ arg min RðHÞ
(6)
converges in probability to actual value H, i.e., m1=2 ðH^ HÞ!d Normalð0; 1=4Þ.
(7)
Therefore, the choice of m is quite important. The larger the value of m, the faster H^ converges to H. On the other hand, if the series also presents short-range behavior, then m should be small. In this paper the values of m ¼ N=8 and m ¼ N=16 are used.6 4
The local Whittle is particularly interesting since Robinson [2] showed that under mild conditions its estimation of the long-range dependence parameter converges asymptotically to the actual value. Moreover, it presents robustness to short-range dependence that may be actually present in the data and also periodicity [12]. 5 See Hurst [13] for the original work on testing for long-range dependence. 6 One should note that using this limiting value of m, larger frequencies are not considered. Therefore, the short-range dependence phenomena are not affecting our conclusions.
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3.2. The Tsallis distribution There is no doubt that one of the most important contributions brought by non-extensive thermodynamics was the so-called non-extensive distributions.7 The most common distribution results from the maximization of the non-extensive entropy (see [15]). R 1 pðxÞq dx Sq ¼ k , (8) q1 where k is a constant, subjected to the usual constraints Z pðxÞ dx ¼ 1, Z hx xi hðx xÞ2 i
ðx xÞpðxÞq dx ¼ 0, Z
ðx xÞ2 pðxÞq dx ¼ s2
(9)
(10)
(11)
i.e., pðxÞ ¼
1 ð1 þ bðq 1Þðx xÞÞ1=ðq1Þ , zq
(12)
where zq ¼
p bðq 1Þ
1=2
Gðð3 qÞ=2ðq 1ÞÞ Gð1=ðq 1ÞÞ
(13)
is the appropriate normalization factor, GðÞ is the gamma function and b is the so-called Lagrange parameter associated to with constraint (11) and satisfies b ¼ 1=ð2s2 Þ. If q ¼ 1, it is obvious that Eq. (12) recoveries the Gaussian distribution. Furthermore, the ordinary variance is given by s2 ¼
1 bð5 3qÞ
(14)
for qo 53 and it diverges for values qX 53. On the other hand, it is easy to show that the kurtosis coefficient K depends only on q and is given by K¼
3ð5 3qÞ . 7 5q
(15)
Eq. (15) is particularly useful to calculate q from real data. A theoretical relationship has been established between q and H. If one considers that the variance is finite, i.e., qo 53 and t ! 1, then Borland [16] showed that hxðbtÞ2 i ¼ b2=ð3qÞ hxðtÞ2 i.
(16)
On the other hand, it is known that (see [17]) hxðbtÞ2 i ¼ b2H hxðtÞ2 i.
(17)
Thus, one gets H¼ 7
1 . 3q
See Borges [14] for a comprehensive presentation of these distributions.
(18)
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However, Cajueiro and Tabak [11] have presented statistical evidence showing that Eq. (18) is in general only valid for fractional Brownian motion returns. While the Hurst exponent is defined as a measure of long-range dependence presented in data, the Tsallis q parameter also measures other types of phenomena that the Hurst exponent H does not take into account. Therefore, in this paper, these two measures of efficiency are used separately to take into account the degree of development of the given markets—while we use the local Whittle method to estimate the long-range dependence phenomenon, we use Eq. (15) to estimate the Tsallis q entropic index. One may notice, as remarked above, that usually these measures do not provide the same rank of efficiency for these countries. 4. Data The data employed in this study consists of daily nominal exchange fates for Australian dollar, Canadian dollar, New Zealand dollar, Japanese yen, British pound, Norwegian kroner, Singapore dollar, Swedish krona, Swiss franc and US dollar, all relative to the euro from 5th January 1999 to 22 April 2005 (more than 6 years of data). The total number of observations is 1644. The exchange rate data were obtained from Bloomberg database. This short historical data availability may impact the results. However, Cajueiro and Tabak [18] suggest that 4 years of data are a sufficiently long time series to perform long-range dependence tests. The authors employ 1008 observations to study the dynamic behavior of emerging and developed countries. We study changes in the natural log of exchange rates and its volatility by evaluating the absolute value of such changes.8 5. Empirical results Table 1 presents Tsallis q entropic index and Hurst exponents for 10 euro bilateral exchange rates. If exchange rates were normally distributed the Tsallis q entropic index would be close to 1. The closer to 1 the more efficient is the time series. The Swiss franc is the most inefficient with a coefficient above 1.3. Hurst exponents are presented with their associated standard errors and a Wald statistic to test for the restriction that these Hurst exponents are close to 0.5. The Wald statistic has been constructed by taking the square difference of the Hurst exponent to 0.5 and dividing it by the square standard error. Such statistic has a w2 distribution with one degree of freedom. Table 2 presents a ranking for euro exchange rates using the Tsallis q entropic index. As we can see the US, Canadian and Singapore dollar rank amongst the most efficient currencies. This result is expected as the euro is traded predominantly against the US dollar. For April 2004 this currency pair represented 76% of foreign exchange turnover involving the euro. Furthermore, the euro/US dollar is the most actively traded currency pair, accounting for approximately 30% of global turnover (see [22]). The Swiss franc has an average daily turnover of 28 billion US dollars (in the euro area) and ranks fifth in terms of turnover, after the US dollar (247 US billion dollars), euro (177 US billion dollars), British pound (45 US billion dollars) and Japanese yen (39 US billion dollars). Therefore, we would not expect to see it as the most weak form inefficient currency, as it has the highest q. In Table 3 we rank exchange rates according to Wald statistics and present their respective p-values. A pvalue under 0.05 suggests that the null of absence of long-range dependence can be rejected at the 5% significance level. We find weak evidence of long-range dependence only for the New Zealand dollar and the Swedish krona. These results do not seem to be driven by liquidity and currency turnover. The BIS [22] report suggests that the average daily turnover for the New Zealand and Swedish krona are around 1 and 7 US billion dollars (daily average for April 2004). However, Canadian and Australian dollar have an average daily turnover of 5 and 6 US billions dollars, and seem to be more efficient than the New Zealand dollar and Swedish krona. 8
The reader is also referred to Cajueiro and Tabak [19–21], in which the authors test for time-varying degree of long-range dependence using approximately 1000 observations.
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Table 1 The first column presents Tsallis q entropic index Currencies
Australian dollar Canadian dollar Swiss franc British pound Japanese yen Norwegian krone New Zealand dollar Swedish krona Singapore dollar US dollar
q
m=8
1.249733 1.149488 1.326573 1.185974 1.257248 1.266709 1.262765 1.25212 1.153443 1.124752
m=16
H
s
W
H
s
W
0.452 0.498 0.454 0.474 0.480 0.487 0.437 0.401 0.499 0.539
0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035
1.88932 0.003758 1.718131 0.537481 0.329005 0.145246 3.226133 8.107876 0.00058 1.259306
0.461 0.455 0.437 0.441 0.451 0.521 0.377 0.419 0.492 0.515
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
0.628453 0.825858 1.644282 1.423184 0.962931 0.185625 6.136701 2.66144 0.023868 0.097138
Next we present Hurst exponents (H) for m=8 and m=16, followed by its standard error (s) and the associated Wald statistic (W), respectively. The Wald statistic is given by W ¼ ðH 0:5Þ2 =ðs2 Þ and has a w12 distribution. It tests the null that the estimated Hurst exponent H is equal to 0.5. The s stands for the standard error associated to the H.
Table 2 Ranking euro exchange rates according to the Tsallis q entropic index Currencies
q
US dollar Canadian dollar Singapore dollar British pound Australian dollar Swedish krona Japanese yen New Zealand dollar Norwegian krone Swiss franc
1.124752 1.149488 1.153443 1.185974 1.249733 1.25212 1.257248 1.262765 1.266709 1.326573
Table 3 Ranking euro exchange rates according to the Wald statistics for changes in exchange rates m=8
Singapore dollar Canadian dollar Norwegian krone Japanese yen British pound US dollar Swiss franc Australian dollar New Zealand dollar Swedish krona
m=16
W
p-value
0.00058 0.003758 0.145246 0.329005 0.537481 1.259306 1.718131 1.88932 3.226133 8.107876
0.980784 0.951119 0.703121 0.566246 0.463478 0.261783 0.189934 0.169279 0.072472 0.004407
Singapore dollar US dollar Norwegian krone Australian dollar Canadian dollar Japanese yen British pound Swiss franc Swedish krona New Zealand dollar
W
p-value
0.023868 0.097138 0.185625 0.628453 0.825858 0.962931 1.423184 1.644282 2.66144 6.136701
0.877221 0.755291 0.666584 0.427923 0.363473 0.32645 0.23288 0.199739 0.102808 0.01324
The Wald statistic is given by W ¼ ðH 0:5Þ2 =s 2 and has a w12 distribution. It tests the null that the estimated Hurst exponent H is equal to 0.5. The s stands for the standard error associated to the H.
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Table 4 We present Hurst exponents (H), estimated for absolute changes of exchange rates (volatility), for m=8 and m=16, followed by its standard error (s) and the associated Wald statistic (W), respectively m=8
Australian dollar Canadian dollar Swiss franc British pound Japanese yen Norwegian krone New Zealand dollar Swedish krona Singapore dollar US dollar
m=16
H
s
W
H
s
W
0.674 0.659 0.747 0.757 0.715 0.670 0.695 0.773 0.686 0.679
0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035
24.954 20.738 50.031 54.300 37.773 23.744 31.280 61.298 28.343 26.171
0.698 0.676 0.829 0.760 0.766 0.717 0.743 0.836 0.749 0.737
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050
15.988 12.570 44.204 27.502 28.942 19.202 24.082 46.086 25.243 22.846
The Wald statistic is given by W ¼ ðH 0:5Þ2 =s 2 and has a w12 distribution. It tests the null that the estimated Hurst exponent H is equal to 0.5. The s stands for the standard error associated to the H.
Table 5 Ranking euro exchange rates according to the Wald statistics for volatility m=8
Canadian dollar Norwegian krone Australian dollar US dollar Singapore dollar New Zealand dollar Japanese yen Swiss franc British pound Swedish krona
m=16
W
p-value
20.738 23.744 24.954 26.171 28.343 31.280 37.773 50.031 54.300 61.298
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Canadian dollar Australian dollar Norwegian krone US dollar New Zealand dollar Singapore dollar British pound Japanese yen Swiss franc Swedish krona
W
p-value
12.570 15.988 19.202 22.846 24.082 25.243 27.502 28.942 44.204 46.086
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
The Wald statistic is given by W ¼ ðH 0:5Þ2 =s 2 and has a w12 distribution. It tests the null that the estimated Hurst exponent H is equal to 0.5. The s stands for the standard error associated to the H.
Belaire-Franch and Opong [9] find evidence of weak form efficiency for most euro exchange rates. The authors employ variance ratio statistics, with a sidack and bootstrap correction for multiplicity, and reject the null of absence of short-term predictability only in two cases: the Canadian and Singapore dollar. Our results are in line with their findings as we find weak evidence of market inefficiency using Hurst exponents for changes in exchange rates. Table 4 present Hurst exponents evaluated for absolute changes of exchange rates, which are used as a proxy for volatility. In this case Hurst exponents are high (all above 0.65), regardless of the choice of m, suggesting a strong degree of long-range dependence. This result is in line with most of the financial literature, which finds strong evidence of long-range dependence in volatility for a variety of markets (see for example [21]). Table 5 presents a ranking for efficiency using Hurst exponents evaluated for the volatility of exchange rates. These results do not depend on liquidity or market turnover, suggesting that persistence in volatility stems from other sources rather than from liquidity. Market microstructure differences may play a role in explaining these results, such as for example, differences in traders that operate in each currency. Understanding this phenomenon in depth is beyond the scope of this paper, but is an important issue to be
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addressed by the literature, although there is very little research focusing on studying for possible causes of long-range dependence. 6. Conclusions This paper examines the dynamics properties of euro bilateral exchange rates. The empirical results suggest that deviations from normality, measured by the q entropic index of the Tsallis distribution, are more pronounced for the Australian dollar, Swedish krona, Japanese yen, New Zealand dollar, Norwegian krone and the Swiss franc, with q above 1.2, while q are all below 1.2 for the US, Canadian and Singapore dollar and the British pound. We develop a Wald statistic to test the null that a specific exchange rate has a Hurst exponent close to 0.5 (random walk behavior). We can reject the null at 1% significance level only for New Zealand (for m=16). Rejection of the null of absence of long memory is strong only for volatility, as we reject the null at the 1% significance level for all exchange rates, regardless of the choice of m. Empirical results suggest that all exchange rates have a strong degree of persistence. The British pound, Swiss franc, Japanese yen and Swedish krona are amongst the most persistent. Pan and Liu [23] have shown that fractional cointegration may be a feature of exchange rate dynamics. However, it could be changing over time, a feature that has been presented before in Refs. [18–21]. Further research could follow along these lines and test for time-varying degrees of long-range dependence. Such tests may be useful in enhancing our knowledge of the dynamics of exchange rates. The role of intervention in exchange rate markets and market microstructure on exchange rate dynamics is an important issue that should be addressed in the literature. Its impact on long-range dependence parameters could provide useful insights to evaluate the impacts of these interventions in the long run. The main question that emerges is whether central banks are able to correct long-term misalignments. References [1] J. Kearns, R. Rigobon, Indentifying the efficacy of central bank interventions: evidence from Australia and Japan, J. Int. Econ. 66 (2005) 31–48. [2] P. Robinson, Gaussian semiparametric estimation of long-range dependence, Ann. Stat. 23 (1995) 1630–1661. [3] Y. Cheung, Long memory in foreign-exchange rates, J. Bus. Econ. Stat. 11 (1993) 93–101. [4] H.J. Jin, J. Elder, W.W. Koo, A reexamination of fractional integrating dynamics in foreign currency markets, Int. Rev. Econ. Finance, forthcoming. [5] Y.W. Han, Long memory volatility dependency temporal aggregation and the Korean currency crisis: the role of a high frequency Korean won (KRW)-US dollar ($) exchange rate, Jpn. World Econ. 17 (2005) 97–109. [6] D. Grech, Z. Mazur, Can one make any crash prediction in finance using the local Hurst exponent idea?, Physica A 336 (2004) 133–145. [7] T. Bollerslev, J.H. Wright, Semiparametric estimation of long memory volatility dependencies: the role of high frequency data, J. Econometrics 98 (2000) 81–106. [8] R.T. Baillie, A.A. Cecen, C. Erkal, Y.-W. Han, Measuring non-linearity, long memory and self-similarity in high-frequency European exchange rates, J. Int. Financial Markets, Inst. Money 14 (2004) 401–418. [9] J. Belaire-Franch, K.K. Opong, Some evidence of random walk behavior of euro exchange rates using rank and signs, J. Banking Finance 29 (2005) 1631–1643. [10] K.V. Chow, K.C. Denning, A simple multiple variance ratio test, J. Econometrics 58 (1993) 385–401. [11] D.O. Cajueiro, B.M. Tabak, Is the expression H ¼ 1=ð3 qÞ valid for real financial data? Working Paper of Chatolic University of Brasilia, 2005. [12] A. Montanary, M.S. Taqqu, V. Teverovsky, Estimating long-range dependence in the presence of periodicity: an empirical study, Math. Comput. Modelling 29 (1999) 217–228. [13] E. Hurst, Long term storage capacity of reservoirs, Trans. Am. Soc. Civil Eng. 116 (1951) 770–799. [14] E.P. Borges, Manifestac- o˜es dinaˆmicas e termodinaˆmicas de sistemas na˜o-extensivos, Tese de Doutorado, Centro Brasileiro de Pesquisas Fı´ sicas, 2004. [15] C. Tsallis, D.J. Bukman, Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E 54 (1996) 2197–2200. [16] L. Borland, Microscopic dynamics of the nonlinear Fokker–Planck equation: a phenomenological model, Phys. Rev. E 57 (1998) 6634–6642. [17] J. Feder, Fractals, Plenum Press, New York, 1988.
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[18] D.O. Cajueiro, B.M. Tabak, Ranking efficiency for emerging markets II, Chaos Solitons Fractals 23 (2005) 671–675. [19] D.O. Cajueiro, B.M. Tabak, The Hurst’s exponent over time: testing the assertion that emerging markets are becoming more efficient, Physica A 336 (2004) 521–537. [20] D.O. Cajueiro, B.M. Tabak, Ranking efficiency for emerging markets, Chaos Solitons Fractals 22 (2004) 349–352. [21] D.O. Cajueiro, B.M. Tabak, Testing for time-varying long-range dependence volatility for emerging markets, Physica A 346 (2004) 577–588. [22] BIS, BIS Triennial Survey, Basel, 2004. [23] M.-P. Pan, Y.A. Liu, Fractional cointegration, long memory, and exchange rate dynamics, Int. Rev. Econ. Finance 8 (1999) 305–316.