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A positive Lyapunov exponent in Swedish exchange rates?
Chaos, Solitons and Fractals 14 (2002) 1295–1304 www.elsevier.com/locate/chaos
A positive Lyapunov exponent in Swedish exchange rates? Mikael Bask
*
Department of Economics, Ume a University, SE-901 87 Ume a, Sweden Accepted 26 March 2002
Abstract In this paper, a statistical framework utilizing a blockwise bootstrap procedure is used to test for the presence of a positive Lyapunov exponent in Swedish exchange rates [M. Bask, R. Gencßay, Physica D 114 (1998) 1]. This is done since a necessary condition for chaotic dynamics is a positive Lyapunov exponent. Daily data for the Swedish Krona against the Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates are examined. In most cases, the null hypothesis that the Lyapunov exponent is zero is rejected in favor of a positive exponent. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Empirical research on nominal exchange rates has shown that a number of important regularities appear to be stable over time. For instance: (1) the behavior of the logarithms of exchange rates over time approximates a random walk; (2) most changes in exchange rates are unexpected; (3) countries with high inflation rates tend to have currencies which depreciate; and (4) actual exchange rate movements appear to overshoot movements in the equilibrium exchange rate [2,3]. These findings are important because one of the purposes of economic theory is to develop models that can explain observed regularities. In this paper, we ask whether a positive Lyapunov exponent is present in nominal exchange rates. This is an important question since a necessary condition for chaotic dynamics is a positive Lyapunov exponent. The characterizing feature of a chaotic dynamic system is that it has a property of ‘‘sensitive dependence on initial conditions’’: any two solution paths with arbitrarily close, but not equal, initial conditions will diverge at exponential rates. Globally, however, the solution paths remain within a bounded set if the dynamic system is dissipative. This means that the predictive ability of the system is strongly limited, especially in long-run predictions, even if it is still possible to make short-run predictions. The local stability properties of a dynamic system can be measured by the Lyapunov exponents. These measure the average exponential divergence or convergence of nearby, but not equal, initial conditions. Accordingly, a positive Lyapunov exponent measures the average exponential divergence of nearby initial conditions, whereas a negative exponent measures the average exponential convergence. Thus, if we refer to the property of ‘‘sensitive dependence on initial conditions’’ and have a deterministic system, a positive Lyapunov exponent can be considered as evidence of chaotic dynamics. Therefore, the estimation of the Lyapunov exponents or, to be more specific, the largest Lyapunov exponent is essential. Another characteristic of a dynamic system is the dimension of the attractor that is associated with the dynamic system. The dimension of an attractor is the amount of information needed to specify its points accurately, and is related to how hyper-volumes scale as a function of a length parameter. For example, areas vary with the square of the length of the side and volumes vary with the cube. For strange attractors, i.e., attractors associated with a chaotic
*
Tel.: +46-90-786 78 77; fax: +46-90-77 23 02. E-mail address: [email protected] (M. Bask).
0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 8 3 - 8
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dynamic system, the dimension typically takes a non-integer value. Therefore, the dimension of a strange attractor is often called the fractal dimension. The dimension of a dynamic system is important because it reveals the number of variables that are sufficient to mimic the systems behavior. Following the early work of Brock [4], there has been an increased interest in detecting chaotic dynamics in economic and financial time series [5]. A few examples are [6–15]. Barnett and Serletis [16] and Jaditz and Sayers [17] review the literature. Most of the empirical papers reached the conclusion that the time series they examined were not generated by chaotic dynamic systems. These authors did not, however, always use the Lyapunov exponents in their empirical research. They estimated, for example, the correlation dimension, i.e., a measure of the fractal dimension, for the time series in order to detect lowdimensional dynamics [18]. This was done because a general difference between deterministic chaos and a stochastic process is that the former is low-dimensional, or at least finite-dimensional, whereas the latter is infinite-dimensional. However, even though the dynamics is low-dimensional, it can still be non-chaotic. Therefore, the use of the Lyapunov exponents is more appropriate than the use of the correlation dimension. However, the second-best method to detect chaotic dynamics would be the estimation of the dimension of the attractor since it typically takes a non-integer value when the attractor is strange. But we will not estimate the dimension in this paper since the author is not aware of any distributional theory to provide a framework for statistical inference. For example, a statistical test with the null hypothesis of an integer valued dimension. The so-called BDS-test statistic was, and is, widely used in empirical research for detecting non-linear dynamics in economic and financial time series [19,20]. Specifically, it tests the null hypothesis that a time series is independent and identically distributed, i.e., that the data generating process is IID. As such, it is not a test for deterministic chaos. It is, however, useful in detecting any remaining ‘‘structure’’ in the estimated residuals of time series models that can be transformed into models driven by independent and identically distributed error terms. Earlier research on detecting chaotic dynamics in exchange rates via Lyapunov exponents has been carried out by Bajo-Rubio et al. [21], Bask [22], Dechert and Gencßay [23] and Jonsson [24]. However, these and other papers have been criticized because they lack a distributional theory to provide a framework for statistical inference. 1 Here, we apply a statistical framework proposed by Bask and Gencßay [1]. This utilizes a moving blocks bootstrap procedure that tests for the presence of a positive Lyapunov exponent in an observed stochastic time series. Bootstrapping uses the available observations to design a sort of Monte Carlo experiment in which the observations themselves are used to approximate the distribution of the error terms or other random quantities [27]. K€ unsch [28] and Liu and Singh [29] extended the idea of bootstrapping to the case where the observations form a stationary sequence, i.e., when the observations constitute a time series. The exchange rate series examined in this paper are the Swedish Krona against the Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates. We use daily data from 17 May 1991 to 31 August 1995. The Swedish Krona was pegged against the ECU between 17 May 1991 and 19 November 1992. The exchange rate series are thus divided into two parts so that the dynamics from the fixed and flexible exchange rate periods can be separated. The remainder of this paper is organized as follows. Section 2 presents the procedure for estimating the largest Lyapunov exponent from an observed scalar time series and for drawing statistical inferences from these estimates. Section 3 contains the results and Section 4 concludes the paper.
2. Theory Several steps are necessary to answer the question whether a positive Lyapunov exponent is present in (nominal) exchange rates. Firstly, the dynamics that govern the evolution of the exchange rates must be reconstructed because the specific equations of motion are not known. In the case where there is a dynamic system hidden in a ‘‘black box’’, Takens [30] has demonstrated that it is possible to reconstruct the unknown dynamics using only an observed scalar time series. Remarkably, the reconstructed dynamics and the dynamic system hidden in the ‘‘black box’’ are ‘‘equivalent’’ in the sense that they share certain dynamic properties. For example, the Lyapunov exponents for the reconstructed dynamics and for the unknown dynamic system are equivalent. The reconstruction of the dynamics using the exchange rate series as the observed scalar time series is presented in Section 2.1. Given a reconstruction of the dynamics, the second step is taken by estimating the Lyapunov exponents. A simple method for estimating the largest Lyapunov exponent from an observed scalar time series was proposed by Rosenstein
1 The consistency and asymptotic normality of an estimator of smooth Lyapunov exponents have been proved under certain regularity conditions by Bask and de Luna [25] and Whang and Linton [26].
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et al. [31] and is presented in Section 2.2. A chaotic dynamic system is characterized by a positive Lyapunov exponent. However, because some stochastic processes would generate a positive Lyapunov exponent and, in theory, the white noise has an infinite Lyapunov exponent, it is important to evaluate whether the estimated Lyapunov exponent is positive but not excessively so. It should be noted that the test utilized in this paper is not a test for deterministic chaos. Instead, it is a test for the presence of a positive Lyapunov exponent, which is a necessary condition for chaotic dynamics. The third and final step in resolving the question whether a positive Lyapunov exponent is present in exchange rates is taken by applying statistical inference to the estimated largest Lyapunov exponent. A statistical framework that utilizes a blockwise bootstrap procedure to test for the presence of a positive Lyapunov exponent in an observed stochastic time series is presented in Section 2.3 [1]. 2.1. Reconstruction of the dynamics Specifically, let f : Rn ! Rn be a dynamic system that, for example, describes the evolution of an economy over time Stþ1 ¼ f ðSt Þ;
ð1Þ
where St is the state of the economy at time t. This economy governs the evolution of the exchange rates and other economic variables such as interest rates, money supplies and price levels. Associate the dynamic system in Eq. (1) with an observer function h : Rn ! R which generates the data points in, for example, an exchange rate series xt ¼ hðSt Þ þ cet ; et IIDð0; 1Þ;
ð2Þ
where c is the noise level and et is the measurement error. Thus, an N-point exchange rate series fx1 ; . . . ; xN g may be observed. Note that observational noise is present in the time series. According to Takens [30], it is possible to reconstruct the dynamics for the system in Eq. (1) using only the scalar time series generated by Eq. (2). Specifically, the data points in an observed scalar time series contain information about unobserved state variables that can be used to define a state at the present time. Therefore, let T ¼ ðT1 ; . . . ; TM Þ0
ð3Þ
be the reconstructed trajectory where Tt is the reconstructed state at time t and M is the number of states on the reconstructed trajectory. Each Tt is given by Tt ¼ fxt ; xtþ1 ; . . . ; xtþm 1 g;
ð4Þ
where t ¼ 1; . . . ; N m þ 1 and m is the embedding dimension. Thus, T is an M m matrix and the constants M, m and N are related as M ¼ N m þ 1. Takens [30] proved that a map exists between the original n-dimensional state St and the m-dimensional reconstructed state Tt (Eq. (4)). This is an embedding if m > 2n, i.e., it is a smooth map that performs a one-to-one coordinate transformation and has a smooth inverse. This means that the map preserves topological information about the unknown dynamic system under the mapping, e.g., the Lyapunov exponents. In particular, the map induces a function g : Rm ! Rm on the reconstructed trajectory Ttþ1 ¼ gðTt Þ;
ð5Þ
which is topologically conjugate to the unknown dynamic system f in Eq. (1). Thus, g in Eq. (5) is a reconstructed dynamic system, e.g., a ‘‘reconstructed’’ economy, which has the same Lyapunov exponents as the unknown dynamic system. 2.2. Estimation of the largest Lyapunov exponent A simple method for estimating the largest Lyapunov exponent k1 from an observed scalar time series was proposed by Rosenstein et al. [31]. As was thoroughly demonstrated in [31], this method works well both in small samples and when the signal-to-noise ratio is low.
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Given a reconstruction of the dynamics, T in Eq. (3), a search is then made for the nearest neighbor of each state on the trajectory that minimizes the distance to the particular reference state dt ð0Þ ¼ min kTt Tt0 kp ; Tt0
where dt ð0Þ, Tt and Tt0 are the distances between the tth state and its nearest neighbor, the reference state and the nearest neighbor, respectively. 2 To consider each pair of neighbors as nearby initial conditions for ‘‘different’’ trajectories, the temporal separation between them should be greater than the mean period of the time series jt t0 j > mean period; which, for example, can be defined as the reciprocal of the mean frequency of the power spectrum. The tth pair of nearest neighbors then diverges at a rate approximated by the largest Lyapunov exponent: dt ðiÞ dt ð0Þ expðk1 iÞ;
ð6Þ
where i is the number of separation steps. Taking the logarithm of both sides of Eq. (6) gives log dt ðiÞ log dt ð0Þ þ k1 i;
ð7Þ
which represents a set of approximately parallel lines each with a slope approximately proportional to k1 . The largest Lyapunov exponent is then estimated using a least-squares fit with a constant to the average line defined by hlog dt ðiÞi, where hi denotes the average over all values of t. Recall, however, that the solution paths to the unknown dynamic system in Eq. (1) remain within a bounded set. This means that the solution paths to the reconstructed dynamic system in Eq. (5) also remain within a bounded set. As a consequence, the approximations given in Eq. (6) and Eq. (7) are more reliable for a limited number of separation steps. This is because nearest neighbors cannot be separated by more than the ‘‘diameter’’ of the reconstructed dynamics. The proper number of separation steps can, therefore, be determined from the plot of the logarithms of the nearest neighbor separations against the number of separation steps. To be more specific, when the plot reaches a plateau, the proper number of separation steps is determined. 2.3. Statistical framework As mentioned above, the statistical framework used in this paper is based on a moving blocks bootstrap procedure. First, the basic ideas behind blockwise bootstrap procedures are presented and then the specific test scheme used in this paper. 2.3.1. Moving blocks bootstrap Consider a sequence fX1 ; . . . ; XN g of weakly dependent stationary random variables. According to K€ unsch [28] and Liu and Singh [29], the distribution of certain estimators can be consistently constructed by using the blockwise bootstrap method. Let Bt denote a moving block of b consecutive observations, i.e., Bt ¼ fXt ; . . . ; Xtþb 1 g. If k satisfies bk N, then resample, with replacement, k blocks from the sequence fB1 ; . . . ; BN bþ1 g. Denote the resulting sampled blocks by n1 ; . . . ; nk and concatenate these blocks into one vector fn1 ; . . . ; nk g which constitutes the bootstrap sample. Let h be the unknown parameter of interest, e.g., the largest Lyapunov exponent, h^ its consideredpestimator and h~ the ffiffiffiffi ~ h^Þ, one can statistic computed from thepbootstrap sample. By obtaining a large number of bootstrap values N ð h ffiffiffiffi estimate the distribution of N ðh^ hÞ. More precisely, the bootstrap values form an empirical distribution which can be utilized in statistical hypothesis testing. With ‘‘certain estimators’’ one generally refer to smooth functions of means, and since both the reconstruction of the dynamics and the estimation of the largest Lyapunov exponent do not contain ‘‘non-smooth’’ operations, the use of a blockwise bootstrap procedure can be justified. The idea behind the test scheme below is simply to let one state on the reconstructed trajectory to be one moving block. 2.3.2. Test scheme The null hypothesis H0 and the alternative hypothesis H1 are formulated as H0 : k1 ¼ 0 vs: H1 : k1 > 0; 2
For a fast search, L1 -norm may be used, i.e., kTt Tt0 k1 ¼ max0 6 i 6 m 1 jxtþi xt0 þi j.
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Table 1 The 1st part of SEK–DEM b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.195 )0.054 )0.041 )0.029
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.115 )0.030 )0.016 )0.002
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.109 0.035 0.041 0.055
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.228 0.085 0.092 0.107
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.084 )0.067 )0.046 )0.036
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.073 )0.055 )0.032 )0.010
Note: The number of data points is 380.
Table 2 The 1st part of SEK–DEM (1st difference)
Note: The number of data points is 379.
where k1 is the unknown parameter, i.e., the largest Lyapunov exponent. The test scheme consists of the following steps: (i) Reconstruct the dynamics from the observed scalar N-point time series where the embedding dimension is equal to the block size, i.e., m ¼ b, and estimate the largest Lyapunov exponent k^1 . Denote the reconstructed states on the reconstructed trajectory by T1 ; . . . ; TM . (ii) Resample, with replacement, k blocks from the sequence fT1 ; . . . ; TM g where k ¼ N mod b. Denote the resulting sampled blocks by n1 ; . . . ; nk . The sequence fn1 ; . . . ; nk g constitutes the bootstrap sample. (iii) Estimate the bootstrap value of the largest Lyapunov exponent k~1 from the bootstrap sample and calculate k~1 k^1 . (iv) Repeat steps (ii) and (iii) a large number of times to construct an empirical distribution for k~1 k^1 . (v) Construct a one-sided confidence interval, e.g., a 99% confidence interval, by calculating the critical value as k^1 qð99%Þ, following from Eq fPrfðk^1 k1 Þ 6 qð99%Þgg ¼ 0:99, where qð99%Þ is the quantile for the distribution in step (iv). (vi) If k^1 qð99%Þ > 0, then the null hypothesis is rejected.
3. Results The exchange rate series we examine in this paper are the Swedish Krona against the Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates. We use daily data from 17 May 1991 to 31 August 1995. The Swedish Krona
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Table 3 The 2nd part of SEK–DEM (1st difference) b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.230 0.182 0.184 0.187
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.089 0.055 0.058 0.060
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.045 0.020 0.021 0.024
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.142 )0.098 )0.089 )0.074
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.075 )0.079 )0.059 )0.046
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.046 )0.078 )0.059 )0.046
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.244 0.055 0.080 0.100
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.076 )0.070 )0.035 )0.024
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.026 )0.124 )0.098 )0.086
Note: The number of data points is 696.
Table 4 The 1st part of SEK–ECU
Note: The number of data points is 380. Table 5 The 1st part of SEK–ECU (1st difference)
Note: The number of data points is 379.
was pegged against the ECU between 17 May 1991 and 19 November 1992. The exchange rate series are, therefore, divided into two parts where the first includes the exchange rates from 17 May 1991 to 19 November 1992, and the second includes the exchange rates from 20 November 1992 to 31 August 1995. This was done to separate the dynamics from the fixed and flexible exchange rate periods, respectively. Note, however, that the Swedish Krona was flexible against the US Dollar and the Yen during both periods.
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Table 6 The 2nd part of SEK–ECU (1st difference) b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.246 0.210 0.212 0.215
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.086 0.047 0.049 0.051
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.041 0.006 0.008 0.011
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.209 0.146 0.153 0.155
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.077 0.027 0.035 0.038
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.043 0.009 0.015 0.018
Note: The number of data points is 696.
Table 7 The 1st part of SEK–USD (1st difference)
Note: The number of data points is 379.
Table 8 The 2nd part of SEK–USD (1st difference) b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.229 0.180 0.182 0.185
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.084 0.048 0.050 0.051
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.035 )0.001 0.001 0.004
Note: The number of data points is 696.
The blockwise bootstrap procedure used in this paper is based on the assumption that the observations form a stationary sequence. This means that it is necessary to ensure that the exchange rate series under consideration are stationary. If they are not, we take the first difference of the specific exchange rate series in order to make it stationary. This was done for all the exchange rate series, except for the Swedish Krona against the Deutsche Mark and the ECU
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Table 9 The 1st part of SEK–YEN (1st difference) b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.227 0.188 0.193 0.196
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.077 0.031 0.037 0.039
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.037 )0.001 0.004 0.006
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.240 0.203 0.205 0.208
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.090 0.058 0.060 0.062
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.045 0.015 0.016 0.019
Note: The number of data points is 379.
Table 10 The 2nd part of SEK–YEN (1st difference)
Note: The number of data points is 696.
exchange rates during the fixed exchange rate period. However, we also present results for the first difference of these exchange rate series for reasons of comparability. Because it was not easy to determine the mean period for the exchange rate series, the temporal separation between nearest neighbors was chosen to be at least 10. The plots of the logarithms of the nearest neighbor separations against the number of separation steps indicated that the proper number of separation steps was 5, i.e., the plot reaches a plateau at 5. The block sizes in each test are 5, 10 and 15, respectively. 400 bootstrap values were used in each test, which is a satisfactory number of values. Tables 1–10 present the test results for a significance level of a ¼ 0:01, a ¼ 0:025 and a ¼ 0:05 for the exchange rate series under consideration. In most cases, the null hypothesis that the Lyapunov exponent is zero is rejected in favor of a positive exponent. The exceptions are the Swedish Krona against the Deutsche Mark and the ECU exchange rates during the fixed period.
4. Discussion In this paper, we asked whether a positive Lyapunov exponent is present in nominal exchange rates. This is an important question since a necessary condition for chaotic dynamics is a positive Lyapunov exponent. The exchange rate series examined were daily data for the Swedish Krona against the Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates. To answer the posed question, the statistical framework proposed by Bask and Gencßay [1] was used. This utilizes a blockwise bootstrap procedure to test for the presence of a positive Lyapunov exponent in an observed stochastic time series. The null hypothesis that the Lyapunov exponent is zero was rejected in most cases in favor of a positive ex-
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ponent. This result is in accordance with the exchange rate series being characterized by deterministic chaos. However, it is also consistent with non-chaotic dynamics with a stochastic component. One should note the different characterization of the Swedish Krona against the Deutsche Mark and the ECU exchange rates during the fixed and flexible exchange rate periods. The null hypothesis could not be rejected during the fixed period, utilizing both the non-differenced and the differenced time series, whereas it could during the flexible period. Further, it should be recalled that the Swedish Krona was flexible against the US Dollar and the Yen during both periods. This means that the null hypothesis that the Lyapunov exponent is zero could only be rejected in those cases where the Swedish Krona was flexible against the aforementioned currencies. It should be noted that taking the first difference of a noisy time series is not problem-free since the operation is an amplifier of high-frequency noise. 3 This raises the question: how will the first difference operation affects the Lyapunov exponent estimate? The author is not aware of any general results on this topic. However, by comparing the results for both the non-differenced and the differenced exchange rate series for the Swedish Krona against the Deutsche Mark and the ECU during the fixed exchange rate period, it seems that no unequivocal answer to the question exists. The results in this paper are not in accordance with those of Jonsson [24]. Jonsson [24] examined daily data for the Swedish Krona against the US Dollar from 20 November 1992 to 30 December 1994, while we examined data for the same exchange rate from 20 November 1992 to 31 August 1995. The estimates of the largest Lyapunov exponent are, however, positive in this paper whereas they are negative in Jonsson [24]. The difference in length of the time series can certainly explain the conflicting results. The specific exchange rate series in this paper is 8 month longer than the single time series in Jonsson [24]. Another explanation for the inconsistency between the papers is that the methods used to estimate the largest Lyapunov exponent are not the same. Jonsson [24] used a method proposed by Nychka et al. [32]. However, one has to remember that the rate of convergence for Nychka’s et al. [32] method is not known and that the number of data points used in Jonsson [24] is small. Further, it should be pointed out that the samples used in this paper are also small which could negatively affect the process of reconstructing the dynamics of the underlying data generating process. Finally, the statistical framework proposed by Bask and Gencßay [1] does not depend on the method used to estimate the largest Lyapunov exponent. However, the utilization of more than one Lyapunov exponent estimation method should be part of further research.
Acknowledgements The author would like to thank Ramazan Gencßay, Cars Hommes, Karl-Gustaf L€ ofgren and an anonymous referee for helpful comments and suggestions. Research grants from the Swedish Council for Research in the Humanities and Social Sciences are also gratefully acknowledged. References [1] Bask M, Gencßay R. Testing chaotic dynamics via Lyapunov exponents. Physica D 1998;114:1–2. [2] Frankel JA, Rose AK. Empirical research on nominal exchange rates. In: Grossman G, Rogoff K, editors. Handbook of international economics, vol. 3. Amsterdam: Elsevier; 1995. p. 1689–729. [3] Mussa M. Empirical regularities in the behavior of exchange rates and theories of the foreign exchange market. In: Brunner K, Meltzer AH, editors. Policies for employment, prices, and exchange rates. Amsterdam: North-Holland; 1979. p. 9–57. [4] Brock WA. Distinguishing random and deterministic systems: abridged version. J Econ Theory 1986;40:168–95. [5] LeBaron B. Chaos and nonlinear forecastability in economics and finance. Philos Trans Roy Soc 1994;348:397–404. [6] Brock WA, Sayers CL. Is the business cycle characterized by deterministic chaos? J Monetary Econ 1988;22:71–90. [7] Eckmann J-P, Oliffson Kamphorst S, Ruelle D, Scheinkman JA. Lyapunov exponents for stock returns. In: Anderson PW, Arrow KJ, Pines D, editors. The economy as an evolving complex system. Reading, MA: Addison-Wesley; 1988. [8] Frank MZ, Gencßay R, Stengos T. International chaos? Eur Econ Rev 1988;32:1569–84. [9] Frank MZ, Stengos T. Some evidence concerning macroeconomic chaos. J Monetary Econ 1988;22:423–38. [10] Frank MZ, Stengos T. The stability of Canadian macroeconomic data as measured by the largest Lyapunov exponent. Econ Lett 1988;27:11–4. [11] Hsieh DA. Chaos and nonlinear dynamics: application to financial markets. J Finance 1991;46:1839–77. [12] Scheinkman JA, LeBaron B. Nonlinear dynamics and stock returns. J Business 1989;62:311–37.
3
The author is thankful to the anonymous referee for pointing out this.
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[13] Abhyankar A, Copeland LS, Wong W. Nonlinear dynamics in real-time equity market indices: evidence from the United Kingdom. Econ J 1995;105:864–80. [14] Abhyankar A, Copeland LS, Wong W. Uncovering nonlinear structure in real-time stock market indexes: the S&P 500, the DAX, the Nikkei 225, and the FTSE-100. J Business Econ Statist 1997;15:1–14. [15] Mizrach B. Determining delay times for phase space reconstruction with application to the FF/DM exchange rate. J Econ Behav Organ 1996;30:369–81. [16] Barnett WA, Serletis A. Martingales, nonlinearity, and chaos. J Econ Dyn Control 2000;24:703–24. [17] Jaditz T, Sayers CL. Is chaos generic in economic data? Int J Bifurcation Chaos 1993;3:745–55. [18] Grassberger P, Procaccia I. Characterization of strange attractors. Phys Rev Lett 1983;50:346–9. [19] Brock WA, Dechert WD, Scheinkman JA. A test for independence based on the correlation dimension. Working Paper, 1987. [20] Brock WA, Dechert WD, Scheinkman JA, LeBaron B. A test for independence based on the correlation dimension. Econ Rev 1996;15:197–235. [21] Bajo-Rubio O, Fernandez-Rodriguez F, Sosvilla-Rivero S. Chaotic behaviour in exchange-rate series: first results for the PesetaUS dollar case. Econ Lett 1992;39:207–11. [22] Bask M. Dimensions and Lyapunov exponents from exchange rate series. Chaos, Solitons & Fractals 1996;7:2199–214. [23] Dechert WD, Gencßay R. Lyapunov exponents as a nonparametric diagnostic for stability analysis. J Appl Econ 1992;7:S41–60. [24] Jonsson M. Studies in business cycles. Institute for International Economic Studies, Stockholm University, 1997. [25] Bask M, de Luna X. Characterizing the degree of stability of non-linear dynamic models. Ume a Economic Studies No. 564. Studies in Nonlinear Dynamics and Econometrics 2002. [26] Whang Y-J, Linton O. The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series. J Econ 1999;91:1–42. [27] Efron B. Bootstrap methods: another look at the jackknife. Ann Statist 1979;7:1–26. [28] K€ unsch HR. The jackknife and the bootstrap for general stationary observations. Ann Statist 1989;17:1217–41. [29] Liu RY, Singh K. Moving blocks jackknife and bootstrap capture weak dependence. In: LePage R, Billard L, editors. Exploring the limits of bootstrap. New York: Wiley; 1992. p. 225–48. [30] Takens F. Detecting strange attractors in turbulence. In: Rand D, Young L, editors. Dynamical systems and turbulence. Berlin: Springer; 1981. p. 366–81. [31] Rosenstein MT, Collins JJ, De Luca CJ. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 1993;65:117–34. [32] Nychka D, Ellner S, Gallant AR, McCaffrey D. Finding chaos in noisy systems. J Roy Statist Soc B 1992;54:399–426.
A positive Lyapunov exponent in Swedish exchange rates? Mikael Bask
*
Department of Economics, Ume a University, SE-901 87 Ume a, Sweden Accepted 26 March 2002
Abstract In this paper, a statistical framework utilizing a blockwise bootstrap procedure is used to test for the presence of a positive Lyapunov exponent in Swedish exchange rates [M. Bask, R. Gencßay, Physica D 114 (1998) 1]. This is done since a necessary condition for chaotic dynamics is a positive Lyapunov exponent. Daily data for the Swedish Krona against the Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates are examined. In most cases, the null hypothesis that the Lyapunov exponent is zero is rejected in favor of a positive exponent. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Empirical research on nominal exchange rates has shown that a number of important regularities appear to be stable over time. For instance: (1) the behavior of the logarithms of exchange rates over time approximates a random walk; (2) most changes in exchange rates are unexpected; (3) countries with high inflation rates tend to have currencies which depreciate; and (4) actual exchange rate movements appear to overshoot movements in the equilibrium exchange rate [2,3]. These findings are important because one of the purposes of economic theory is to develop models that can explain observed regularities. In this paper, we ask whether a positive Lyapunov exponent is present in nominal exchange rates. This is an important question since a necessary condition for chaotic dynamics is a positive Lyapunov exponent. The characterizing feature of a chaotic dynamic system is that it has a property of ‘‘sensitive dependence on initial conditions’’: any two solution paths with arbitrarily close, but not equal, initial conditions will diverge at exponential rates. Globally, however, the solution paths remain within a bounded set if the dynamic system is dissipative. This means that the predictive ability of the system is strongly limited, especially in long-run predictions, even if it is still possible to make short-run predictions. The local stability properties of a dynamic system can be measured by the Lyapunov exponents. These measure the average exponential divergence or convergence of nearby, but not equal, initial conditions. Accordingly, a positive Lyapunov exponent measures the average exponential divergence of nearby initial conditions, whereas a negative exponent measures the average exponential convergence. Thus, if we refer to the property of ‘‘sensitive dependence on initial conditions’’ and have a deterministic system, a positive Lyapunov exponent can be considered as evidence of chaotic dynamics. Therefore, the estimation of the Lyapunov exponents or, to be more specific, the largest Lyapunov exponent is essential. Another characteristic of a dynamic system is the dimension of the attractor that is associated with the dynamic system. The dimension of an attractor is the amount of information needed to specify its points accurately, and is related to how hyper-volumes scale as a function of a length parameter. For example, areas vary with the square of the length of the side and volumes vary with the cube. For strange attractors, i.e., attractors associated with a chaotic
*
Tel.: +46-90-786 78 77; fax: +46-90-77 23 02. E-mail address: [email protected] (M. Bask).
0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 8 3 - 8
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dynamic system, the dimension typically takes a non-integer value. Therefore, the dimension of a strange attractor is often called the fractal dimension. The dimension of a dynamic system is important because it reveals the number of variables that are sufficient to mimic the systems behavior. Following the early work of Brock [4], there has been an increased interest in detecting chaotic dynamics in economic and financial time series [5]. A few examples are [6–15]. Barnett and Serletis [16] and Jaditz and Sayers [17] review the literature. Most of the empirical papers reached the conclusion that the time series they examined were not generated by chaotic dynamic systems. These authors did not, however, always use the Lyapunov exponents in their empirical research. They estimated, for example, the correlation dimension, i.e., a measure of the fractal dimension, for the time series in order to detect lowdimensional dynamics [18]. This was done because a general difference between deterministic chaos and a stochastic process is that the former is low-dimensional, or at least finite-dimensional, whereas the latter is infinite-dimensional. However, even though the dynamics is low-dimensional, it can still be non-chaotic. Therefore, the use of the Lyapunov exponents is more appropriate than the use of the correlation dimension. However, the second-best method to detect chaotic dynamics would be the estimation of the dimension of the attractor since it typically takes a non-integer value when the attractor is strange. But we will not estimate the dimension in this paper since the author is not aware of any distributional theory to provide a framework for statistical inference. For example, a statistical test with the null hypothesis of an integer valued dimension. The so-called BDS-test statistic was, and is, widely used in empirical research for detecting non-linear dynamics in economic and financial time series [19,20]. Specifically, it tests the null hypothesis that a time series is independent and identically distributed, i.e., that the data generating process is IID. As such, it is not a test for deterministic chaos. It is, however, useful in detecting any remaining ‘‘structure’’ in the estimated residuals of time series models that can be transformed into models driven by independent and identically distributed error terms. Earlier research on detecting chaotic dynamics in exchange rates via Lyapunov exponents has been carried out by Bajo-Rubio et al. [21], Bask [22], Dechert and Gencßay [23] and Jonsson [24]. However, these and other papers have been criticized because they lack a distributional theory to provide a framework for statistical inference. 1 Here, we apply a statistical framework proposed by Bask and Gencßay [1]. This utilizes a moving blocks bootstrap procedure that tests for the presence of a positive Lyapunov exponent in an observed stochastic time series. Bootstrapping uses the available observations to design a sort of Monte Carlo experiment in which the observations themselves are used to approximate the distribution of the error terms or other random quantities [27]. K€ unsch [28] and Liu and Singh [29] extended the idea of bootstrapping to the case where the observations form a stationary sequence, i.e., when the observations constitute a time series. The exchange rate series examined in this paper are the Swedish Krona against the Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates. We use daily data from 17 May 1991 to 31 August 1995. The Swedish Krona was pegged against the ECU between 17 May 1991 and 19 November 1992. The exchange rate series are thus divided into two parts so that the dynamics from the fixed and flexible exchange rate periods can be separated. The remainder of this paper is organized as follows. Section 2 presents the procedure for estimating the largest Lyapunov exponent from an observed scalar time series and for drawing statistical inferences from these estimates. Section 3 contains the results and Section 4 concludes the paper.
2. Theory Several steps are necessary to answer the question whether a positive Lyapunov exponent is present in (nominal) exchange rates. Firstly, the dynamics that govern the evolution of the exchange rates must be reconstructed because the specific equations of motion are not known. In the case where there is a dynamic system hidden in a ‘‘black box’’, Takens [30] has demonstrated that it is possible to reconstruct the unknown dynamics using only an observed scalar time series. Remarkably, the reconstructed dynamics and the dynamic system hidden in the ‘‘black box’’ are ‘‘equivalent’’ in the sense that they share certain dynamic properties. For example, the Lyapunov exponents for the reconstructed dynamics and for the unknown dynamic system are equivalent. The reconstruction of the dynamics using the exchange rate series as the observed scalar time series is presented in Section 2.1. Given a reconstruction of the dynamics, the second step is taken by estimating the Lyapunov exponents. A simple method for estimating the largest Lyapunov exponent from an observed scalar time series was proposed by Rosenstein
1 The consistency and asymptotic normality of an estimator of smooth Lyapunov exponents have been proved under certain regularity conditions by Bask and de Luna [25] and Whang and Linton [26].
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et al. [31] and is presented in Section 2.2. A chaotic dynamic system is characterized by a positive Lyapunov exponent. However, because some stochastic processes would generate a positive Lyapunov exponent and, in theory, the white noise has an infinite Lyapunov exponent, it is important to evaluate whether the estimated Lyapunov exponent is positive but not excessively so. It should be noted that the test utilized in this paper is not a test for deterministic chaos. Instead, it is a test for the presence of a positive Lyapunov exponent, which is a necessary condition for chaotic dynamics. The third and final step in resolving the question whether a positive Lyapunov exponent is present in exchange rates is taken by applying statistical inference to the estimated largest Lyapunov exponent. A statistical framework that utilizes a blockwise bootstrap procedure to test for the presence of a positive Lyapunov exponent in an observed stochastic time series is presented in Section 2.3 [1]. 2.1. Reconstruction of the dynamics Specifically, let f : Rn ! Rn be a dynamic system that, for example, describes the evolution of an economy over time Stþ1 ¼ f ðSt Þ;
ð1Þ
where St is the state of the economy at time t. This economy governs the evolution of the exchange rates and other economic variables such as interest rates, money supplies and price levels. Associate the dynamic system in Eq. (1) with an observer function h : Rn ! R which generates the data points in, for example, an exchange rate series xt ¼ hðSt Þ þ cet ; et IIDð0; 1Þ;
ð2Þ
where c is the noise level and et is the measurement error. Thus, an N-point exchange rate series fx1 ; . . . ; xN g may be observed. Note that observational noise is present in the time series. According to Takens [30], it is possible to reconstruct the dynamics for the system in Eq. (1) using only the scalar time series generated by Eq. (2). Specifically, the data points in an observed scalar time series contain information about unobserved state variables that can be used to define a state at the present time. Therefore, let T ¼ ðT1 ; . . . ; TM Þ0
ð3Þ
be the reconstructed trajectory where Tt is the reconstructed state at time t and M is the number of states on the reconstructed trajectory. Each Tt is given by Tt ¼ fxt ; xtþ1 ; . . . ; xtþm 1 g;
ð4Þ
where t ¼ 1; . . . ; N m þ 1 and m is the embedding dimension. Thus, T is an M m matrix and the constants M, m and N are related as M ¼ N m þ 1. Takens [30] proved that a map exists between the original n-dimensional state St and the m-dimensional reconstructed state Tt (Eq. (4)). This is an embedding if m > 2n, i.e., it is a smooth map that performs a one-to-one coordinate transformation and has a smooth inverse. This means that the map preserves topological information about the unknown dynamic system under the mapping, e.g., the Lyapunov exponents. In particular, the map induces a function g : Rm ! Rm on the reconstructed trajectory Ttþ1 ¼ gðTt Þ;
ð5Þ
which is topologically conjugate to the unknown dynamic system f in Eq. (1). Thus, g in Eq. (5) is a reconstructed dynamic system, e.g., a ‘‘reconstructed’’ economy, which has the same Lyapunov exponents as the unknown dynamic system. 2.2. Estimation of the largest Lyapunov exponent A simple method for estimating the largest Lyapunov exponent k1 from an observed scalar time series was proposed by Rosenstein et al. [31]. As was thoroughly demonstrated in [31], this method works well both in small samples and when the signal-to-noise ratio is low.
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Given a reconstruction of the dynamics, T in Eq. (3), a search is then made for the nearest neighbor of each state on the trajectory that minimizes the distance to the particular reference state dt ð0Þ ¼ min kTt Tt0 kp ; Tt0
where dt ð0Þ, Tt and Tt0 are the distances between the tth state and its nearest neighbor, the reference state and the nearest neighbor, respectively. 2 To consider each pair of neighbors as nearby initial conditions for ‘‘different’’ trajectories, the temporal separation between them should be greater than the mean period of the time series jt t0 j > mean period; which, for example, can be defined as the reciprocal of the mean frequency of the power spectrum. The tth pair of nearest neighbors then diverges at a rate approximated by the largest Lyapunov exponent: dt ðiÞ dt ð0Þ expðk1 iÞ;
ð6Þ
where i is the number of separation steps. Taking the logarithm of both sides of Eq. (6) gives log dt ðiÞ log dt ð0Þ þ k1 i;
ð7Þ
which represents a set of approximately parallel lines each with a slope approximately proportional to k1 . The largest Lyapunov exponent is then estimated using a least-squares fit with a constant to the average line defined by hlog dt ðiÞi, where hi denotes the average over all values of t. Recall, however, that the solution paths to the unknown dynamic system in Eq. (1) remain within a bounded set. This means that the solution paths to the reconstructed dynamic system in Eq. (5) also remain within a bounded set. As a consequence, the approximations given in Eq. (6) and Eq. (7) are more reliable for a limited number of separation steps. This is because nearest neighbors cannot be separated by more than the ‘‘diameter’’ of the reconstructed dynamics. The proper number of separation steps can, therefore, be determined from the plot of the logarithms of the nearest neighbor separations against the number of separation steps. To be more specific, when the plot reaches a plateau, the proper number of separation steps is determined. 2.3. Statistical framework As mentioned above, the statistical framework used in this paper is based on a moving blocks bootstrap procedure. First, the basic ideas behind blockwise bootstrap procedures are presented and then the specific test scheme used in this paper. 2.3.1. Moving blocks bootstrap Consider a sequence fX1 ; . . . ; XN g of weakly dependent stationary random variables. According to K€ unsch [28] and Liu and Singh [29], the distribution of certain estimators can be consistently constructed by using the blockwise bootstrap method. Let Bt denote a moving block of b consecutive observations, i.e., Bt ¼ fXt ; . . . ; Xtþb 1 g. If k satisfies bk N, then resample, with replacement, k blocks from the sequence fB1 ; . . . ; BN bþ1 g. Denote the resulting sampled blocks by n1 ; . . . ; nk and concatenate these blocks into one vector fn1 ; . . . ; nk g which constitutes the bootstrap sample. Let h be the unknown parameter of interest, e.g., the largest Lyapunov exponent, h^ its consideredpestimator and h~ the ffiffiffiffi ~ h^Þ, one can statistic computed from thepbootstrap sample. By obtaining a large number of bootstrap values N ð h ffiffiffiffi estimate the distribution of N ðh^ hÞ. More precisely, the bootstrap values form an empirical distribution which can be utilized in statistical hypothesis testing. With ‘‘certain estimators’’ one generally refer to smooth functions of means, and since both the reconstruction of the dynamics and the estimation of the largest Lyapunov exponent do not contain ‘‘non-smooth’’ operations, the use of a blockwise bootstrap procedure can be justified. The idea behind the test scheme below is simply to let one state on the reconstructed trajectory to be one moving block. 2.3.2. Test scheme The null hypothesis H0 and the alternative hypothesis H1 are formulated as H0 : k1 ¼ 0 vs: H1 : k1 > 0; 2
For a fast search, L1 -norm may be used, i.e., kTt Tt0 k1 ¼ max0 6 i 6 m 1 jxtþi xt0 þi j.
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Table 1 The 1st part of SEK–DEM b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.195 )0.054 )0.041 )0.029
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.115 )0.030 )0.016 )0.002
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.109 0.035 0.041 0.055
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.228 0.085 0.092 0.107
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.084 )0.067 )0.046 )0.036
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.073 )0.055 )0.032 )0.010
Note: The number of data points is 380.
Table 2 The 1st part of SEK–DEM (1st difference)
Note: The number of data points is 379.
where k1 is the unknown parameter, i.e., the largest Lyapunov exponent. The test scheme consists of the following steps: (i) Reconstruct the dynamics from the observed scalar N-point time series where the embedding dimension is equal to the block size, i.e., m ¼ b, and estimate the largest Lyapunov exponent k^1 . Denote the reconstructed states on the reconstructed trajectory by T1 ; . . . ; TM . (ii) Resample, with replacement, k blocks from the sequence fT1 ; . . . ; TM g where k ¼ N mod b. Denote the resulting sampled blocks by n1 ; . . . ; nk . The sequence fn1 ; . . . ; nk g constitutes the bootstrap sample. (iii) Estimate the bootstrap value of the largest Lyapunov exponent k~1 from the bootstrap sample and calculate k~1 k^1 . (iv) Repeat steps (ii) and (iii) a large number of times to construct an empirical distribution for k~1 k^1 . (v) Construct a one-sided confidence interval, e.g., a 99% confidence interval, by calculating the critical value as k^1 qð99%Þ, following from Eq fPrfðk^1 k1 Þ 6 qð99%Þgg ¼ 0:99, where qð99%Þ is the quantile for the distribution in step (iv). (vi) If k^1 qð99%Þ > 0, then the null hypothesis is rejected.
3. Results The exchange rate series we examine in this paper are the Swedish Krona against the Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates. We use daily data from 17 May 1991 to 31 August 1995. The Swedish Krona
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M. Bask / Chaos, Solitons and Fractals 14 (2002) 1295–1304
Table 3 The 2nd part of SEK–DEM (1st difference) b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.230 0.182 0.184 0.187
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.089 0.055 0.058 0.060
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.045 0.020 0.021 0.024
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.142 )0.098 )0.089 )0.074
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.075 )0.079 )0.059 )0.046
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.046 )0.078 )0.059 )0.046
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.244 0.055 0.080 0.100
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.076 )0.070 )0.035 )0.024
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.026 )0.124 )0.098 )0.086
Note: The number of data points is 696.
Table 4 The 1st part of SEK–ECU
Note: The number of data points is 380. Table 5 The 1st part of SEK–ECU (1st difference)
Note: The number of data points is 379.
was pegged against the ECU between 17 May 1991 and 19 November 1992. The exchange rate series are, therefore, divided into two parts where the first includes the exchange rates from 17 May 1991 to 19 November 1992, and the second includes the exchange rates from 20 November 1992 to 31 August 1995. This was done to separate the dynamics from the fixed and flexible exchange rate periods, respectively. Note, however, that the Swedish Krona was flexible against the US Dollar and the Yen during both periods.
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Table 6 The 2nd part of SEK–ECU (1st difference) b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.246 0.210 0.212 0.215
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.086 0.047 0.049 0.051
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.041 0.006 0.008 0.011
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.209 0.146 0.153 0.155
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.077 0.027 0.035 0.038
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.043 0.009 0.015 0.018
Note: The number of data points is 696.
Table 7 The 1st part of SEK–USD (1st difference)
Note: The number of data points is 379.
Table 8 The 2nd part of SEK–USD (1st difference) b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.229 0.180 0.182 0.185
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.084 0.048 0.050 0.051
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.035 )0.001 0.001 0.004
Note: The number of data points is 696.
The blockwise bootstrap procedure used in this paper is based on the assumption that the observations form a stationary sequence. This means that it is necessary to ensure that the exchange rate series under consideration are stationary. If they are not, we take the first difference of the specific exchange rate series in order to make it stationary. This was done for all the exchange rate series, except for the Swedish Krona against the Deutsche Mark and the ECU
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Table 9 The 1st part of SEK–YEN (1st difference) b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.227 0.188 0.193 0.196
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.077 0.031 0.037 0.039
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.037 )0.001 0.004 0.006
b¼5
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.240 0.203 0.205 0.208
b ¼ 10
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.090 0.058 0.060 0.062
b ¼ 15
k^1 Critical value ða ¼ 0:01Þ Critical value ða ¼ 0:025Þ Critical value ða ¼ 0:05Þ
0.045 0.015 0.016 0.019
Note: The number of data points is 379.
Table 10 The 2nd part of SEK–YEN (1st difference)
Note: The number of data points is 696.
exchange rates during the fixed exchange rate period. However, we also present results for the first difference of these exchange rate series for reasons of comparability. Because it was not easy to determine the mean period for the exchange rate series, the temporal separation between nearest neighbors was chosen to be at least 10. The plots of the logarithms of the nearest neighbor separations against the number of separation steps indicated that the proper number of separation steps was 5, i.e., the plot reaches a plateau at 5. The block sizes in each test are 5, 10 and 15, respectively. 400 bootstrap values were used in each test, which is a satisfactory number of values. Tables 1–10 present the test results for a significance level of a ¼ 0:01, a ¼ 0:025 and a ¼ 0:05 for the exchange rate series under consideration. In most cases, the null hypothesis that the Lyapunov exponent is zero is rejected in favor of a positive exponent. The exceptions are the Swedish Krona against the Deutsche Mark and the ECU exchange rates during the fixed period.
4. Discussion In this paper, we asked whether a positive Lyapunov exponent is present in nominal exchange rates. This is an important question since a necessary condition for chaotic dynamics is a positive Lyapunov exponent. The exchange rate series examined were daily data for the Swedish Krona against the Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates. To answer the posed question, the statistical framework proposed by Bask and Gencßay [1] was used. This utilizes a blockwise bootstrap procedure to test for the presence of a positive Lyapunov exponent in an observed stochastic time series. The null hypothesis that the Lyapunov exponent is zero was rejected in most cases in favor of a positive ex-
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ponent. This result is in accordance with the exchange rate series being characterized by deterministic chaos. However, it is also consistent with non-chaotic dynamics with a stochastic component. One should note the different characterization of the Swedish Krona against the Deutsche Mark and the ECU exchange rates during the fixed and flexible exchange rate periods. The null hypothesis could not be rejected during the fixed period, utilizing both the non-differenced and the differenced time series, whereas it could during the flexible period. Further, it should be recalled that the Swedish Krona was flexible against the US Dollar and the Yen during both periods. This means that the null hypothesis that the Lyapunov exponent is zero could only be rejected in those cases where the Swedish Krona was flexible against the aforementioned currencies. It should be noted that taking the first difference of a noisy time series is not problem-free since the operation is an amplifier of high-frequency noise. 3 This raises the question: how will the first difference operation affects the Lyapunov exponent estimate? The author is not aware of any general results on this topic. However, by comparing the results for both the non-differenced and the differenced exchange rate series for the Swedish Krona against the Deutsche Mark and the ECU during the fixed exchange rate period, it seems that no unequivocal answer to the question exists. The results in this paper are not in accordance with those of Jonsson [24]. Jonsson [24] examined daily data for the Swedish Krona against the US Dollar from 20 November 1992 to 30 December 1994, while we examined data for the same exchange rate from 20 November 1992 to 31 August 1995. The estimates of the largest Lyapunov exponent are, however, positive in this paper whereas they are negative in Jonsson [24]. The difference in length of the time series can certainly explain the conflicting results. The specific exchange rate series in this paper is 8 month longer than the single time series in Jonsson [24]. Another explanation for the inconsistency between the papers is that the methods used to estimate the largest Lyapunov exponent are not the same. Jonsson [24] used a method proposed by Nychka et al. [32]. However, one has to remember that the rate of convergence for Nychka’s et al. [32] method is not known and that the number of data points used in Jonsson [24] is small. Further, it should be pointed out that the samples used in this paper are also small which could negatively affect the process of reconstructing the dynamics of the underlying data generating process. Finally, the statistical framework proposed by Bask and Gencßay [1] does not depend on the method used to estimate the largest Lyapunov exponent. However, the utilization of more than one Lyapunov exponent estimation method should be part of further research.
Acknowledgements The author would like to thank Ramazan Gencßay, Cars Hommes, Karl-Gustaf L€ ofgren and an anonymous referee for helpful comments and suggestions. Research grants from the Swedish Council for Research in the Humanities and Social Sciences are also gratefully acknowledged. References [1] Bask M, Gencßay R. Testing chaotic dynamics via Lyapunov exponents. Physica D 1998;114:1–2. [2] Frankel JA, Rose AK. Empirical research on nominal exchange rates. In: Grossman G, Rogoff K, editors. Handbook of international economics, vol. 3. Amsterdam: Elsevier; 1995. p. 1689–729. [3] Mussa M. Empirical regularities in the behavior of exchange rates and theories of the foreign exchange market. In: Brunner K, Meltzer AH, editors. Policies for employment, prices, and exchange rates. Amsterdam: North-Holland; 1979. p. 9–57. [4] Brock WA. Distinguishing random and deterministic systems: abridged version. J Econ Theory 1986;40:168–95. [5] LeBaron B. Chaos and nonlinear forecastability in economics and finance. Philos Trans Roy Soc 1994;348:397–404. [6] Brock WA, Sayers CL. Is the business cycle characterized by deterministic chaos? J Monetary Econ 1988;22:71–90. [7] Eckmann J-P, Oliffson Kamphorst S, Ruelle D, Scheinkman JA. Lyapunov exponents for stock returns. In: Anderson PW, Arrow KJ, Pines D, editors. The economy as an evolving complex system. Reading, MA: Addison-Wesley; 1988. [8] Frank MZ, Gencßay R, Stengos T. International chaos? Eur Econ Rev 1988;32:1569–84. [9] Frank MZ, Stengos T. Some evidence concerning macroeconomic chaos. J Monetary Econ 1988;22:423–38. [10] Frank MZ, Stengos T. The stability of Canadian macroeconomic data as measured by the largest Lyapunov exponent. Econ Lett 1988;27:11–4. [11] Hsieh DA. Chaos and nonlinear dynamics: application to financial markets. J Finance 1991;46:1839–77. [12] Scheinkman JA, LeBaron B. Nonlinear dynamics and stock returns. J Business 1989;62:311–37.
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The author is thankful to the anonymous referee for pointing out this.
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[13] Abhyankar A, Copeland LS, Wong W. Nonlinear dynamics in real-time equity market indices: evidence from the United Kingdom. Econ J 1995;105:864–80. [14] Abhyankar A, Copeland LS, Wong W. Uncovering nonlinear structure in real-time stock market indexes: the S&P 500, the DAX, the Nikkei 225, and the FTSE-100. J Business Econ Statist 1997;15:1–14. [15] Mizrach B. Determining delay times for phase space reconstruction with application to the FF/DM exchange rate. J Econ Behav Organ 1996;30:369–81. [16] Barnett WA, Serletis A. Martingales, nonlinearity, and chaos. J Econ Dyn Control 2000;24:703–24. [17] Jaditz T, Sayers CL. Is chaos generic in economic data? Int J Bifurcation Chaos 1993;3:745–55. [18] Grassberger P, Procaccia I. Characterization of strange attractors. Phys Rev Lett 1983;50:346–9. [19] Brock WA, Dechert WD, Scheinkman JA. A test for independence based on the correlation dimension. Working Paper, 1987. [20] Brock WA, Dechert WD, Scheinkman JA, LeBaron B. A test for independence based on the correlation dimension. Econ Rev 1996;15:197–235. [21] Bajo-Rubio O, Fernandez-Rodriguez F, Sosvilla-Rivero S. Chaotic behaviour in exchange-rate series: first results for the PesetaUS dollar case. Econ Lett 1992;39:207–11. [22] Bask M. Dimensions and Lyapunov exponents from exchange rate series. Chaos, Solitons & Fractals 1996;7:2199–214. [23] Dechert WD, Gencßay R. Lyapunov exponents as a nonparametric diagnostic for stability analysis. J Appl Econ 1992;7:S41–60. [24] Jonsson M. Studies in business cycles. Institute for International Economic Studies, Stockholm University, 1997. [25] Bask M, de Luna X. Characterizing the degree of stability of non-linear dynamic models. Ume a Economic Studies No. 564. Studies in Nonlinear Dynamics and Econometrics 2002. [26] Whang Y-J, Linton O. The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series. J Econ 1999;91:1–42. [27] Efron B. Bootstrap methods: another look at the jackknife. Ann Statist 1979;7:1–26. [28] K€ unsch HR. The jackknife and the bootstrap for general stationary observations. Ann Statist 1989;17:1217–41. [29] Liu RY, Singh K. Moving blocks jackknife and bootstrap capture weak dependence. In: LePage R, Billard L, editors. Exploring the limits of bootstrap. New York: Wiley; 1992. p. 225–48. [30] Takens F. Detecting strange attractors in turbulence. In: Rand D, Young L, editors. Dynamical systems and turbulence. Berlin: Springer; 1981. p. 366–81. [31] Rosenstein MT, Collins JJ, De Luca CJ. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 1993;65:117–34. [32] Nychka D, Ellner S, Gallant AR, McCaffrey D. Finding chaos in noisy systems. J Roy Statist Soc B 1992;54:399–426.