=Paper=
{{Paper
|id=Vol-2177/paper-12-c002
|storemode=property
|title=
Application of the Phase Analysis of Time Series for the
Identification of Macroeconomic Cycles Based on the Dynamics of the
Exchange Rates
|pdfUrl=https://ceur-ws.org/Vol-2177/paper-12-c002.pdf
|volume=Vol-2177
|authors=Anastasiya V. Demidova,Elena A. Teveleva,Nikolay P. Tretyakov,Alexandre Ya. Terletsky
}}
==
Application of the Phase Analysis of Time Series for the
Identification of Macroeconomic Cycles Based on the Dynamics of the
Exchange Rates
==
https://ceur-ws.org/Vol-2177/paper-12-c002.pdf
83
UDC 519.246.8
Application of the Phase Analysis of Time Series for the
Identification of Macroeconomic Cycles Based on the Dynamics
of the Exchange Rates
Anastasiya V. Demidova* , Elena A. Teveleva† ,
Nikolay P. Tretyakov†‡ , Alexandre Ya. TerletskyS
*
Department of Applied Probability and Informatics
Peoples’ Friendship University of Russia (RUDN University)
6 Miklukho-Maklaya st., Moscow, 117198, Russian Federation
†
Department of Applied Information Technologies
Russian State Social University
4-1 Wilhelm Pieck str., Moscow, 119571, Russian Federation
‡
Department of Informatics and Applied Mathematics
Russian Presidential Academy of National Economy and Public Administration
82-2 Prospect Vernadskogo, Moscow, 129226, Russian Federation
S
Institute of Physical Research and Technology
Peoples’ Friendship University of Russia (RUDN University)
6 Miklukho-Maklaya str., Moscow, 117198, Russian Federation
Email: demidova_av@rudn.university, eteveleva@yandex.ru, trn11@rambler.ru
The article deals with one of the methods for identifying macroeconomic cycles of economic
indicators using the example of the dollar and the euro exchange rates for the period 2000-2016.
Time series of economic indicators, especially over a long period of time, contain irregular
cyclical fluctuations with unstable amplitudes and periods. The use of traditional methods
to study such oscillations, generally speaking, is not suitable. The method of phase analysis
of time series allows one to identify hidden long-term macrocycles and, in some cases, make
predictions. Since the end of 2008, both currencies started a new cycle, and their phase
diagrams make a simultaneous jump up. Then they go into the negative area. From 2010 to
early 2015, both phases coincide and are below the trend line of both currencies. But from
the beginning of 2015 they make a sharp jump upward, when both currencies have risen above
the expected trend value by almost 20 points, which indicates a new strongest wave of the
economic crisis.
Key words and phrases: exchange rates, time series analysis, macroeconomic cycles.
Copyright © 2018 for the individual papers by the papers’ authors. Copying permitted for private and
academic purposes. This volume is published and copyrighted by its editors.
In: K. E. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the VIII Conference
“Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”,
Moscow, Russia, 20-Apr-2018, published at http://ceur-ws.org
84 ITTMM—2018
1. Introduction
As a rule, time series of economic indicators, especially over a long period of time,
contain irregular cyclical fluctuations with unstable amplitudes and periods. Outwardly,
the stochastic nature of these phenomena reflects the cyclical development of the economy
under the influence of many random and nonrandom, market and volitional influences,
that is, many hidden factors that cannot always be taken into account. Thus, such
fluctuations can be a reflection of macrocycles of the development of economic processes.
The use of traditional methods to study such oscillations, generally speaking, is not
suitable. For example, the spectral analysis models the motion of a time series by the
sum of regular sinusoids. However, it is unlikely that economic indicators have a strict
periodicity and constancy of amplitudes due to interference of a huge number of external
economic and political influences. Regression analysis approximates the entire series
as a whole, not taking into account the local properties of the series. Meanwhile, in
the economy, each cycle has its own characteristics, since it is generated by a variety of
causes of a very different nature, which can only be inherent in certain time intervals
and, as a rule, do not repeat [1]. Therefore, other methods are needed to investigate
irregular cyclic oscillations. One such method is phase statistics approach to time series
analysis. As mentioned in [2], many studies have indicated that phase patterns can code
more information than the amplitude [17–19].
There are different approaches to the phase analysis of time series [1, 2]. Some
phase statistics approaches were introduced to study physiological [3] and financial
time series [4]. The approach mainly consists of application of the Hilbert-Huang
method [5] to decompose an empirical time series into a number of intrinsic mode
functions (IMFs). Cross-disciplinary studies on financial systems have attracted much
attention in recent decades [1, 6–10]. Note, for example, the wavelet transform modulus
maxima approach [11–16].
2. Main section
To present the proposed approach, some definitions are needed. The fluctuation is
the amount of deviation of the values of a series from a certain fixed level. As a rule, this
is a trend, or a deviation from the average value in case the trend is not significant. The
fluctuation power is the absolute value of the fluctuation |𝐷𝑦𝑡 |. The phase is the period
of positive or negative fluctuations of the series. When, for example, the dollar or euro
exchange rate is above the trend line, this is a positive fluctuation. Otherwise, there is a
negative fluctuation. The duration of the phase is the time interval of the corresponding
phase. Thus, irregular cyclic oscillations mean the presence of a number of differently
directed fluctuations. The power of the i-th phase is the sum of the absolute fluctuations
of the series inside the phase:
𝑡𝑖+𝑙
∑︁𝑖
𝑃𝑖 = |𝐷𝑦𝑡 |,
𝑡=𝑡𝑖
where 𝑡𝑖 is the moment of the beginning of the i-th phase, 𝑡𝑖+𝑙𝑖 is the phase-ending
moment, 𝑙𝑖 is the phase duration. The average value of the phase is the power averaged
over the interval 𝑡𝑖 ; 𝑡𝑖+𝑙𝑖 .
The economist, as a rule, deals with a time series containing random fluctuations. In
the initial series, each such fluctuation or several neighboring ones can form low-power
phases that have no essential content. Therefore, it is desirable to clear the series of
random fluctuations and their corresponding phases in order to obtain some significant
motions of the series movements that can be interpreted in some way. Typically, this
is an iterative process, where at each step there is an aggregation of low-power phases
with two neighboring phases with a more significant power. Therefore, it is necessary
to specify a criterion for stopping the absorption of low-power phases. What kind of
criterium is this? This can be the level of power lost in the series, i.e. a predetermined
Demidova Anastasiya V. et al. 85
percentage of the aggregate power of the series that is allowed to lose during the phase
aggregation process. In this case, the sum of absolute values is calculated, which is taken
as 100 percent.
As a second option, such a criterion can be a predetermined number of phases. The
choice depends on the nature of the problem being solved [1].
After the iterations are completed, a phase diagram is constructed, where each
moment of time corresponds to the mean value of the phase with the corresponding sign.
Let’s consider an example. Figure 1 shows the dynamics of the dollar for the period
2000-2016 citekurs.As fluctuations, we use deviations of the initial values from the trend.
The trend of the ruble exchange rate (in Fig. 1 it is depicted by a dotted line) for
the period 2000–2016 is statistically insignificant, therefore the 0X axis corresponds
approximately to the average value of the dollar for the indicated period, which was
28.75 rubles.
Figure 1. The dollar exchange rate for the period 2000-2016
Figure 2. The phase diagram of the dollar exchange rate for the period
2000-2016
86 ITTMM—2018
Figure 2 shows the phase diagram of the dynamics of the dollar exchange rate for the
specified period. Each phase is matched with the average dollar rate corresponding to
each phase. As a criterion for stopping the iterative process, the level of the lost power
of the elements of the series, 5%, was adopted here. However, after the third iteration,
there was no point in continuing the process of combining low-power phases. The level
of power loss of the series was only 4.5%.
Four macrocycles are clearly distinguished here: 2001–2004, 2004–2008, 2008–2014
and a cycle beginning in early 2015.
Similarly, the euro was analyzed for the period 2000-2016. The dynamics of the euro
is shown in Figure 3. Unlike the dollar, the euro’s time series contains a significant trend.
As fluctuations, deviations of the initial values from the trend were also considered
here. The lost power level of the series is 2.75%. Figure 4 shows the phase diagram of
the dynamics of the euro exchange rate and the same four macrocycles are also clearly
visible here.
Figure 3. The euro exchange rate for the period 2000-2016
Figure 4. The phase diagram of the euro exchange rate for the period 2000-2016
Demidova Anastasiya V. et al. 87
To compare the phase diagrams of the dynamics of the euro and the dollar, they
were plotted on a single graph (Fig. 5). Note that since the end of 2008, both currencies
started a new cycle, and their phase diagrams make a simultaneous jump up. Then they
go into the negative area. From 2010 to early 2015, both phases coincide and are below
the trend line of both currencies. But from the beginning of 2015 they make a sharp
jump upward, both currencies have risen above the expected trend value by almost 20
points, which indicates a new strongest wave of the economic crisis. Such a situation
remains unchanged for two years, which indicates the possible duration of this economic
crisis.
Figure 5. Phase diagrams of the dollar and euro exchange rates for the period
2000-2016
Similar calculations were made for the exchange rates of the yuan and the yen (Fig. 6
and Fig. 7). The corresponding phase diagrams are shown in Fig. 8 and Fig. 9. Here you
can watch macrocycles that are more similar to each other than to the corresponding
diagrams for the dollar and euro.
Figure 6. The yuan exchange rate for the period 2000-2015
88 ITTMM—2018
Figure 7. The yen exchange rate for the period 2000-2015
Figure 8. The phase diagram of the yuan exchange rate
Figure 9. The phase diagram of the yen exchange rate
Demidova Anastasiya V. et al. 89
3. Conclusions
Thus, the method of phase analysis of time series allows one to identify hidden
long-term macrocycles in them and, in some cases, make predictions. Of great interest
are multidimensional generalizations of the method, which are supposed to be carried
out in subsequent works.
References
1. M.-C. Wu, Phase Statistics Approach to Time Series Analysis, Journal of the Korean
Physical Society 50 (1) (2006) 304–312.
2. P. Yang, P. Shang, A. Lin, Financial time series analysis based on effective phase
transfer entropy, Physica A (2016), URL: http://dx.doi.org/10.1016/j.physa.
2016.10.085
3. M.-C. Wu and C.-K. Hu, Empirical mode decomposition and synchrogram approach
to cardiorespiratory synchronization, Phys. Rev. E 73 (2006) 051917.
4. M.-C. Wu, M.-C. Huang, H.-C. Yu and T. C. Chiang, Phase distribution and phase
correlation of financial time series, Phys. Rev. E 73 (2006) 016118.
5. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen,
C.-C. Tung and H. H. Liu, The empirical mode decomposition and the Hilbert
spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Lond.
A 454, (1998) 903.
6. L. Laloux, P. Cizeau, J.-P. Bouchaud and M. Potters, Noise Dressing of Financial
Correlation Matrices, Phys. Rev. Lett. 83 (1999) 1467.
7. V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. Nunes Amaral and H. E. Stanley,
Random matrix approach to cross correlations in financial data, Phys. Rev. Lett.
83 (1999) 1471.
8. W.-J. Ma, C.-K. Hu and R. E. Amritkar, Stochastic dynamical model for stock-stock
correlations, Phys. Rev. 70 (2004) 026101.
9. R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics, Correlations
and Complexity in Finance, Cambridge University Press, Cambridge, 2000.
10. J. Voit, The Statistical Mechanics of Financial Markets, 2nd ed., Springer Verlag,
New York, 2003.
11. K. Ohashi, L. A. H. Amaral, B. H. Natelson and Y. Yamamoto, Asymmetrical
singularities in real-world signals, Phys. Rev. E 68 (2003) 065204(R).
12. J. F. Muzy, E. Bacry and A. Arneodo, The multifractal formalism revisited with
wavelets, Int. J. Bifurcation Chaos Appl. Sci. Eng. 4 (1994) 245.
13. S. Thurner, M. C. Feurstein and M. C. Teich, Multiresolution wavelet analysis of
heartbeat intervals discriminates healthy patients from those with cardiac pathology,
Phys. Rev. Lett. 80 (1998) 1544.
14. P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum,
Z. R. Struzik and H. E. Stanley, Multifractality in human heartbeat dynamics, Nature
(London) 399, (1999) 461.
15. A. Marrone, A. D. Polosa, G. Scioscia, S. Stramaglia and A. Zenzola, Multiscale
analysis of blood pressure signals, Phys. Rev. E 60 (1999) 1088.
16. L. A. Nunes Amaral, P. C. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Gold-
berger, H. E. Stanley and Y. Yamamoto, Behavioral-independent features of complex
heartbeat dynamics, Phys. Rev. Lett. 86 (2001) 6026.
17. M.-C. Wu, Phase correlation of foreign exchange time series, Physica A 375 (2007)
633.
18. P. G. Schyns, G. Thut, J. Gross, Cracking the code of oscillatory activity, PLoS Biol
9 (5) (2011) e1001064.
19. B. S. W. Ng, N. K. Logothetis, C. Kayser, Eeg phase patterns reflect the selectivity
of neural firing, Cereb. Cortex 23 (2) (2013) 389–398.
20. URL: https://www.bloomberg.com/quote/EURUSD:CUR