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Segmentation Study of Foreign Exchange Market

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Abstract

This chapter explains a recursive segmentation procedure under normal distribution assumptions. The Akaike information criterion between independently identically distributed Gaussian samples and two successive segments drawn from different Gaussian distributions is used as a discriminator to segment time series. The Jackknife method is employed in order to evaluate a statistical significance level. This chapter shows univariate and multivariate cases. The proposed method is performed for artificial time series consisting of two segments with different statistics. Furthermore, log-return time series of currency exchange rates for 30 currency pairs for the period from January 4, 2001 to December 30, 2011 are divided into 11 segments with the proposed method. It is confirmed that some segment corresponds to historical events recorded as critical situations.

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Notes

  1. 1.

    The selected currency pairs are listed as AUD/JPY, BRL/JPY, CAD/JPY, CHF/JPY, EUR/AUD, EUR/BRL, EUR/CAD, EUR/CHF, EUR/GBP, EUR/JPY, EUR/MXN, EUR/NZD, EUR/SGD, EUR/USD, EUR/ZAR, GBP/JPY, MXN/JPY, NZD/JPY, SGD/JPY, USD/AUD, USD/BRL, USD/CAD, USD/CHF, USD/GBP, USD/JPY, USD/MXN, USD/NZD, USD/SGD, USD/ZAR, and ZAR/JPY.

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Acknowledgments

The author would like to express his sincere gratitude to Prof. Zdzislaw Burda of Jagiellonian University for constructive comments and stimulating discussions.

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  1. Graduate School of Informatics, Kyoto University, Kyoto, Kyoto, Japan

    Aki-Hiro Sato

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Appendix A: Derivation of the Likelihood Function

Appendix A: Derivation of the Likelihood Function

Firstly, let us derive the likelihood function of the i.i.d \(M\)-dimensional Gaussian distribution \(p(\varvec{x};\varvec{\mu },\varvec{C})\). The log–likelihood value is calculated as follows:

$$\begin{aligned} \ln L_1&= \sum _{s=1}^T\ln p(\varvec{x}(s); \varvec{\mu }, \varvec{C}) \nonumber \\&= T \times \frac{1}{T} \sum _{s=1}^T\ln p(\varvec{x}(s); \varvec{\mu }, \varvec{C})\nonumber \\&\approx T \int \limits _{-\infty }^{\infty }\text{ d }x_1 \cdots \int \limits _{-\infty }^{\infty } \text{ d }x_M p(\varvec{x}; \varvec{\mu }, \varvec{C}) \ln p(\varvec{x}; \varvec{\mu }, \varvec{C}) \nonumber \\&= -\frac{T}{2}\ln |\varvec{C}| - \frac{TM}{2}\ln (2\pi ) - \frac{TM}{2}. \end{aligned}$$
(6.39)

Replacing true parameters \({\varvec{C}}\) as its maximum likelihood estimators \(\hat{\varvec{C}}\), one has

$$\begin{aligned} \ln L_1 = -\frac{T}{2}\ln |\hat{\varvec{C}}| - \frac{TM}{2}\ln (2\pi ) - \frac{TM}{2}. \end{aligned}$$
(6.40)

The log-likelihood value \(\ln L_2(t)\) of the alternative model expressed in Eq. (6.15) is similarly computed as

$$\begin{aligned} \ln L_2(t) =&\sum _{s=1}^t\ln p(\varvec{x}(s); \varvec{\mu }_L, \varvec{C}_L) + \sum _{s=t+1}^T\ln p(\varvec{x}(s); \varvec{\mu }_R, \varvec{C}_R) \nonumber \\ \approx&t \int \limits _{-\infty }^{\infty }\text{ d }x_1 \cdots \int \limits _{-\infty }^{\infty } \text{ d }x_M p(\varvec{x}; \varvec{\mu }_L, \varvec{C}_L) \ln p(\varvec{x}; \varvec{\mu }_L, \varvec{C}_L)\nonumber \\&+\, (T-t) \int \limits _{-\infty }^{\infty }\text{ d }x_1 \cdots \int \limits _{-\infty }^{\infty } \text{ d }x_M p(\varvec{x}; \varvec{\mu }_R, \varvec{C}_R) \ln p(\varvec{x}; \varvec{\mu }_R, \varvec{C}_R) \nonumber \\ =&-\frac{t}{2}\ln |\varvec{C}_L| - \frac{tM}{2}\ln (2\pi ) - \frac{tM}{2} \nonumber \\&-\, \frac{T-t}{2}\ln |\varvec{C}_R| - \frac{(T-t)M}{2}\ln (2\pi ) - \frac{(T-t)M}{2} \nonumber \\ =&-\frac{t}{2}\ln |\varvec{C}_L| -\frac{T-t}{2}\ln |\varvec{C}_R| - \frac{TM}{2}\ln (2\pi ) - \frac{TM}{2}. \end{aligned}$$
(6.41)

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Sato, AH. (2014). Segmentation Study of Foreign Exchange Market. In: Applied Data-Centric Social Sciences. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54974-1_6

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