Recently, the economists and financial researchers have concentrated on many
features in financial and economic data. Among the main features they are focusing
now are regime shift or structure break, long memory and volatility clustering.
These three features are the main concerns to them because these are the usual
observed behaviors that occur in financial time series. Furthermore, by monitoring
these main features frequently, the researchers hope to understand more about
a series and the probable development in the future. A comprehensive overview
on the recent development of modelling structural breaks, the analysis of long
memory and stock market volatility can be found by Banarjee
and Urga (2005).
Structural breaks are the main highlight that will be discussed in this study.
According to Brooks (2002), structural breaks is define
as the behaviour of a series that may change for a period of time before reverting
back to its original behaviour or switching to another style of behaviour. Structural
break has been a major concern especially for economists. Various theories of
economic assume that economic relationship changes over time. Such a change
has been explained in descriptive way without being use a statistical test.
With the introduction of regression analysis as the principle tool of economic
data processing in the 1950s and 1960s, attempts were made to describe changes
of economic relationship in regression framework. A detail discussion of structural
break in economic and financial data can be found by Hackl
In this study, we will try to detect the structure break or regime shift in Amman stocks market from Bursa Jordan. This index is the leading stock market indicator in Bursa Jordan. We want to obtain statistical and financial results about the structure break in Amman stocks market by using two approaches. Firstly, the traditional technique Fast Fourier Transform (FFT) and the second approach is Discrete Wavelet Transform (DWT) by using Daubechies wavelet function. Both of these methods are designed to analyze the financial time series data and detect its behavior.
FFT is a spectral filtering method that has been used widely in sciences and
engineering applications. Wavelet transform has a property to Zoom in on short
lived frequency phenomena. This property gives us a tool to learn quickly localized
changes in a financial time series. More generally wavelet transform needs a
series to be presented by some wavelet functions. Wavelet transform is localized
in both time (position) and frequency (scale) domain, while FFT is only localized
in frequency domain not in time. Refer to Karim et al.
(2008), Karim and Ismail (2008), Gencay
et al. (2002), Daubechies (1992) and Chui
(1992) for more detail on these topics.
Wavelet transforms have used in many scientific fields and areas; such that:
signal processing, approximation techniques, quantum field, decomposition, digital
watermarking algorithm, geotechnical engineering, forecasting, pattern recognition
and other fields, for more details refer to Jin and Peng
(2006), Chik et al. (2009), Rizzi
et al. (2009), Rahnama and Noury (2008).
DEFINITIONS AND CONCEPTS
Fourier transforms: It is an operation to transfer the set of complex
valued function to other function; which is known as frequency domain. Consequently,
the Fourier transforms is similar to the other operation in mathematics. We
discuss one type of Fourier transform which is the Discrete Fourier Transform
(DFT) (Janacek and Swift, 1993).
Definition: Discrete Fourier Transforms (DFT) was defined for discrete
points N (Oraintara et al., 2001) as follow:
where, X (n) represents time series data and .
Moreover, the Inverse Discrete Fourier Transform (IDFT) was defined by:
Consequently, FFT and IFFT directly depend on the DFT and IDFT respectively,
FFT and IFFT are two algorithms which is designed from the previous equations
DFT and IDFT respectively (Gencay et al., 2002).
Wavelet transform: The wavelet transform function is constructed by
dilation and translation operations on the scaling function by using the Multiresolution
Analysis (MRA) for more details refer to Mallat (1989),
Daubechies (1992), Nadhim (2006)
and Chui (1992). For a signal C0, its Fast
Wavelet Transform (FWT) or Discrete Wavelet Transform (DWT) can be applied by
using Eq. 3 and 4:
Wavelet transforms by daubechies function: Haar wavelet is the simplest
wavelet transform and then it was improved by Daubechies (1992).
She developed the frequency-domain characteristics of the Haar wavelet. However,
we do not have specific formula for this method of wavelet transform. Therefore,
we tend to use the square gain function of their scaling filter. The square
gain function was defined by Gencay et al. (2002).
where, l is positive number and represents the length of the filter, for more
details and examples see Chiann and Moretin (1998),
Yago (2000) and Struzik (2001).
RESULTS AND DISCUSSION
Here, we start by giving some description for the data. Then we show the analysis of structure change using Fast Fourier transform and Discrete Wavelet Transform (DWT). Finally, we compare the results for the two models.
Data: The data under study are monthly Amman stock market from Bursa Jordan. The estimation period for the monthly data are from December 1998 until July 2009. We utilize the monthly series because we believed that regime shift can be observed specifically across time if low frequency data is used. The total data is 128.
Identifying structure breaks by using fast fourier transform: Figure
1 shows the distribution for the financial time series data with 128 observations,
while Fig. 2 shows the periodogram or the plot of the estimation
power spectrum versus frequency. It appears from Fig. 2, Fast
Fourier Transform (FFT) is not sufficient to capture the regime shift or structure
break since it represents the data as a function of position or frequency domain.
|| Original monthly data
|| Power spectrum estimation
Moreover, a plot of the FFT (Fig. 2) of this signals show
nothing particular interesting. Since FFT represents the data as a curve. This
curve approach or inherent on the X-axis and Y- axis, which hardly capture the
regime shift or structure breaks and interpret the financial behavior. Hence,
it is impossible to obtain the residual data, denoised data, statistical analysis
and the compressed data from Fig. 2.
Identify structure break by using wavelet transform: Now we transform the data using Daubechies function. Figure 3 present the decomposition of the data until level 7. It shows the fluctuations, magnitudes and phases for the monthly data. As the decomposition of the data move from details d1 until d7, all the abnormal values had been eliminated and we have smooth series. While Fig. 4 shows the descriptive statistics and the distribution of the data. From Fig. 4 (X- axis shows the observations, Y-axis is the observation values) it appears the variation among data are very high where the range is 9164 and standard deviation is 2682. Thus, by denoising we hope to reduce these two values.
Figure 5 exhibits the construction of denoise series, The
soft thresholding has been used and in general it produced a smother estimation,
so that it is a good way to decide the convenient model, as well as it has a
good effect on the structure break. If the model has a smooth denoising, then
this model will be more suitable to capture the structure break. Whereas, Fig.
6 displays the residuals series of the denoise series. It seems that the
residuals fluctuated around zero value quite constantly which indicates white
||Wavelet decomposition by using daubechies wavelet
Nevertheless, after around point 70 the fluctuation of the residuals are more
erratic which implies an abnormal behavior happened. Moreover, this figure shows
the autocorrelation function which measured the correlation coefficients of
a signal with itself (Phillips et al., 1999). Thus
the autocorrelation function appears after analyze the transform data by using
fast Fourier transform around zero, it means the data columns have no positive
linear relationship, hence we could not detect the structure breaks via FFT
The process of detecting regime shift or structure break by using Daubecies
wavelet starts by computing the wavelet transformation of the noisy Amman stocks
market index data. Then we compare the wavelet coefficient with the estimate
thresholding values. Thus, it has the spatial positions at which the wavelet
transformation across fine scale levels exceeds the threshold to detect the
regime shift. For the purpose of Amman stocks market analysis by using DWT we
used Daubechies 7 (14 filters) up to level 7. For more details, refer to 8.
The Daubechies 7 wavelet is relatively smooth as compared with the Haar wavelet
filter. By using DWT, we are able to understand and compare between all levels
of the analysis.
|| Statistical analysis results
|| Distribution of the denoising data
Figure 3 shows the result when we analyze the Amman stocks
market series up to the level 7.
There are many notations about the structure breaks. It can be seen that most of the financial crises happened after the months of 70 through 90 which means after the year 2004. This is because we can notice that there are very high fluctuations around this period. This high volatility period indicated that structural breaks happened during this period and it continues until 2009.
From our inspection, the main reasons why the Jordan stock market highly fluctuate
from 2004-2009 because of the increase the numbers of non Jordanian investments.
Therefore, we notice that precisely in 2004-2006, the investments are unbalance
(sometimes positive and sometime negative). Moreover, in February 2006 the investment
be more balance and continued until August 2006. However, in August 2006 the
investment showed a negative balance, but in September the non Jordanian investments
already increased and the investments fluctuated from time to time (http://www.ase.com.jo/).
This instability in the investments affected the stocks market during all the
time from 2004 until 2009. Consequently, the investment was the main variable
which affected the Amman stocks market and we also notice that before 2004 the
investment was very low and there are very small structure breaks or non structure
|| Residuals Distribution of denoising data
In recent years, sudden changes in financial and economic time series data have gained much attention from economic and financial scholars. These sudden changes are called structural breaks. In this study, we discussed two different methods in order to study the existence of structure breaks in Amman stocks market. The two methods are Fast Fourier Transform (FFT) and Discrete Wavelets Transform (DWT). Overall, results indicated that FFT are unable to capture structural breaks because FFT localized in frequency domain only and not in time domain. However, information which contained in the volatility series is perfectly captured by using the DWT method. No anomalies have been introduced by DWT. In addition, by using DWT, we also found the period of structural breaks between 2004 until 2009.
The authors would like to thanks USM for financial support (FRGS:203/PMATHS/6711121) and USM fellowship scheme. Special thanks also due to the anonymous referees for the constructive comments that improved the study.