2010 International Conference on Science and Social Research (CSSR 2010), December
5
-
7,
2010, Kuala Lumpur, Malaysia
Forecasting volatility data based on Wavelet transforms and ARIMA model
d b as. AI Wadi, Mohd Tahir Ismail and Alsaidi M. AItaher bd a, , School of Mathematical science, Universiti sains Malaysia, 11800 Minden, Penang, Malaysia E-mail:
[email protected]@cs.usm.my
[email protected] cSamsul Ariffin Addul Karim, cUniversiti Teknologi Petronas, 31750 Bandar Seri Iskandar, Tronoh, Perak, Malaysia E-mail:
[email protected]
Abslracl- this article suggests a novel technique for forecasting
potential future losses of a portfolio of assets, and in order to
the volatility data based on Wavelet transforms and ARIMA
measure these potential losses, estimates must be made of
model.
The
volatility
data
are
decomposed
via
Wavelet
future volatilities and correlations [21].
transforms. Then, the future observations of this series are
Recently, wavelet transforms are used for filtering time series
forecasted using a suitable and best fitted ARIMA model. Daily
representing. Wavelet analysis has grows very quickly in the
prices from Amman Stocks Market (Jordan) from 1993 until
recent years and more recently Wall Street analysts have
2009 are used in this study.
Keywords- Wavelet transform; forecasting. I.
ARIMA
started using mathematical models to analyze their financial
model; volatility data;
data. Wavelet analysis also has been used in signal processing (time
INTRODUCTION (HEADING 1 )
(typically three to six) of constitutive series. These series show
has gotten high attention in financial time series and financial
a better behavior than the original price series. Besides, it is
researchers. Forecasting stocks market is difficult because
more stable in variance and no outliers. Wavelet transform is
unlike demand series, price series present such characteristics
more efficient than Fourier transform ([3, 5, 6, 8, 1 1 , 12, 13]).
as inconstant mean and variance and significant outliers.
For this, wavelet transform can be used to analyze nonlinear
The developing economies are facing many impediments in
and non-stationary time series signals, useful in identifying
their financial markets, and with many other factors, high
transient events, used to filtering the denoise data to get more
volatility in prices which also considered as high risk or
accurately data, providing decomposition of a time series into
uncertainty is a major factor of erosion of capital from
several components from different scale and appears their
markets. As due to this the investors become fearful and run
correlation as a function on scale and time (localized in both).
away from the market. Though it is not the sign of inefficiency
Therefore, to show the efficiency of wavelet transform, this
of market, it poses a threat to 'crash' the market due to high
paper will introduce the ARIMA wavelet transform (AWT)
volatility. High volatility creates a high uncertainty in a stock
and ARIMA models then demonstrate that ARIMA wavelet
market and individual security prices and these may curtail
transforms is better than ARIMA models in forecasting
down the prices and associated returns. The stock market
volatility data. We will use MATLAB and SAS programming
volatility caused by number of factors such as: credit policy,
to obtain some of numerical and statistical results. In order to
inflation rate, interest, financial leverage, corporate earnings, yield
macroeconomic
policies, where
bonds
social
prices
and
and
political
recognition, decomposition,
the wavelet transform converts the financial series in a set
Stocks markets forecasting i s required for the investors and it
dividends
scale analysis), pattern
approximation techniques and quantum field [12]. Moreover,
many
illustrate the effectiveness of wavelet transforms, the Amman
other
variables
Stocks Market data sets are selected for discussion. We
are
consider a daily volatility for time period from 1993 (the days
involved. Madhavan (1992) [16] defines volatility in terms of
when stocks exchanges were open) until 2009 with a total of
price variance. Low volatility is preferred as it reduces
4096 observations. The total number of observations for ) mathematical convenience is suggested to be divisible by 2 .
unnecessary risk borne by investors thus enables market traders to liquidate their assets without large price movements.
It
Glen (1994) defines volatility as the frequency and magnitude
means
that
the
data
should
satisfy
the
condition of
of price movements and comparing the various microstructure
) observations= 2 ,j
attributes argues that liquid and efficient markets have less
the data set until level 12 (4096=212 observations) [14 , 10].
=
0 ,1,2, ... . Therefore, we can decompose
volatility than illiquid and inefficient markets. [15]Amihud et al. (1990) finds a reduced volatility in the Tel Aviv market as the market adopted a more continuous trading system.
II.
The three main purposes of forecasting volatility are for risk
The main aim of using Fourier transform, wavelet transform
management, for asset allocation, and for taking bets on future
and the other filtering methods is to represent the original time-
volatility. A large part of risk management is measuring the
978-1-4244-8986-2/10/$26.00 ©2010 IEEE
BASIC CONCEPT
86
series as a new sequence, which appears and explains the
l =J2 'f ¢(/)¢(2/-k)dl k -00
importance of each frequency components, trends, magnitudes, fluctuations and deferent requirements in the dynamics of the
=
{
1
O:o::t:O::l
'
0,
otherwise
original series. We will start this section by explaining the Haar And forN=2,
wavelet transforms and Daubechies wavelet transform. For more details and examples refer to [4].
Note: the mother wavelet satisfies the following conditions: 2.1 Wavelet transforms 00
� If(t)dt=0,
Consider the following function:
_
1
(t)=22 1f(2jt-k), j,kEZ; z={O,1,2,.... }. (1) If k j, Where;
If/
d(O
_
< 00.
Coiflets Wavelet. Haar wavelet is the simplest and oldest
For examples and details; refer
If/ is
lfl «(0)
example for the wavelet transform then this method was improved by Daubechies transformation, Daubechies (1992)
-00
to [7, 10]. The oldest and simplest example of
_
I 1 � 1(01
00 < 00,
The most popular types of orthogonal wavelets transform are:
00
f 1f(/)d1 =0 .
� 1 1f(t)1
Haar Wavelet, Daubechies Wavelet, Symmlets Wavelet and
is a real valued function having a compactly
supported, and
00
2
[9] developed the frequency - domain characteristics of the the Haar
Haar Wavelet. Actually there was no specific formula for this
wavelet, defined as:
type of wavelet. Thus, we have to use the square gain function
[
of their scaling filter (Iow- pass) by the following equation to determine the class [12].
1 0:0:: /:0::-
I,
2
H
If (I)= -L 0,
1 -:0::1 :0::1
g(j) =2 cos
(2)
2
1
LI 1 (trf) 2� -2-1 + I / o I
Qkrwise
J
calculate
the
wavelet
fimction,
we
use
the
dilation
This article includes some application of ARMA models in
¢(/)=fiI1d(21 -k), k If(t)=J2I hk¢(2t-k). k
is the father wavelet,
forecasting
(3)
economic
variables,
its
merits,
demerits
and
advantages of the model in comparison with conventional time series models. ARMA is the acronym for "Autoregressive Moving Average", invented by two great Statisticians George
(4)
Previously, we defined father and mother wavelets in
¢( 21 -k)
([12,18, 19,20]).
2.2 ARIMA Model:
equations, given as:
(4) where
(trf)·
I: Represents the length of the filter and should be a positive number. For more details and examples see
To
1
sin 2
1f(/) is
(3)
Box and Gwilym Jenkins and hence ARMA models are also
and
known as Box and Jenkins models [17].
the mother
ARMA models are
suitable for high frequency data. Since most of the economic
wavelet. Father is used to represent the high scale smooth
time
components of the signal, while the mother wavelets display
differencing is employed to convert non stationary data into
series
data
are
non-stationary,
a
method
called
the deviations from the smooth components. In other words,
stationary. The differenced series is regressed on to the original
the father wavelet generates the scaling coefficients and mother
series
wavelet gives the differencing coefficients. We define the
Regressive Integrated Moving Average).
father wavelets as lower pass filter coefficients mother wavelets as high pass filters coefficients
(hk)
then
ARMA
model
becomes
ARIMA (Auto ARIMA models
produce accurate forecasts based on the historical patterns of
and the
the time series data. ARIMA belongs to the class of linear
(lk ) [9].
l =J2 'f ¢«('I/J(2t-k)dt, k -00
and
models and can represent both stationary and non-stationary data. ARIMA models do not involve the dependent variable;
(5)
instead they make use of information in the series to generate the series itself. Stationary series is the one which vary about a fixed value whereas non-stationary series do not vary about a
(6)
fixed value. For more details; refer to ([1,
-00
2,17]).
However, quadrature mirror filters (QMF) is other concept There are a huge Variety of ARIMA models. The general non
related to the wavelet transform, it has some a good ability to reconstruct the signals perfectly without aliasing effects.
seasonal model is known as ARIMA (p, d, q) [ 22]:
For the Haar wavelet:
AR: P= order of the autoregressive part. I
: d= degree of first degree involved.
MA: q= order of the moving average part.
87
If non- stationary is added to a mixed ARMA model, and then the general ARIMA (p, d, q) is obtained. The equation for the simplest case ARIMA (1,1,1) is as following:
Once the model adequacy is established, the series in question shall be forecasted for specified period. It is always advisable to keep tracking on the forecast errors and depending on the magnitude of errors, the model shall be re-evaluated. 2.3. Volatility: Researchers have already improved a lot of the definitions about volatility. Firstly, the most popular method is the standard deviation as a traditional technique. Secondly, the difference in the price results between up and down. Practically, both of them have the similar results. Therefore, in this paper we use the second definition since it is considered as a modern technique. In other words, it is defined as the absolute value of the daily return. It is mathematically expressed as [12]:
The model building process involves the following steps [22]: Model identification: The first step is to determine whether the time series data is stationary or non- stationary. The stationarity can be assessed either using Dickey Fuller test or run sequence plots. If the original series has no trend (stationary) then the series is an ideal candidate for ARIMA. If the original series has trend (non-stationary), the series can be converted to stationary by differencing the series. The order of differencing is zero for a stationary series and greater than zero for non- stationary series. •
vt = III.
PROCEDURE AND RESULTS
The Prediction technique for the volatility time series data taken from Amman stocks market works as follows. First step: transform the original data through the wavelet transform based on Haar wavelet transform and Daubechies wavelet transform (db2). Second step: evaluate the volatility data for the transformed data. Third step: select the fitted ARIMA models for the approximation Haar wavelet series and Daubechies wavelet series, after that, make the forecasting for the future data for each approximation series. Fourth step: select the fitted ARIMA model to the original volatility series (before the transformation). Fifth step: compare all of these results and decide the best model.
Model parameter estimation: The estimation of parameters is very importance in the model building. The parameters thus obtained are estimated statistically by the method of least squares. A t-statistic shall be employed to test the parameters significance. •
Model Diagnostics: Once the parameters are statistically estimated, before forecasting the series, it is necessary to check the adequacy of the tentatively identified model. The model is declared adequate if the residuals cannot improve forecast anymore. In other words, residuals are random. To check the overall model adequacy,the Ljung-Box Statistic is employed which follows a Chi-Square distribution. The null hypothesis is either rejected or not rejected based on the low or high p-value associated with the Statistic. •
•
h l = l log(xt)-log(xt_I) I ·
Forecasting:
Prediction errors for F2 D,D2D D,D 15 D,DID D,DD5 D,DDO -0,005 - D,0 10 -0,0 15 500
1500
1DDD
88
2500
JDDD
3500
�D95
Fig.I. forecasting error before the transformation
Prediction err 0 rs for VDLCA1 o , 10 O,OB o ,0 6 0, D4 0,02 0,00 -0,02 -0, D4 •
1
•
200
400
600
800
1000
1200
•
•
1400
1600
•
1800
200047
Fig.2. forecasting errors after transforming the data via Haar Wavelet transform.
Prediction errors for VDL082 o ,0 6 0, D4 0,02 0,00 +-....--------------------------1 ,. -0,02 -O,H -0,0 6 -O,OB
.
1
200
400
.
600
800
1000
.
1200
1400
1600
.
1800
Fig.2. forecasting errors after transforming the data via Daubechies Wavelet transform.
89
.
200048
the fiItted ARIMA modeI be r.ore and after the trans ormatIOn Value after transforms Value before transformation Daubechies wavelet transform Haar wavelet transform
Table. 1 StatlstlcaIPropertI es
Statistics of fit
0f
Mean square error
7.41951E-6
5.675E-6
7.82381E-6
Root mean square error
0.0027239
0.00238
0.0027971
Mean absolute error
0.0002457
0.0002200
0.0019719
[6]
The volatility data for Amman stocks market has been used as case study. Price forecasting is performed using daily data.
wavelet pairs, Signal Processing 85. 2005. pp. 2065-2081.
Moreover, for the sake of fair comparison the same sample
[7]
data are selected. (From 1993-2009). The fitted ARIMA model
for
the
ARIMA(2,0,2)
original with
volatility
root
mean
data
is
square
considered error
equal
[8]
as
D.E. Newland, , An Introduction to Random Vibrations Spectral and Wavelet Analysis (third ed). Prentice-Hall. Englewood Cliffs, NJ. 1993.
to
[9]
I. Daubechies, Ten Lectures on Wavelets, PA. SIAM and Philadelphia. 1992.
model for the transform data by using Haar wavelet transform
[10] Philippe Masset. Analysis of Financial Time-Series Using Fourier and Wavelet Methods. University of Fribourg (Switzerland) - Faculty of
with root mean square error
Economics and Social Science. 2008.
equal to (0.00238)as presented in table 1 also. Although the fit
[11] P.Manchanda,
ARIMA model for the transform data by using Daubechies wavelet transform is selected as ARIMA (2,1,0)
Chang Chiann and Pedro A. Moretin. A wavelet analysis for time series, Nonparametric Statistics. 1998. 10 pp: 1-46.
(0.0027971) as presented in table 1, while the fit ARIMA is selected as ARIMA (1,0,0)
Brandon Whitchera, Peter F. Craigmileb and Peter Brownc. Time varying spectral analysis in neurophysiologic time series using Hilbert
lKumar,
A.H.Siddiqi
Mathematical
methods
for
modeling price fluctuations of financial time series. Journal of Franklin
with root
institute. 2007. 344, pp: 613-363. [12] Gencay, R., F. Seluk and B. Whitcher. An Introduction to Wavelets and
mean square error equal to (0.0027239). Table 1 shows some
Other Filtering Methods in Finance and Economics, Academic Press,
other criteria about these results. All of these criteria explain
New York. 2002.
that the Wavelet-ARIMA model is better than the ARIMA
[13] S.Mallat. A Wavelet Tour of Signal Processing. Academic Press, San Diego.200 I.
model. Moreover, the Haar wavelet transform gives more
[14] Todd Wittman. Time-Series Clustering and Association Analysis of
sufficient result and better than Daubechies wavelet transform
Financial Data, CS 8980 Project.2002.
in the forecasting.
[15] Y. Amihud, H. Mendelson and M. Murgia. Stock market microstructure and return volatility: evidence from Italy. Journalof
VI.
Banking and Finance. 1990, 14: 423-440.
CONCLUSION
[16] A. Madhavan. Trading mechanisms in securities markets. Journal of
As conclusion for this article, if the Wavelet transform is used
Finance.1992, XLVIl(2), 607-641.
for the volatility data, then there is no outlier, seasonal effects and
other
irregular
effects.
Generally
the
result
of
[17] G. Janacek, and L. Swift. Time series forecasting, simulation and
the
applications. Ellis Hoewood limited. England. 1993. [18] Y. Motohiro. Measure business cycle: A wavelet analysis of economic
approximation series under the wavelet transforms is better
time series. Economics letter. 2008, 100:208
than the original series and more stable in variance, mean and
212.
no outliers. Furthermore, the forecasting by using ARIMA (p, d, q)
[19] Zbigniew
under the transforms is better than the forecasting
series
International
Finance
Corporation Discussion. 1994. Paper
No. 2
Washington D.e. IFe. [22]Spyros Makridakis, Steven e. Wheelwright and Rob J. Hyndman. Forecasting methods and applications. Third edition. John Wiley
J.Contreras, R. Espinola, F. J. Nogales, and A. J. Conejo. "ARIMA
Sons, Inc. New York. 1998.
models to predict next-day electricity prices," IEEE Trans. Power Syst,2003.vol. 18, no. 3, pp. 1014-1020. F. J. Nogales, J. Contreras, A. J. Conejo, and R. Espinola. "Forecasting next-day electricity prices by time series models," IEEE Trans. Power Syst. 2002 , vol. 17, no. 2, pp. 342-348. Arneodo, A., B. Audit, N. Decoster, J.F. Muzy and e. Vaillant. Wavelet based multiracial formalism: applications to DNA sequences. Springer, Berlin pp. 27-102. 2002. A.H. Siddiqi. Applied Functional Analysis, Marcel Dekker, New York. 2004. A.Razdan. Wavelet correlation coefficient of strongly correlated time series, Physics A.2004,
time
learning and cybernetics, Dalian, 13-16 August 2006.
REFERENCES
[5]
(financial)
[21] J., Glen. An introduction to the microstructure of emerging markets.
and USM fellowship scheme
[4]
in
Sui. The research of insurance income forecast based on wavelet
ACKNOWLEDGEMENT
[3]
methods
analysis. Proceedings of the fifth international conference on machine
The authors would like to thanks USM for financial support
[2]
Wavelet
[20] Bin-Sheng Liu, Yi-Jun Li, Hai-Tao Yang, Yu-Peng Hou and Xue-Shen
directly as well as it gives more accuracy results.
[1]
R.Struzik.
processing. Physica A. 2001, 296:307-319.
333, pp: 335-342.
90
&
