Predicting Malaysia Business Cycle using Wavelet Analysis

Samsul Ariffin Abdul Karim

Fundamental and Applied Sciences Department, Universiti Teknologi Petronas, Bandar Seri Iskandar, 31750 Tronoh, Perak Darul Ridzuan, Malaysia.

E-mail: samsul_ariffin@petronas.com.my Bakri Abdul Karim

Fakulti Ekonomi dan Perniagaan, Universiti Malaysia Sarawak, 94300 Kota Samarahan, Sarawak, Malaysia. E-mail: akbakri@feb.unimas.my

Fredrik NG Andersson Department of Economics, Lund University

P.O. Box 7082 S-220 07 Lund, Sweden NGF.Andersson@nek.lu.se Mohammad Khatim Hasan

Jabatan Komputeran Industri, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia. khatim@ftsm.ukm.my

Jumat Sulaiman Program Matematik dengan Ekonomi, Universiti Malaysia Sabah, Beg Berkunci 2073, 88999 Kota Kinabalu, Sabah,

Malaysia. jumat@ums.edu.my

Radzuan Razali Fundamental and Applied Sciences Department, Universiti Teknologi Petronas, Bandar Seri Iskandar,

31750 Tronoh, Perak Darul Ridzuan, Malaysia. E-mail: radzuan_razali@petronas.com.my

Abstract-Wavelet transforms are capable to decompose time series at various level which corresponds to the resolution of the decomposition. We can find the trend, cycle, noise, structural break etc. This is where wavelets are so efficient in studying characteristics of the any time series. In this present article, we study the use of wavelet (symlet 16) to detect the business cycle in Malaysia. Firstly we decompose the time series then we study the long-run trend and we filtered the high frequency components and finally we find the business cycle in Malaysia. The results indicated the existence of business cycles for GDP data in Malaysia which is strongly counter-cyclical.

I. INTRODUCTION

Wavelets are successfully being applied in various disciplines e.g. engineering, mathematics and sciences. One of the main advantages of wavelets is there exist fast algorithm to compute the coefficients. Furthermore with multi-resolution analysis (MRA), we can study the characteristics of the data at any level, hence this is why wavelets potentially being used in various economics and finance applications. For instance, [1] has studied the use of wavelets in financial applications. Reference [2] has utilized the multi-resolution property to show the business cycle for USA GDP data. He found that wavelets analysis provides better resolution in the time domain as compared with other band-pass filters that is always being used in economics. Reference [3] has studied the use of wavelet transform in stock exchange problem. They found that wavelets (symlet 8) are capable to analyze Kuala Lumpur Composite Index (KLCI) even at level 7. They also show that the Minimax and fixed form threshold give the better result as compared to the heuristic SURE and SURE options in terms of Root

Mean Square Error (RMSE) calculation. Reference [4] has studied the applications of DWT in compressing the temperature data. In this paper they use Daubechies 4 (D4) to compress the original data at level 5. Meanwhile [5] has used wavelets together with Box-Jenkins ARIMA model to study the time series forecasting. References [6], [7], [8] has applied WT into various problem in finances and they also show that wavelets are really suitable to study the time series behavior. For more details on wavelet theory and its applications in various disciplines, the reader are encourage to refer [9], [10], [11], [12] – this books discuss about wavelets and other filtering method and its applications in finances and economics, [13] and [14]. Of course there exist various textbooks on wavelets in the markets.

Noise is extraneous information in a signal or time series that can be filtered out via the computation of averaging and detailing coefficients from wavelet transforms. Hence we can filter out the low frequency (approximation coefficients) and high frequency (details coefficients). In fact many statistical phenomena have wavelet structure [3].

In this paper we will discuss the applications of wavelets in detecting the business cycle (BC) in Malaysia using Malaysia GDP data. We use the data from Quarter 1 year 1980 until Quarter 2 year 2007. Total we have 110 data. We utilized symlet 16 to filter the data and study the business cycle from the decomposition of the data via MRA. The reason we apply symlet 16 because it has 32 filters and after we have done several numerical simulation we noticed that symlet 16 is really suitable to find the business cycle in Malaysia. Furthermore, symlet 16 has higher vanishing

2011 IEEE Symposium on Business, Engineering and Industrial Applications (ISBEIA), Langkawi, Malaysia

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moments (VM) which is 15. Symlet wavelet is also nearly symmetric. For more detail please refer to [11].

Business cycles refer as fluctuations found in the aggregate economic activity of nations. The cycles consist of expansions occurring at about the same time in many economic activities, followed by similarly general recessions, contractions, and revivals which merge into the expansion phase of the next cycle. In general, the cycles were cyclical components of between six quarters to thirty quarters [15]. Business cycles in emerging markets are characterized by strongly counter-cyclical than developed markets [16]. According to [17] understanding business cycles is the first step in designing appropriate stabilization policies.

II. WAVELET ANALYSIS

In this section, we will review the basic definition of Wavelet theory by using Multiresolution Analysis (MRA) approach. For more detailed the reader can refer the books on wavelet by [9-11], [13] and [18-31].

Until recently wavelet analysis via MRA approach has been found to be a reliable method in financial and economic analysis, in particular for stock market and foreign exchange. Applications of wavelets in finance can be seen in the study of non-stationary and non-linearity property of financial time series because of structure change, volatility and long-memory process. Furthermore, wavelet methods have also been used as a tool for forecasting. Nevertheless, wavelets decompositions of a signal or data can be adopted to improve the hypothesis testing on existing theories and can also provide insights of financial phenomena and enhance the development of theories. In addition, wavelets decompositions are being used as a measurement of co-movement among economic and financial variables.

Based on [13], [19] and [20], suppose that there exists a function ( ) ( )2t L Rφ ∈ such that the family of functions

( ) ( )22 2 , , ,j

jt t k j k Zφ = − ∈ (1)

is an orthonormal basis of jV hence 1φ = . Definition 1 ([13])

We define a MRA in ( )RL2 as a sequence of closed

subspaces ,jV ,Ζ∈j of ( )RL2 and satisfying the following properties: (M1) ;1+⊂ jj VV

(M2) jVf ∈ if and only if, ( ) .2 1+∈ jVtf

(M3) { }.0Vj j =Ζ∈∩

(M4) ( ).RLV 2j j =Ζ∈∪

(M5) There exists a function 0V∈φ such that the set ( ){ }Ζ∈− kkt ;φ is an orthonormal basis of .0V

( )2 ,jjL R W j

+∞

=−∞= ⊕ ∈ Ζ (2)

Eq. (2) means that any function, ( )2f L R∈ can be represented as a wavelet series as follow

( ) ( ) ( )00

,k k jk jkk j k

f x x xα ϕ β ψ∞

=

= +∑ ∑∑ (3)

where jkk βα , are coefficients defined by (4), and

{ } Ζ∈kjk ,ψ is a basis for jW . The relation (3) is called a multiresolution expansion of f where we have

( ) ( )22 2j

jjk

x x kψ ψ= − , ,j k ∈ Ζ .

Basically, the function ( )jk xϕ and ( )jk xψ are called the scaling function (father wavelet) and the mother wavelet respectively, where the coefficients

( ) ( )0k kf x x dxα ϕ= ∫ , ( ) ( )k jkf x x dxβ ψ= ∫ (4) The mother wavelet satisfies

( ) ( )2 2kk

x g x kψ ϕ= −∑ , (5)

where ( ) 11 k

N kkg h+−= − . For the father wavelet we have the

following relations

2 0k k l lk

h h δ+ =∑ and 1 12 k

kh =∑ . (6)

In this paper, we use the symlet 16 wavelet (32 filter

coefficients). To show the power of DWT, we apply the symlet 16 to decompose the GDP data hence later we show the existence of business cycle in Malaysia.

Figure 1. Symlet 16 scaling function and wavelet function

Figure 1. Symlet 16 scaling function and wavelet function Figure 1 shows the example of symlet 16 scaling function

and its corresponding wavelet function.

III. WAVELET DECOMPOSITION OF GDP DATA

Meanwhile Figure 2 shows the wavelet decomposition for the original GDP data. For this data set, the best level of decomposition is 7. We can see clearly from Figure 2, even at level 7, the wavelet decomposition already contains the main patterns of the original time series that is appear very volatile and various point of spike. We discuss the detection business cycle in Section IV.

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2 (a)

2 (b)

Figure 2. Wavelet Decomposition to the original GDP data up to level 5 (a) Approximation (b) Detail

Based on Figure 2 (a) and 2 (b), level 1 are 1-2 quarter and level 5 are 16-32 quarter for GDP data. From level 1 until level 5 in details parts, we can see clearly that the high frequency coefficients have been filtered out until we get the trend more smooth at level 5.

Figure 3. The detail combination of 54 dd +

Data at 1=x and at 110=x correspond to the GDP data at Quarter 1 Year 1980 and Quarter 2 Year 2007, respectively.

Table 1: Business Cycle in Malaysia

Data on X-axis Business Cycle 14 1983 (Q2) 39 1989 (Q3) 66 1996 (Q2) 92 2002 (Q4)

110 2007 (Q2)

Figure 4. Real Malaysia GDP and Long-Run Trend

IV. RESULTS AND DISCUSSIONS

In this section we discuss the main results that we achieved from the numerical experiment that we have done in section III above. Figure 3 shows the correlation of GDP data. Based on the figures and statistical results we have the following observations:

• From the approximations part in Figure 2(a) at level 5 we can notice that the time series is non-stationary and its show the long term trend of GDP in Malaysia.

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• Even at level 5 (16-32 data) we already recapture the trend of the time series.

• We also observed that the relative contributions of 32 , dd are smaller than the contribution of 54 , dd .

Hence, at level 5 the decomposition already recapture the nonlinear trend.

• Figure 3 shows the results when we combined two details: 54 dd + (both details have 110 coefficients), therefore we still obtained 110 coefficients. Here we follow the idea in Yamada and Honda [32].

• There exist business cycles for GDP data in Malaysia. In line with [16], we found that the Malaysia business cycles are strongly counter-cyclical.

As mentioned by [2], wavelet analysis is a natural way to

decompose the economic time series into long-run trend which is periodicity greater than 32 quarters and the business cycle component (periodicity between 4 and 32 quarters). This is corresponding with our findings in this paper. At level 4 and 5 we could find the business cycle in Malaysia. Table 1 summarizes the result. From long-run trend in Figure 4, we noticed that Malaysian GDP has not largely deviated far from its long-run trend.

The existence of business cycles in Malaysia, perhaps the susbtantial component is due to the world factor. In addition, it may be driven by country and idiosyncratic factors. Thus as noted by Lucas [17] understanding business cycles is the first step in designing appropriate stabilization policies.

From Table 1, the period of second quarters of 1983 is where Malaysia has just recover from recession because of the restructuring in the industrial sector in major industrial countries such as the US. Then the third quarters 1989 is the expansion period after world stock market crash in 1987. While the second quarters of 1996 is also the growth period just before the financial crisis in 1997. Finally for the fourth quarters of 2002 and second quarters of 2007 are refer to the recession periods because of recession period in US. Malaysia economic was affected because the US was Malaysia main trade partner.

CONCLUSIONS

In this paper we have discussed the application of wavelet in detecting the business cycle event in Malaysia from 1980 until 2007. We have utilized symlet 16 (32 lowpass and highpass filters respectively). Firstly we decompose the GDP data up to level 5 (this is an optimum level for the data set). Then we study the characteristics of the data. Table 1 summarizes the business cycle event in Malaysia with its period or quarter and year. There exist business cycles for GDP data in Malaysia with strongly counter-cyclical. Future work will focus on filtered GDP data by using the approximate bandpass filter [33] and do numerical comparison between wavelet and bandpass filter. The authors are keen to discuss it in a subsequent paper.

ACKNOWLEDGMENT

The first author gratefully acknowledges Universiti Teknologi PETRONAS (UTP) for the financial support

received in the form of a research grant: Short Term Internal Research Funding (STIRF) No. 76/10.11 and computing facilities including MATLAB software.

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