A Wavelet-Based Method to Measure Stock Market Development

In this paper, we introduce a novel algorithm, based on the wavelet transform, to measure stock market development. This algorithm is applied to the return series of fourteen worldwide market indices from 1996 to 2005. We find that a comparison of the return series in terms of the quantity of fractional Gaussian noise (fGn), for different values of Hurst exponent (H), facilitates the classification of stock markets according to their degree of development. We also observe that the simple classification of stock markets into “emerging” or “developing” and “mature” or “developed” is no longer sufficient. However, stock markets can be grouped into three categories that we named emerging, intermediate and mature.


Introduction
There is no precise set of criteria which clearly distinguishes between different stock market types.Therefore, different institutions use different criteria to group countries (or stock markets) by their development level.The World Bank, for example, classifies stock markets into emerging and mature, depending on their national economies using GNP per capita 1 .This classification is, however, disputable for several reasons not least the fact that most developed countries are still undergoing development and some countries, still considered as "developing", have graduated to a further stage over time.
In relation to ways in which stock market development can be measured, there is neither a common concept nor a common indicator agreed by Economists.For example, Demirguc-Kunt and Levine [1] compared many different developmental measures, including market size, liquidity, concentration, volatility, institutional development and international integration, across forty-one countries.Their findings indicated that: 1) small stock markets are less liquid, more volatile and less internationally integrated than larger markets.2) Richer countries generally are more developed than poorer ones.Exceptions include some stock markets defined as "developing" on the basis of national economy, (e.g.those of the Republic of Korea, Malaysia and Thailand).These show indications of maturity stronger than many "mature" markets (e.g.those of Australia, Canada and many European countries).
Recently, Di Matteo et al. ([2] and [3]) studied the scaling properties of different global stock market indices by using the generalized Hurst approach.They found in particular that a) Deviations from pure Brownian motion behavior are associated with the degree of the market's development.b) The generalized Hurst exponent (H) is a powerful tool in distinguishing between the degree of development of stock markets with emerging and mature markets having 0.5 H > and 0.5 H < respectively.In [4], we have attempted to differentiate, qualitatively, between emerging and mature markets using Principle Component Analysis.In particular, we investigated the behaviour of the first three eigenvalues of the covariance matrices of the return series ( ) , , λ λ λ and the ratios using the wavelet transform and eigenvalue analysis.This, in order to study the reaction of emerging and mature markets to crashes and events, and also to measure the recovery time for these two market types.Our results indicated that mature markets respond to crashes differently to emerging ones, in that emerging markets may take up to two months to recover while major markets take less than a month to do so.In addition, the results showed that the subdominant eigenvalues ( ) , λ λ give additional information on market movement, especially for emerging markets and that a study of the behavior of the other eigenvalues may provide insight on crash dynamics.We also found that emerging markets show evidence of persistent behaviour, while mature markets exhibit anti-persistent behaviour (see also [5]).
In our most recent work [5], we suggest a new method based on a time-scale extension of Detrended Fluctuation Analysis (TSDFA) in order to study the behaviour of different stock markets in different time periods and at different scale points.Our results implied that there are three groups of stock market: 1) Very mature markets, e.g.UK, US and Japan, which behave in a mature manner for all time and scale points.2) Emerging markets, e.g.India, Egypt and Sir Lanka, which act as emerging at all time and scale points.3) Other stock markets, e.g.Canada, which has behaviour which is varied depending on the time period and scale (i.e.emergent for some, mature for others).
Since the Hurst exponent (H) is very sensitive to the stock market stage of development, we, therefore, suggest a new algorithm, based on the discrete wavelet transform (DWT) and fractional Gaussian noise (fGn) assessed for different values of H, and evaluate its performance here.Investors are interested in knowing market type in order to make the right investment decisions; several studies notably (e.g.[4,[6][7][8][9]) have reported that emerging markets consistently behave differently to mature ones.Moreover, for foreign investors, emerging markets are more attractive because of their investment opportunities for making higher returns.However, they are riskier and more volatile due to some of their structural issues, such as foreign debt and political instability, while mature markets are safer, more solid and more stable.Therefore, the investors' goal is to find a risk-return balance which generates some returns (or profit) with acceptable risk.
The remainder of this paper is organized as follows: the methodology used in this study is described briefly in Section 2, with data and results given in Section 3. The final section provides a discussion of the results and gives principal conclusions.

Fractional Gaussian Noise
Fractional Gaussian noise (fGn) series { } with zero mean, where the auto-covariance function ( ) ( ) is given by ( ) ( ) An important point about γ is that it satisfies For 0.5 H < , fractional Gaussian noise (fGn) demonstrates anti-persistent behaviour, where this implies that if a series is down in one period then it is more likely to rebound in the next period.For fGn with 0.5 H > , long memory or persistent behaviour is demonstrated.If, e.g., a series is down in a given period, then it is likely that in the next period this behaviour will be sustained.The special case of fGn with 0.5 H = corresponds to Gaussian white noise, representing randomness and implying that the values are uncorrelated.Fractional Gaussian 91 noise series (fGn), corresponding to different values of H (0.3, 0.4, 0.5, 0.6 and 0.7), have been simulated here using the S-plus function 3 in order to compare behaviour with that of the return series of stock market indices.

Our Wavelet-Based Classification
Several studies (such as: [2][3][4]) conclude that developing and developed markets exhibit persistent ( ) and anti-persistent ( ) behaviour respectively, indicating that the development of a stock market is associated with the change in its behaviour from persistence to anti-persistence.Based on this, we have developed an extended DWT technique, which is described by the following steps: 1) We simulate a set of one hundred series of 3000 (each) points of fractional Gaussian noise (fGn) with each { } 0.3,0.4,0.5,0.6,0.7 H ∈ .2) For each set, we apply the DWT in order to compute the energy percentage explained by each wavelet component for the 100 series in each set and take the average of these percentages (Table 1).
3) We estimate the energy percentage explained by each wavelet component for the return series of stock market indices (Table 1).
4) The logarithm to base two of the energy percentages4 , (log 2 (energy%)), explained by the first six components(d 1 -d 6 ), are calculated and plotted in Figure 1.
5) The behaviour of the return series (or that of the linear fit of the returns) is compared with that of the fGn series, for different values of H, in order to group the stock markets.

Data Description
Our empirical analysis is performed on the daily returns of fourteen worldwide market indices which are listed in Table 2, where Daily Return = ln(P t /P t-1 ), where P t and P t-1 are the closing price of the index at day t and t -1 respectively.
The World Bank classification of stock markets is given in Table 3, with the two basic groups consisting of: • Emerging: Argentina, Czech Republic, Ireland, Mexico, Portugal, Russia and Singapore.
• Mature: Australia, Canada, Germany, Hong Kong, Japan, the UK and the US.
The energy percentages described by each wavelet component for the daily returns of the fourteen market indices are given also in Table 1.This shows that the first two high frequency components (d 1 and d 2 ) explain more than 65% of the energy of these series, implying that movements are mainly caused by short-term fluctuations.

Empirical Results
The algorithm of Section 2.2, which is designed to measure the degree of development of a stock market, has been applied to fourteen global stock markets listed in Table 1.The results are given in Figure 1.Firstly, we need to clarify the following key points: • Developing and Developed markets demonstrate persistent and anti-persistent behaviour respectively (with correspondingly, 0.5 H > and 0.5 H < ).The expectation, therefore, is that the stock market should move from persistence to anti-persistence as it develops.
• Our new approach allows for variation in H, but a market will fall on one side or another of the well-defined threshold of 0.5 H = (Gaussian noise) when it is exhibiting clear persistent or anti-persistent behaviour.Note that these are fixed values of ( ) with τ = length of series (number of observations) and θ = number of trading days =1.
• The behaviour of the linear fit of logarithms of stock market returns is compared with that of the generated fractional Gaussian noise (fGn) series for different values of H.The straight line fit for the fGn log series versus the wavelet components indicates that the d 1 doublet explains the largest percentage of energy, d 2 the next largest and so on.
For classification purposes, we compare the behaviour of the linear fit5 of the returns of stock market indices to that of the fractional Gaussian noise (fGn) series with different values of H. From Figure 1, we can see that:    2) The Australian market behaves like to fGn with 0.5 H ≅ (or Gaussian noise), meaning that this market has graduated from the emerging group, but is not yet in the mature one (Similarly, Canada, Hong Kong and Singapore).
3) However, the UK market fit is close to that for fGn with 0.5 H < (anti-persistent), indicating that it is a mature market, (similar behaviour is demonstrated for the German, Japanese and US markets).
It is widely known that emerging and mature stock markets behave in a persistent (or long memory) and anti-

Table 2 . The stock market indices.
(persistent), indicating that it is essentially an emerging market (Similarly, this can be shown for the Czech, Irish, Mexican, Portuguese, Russian markets).