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The Random Walk Theory And Stock Prices: Evidence From Johannesburg Stock Exchange

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  • University of Mpumalanga

Abstract

In this paper, we test the Johannesburg Stock Exchange market for the existence of the random walk hypothesis using monthly time series of the All Share Index (ALSI) covering the period 2000 2011. Traditional methods, such as unit root tests and autocorrelation test, were employed first and they all confirmed that during the period under consideration, the JSE price index followed the random walk process. In addition, the ARIMA model was constructed and it was found that the ARIMA (1, 1, 1) was the model that most excellently fitted the data in question. Furthermore, residual tests were performed to determine whether the residuals of the estimated equation followed a random walk process in the series. The authors found that the ALSI resembles a series that follow random walk hypothesis with strong evidence of a wide variance between forecasted and actual values, indicating little or no forecasting strength in the series. To further validate the findings in this research, the variance ratio test was conducted under heteroscedasticity and resulted in non-rejection of the random walk hypothesis. It was concluded that since the returns follow the random walk hypothesis, it can be said that JSE, in terms of efficiency, is on the weak form level and therefore opportunities of making excess returns based on out-performing the market is ruled out and is merely a game of chance.
International Business & Economics Research Journal November/December 2014 Volume 13, Number 6
Copyright by author(s); CC-BY 1241 The Clute Institute
The Random Walk Theory
And Stock Prices: Evidence
From Johannesburg Stock Exchange
Tafadzwa T. Chitenderu, University of Fort Hare, South Africa
Andrew Maredza, North West University, South Africa
Kin Sibanda, University of Fort Hare, South Africa
ABSTRACT
In this paper, we test the Johannesburg Stock Exchange market for the existence of the random
walk hypothesis using monthly time series of the All Share Index (ALSI) covering the period 2000
2011. Traditional methods, such as unit root tests and autocorrelation test, were employed first
and they all confirmed that during the period under consideration, the JSE price index followed
the random walk process. In addition, the ARIMA model was constructed and it was found that the
ARIMA (1, 1, 1) was the model that most excellently fitted the data in question. Furthermore,
residual tests were performed to determine whether the residuals of the estimated equation
followed a random walk process in the series.
The authors found that the ALSI resembles a series that follow random walk hypothesis with
strong evidence of a wide variance between forecasted and actual values, indicating little or no
forecasting strength in the series. To further validate the findings in this research, the variance
ratio test was conducted under heteroscedasticity and resulted in non-rejection of the random
walk hypothesis. It was concluded that since the returns follow the random walk hypothesis, it can
be said that JSE, in terms of efficiency, is on the weak form level and therefore opportunities of
making excess returns based on out-performing the market is ruled out and is merely a game of
chance.
Keywords: Random Walk Hypothesis; ARIMA; Johannesburg Stock Exchange (JSE); Variance Ratio Test
INTRODUCTION
he theory of market competence (or efficiency) dominates financial literature due to the scarcity of
financial resources. As such, how stock market prices behave plays a very significant part in the share
of the scarce monetary resources. Market efficiency explains the relationship that exists amid
information and stock price in financial markets; that is, whether or not proceeds in a market pursue a random walk
process. Regulators now and again try to improve the condition of the Johannesburg Stock Exchange (JSE) by
imposing different rules and regulations because price trends are important to investors and/or companies at the time
when they are deciding on spreading their investment funds and risk. Stock prices also provide a benchmark against
which profits on investment projects can be evaluated (Green et al., 2005). An informationally efficient market
implies that capital and risks are appropriately priced without any distortions.
Following the developments in the JSE, it is necessary to add to the existing literature concerning the
randomness of the All Share Index (ALSI) using current information and see if the results have changed or not. Over
the years, it has become a major interest to financial analysts to come up with theories and models that explain how
stock market prices behave or how they can be determined. One such model is the Random Walk Hypothesis
(RWH), which is an economic theory that stipulates that prices of stocks move in accordance to a random procedure
and, consequently, the prices of the stocks in the market cannot be foretold. Viney (2007; 309) defines random walk
T
International Business & Economics Research Journal November/December 2014 Volume 13, Number 6
Copyright by author(s); CC-BY 1242 The Clute Institute
hypothesis as a theory that contends that each observation in a time series, such as share prices, is dependent on the
previous observation. Put differently, the hypothesis says price sequences do not demonstrate predictive patterns
over a period of time but can be best described by a random walk. According to the RWH, the definite lack of
correlation between the precedent and current prices can easily be seen; hence, if a share increases at a particular
time, no market partaker is able to precisely foresee that it will go up again the following day (Fama, 1965).
In order to understand the RWH, it is vital to understand the theories which describe how one can predict
the stock market price. There are basically two approaches, common to market professionals, which are used to
predict stock prices - the chartist or technical theories and the theory of fundamental or intrinsic value analysis
(Fama, 1965). The chartist theories hinges on the basic assumption that past/history repeats itself and therefore the
system to forecast stock prices, as well as increase one’s possible gains, is by becoming familiar with history
patterns of price behaviour as well as situations of likely repetition. The fundamental or intrinsic analysts, on the
other hand, hold the assumption that at every point in time, a single security has what is called an intrinsic value
(equilibrium) which rests on the earnings probable of that security. Earnings potential, in turn, relies on crucial
factors such as worth of management, position of the trade and the financial system, to name but a few. What this
then means is that a shareholder can, through a watchful study of the fundamental factors, be able to establish
whether the real price is higher than or below the intrinsic value. If real prices of stocks have a tendency of moving
towards basic values, an attempt to establish this value is the same as creating predictions of prospect price which
forms the heart of the analytical procedure implied by the fundamental analysis (Mishkin, 2010). In contrast, the
RWH starts from the ground that the market for securities is a good example of an efficient market and in an
efficient market, normal profit-maximizers actively compete and try to forecast prices. This leads to actual prices
reflecting all information and becomes fine estimates of the intrinsic value of the security. According to the RWH,
the dealings of numerous participants will lead to the real price of a stock wandering randomly around its intrinsic
value. If the variance amid real prices and intrinsic values are methodical rather than haphazard in nature, awareness
of this ought to aid intellectual market partakers to better forecast the trail through which real prices will travel
toward intrinsic values. Whilst numerous clever traders try to gain using this awareness, they have a tendency of
counteracting such methodical behavior in price sequence; and even though ambiguity regarding intrinsic values
will stay, genuine stock prices will stroll haphazardly about their basic values (Fama, 1965). The self-determination
supposition of the random-walk model is applicable as long as information of previous performance of the sequence
of changes in price cannot be used to enhance anticipated gains. The implication for investment purposes is that the
independence assumption acts as a sufficient explanation of actuality if actual level of reliance in series of price
changes is not sufficient to create more returns than the probable proceeds under a naive buy-and-hold plan (Brooks,
2008).
The main concern of this research is to investigate the random walk hypothesis in JSE price trends. Since
stock markets are important, the government, commerce, and the reserve banks of countries maintain a close look at
the activities of the stock market price index. The stock market price changes now and again and sometimes a
market's rise can be attributed to pure speculation or to changes in the economic variables. Hence, at times, trying to
anticipate stock market movements by analyzing traditional economic and financial indicators can lead to incorrect
forecasts. Put differently, the idea or assumption of price dependency or serially independent price increments is
supported by the act of different investors competing in the market. If there is correlation amid different prices in
different time periods, clever investors might gamble on it and override the market (arbitrage). The process of trying
to outperform the market would subsequently destroy the foundation of their personal investment plan and force the
correlations they utilised backside to zero. As a result, the numerical random walk model postulates that at any given
instant, it is not possible to approximate where in the trade cycle an economy is and utilize such knowledge for
investment purposes (Fama, 1970). As noted by Ko and Lee (1991), when the RWH holds, the weak form efficiency
also holds, but the vice-versa is not true. This, then, means that the evidence supporting the random walk hypothesis
can be used to also support evidence of market efficiency; however, it must be noted that infringement of the RWH
must not necessarily be used as evidence of market inefficiency in the weak form.
It is against this background that this research looks at the behaviour of South Africa’s stock prices and,
more precisely, the independence of the South African stock market prices. The RWH says that changes in stock
prices contain similar distribution and are sovereign, so a precedent tendency of a stock price cannot be effectively
used to forecast its upcoming movements. Given that the authorities thrive now and again to improve the stock
International Business & Economics Research Journal November/December 2014 Volume 13, Number 6
Copyright by author(s); CC-BY 1243 The Clute Institute
market, particularly in the areas of efficiency as an efficient stock market attracts even international investors, it
becomes apparent to seek evidence for or against the random walk hypothesis of the JSE so as to determine whether
it is efficient in terms of behaviour and forecastability of the stock prices. Having understood how the ALSI
behaves, policymakers and regulators of the JSE may have a better understanding of the factors that may drive in the
much needed capital from both local and international investors into the financial system.
LITERATURE REVIEW
RWH is an investment theory which states that market prices trail randomly up and down, lacking any
power from precedent price movements, making it unfeasible to forecast with any exactness as to which route the
market will go at any point (Mishkin, 2010). The RWH asserts that the path a stock's price follows is random; hence,
it cannot be determined from past price information. According to Keane (1983), investors that agree with the RWH
believe they cannot surpass the market unless they incur additional risk, thus to them, fundamental or technical
analysis can be used fruitfully. Therefore, trying to make use of the theories will be a mere waste of time, which
does not yield any extra returns.
Researches have been carried out on RWH using data from both developed countries and in developing
economies with those on developed countries being prevalent. This section briefly presents various researches that
have been conducted on properties of stock market prices in different countries with their various findings and
conclusions. Although there have been numerous studies on whether stock markets are characterized by RWH or
not, it has been shown that there still exists some inconclusiveness on the matter. This section is divided into
empirical literature from developed economies, developing economies and emerging markets. It is generally
acknowledged that financial markets within countries that are more developed tend to be weak-form efficient.
Hence, share prices in developed markets are more inclined to follow a random walk process in comparison to
developing markets.
Developed Economies
In developed economies, studies on the RWH are numerous. Kleman (2002) conducted a study to
investigate market efficiency as well as the existence of random walk using indices for geographical regions of
Europe, Asia, and North America. Using monthly series for the period 1983-1997, they employed the ADF and
Phillips-Perron unit root test and Cochrane variance test. Their results showed that all the markets under study
exhibited random walk behaviour. In addition, they further employed the non-parametric runs tests that also reached
the same conclusion.
Another study was undertaken by Worthington and Higgs (2006), who examined the weak-form market
efficiency of Asian equity markets. Their findings showed that no emerging markets were characterized by random
walk and were therefore considered inefficient in the weak-form. However, developed markets, such as Hong Kong,
New Zealand and Japan, showed consistency with RWH.
Mwamba (2011) undertook a study to investigate the predictability of stock prices in the USA and the UK.
The study used daily stock prices of the S & P 500, Dow Jones and FTSE 100 for the entire year of 2012. Their
study used both parametric and non-parametric methods and the results showed that forecasts obtained from non-
parametric method were nearer to actual or observed prices as compared to results obtained from the parametric
model. Thus, it was concluded that both markets were characterized by predictable stock prices.
Developing And Emerging Market Economies
Evidence of random walk process in stock prices is weak in developing economies. Butler and Malaikah
(1992) conducted a study in which they established the performance of separate stock returns in Saudi Arabia and
Kuwait for the period 1985-1989. The serial correlation method and runs tests were employed. They concluded that
Kuwait stock market price index followed the RWH, whereas the Saudi Arabia stock market showed a considerable
departure from RWH.
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In Hungary, Dockery and Vergari (1996) investigated the RWH using the Budapest Stock Exchange (BSE)
weekly data for the period 1991-1995. They used both homoscedastic and heteroscedastic error variances of the
variance ratio tests. Their results showed evidence of the random walk.
Another study was conducted for Bahrain, Kuwait, Oman and Saudi Arabia by Dahel and Laabas (1999)
for the period 1994-1998, indicating that the Kuwait market is weak-form efficient. However, only the regression
test discarded the weak-form efficiency for the other three markets. When the sample was split into two sub-periods,
the efficiency hypothesis was not rejected for the second sub-period in two of the markets and only by a small
margin in the case of the Saudi Arabian market.
Yilmaz (2001) tested for weak-form efficiency by employing the variance-ratio-based multiple comparison
test (MCT) using daily and weekly returns data for twenty-one emerging stock markets for the period 1988-2000.
The results indicated that over time there was a move towards market efficiency for all the countries included in the
sample. The study further showed that countries with developed stock markets and features like Japan exhibited
random walk process behaviour in their stock prices. Financial crises were found to exert a negative effect on the
capacity of emerging markets to efficiently price their stocks, particularly in Mexican and East Asian economies. In
the case of Malaysia, the imposition of capital controls was found to cause stock prices to deviate from the random
walk.
In a study of selected African stock markets, Smith et al. (2002) tested the RWH using the multiple
variance test covering the period 1990-1998. Based on the obtained results, the random walk theory was discarded in
all the markets, except for South Africa whose stock market was found to follow a random walk process.
Asiri (2008) applied the Dickey Fuller unit root tests and the ARIMA model, as well as exponential
smoothing techniques, to measure performance of the Bahrain Stock Exchange (BSE). Their results showed
evidence that stock returns followed a random walk process with no drift and trend. Ten years earlier, similar results
were obtained by Khababa (1998) for the BSE.
In another research, Mobarek et al. (2008) investigated the return series on Bangladesh's Dhaka Stock
Exchange (DSE) to see if they are sovereign and resemble the RWH. They used both non-parametric and parametric
tests with daily data from 1988 to 2000. Their results showed that the returns did not trail the RWH and the
important auto-correlation coefficient at dissimilar lags rejected the weak-form efficiency.
In a study of Zimbabwe’s Stock Exchange (ZSE), Sunde and Zivanomoyo (2008) applied the ADF unit
root technique to test the random walk hypothesis using monthly data covering the period 1998-2006. Their
findings indicated that ZSE stock prices were following a non-random walk process. In other words, they found
previous stock prices to influence future prices.
Another study was conducted by Okpara (2010) who investigated whether or not Nigerian security prices
follow a random walk. The non-parametric run test and the Ljung-Box Q-statistic where employed on annual data
covering the period 1984-2006. The obtained results confirmed the Nigerian stock market to be weak-form efficient
and following a random walk process.
In Bangladesh, Sharmin and Charity (2011) conducted a study to determine the level of market efficiency
of the Dhaka Stock Exchange (DSE) for the period 1993-2011. Their study showed evidence that the DSE was not
weak-form efficient and not following the random walk.
Empirical evidence on South Africa provides mixed results and many of these studies applied traditional
methods, such as autocorrelation and unit root tests. Studies that refuted the RWH include those conducted by
Jammine and Hawkins (1974), Mabhunu (2004), and Cubbin et al. (2006) while studies undertaken by Jefferis and
Okeahalam (1999), Smith et al. (2002), Hamman et al. (2006), and Smith and Rodgers (2006) supported the RWH.
Apart from the simplistic methodological techniques applied, the reviewed studies conducted in South Africa were
undertaken in the early 2000s. It was argued that the global financial crisis may have significantly influenced the
behaviour of the financial sector, particularly stock market prices.
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METHODOLOGY AND DATA ANALYSIS
The model is developed following the random walk theory presented in the preceding section. To establish
whether the JSE follows a RWH, this research will modify the model developed by Box-Jenkins (1986) specified as
follows:
tnnttnnttt
Y...YYY
2221112211
(1)
The BJ-type time series model allows the dependent variable to be explained by its past/lagged values and
the stochastic error terms (Gujarati, 2004). The model postulates that the present value of the ALSI depends linearly
on the previous values of that variable plus a mixture of present and past values of white noise error terms (Brooks,
2008). In this paper, the model above is modified and specified as below:
tqtqttptpttt
...ALSI...ALSIALSIALSI
2212211
. (2)
where,
ALSI
t
= All Share Index in the current period t
ALSI
t-1
= ALSI of previous period
= autoregressive parameter
= moving average parameter
t
= error term
p = the number of autoregressive terms and q is the number of moving average terms
In order to check for robustness of the results, the stock indices are tested using four methodological
approaches; namely, the Augmented Dickey Fuller (ADF) test, Correlograms, the Auto-Regressive Integrated
Moving Average (ARIMA), and the Variance Ratio test (VRT). Based upon the work of Sultana and Sharmin
(2011), the ADF test is traditionally used for testing randomness of price changes. The ADF is employed first in this
study, followed by two autocorrelation tests - the graphical analysis (Correlogram) and the Q-statistic (Ljung-Box
test). The ARIMA constitutes the authors’ main methodology. In addition, the variance ratio test is also performed.
This study used monthly closing indices of the Johannesburg Stock Exchange (JSE), ALSI, covering the
period 2000:1-2011:12 and generating a total of 144 observations. The ALSI is capitalization-weighted average of
the market prices of all listed shares on the JSE. It is an index that shows the best signal of general market route as it
encompasses shares from all sectors. It is an index figure based on the current market prices of shares (JSE, 2012).
In this paper, the ALSI was used as the only variable and the series was transformed to natural logarithms.
DISCUSSION OF RESULTS
Results showed that price changes are independent, thereby leading to the conclusion that the JSE supports
the RWH. Firstly, the ADF test for unit root was employed on the ALSI and results obtained showed that the critical
values at 1% significant level with an intercept (-3.476472), trend and intercept (-4.023506), and none (-2.581233)
are greater than their respective t-statistics in absolute terms, which are -0.664657 (0.8510), -1.869282 (0.6652), and
1.903195 (0.9863), respectively. The italicized values in parentheses represent the p-values which were all above
0.05, leading to non-rejection of the null hypothesis of unit root. Hence, the JSE price indices for the covered period
followed a random walk process.
Having performed the autocorrelation tests, the results confirmed that the ALSI was characterized by price
independence. The ACF of the series died away slowly and the PACF died away after the first lag, showing that the
series was non-stationary and resembling a random walk. Also, the Q- statistics supported the fact that the JSE
follows the RWH. The probability of the sum of 36 squared estimate autocorrelations coefficient was insignificant,
leading to non-rejection of the null hypothesis of no autocorrelation. Therefore, it was concluded that price changes
in the JSE during 2000-2011 were independent of each other.
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The Box-Jenkins’s approach to time series modelling ARIMA - was also employed in the analysis. For
robustness, two ARIMA models were estimated - ARIMA (1.1.1) see Appendix 1 - and ARIMA (2.1.2) - see
Appendix 2. The model that was chosen to best suit the series was ARIMA (1, 1, 1), and the diagnostic tests
conducted on this model showed that the series under consideration exhibited trends that were random, so strongly
supported the results obtained by the other test statistics. Furthermore, the outcome of the variance ratio test
conducted under heteroscedasticity also confirmed the random walk behaviour of JSE stock indices (see Appendix
3). The findings were consistent with those of Bonga-Bonga (2012) who conducted a research on the JSE and
confirmed weak-form efficiency using the GARCH model. Our results also conform to those of Jefferis and
Okeahalam (1999), Smith et al. (2002), Hamman et al. (2006), and Smith and Rodgers (2006) who found evidence
of random walk behaviour of the JSE stock prices. However, the results run contrary to the findings obtained by
Jammine and Hawkins (1974), Mabhunu (2004), Cubbin et al. (2006) who found the JSE’s stock price indices to be
predictable.
CONCLUSIONS
In this paper, it has been established that the JSE stock prices are uncorrelated and therefore follow a
random walk. This finding has major implications to investors, policymakers, and researchers which means, inter
alia, that the inability to predict future stock prices implies that investors cannot beat the market trading rules. This
can only be possible where there is information asymmetry. Therefore, policymakers must ensure that the stock
market is continuously monitored so as to increase the level of efficiency. A number of extra measures, other than
those by the JSE in improving information dissemination, can be taken into consideration. These include improving
financial reporting procedures, embracing legislations and risk management measures as they are aimed at making
the investors better informed, well-protected and confident. As indicated before, the efficiency of the stock market is
of paramount importance to issuers of equity and portfolio investors. It has the ability to draw foreign investors and
persuade home savings, thereby increasing the movement of financial capital. Analyzing the performance of stock
market prices is also of importance to policymakers of any country since the stock market is the major index of
economic conditions. If a market trails a random walk process, it means that prices in that market provide adequate
and appropriate information and it also acts as an indicator of an efficient allocation of wealth in that country.
Moreover, since the movement of stock prices is random, investors need not worry about timing the market. In this
case, an investor’s ability to perform the market is just about luck and not analytical skill. Investors will therefore do
better with a strategy of buy-and-hold as compared to a strategy that aims at outperforming the market, as this will
not succeed. Having established that the stock market price behavior is attractive, it remains necessary to try and
ensure that the country has a stable environment that will aid in attracting investors. The focus of policymakers
should then be on continuously improving the overall macro-economic conditions so as to maintain a desirable
environment that encourages savings and investments.
AUTHOR INFORMATION
Tafadzwa Chitenderu obtained her B.Com Honours (Economics) degree, cum laude and M.Com (Economics)
degree from University of Fort Hare. Her research interests are financial markets, monetary economics and
econometric modelling. Email: chitenderuf@gmail.com.
Dr. Andrew Maredza is a lecturer in the School of Economic & Decision Sciences at North-West University,
South Africa. His research focus is efficiency and productivity analysis of the banking sector and its linkage with the
rest of the economy, financial sustainability and financial inclusion. His other interests include financial markets,
monetary economics, and applied panel and time series econometrics. He published several papers in international
journals. E-mail (Corresponding Author): Andrew.Maredza@nwu.ac.za.
Kin Sibanda is a Ph.D. (Economics) student in the School of Business and Enterprise, University of Fort Hare,
South Africa. E-mail: keith08.kin@gmail.com.
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APPENDIX 1
Regression Results For ARIMA (1, 1, 1)
Variable
Coefficient
t-Statistic
Prob.
C
0.004137
1.967751
0.0511
AR(1)
-0.944977
-49.17726
0.0000
MA(1)
0.986428
114.1207
0.0000
R-squared
0.058296
Mean dependent var
0.004314
Adjusted R-squared
0.044747
S.D. dependent var
0.025107
S.E. of regression
0.024538
Akaike info criterion
Schwarz
-4.556249
Sum squared resid
0.083697
Schwarz
criterion
-4.493802
Log likelihood
326.4937
Hannan-Quinn criter.
-4.530873
F-statistic
4.302402
Durbin-Watson stat
1.980862
Prob(F-statistic)
0.015383
Inverted AR Roots
-.94
Inverted MA Roots
-.99
APPENDIX 2
Regression Results For ARIMA (2, 1, 2)
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
0.003969
0.002033
1.952521
0.0529
AR(1)
-0.033246
0.029280
-1.135451
0.2582
AR(2)
-0.904056
0.028411
-31.82078
0.0000
MA(1)
-0.033675
0.023254
-1.448150
0.1499
MA(2)
0.989733
0.010871
91.04242
0.0000
R-squared
0.109830
Mean dependent var
0.004226
Adjusted R-squared
0.083648
S.D. dependent var
0.025174
S.E. of regression
0.024098
Akaike info criterion
-4.578532
Sum squared resid
0.078979
Schwarz criterion
-4.473966
Log likelihood
327.7865
Hannan-Quinn criter.
-4.536040
F-statistic
4.194934
Durbin-Watson stat
1.889112
Prob(F-statistic)
0.003104
Inverted AR Roots
-.02+.95i
-.02-.95i
Inverted MA Roots
.02+.99i
.02-.99i
APPENDIX 3
Regression Results For Variance Ratio Test
Joint Tests
Value
df
Probability
Max |z| (at period 16)*
0.505235
143
0.9777
Individual Tests
Period
Var. Ratio
Std. Error
z-Statistic
Probability
2
0.991701
0.105763
-0.078465
0.9375
4
1.058904
0.189232
0.311278
0.7556
8
1.140317
0.283120
0.495611
0.6202
16
1.206503
0.408727
0.505235
0.6134
*Probability approximation using studentized maximum modulus with parameter value 4 and infinite degrees of freedom.
Test Details (Mean = 0.0040875771074)
Period
Variance
Var. Ratio
Obs.
1
0.00063
--
143
2
0.00063
0.99170
142
4
0.00067
1.05890
140
8
0.00072
1.14032
136
16
0.00076
1.20650
128
International Business & Economics Research Journal November/December 2014 Volume 13, Number 6
Copyright by author(s); CC-BY 1250 The Clute Institute
NOTES
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