INTRODUCTION
Financial time series exhibit some apparently puzzling empirical (statistical)
regularities. These are known as stylized facts. Stylized facts are empirical
observations that are so consistent and have been made in so many contexts (e.g.,
across a wide range of instruments, markets and time periods) that they are
accepted as truths, to which theories must fit (Cont, 2001,
2008). Thus stylized facts are universal regularities,
independent of time, place and specific compositional details. Stylized facts
are obtained by taking a common denominator among the properties observed in
studies of different markets and instruments. Due to their generality, stylized
facts are often qualitative and not quite precise enough to distinguish among
different parametric models (Coolen, 2004; Ding
et al., 1993).
The universality of the regularities indicates that the stylized facts can
be thought to have some common origin. Some of the commonly occurring stylized
facts are given below (Brock and de Lima, 1995; Campbell
et al., 1996):
• 
Absence of autocorrelation(linear) autocorrelations of asset
returns are not at all significant; analyses of high frequency data show
that correlation times can be as short as a few minutes in highly traded
stocks or indices 
• 
Slow decay of autocorrelation in absolute returnsthe autocorrelation
function of absolute returns decays slowly as a function of the time lag,
approximately as a power law; this fact is often interpreted as a sign of
longrange dependence 
• 
Fat tailsthe (unconditional) distribution of returns appears
to display a powerlaw or paretolike tail, with a tail index which is finite
and greater than two (positive excess kurtosis). The distributions are approximately
bellshaped but assign more than normal probability to events in the center
(more peaked) and at the extremes (exhibit heavy tails) 
• 
Excess volatilityseveral empirical studies indicate that
it is not easy to justify the observed level of variability in asset returns
by taking into account variations in the fundamental economic variables.
The fact that large (positive or negative) returns occur cannot always be
explained by the arrival of new information on the market 
• 
Volatility clusteringdifferent measures of volatility display
a positive autocorrelation over several days, which quantify the fact that
high and low volatility events tend to cluster in time; in other words,
periods of intense fluctuations and mild fluctuations tend to cluster together 
• 
Volatility persistence (long memory)there is dependence between
stock market returns at different times; technically, the volatility has
slowly decaying autocorrelations and there is nonlinear dependence 
The stylized facts show that the normal distribution is inadequate in describing
the distribution of the financial time series. It is necessary to invoke a distribution
which assigns more probability to the tails of the distribution and thus, immediately
alludes to powerlaw type distributions (a strong signature of complexity).
The appearance of these facts in apparently unrelated and diverse systems suggests
the existence of some common underlying mechanism responsible for the appearance
of these facts (collective behaviour and critical phenomena typical of complex
systems). Thus the existence of stylized facts in financial markets reinforces
the idea of the latter as complex systems (Borland, 2005;
Johnson et al., 2003). Furthermore, the dependence properties of
asset returns and the phenomenon of volatility clustering have led to the development
of stochastic models in financethe GARCH School of models is used essentially
to model these phenomena.
In this communication, we address the issue of stylized facts by using data from the Indian stock market. While descriptive statistics are studied briefly, a great deal of emphasis is put on the study of volatility by applying several models from the GARCH school like GARCH, EGARCH, TARCH and Asymmetric Component GARCH model. We also examine the probable existence of long memory in financial time series. This is an empirical investigation using statistical tools to show the relevance of the stylized facts and other related issues.
This study is unique for the following reasons:
• 
Several authors have carried out studies on stylized facts,
but there has been little study using data from the Indian financial market 
• 
This is probably the first systematic and exhaustive study
to verify the existence of stylized facts for the Indian financial market.
The Indian financial market is an emerging one (the Indian economy opening
up essentially from the midnineties upwards) and this study captures the
entire spectrum from 1997 to 2009 (when the study was concluded) by using
data from the BSE SENSEX. Existence of several stylized facts are corroborated
as discussed in the study 
• 
His study is enriched by the application of several sophisticated
analytical techniques including extensive usage of models from the GARCH
school as well as their comparisons, the HannanQuinn criterion, the different
tests for long memory etc. 
MATERIALS AND METHODS
The study presented below considers SENSEX as a proxy for the Indian market. The period under consideration is from 1997 to March 2009. The daily return is used for all the analyses to make the series unit root free. The daily return is defined as:
where, P_{t} is the closing price on day t. The analysis considers the raw return series (with sign) for the study of descriptive statistics, volatility and related topics while the absolute return series (without sign) is used for the study of long memory.
The next few subsections introduce the concepts and tools necessary to carry out the subsequent analysis concerning volatility and long memory and obtain the results.
Checking for unit root: Once the (raw) daily return series is constructed, the Augmented Dickey Fuller test and the Phillips Perron test are performed to check the presence of unit root.
Computation of the descriptive statistics: The basic understanding about the raw return series can be found from the descriptive statistics. Two most important statistical measures for the series are the kurtosis and the Jarque Bera statistic. Kurtosis of a series is a measure of the flatness of the distribution. The Jarque Bera test statistic shows whether the series follows the normal distribution.
Study of volatility using models from the GARCH school: To understand the embedded volatility in the series, models from the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) school are employed because of their ability to capture the conditional volatility and also the asymmetry present in it, if any. It is possible to use GARCH models when the series is nonnormal. The models that are used in this analysis are explained briefly.
GARCH (1,1): This is the simplest form of GARCH process. First used
by Engle (1982) and later modified by Bollerslev
(1986), it has a first order autoregressive GARCH term and a first order
moving average ARCH term and hence the nomenclature (1,1). This model is specified
by a mean equation and a variance equation.
Where:
X_{t}′ 
= 
Exogenous variable 
σ^{2} 
= 
The conditional variance 
ω 
= 
Constant 
ε_{t1}^{2} 
= 
News about volatility from the previous period, measured as the squared
residual from the previous periodthis is the ARCH term 
σ_{t1}^{2} 
= 
Previous period's forecast variancethis is the GARCH term 
GARCH (q, p): This is a more generalized form and denotes higher order
GARCH model with q as the GARCH term and p as the ARCH term.
The conditional variance for a GARCH (p, q) process is specified as:
ε^{2}_{t1} is the ARCH term, denoting news about volatility from the previous period and σ^{2}_{t1} is the GARCH term, denoting forecast variance of the previous period.
Models with asymmetric information shock: However, the above models
cannot capture the phenomena of asymmetry of volatile response of the market
towards positive and negative announcements (Engle and Ng,
1993). The asymmetry is characterized by the fact that an unexpected bad
news or a negative shock has a greater impact on the market than the effect
of a good news or positive shock of the same magnitude (gain/loss asymmetry
in the language of stylized facts). The Exponential GARCH (EGARCH) and Threshold
GARCH (TGARCH) models allow for such possibility.
EGARCH: The EGARCH (Exponential GARCH) model was proposed by Nelson
(1991). Here the conditional variance is expressed as a logarithmic term,
which shows an exponential leverage effect. If the hypothesis that γ_{k}<0
is true, a leverage effect is said to be present. Presence of asymmetry is confirmed
if γ_{k}≠0.
TARCH: The TARCH (Threshold GARCH) model is also used to study the asymmetric
nature of the volatility (Zakoian, 1994; Glosten
et al., 1993). The TGARCH model is often called GJR model after Glosten,
Jagannathan and Runkle.
In this model, good news, ε_{ti}>0 has an impact α_{i} while bad news ε_{ti}<0, has an impact α_{i} + γ_{i}. If γ_{i}>0, bad news increases volatility and there exists a leverage effect. If γ_{i}≠0 then the impact is asymmetric. This model takes into account a quadratic leverage effect term, rather than an exponential one.
Asymmetric component GARCH model: This model can capture both short run and long run volatility and also the asymmetry in the volatility. This model allows for a mean reversion to a varying level.
Study of long memory: The study now tries to unravel whether there is any long term dependence or long memory in the market. A random process is said to possess longmemory if it has an autocorrelation function which is not integrable. This implies that the autocorrelation function decays asymptotically as a function of the time lag. Such a strong autocorrelation implies a high degree of longterm predictability, meaning thereby that present stock prices can be significantly affected by prices from the distant past. If long memory is present in the system, a shock once propagated in the system continues to affect future outcomes even after significant amount of lags. This increases risk in the market and hence it has a profound effect on international trade and investment policies of different countries.
The most widely used test to investigate the nature and extent of longterm
memory in a time series, is the Rescaled Range (R/S) analysis. This test was
first used by Hurst (1951) to test for long memory in
the pattern of flooding by the river Nile. It was later modified by Mandelbrot
(1972, 1975), Mandelbrot and
Taqqu (1979) and Mandelbrot and Wallis (1968, 1969).
The basic idea underlying the classical R/S test is to compare the minimum and maximum values of running sums of deviations from the sample mean, renormalized by the sample standard deviation. For longmemory processes the deviations are larger than for processes which do not have longmemory.
Consider a time series y_{t} for t = 1, 2,.........T. The classical R/S statistic is defined as:
where y_{1}…..y_{T} are sample observations and is
the sample mean given by:
The term within bracket shows the range of partial sum of deviations and it is rescaled by dividing with s_{T}, the standard deviation which is defined as:
The classical R/S test has been proven to be too weak, i.e., it tends to indicate
that a time series has long memory when, it does not really have so. Lo
(1991) criticized the R/S statistic for being unable to distinguish between
short term and long term dependence. In order to remove this shortcoming Lo
modified the classical R/S statistic in such a way that its statistical behaviour
is invariant over a general class of short memory processes, but deviates for
long memory ones. This is done by replacing the denominator which is the square
root of a consistent estimator of the variance of the partial sum. The estimator
involves not only sums of squared deviations of y_{i}, but also its
weighted autocovariances up to lag q.
The present study uses both these models (R/S statistic and modified R/S statistic) to test for long term memory.
The study on long memory concludes with an examination of whether the series is characterized by fractional integration. A fractionally integrated process is neither a perfectly short memory I(0) process, nor a I(1) but falls between these two processes, thus, possessing weak correlation between observations situated far apart in the series. To test for fractional integration, the GPH test is used.
RESULTS
The total sample size considered in this investigation was T = 2777.
The results obtained could be classified under three categories:
• 
Results from the unit root tests and descriptive statistics 
• 
Results from the study of volatility 
• 
Results from the tests for long memory 
Unit root tests: The filtered raw return series (ARGARCH filtered) were checked for possible presence of unit root. Augmented Dickey Fuller test and Philips Perron tests were carried out to check the stationarity of the raw return time series. The results were summarized in Table 1.
The null and alternative hypotheses were as follows:
• 
H_{0} : Unit root is present 
• 
H_{1} : Unit root is not present 
The tests were carried out at 1% level of significance.
Values of tstatistic were computed and the pvalues were determined. The extremely low pvalues indicated that there was sufficient statistical evidence to reject the Null hypothesis. In other words, it could be inferred that the raw return series was free from unit root.
Descriptive statistics: The results for the descriptive statistics were displayed in Table 2 where values of the mean, median, maximum and minimum values, standard deviation, skewness and kurtosis were shown. The kurtosis (value>3) for the raw return series revealed that the series had a fat tail and that it was leptokurtic (more peaked than normal). The series was also found to be negatively skewed.
JarqueBera test: The above test was carried out under the following Null and Alternative hypotheses:
• 
H_{0}: The series follows a normal distribution 
• 
H_{1}: The series is nonnormal 
The computed value of the JB test statistic was 1252.12 and the pvalue was found to be 0. The extremely low pvalue indicated that there was sufficient statistical evidence to reject the null hypothesis and conclude that the series was nonnormal.
Table 1: 
Result of unit root tests 

*Indicates significance at 1% level (1% critical value = 3.961) 
Table 2: 
Descriptive statistics for raw SENSEX daily return 

Results from application of the GARCH models: In this analysis, first a GARCH model was employed to test for the presence of volatility. Then TARCH and EGARCH models were employed and the best fit model among the two, based on the lowest value of the selection criterion (HannanQuinn for present study) was analyzed to examine possible presence of asymmetry and leverage effect in the series. Lastly, the asymmetric component GARCH model was used to check for the mean reverting property of the series.
The results from the GARCH School of models were summarized below in Table 3.
The lags were chosen according to the HannanQuinn criterion. As the series
had a large number of observations, HannanQuinn criterion was a better selection
criterion than the Schwartz or Akaike criterion (Shittu
and Asemota, 2009). From Table 3, it could be seen that
news about volatility from past periods had a significant effect on the current
period’s volatility. The coefficients of the GARCH term were significant
at 10% level in all lags except the third. So, it could be said that in the
first, second, fourth and fifth lags, the current period’s volatility was
dependent on the past period’s volatility.
Among the TARCH and EGARCH models, EGARCH was the better model because the HannanQuinn criterion was lower for EGARCH and the term indicating asymmetry was significantly negative. Hence it could be said that the series showed significant amount of asymmetry. Finally, the component GARCH model showed that the coefficient of the term indicating long run persistence was 0.99 and therefore there existed strong volatility persistence.
Finally, the ARCHLM test was employed to examine whether any ARCH effect remained in the residual. Ignoring the presence of ARCH effect might lead to inefficient results. However, upto 5 lags were taken and there was no evidence of any remaining ARCH effect.
Table 3: 
Summary of results from the models of the GARCH School for
Raw SENSEX daily return series 

When the raw returns were plotted against time, another distinct feature of the SENSEX return series could be identified as shown in Fig. 1. The series showed sustained periods of high and low volatilities. In other words, volatility clustering could be said to be present in the series.
Results from the tests of long memory: Absolute return was considered for the long memory tests because the raw returns tended to have a mean very close to zero and hence might indicate the absence of long memory when it was actually present.
The R/S and modified R/S tests were done using the following Null and Alternative hypotheses:
• 
H_{0}: There is no long term dependence in the return
series 
• 
H_{1}: There exists long term dependence in the return
series 

Fig. 1: 
Plot of raw daily return on SENSEX against time for the period
extending from 1997 to 2009 

Fig. 2: 
Plot of Autocorrelation Function (ACF) for absolute daily
return on SENSEX for the period extending from 1997 to 2009 
Table 4: 
Results from tests of long memory and fractional integration
for absolute daily return series 

**indicates significance at 1% level; ^{#}GPH
test critical value = 3.7885 
The tests were carried out at the 1% level of significance.
The following values were obtained:
• 

Values in both cases were greater than the corresponding critical values at the 1% level of significance which indicated that there was sufficient statistical evidence to reject the Null hypothesis and conclude that there was significant presence of long term memory in the absolute return series.
The d value from the GPH test suggested that the series was fractionally integrated; all these results were summarized in Table 4.
The long memory property could also be illustrated with help of a diagram. Figure 2 showed the autocorrelation function upto 200 lags. As could be seen from the figure, the autocorrelations of the absolute returns were highly persistent and although slowly decaying, clearly indicated presence of long memory in the SENSEX return series.
DISCUSSION
The stylized facts occurring in financial markets have been reviewed by different
authors (Cont, 2001; Coolen, 2004).
Several different stylized facts like fat tails, volatility clustering, volatility
persistence and long memory have been discussed (Beran, 1994;
Ding et al., 1993; Greene
and Fielitz, 1977). Attempts have been made to interpret the stylized facts
as emergent properties of a complex system, viz., the financial market (Rickles,
2008). Stylized facts including the leverage effect have been discussed
by Bouchaud and Potters (2001), while they have been
related to minority games by Challet et al. (2001).
The dynamics of financial markets including stylized facts have been discussed
by Reimann and Tupak (2010).
Long term dependence (or long memory) has also been a topic of intense research.
Greene and Fielitz (1977) took 200 daily stock return
series and used the R/S analysis to show the presence of long term memory in
many of them. They concluded that in the presence of long term dependence, martingale
process does not hold. Ding et al. (1993) discovered
long memory stochastic volatility in stock returns. Baillie
et al. (1996) in their seminal paper introduced the Fractionally
Integrated Exponential GARCH model (FIEGARCH) to capture the long run dependence
in the US stock market.
The objective of the present study was to highlight the importance of the stylized facts and provide empirical evidence for their existence by using data from the Indian stock market and subjecting them to statistical analysis. The raw daily return was found to be a nonnormal distribution having fat tail. Volatility clustering, volatility persistence as well as asymmetry in volatility were all observed in the raw return. Impact of past news was also observed. Further, the absolute return series displayed long memory features.
Thus, the foregoing analysis showed ample justification for the existence of the stylized facts. The stylized facts are intimately related to the complexity argument; hence justification of these facts indicates that a financial market may be regarded as a complex system.
ACKNOWLEDGMENTS
The authors acknowledge the support of West Bengal University of Technology (WBUT) under Grant at WBUT/UGC/F.5216/2006 (HRP), January 2007.