Some Stylized Facts of Short-Term Stock Prices of Selected Nigerian Banks

This paper examines presence of some stylized facts of short-term stock prices in the banking sector of the Nigerian Stock Market (NSM). Non-normality, lack of autocorrelation in the returns at first lag and significant positive autocorrelation in higher magnitude returns, widely studied in other markets, are investigated using daily closing stock prices of the four major Nigerian banks (Access, First, Guaranty Trust and United Bank for Africa (UBA)), from 2001 to 2013; encompassing periods of different financial scenarios. Jarque-Bera (JB), Doonik-Hansen, Kolmogrov-Smirnov and Ljung-Box (Q) test statistics are applied. Our findings reveal that the four banks stocks behave slightly different, but generally possess the stylized facts found in other markets. Observed is that, while the distributions of the returns for two of these banks (First and UBA) are approximately symmetric and leptokurtic; those of Access and Guaranty Trust banks are significantly non-symmetric and leptokurtic, thus non-normally distributed. Also established is that, while autocorrelation functions of daily returns are either negative or zero, those of both absolute returns and the squared returns are mostly positive. The autocorrelations of absolute returns are found to be predominantly positive and more persistent than those of the squared returns; indicating volatility clustering. Consequently, we conclude that the short-term stock prices of these banks behave like those of other markets. Some implications of the results for financial investment and stock market behaviour in the banking sector of NSM are discussed.


Introduction
Following the 2004 bank restructuring, related financial reforms in Nigeria, and the 2007-09 global financial crisis, the Nigerian financial market experienced near bank failures which were prevented by proactive interventions by the Central Bank of Nigeria (CBN).There has also been a dramatic fall (more than 60%) in the market capitalisation of the Nigerian Stock Market (NSM) between 2008 and 2009.These fluctuations make the study of volatility in NSM returns topical, especially in light of financial reforms, investment and risk management [1], financial policy [2], economic development, and stock market development, linked to GDP growth, competitiveness, market microstructure, and macroeconomic policy setting [3]- [10].
Despite the ubiquity of volatility studies such as these in Nigeria and elsewhere, there is a dearth of studies which are focused specifically on the link between stylized facts of bank stock returns and some aspects of volatility.This link is particularly important because the distributions of returns underpin volatility and financial modelling of investment assets.This paper fills this empirical gap and anticipates related future work in systematic stock market characterisation and development (SSMCD) [11] [12], for example, argue that Nigerian policy makers need to understand the volatilities in the financial system at overall market, sector-and company-specific levels, in order to gear monetary, macroeconomic and fiscal policies towards enhancement of economic performance, GDP growth, competitiveness and asset price stability; see also Aliyu [4].Therefore, the focal point of SSMCD work to which this study belongs is the need to model the underlying distributions of stock returns in key sectors of the NSM, for example, and their implications for financial policy.The choice of the banking sector in this paper is because it is by far the most dominant sector of the NSM.Further rationale for the paper and its focus on Nigeria as a developing country is provided later in this introduction.
We note that even though the paper is based on four main Nigerian banks, the results would seem to provide a good insight into the nature of stylized facts of bank returns generally.This is because bank stocks typically correlate significantly since they use interbank lending rates in their investment decisions and are subject to similar CBN financial and corporate governance regulations.Moreover, the selected banks have been in the market for fairly long periods and generally monitor each other's financial innovations, for which reason their performance would indicate the general pulse of the banking sector of the NSM.
Based on insights from this paper, future work will explore possible differences in stylized facts of the entire sector.
With particular reference to stylized facts, the purpose of modelling financial data is to approximate the behaviour of the unobservable data generating process (DGP) that determines observed stock prices [13].Thus, the process of examining how fit this approximation is to the data leads to identifying the "stylised facts" of stock returns.A stylised fact is a statistical property that is expected to be found in any series of observed stock prices or returns [14] [15].Hence, if a model is set to approximate the behaviour of asset returns, then it should as much possible capture these facts.
Several studies have investigated the dynamic nature of major stock markets (at both developed and emerging level), with the discovery of some number of stylized facts.It has been observed that the empirical distributions of stock returns appear to be excessively leptokurtic, thereby making the returns non-normally distributed (see: [16] [17] [18] [19]).Also noticed is that short-term stock returns exhibit volatility clustering; this according to Mandelbrot [16] implies that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes".The characteristic that has successfully been modelled using ARCH-GARCH type models (see: [20] [21] [22] [23]).Another stylized fact is that changes in stock prices tend to be negatively related to changes in volatility [24] [25] [26]; better put, it means negative returns (fall in prices) are followed by greater volatility than do positive returns (increase in prices) of the same magnitude.In some contexts this is referred to as "Asymmetric effect".First-order autocorrelation of stock index was also identified to be negatively related to the level of volatility [27] [28] [29].
Meanwhile, most of the works thus far have focused on developed financial markets with few exceptions (see [30]).For investors seeking opportunities in emerging markets however, a better understanding of the stock return behaviour is of great interest.One such promising market that has received less than expected attention (as far as we know) is the Nigerian Stock Market (NSM).The NSM with its position and potentials in African and World markets has progressed tremendously in recent years, especially after surviving the 2007-2009 financial crises.Consequently, a better understanding of its stock market should be of interest to both local and foreign investors.Specifically, it would be useful to examine whether stylized facts, which are present in the developed markets also characterize the behaviour of the NSM.More importantly, returns on the NSM, like many other emerging markets have been found to have low (in some years even negative) correlation with the returns of major developed markets, providing opportunities for substantial risk reduction through diversification to international investors [31].
Also noted is that the daily returns from most emerging markets (including Nigeria) are found to depart from normal distribution [32].Thus, understanding the stochastic behaviour of the NSM is vital for both international as well as local investors and may point out interesting similarities and differences between the NSM and developed stock markets.Thus far, there has been no empirical study that methodically investigates the short-term dynamics of stock returns in the NSM despite the potential importance of this market to investors.This research investigates if the stylized facts which have been associated with asset returns behaviour in most climes, especially the developed markets and some emerging markets, also characterise the behaviour of stocks in Nigeria.
Of particular interest are the three major facts identified by Taylor [33] and a few other researches, which are more peculiar to daily returns, namely: 1) the distribution of returns is non-normal; 2) correlations between the magnitudes of returns over short-time periods are positive and statistically significant; 3) and lack of correlation between the returns of different days.To fulfil the set targets, four major and investing banks have been chosen for our study.
The main objectives of the paper are: 1) to explore the distributional characteristics of the stylized facts of bank stock returns, namely normality tests, serial correlations in stock returns, and serial correlations in volatility; and 2) to link the results to possible differences in banks' returns stability and/or performance in periods of bank reforms and global financial crisis.These basic stylised facts will foreshadow detailed distributional characterisations envisaged in future work, which will look at the whole banking sector and therefore enable the researchers to identify possible differences in underlying returns distributions across banks with different banking mandates and business models.The wider SSMCD line of work to which the paper belongs is described in more detail in the following literature review.

Literature Review
The focus of this literature review is to delineate current knowledge on stylized facts, clarify the aspects of key interest in this paper, and the gaps in knowledge they relate to.It has been observed that the empirical distribution of stock returns appears to be excessively leptokurtic, thereby making the returns to be described via non-normal distribution (see: [16] [17] [18] [19] [34]).Particularly, the empirical distributions of most daily stock return series tend to be leptokurtic and skewed than would be of a normal distribution ( [14] [35] [36]).That is, daily return series are known for having heavy tails and highly peaked centre.
Fama [17] by examining the daily returns data spanning from 1957 to1962 on 30 stocks obtained from DJIA (Dow Jones Industrial Average) concluded that daily stock returns are undoubtedly leptokurtic; with the same view shared by Pagan [35] and Cont.[15].The strength of the leptokurtosis in monthly series was rather found to be weaker [14] [15] [36].
It is, however, noted that the lack of significant autocorrelation in the daily returns does not make the series to be independent.This is simply because of the presence of significant autocorrelation in the series of absolute and squared returns for several lags, especially for most daily stock returns.Ding et al. [37] found positive autocorrelation in the returns of both squares and absolute returns for lags running to thousands; the situation that is referred to as Volatility Clustering.Specifically, short-term interval stock returns (e.g.intra-daily, daily and weekly) exhibit volatility clustering; according to Mandelbrot [16], this implies that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes".Taylor [33] presents the observation made by Mandelbrot [16], which was confirmed by Fama [17] that compared to small absolute returns, large absolute returns are likely to be followed by large absolute returns.The finding was however not alone satisfactory enough in predicting the direction of price changes.
It is claimed that daily returns for liquid stocks exhibit non-significant (linear) autocorrelation at various lags [14] [15] [35] [37]; Aggarwal et al., (38) however notice the significant autocorrelation in daily returns of some markets (whose stocks are known to be illiquid) in Latin America and Asia between 1985 and 1995.Cont.[15] and Taylor [39] found that intra-daily returns of liquid stocks for periods more than 20 min.do not exhibit significant autocorrelations, whereas those of shorter periods reflected negative autocorrelation.This according to them might be due to the effect of market microstructure.It is believed that while very little autocorrelation is present in the series of returns { } t r itself, substantially more autocorrelation is found in the series of absolute returns { } t r .The autocorrelation of absolute returns are always positive at a lag of one day and positive dependence continues to be found for several lags [33].It is also noted that the dependency level displayed by { } 2 t r is generally less in degree compared to that which is attributed to absolute returns.
Taylor [33], however, concludes that all the aforementioned features apply to any long series of the daily returns for financial assets that are traded frequently.
In general, the distributions of daily returns especially are found to be approximately symmetric, but with high kurtoses, fat tails, and peaked centres compared to the normal distribution.Also, while autocorrelations of daily returns are close to zero (or mostly negative), those of absolute (and probably squared) returns are positive for several lags, which substantially indicates more linear dependence in the autocorrelations of the latter than do the former.This level of dependence in the absolute as well as the squared returns, according to Taylor [14], is responsible for slow changes in the returns volatility.Furthermore, according to many studies, periods when the returns are relatively more variable are followed by relatively high expected absolute returns, whereas those periods with relatively less variable returns result into having relatively low expected absolute returns.
In the light of the above and in line with the objectives of this study, this paper considers the foundational distributional characteristics of the stylized facts of the bank stock returns, namely normality tests, serial correlations in stock returns, and serial correlations in volatility which complement the directions.It also attempts to relate to results to share price behaviours of the banks in different periods of financial reforms and global financial crisis.These basic stylised facts will inform detailed distributional characterisation envisaged in future work which will look at the whole banking sector and therefore enable the researchers to identify possible differences in underlying returns distributions across banks with different banking mandates and business models.
The paper builds on the results obtained by Omar [11] and Ezepue and Omar [12] which use the All-Share Market Index data from the NSM for periods encompassing the 2004 reforms and the 2007 global financial crisis.The results of which should have important implications for investment analysis, financial policy, stock market development, and financial engineering within the banking sector and the NSM generally [11] [12].

Methodology
This section presents brief account on the data source and our choice of the four banks; and the techniques used in achieving the objective of this research.Firstly, a brief discussion on how returns are being generated is provided and secondly, various statistical tests and methods adopted in achieving the aim of this research are presented.

Data Presentation
The data used in this research were obtained from the Cash craft's website (http://www.cashcraft.com/), a subsidiary of the Nigerian stock Exchange (NSE), are daily closing stock prices for four banks frequently trading with NSE, ranging from 2 nd January, 2001 to 31 st December, 2013, making up a total of 3201 (Access), 3200 (First), 3203 (Guaranty) and 3202 (UBA) observations.Our choice of these banks was informed by the fact that they are among the leading trading banks with very high market capitalizations, and are banks which in spite of various difficulties experienced in the Nigerian financial system have maintained their original name.Also, while two of these banks, Access and UBA, have at one time merged with other banks, First Bank and Guaranty Trust never merged with any other banks.
Meanwhile, Tables 1-4 below present sample data on the first 10 trading days across the four banks-Access, First, Guaranty and UBA respectively.
Having obtained the data continuously compounded (or geometric or log) returns, absolute, mean-adjusted and squared returns are then computed from the closing prices of the four banks and used in the analyses.
The returns are generated from the stock prices using the formula: where t R , is the simple return at time t, derived as: P is the stock closing price at time t and r is called geometric or continuously compounded return; used for analysing stock data, t p is the log of price at time "t" and 1 t p − is the log of price at time " 1 t − " period preceding time "t".

Mean-Adjusted Returns
Suppose the mean returns is obtained as where T is the total number of observations.
Then the mean-adjusted return is obtained as: From (3.2), we obtain absolute returns as: t r  , which is the absolute value of the mean-adjusted returns.
Squaring both sides of (3.2), we have the squared returns: ( ) Having obtained the (log) returns from the closing stock price of each of the banks and derived the mean-adjusted returns, absolute and squared returns from the (log) returns; we on proceed to discussing the different tests to be implemented in addressing the set objectives for the study.

Remark
Note that the "log()" function used in (3.1) is logarithm to base "e", mostly referenced as "ln()" function; which is called "natural logarithm" function.

Normality Tests
Mandelbrot [16] considered two related issues as regards the returns distribution in finance.These are:

Tests based on Moments
The most widely used test due to Jarque and Bera [40] and Bera and Jarque [41], relies on the fact that for a normal distribution both the skewness ( ŝ ) and excess kurtosis ( ˆ3 k − ) should be equal to zero.Skewness The sample coefficient of skewness, which is a measure of symmetry of the return series, is computed using the formula:

Kurtosis
The sample coefficient of kurtosis, which determines the tail thickness of the return's distribution, is obtained as: Jarque and Bera [40] propose the following asymptotic distributions for the two moments: ( ) Subject to this, they propose the JB test statistic defined as: For testing normality, where Ŝ is the sample skewness, K the sample kurtosis and T the sample size.Null hypothesis (H 0 ) of normality is rejected in favour of Alternative hypothesis (H where 1 z and 2 z are respectively denoted as the finite-sample skewness and kurtosis, and computed as: With the following correction factors for finite sample: 6 Under the hypotheses: With θ assumed to be known.However, when θ is unknown, Lilliefors test is used.According to Kolmogorov [43], the simplest measure of normality is the largest distance between the two functions Practically, this test is implemented as follows: 1) The data set is sorted in ascending order, with the new sample denoted as We then generate the assumed cdf ( ) If the referenced distribution is normal, then we assume mean µ and standard deviation σ are known 3) Finally, KS test statistic is computed as: The major limitation of KS however is that the mean and standard deviation are unknown, and have to be estimated from the sample, thereby leading to sampling errors.To adjust for this, the modified Lilliefors L KS test is applied.

Lilliefors Test for Normality
The Lilliefors test [44] tests the following null and alternative hypotheses for data sets up to n (say, n = 1000) H 0 : The data have been drawn from a normal distribution H 1 : The data have been drawn from a non-normal distribution Test Procedures 1) Compute the sample mean, and the sample standard deviation, ( ) 3) Compute the Lilliefors test statistic T as follows: where T is the supremum over all x, of the absolute value of the difference 4) Reject H 0 and accept H 1 at the α significant level if T exceeds the critical value for the test which can be obtained from Table A14 1 [44].

Testing Dependency in Stock
Time Dependence in Returns is very crucial in finance for the following reasons: 1) If the returns distribution is time dependent, then the statistical tests using unconditional statistics and inferences drawn from then could be misleading; 2) If time dependence can be fully exploited, one stands the better chance of making reliable forecasts of the level, volatility and higher moments of the re- i ρ is non-zero for the first p-lag auto-correlation.Where correlation ( J ρ ) of order j is computed using the formula: The appropriate test statistic subject to H 0 is Ljung-Box (Q) statistic, given as: ( )

Testing for Serial Correlation in Volatility
To test for existence of time dependence in volatility requires obtaining a time-varying measure of volatility.The two possible measures here are: 1) use of mean-adjusted squared returns or 2) use of absolute returns.

Assumptions
where µ , is the constant mean, t ε is the mean adjusted returns, t σ repre- sents the time varying volatility and ( ) innovation.Given the information set, 1 t F − at time, 1 t − , the expected value of the squared residuals is given as: t z χ .Hence, 2 t ε , can be viewed as proxy for the volatility at time t.
Alternatively, omitting µ for the moment, we have that,

Results and Discussion
This section presents some results of the analyses carried out in this research.
We divide the presentation into three aspects, namely preliminary results on exploratory data analyses, quantitative results, and implications for theory and practice.

Preliminary Results: Exploratory Data Analysis
In this section we present exploratory results in terms of graphs, for instance,         the returns volatility.Though there was significant rise in price recorded around February, 2006, there was no relative effect on the volatility level for the bank.
These contrasting effects of both drop and rise in price level on the returns volatility may be likened to the asymmetric behaviour of the returns widely established in most researches as an important stylized fact.Apart from the two periods one could see that there is relatively low spikes level in the returns volatility across the years.
For Access bank (Figure 3           In general, from these results it could be deduced that uncommon events and or reforms in the banking sector often affect the returns volatility.With these findings, one could say that there is significant level of dependence or serial auto-correlation in the returns of these banks.However, findings based on this exploratory approach are not always full reflection of the reality; it provides a pictorial reflection which requires further confirmatory statistical tests.

Normality Test Results
In this section the normality plots for both returns and mean-adjusted returns, as well as the q-plots for each of the banks are presented.While Figures A1-A4 From the plot it could be observed that virtually all the graphs display highly                      C6 (see Appendix C) explore the extent of serial auto correlation in returns and in volatility.The plots have auto-correlations for daily returns, absolute returns and mean-adjusted returns for each of the banks.While that of daily returns is to ascertain how auto-correlated each return series is, those of absolute and mean-adjusted returns determine the extent of dependence in volatility.For the four banks, while auto-correlations for both absolute and mean-adjusted returns are positive for substantial number of lags, those of the returns exhibit negative or near zero autocorrelations.It could also be noticed that Access bank has auto-correlation in absolute returns for several lags the same time the positive correlations persist and refuse to die off for many lags, indicating the probability of the returns for this bank to have long memory; which is another very important stylized fact.The presence of significant positive autocorrelation in absolute returns also a sign of existence of volatility clustering and persistence in the returns for these banks.
These findings, as reported by many researchers across different markets, for various stock returns, note that the ACF of absolute (or squared) returns remains positive and decays slowly over several lags.This feature is known to be due to the autoregressive conditional heteroscedasticity (ARCH) effect which characterises stock returns [45].

Quantitative Results
In this section the outcomes of the following confirmatory statistical tests are presented: 1) Normality tests based on moments and those based on density function of the distribution; and 2) dependence tests, either in returns or in volatility.

Normality test
Testing for normality in the returns using moments We use Jarque-Bera statistic joint test built around skewness and kurtosis as well as Doornik and Hansel statistic.Table 5 and Table 6 contain the summary statistics and the relevant test statistics.Specifically, from Table 6, we find that while the two other banks have skewness not significantly different from zero, Access and Guaranty trust banks are significantly skewed, with GTB having the highest skewness.Looking at the mean returns of the four banks, one would say that they are approximately equal to zero, which is in line with the standard Random Walk assumption.Another notable observation here is that GTB with relatively highest variance produces the highest mean; this is another important feature of short term stock returns behaviour.Also, First bank with negative skewness is prone to effects of a negative shock than a positive shock compared to others which are likely to be influenced by good news than negative news.
And finally, compared to other banks, GTB with the highest coefficients of skewness and kurtosis would seem to be excessively favoured by good news for prolonged periods within the considered periods.However, the four banks have excess kurtosis far above "3", the benchmark for a normal distribution.Further, the results from "JB" and "DH" tests reveal that the returns for these banks cannot be described by a normal distribution.
The excess kurtosis observed in the returns of the four banks might be traced to the presence of conditional heteroscedasticity that is, time varying nature of the volatility (same view shared by Corhay and Tourani) [46].

Using Distributional Approach
Table 7 contains the details on Kolmogorov-Smirnov (KS), according to the p-value (0.000), it could be deduced that none of the banks return series is distributed normally.

Dependency in Returns
This is achieved by computing and determining the significance of serial autocorrelation in daily returns using Box Ljung (Q) statistic.Tables 8-11 contain the results of the computations as well as the p-values for Access, UBA, First and Guaranty Trust bank respectively.It could be observed that though all the auto-correlation functions for the four banks are significant at first lag up till 16th lag, all remain negatives at least at lag 1 across the four banks; this according to most researches is an important stylized fact of daily stock returns (see [33] [47]).Meanwhile the significant level of ACF for the four banks at lag 1 is a sign of persistence or long term dependence in their returns.

Dependency in Volatility
Tables 12-15 contain the outcome of the serial dependence in absolute returns for Guaranty, First bank, Access bank and UBA respectively, as a measure of dependence in volatility.It is found that all the autocorrelation functions for the four banks are significant at 5% level of significance and all positives at lag 1.
This is a confirmation of the presence of long term dependence in absolute for example.The results generally follow the non-normal, leptokurtic and sometimes asymmetric character of financial returns in most markets, especially emerging markets discussed in the literature review.
As noted earlier, the results obtained in this paper are based on the banking sector and complement to some extent overall market-based volatility results obtained by Ezepue and Omar [12].The combined results from Ezepue and Omar [12], Omar [11] and this paper contribute useful new knowledge on the foundations of systematic stock market characterisation work, earlier anticipated in Ezepue and Omar [12], as opposed to the more traditional volatility modelling which dominate the received literature.

Key Findings
Daily stock returns of four major banks, Access, First bank, Guaranty Trust and United Bank for Africa (UBA) in Nigeria have been examined to see if the well documented and established stylized facts commonly are associated with the behaviour of short-interval asset returns, as reported for most major markets in the world, characterise the behaviour of the stocks traded by these banks within the NSM.From the price series, we obtain log returns as proxy for the asset return series for each bank, from which we computed absolute returns, squared returns mean-adjusted returns and mean-adjusted returns squared for the four banks.Our findings reveal that while the First bank and UBA returns are approximately symmetric, given their near zero-value for the coefficient of skewness, Access bank (with skewness of 1.597) could be said to be fairly symmetric, and Guaranty trust tends to be highly skewed (with skewness coefficient of 52.203; see Table 6).
Meanwhile, given high values of kurtosis, the banks could be said to be highly peaked, excessively leptokurtic with heavy tails (see Table 6), hence, non-normally distributed.The tests of Normality, those based on moments as well as that which is based on distribution reveal further the extent of deviation from normal distributions by the returns for these banks.
By these findings, we affirm that daily stock returns of Nigerian banks could not be described by a normal distribution.According to Koutmos et al. [52], the non-normality of the return distribution may be attributed partly to the presence of second-moment temporal dependencies in the stock returns.Additionally, Tables 8-11 as presented show that the autocorrelation functions (ACFs) in the daily returns for each of the banks is either negative or near-zero, which establishes the stylized fact that the series of daily returns are either negative or zero.
However the tests of linear dependence in returns for these banks show some level of significance; also testing auto-correlation in volatility using absolute returns showed that there is high significant level of auto-correlation in volatility for the banks returns.The possible causes for the presence of dependence in the returns and its absolute value may be: 1) non-synchronous trading [53] and 2) time-varying short-term expected returns [54] [55].The presence of serial auto-correlation in either absolute returns or mean-adjusted squared returns indicates the presence of long-range serial correlation in the returns volatility, which justifies our subsequent adoption of ARCH-GARCH family models for describing the volatility [52].We also found that there exist high levels of volatility clustering in the return series of these banks, which became more pronounced during the global financial crises of 2007-2008 (see . Further empirical results include: 1) the returns of the four banks are leptokurtic, with fat tails, thus non-normal; 2) autocorrelations of daily returns are either negative or almost zeros at several lags; and 3) there is a positive and significant level of serial correlation in the absolute returns as well as in the higher order daily stock returns of each of the four banks.These serve to explain that statistical tests obtained via unconditional approach as well as the inferences drawn based on them might be misleading.Also, the presence of time dependence in the returns will help to produce reliable forecasts of the level, volatility and higher moments of the returns.Thus our findings have succeeded in establishing that stylized facts which are generally found to characterise daily returns of most stock markets are also present in the banking sector of NSM, thereby indicating that NSM is not a stand-alone market from the rest of the global market.
Wider implications of the research results were discussed in Section 4.3 of the paper.
Some interesting lines of future work which will extend the initial results in this paper, and some of which are suggested by the reviewers, include: 1) more detailed analyses of returns distributions of Nigerian banks, using different types of univariate and multivariate probability models and their mixtures, including distributions that explore tail behaviours of returns and related risk factors; 2) discussion of the implications of these distributional characteristics for portfolio and risk management involving bank assets; and 3) doing these analyses and further empirical finance analyses of the banking sector (volatility, efficiency, bubbles, anomalies, valuation, and predictability), for different periods of reforms, financial policies and global financial crisis.
Related further work on the stability of the various return series over the different time regimes should involve the use of tail indices, similar to the analyses conducted by Hols and de Vries [56], Koedijk and Kool [57] and Loretan and Phillips [58].It will be important to examine how all these characterisation results relate to the business models of the different banks, linked to their relative financial performance in these periods.The researchers are currently addressing some of these lines of analyses.
In conclusion, the paper presents important foundational insights on the links between volatility of bank returns, the basic stylized facts, and anticipates more detailed characterisation of Nigerian banks' stock market performance in future studies, in a way that departs from the traditional approach of fitting volatility models to observed stock market data.

1 )
Does the normality assumption hold for equity returns?2) To what extents are returns non-normal?Various tests of normality have been developed based on, moments of the return series, the density function of the distribution, or properties of ranked series.To consider tests based on the properties of the original distribution of return, we restrict ourselves to tests based on moments and density function of the returns, with a view to accomplishing the first objective of this research; further details of which are presented as follows.

FF
Tests based on the Density functions In this case, the Kolmogorov-Smirnov (KS) that compares the empirical cumulative density function (cdf), of the returns with the cdf of normal distribution (or any other assumed distribution),( ).F * (with the parameter vector θ ) is used.Since the true distribution of ( ) .r is unknown, it is approximated using the empirical cumulative density function (cdf)

F
x is the cumulative distribution function of a normal distribution with mean zero (0) and standard deviation one (1); and ( ) S x is the empirical distribution function of the values of i z computed using (**) above.

Figures 1 -Figure 1 .
Figures 1-8 contain both daily price and return series for the four banks.For First bank (Figure 1 and Figure 2), there was substantially high volatility noted around September 2003 as well as June, 2004 due to sharp drops in the stock prices, which was big enough to have caused relatively high spikes in

Figures 9 -
Figures 9-16 present both price and price return series for the four banks from Jan, 2001 to May, 2004.It is noticed that while the fluctuations in the price series appear more pronounced, the return series appear similar as it was when the longer length of time was used.For Access bank the spikes in both price and return series that appear relatively flattened in the longer horizon become well pronounced.The price experiences gradual rise from around May, 2002 reaches
Figures 25-32 contain both the price and return series from January, 2007 to December, 2008.It could be ascertained that virtually all the four banks have records of thick, clustered and well pronounced volatility.For instance, considering Guaranty bank, the volatility rate in the return series is highly pronounced with a very big spike observed around May, 2007.First bank has two big and sharp volatility levels around August, 2007 and August, 2008, with other periods having thick and clustered volatility except for the periods between April-July, 2007 as well as August-October, 2008, which could be regarded as the periods of calmness or stable fluctuations.Access bank witnesses high volatility throughout these periods except for periods falling between May-September, 2007 with minimal or stable fluctuations in the series.UBA on the other hand has the records of three big volatility levels around December, 2007; June, 2008 and December, 2008; with the remainder of the periods witnessing highly clustered series Periods of Second Banking Reforms (2009-2013) Figures 33-40 contain both the price and return series for the four banks.These periods represent periods of intense volatility.That is periods when the returns are highly clustered together.More jumps or spikes are noticed around most of these series.For example, First bank records four pronounced jumps noticed around July, 2009; May, 2010; May, 2011 as well as January, 2013.The returns for this bank could be said to be noisy.Access bank experiences biggest jump around May, 2011, with the rest of the periods having thick and highly clustered returns.Guaranty Trust bank witnesses pronounced spikes around April, 2010; April, 2011 and October, 2012.Lastly, UBA though records serious spike around May, 2011, the rest of these periods are characterized by intense and high volatility level.

(
Figures B1-B4 (see Appendix B) have normal Q-Q plots for the four banks.

Figure A3 .
Figure A3.Normal plot for guaranty trust bank.

Figure
Figure B3.Q-Q plot for guaranty trust.

Figure
Figure B4.Q-Q Plot for UBA.

Figure C4 .
Figure C4.ACF of FIRST bank daily ABSOLUTE returns.

Figure C5 .
Figure C5.ACF of GUARANTY TRUST bank absolute daily returns.

Figure C6 .
Figure C6.PACF of GUARANTY TRUST bank daily returns at lag 16.
1 ) of Non-normality if the p-value of the JB [42] is most widely used in finance, it does have two limitations: 1) it only holds for very large samples; to correct this bias, Doornik and Hansen[42]provide an omnibus test for normality.2)Theempirical skewness and kurtosis are computed for given values of mean and variance, which are both subject to sampling errors.Doornik and Hansen[42]first obtain approximations for the finite sample distributions of skewness and kurtosis under normality assumption, coupled with assumptions that the kurtosis follows a Gamma distribution and that

Table 8 .
Auto-correlation for ACCESS BANK.

Table 10 .
Auto-correlation for first BANK.

Table 11 .
Auto-correlation for guaranty trust BANK.

Table 12 .
ACF test on absolute returns for guaranty trust.

Table 13 .
ACF for absolute returns of first bank.

Table 14 .
ACF for absolute returns of access bank.

Table 15 .
ACF of absolute returns for uba.

Wider Implications of the Results for Theory and Practice
We further remark that most of the results obtained here are new in the sense that they are connected mainly to the stylized facts of bank returns, which have not been studied in this way in the volatility modelling of the NSM.Existing Nigerian studies are focused more on core volatility modelling which involves fit- [11]emics, for different and sometimes related reasons[10][12][38].Investors search for arbitrage opportunities which relate to the observed or implied volatilities associated with key financial quantities and their derivatives, basically because high volatility means high risk and uncertainty, and therefore signals potentials for excessive gains or losses.These financial quantities include stock market indices, stock price returns, interest and exchange rates and commodity prices, for example oil prices in Nigeria, which correlate with key financial variables, given the country's heavy dependence on oil revenues, Omar[11].Nigerian policy makers need to understand the volatilities in the financial system at overall market, sector-and company-specific levels, in order to gear monetary, cal finance features mentioned above.These results will overall help to more systematically characterise the returns profiles of the banking sector which is the most important sector of the NSM.