“Day-of-the-week effect in Nigerian stock exchange: adaptive market hypothesis approach”

The problems that this study informed are rooted in the uncertainty surrounding the presence of calendar anomalies in the Nigerian stock market and the need to ascertain whether calendar anomaly is changing with time and market condition according to the adaptive market hypothesis. This study evaluates how calendar anomaly behaves over time in the Nigerian stock market through the day-of-the-week effect since the latest trend is to examine time-changing anomaly. The general All Share Index returns of the Nigerian Stock Exchange between 2000 and 2017 are used in the analysis. Secondary daily index returns data for the period are sourced from the NSE Fact Book. The major estimation techniques employed in the study are the mean equations of the generalized autoregressive conditional heteroscedasticity (GARCH) and overlapping sub-period methodology. Moreover, returns are grouped into Up and Down periods depending on the periods that generate positive and negative returns, respectively. This study found out that Monday (MON), Tuesday (TUE), and Friday (FRI) effects are the only adaptive day-of-the-week effects. Thus, three (MON, TUE, FRI) day of the week effects found in the full sample are time-varying in subsample and are affected by market condition. On the whole, MON and Thursday (THUR) effects are found in Bull, while TUE and FRI are found in Bear.The investor must be careful to take time- variation into consideration; otherwise, they may incur a loss by thinking that the day-of-the-week effect is present every time. hypotheses imply that the Nigerian stock market, like many other emerging markets, is not efficient. The implication of this is that stock return anomalies could be present in the market so that investors can consistently earn abnormal returns by tar-geting certain periods for trading or using a trading strategy based on past information. In the presence of this inefficiency, the activi-ties of sharp speculators are prevalent, and speculation is done at the expense of uninformed investors, which is not good for the nation’s financial market. Among anomalies, calendar anomaly is prominent.


INTRODUCTION
The presence of calendar anomalies has been documented extensively for the last four decades in financial markets. However, the subject is yet to receive adequate attention from developing markets such as Nigerian stock markets. Submissions from most studies of weak and semi strong form hypotheses imply that the Nigerian stock market, like many other emerging markets, is not efficient. The implication of this is that stock return anomalies could be present in the market so that investors can consistently earn abnormal returns by targeting certain periods for trading or using a trading strategy based on past information. In the presence of this inefficiency, the activities of sharp speculators are prevalent, and speculation is done at the expense of uninformed investors, which is not good for the nation's financial market. Among anomalies, calendar anomaly is prominent. Osarumwense (2015) examined the day-of-the-week calendar effect, while Efayena (2014) and Olowe (2011) investigated the month-of-theyear calendar effect in Nigeria with the latter focusing on oil price returns as opposed to stock price returns.
More so, the advent of adaptive market hypothesis (AMH) has caused a revision of calendar anomalies in recent literature. Furthermore, the Akkaya and Çimen (2013), Güler and Çimen (2014) opined that financial anomaly is synonymous with abnormal return, which implies a deviation from the average return. An anomaly is defined as a price or profit distortion, which is evidence of inefficiency within the financial markets (Kroon, 2008). It is usually a result of structural factors such as unfair competition, lack of transparency in the market or behavioral biases by various economic agents, which forms the bedrock of behavioral finance. It is used to describe a situation where changes in stock returns display high or low patterns at certain calendar periods (Yousop, Sipon, & Yoke, 2014). Applying descriptive analysis, decomposition method, dummy variable regression and binary logistic regression, Gulseven (2014) used stock market indices spanning from 1996 to 2014 to check for the monthly market anomalies in Turkish and American Stock Exchanges. The study revealed a strikingly negative May effect on the Turkish stocks and a positive return in April. Stocks tend to be bullish in December in both markets, and there is no significant January effect. Patel (2016) evaluated the January effect anomaly in stock returns using U.S. stock, Russell developed stock, Russell Asia-Pacific stock, Russell Europe stock, Russell Latin-America stock and the Russell Emerging stock indices from 1997 to 2014. OLS dummy regression and descriptive statistics analyses revealed that January effect in stock returns does not exist as January returns are negative and statistically insignificant, while December returns are positive and statistically significant. Kampman (2012)  Using All Share Index of Nigeria, Kenya, Tunisia, Morocco, South Africa, Egypt, and Zimbabwe, Alagidede (2013) evaluated the implication of month of the year and the pre-holiday effects for efficiency in African stock market. The descriptive statistics of the mean and variance of returns and dummy regression found high and significant returns in days preceding a holiday in South Africa; a February effect for Morocco, Kenya, Nigeria, and South Africa; a January effect in Egypt and Zimbabwe and monthly seasonality is pronounced for Nigeria. GARCH model employed due to the presence of conditional heteroscedasticity showed that Zimbabwe exhibits more significant negative returns virtually throughout the year with only September and April showing significant negative returns in Nigeria. Overall, results indicate that the month-of-the-year effect is prevalent in African stock returns. Osarumwense (2016) concluded that the day-of-the-week anomalies in foreign exchange market lies on the choice of model specified. Ajao and Wemambu (2012) examined the volatility and prediction of Nigerian stock market using monthly data from 2005 to 2009. Result of autoregressive conditional heteroscedasticity (ARCH) showed the presence of vola-tility and that stock market prices may indeed be predicted.

LITERATURE REVIEW
In addition, Tan and Tat (1998) show that the January effect and daily effects, among others, are diminishing in Singapore using two sub-periods, while WingKeung, Aman, and NeeTat (2006) revealed that calendar effects are weakening across the sub-periods. New growing trend in literature is the changing behavior of calendar anomalies in reaction to different market conditions (i.e., Bull and Bear markets, and market crashes). Such proof, according to Urquhart and McGroarty (2014), would be in line with AMH as opposed to the EMH. Urquhart and McGroarty (2014) found that calendar anomalies perform in an adaptive style by estimating them in six subsamples and overlapping windows. The review also disclosed that, apart from the stock market, seasonal effect has also been investigated for other commodity markets, such as the oil market and foreign exchange market. For instance, Olowe (2011) and Osarumwense (2016) in Nigeria studied the month-of-the-year and day-of-the-week, respectively, in oil market and foreign exchange market. That is to say, some of the calendar anomalies have been investigated in Nigeria. However, calendar anomaly has not been investigated in line with the AMH in Nigeria. To the best of our knowledge, there seems to be scarcity of study investigating whether calendar anomalies change over time in Nigerian stock market.

Source and description of data
The study employs secondary data. The Nigerian Stock Exchange (NSE) daily index was obtained from the Capital Asset Section of NSE Fact Book available online. Daily return data are calculated from the index for eighteen years (2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017) during which the data are available. Daily return is obtained based on the formula in equation (1): where P t and P t-1 indicate stock indices on day t and t-1, respectively, and R t is the stock return on day t say Monday price relative to Friday price.

Model specification
The main objective of this study is to establish various calendar effects in mean returns, ranging from the weekdays effect and monthly/January effect. Borges  where t is the daily return of index on day t, D it are the dummy variable, which is equal to 1 if the corresponding return for day t is Tuesday, Wednesday, Thursday, and Friday, respectively and 0 if not. i = 2,3,4,5 and ε t is the error term. The first equation (2) or specification can only reveal through β 2 whether Tuesday returns differ significantly from Monday's but cannot compare Tuesday with other days. In equation (3), if the sample size is considerably long, mean daily return tends to be positive. Consequently, the t-tests are biased towards accepting positive excess returns, and against accepting negative excess returns (Borges, 2009). Borges (2009) argues that the alternative to the above equations is to adopt a simpler approach, which surmounts all these limitations, which is to estimate five equations separately, each with to spot a particular day-of-the-week effect: Based on equation (4), if dummy variable is included for Mondays alone, α accounts for the mean daily return of remaining four days, and β 1 is the excess return of Mondays, in comparison to remaining days. The t-test of β 1 reveals whether Monday effect is significant, and the same arguments hold for β 2 , β 3 , β 4 and β 5 , for detecting other day-of-the-week effects. Thus equation (4) is adopted in this study as recent studies such as Urquhart and McGroarty (2014) have embraced it.
For weekend effect to exist, β 1 must be low or negative and β 5 must be high and positive and statistically significant.

Overlapping sub-period analysis
To evaluate how calendar anomalies, vary (behave) over time in the Nigerian stock market, the study will employ overlapping sub-period analyses in the estimation of specified calendar anomaly equations (4). This is also known as rolling window analysis, which is used by scholars (Urquhart & McGroarty, 2014;Obalade & Muzindutsi, 2018) in the investigation of AMH. Attention is paid to the behavior of the coefficient of calendar effect of interest between positive and negative coefficient and between statistical significance and insignificance of the coefficient. If the behavior changes over time as the rolling sub-period changes, then, it will be concluded that the behavior of calendar anomaly is consistent with AMH. This study adopts 4-year overlapping sub-period, moved one year forward to account for the changes in behavior of calendar anomalies over time.

Estimation techniques
The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) estimation technique is used to estimate the equation (4). GARCH (1,1) is the best model in the estimation of calendar anomalies (Urquhart & McGroarty, 2014). The simplest GARCH (1,1) specification is presented as follows: Equation (5) represents the mean equation, while equation (6) represents the variance equation. The former is given as a function of exogenous variables and stochastic error term, while the latter depends on three components, namely the constant term (w), the squared residual in the preceding period ( 2

RESULTS AND INTERPRETATIONS
In this section, results are presented in the order of the objectives. In other words, the first subsection of this section accommodates the results of test of calendar anomalies in mean returns in absolute form. Further, the results of time-changing calendar anomalies are presented and, lastly, the results of calendar anomalies under Up (Bull) and Down (Bear) conditions are presented. The section is concluded by the discussion of findings with reference to exiting studies and theory, while implications of findings are highlighted for the stakeholders.

Calendar anomalies (full sample)
This study ran calendar effect model for each dayof-the-week against all other days. Hence the coefficient of a particular day, say, MON, reflects the excess or deficit of MON compared to (C), which is the mean daily return of non-Monday days, etc. The full sample results of the five equations for each day-of-the-week effect is summarized in Table 1. It can be seen that the popular negative Monday effect is present with the day having negative return. The t-test of the coefficient shows that the negative MON effect is significant since the probability value is less than 5% significance level. Similarly, the results of the TUE model reveal that there is negative TUE effect evidenced by negative value of TUE coefficient, relative to other days. Conversely, the coefficients of returns associated with the WED and THUR are positive and higher relative to the average returns on other days. However, WED and THUR effect cannot be said to exist since the t-statistic tests of the relevant coefficients are not statistically significant. From the results of FRI effect model, the FRI coefficient is positive and higher than other days average returns. It means that the FRI effect is positive since the t-statistics is equally statistically significant.
Based on the examination of the day-of-the-week effect in absolute form, it can be seen that there is negative MON and TUE effects, as well as the positive FRI effect in the Nigerian stock market. These effects translate to what is theoretically referred to as the weekend effect characterized by positive and significantly high FRI returns and negative and significantly low MON returns.

Time-varying behavior of calendar anomalies
The examination of calendar anomalies in stock returns in absolute form as done in subsection 3, has been challenged with the emergence of AMH in recent times. AMH suggests that the best way to follow is to anomalies over time. Consequently, the results of the calendar anomalies based on overlapping sub-period analyses are discussed in this subsection.
The results of overlapping sub-period estimations for all days of the week are presented in Table 2   Note: significance of estimated coefficients is taken at prob. < 5%. *, **, *** correspond to 10, 5, 1%. In terms of the signs of the return's coefficients, WED and THUR returns vary over time between positive and negative coefficients. As with the full sample results, the results of sub-periods analyses on WED and THUR effect does not show any observable anomalies. For all the fifteen win-dows, there is no trace of WED and THUR effect except for one window of positive WED effect in 2009-2012. The last day-of-the-week effect result is presented for FRI effect. This effect is found in full sample (Table 1). In overlapping the sub-period analyses, it is clear that positive FRI is not found at all times. In fact, there are two sub-periods of negative FRI effect in the first three sub-periods, though they are not statistically significant. Basically, there are nine (9) sub-periods of statistically significant (some at 10%) FRI effects out of fif-teen (15) total sub-periods. It implies that the FRI effect fluctuates over time according to AMH.
Overall, while the full sample reveals that the MON, TUE and FRI effect exist, the overlapping sub-period estimations show that they only exist in some and not all sub-periods, hence, time-varying and adaptive. For better understanding of the time-varying behavior, this study presents the coefficients of MON, TUE and FRI effect in Figure 1. The bluish line indicates the coefficient, while the reddish line stands for the p-values. It can be seen from the figure that significant MON effect is only found in 4 th , 9 th , 10 th , 11 th , and 12 th sub-periods, while other periods are not associated with significant MON effect. Similarly, Figure 1 shows that significant negative TUE effect is found in 7 th , 8 th , 10 th , and 15 th sub-sample. Lastly, positive FRI effect is associated with 5 th , 6 th , 7 th , 8 th , 9 th , 10 th , 12 th , 14 th , and 15 th .

Calendar effect and market condition results
Since the previous section reveals that MON, TUE, and FRI effects behave in a changing version, hence, adaptive. This study also examines under which market conditions these recognized calendar effects can be found. After separating the data into a positive and negative (months) condition, the study then estimates the behavior of the calendar effects during each condition using the previously described GARCH (1,1) regression model. The study has the data separated into Up and Down months similar to Fabozzi and Francis (1977) and Urquhart and McGroarty (2014) by identifying the months with positive return as Up condition and months with negative return as Down period.
Results of the estimation of calendar anomalies under Bull and Bear condition are presented in Surprisingly, the THUR effect, which was not found in full sample and sub-periods appear under Bear periods. Table 3 shows that THUR coefficient changes from insignificant negative in Bull to significant positive in Bear. Moreover, the establishment of time-variation in FRI effect in previous section can now be traced to market condition that changes. The positive Friday effect is found in both Bull and Bear periods, but the effect is higher in Bear (0.068197) as compared to Bull, relative to other days (C). Thus, the three (MON, TUE, FRI) Note: significance of estimated coefficients is taken at prob. < 5%. *, **, *** correspond to 10, 5, 1%.
day-of-the-week effects found in the full sample, found to be time-varying in subsample, are affected by market condition. However, the Thursday effect is also found in Bear condition. On the whole, MON and THUR effects are found in Bull, while TUE and FRI are found in Bear.

DISCUSSION AND IMPLICATIONS
The study examines whether day-of-the-week effect conforms to AMH in the Nigerian stock market. Based on the examination of the day-of- Lastly, the study analyzes how calendar anomalies respond to changing regimes or market conditions in the Nigerian stock markets. Generally, the day-of-the-week effects, especially the negative MON and TUE effects, as well as the positive FRI effect is time varying due to changing Bull and Bear market condition. Specifically, Monday effect is associated with Up condition. This finding contradicts Urquhart and McGroarty (2014) who held that Monday effect is significant in Bear, Down and crashes. It is recommended that investors should not be dogmatic in the exploitation of the identified calendar effects since the effect is time-varying.

CONCLUSION
This study aims at evaluating the presence of day-of-the-week effect and determining the relevance of AMH in describing the behavior of the identified calendar effect. To achieve the objectives, the study evaluates the calendar effect in full sample and overlapping sub-periods. Based on the test of hypotheses and the findings of the study, the study concludes that there is a significant weekend effect evidenced by significant negative Monday effect and significant positive Friday effect. Secondly, there is evidence of stock market calendar anomaly in the Nigerian stock returns and violation of efficient market. Thirdly, the mentioned anomaly is not present every time as previous studies have suggested. Finally, some of the daily anomalies are present in Bull periods, while some are present in Bear periods. In Nigeria, just few papers are examining AMH. The current study contributes to literature by showing that dayof-the-week effect depends on time and that Up and Down conditions affect the calendar effect in the Nigerian stock market. In general, AMH seems to be a more realistic theory when it comes to the behavior of stock market returns and calendar anomalies. Therefore, stakeholders must consider possibility of adaptive behavior while interacting with the stock market. Investors must be careful to take time variation into consideration; otherwise, they may incur a loss by believing that day-of-the-week effect is unvarying. Going forward, other anomalies such as holiday and monthly effect can also be examined using AMH approach. Future researchers can investigate AMH in other markets such as Nigerian foreign exchange and oil markets.