Weak-form Efficiency After Global Financial Crisis: Emerging Stock Market Evidence

Abstract

This article investigates weak-form efficiency of the Nigerian Stock Exchange (NSE) and its sectors for the post-global financial crisis period using autocorrelation test, Ljung–Box Q test, McLeod-Li portmanteau test and ARCH-LM test. The descriptive statistics show that the returns of NSE and its sectors are positive. The results show that (i) investors can only predict banking sector return using superior fundamental analysis of their intrinsic values; (ii) prediction of the NSE 30 and Shari’ah equities sector returns require nonlinear model and fundamental analysis and (iii) consumer goods sector and oil and gas sector may be predicted using both technical and fundamental analyses.

JEL Classification: G11, 14

Keywords

Weak-form EMH stock returns sectors of economy global financial crisis Nigeria

1. Introduction

The global financial crisis (hereafter GFC) not only destabilized global financial system which led to a major economic crisis but also had devastating effect on the Nigerian Stock Exchange (NSE). Emenike (2009) notes that the All Share Index (ASI), which is an indicator of the average price level on the exchange, ‘skyrocketed’ from 1113.4 points in January 1993 to 57,990 points in December 2007, and that the market capitalisation grew by 27989 per cent in the same period. The ASI attained its peak (65652.38 points) in February 2008, before falling to 23377.14 points a year after and to 19851.89 points in March 2009. According to Sanusi (2012), the stock market collapsed by 70 per cent and many Nigerian banks have to be rescued. Consequently, so many retail investors lost confidence in the stock market and exited, institutional investors adjusted their portfolios towards money market instruments and real estate. Indeed, the NSE could not perform its major functions of price transparency, price discovery, reduced transaction costs and exchange regulation to the investors in particular and the economy in general during the GFC (Adewale & Eromosele, 2009; Apampa, 2008).

Performance of price transparency, price discovery, reduced transaction costs and exchange regulation, strengthens investors’ confidence in a stock exchange. Price transparency, for example, ensures that every stock exchange participants have access to a neutral reference price. Price discovery ensures that demand and supply developments in the stock exchange are readily reflected in stock prices. Reduced transaction costs result from the easy access to buyers and sellers through a centralised market place—the stock exchange. Exchange regulation involves ensuring safety and soundness of the stock trading system as well as promoting growth and development of the exchange. Each of these functions can be inconvenient for certain market participants, without an efficient stock exchange. Any stock market that provides these basic functions may be described as an efficient stock market.

According to Fama (1970), a market in which prices always ‘fully reflect’ all available information, is efficient. Similarly, Kolb and Rodriguez (1992, p. 288) describe efficient stock market as a market in which, on average, prices adjust quickly after the arrival of new information and the price changes reflects the economic value of the information. Efficient stock markets, therefore, exhibit transparent price discovery process such that prices accurately reflect the trade-off between the relative risk and potential returns associated with the stock. Strong (2003) posits that stock markets are kept reasonably efficient because of the vast number of market participants who are quick to take advantage of security mispricing. This implies that market efficiency is related to the quality of information analysis by stock market participants which are instantaneously reflected in stock prices. Thus, no profits beyond normally required returns can be made by trading on available information.

Fama (1970) categorizes market efficiency into three different levels¹ according to the information item reflected in the prices: weak-form efficiency (WFE), semi-strong-form efficiency and strong-form efficiency. According to the efficient market hypothesis (EMH), a market is efficient in the weak-form version if prices of traded assets already reflect all information contained in the history of past prices, trading volume or short interest. Semi-strong EMH holds when stock prices already reflect all the publicly available information regarding the prospects of a firm. Lastly, the strong form posits that the prices of financial assets reflect, in addition to information on past prices and publicly available information, information available only to company’s insiders.

Early empirical studies from developed stock market found support for WFE based on low degree of serial correlation and transaction costs observed in the series (see, for example, Cootner, 1962; Fama, 1965; Kendal, 1953). These studies generally agree that stock price changes are random and that past prices are not useful in predicting future price changes particularly after transaction costs have been taken into consideration. Some recent studies that support WFE in developed markets include Konak and Şeker (2014). On the other hand, majority of the recent studies of WFE from emerging markets show evidence of weak-form inefficiency (see, for example, Afego, 2012; Alkhatib & Harasheh, 2014; Arora, 2013; Harrison & Moore, 2012; Ogege & Mojekwu, 2013; Owido, Onyuma, & Owuor, 2013). A few studies, however, provide evidence of WFE (see, Ajao & Osayuwu, 2012; Kulikova & Taylor, 2014; Lim, Huang, Yun, & Zhao, 2013).

Majority of the existing studies of weak-form EMH generally tests absolute efficiency. Absolute efficiency involves applying different types of past return-based tests on one index or sector for the full sample period. The notion of absolute efficiency does not seem to be an informative method of gauging the efficiency of a given market (Campbell, Lo, & MacKinlay, 1997, p. 25). Relative efficiency, the efficiency of one market or sector measured against another, is more useful method than the absolute efficiency. Relative efficiency may also be described as the efficiency of a market or sector of a market measured across time, sectors, or markets. Unlike absolute efficiency, relative efficiency is more informative and provides a basis for comparing WFE between different markets, sectors, or across time. Despite the glaring superiority of relative efficiency in providing comparative evidence on stock market efficiency, little attention has been devoted to it in empirical studies.

More so, given that the NSE has overcome the GFC,² it is meaningful to ask: what is the nature of the WFE of the NSE and its sectors after the GFC? An answer to this question is needed urgently because investors are interested in understanding nature of WFE of the bourse. WFE reveals the nature of price discovery, quality of information analysis by market participants, safety of investment and investors’ confidence in a stock market. In addition, answer to the question needs empirical analysis.

The objective of this study therefore is to answer the question. The findings will not only show the nature of WFE in each of the NSE sectors, but will also highlight the sector(s) that require improvement in the price discovery process. Again, unlike absolute efficiency of the NSE based on NSE ASI that is not investible,³ the relative efficiency of the NSE sectors will be useful to investors for making optimal investment decision and portfolio management based on sector indexes that are investible.⁴ Regulators of the Nigerian capital market will equally benefit from the findings of this study as they will channel policy efforts to enhance the process of price discovery and price transparency in inefficient sectors. The remainder of the article is organized as follows: after this introduction is Section 2, which contains theoretical basis and brief review of recent empirical findings. Section 3 describes data for analysis and methodology, Section 4 presents empirical results and discussions, and Section 5 concludes.

2. Theoretical Basis and Brief Review of Recent Empirical Findings

2.1. Theoretical Basis: Random Walk Weak-form EMH

The theoretical basis for analysing NSE and the sectors indexes for evidence of WFE is the random walk hypothesis (hereafter, RWH). The weak-form EMH posits that stock prices already reflect all information that can be derived from examining market trading data such as the history of past prices, trading volume or short interest (Bodie, Kane, & Marcus, 1999, p. 331). If markets are weak-form efficient, unanticipated return is not correlated with previous unanticipated returns and all such information is clueless for directing a trading strategy (Kolb & Rodriguez, 1992, p. 288). The unanticipated portion of return, by definition, follows a random walk. RWH holds that news arrives randomly, and, because markets are efficient, security prices adjust to the arrival of news (Strong, 2003, p. 244). In other words, the direction as well as the size of change in a stock price are random and cannot be predicted from past information about share prices.

Campbell et al. (1997, pp. 31–33) summarize three versions of random walk model based on the distributional characteristics of increments. Random walk 1 (hereafter RW1) implies that price increments are independent and identically distributed (hereafter iid), in which case the process P_t is given by

P_{t} = μ + P_{t - 1} + ε_{t}, ε_{t} i i d ~ (0, σ^{2})

(1)

where μ is the drift parameter or the expected price change, and iid (0,σ²) denotes that ε_t is independent and identically distributed with mean 0 and variance σ². The independence of increments (ε_t) implies not only that ε_t is uncorrelated but any nonlinear functions of the increments are also uncorrelated (Campbell et al., 1997, p. 32). Though the RW1 appear elegant and simple, the assumption of identically distributed increments is not plausible for financial assets prices over long periods of time spans because of changes in probability distributions of financial assets returns resulting from changes in economic, technological, institutional and regulatory environment surrounding the asset prices.

As a result of implausibility of identically distributed increments, random walk 2 (RW2) assumes independence but not identically distributed (inid) increments and thus allows for heteroscedasticity in ε_t. RW2 is estimated as follows:

P_{t} = μ + P_{t - 1} + ε_{t}, ε_{t} i n i d ~ (0, σ^{2})

(2)

where ‘inid’ denotes that the error term is independent but not identically distributed. The RW2 therefore allows for unconditional heteroscedasticity which is particularly useful feature of time variation in volatility of many financial assets. Relaxing the identical distribution assumption in RW2 does not change the main economic property of ε_ts, that is, prediction of future price increments cannot be estimated using past price increments (Campbell et al., 1997, p. 33).

Random walk 3 (RW3) is obtained by relaxing the independence assumption of RW2 to include processes with dependent but uncorrelated increments. It only imposes lack of correlation between subsequent ε_ts. A case in which RW3 will hold but not RW1 and RW2 is any process where Cov[ε_t, ε_t+k] = 0 for all K, but where Cov[ε_t², ε_t²_+k ] ≠ 0 for some K, in both cases K ≠ 0. This process has uncorrelated increments but is evidently not independent because its squared increments are correlated.

2.2. Brief Review of Recent Empirical Findings

Numerous empirical studies have been conducted to ascertain weak-form EMH in both developed, emerging and frontier stock markets in the world. The earliest empirical studies, which started in developed market, include Kendal (1953), Cootner (1962), Fama (1965), among others. Kendal (1953) examines the behaviour of weekly changes in 19 indices of British industrial share prices and in spot price for cotton in New York and wheat in Chicago from 1928 to 1938. After extensive analysis of serial correlation, he conclude: ‘the series look like a wandering one, almost as if once a week the Demon of chance drew a random number from a symmetrical population of fixed dispersion and added it to the current price to determine the next week’s price’. This conclusion implies that the common stock price changes are not serially correlated but are random. Recent studies of weak-form EMH in developed markets also provide support for WFE (Konak & Şeker, 2014; Kulikova & Taylor, 2014). Konak and Şeker (2014), for example, examine the Financial Times Stock Exchange 100 (FTSE 100) for WFE using random walk model during the period from January 2001 to November 2009. Their empirical results from Unit root tests and GARCH (1,1) model provide support for random walk and hence WFE in London FTSE 100.

In emerging and frontier markets, the recent evidence on WFE is mixed. Magnus (2008) examines the weak-form EMH in the Ghana Stock Exchange using daily returns from 1999 to 2004. The results of random walk and GARCH (1,1) models used in the analysis show that the GSE DSI returns series exhibit volatility clustering, which he interpreted as an indication of inefficiency on the GSE. He therefore rejects weak-form efficient market (random walk) hypothesis for the GSE and concluded that the market is inefficient. In Middle East and North Africa (MENA) region, Harrison and Moore (2012) investigate stock market efficiency in a group of emerging markets using test for evolving market efficiency. They find that the MENA region markets are inefficient despite growth in size and the implementation of reforms designed to improve the operation of markets in the region. In Pakistan, Zahid, Ramzan and Ramzan (2012) test WFE of Karachi Stock Exchange (KSE) using different parametric and nonparametric tests of random walk. The results of runs test and autocorrelation show that KSE return series do not follow random walk and are serially correlated. They therefore reject WFE. In India, Arora (2013) assesses the daily indices of S&P CNX Nifty (Index of National Stock Exchange India) for evidence of weak-form EMH and RWH using a battery of econometric tests. His results show that the S&P CNX Nifty returns are characterised by linear as well as nonlinear dependences and a high persistence of volatility clustering over the sample period. He therefore rejects null hypothesis of random walk for the series and concludes that Indian Stock Market does not show evidence of weak form of market efficiency. In Kenya, Owido et al. (2013) examine the efficiency of the Nairobi Securities Exchange using GARCH model. Their results show market inefficiency of the weak-form. In Palestine, Alkhatib and Harasheh (2014) apply random walk model to empirically examine the weak-form market efficiency of Palestine Exchange. The results of their regression analysis, serial correlation, ADF and runs tests reveal that the stock market is inefficient at the weak-form. They conclude that their findings do not support the random walk model.

Other emerging market studies have support for WFE, or at least, are not able to reject it. Ntim, Opong, Danbolt and Dewotor (2011) investigate and compare the WFE of a set of 24 African continent-wide stock price indices and those of eight individual African national stock price indices using variance-ratio tests. Their results show, among others, that the African continent-wide stock price indices have significantly better weak-form informational efficiency than their national counterparts. In China, Lim et al. (2013) investigate the efficiency of the daily closing prices of A-share and B-share indexes in both the Shanghai and Shenzhen stock exchanges using three different techniques namely serial correlation test, runs test and variance ratio test. Though they find mixed results, they still conclude that China’s stock market is still weak-form efficient. In South Africa, Kulikova and Taylor (2014) investigate the change in the level of market efficiency in three markets, chosen to reflect a developed (London LSE), mature emerging (Johannesburg JSE) and immature emerging (Nairobi NSE) market perspective, over a period that includes the financial markets crisis of 2007/2008. Their empirical finding from multi-factor model with time-varying coefficients and GARCH errors suggests that in spite of the financial crisis, all three markets maintained their pre-crisis level of WFE.

Recent empirical evidence on the WFE of the NSE has equally been mixed but tends towards rejection than acceptance of WFE for Nigeria capital market. Agwuegbo, Adewole and Maduegbuna (2010) analyse the behaviour of daily return of all securities listed in the Nigeria Stock Exchange. Their results provide evidence showing that the NSE is not efficient even in weak-form. Similarly, Ogege and Mojekwu (2013) investigate weak-form EMH of the Nigerian Stock Market by searching for evidence of serial correlation in monthly time series data of the NSE. Their results show that investors can use past data to predict the future prices of the market and they conclude that the NSE is inefficient in the weak from. In the same vein, Afego (2012) finds evidence against weak-form EMH for the Nigerian stock market by testing for random walks in the monthly index returns. Results of the nonparametric runs test show that index returns on the NSE display a predictable component, thus suggest that traders can earn superior returns by employing trading rules. In contrast to studies that find evidence against WFE, Ajao and Osayuwu (2012) provide evidence in support of WFE by applying serial correlation technique and runs test to monthly ASI of the NSE.

3. Data Description and Methodology

3.1. Data

The analysis presented in this study is based on post-GFC daily indexes of the NSE. The sample period ranges from 04 January 2010 to 30 April 2014. A set of five indexes are analysed namely NSE 30 index (hereafter denoted NI), NSE banking index (BI), Oil & Gas index (OG), NSE consumer goods index (CG) and NSE Lotus Islamic Index (LII). The BI, which captures the banking sector, is designed to provide an investable benchmark to capture the performance of the banking sector, this index comprises the most capitalized and liquid companies in banking sector of the Nigerian economy. The OG captures the oil and gas sector and is designed to provide an investable benchmark to capture the performance of the oil and gas sector, this index comprises the most capitalized and liquid companies in oil and gas marketing. The CG provides an investable benchmark to capture the performance of the consumer goods sector, this index comprises the most capitalized and liquid companies in food, beverage and tobacco. The LII tracks the performance of Shari’ah compliant equities which have met the eligibility requirements of a renowned Shari’ah Advisory Board. All the indexes were obtained from the NSE and converted into log-returns as follows:

r_{t} = \ln (P_{t} / P_{t - 1}) * 100

(3)

where r_t is a vector of daily returns of the sector indexes, P_t is a vector of closing indexes t, P_t-₁ is the previous day closing indexes and ln is natural logarithm. Subsequent analyses are based on r_t and constitute a series of 1,070 observations. Descriptive statistics of r_t are presented in Table 1.

3.2. Methodology

3.2.1. Model

RWH is examined in this study for the NSE and its sectors using AR-GARCH (1,1) residuals. The ARCH family models allow for testing nonlinear dependence in return series by relaxing the assumption that the second moment must be constant (Tsay, 2005, p. 100). The AR model, which serves as the mean model, is specified in accordance with Campbell et al. (1997, p. 33) and the GARCH (1,1) model, which is our variance model, is specified following the ARCH model of Engle (1982) as generalised by Bollerslev (1986). The AR-GARCH (1,1) model is specified as follows:

r_{t} = φ + θ r_{t} - i + ε_{t}, ε_{t} ~ i n i d (0, σ_{t}^{2})

(4)

σ_{t}^{2} = ω + α_{1} ε_{t - 1}^{2} + β_{1} σ_{t - 1}^{2}

(5)

where r_t is the vector of daily return described in equation (3), ϕ is the vector of the drift parameter, θ is the AR (p) parameter to account for time dependence in the returns, and ε_t is the residual term in the mean equation. In equation (5), $σ_{t}^{2}$ is the conditional variance equation (i.e., the volatility at time t), ω is the constant, α₁ and β₁ refer to a first order ARCH term (i.e., news about volatility from the previous period) and a first order GARCH term (i.e., persistent coefficient). The expectations of parameters of weakly stationary conditional variance are ω > 0, α₁ ≥ 0, β₁≥ 0 and α₁ + β₁ < 1 (Bollerslev, 1986; Emenike, 2010; Tsay, 2005, p. 114). The models were estimated using the BFGS maximum likelihood algorithm (described in Press, Flannery, Teukolsky, & Vettering, 2007) with correction for heteroscedasticity and misspecification. A major importance of modelling RWH with GARCH model is that the RWH2 allows for heteroscedasticity. The GARCH model therefore is superior for modelling RWH, except in cases where only RWH3 is considered, which is based only imposes lack of correlation between the increments.

The a priori expectation is that those sectors which are frequently traded should more closely follow random walk model of WFE.

3.2.2. Tests for Serial Dependence

Examining RWH involves both linear and nonlinear tests for serial dependence in a time series. Testing for linear and nonlinear serial dependence has important implications for WFE and return predictability. Evidence of nonlinear serial dependence in a series, for instance, suggests, according to Brooks (1996) that, at least in the short term, forecast may be improved by switching from a linear to a nonlinear modelling strategy. Therefore, any evidence of linear and/or nonlinear dependence in the NSE and/or its sectors returns will be viewed as evidence against WFE. The serial dependence tests applied in this study include autocorrelation test, Ljung–Box Q test, McLeod-Li portmanteau test and ARCH-LM test.

3.2.2.1. Autocorrelation Test

One method to testing RWH adopted in numerous study of WFE is the autocorrelation (or serial correlation) test. Autocorrelation function (ACF) measures the linear dependence between returns at current period and its past values. The lag-i sample autocorrelation of r_t is

ρ ℓ = \frac{\sum_{​}^{T} (r_{​} - \overset{─}{r}) (r_{​} - \overset{─}{r})}{\sum_{​}^{T} {(r_{​} - \overset{─}{r})}^{2}}, 0 \leq ℓ < T - 1,

(6)

where ρℓ is the serial correlation coefficient of the returns of lag ℓ, T is the number of observations, r_tis the return for period t specified in equation (3), $\overset{─}{r}$ is the sample mean of return and ℓ is lag of the period. The ACF is used to detect whether the serial correlation coefficients are significantly different from zero under the null hypothesis ρ₁=0 versus the alternative hypothesis ρ₁≠0. If r_t is uncorrelated sequence, the p value is greater than α, the significance level. Similarly, the null hypothesis of RW3 should be rejected if r_t is serially correlated.

3.2.2.2. Ljung–Box Q-test

To test jointly, that several autocorrelations of r_t are zero, the Ljung–Box (1978) modification of Box and Pierce (1970) portmanteau (Q) test is applied. Ljung–Box Q involves subjecting the squared error series to standard tests of serial correlation based on autocorrelation structure using portmanteau tests as follows:

Q_{L B} (m) = T (T + 2) \sum_{ℓ = 1}^{m} \frac{{\hat{ρ}}_{ℓ}^{2}}{T - ℓ},

(7)

where T is the sample size, m is the number of autocorrelation used in the test. Under the condition that r_t is an inid sequence, the Q-statistic is asymptotically a chi-square random variable with degrees of freedom equal to the number of autocorrelation (m). The null hypothesis is that the first m lags of ACF of $ε_{t}^{2}$ are zero (Tsay, 2005, p. 101). The decision rule is therefore to reject null hypothesis if the p value is less than or equal to α, the significance level.

3.2.2.3. McLoed and Li Test

McLeod and Li (1983) show that autocorrelation coefficients and Ljung–Box Q-statistics of the squared residual of an ARMA model can be used to test for nonlinear dependence. The McLeod–Li test is used to examine the squared residual series of r_t for inid. The test statistic is

Q (m) = T (T + 2) \sum_{ℓ = 1}^{m} \frac{{\hat{ρ}}_{ℓ}^{2} (ε_{t}^{2})}{T - ℓ}

(8)

where ε_tdenotes the residual series, and ${\hat{ρ}}_{ℓ}^{2} (ε_{t}^{2})$ is the lag-i ACF of ε_t. Under the null hypothesis of iid, the statistic is asymptotically a random variable with m – p – q degree of freedom. Similar to the Ljung–Box Q test, the decision rule is to reject null hypothesis if the p value is less than or equal to α, the significance level. Any departure from inid case implies that r_t is not a random walk (2) series.

3.2.2.4. ARCH-LM Test

ARCH-LM test is the Lagrange multiplier test of Engel (1982). The basic idea of ARCH model is that shock ε_t of an asset return is serially uncorrelated but dependent (Tsay, 2005, p. 102). Bollerslev, Chou and Kroner (1992) observe that LM test for the null hypothesis of α₀= … α_q= 0 can be calculated as TR2 from the regression of $ε_{t - 1}^{2}, \dots ε_{t - q}^{2}$ . I therefore, apply the ARCH-LM as a test for nonlinear dependence in residuals of r_t thus:

ε_{t}^{2} = α_{0} + α_{1} ε_{t - 1}^{2} - i + \dots . + α_{m} ε_{t - m}^{2} + ε_{t}, t = m + 1, ... T

(9)

The decision rule is to reject the null hypothesis of no ARCH effect (i.e., inid) if p value is less than the level of significance.

4. Empirical Results and Discussions

4.1. Time Series Graph of the NSE 30 and NSE Sector Indexes

Figure 1 shows behaviour of log levels and daily returns of the NI, BI, CG, OG and LII series for the period ranging from 04 January 2010 to 30 April 2014. The log level of all the five series shows trending behaviour from the beginning of the period, albeit minor fluctuations, till the visible decline in last quarter of 2011. This decline was in response to the massive sell-offs sparked by the sovereign debt crisis in Europe and USA in 2011, which affected the global equity markets. In response to the global trend, the average daily value traded on the NSE, for instance, dropped from N3.4 billion in first quarter of 2011 to N2.7 billion in second quarters and further to N1.9 billion by the end of October. Again, the negative spike in the OG in the first quarter of 2012 resulted from losses incurred by the sector during the six-day strike embarked to protest removal of subsidy on premium motor spirit. However, from the first quarter of 2012, the series regained their northward movement. Also noticeable from Figure 1 is the mean reversion tendency of the return series of the indexes. While trending series suggest that the underlying series are non-stationary, mean reverting series indicate stationarity.

Figure 1.

Source: Author’s plot.

4.2. Descriptive Statistics

Descriptive statistics for daily NI, BI, OG, CG and LII are presented in Table 1. As this table shows annualised mean returns are 17.28, 3.36, 2.42, 13.70 and 22.88 per cent for the NI, BI, OG, CG and LII returns, respectively. The annualised mean estimates show that the NI and the sectors under study have started yielding positive returns after the GFC. The annualised volatility of returns are 14.68, 20.01, 21.21, 17.98 and 15.33 per cent for the NI, BI, OG, CG and LII returns, respectively. The annualized mean and volatility show that the OG has the lowest return and highest volatility, whereas the LII has the highest return and lowest volatility for the study period. The skewness of a normal distribution is zero (0). The returns of the NI appear not to be skewed, whereas the return series of the BI exhibit most negative skewness (–3.20). The negative skewness suggests that there are more negative observations in the three sectors than in standard normal distribution. The excess kurtosis of a normal distribution is zero (0). But the excess kurtosis for the return series indicates that they are all leptokurtic, with the NI exhibiting least leptokurtosis. In addition, the Jarque–Bera estimates for the returns are significant at conventional levels suggesting that all the return series are not normally distributed. But the NI returns appear closer to normality than the sector returns.

4.3. Unit Root Test Results

Table 2 shows the results of the Dickey–Fuller (from Dickey & Fuller, 1979) and Phillips–Perron (from Phillips & Perron, 1987, 1988). The difference between the two is in their treatment of serial correlation and heteroscedasticity in the errors. While the ADF tests use a parametric autoregression to approximate the ARMA structure in the errors of the test regression, PP tests ignore any serial correlation in the test regression. Thus, the PP non-parametric test is robust to general forms of heteroscedasticity in error term and is used to confirm the result of the ADF test. The null hypotheses of the ADF and the PP tests are that the time series contains a unit root.

Table 1.
Descriptive Statistics for NSE and Sectors Indexes Returns

NI BI OG CG LII

Observations 1070 1070 1070 1070 1070

Mean 0.0694 0.0135 0.0097 0.0550 0.0919

Std. Dev. 0.8649 1.6086 1.8074 1.2987 0.9434

Skewness 0.0023 –3.2016 0.1474 –0.6877 –0.0819

Kurtosis 1.1370 49.9295 143.550 32.387 2.5234

Jarque–Bera 57.6461 112972.4 918715.12 46850 285.092

	NI	BI	OG	CG	LII
Observations	1070	1070	1070	1070	1070
Mean	0.0694	0.0135	0.0097	0.0550	0.0919
Std. Dev.	0.8649	1.6086	1.8074	1.2987	0.9434
Skewness	0.0023	–3.2016	0.1474	–0.6877	–0.0819
Kurtosis	1.1370	49.9295	143.550	32.387	2.5234
Jarque–Bera	57.6461	112972.4	918715.12	46850	285.092

Source: Author’s computation.

Note: Abbreviations are defined in the text.

As shown in Table 2, the calculated values of the ADF test statistics indicate that the NI, BI, CG, OG, and LII level series contain unit root at the 1 per cent significance level. However, in the case of their returns series, the statistics reject the null hypotheses of unit root at the 1 per cent significance level. Similarly, the results of the PP tests show that the level series are nonstationary, whereas their returns are stationary.

Table 2.

Unit Root Tests Results

Variables	ADF	Computed	PP	Computed	ADF	Computed	PP	Computed
Level Series					Return Series
NI	–3.41	–1.218	–2.86	–0.699	–3.416	–20.87**	–2.86	–23.99**
BI	–3.41	–2.126	–2.86	–2.214	–3.416	–19.275**	–2.86	–24.45**
OG	–3.41	–0.132	–2.86	–0.862	–3.416	–34.72**	–2.86	–34.69**
CG	–3.41	–1.509	–2.86	–1.801	–3.416	–22.25**	–2.86	–34.09**
LII	–3.41	–0.979	–2.86	–0.546	–3.416	–27.68**	–2.86	–27.70**

Source: Author’s computation.

Note: * and ** refer to 1 and 5 per cent statistical significance levels, respectively.

4.4. Results of Random Walk Tests for NSE and Its Sectors

Table 3 displays the ACFs, Ljung–Box Q and Q² statistics, McLeod–Li and ARCH-LM statistics as well as their p values for the returns, residuals and squared residuals of NI, BI, CG, OG and LII. As Table A.1 shows the lag 1 of sample ACFs of the residuals of NI, BI and LII, from the AR-GARCH (1,1) model specified in equation (4), are not significantly different from zero at 1 per cent significance level. In contrast, CG and OG are significantly different from zero at all conventional significance levels. Figures B.1–B.5 in Appendix B, show the ACFs and PACFs of the residuals from the NSE and its sectors’ return series. It is visible that the residual series of the LII, BI and NI returns have no serial correlation. On the other hand, there are big spikes in the ACFs and PACFs of CG and OG residual series. Such noticeable spikes suggest that the residual series may be serially correlated. The Ljung–Box Q statistics supports evidence against serial correlation in BI at 1 per cent level, LII at 5 per cent level and the NI at 10 per cent level, for lags 1, 5 and 10. On the other hand, the coefficients of all lags are significant for the residuals of CG and OG. Clearly, the ACF and Ljung–Box Q statistics provides evidence of RW3 for NI, BI and LII within the conventional levels.

The ARCH-LM statistics also reported in Table 3 suggest that the NI, BI, CG, OG and LII are second-order dependent at lags 1 to 5. Specifically, the p values of lags 1 and 5 of the ARCH statistics for squared residual series of the variables are significantly different from zero at the conventional significance levels, except BI which exhibit second order independence at the 10 per cent level. The ARCH-LM statistics, therefore, reject the null hypothesis of inid in the NSE and its sectors with 95 per cent confidence as the conditional variances appear to be predictable and can be described by an autoregressive volatility model. In other words, ARCH-LM statistics provide evidence against RWH in NSE and its sectors.

The statistics of the Ljung–Box Q² and McLeod–Li are similar. Notice from that the p values of NI, CG, OG and LII are all significant at 1 per cent level at lags 1, 5 and 10. These indicate evidence against inid in their residuals at lags 1–10. On the other hand, the Ljung–Box Q² and McLeod–Li statistics suggest that BI is second-order independent with 99 perc ent confidence at lag 10 and 10 per cent significance level at lag 5. The inid hypotheses is therefore rejected for NI, CG, OG and LII but accepted for the BI based on the decision rules set in Section 3. In other words, the Ljung–Box Q² and McLeod–Li statistics indicate evidence of RW2 for the banking sector in Nigeria. These results are in concord with the a priori expectation of sector with more frequent trading closely following random walk model of WFE. Nigeria Securities and Exchange Commission (SEC) (2010, February), for instance, reports that about 4.04 billion banking sector shares valued at N32.87 billion were exchanged in 70,307 deals, representing 51.35, 60.80 and 52.81 per cent of the month’s total volume, value and number of deals, respectively, whereas trading value of consumer goods sector was N4.81 billion for the same period. More recently, SEC (2013, January) reports that the financial services sector (comprising banking and insurance sectors) accounted for 62.62, 77.17 and 68.04 per cent of the month’s total deals, volume and value, respectively, while the consumer goods sectors controlled 5.43 and 18.47 per cent of the month’s volume and value traded. Again, Emenike and Ani (2014) identify lack of depth and breadth for the other sectors of the NSE as well as inactivity for majority of the listed firms, given that banking companies, made up of about 18 out of 217 listed equities, control over 50 per cent of share traded on the NSE. If the other sectors are as vibrant as the banking sector, it will be difficult for one sector to account for 50 per cent of trades.

Table 3.
Serial Correlation Results for NSE 30 and Sectors from GARCH Residuals

Lags ACF Q Q² LM M-L

Panel a: NSE 30 Returns Series

1 0.042 1.929{0.164} 59.71{0.000} 59.49{0.000} 59.21{0.000}

5 0.054 12.37{0.030 } 104.67{0.000} 75.48{0.000} 101.9{0.000}

10 0.055 22.51{0.012} 139.06{0.000} 134.1{0.000}

Panel b: Banking Sector Returns Series

1 0.040 1.756{0.185} 9.560{0.001} 9.524{0.002} 9.562{0.002}

5 0.056 5.768{0.329} 9.673{0.084} 9.567{0.088} 9.675{0.085}

10 0.040 9.207{0.512} 9.790{0.459} 9.790{0.459}

Panel c: Consumer Goods Sector Returns Series

1 –0.188 37.83{0.000} 257.0{0.000} 256.1{0.000} 257.1{0.000}

5 0.039 44.16{0.000} 257.7{0.000} 352.6{0.000} 257.7{0.000}

10 0.049 50.19{0.000} 257.8{0.000} 257.9{0.000}

Panel d: Oil & Gas Sector Returns Series

1 –0.433 201.0{0.000} 206.9{0.000} 206.1{0.000} 206.8{0.000}

5 0.015 202.5{0.000} 208.3{0.000} 244.9{0.000} 208.3{0.000}

10 0.039 207.6{0.000} 208.4{0.000} 208.4{0.000}

Panel e: Shari’ah Equities Sector Returns Series

1 –0.010 0.100{0.751} 86.80{0.000} 86.47{0.000} 86.739{0.000}

5 0.032 3.043{0.693} 111.08{0.000} 102.6{0.000} 110.9{0.000}

10 0.072 17.763{0.059} 175.8{0.000} 175.7{0.000}

Lags	ACF	Q	Q²	LM	M-L
Panel a: NSE 30 Returns Series
1	0.042	1.929{0.164}	59.71{0.000}	59.49{0.000}	59.21{0.000}
5	0.054	12.37{0.030 }	104.67{0.000}	75.48{0.000}	101.9{0.000}
10	0.055	22.51{0.012}	139.06{0.000}		134.1{0.000}
Panel b: Banking Sector Returns Series
1	0.040	1.756{0.185}	9.560{0.001}	9.524{0.002}	9.562{0.002}
5	0.056	5.768{0.329}	9.673{0.084}	9.567{0.088}	9.675{0.085}
10	0.040	9.207{0.512}	9.790{0.459}		9.790{0.459}
Panel c: Consumer Goods Sector Returns Series
1	–0.188	37.83{0.000}	257.0{0.000}	256.1{0.000}	257.1{0.000}
5	0.039	44.16{0.000}	257.7{0.000}	352.6{0.000}	257.7{0.000}
10	0.049	50.19{0.000}	257.8{0.000}		257.9{0.000}
Panel d: Oil & Gas Sector Returns Series
1	–0.433	201.0{0.000}	206.9{0.000}	206.1{0.000}	206.8{0.000}
5	0.015	202.5{0.000}	208.3{0.000}	244.9{0.000}	208.3{0.000}
10	0.039	207.6{0.000}	208.4{0.000}		208.4{0.000}
Panel e: Shari’ah Equities Sector Returns Series
1	–0.010	0.100{0.751}	86.80{0.000}	86.47{0.000}	86.739{0.000}
5	0.032	3.043{0.693}	111.08{0.000}	102.6{0.000}	110.9{0.000}
10	0.072	17.763{0.059}	175.8{0.000}		175.7{0.000}

Source: Author’s computation.

Notes: p value is displayed as {.}. Q and Q² are the Ljung–Box Q statistic for null hypotheses of no serial correlation in error term and squared error term of the NSE and the sectors’ returns. LM is the ARCH-LM statistics for null hypotheses of inid in the error term.

The implications of the results are that the banking sector returns are not correlated with past returns because changes in returns are in response to new information which is independent of past information. Similarly, NSE and LII returns are uncorrelated but are not independent of past returns. Lack of independence in returns of NSE and LII results from evidence of nonlinearity in their residual and implies that, at least, in the short-term forecast of their returns may be improved by applying nonlinear modelling strategy. Again, prediction of NI, BI and LII returns require superior fundamental analysis of their intrinsic values because those that use the same data and make the same interpretations earn average returns, whereas the returns of CG and OG may be predicted using both technical and fundamental analyses.

Onyema (2013) highlights some of the major achievements of the NSE in 2012 to include strong regulatory environment, equity primary market making, securities lending and short selling, VAT and stamp duty elimination, margin debt relief, among others. Apart from these achievements, two key factors will enhance efficiency of the NSE and its sectors. The first is to enhance research capacity of the market operators and investors through training on the use of information technology to analyse security prices. This will strengthen robust analysis of the information flowing into the market and result in all available information reflecting instantly in share prices. It will also broaden chances of trading based on robust fundamental analysis and not as Adewale and Eromosele (2009) report—trading based on a stock broker’s moves. In addition, it will enhance professionalism and competence of market participants. The second is to reduce equity dominance of the market by increasing the number of asset classes to include derivative instruments such as options, futures, among others. This will boost the depth, breadth and sophistication of the NSE as well as serve as risk mitigation mechanism to protect investors.

5. Conclusions

This article investigates the nature of random walk weak-from efficiency of the NSE and its sectors, namely Banking, Consumer Goods, Oil & Gas and Shari’ah Equities, for the period ranging from 04 January 2010 to 30 April 2014, using autocorrelation test, Ljung–Box Q test, McLeod–Li portmanteau test and ARCH-LM test. The descriptive statistics shows that the daily returns of the NSE and its sectors are positive for the study period, thus imply that the market has become profitable after the GFC. Results of the ACF and Ljung–Box Q statistics indicate evidence of random walk (3) WFE for NSE 30, banking sector and Shari’ah equities sector, but fail to show such evidence for consumer goods sector and oil and gas sector. The results of ARCH-LM provide evidence against random walk (2) within conventional level in NSE and its sectors returns, except the banking sector. The results of Ljung–Box Q² and McLeod–Li tests indicate that the banking sector is random walk (2) weak form efficient, but the NSE, consumer goods, oil and gas and Shari’ah equities sectors are not. The key implications are investors can only predict banking sector return using superior fundamental analysis of their intrinsic values, prediction of the NSE 30 and Shari’ah equities sector returns require nonlinear model and fundamental analysis, and consumer goods sector and oil and gas sector may be predicted using both technical and fundamental analyses

Footnotes

Appendix A

Table A.1.

NSE 30 Companies and Sectors Indexes

S/N	NSE 30	Oil & Gas	Banking	Consumer Goods	Lotus Islamic Index
1	Access	Conoil	Access	Nb	Ashakacem
2	Cadbury	Eterna	Diamondbnk	Vitafoam	Cadbury
3	Dangcem	Fo	Eti	7up	Cap
4	Dangsugar	Mobil	Fidelitybk	Intbrew	Ccnn
5	Diamondbnk	Mrs	Guaranty	Nnfm	Dangcem
6	Eti	Oando	Skyebank	Nestle	Glaxosmith
7	Fbnh	Total	Ubn	Unilever	Jberger
8	Fcmb		Uba	Flourmill	Wapco
9	Fidelitybk		Wemabank	Honyflour	Nascon
10	Flourmill		Zenithbank	Dangsugar	Nestle
11	Fo			Guinness	Nahco
12	Glaxosmith			Pz	Okomuoil
13	Guaranty			Nascon	Pz
14	Guinness			Cadbury	Presco
15	Intbrew			Dangflour	Unilever
16	Jberger
17	Wapco
18	Nestle
19	Nb
20	Oando
21	Pz
22	Skyebank
23	Stanbic
24	Total
25	Transcorp
26	Uacn
27	Unilever
28	Ubn
29	Uba
30	Zenithbank

ACFs and PACFs of the NSE and Its Sectors’ Residuals

Notes

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