P.P. Njiforti and Akaolisa Chidiogo

Department of Economics, Ahmadu Bello University, Zaria




This paper investigated the characteristic growth in the NSE by considering the share prices of selected banks and insurance companies.  Time series data on daily basis for price-dividend ratio, share prices, dividend from 1st  quarter of 2008 to 4th quarter of 2009 were used for analysis. The Augmented Diker Fuller (ADF) test, Augmented Engel Granger (AEG) cointegration test and the Autoregressive Conditional Heteroskedascity (ARCH) are the econometric techniques used to investigate the characteristic bahaviour of the NSE. The ADF result suspected speculative bubbles in most of the banks and insurance companies (i.e. the price-dividend ratio, share prices and dividend were non stationary). The AEG result confirmed that the ADF conducted on the  residuals for most of the banks and insurance companies were non stationary The GARCH estimates suggested  volatility  clustering meaning that the shock in these stocks were persistent. TARCH estimate indicated that positive and negative shocks /news are asymmetry and have asymmetric effect on volatility.  Therefore, it is  concluded that bubble existed in the NSE in the period under review. Information should be well circulated about the stocks in the market and Investors especially illiterate ones should be tutored before they make their investment decision


Key words: Speculative, bubble, stock market, share price, dividend



Bubbles can be defined as increase in share prices and volumes that are far from intrinsic values. It is described as a steep and persistent increase in the price of an asset which is followed by a sharp fall, whatever the causes of price movements (Pratten 1993:29). It is also described as a situation when the price for an asset exceeds its fundamental price by a large margin (WEO.2003). Prices rise or bubble when investors become greedy and act as risk lovers. The bubble is not completed until prices fall back down to normalized levels; this usually involves a period of steep decline in price during which most investors panic and sell out of their investments. While each speculative bubble has its own driving factors and variables, most involve a combination of fundamental and psychological forces. In the beginning, attractive fundamentals may drive prices higher, but over time behavioral finance theories suggest that people invest so as to not "miss the boat" on high returns gained by others. When the artificially high prices inevitably fall, most short-term investors are shaken out of the market after which the market can return to being driven by fundamental metrics (Investopedia). Moreover, a bubble is not indefinitely sustainable. Prices cannot go up forever, and when price increases end, then the increased demand that the price increases generated ends too. Then, a downward feedback can replace the upward feedback.


Nigeria, from a market capitalization of 2.94billion in 1999, the Nigerian Capital Market as at June 2007 had reached a stock value of 63 billion and by March 6, 2008 reached a peak of 12.6 trillion. We see that the percentage increase is quite high and that from history a huge and rapid growth in the stock market will most times lead to a feedback correction mechanism owing to the fact that the growth or increase may not have been genuine, efficient or realistic in the case of those nations. However in the case of NSE we see that this growth (sudden rise in prices of stock) was not sustained for a long period of time such that by the month of March 2008, the market started declining. It was like a joke to the stock market brokers and the NSE body as a whole because they were busy telling people then that it was just a correction that will quickly take place and not affect their investments such that within a little time the market will bounce back. This seems not to be true because  as at March 2009 the market has lost over 50 % from that high peak of March 2008.  It is necessary to see if the growth that was in the market between 2004 and March 2008 could be explained by fundamentals or not, and its impact on the economy. Also relating what is happening globally in the economy it is seen that the melt down or sudden decrease have been experienced might be as a result of domestic factors or foreign factors.Having seen these  experiences by others, it becomes expedient to look at this financial term “bubble” in the case of Nigeria Stock Market knowing that countries that had this experience never remained the same in that despite the fact that their income grew around that period the negative effect was undesirable and disastrous. As believed by most authors that a bubble will always be followed up by a crash due to the imperfections in the financial market it then becomes necessary to consider the NSE and its behavior to observe the following to verify if the NSE has experienced a speculative bubble, and whether the bubbles had impact on the economy. 


Theoretical literature

The efficient market theory assumes rational behaviour. It states that the price of a stock at any given time is equal to the expected present value of the stream of future dividends that will accrue on the stock. This theory can be seen as theory of competitive equilibrium applied to assets market. EMH evolved in the 1960s from the PhD dissertation of Eugene Fama in which he defined an efficient market as a market where there are large numbers of rational profit maximizers, actively competing with each other trying to predict future market values of individual securities and where important current information is almost freely available to all participants. If a market is perfectly efficient, price at all times will reflect consensus of value determined by buyers and sellers acting upon their assessment of all pertinent information (i.e. unexpected news) will cause prices to change quickly until a new consensus of value is reached too quickly for traders to profit from the news unexpected events randomly occur thereby making the market either bullish or bearish (Herbst 1992).



Arch and garch

The ordinary least square (OLS) lacks the ability of estimating fat fail, clustered volatility and large effect nature of financial data because of its assumptions of constant variance and normal Gaussian distribution, this therefore makes it insufficient to handle financial data.


However, this led to the discovery of ARCH – type modes by Engle with a new class of stochastic processes that model time varying conditional variances by relating them to variables known from the previous periods.


The ARCH type-models fit in well with the financial time series data for some reasons like.

I.    Probability distributions for asset returns often exhibit fatter tails than the standard normal or Gaussian distribution.

II.   To explain the volatility clustering that is usually exhibited by financial time series.  This volatility clustering is when large changes tend to follow large charges and small changes follow small changes.

III. The ARCH type-model helps to capture leverage effects that are evident in financial data or assets. (in which asset returns are often observed to be negatively correlated with changes in volatility.

The ARCH model was first introduced by Engle (1982) and as GARCH (Generalised ARCH) by Bollerslev (1986). They have proven to be useful in financial time series analysis.  ARCH which is the same as Autoregressive conditional Hetro-schedasticity is specifically designed to model and forecast conditional mean and conditional variance. Conditional here implies a dependence on the observations of the immediate past while autoregressive emphasizes that there is a feedback mechanism involved which incorporates past observations into the present to explain future variances. Standard econometric tests applied to simulated data confirm that the extent of ARCH effects depends on agent aggressiveness and on the variance of the potential extraneous element that might enter the mystical forecast. If the latter variance is small relative to the variance of the fundamentals or if agents are not very aggressive, then the asset price tends to follow fundamentals nearly all the time. If the variance of the extraneous element is larger and agents are more aggressive, then asset prices show occasional bubble behaviour and both Engle’s (1982) test for ARCH and estimates of a GARCH(1,1) model support the conclusion that the data can be described as ARCH/GARCH for many of the simulations.


GARCH which is Generalized Autoregressive conditional Hetro-schedasticity enables you to take care of the declining effect of information on volatility ie it allows the user to model the serial dependence of volatility.


Two specification are required in the application of ARCH model and are

yt= a+XtY +єt           ------------------------------ (1)

2t=  + є2t-1 + β2t-1   ------------------------ (2)


Where eq (1) is the conditional mean and eq (2) is the conditional variance.

XtY__ exogenous variables

Ф__ mean

є2t-1 __News about volatility from the previous period, measured as the lag of the squared residual from the mean equation, which is the ARCH component.

2t-1____ last periods forecast variance which explains the GARCH term

 & β___ _parameters to be estimated

While eqn. (1) is a function of exogenous variables with an error term, eqn. (2) is a function of mean, lag of the squared residual from the mean equation and last periods forecast variance.

Variables that have been shown to help predict volatility are trading volume macroeconomic news  announcements, implied volatility from option prices and realized volatility, overnight returns, and after hours realized volatility. (Ziovt, 2008).

Volatility clustering and non-Gaussian behave often financial returns is typically seen in weekly, daily or intraday data.

We test for ARCH effects in daily returns using modified Q-statistic or Ljung Box.


Asymmetric ARCH and GARCH models

ARCH/GARCH models thus far has ignored information on the direction of returns, only the magnitude matters but it has been shown that for broad based equity indices and bond market indices, it appears that market declines forecast higher volatility than comparable market increases do. A stylized fact of financial volatility is that bad news (negative shocks) tends to have a larger impact on volatility than good news (positive shocks).  That is volatility tends to be higher in a falling market than in a rising market, this has been attributed to the fact that bad news tends to drive down the stock price.  Thus increasing the leverage (i.e. the debt equity ratio) of the stock and causing the stock to be more volatile.

This asymmetric news impact on volatility is commonly referred to as the leverage effect and can be tested for using the GARCH, EGARCH, TGARCH and PGARCH models are all capable of modeling leverage effect. Zivot (2008)

E-GARCH model is an asymmetric ARCH model and was proposed by Nelson (1991). It stands for exponential Generalized Autoregressive Conditional Hetroschedasticity. It allows for the asymmetry in the responsiveness of returns to the sign of shocks to policy change and is specified in logarithms thereby not imposing the non negativity constraints on parameters. Specified as:


       -           -(3)


Where ht= log2t

An advantage of the E-GARCH model over the basic GARCH model is that the conditional variance 2t is guaranteed to be positive regardless  of the value of the coefficient in eqn. (3), because the log of 2t instead of 2t itself is modeled.

 GARCH-M model allows the conditional variance to affect the mean. Engle, Lilien and Robins (1987) proposed to extend the basic GARCH model so that the conditional volatility can generate a risk premium which is part of the expected returns.  This extended GARCH model is often referred to as GARCH in the mean or GARCH-M model. The estimated coefficient on the expected risk is a measure of the risk return trade-off and is specified thus:

yt = XtY + β2t + єt

yt ____conditional mean return

2t____conditional variance as previously defined

XtY____exogenous variable included in the mean deviation

єt _____error term

TGARCH model is the threshold GARCH model and was proposed by Zakoian (1990) and Glosten, Jaganathan & Runkle (1993). It is specified as follows:


Where St-i denote a dummy variable equal to unity when St−i = 1 if єt−i < 0

0 if єt−i ≥ 0

This then means that depending on whether єt−i is above or below the threshold value of zero є2t-I has different effects on  the conditional variance 2t such that when єt−i is positive, the total effects are given by aiє2t-I when  єt-I is negative, the total effects are given  by (ai + γi2t-I. So, one would expect γi to be positive for bad news to have larger impact.



Methodology and analytical techniques

Unit root test: This is a stationarity test that is necessary in time series data. According to Kasmir& Koskinen (2005) a characteristic property of rational bubbles is that the price-dividend ratio has a unit root. It is carried out using the ADF test. In this case we will investigate univariate time series of price-dividend ratio using unit root tests of ADF test. The ADF test constructs a parametric correction for higher order correlation by assuming that the time series of price-dividend ratio follow an AR (p) process and adding ‘p’ lagged difference terms of the dependent variable –price-dividend ratio to the right hand side of the test regression. So that


Where   yt   -time series of (price-dividend ratio)

                -difference operator

                -assumed to be white noise

y is a stationary series if -1<ρ<1 if the absolute value of ρ is greater than 1, the series becomes explosive and doesn’t make economic sense and so the null hypothesis is tested against the one sided alternative i.e.

                    Null hypothesis H0:=0

          Alternative hypothesis H1:<0

Also if  =0 then ρ =1which implies non stationary

       If <0 then ρ<1 which implies stationary

Running the above model in the E-view, the evidence of a unit root in the price-dividend ratio will be consistent with rational bubbles, this then means that non-stationary price-dividend ratio are consistent with existence of rational speculative bubbles while stationarity implies that deviations from market fundamentals are short lived therefore showing absence of bubbles.


In the stock market if the prices truly reflect the value of the expected future flow of dividends, we should be expecting a cointegration between dividends and stock prices in the long run despite the fact that they both follow random walks. Shiller (2001) quoted Campbell and Shiller (1987) to have argued that if dividend and stock prices fail to co integrate, then there is evidence of a bubble.  If a cointegration exists between the stock price and dividends, it will be suggesting absence of bubbles meaning that there was no serious deviation from fundamentals. In a case where it shows no co-integration it will then suggest that speculative bubble is present and that there is actually a serious deviation from price.   After we must have tested for random walks in the both variables using the unit roots test and found out if ∆dividends and ∆prices are stationary, we can then test for their co-integration by running the OLS regression i.e.





We then test whether the residuals, єt from this regression are stationary. If dividends and stock prices are not co-integrated, any linear combination of them will be non-stationary and hence the residuals єt will be non-stationary. So we test the hypothesis that єt is not stationary which is the hypothesis of no co-integration. This test of the hypothesis єt is non stationary will be done in two ways.


1.      Using the ADF unit root test on the residuals estimated from the cointegrating regression. Then we check the significant values based on the Engle Granger (EG) and augmented Engle Granger tests (AEG).


2. Using CRDW (Cointegrating Regression Durbin Watson) test. This will be done by using the Durbin Watson Obtained from the cointegrating regression and testing it against the null hypothesis that d=0 instead of the standard d=2. Such that if the computed d-value is smaller than the critical values then we reject the null hypothesis of cointegration depending on the level of significance we decide to use.

Arch & garch model: The stock market (price) is one of the financial time series that exhibit volatility clustering. From Singh Ajit(1996),Yartey & Adjasi(2007), Zivot (2008),Xavier (2006), Gurkaynak (2005) It is therefore important for investors in the stock market to know about volatility because high volatility which is a character of bubbles could mean huge losses or gains & hence greater uncertainty, which suggests why we are testing for it in this work. While the ARCH model is a mechanism that includes past variances in the explanation of future variances, the GARCH model takes care of the declining effects of information on volatility.

In constructing an ARCH model, two specifications are needed i.e. one for the conditional mean and the one for conditional variance so that the standard GARCH (1, 1) specification will be

----------------------------------- (7)

----------------- (8)

--------------------------- (9)

-------------------- (10)


Sources of data

Data were sourced from the Daily Price Listings from NSE, NSE fact book, Securities and Exchange Commission quarterly magazine and Central Bank of Nigeria (CBN) statistical bulletin. The banks considered are Access bank, GTB, Intercontinental, UBA and UBN and the Insurance companies are Aiico insurance, Cornerstone insurance, Lasaco insurance, Law union Rock and Niger insurance. The variables considered are the price-dividend ratio, share price  and dividend for the selected banks and insurance companies.


Results and discussions

Stationarity test of price-dividend ratio using adf test

The price-dividend ratio  for all the banks were integrated of order 1 I(1) when the Mackinon critical values were considered at 1% and 5%. (appendix 1.1 and 1.2) The price-dividend ratio for the insurance companies were equally integrated of order one [I(1] except for Niger insurance which showed stationarity at levels (Appendix 2.1 and 2.2). Consequently, the result attested that bubbles existed  according to Kasmir & Koskinen(2005).

The ADF test for bank share price and dividend (appendix 3.1 and 3.2) were stationary at first difference [I(1)].    

The AEG cointegration test on the residuals of banks share price and dividend indicated that bubbles existed in three out of the 5 selected banks (appendix 3.3 and 3.4).    Access, GTB, & Intercontinental banks showed the existence of bubbles while UBA and UBN showed the absence of bubbles.


 For the insurance companies,  the ADF result for the share price and dividend were integrated of first order for test for Cornerstone, Lassaco & Lawunion And The Aeg Cointegrated Test  indicated the existed of bubbles for these insurance companies.  However the result for   AIICO and Niger  was inclusive because one was integrated at level and the other at first difference (appendix 4.1, 4.2, 4.3 and 4.4)


For the 5 companies under the banking sector, ARCH and GARCH estimates,  all but PGTB reflected persistent volatility.  For the insurance companies, only PNIGERIN did not indicate volatility clustering.  The rest suggests persistent volatility meaning that the shock in these stocks were persistent (appendix 5.1 and 5.2).


TARCH estimate (appendix 6.1 and 6.2) indicated that positive and negative shocks /news are asymmetry and have asymmetric effect on volatility.  PINTER, PUBA and PUBN are negative but significant, which implies that volatility tends to fall when the returns surprises are negative i.e. when they come as bad news. In other words negative shocks in these three banks cause less volatility than the positive shocks which means that they contradict the theoretical expectation that negative shocks cause greater volatility than the positive shocks. In essence the effects of bad news on these three banks led to less volatility. However, the other two banks show positive relationship meaning that bad news brings about more volatility. This then suggested that the market had both stabilizing agents and destabilizing agents since most stock markets in reality do not adjust too quickly nor do they persist in their instability. The destabilizing agents are those who follow a behavior that will prolong the unexpected shock and cause persistent volatility while stabilizing agents are those that would see that when there was an unexpected (rise) fall in prices they would expect the prices to (fall) rise.  Therefore,  disturbance from previous mean and variance varied and contributed to the volatility in the stock market because of information asymmetry.  TARCH estimates for insurance companies (appendix 6.2) was similar to that  of the banks i.e. two were negative and two were positive while PNIGERIN had a near singular matrix and so no result could be given for it. 


Therefore, it is can concluded that bubble existed in the NSE in the period under review since both sectors exhibited bubbles except two companies in the insurance sector. In the banking sector the residuals of two banks UBA and UBN were stationary.  However because the banking sector has been observed as the main contributor to the market capitalization of the NSE it can be said that the existence of bubbles in 3 out of the 5 banks sampled influenced the market and made others to also reflect the presence of this bubbles. The ARCH and the GARCH estimates equally manifested volatility clustering and the TARCH confirmed information asymmetry.


Summary and recommendations

The pd-ratio of the insurance and banking sector showed the existence of bubbles.  That the prices and dividends for most of them did not co integrate i.e. they violated the EMH that expected the price and dividends ratio to co integrate at the long run therefore showing an imperfect market that was run by things outside fundamentals.   The fact that most of them violated the EMH means that information is not complete i.e. there is information asymmetry.  Since bubbles have been tested to be in existence it means that the crash that was being experienced might not be out of place since a feedback correction mechanism was expected after bubbles to bring the market back to fundamentals.  6) Since banking & insurance sectors of the stock market contribute more to the market capitalization and it has been seen that bubbles existed in a greater percentage of the ones examined, therefore, bubbles existed in the NSE as a whole.  It was equally found that iInformation asymmetry and other factors contributed to the market deviating from its fundamentals.


Therefore, information should be well circulated about the stocks in the market so that investors will have every necessary information on what they want to invest into. Investors’ especially illiterate ones should be tutored before they make their investment decision.  The NSE body should have a way of moderating the market so that it is not falsified by activities of greedy brokers and investors.   The SEC (Security Exchange Commission) and NSE directors should be at alert when they see unusual purchases or sales going on in the market and should put a limit to the activities of the brokers so that they don’t do whatever they feel like doing thereby affecting the market adversely.   To improve efficiency they should carry out measures that will develop systems that facilitate smooth dissemination of important information to potential investors and rules should be set out to ensure that information is made known at the same time to all and on time.



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