Modeling Volatility of Nigeria Stock Exchange Using Multivariate GARCH Models

The aim of this research work was to provide model for predicting stock volatility in Nigeria Stock market. To achieve this, monthly data for Nigerian stock exchange, Exchange rate, Share index and inflation rate was collected for a period of January 1990 to December 2016.The descriptive statistics revealed these variables to exhibit volatility as a characteristics of financial time –varying series. DCC Model was fitted, were the coefficients for all the parameters and that of the correlation-Targeting (rho_21) are both negative and positive and tend very close to 1 and -1, indicating that high persistence in the conditional variances. The Model DCC, satisfied the properties of a good model of conditional mean and variance of the confidential Interval (C.I) of 1 and -1, that is, the conditional variances are finites and their series are strictly stationary. This therefore implies that the Nigerian Stock Exchange, Exchange rate, share index and Inflation rate will experience a non-steady shock in the Stock market. However Each of these variables have different length of recovery (volatility halflife) ranging from 1.5month, 6.5months, 6months to 2,4months for stock exchange, exchange rate, share index and inflation rate respectively. By implication, the volatility of these variables had a long memory, persistence and mean-reverting.


Introduction
Multivariate GARCH models were initially developed in the late 1980s and the first half of the 1990s, and after a period of tranquility in the second half of the 1990s, this area seems to be experiencing again a quick expansion phase. The univariate ARCH model introduced in Engle (1982) has been extended in several directions by allowing formultiple time series, conditional covariance terms in the mean, and own past conditionalcovariances in each of the covariance equations. Bollerslev et al., (1988) originally proposed the basic framework of MGARCH which extends the univariate GARCH into the vectorized conditional-variance matrix. This VECH model involves a large number of parameters estimation. In order to make estimation more tractable, Bollerslev et al., (1988) proposed the diagonal VECH model. However, this type of MGARCH model could not be used to examine spill-over effects since it simplified the correlation between parameters. The factor GARCH introduced by Engle et al., (1990) reduces the number of parameters to O(k2) but empirical studies reveal its poor performance on low land negative correlations. Alexander (2000) further demonstrates how to apply factor (Orthogonal) GARCH models, which limit the factors accounting for the amount of volatility. The most attractive feature of this kind of MGARCH model is generous enough to provide a method for estimating any variance-covariance matrix using univariate GARCH models. However, Sheppard (2003) criticized this approach in that it is hard to interpret the coefficients on the univariate GARCH model and that it performs poorly for less correlated systems such as individual equities since it reduces the number of parameters to O(k). Engle and Kroner (1995) made improvements based on the work of Baba Engle, Kraft and Kroner and created a general quadratic form for the covariance equation which successfully eliminated the positive definiteness problem of the original VECH model. In the full general BEKK model, the number of parameters needed to be estimated is O(k4), the standard BEKK estimation will involve O(k2) parameters. Other more plausible formulations of BEKK model include diagonal and scalar BEKK where the parameters are restricted to be either diagonal matrices or to be scalars. The most obvious shortcoming of those simplified BEKK models is that some information such as volatility spill-over effects are missing in the variance covariance matrix since the parameters have been reduced. Bollerslev (1990) proposed the Constant Correlation (CC) model; although it still allows volatility time-varying, the conditional correlations are restricted to be time invariant. Tsui and Yu (1999) have found out that the constant correlation assumption can be rejected for certain assets which indicate that the CC model may not be generous enough. In the light of the prementioned limitations of various MGARCH models, Engle (2002) advocate a new class of MGARCH model which is named as Dynamic Conditional Correlation (DCC). Intuitively, the DCC model maintains the plausibility of the CCC model whilst still allowing for time-varying conditional correlation. Sheppard (2003) made a great contribution to the DCC model estimation by reducing the estimation of MGARCH to a series of univariate GARCH process plus an additional correlation estimator. According to Ling and Dhesi (2010), the specification of the univariate GARCH is generous to any GARCH process with normal distribution that satisfies the non-negative constraints and the stationary condition. This recent development is motivated by the usual phenomenon in multivariate modeling of the unequal or mismatching durations of different datasets. Patton (2006) proposed two maximum likelihood estimators (MLEs) of parameters of a multivariate model for time series with histories of different lengths. In comparing DCC with the BEKK model, it is found that, the prominent strength of the DCC model is that it does not suffer dimension hindrance and could be applied to any dimension. This is because the estimation can be decomposed into two steps: estimating the univariate GARCH and subsequently constructing a maximum likelihood function which has only two parameters.
However, the DCC model imposes more restrictions on the type of dynamic effects than the BEKK model. In particular, the conditional variance of returns only depends on the past squared returns, some of which can cause the "volatility spillovers" to be excluded. Similarly, feedback from past volatilities or squared returns on correlations is severely limited in the DCC model (Micheal,2010) and (Ling & Dhesi,2010).). Most scholars chooses the BEKK model to capture the volatility spillover effects and the DCC model to measure the dynamic conditional correlations.
This study aimed at modeling stock volatility in Nigeria Stock market using Multivariate GARCH.

Empirical Review of Nigeria Stock Exchange
Stock Exchange can simply be defined as an organized and regulated financial market where securities (bonds, notes, shares) are bought and sold at prices governed by the forces of demand and supply. Stock exchanges basically can be categorising into two which are primary and secondary markets (Friedmann & Sanddorf, 2002). Stock market volatility can be said to be a measure for variation of price of a financial asset over time. It is essentially, concerned with the dispersion and not the direction of price changes. Issues of volatility in stock market behaviour are of importance as they shed light on the data generating process of the returns (Hongyu & Zhichao, 2006). As a result, such issues guide investors in their decision making process because not only are the investors interested in returns, but also in the uncertainty of such returns and efforts toward financial sector reforms would be an exercise in futility if volatility of stock market is not addressed.
Stock exchange issue has attracted the attentions of researchers and statisticians all over the world. The ability to predict the stock price to meet the fundamental objectives of investors and operators of stock market for gaining more benefits cannot be overemphasized. Stock markets are influenced by numerous factors and this has created a high controversy in this field. Many methods and approaches for formulating forecasting models are available in the literature which comprises of both the empirical and theoretical literature on the Nigeria Stock exchange.
Several research and approaches have been carried out or used to model monthly returns from the Nigeria Stock Exchange in which are discussed below. Bala, & Takimoto (2017) investigated stock returns volatility spillovers in emerging and developed markets (DMs) using multivariate-GARCH (MGARCH) models and their variants.
They analysed the impacts of global financial crisis (2007)(2008)(2009) on stock market volatility interactions and modify the BEKK-MGARCH-type models by including financial crisis dummies to assess their impact on volatilities and spillovers. Their major findings reveal that correlations among emerging markets (EMs) are lower compared with correlations among DMs and increase during financial crises. Furthermore, they detected evidence of volatility spillovers and observed that own-volatility spillovers are higher than cross-volatility spillovers for EMs suggesting that shocks have not been substantially transmitted among EMs compared to DMs.
They also found significant asymmetric behaviour in DMs while weak evidence is detected for EMs. Finally, the DCC-with-skewed-t density model provided improved diagnostics compared to other models partly due to its taking into account fat tails and skewed features often present in financial returns. Eke (2016) examined the statistical modeling of monthly returns from the Nigerian stock exchange using probability distributions (normal and logistic distributions) for the period, 1995 to 2014. He analyzed the data to determine the returns from the stock exchange, its mean and standard deviation in order to obtain the best suitable distribution of fit. Furthermore, comparative analysis was done between the logistic distribution and normal distribution. The findings proved that the logistic distribution is better than the normal distribution in modeling the returns from the Nigeria stock exchange due to its p -value which is greater than the 5% significance level of alpha and also because of its minimum variance. Yaya et al. (2016) estimates the dynamic pattern of the Nigeria All Share Index (ASI) from January 3, 2006 to July 22, 2014. Parameter estimates of the models were obtained using the Quasi Maximum Likelihood (QML) approach, and in-sample conditional volatility forecasts from each of the models were evaluated using the minimum loss function approach. Among the classical volatility models, the initial results detected IGARCH-t as the best model for predicting volatility in the ASI. However, in estimating the GAS variants, the Beta-t-EGARCH model proves to predict the volatility in the stock returns better than the IGARCH-t. The estimates could not improve further when the skewed version of the Student-t distribution was considered. We therefore recommend the GAS, EGAS and AEGAS family models in predicting jumps, outliers and asymmetry in financial time series modelling. Olayiwola et al. (2016) focused their studies on the returns on the All-Share Index of the Nigerian stock exchange on the Nigerian economy. A time plot was used to examine the trend, seasonality, and discontinuity of the data. Box Jenkins approach with R statistical package was used to examine the stationary of Nigeria All-Share Index. The parameters of the model were estimated using the maximum likelihood estimation techniques and Akaike Information Criteria (AIC) was used to check for the goodness of fit of the fitted model. Murekachiro (2016) explored the comparative ability of different statistical and econometric volatility forecasting models in the context of Zimbabwe stock market. He considered two different models in his study. The volatility of the ZSE industrial index returns have been modeled by using a univariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models including both symmetric and asymmetric models that captures most common stylized facts about index returns such as volatility clustering, fat tails and leverage effect, these models are GARCH (1,1) and exponential GARCH (1,1). The first model he used for capturing the symmetry effect whereas the second model is for capturing the asymmetric effect. The study used a stock market average daily industrial index from Zimbabwe (ZSE index), a country not previously considered in the volatility literature, for the period 19 February 2009 to 31 December 2014. Basing on the empirical results presented, the following can be concluded. ZSE data showed a significant departure from normality and existence of conditional heteroskedasticity in the residuals series. For all periods specified, the empirical analysis evidenced that asymmetric EGarch (1,1) model outperform the symmetric Garch (1,1) model on forecasting future volatility after using two different evaluation techniques of Error measures statistics and regression based analysis. Future studies should consider the general extensions of Garch models, in particular symmetric and asymmetric Garch (p; q) rather than symmetric and asymmetric Garch (1; 1) model. This general extension is helpful because such higher order models are often useful when long span of data is used. Therefore, with additional lags such models allow both fast and slow decay of information. models of heteroscedastic processes, namely: GARCH (1,1), EGARCH (1,1) and GJR-GARCH models respectively. The four firms whose share prices were used in this analysis are UBA, Unilever, Guinness and Mobil. All the return series exhibit leverage effect, leptokurtosis, volatility clustering and negative skewness, which are common to most economic financial time series. Except for Guinness, other series display significant level of second-order autocorrelation, satisfying covariance-stationary condition. These models were estimated assuming a Gaussian distribution using Brendt-Hall-Hall-Hausman (BHHH) algorithm's program in Eview software platform. The estimation results reveal that the GJR-GARCH (1, 1) gives better fit to the data and are found to be superior both in-sample and out-sample forecasts evaluation.
Stelzer (2008) established that all VEC models not representable in the simplest BEKK form contain matrices as parameters which map the vectorised positive semi-definite matrices into a strict subset of themselves. Moreover, a general result from linear algebra is presented implying that in dimension two the models are equivalent and in dimension three a simple analytically tractable example for a VEC model having no BEKK representation is given. Okafor (1985) & Anyafo (1994 identify securities traded based on the nature of rights and control exercised by US, security-holder. Hence, this ultimately depends on the type of security held. In that regard, Okafor (1996) identifies classification of securities that determine the right of a holder.

Multivariate GARCH (MGARCH) model
Multivariate GARCH models have been studied intensively in recent years and many different specifications have been used in the literature (Bauwens, Laurent &Rombouts, 2006) for a comprehensive overview and Boussama (1998,2006) for a detailed discussion on strict stationarity and geometric ergodicity. In this paper, the multivariate GARCH (MGARCH) models to be considered include BEKK, CCC, DCC

Baba, Engle, Kraft and Kroner (BEKK) Model
For an estimated multivariate GARCH model to be plausible, the parameter t  is required to be positive definite for all values of the disturbances. Engle & Kroner (1995) proposed a quadratic formulation for the parameters that ensured positive definiteness. This became known as the BEKK model (Brooks et al., 2003). This model is relatively parsimonious and suitable for a large set of assets because its number of parameters grows linearly with the number of assets (De Goeij et al., 2004). The BEKK model is given as shown in the form below.
Based on the symmetric parameterization of the model, t  is almost surely positive definite provided Engle and Kroner (1995) proved that the necessary condition for the covariance stationarity of the BEKK model is that the eigenvalues, that is the characteristic roots of,

 
should be less than one in absolute value. Thus, the process can still render stationary even if there exists an element with a value greater than one in the matrix. Obviously, this condition is different from the stationarity condition required by univariate GARCH model: that the sum of ARCH and GARCH terms has to be less than one (Pang et al, 2002).
The BEKK (1, 1, K) model is defined as: The diagonal and scalar BEKK models can be defined as follows: i. The diagonal BEKK model. Take, k A and k B as diagonal matrices. For this case, the BEKK model is a restricted version of the VEC model with diagonal matrices (Bauwens, 2005;Franke et al, 2005).
ii. The scalar BEKK model. , , Where a and b scalars and U is a matrix of ones.
smaller than one in modulus, and thus

Constant Conditional Correlation (CCC) model
Correlation models are based on the decomposition of the conditional covariance matrix into Conditional standard deviations and correlations. The simplest multivariate correlation model that is nested in the other conditional correlation models, is the Constant Conditional Correlation (CCC-) GARCH model of Bollerslev (1990). In this model, the conditional correlation matrix is time-invariant, so the conditional covariance matrix can be expressed as follows: Positivity of t S follows from the positivity of t  and that of each , ii t  (Bauwens, 2005). As we said before, it is often difficult to verify the condition that the conditional variance matrix of an estimated multivariate GARCH model is positive definite. Furthermore, such conditions are often very difficult to impose during the optimization of the log-likelihood function. However, if we postulate the simple assumption that the correlations are time invariant, these difficulties elegantly disappear (Tse, 2000). Bollerslev (1990) (Bauwens, 2005).
Because of its simplicity, the CCC model has been very popular in empirical applications. A specific member of the group of CCC models is obtained by further constraining the correlations to be zero. This model is denoted as the no correlation (NC) model. Thus the CCC model is given by and the NC model is its special case with 0 i j   (Tse, Tsui,1999). The restriction that the constant conditional correlations, and thus the conditional covariances, are proportional to the product of the corresponding conditional standard deviations highly reduces the number of unknown parameters and thus simplifies estimation (Bauwens et al., 2006).

The Dynamic Conditional Correlation Model
This model was developed by Engle et al (2011). The Dynamic Conditional Correlation models is t = t t t 3.9 Where t is the covariance matrix and Dt is an n x n matrix of the conditional correlation of the returns. The diagonal matrix Dt is expressed as This matrix consists of the univariate GARCH models. Furthermore, t has to be positive definitive, which is automatically obtained while t is a correlation matrix that is symmetric by definition. When this matrix is defined, two requirements are needed. Firstly, tneed to be positive definite since it is a covariance matrix. Secondly, the parts that belong to t need to be less than one. This requirements are met through a decomposition: t= t= --1 for I = 1 ...n where Moreover, the scalars a and b must be larger than zero, but the sum has to be less than one. One may note that these are conditions of the univariate GARCH to be stationary, but which is applied in the DCC model. (Orskaug, 2009)

Estimation of Multivariate GARCH model
Suppose the vector stochastic process   t r (for t = 1,..., T) has conditional mean, conditional variance matrix and conditional distribution The likelihood function for the i.i.d. case can then be viewed as a quasi-likelihood function (Bauwens et al., 2006). Consequently, one has to make an additional assumption on the innovation process by choosing a density function, denoted

Distribution forms and Estimation of GARCH models
Estimation of GARCH models is based on the assumption of normality, Students t and Generalized Error Distribution (GED) for the innovations series t  . The log-likelihood from the normal distribution is Where v is the degrees of freedom to be estimated and   .
Where v is the tail thickness parameter.

Model Selection Criteria
The GARCH variants will be evaluated by Akaike's (1994) Information Criterion (AIC) and Schwarz (1978) Bayesian Information Criterion (SBIC), even though the statistical properties of the criteria in the GARCH context are yet to be known. The two criteria are given as, Where   t l  is the maximum likelihood function conditioned on the parameter set.

Mean Reversion and Calculation of half-life of volatility shock for a stationary
The high or low persistence in volatility is generally captured in the GARCH coefficient(s) of a stationary GARCH model. For a stationary GARCH model, the volatility mean reverts to its long run level, at the rate given by the sum of ARCH and GARCH coefficients, which is generally close to one for a financial time series.
The average number of time periods for the volatility to revert to its long run level is measured by the half-life of the volatility shock. The mean reverting rate implied by most fitted model is usually very close to 1. The magnitude of controls the speed of mean reversion.
The half-life of a volatility shock is given by the formula: 3.17 The closer is to one the longer is the half-life of a volatility.
If ,the GARCH model is nonstationary and the volatility will eventually explode to infinity. (Banerjee and Sarkar, 2006) 4.0 Source of Data The data for this research work being a secondary data were collected from Table A

Data Presentation and Analysis
This section which is the most important part of this paper is made up of the data presentation, analysis and discussions of data from secondary sources. The volatility test using multivariate  Exchange rate has the probability value of 0.3360 which indicated normality.        indicates that a GARCH model can be used to fit the data.

Stock Exchange Log Return Exchange Rate Log Return
Inflation Rate Log Return Share Index Log Return It was observed in Figure 5 that the log returns of Stock exchange, Exchange rate, Inflation rate and Share index offer evidence of the well-known volatility clustering effect. It is a tendency for volatility in financial markets to appear in bunches. Large returns (of either sign) are expected to follow large returns and small returns (of either sign) to follow small returns (Brooks, 2002).      The kurtosis of stock exchange, Exchange rate and inflation are just like the kurtosis of most financial asset returns are larger than three which means that they have too many extreme value to be normally distributed while that of Share Index that is less than three (3) is the reverse.      -4.4679 -4.4679 -8.130043 -8.116055 persistence in the conditional variances. The Model DCsC, satisfied the properties of a good model of conditional mean and variance of the confidential Interval (C.I) of 1 and -1 that is the conditional variance is finite and the series are strictly stationary. The Nigerian Stock Exchange will experience a non-steady shock in the Stock market. (Samson &Akinwande, 2017).  Table 4.11 shows that the coefficients of all the parameters and correlation-Targeting value are all positive at 5% level. The DCC Model shows that alpha1 + beta1 = 0.000000 + 0.900000 < 1 and rho_21 (correlation Targeting) =0.044103>-1,both of them are very close to 1 and -1, indicating that high persistence in the conditional variance. This also means that conditional variance is finite and the series are strictly stationary, thus the Exchange Rate will experience a volatility in Nigeria stock market. (Samson & Akinwande,2017).   Table 4.13 shows that the coefficients of all the parameters are positive at 5% level. DCC Model was fitted, were the coefficients for all the parameters and that of the correlation-Targeting (rho_21) are positive and they are very close to 1 and -1, indicating that high persistence in the conditional variances. The Model DCC, satisfied the properties of a good model of conditional mean and variance of the confidential Interval (C.I) of 1 and -1 that is the conditional variance is finite and the series are strictly stationary. The Nigerian Inflation rate will experience a nonsteady shock in the Stock market. (Samson &Akinwande, 2017).

Discussion of Findings
The relevant descriptive statistics in the context of this study are mean, median, maximum, minimum, standard deviation, skewness, kurtosis, jarque-bera test and p-values. These statistics were computed for the monthly data set of inflation rate, exchange rate, share index, stock exchange from January, 1990 to December, and 2016. exchange. The data for exchange rate are approximately symmetric since its skewness is between -0.5 and +0.5 (Atoi, 2014). The distribution of data set for share index and stock exchange are moderately skewed. The kurtosis values of exchange are positive and appear to be smaller than 3 implying that they variables are all platykurtic in nature while the kurtosis of inflation rate is greater than 3 implying that it is leptokurtic in nature (highly peaked than the normal). This in line with the work of Engle and Patton (2001), while in the case of inflation rate, share index and stock exchange, the probabilities of the Jarque-Bera (JB) statistics are zeros less than the 5% level of significance. Thus, null hypothesis which state that the data are normally distributed is rejected at 5% level of significance.
However, the probability of the JB statistics for exchange rate is 0.3360 greater than 5% level of significance since (P-value) hence, exchange rate is normally distributed. Figure5 represents graphs of the return series of the variables. Table 4.3. and Table 4.5 compare the four variables under the scalar BEKK(1,1) model and diagonal BEKK(1,1) model and clearly revealed that Nigeria stock exchange has the best performance among the variables with highest log-likehood and minimum information criteria. Tables4.6 and 4.8 .revealed that theuniverate models are unfit and non-stationary according the assumption that if the mean reverting rate ( ) is greater than one and since the volatility will eventually explode to infinity (Banerjee and Sarkar, 2006).
Thus it becomes impossible to forecast with the univariate GARCH models due to its instability. Exchange, Exchange rate, share index and Inflation rate will experience non-steady shock (volatility) in the Stock market. (Samson &Akinwande, 2017). In each of these variables, the mean reverting rate ( ) and volatility half-life was calculated using the formula described in section 3.8. For stock exchange ,the reverting rate ( ) was 0.6364 and the volatility half-life was calculated to be 1.5 month (one month and fifteen days). For exchange rate, the mean reverting rate ( ) was 0.9000 and the volatility half-life was calculated to be 6.5 months (six months and fifteen days).For share index, the mean reverting rate ( ) was 0.8904 and the volatility half-life was calculated to be 6months.For inflation rate, the mean reverting rate ( ) was 0.7467 and the volatility half-life was calculated to be 2.4months (two months and twelve days). By implication, in all these four cases, the volatility had a long memory, persistence and mean-reverting (Banerjee and Sarkar, 2006).
In summary, the univariate models revealed instability and explosion to infinity of volatility. As a result, making it impossible to make forecast on the variables in Nigeria stock market. DCC Model was fitted, the coefficients for all the parameters and that of the correlation-Targeting (rho_21) are positive and they are very close to 1 and -1, indicating that high persistence in the conditional variances. The Model DCC, satisfied the properties of a good model of conditional mean and variance of the confidential Interval (C.I) of 1 and -1 that is the conditional variances are finites and their series are strictly stationary. The Nigerian Stock Exchange, Exchange rate, share index and Inflation rate will experience non-steady shock (volatility) in the Stock market. (Samson &Akinwande, 2017) .Each of these variables have different length of returning to normalcy after shock ranging from 1.5month, 6.5months, 6months to 2,4months, accordingly.

Conclusions and Recommendations
The return series of the variables also shows periods of low and high volatilities, which signify volatility clustering. The parameters of the four variables are estimated and compared using Multivariate Garch models. The Model DCC, satisfied the properties of a good model of conditional mean and variance of the confidential Interval (C.I) of 1 and -1 that is the conditional variances are finites and their series are strictly stationary. The Nigerian Stock Exchange, Exchange rate, share index and Inflation rate will experience non-steady shock (volatility) in the Stock market (Samson & Akinwande, 2017) .Each of these variables have different length of returning to normalcy after shock ranging from 1.5month, 6.5months, 6months to 2,4months, respectively. The findings from this study raise some policy issues and recommendations, which will reinforce the link between the stock exchange, exchange rate, share index and inflation rate in Nigerian Stock market as follows: (i) That market information should be allowed to flow unhindered and aggressive trading on a wide range of securities be encouraged so as to increase market depth.
(ii) Investors should be wary of period of instability in the Nigeria Stock market as volatility is bound to be sustained for relatively long period of time.
(iii) Investors and other players in the Nigerian Stock market should be able to sift and distinguish good news from bad news as market return discriminates the response to each. Thus investors should refrain from investing just before the market receives bad news as the momentum of the effect (loss) will take relatively longer time to subdue.
(iv) Investor can invest immediately before the announcement of good news and divest not long afterwards as the positive effect of the market will only last a relatively shorter period of time.
(v) Extensions of this work using a combination of concepts from half-life and the unconditional variance extracted from the fitted variance model would be worthwhile. It will improve upon investors' knowledge on their investing ability and chances.