Assessing the dynamics of financial assets: MGARCH modelling of Nigerian stock returns, all share index, and crude oil prices

Introduction

The Nigerian financial landscape is intricately shaped by the interplay between its stock market and the global oil market. As one of Africa's largest oil producers, Nigeria's economic health is closely tied to crude oil prices. The Nigerian Stock Exchange (NSE), particularly the All Share Index (ASI), serves as a barometer for investor sentiment and economic stability. Understanding the volatility of these financial assets is crucial for investors, policymakers, and economic analysts. Recent studies have employed Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models to analyze the volatility dynamics of Nigerian financial assets. For instance, Atoi (2015) [1] utilized various GARCH models to assess the volatility of the ASI, highlighting the significance of error distributions in enhancing model efficiency. Similarly, Yaya et al. (2016) [8] applied Beta-t-GARCH variants to model volatility in the Nigerian stock market, emphasizing the importance of accounting for jumps and asymmetry in financial time series. The relationship between crude oil prices and stock market volatility has also been a focal point of research. Studies by Tumala et al. (2023) [6] and Okorie (2022) [3] have explored the impact of oil price shocks on stock market volatility in Nigeria, employing advanced GARCH-MIDAS and asymmetric models to capture the nuances of this relationship. These studies underscore the sensitivity of the Nigerian stock market to global oil price fluctuations, reflecting the country's dependence on oil revenues. Furthermore, the application of Multivariate GARCH (MGARCH) models has provided deeper insights into the interconnectedness of financial markets. Research by Bagirov (2025) [2] and Salisu & Oloko (2018) [4] demonstrated the effectiveness of MGARCH models in capturing volatility spillovers between crude oil prices and stock market returns, offering a comprehensive view of market dynamics. This study aims to extend the existing literature by employing MGARCH models to assess the volatility dynamics of Nigerian stock returns, the ASI, and crude oil prices. By doing so, it seeks to provide a nuanced understanding of the interrelationships among these financial assets, thereby contributing to more informed investment and policy decisions in Nigeria's financial markets.

Literature Review

The relationship between crude oil prices and the Nigerian stock market has been extensively studied, with various methodologies employed to understand the dynamics of this interaction. Early research by Tule et al. (2017) [5] utilized the VARMA-AGARCH model to examine volatility spillovers between oil prices and the Nigerian sovereign bond market. Their findings indicated significant cross-market volatility transmission, highlighting the sensitivity of financial markets to oil price shocks. Similarly, Usoro and Ekong (2022) [7] applied MGARCH models to assess the volatility of crude oil production and price in Nigeria, revealing a bidirectional causality between production quantity and price, which underscores the interconnectedness of these variables. In the context of the Nigerian stock market, studies have focused on the volatility dynamics of the All Share Index (ASI). Atoi (2015) [1] employed various GARCH models to analyse the volatility of the ASI, emphasizing the importance of error distributions in enhancing model efficiency. Yaya et al. (2016) [8] applied Beta-t-GARCH variants to model volatility in the Nigerian stock market, highlighting the significance of accounting for jumps and asymmetry in financial time series. These studies contribute to a deeper understanding of the factors influencing stock market volatility in Nigeria. Despite the extensive research, there remains a gap in the literature regarding the simultaneous modelling of volatility spillovers between Nigerian stock returns, the ASI, and crude oil prices using Multivariate GARCH (MGARCH) models. This study aims to fill this gap by employing MGARCH models to assess the volatility dynamics of these financial assets, providing a comprehensive analysis of their interrelationships. Understanding these dynamics is crucial for investors and policymakers to make informed decisions in the face of global economic uncertainties and oil price fluctuations.

Methodology 3.1 The Multivariate GARCH Models

Multivariate GARCH (MGARCH) models are extensions of the univariate GARCH models, enabling the modelling of volatility dynamics across multiple interrelated time series. Unlike univariate GARCH models, which focus on modelling the conditional volatility of a single series, MGARCH models account for the conditional covariances and correlations between multiple assets or variables. A well-known MGARCH model is the Constant Conditional Correlation (CCC) model, introduced by Bollerslev in 1990. The CCC model assumes that the conditional correlations between the residuals of different series remain constant over time, while allowing for time-varying conditional variances. Despite its relative simplicity, the CCC model is effective in capturing the co-movement and volatility structure across financial time series and other multivariate datasets. Another important MGARCH model is the Dynamic Conditional Correlation (DCC) model, proposed by Engle in 2002. Unlike the CCC model, the DCC model allows for time-varying conditional correlations between the residuals of different series, offering greater flexibility in modelling dynamic interdependencies. This feature makes the DCC model particularly suitable for capturing changes in the strength and direction of correlations between assets over time. Additionally, there are further variations and extensions of MGARCH models, such as the Vector Autoregressive Moving Average GARCH (VARMA-GARCH) model and the factor MGARCH model. These models incorporate additional structures, like vector autoregressive dynamics or common latent factors, which drive the volatility across multiple series. MGARCH models are crucial in capturing the complex dependencies and volatility transmission across multivariate time series. They are widely used in applications such as risk management, portfolio optimization, and forecasting in finance, economics, and macroeconomics, as they allow for the analysis of the conditional variance-covariance structure and the interrelationships among multiple time series simultaneously. The MGARCH models handles multiple time series. It is used to analyse and forecast the conditional volatility and correlation between multiple financial time series. MGARCH models can handle multiple time series, allowing for the analysis of relationships between different financial instruments. In MGARCH models, the conditional variance and covariance matrices evolve over time, depending on past values of the time series. The model assumes that the data follows a multivariate distribution and that conditional variances and covariances are functions of lagged innovations (shocks or residuals) and past conditional variances. The general form of an MGARCH model can be represented as:

Where;  𝑟 𝑡 =n × 1 vector of log returns of n assets at time t.  𝜇 𝑡 = is the vector of mean returns (or a constant)  𝜀 𝑡 = n × 1 is the vector of residuals (errors) at time t,  𝐸(𝜀 𝑡 ) = 0 𝐶𝑜𝑣(𝑎 𝑡 ) = 𝐻 𝑡 .

 𝜇 𝑡 = n × 1 vector of the expected value of the conditional 𝑅 𝑡 .  𝐻 𝑡 = n × n matrix of conditional variances at time t.

 𝐻 𝑡 1 2 = Any n × n matrix at time t such that 𝐻 𝑡 is the conditional variance matrix of 𝑎 𝑡 . 𝐻 𝑡 1 2 could be gotten by Cholesky factorization of 𝐻 𝑡  𝜀 𝑡 = 𝐻 𝑡 1 2 𝑍 𝑡 where 𝑍 𝑡 is a vector of independent and identically distributed errors (often assumed to be normal or t-distributed), and 𝐻 𝑡 is the conditional covariance matrix at time ttt.  𝑍 𝑡 is an n × 1 vector of iid errors such that 𝐸(𝑍 𝑡 ) = 0 and 𝐸(𝑍 𝑡 𝑍 𝑡 ′ ) = 1

Just like in the single-variable case, the residuals (𝑎 𝑡 appear to be uncorrelated over time. However, this doesn't imply a complete lack of serial dependence; rather, any existing dependence is non-linear in nature MGARCH models are used to analyse the conditional variance and covariance structure of multivariate time series data. There are numerous ways to specify the conditional covariance matrix 𝐻 𝑡 . However, as the number of variables (dimension of at) becomes larger, the number of parameters in 𝐻 𝑡 increases exponentially, making estimation computationally challenging, especially for large datasets. To overcome this, MGARCH models aim to strike a balance between parsimony and flexibility. Additionally, ensuring that the conditional covariance matrix remains positive definite is crucial, as this guarantees that the matrix is valid and can be inverted efficiently. The conditional covariance matrix 𝐻 𝑡 in an MGARCH model depends on past residuals and past covariance terms, and it models how the volatility and correlation evolve over time. There are different types of MGARCH models, such as: CCC-GARCH (Constant Conditional Correlation GARCH) which assumes constant correlations between assets while allowing variances to change over time; DCC-GARCH (Dynamic Conditional Correlation GARCH) which allows time-varying correlations between the assets, providing greater flexibility to capture dynamic interdependencies; The VARMA-GARCH which Incorporates vector autoregressive moving average dynamics into the volatility modelling. This study utilized the Dynamic Conditional Correlation (DCC) model which is particularly suitable for analysing the volatility and interdependence of multiple financial assets because it allows for time-varying correlations. Unlike constant-correlation multivariate models, the DCC model captures the evolving relationship between asset returns, such as Nigerian stock returns, the All Share Index (ASI), and crude oil prices, over time. This is crucial in financial markets that are highly sensitive to shocks from both domestic and global factors, including oil price fluctuations and economic policy changes. Additionally, the DCC model efficiently accommodates large datasets and provides more accurate estimates of volatility spillovers and co-movements compared to simpler multivariate GARCH specifications. Its flexibility in modeling conditional correlations dynamically enables researchers and investors to monitor changing risk structures, optimize portfolio allocation, and implement more informed hedging strategies. Consequently, employing the DCC model provides a robust framework for understanding the interconnectedness of Nigerian financial markets and the impact of exogenous shocks on asset volatility.

Dynamic Conditional Correlation (DCC) Model (DCC-) GARCH belongs to the class of models the take models of conditional variances and correlations. It was introduced by Engle and Sheppard in 2001. The Dynamic Conditional Correlation (DCC-) GARCH model is a popular econometric model used to estimate time-varying correlations between multiple financial time series. It is an extension of the GARCH (Generalized Autoregressive Conditional

Heteroskedasticity) model, which itself models time-varying volatility. Models in this category are based on the approach of modelling conditional variances and correlations, rather than directly focusing on the conditional covariance matrix. The conditional covariance matrix is instead expressed as a combination of conditional standard deviations and a correlation matrix, as follows:

Where;  𝑅 𝑡 is the time-varying correlation matrix

Models in this class can be classified in two groups; those with a constant correlation matrix and those when the correlation matrix is time-varying. The DCC-GARCH model combines univariate GARCH models for individual series with a dynamic correlation structure. Each financial time series is modelled using a univariate GARCH model to capture its own time-varying volatility:

𝜀 𝑡 = 𝐻 𝑡  𝑍 𝑡 could also be defined as a Vector of standardized residuals. 𝑍 𝑡 ~𝑁(0,1)

The idea of the models in this class is that the covariance matrix, 𝐻 𝑡 =, can be decomposed into conditional standard deviations, 𝐷 𝑡 , and a correlation matrix, 𝑅 𝑡 . The elements in the diagonal matrix 𝐷 𝑡 are standard deviations from univariate GARCH models.

Where 𝜎 1𝑡 is the time-varying conditional standard deviation (square root of variance) of the 𝑖𝑡ℎ time series at time 𝑡, typically obtained from the univariate GARCH models. The overall conditional covariance matrix 𝐻 𝑡 is constructed as 𝐻 𝑡 = 𝐷 𝑡 𝑅 𝑡 𝐷 𝑡 Where: 𝑅 𝑡 is the time-varying correlation matrix (from the DCC part of the model). In the DCC-GARCH model, the matrix 𝑅 𝑡 is the time-varying conditional correlation matrix. It represents the dynamic correlations between the standardized residuals of the time series at time t. The residuals 𝑍 𝑡 = 𝐷 𝑡 -1 𝜀 𝑡 are standardized with unit variances, which are used to estimate 𝑅 𝑡 .

Since 𝑅 𝑡 is a correlation matrix it is symmetric, thus we have: (7) After estimating the conditional variances, the model estimates the conditional correlation matrix 𝑅 𝑡 , which is dynamic over time. 𝐻 𝑡 has to be positive definite because it is a covariance matrix. To ensure 𝐻 𝑡 to be positive definite, Rt has to be positive definite (Dt is positive definite since all the diagonal elements are positive). All the elements in the correlation matrix Rt have to be equal to or less than one. The standardized residuals 𝑍 𝑡 are used to estimate the dynamic correlation matrix 𝑅 𝑡 . To ensure both of these requirements in the DCC-GARCH model, Rt is decomposed into:

Likelihood Function for 𝝐 𝒊,𝒕

The conditional distribution of the residuals 𝜖 𝑖,𝑡 is: 𝜖 𝑖,𝑡 / ℱ 𝑡-1 ~𝑁(0, 𝜎 𝑖,𝑡 2 ), where ℱ 𝑡-1 signifies the information set available up to time t-1. The probability density function (PDF) for a normal random variable 𝜖 𝑖,𝑡 with mean 0 and variance 𝜖 𝑖,𝑡 is given by:

The log-likelihood for a single observation is the natural logarithm of the PDF:

For N observations, the total log-likelihood is:

Where; 𝜖 𝑖,𝑡 = 𝑦 𝑖,𝑡 -𝜇 𝑖,𝑡 are the residuals

DCC Likelihood

The joint likelihood of the multivariate DCC model is based on the standardized residuals 𝑍 𝑡 = (𝑧 1,𝑡 , 𝑧 2,𝑡 , 𝑧 3,𝑡 , … , 𝑧 𝑁,𝑡 ) 𝑇 .

The log-likelihood for the DCC part is: Where; 𝑃 𝑡 is the price at time t 𝑃 𝑡-1 is the price at time, 𝑡 -1 These results suggest that the series exhibit non-normality and may require robust statistical methods for analysis. The Jarque-Bera test confirms the non-normality of the series, rejecting the null hypothesis of normality for all three series. This implies that traditional statistical methods assuming normality may not be suitable for these series. Overall, the results highlight the importance of considering the unique characteristics of each series when selecting statistical models or methods for analysis. By acknowledging and addressing these characteristics, researchers can develop more accurate and reliable models that capture the complexities of the data. Figure 3 shows the time series plot of the Nigerian All Share Index from 2017-2025. The graph shows while the price level does not shift drastically overall, the graph displays short bursts of increased variability (e.g., late 2018 and early 2020), followed by relatively quieter periods. These visible fluctuations suggest the presence of short-term volatility clustering. The early 2020 segment, during the COVID-19 pandemic onset, shows a sudden sharp decline followed by a quick rebound. This is a classic example of volatility clustering a high-volatility period (market crash and recovery) concentrated in a short time span. From 2021 to early 2024, the ASI trends upward sharply, but not smooth. There are noticeable stretches where prices rise rapidly followed by abrupt pullbacks (e.g., sharp corrections in late 2022 and mid-2023), again indicating periods of intensified volatility grouped together. Table 2 shows the unit root test for the Nigeria daily stock price, brent crude oil price and all share. The Augmented Dickey-Fuller (ADF) test statistic for Nigerian stock prices is 0.964348, which is significantly higher than all the critical values at the 1%, 5%, and 10% levels (i.e., -3.432, -2.862, -2.567, respectively). Furthermore, the p-value is 0.9963, far above the critical 0.05 value. Therefore, the stock price series is non-stationary in levels thus, must be differenced (usually converted to returns) to achieve stationarity before any further modelling.

The ADF test statistic for Brent crude oil price is -1.574216, still less negative than any of the critical values. The associated p-value of 0.4957 also exceeds levels of significance. Again, we fail to reject the null hypothesis of a unit root, implying that the Brent crude oil price series is also non-stationary. Like the stock prices, it requires transformation typically taking log differences or returns to stabilize its statistical properties for reliable volatility modelling and forecasting using time series techniques. For the All Share Index, the ADF test statistic is 2.091968, which is not only higher than all critical values but also positive, which is unusual for a stationary series. The p-value of 0.9999 confirms that there is overwhelming evidence against stationarity. This result further supports the conclusion that the ASI is non-stationary in its current form. Thus, just like the other two series, the ASI must be differenced or converted to returns to ensure stationarity prior to fitting MGARCH models or any other statistical or econometric analysis.

Clearly, none of the three series is stationary in levels, as all their ADF test statistics are insufficiently negative and their pvalues are far above standard significance thresholds. Stationarity is a prerequisite for MGARCH modelling, so all three must be transformed into returns to proceed with meaningful volatility and correlation analysis. The results of the ARCH LM tests for stock returns, the All Share Index (ASI), and crude oil prices all reveal highly significant ARCH effects, with Chi-squared statistics of 349.62, 375.02, and 365.97 respectively, each having p-values below 0.0000. These findings strongly reject the null hypothesis of no ARCH effects, indicating that the conditional variances of the return series are time-varying and exhibit volatility clustering-a common feature of financial time series. This means that periods of high volatility tend to follow high volatility, while calm periods follow calm periods.

Therefore, the presence of significant heteroskedasticity across all three return series justifies the use of ARCH or GARCH-type models to appropriately capture and model their dynamic volatility behaviour. correlations confirm that the Nigerian stock market and crude oil market are interlinked, implying that portfolio diversification opportunities may be limited during periods of market turbulence.

Conclusion

The spillover effects from the MGARCH-DCC (1,1) estimation reflect the extent to which volatility in one financial market transmits to others in this case, between Nigerian stock returns (RSP), the All Share Index (RASI), and crude oil prices (ROIL). The significant ARCH and GARCH coefficients across all series indicate that volatility shocks are not isolated within individual markets but tend to propagate through the financial system. Specifically, the persistence of volatility in crude oil returns (β₁ = 0.866) implies that fluctuations in the oil market driven by global supply-demand imbalances or geopolitical developments can spill over into the Nigerian stock market, influencing both stock returns and the ASI. This interconnected behavior suggests that investors and policymakers should monitor the oil market closely, as shocks originating there can affect the overall financial stability of the Nigerian economy. The DCC parameters further support the presence of timevarying and persistent correlations among these assets, which provide evidence of volatility spillovers. The significant DCC β₁ value (0.679, p<0.01) indicates that correlations among crude oil, stock returns, and the ASI are not constant but evolve gradually over time. This means that periods of high volatility in the oil market are likely to coincide with heightened volatility in the Nigerian stock market. Such comovements are particularly pronounced during global financial or commodity market disturbances, implying that shocks in one asset class tend to amplify risk levels in others. This persistence of dynamic correlations underscores the integrated nature of Nigeria's financial and commodity markets. Conclusively, the observed spillover effects highlight the vulnerability of the Nigerian stock market to external shocks, particularly those emanating from crude oil price volatility. As the economy remains heavily reliant on oil revenue, disturbances in the global oil market can transmit rapidly to domestic financial assets, influencing investor sentiment and market performance. These results emphasize the importance of diversification and the implementation of sound macroeconomic and risk management policies to cushion the Nigerian economy against adverse volatility spillovers from the oil sector into the broader financial market.

References

1. Atoi NV. Testing volatility in Nigeria stock market using GARCH models. Cent Bank Niger J Appl Stat. 2015;5(2):65-93.
2. Bagirov M. Modelling and forecasting of Nigeria stock market volatility. Financial Innovation. 2025;11(1):1-20.
3. Okorie DI. Crude oil market and Nigerian stocks: An asymmetric multivariate GARCH approach. Int J Finance Econ. 2022.
4. Salisu AA, Oloko TA. Volatility transmission in the Nigerian financial market. J Afr Bus. 2018;19(2):1-17.
5. Tule MK, Ndako UB, Onipede SF. Oil price shocks and volatility spillovers in the Nigerian sovereign bond market. Rev Financ Econ. 2017;35(1):57-65. doi:10.1016/j.rfe.2017.03.003.
6. Tumala MM, Salisu AA, Gambo AI. Disentangled oil shocks and stock market volatility in Nigeria and South Africa: A GARCH-MIDAS approach. Econ Anal Policy. 2023;78:707-717.
7. Usoro AE, Ekong A. Modeling Nigeria crude oil production and price volatility using multivariate generalized autoregressive conditional heteroscedasticity models. Afr J Math Stat Stud. 2022;5(1):33-54. doi:10.52589/AJMSS-L4FI9DW6.
8. Yaya OS, Bada AS, Atoi NV. Volatility in the Nigerian stock market: Empirical application of Beta-t-GARCH variants. CBN J Appl Stat. 2016;7(2):27-48.