Characterization and Prediction of the Ghana Stock Exchange Composite Index Utilizing Bayesian Stochastic Volatility Models

Abstract

This study delves into the dynamics of the Ghana Stock Exchange Composite Index (GSE-CI) over the period from 2011 to 2022, a symbolic emerging market index that presents unique challenges and opportunities for financial analysis. We characterize the GSE-CI using advanced analytical tools such as the Hurst exponent and R/S analysis to uncover its fractal properties and complex dynamics. The paper then advances to predictive modeling, employing an innovative approach with four variations of Stochastic Volatility (SV) models: SV with linear regressors, SV with Student’s t errors, SV with leverage effects, and a hybrid model combining Student’s t errors with leverage. Each model offers a unique perspective on forecasting the behavior of the GSE-CI, with the SV model incorporating Student’s t errors emerging as the most effective, as evidenced by the lowest Root Mean Square Error (RMSE) in our comparative evaluation. The integration of these models highlights their robustness in capturing the intricate volatility patterns of the GSE-CI, making a compelling case for their applicability to similar financial markets in other emerging economies. This research also paves the way for future investigations into other market indices and assets within and beyond the borders of emerging markets.

1. Introduction

The financial landscape of emerging markets presents a distinctive set of challenges and opportunities that are often underrepresented in financial modeling literature. This paper focuses on the Ghana Stock Exchange Composite Index (GSE-CI), an exemplar of the dynamic and complex nature of financial markets in developing economies. The GSE-CI, as the benchmark index, encapsulates the economic pulse of Ghana, offering a unique perspective into the interplay of local and global economic forces within an emerging market.

Emerging markets like Ghana are particularly sensitive to global economic shifts, such as changes in international interest rates and commodity prices, which can lead to increased financial volatility (International Monetary Fund 2024). These markets face distinct challenges, including regulatory complexities and significant reactions to global economic policies, which often result in pronounced effects on their financial markets (World Bank 2021). For instance, the response to the COVID-19 pandemic in emerging markets involved significant monetary interventions that stabilized financial markets but also highlighted the susceptibility of these economies to external shocks (Gudmundsson et al. 2022).

Furthermore, the structural peculiarities of emerging markets–such as heightened inflationary pressures and vulnerability to currency depreciation–necessitate robust financial modeling to anticipate and mitigate potential risks (Gudmundsson et al. 2022; Tabash et al. 2024).

The importance of studying such markets lies not only in the academic pursuit of extending financial theories but also in the practical implications for investors and policymakers who navigate these turbulent waters.

The GSE-CI offers a valuable case study to explore these dynamics, employing advanced stochastic volatility models to capture the intricate financial fluctuations faced by emerging markets. Moreover, the GSE-CI provides a fertile ground for the application and validation of advanced stochastic volatility models, which have predominantly been tested in developed markets. Research has identified that emerging market equities have historically underperformed compared to developed markets, which can be attributed to several factors. Emerging markets are typically more sensitive to global economic shifts and have domestic economies that may act as performance headwinds. The structural differences and lesser transparency in these markets also contribute to the unpredictability and difficulty in achieving expected returns, leading to frequent revisions of financial theories and investment strategies that were originally based on these markets’ potential for rapid growth and convergence with developed economies (Onifade 2024).

This research focuses on the application of Bayesian Stochastic Volatility models to the Ghana Stock Exchange Composite Index (GSE-CI), particularly on their effectiveness within the volatile environment of an emerging market. The primary research question investigates how these models can enhance predictive accuracies and risk assessment. The gap in existing literature mainly revolves around the insufficient exploration of advanced volatility models like Bayesian Stochastic Volatility in emerging markets, which often exhibit different dynamics compared to more developed markets. This study significantly contributes to the existing body of knowledge by providing empirical evidence on the suitability and enhanced performance of these models in capturing the unique characteristics of the GSE-CI. Through this approach, the research offers broader insights applicable to similar financial markets in emerging economies in Africa, thus enriching the dialogue on global economic resilience and providing a framework that can aid investors, policymakers, and academic researchers in understanding and navigating the complexities of these markets.

The paper is structured as follows: The theoretical foundation and review of relevant literature are discussed in Section 2. Section 3 provides a detailed overview of the data utilized in this study. The core analytical methodologies are discussed in Section 4, where we elaborate on the application of the Hurst Exponent and R/S Analysis, as well as the Stochastic Volatility models. The results and discussion derived from these analytical techniques are thoroughly presented in Section 5 and Section 6, respectively. The conclusion and proposed potential avenues for future research are detailed in Section 7.

2. Literature Review

The study of financial markets in emerging economies, particularly through the lens of stochastic volatility, occupies a significant space in economic research due to the unique characteristics and challenges presented by these markets. This literature review section explores the theoretical frameworks, key hypotheses, and related studies that underpin our understanding of financial volatility and modeling approaches in contexts similar to the Ghana Stock Exchange Composite Index (GSE-CI).

Financial markets in emerging economies are often characterized by higher volatility compared to their developed counterparts (Maharaj et al. 2011). The theory of financial volatility asserts that emerging markets are more susceptible to external shocks and display more pronounced reactions to global financial instability (Bekaert and Harvey 1995). This is partly due to their less mature financial systems and the evolving nature of their market infrastructures, which can intensify the effects of global financial market contagion.

Based on the theoretical frameworks reviewed, our study postulates the following hypotheses:

H1. 

Stochastic Volatility (SV) models equipped with leverage effects provide superior predictive accuracy for the GSE-CI compared to models without leverage effects due to the pronounced impact of external financial shocks observed in emerging markets.

H2. 

The inclusion of Student’s t-distribution in SV models will better capture the heavy tails and excess kurtosis typically observed in the return distributions of emerging market indices like the GSE-CI.

Over the years, a variety of models have been developed to simulate the dynamics of stock markets, applicable across both developed and emerging markets. These models range from deterministic models such as the Autoregressive Conditional Heteroskedasticity (ARCH) introduced by (Engle 1982), and its extension, the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) proposed by (Bollerslev 1986), to more complex frameworks like Stochastic Volatility models (Taylor 1986). These models aim to encapsulate essential market behaviors, including volatility, fat tails, and skewness.

More recent studies have focused on enhancing SV models with features like leverage effects and heavy-tailed distributions. For example, the authors in (Takaishi 2009) demonstrated the effectiveness of SV models with leverage in capturing the asymmetrical volatility observed during financial crises. Meanwhile, (Lafosse and Rodríguez 2018) utilized SV models with Student’s t-distributions to better model the fat tails observed in stock return distributions from emerging markets. The authors analyze daily stock returns from Argentina, Brazil, Chile, Mexico, and Peru, applying Bayesian estimation methods to assess the performance of the SV models. Their findings indicate that models incorporating skew Student’s t-distribution provide a better fit for the heavy-tailed nature of stock returns in these emerging markets compared to models assuming normality. This suggests that SV models with heavy-tailed distributions, such as the skew Student’s t-distribution, are more adept at modeling the volatility characteristics of emerging market stock returns.

These studies provide a strong theoretical and empirical basis for our investigation into the GSE-CI using advanced SV models, suggesting that such models may offer significant improvements in forecasting accuracy and model robustness in the face of the financial anomalies typical of emerging markets in Africa, such as Ghana.

Although these models are extensively applied in the robust, well-documented markets of developed economies, their adoption in emerging markets like Ghana is notably less prevalent. Research focusing on the Ghana Stock Exchange predominantly employs ARCH/GARCH models, with little exploration into Stochastic Volatility models. For instance, Frimpong and Oteng-Abayie (2006) utilized Random Walk, GARCH(1,1), EGARCH(1,1), and TGARCH(1,1) models to analyze volatility clustering and kurtosis, concluding that GARCH models, under the assumption of a normal distribution, outperformed other models. Antwi et al. (2012) employed the Jump Diffusion model, based on Geometric Brownian Motion, to capture sudden market shifts within the Ghanaian stock market.

These methodologies, rooted in frequentist statistics, contrast with the scarce but insightful Bayesian approaches applied to similar datasets. Notably, ARCH/GARCH models often exhibit significant persistence in volatility clustering, a phenomenon effectively addressed by the adaptive nature of Stochastic Volatility models, which manage variance mixing with greater efficacy (Jacquier et al. 1994). The Bayesian framework was notably advanced by (Jensen and Maheu 2010), who analyzed return data using a semiparametric autoregressive stochastic volatility model designed to accommodate the leptokurtic and negative attributes of returns alongside evolving market volatilities.

The synthesis of theoretical insights with empirical findings from the literature suggests that advanced SV models, particularly those incorporating leverage and heavy-tailed distributions, are well-suited for modeling the volatility of indices like the GSE-CI in emerging markets. This review supports the relevance of our hypotheses and frames our research within the broader discourse on financial market dynamics in emerging economies.

3. Materials

In this section of the paper, we briefly describe the dataset used for the modeling and analysis.

3.1. Data Collection and Analysis

The dataset used in this study was obtained from the Ghanaian Stock Exchange Composite Index (GSE-CI), which serves as a performance indicator for the stock market. It is an index that measures the performance based on the market capitalization of all listed stocks. The daily closing index values from a period spanning twelve years, specifically from 2011 to 2022, were employed for our analysis.

For the purpose of our study, we utilized the logarithm of daily returns calculated from the GSE-CI closing index. The rationale behind this decision stems from the mathematical convenience provided by log returns. More specifically, log returns have the distinct advantage that over n periods, they can be simply summed up to yield the total return over those periods. Additionally, in cases where simple returns are near zero, log returns closely approximate these simple returns.

The daily log returns are calculated using the following formula:

$$\begin{array}{cc}\hfill & R_t=ln\left({\displaystyle \frac{C{I}_t}{C{I}_{t-1}}}\right)\hfill \end{array}$$

(1)

where,

$R_t$ denotes the return at time t, $C{I}_t$ represents the Composite Index at time t, and $C{I}_{t-1}$ signifies the Composite Index at time $t-1$.

3.2. Data Inspection and Visualization

Upon analysis, certain descriptive statistics pertaining to the daily returns of the GSE-CI were established. The descriptive statistics provide a concise summary of the central tendency, dispersion, and shape of the distribution of the daily returns, contributing to an initial understanding of the data behavior. The skewness and kurtosis values, in particular, offer insights into the asymmetry and ’tailedness’ of the distribution. These are illustrated in

Table 1. Descriptive statistics of daily returns of the GSE-CI.

	Estimate		Estimate
Mean	0.000301	Standard Deviation	0.006702
Maximum	0.59223	Skewness	0.518391
Minimum	−0.050497	Kurtosis	11.925697

below.

The daily returns of the GSE-CI were characterized using descriptive statistics, which are presented in Table 1. It shows a mean return of 0.03%, indicating an average return of 0.03% per day over the 12-year period from 2011 to 2022. The standard deviation of 0.00670 demonstrates a high volatility in the GSE-CI. This aligns with the financial principle that higher risk (volatility) is often associated with higher expected returns.

The skewness of 0.5 reveals a nearly symmetrical distribution of the series (refer to Figure 1). However, the high kurtosis of 11.9 suggests the presence of heavy tails, pointing towards the occurrence of extreme outcomes with a larger frequency than what is predicted by the normal distribution. This justifies our decision to utilize a stochastic volatility model that is adept at capturing the effects of outliers in the series.

Figure 2 and Figure 3 present the time series plots of the daily GSE-CI and its corresponding returns, respectively. We observe significantly greater volatility from 2018 to 2022 compared to the earlier period (2011 to 2017). This period of higher volatility coincides with increased market liquidity, as reported by the Ghana Stock Exchange. This trend might be attributed to a decline in interest rates that shifted investors’ focus to the capital markets.

Historically, the return on the GSE-CI reached an all-time high of 6% in August of 2022 and an all-time low of −5% in September 2021; as a matter of fact, the top 5 all-time high and top 5 all-time low within the period under study occurred from 2018 to 2022. It is worth noting that the period within which the series was characterized by higher volatilities was the same period that liquidity pushed higher. For instance, empirical evidence from the Ghana Stock Exchange showed that liquidity moved 18% higher in 2019 from the previous year of 200 million shares traded.

The return series shows random variation around a constant mean of 0%, without discernible patterns or cycles, suggesting stationarity. This assumption is confirmed by the Augmented Dickey–Fuller (ADF) test for stationarity shown in

Table 2. Results of Augmented Dickey–Fuller test.

Test Statistics	−14.51730
p-value	$5.60\times {10}^{-27}$
Number of lags used	7
Number of observations used	2965

.

The negative value of the Augmented Dickey–Fuller (ADF) test statistic, along with the very small p-value, provides strong evidence against the null hypothesis of a unit root in the series. Specifically, the test statistic of −14.5173 and a p-value of approximately $5.60\times {10}^{-27}$ strongly reject the null hypothesis that the GSE-CI has a unit root, indicating that the series is stationary.

Stationarity in the context of the GSE-CI suggests that the index values are mean-reverting over time, which is an essential characteristic for certain financial models, particularly those predicting future values based on past data. The stationarity implies that the series has a constant mean, variance, and autocorrelation structure over time, which simplifies both modeling and forecasting efforts. This makes it economically sensible to use stochastic volatility models that assume or require the underlying series to be stationary for accurate volatility estimation and forecasting.

This result can be compared with findings from other emerging markets where studies might show varying degrees of stationarity due to different market dynamics, regulatory environments, and economic conditions. For example, studies on other emerging market indices may find non-stationarity due to structural breaks or significant market events that the ADF test could fail to account for without modifications. A study by (Narayan and Popp 2010) found that emerging market indices frequently exhibit breaks in variance, which can affect stationarity tests and their implications for financial modeling.

4. Methods and Modeling

This study employs a two-stage modeling approach refined to address specific characteristics of the Ghana Stock Exchange Composite Index (GSE-CI). First, the Hurst Exponent is utilized to characterize the financial time series, providing insights into the degree of persistence and correlation. This method is particularly chosen for its ability to reveal long-term memory and the fractal nature of the financial time series, which is critical for understanding market dynamics in emerging markets such as Ghana.

In the second stage, we proceed with the application of Stochastic Volatility (SV) models, assuming the presence of fractal market characteristics in the GSE-CI. This assumption allows for the exploration of complex market dynamics through models that can accommodate sudden market shifts and large movements in prices, which are typical in less mature financial markets. The SV models are selected based on their robustness in handling data complex financial time series and their efficacy in forecasting under conditions of irregular volatility, which are often observed in emerging market indices.

4.1. Hurst Exponent and the R/S Analysis

The Hurst Exponent (H) is a measure of long-term memory of time series. It relates to the autocorrelations of the time series and the rate at which these decrease as the lag between pairs of values increases. Ranges of the Hurst exponent indicate the presence or absence of trends, mean reversion, or random walks in the time series.

The value of the Hurst Exponent (H) can be between 0 and 1. A Hurst exponent value $H<0.5$ suggests that the time series has a tendency to revert to its long-term mean value, a value $H>0.5$ suggests that the time series has a persistent trend, while $H=0.5$ indicates a random walk.

The R/S analysis, which stands for Rescaled Range analysis, was developed by the hydrologist Harold Edwin Hurst (1880–1978) to deal with a long series of data related to natural phenomena, such as river flows and lake levels (Hurst 1951). The R/S analysis estimates the Hurst exponent, which measures the extent of the variability or statistical dispersion of a time series.

In this paper, the Hurst exponent is calculated using R/S analysis. The steps involved in the calculation of the Hurst exponent are as follows:

• The time series of length N is divided into non-overlapping sub-series of length n.
• For each sub-series of length n, calculate the mean, create the mean-adjusted series, create the cumulative deviation series, and calculate the range and standard deviation.
• The rescaled range $R/S$ is calculated by dividing the range by the standard deviation for each sub-series.
• Finally, the Hurst exponent H is estimated by performing a least-squares regression fit on $log(R/S)$ as a function of $log\left(n\right)$.

From

Table 3. Scaling exponents of the daily returns of the GSE-CI.

Index	$R/S\left(H\right)$
GSE-CI	0.6058

, since the Hurst exponent R/S(H) = $0.6058>0.5$, it suggests that the daily returns of the GSE-CI exhibit long-term memory. The presence of long-term memory indicates that the Ghanaian market does not immediately react to the information coming into the financial market; however, it responds to such information gradually over a period of time. Thus, past price changes can be used as significant information for the prediction of future price changes (Stanley and Mantegna 2000).

4.2. Bayesian Stochastic Volatility Modeling

Following the characterization of the time series, the Stochastic Volatility (SV) model is used to forecast the daily returns. The SV model allows for volatility clustering, wherein periods of high volatility tend to be followed by periods of high volatility, and periods of low volatility tend to be followed by periods of low volatility.

The SV model assumes that the log returns follow a normal distribution, with a mean of zero and a time-varying standard deviation. The logarithm of the square of the standard deviation, referred to as the log-variance, is modeled as a random walk. The model allows for the log-variance to slowly evolve over time, capturing the persistence in volatility observed in financial returns data.

Bayesian methods are used to estimate the parameters of the SV model. In the Bayesian framework, prior distributions are assigned to the parameters, and the posterior distributions of the parameters are obtained via Markov chain Monte Carlo (MCMC) methods. The posterior distributions of the parameters provide insights into the uncertainty in the parameter estimates and are used to make forecasts of future returns.

Employing a stochastic and time-varying variance specification in the SV model can lead to more precise estimations of asset return volatility and help to capture crucial aspects of financial markets, such as volatility clustering and risk premiums that vary with time.

We present the following models as discussed in (Hosszejni and Kastner 2021).

4.2.1. Stochastic Volatility Incorporating Linear Regressors

Let us denote a vector of observations as $y={(y_1,\cdots ,y_n)}^T$. The structure assumed for the Stochastic Volatility (SV) model with linear regressors is as follows:

$$\begin{array}{cc}\hfill y_t& =x_t\beta +exp\left({\displaystyle \frac{h_t}{2}}\right){\epsilon }_t,\hfill \\ \hfill h_{t+1}& =\mu +\phi (h_t-\mu )+\sigma {\eta }_t,\hfill \\ \hfill {\epsilon }_t& \sim \mathcal{N}(0,1),\hfill \\ \hfill {\eta }_t& \sim \mathcal{N}(0,1),\hfill \end{array}$$

(2)

Here, ${\epsilon }_t$ and ${\eta }_t$ are independent, with each following a normal distribution with a mean of 0 and variance of 1. The vector of K regression coefficients is represented by $\beta ={({\beta }_1,\cdots ,{\beta }_K)}^T$. The log-variance process, denoted as $h={(h_1,\cdots ,h_n)}^T$, is initiated by $h_0\sim (\mu ,{\sigma }^2/(1-{\phi }^2))$. We define $\vartheta =(\mu ,\phi ,\sigma )$ as the SV parameters, where $\mu$ signifies the level, $\phi$ represents the persistence, and $\sigma$, also known as volvol, is the standard deviation of the log-variance.

4.2.2. Stochastic Volatility with Student’s t Errors

The Students’s t errors equation is formulated as below, in line with the implementation in the R package stochvol:

$$\begin{array}{cc}\hfill y_t& =x_t\beta +exp\left({\displaystyle \frac{h_t}{2}}\right){\epsilon }_t,\hfill \\ \hfill h_{t+1}& =\mu +\phi (h_t-\mu )+\sigma {\eta }_t,\hfill \\ \hfill {\epsilon }_t& \sim t_v(0,1),\hfill \\ \hfill {\eta }_t& \sim \mathcal{N}(0,1),\hfill \end{array}$$

(3)

In this model, ${\epsilon }_t$ and ${\eta }_t$ are independent, with $t_v(0,1)$ representing the Student’s t distribution having v degrees of freedom, mean 0, and variance 1. This model, while similar to the previous equation, posits that observations are, conditionally, t-distributed. This is a more generalized model that introduces a new parameter v, enabling the Student’s t-distribution to converge to the standard normal distribution as v approaches infinity.

4.2.3. Stochastic Volatility with Leverage

The leverage effect is formulated as follows in the R package stochvol:

$$\begin{array}{cc}\hfill y_t& =x_t\beta +exp\left({\displaystyle \frac{h_t}{2}}\right){\epsilon }_t,\hfill \\ \hfill h_{t+1}& =\mu +\phi (h_t-\mu )+\sigma {\eta }_t,\hfill \\ \hfill {\epsilon }_t& \sim \mathcal{N}(0,1)\hfill \\ \hfill {\eta }_t& \sim \mathcal{N}(0,1)\hfill \end{array}$$

(4)

The correlation matrix of (${\epsilon }_t,{\eta }_t$) is represented as

$${\Sigma }^{\rho }=\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)$$

(5)

The SV parameters are collected in the vector $\zeta =(\mu ,\phi ,\sigma ,\rho )$. Compared to Equation (2), this model introduces a new correlation parameter $\rho$, which links the residuals of the observations with the innovations of the variance process. Thus, Equation (2) is a specific case of this model where $\rho =0$ is predefined.

4.2.4. Stochastic Volatility with Student’s t Errors and Leverage

The combined model of t errors with the leverage effect is formulated as follows:

$$\begin{array}{cc}\hfill y_t& =x_t\beta +exp\left({\displaystyle \frac{h_t}{2}}\right){\epsilon }_t,\hfill \\ \hfill h_{t+1}& =\mu +\phi (h_t-\mu )+\sigma {\eta }_t,\hfill \\ \hfill {\epsilon }_t& \sim t_v(0,1),\hfill \\ \hfill {\eta }_t& \sim \mathcal{N}(0,1),\hfill \end{array}$$

(6)

Here, the correlation matrix of (${\epsilon }_t,{\eta }_t$) is ${\Sigma }^{\rho }$ as in Equation (5).

4.3. Prior Distributions for the SV Models

In Bayesian analysis, we combine prior knowledge, expressed as a prior distribution, with the distribution determined by the data to form a posterior distribution. Inferences are made from this posterior distribution. For building the model, we used the “stochvol” package in R, which presents a comprehensive Bayesian approach for modeling heteroskedasticity using the stochastic volatility framework. It utilizes Markov chain Monte Carlo (MCMC) samplers to facilitate inference by generating samples from the posterior distributions of parameters and latent variables (Kastner 2019).

In this study, we assume a priori, $\beta \sim N_k(b_{\beta },B_{\beta })$, where $N_l(b,B)$ represents the l-dimensional normal distribution with mean vector b and variance-covariance matrix B. For the variance process to be stationary, a restricted persistence $\phi \in (-1,1)$ is needed. However, when stationarity is not assumed, the untruncated prior $\phi \sim N(b_{\phi },B_{\phi })$, please refer to (Hosszejni and Kastner 2021) for details of the prior distribution as implemented in the stochvol package.

5. Results

In this section, we discuss the estimation, visualization, and prediction of the Ghana Stock Exchange Composite Index (GSE-CI) using four variants of the Stochastic Volatility (SV) models, namely, SV with linear regressors, SV with Student’s t errors, SV with leverage, and SV with Student’s t errors and leverage. Our analysis is based on the daily closing index of the GSE-CI from 4 January 2011 to 30 December 2022, totaling 2973 data points.

We divided our financial time series into a 60% training set (comprising data from 4 January 2011 to 3 March 2018, a total of 1783 data points) and a 40% testing set (consisting of data from 4 March 2018 to 30 December 2022, amounting to 1190 data points). The training set was used to estimate our model, and predictions were made using the testing set.

5.1. Stochastic Volatility with Linear Regressors

The first approach involves employing the SV model with linear regressors to investigate whether the GSE-CI adheres to a random walk characterized by SV. This model’s adequacy is suggested by the alignment of the posterior estimates of ${\beta }_0$ and ${\beta }_1$ around 0 and 1, respectively. The AR(1) specification was adopted for this analysis, as configured in the designmatrix parameter. The estimated volatility is shown in Figure 4.

5.2. Stochastic Volatility with Student’s t Errors

The SV with Student’s t errors model is used to examine whether the daily closing index of the GSE-CI exhibits heavy-tailed returns. If the returns are indeed heavy-tailed, most of the posterior mass of the model will concentrate on lower values, such as those less than 20, suggesting high kurtosis (Hosszejni and Kastner 2021). The estimated volatility is shown in Figure 5.

5.3. Stochastic Volatility with Leverage

Employing the SV model with leverage, we tested for the presence of asymmetric volatility, known as the leverage effect. This effect reflects how negative market shocks might have a more pronounced impact on index volatility than positive ones. If this asymmetric feature is present and not accounted for in the SV model, the GSE-CI could be considerably biased (Hull and White 1987). The estimated volatility is shown in Figure 6.

5.4. Stochastic Volatility with Student’s t Errors and Leverage

The SV model with Student’s t errors and leverage combines the properties of heavy-tailed distributions with leverage effects to examine the nuanced dynamics of the GSE-CI’s volatility. Such a combination aims to provide a more comprehensive understanding of the underlying volatility patterns. The estimated volatility is shown in Figure 7.

5.5. Prediction with Stochastic Volatility models

The predictive performance of each model was evaluated by forecasting the daily log returns of the GSE-CI for the test set. Predictive accuracy was visually assessed (See Figure 8, Figure 9, Figure 10 and Figure 11) through the comparison of predicted versus observed values across all models.

A comparative analysis of prediction errors is provided in

Table 4. Comparison of Errors.

Model	Root Mean Square Error (RMSE)
SV model with linear regressors	0.008199035
SV model with Student’s t errors	0.008195911
SV model with leverage	0.008199373
SV model with Student’s t errors and leverage	0.008201156

highlighting the model with the least Root Mean Square Error (RMSE), which indicates the best predictive accuracy among the tested models.

6. Discussion

Our findings presented in Figure 4, Figure 5, Figure 6 and Figure 7 offer a comprehensive view of the comparative performance of four distinct Stochastic Volatility (SV) models applied to the Ghana Stock Exchange Composite Index (GSE-CI). These models include the SV model with linear regressors, the SV model with Student’s t errors, the SV model with a leverage effect, and the combined SV model with both Student’s t errors and a leverage effect.

Each of these figures presents several essential features. The top row in each figure is a representation of the posterior distribution of the daily closing index of the GSE-CI (in percent), specifically $100exp(h/2)$. Here, the 50% quantile (the median) is shown in black. This provides an estimate of the most likely daily closing index at each point in time. To demonstrate the range of uncertainty in these estimates, we also provide the 5% and 95% quantiles. This gives us an idea of the plausible range of variation around the median.

The middle and bottom rows of each figure display a detailed view of the Markov chains for the parameters $\mu ,\varphi ,\sigma ,\nu$, and $\rho$. The trace plots in the middle row are useful for diagnosing the convergence of our models, while the prior (gray, dashed) and posterior (black, solid) densities in the bottom row provide insights into how our data have updated our prior beliefs about these parameters. Notably, in Figure 6, we observe a high persistence and significant leverage, indicating that the SV model with leverage has captured a crucial aspect of the GSE-CI dynamics.

According to the results tabulated in Table 4, the SV model with Student’s t errors offers the most accurate predictions for the GSE-CI’s daily closing index, as evidenced by its smallest RMSE value of 0.008195911. This suggests that the SV model incorporating Student’s t errors is notably effective. This model specifically captures the pronounced heavy tails and excess kurtosis characteristics of the GSE-CI returns, which are prevalent in the financial series of many emerging markets. Such markets often exhibit higher volatility and atypical return distributions due to various factors like economic instability, policy changes, and sensitivity to global financial shifts, which are effectively accommodated by the Student’s t distribution.

In contrast, the other models demonstrated certain limitations. For example, the SV with Linear Regressors and SV with Leverage generally underperformed as they assume normal error distributions, which are less capable of handling the extreme values typically observed in the GSE-CI data. The SV with both Leverage and Student’s t Errors, despite its theoretical robustness, did not show a significant improvement over the Student’s t model alone. This might suggest that the added complexity of combining these features does not necessarily translate into better predictive performance for the GSE-CI.

The superior performance of the Student’s t errors model is rooted in its ability to more accurately reflect the empirical characteristics of the GSE-CI’s volatility, particularly in accommodating the heavy tails of the distribution. This finding underscores the importance of model selection based on the specific attributes of the financial series being analyzed. It highlights how the Student’s t errors model can offer more reliable predictions and a better understanding of the risk dynamics in emerging market indices like the GSE-CI.

The models and findings presented in this study, while grounded in theoretical analysis, have significant practical implications for various stakeholders in emerging markets like the Ghana Stock Exchange Composite Index. For investors, the enhanced understanding of stochastic volatility models, especially those incorporating Student’s t errors, offers a more robust tool for assessing risk and forecasting market volatility. This can aid in better portfolio management and investment strategies, particularly in environments known for their unpredictability.

Policymakers can leverage insights from this study to better understand the market dynamics that influence financial stability in emerging markets. By recognizing the patterns and potential triggers of market volatility identified through our models, policymakers can formulate more effective regulatory measures to mitigate excessive market fluctuations and enhance investor confidence.

Market analysts can use the findings to refine their forecasts and advisory services to clients. The ability to predict market movements more accurately, especially in using the SV model with Student’s t errors, provides a competitive edge in financial analysis and consultancy.

In addition, the detailed exploration of how different stochastic volatility models perform, including the comparative analysis, which highlights the superiority of the Student’s t errors model in capturing the nuances of the GSE-CI, helps all stakeholders better understand the specific characteristics of emerging market indices. This knowledge is crucial for developing tailored financial products and investment strategies that align with the risk profiles and return expectations in these markets.

7. Conclusions and Future Work

This study has conducted an in-depth exploration of the Ghana Stock Exchange Composite Index (GSE-CI), employing advanced Bayesian Stochastic Volatility (SV) models to unravel the complex dynamics characteristic of this emerging market index. Initial analysis leveraging the Hurst exponent and R/S analysis has deepened our understanding, revealing intricate patterns often obscured in simpler analytical approaches.

Among the various SV models tested, the model incorporating Student’s t errors emerged as the most effective, demonstrating superior performance with the lowest Root Mean Square Error (RMSE). This finding highlights the model’s capability to efficiently navigate the complexities of the GSE-CI, affirming the potency of integrating SV models with the Hurst exponent and R/S analysis in forecasting and decoding financial indices like the GSE-CI. Such tools prove essential for investors and policymakers reliant on accurate market predictions to guide their strategic decisions.

However, this research is not without its limitations. The study’s focus on a single financial index might not fully reflect the broader market dynamics or the influences of global economic factors that affect other indices. Furthermore, the exclusion of direct global economic impacts in our models could overlook significant variables that shape the behavior of emerging market indices.

Future research should aim to broaden this investigative framework by incorporating multiple indices from various emerging markets. This expansion would enhance our comprehension of regional disparities and commonalities in financial volatility. Moreover, the integration of machine learning techniques within a Bayesian context, such as Bayesian Deep Learning and Gaussian Processes, promises to refine predictive accuracy and accommodate more complex data interrelations.

Pursuing these avenues could substantially enrich our understanding of financial time series forecasting. Such advancements would bolster the analytical tools available for navigating stock market dynamics, ultimately supporting more informed economic planning and investment strategies across emerging markets in Africa. These future efforts will not only mitigate the current study’s limitations but also extend its applicability and relevance in the ever-evolving landscape of global finance.