Empirical evidence suggests that the traditional GARCH-type models are unable to accurately estimate the volatility of financial markets. To improve on the accuracy of the traditional GARCH-type models, a hybrid model (BSGARCH (1, 1)) that combines the flexibility of B-splines with the GARCH (1, 1) model has been proposed in the study. The lagged residuals from the GARCH (1, 1) model are fitted with a B-spline estimator and added to the results produced from the GARCH (1, 1) model. The proposed BSGARCH (1, 1) model was applied to simulated data and two real financial time series data (NASDAQ 100 and S&P 500). The outcome was then compared to the outcomes of the GARCH (1, 1), EGARCH (1, 1), GJR-GARCH (1, 1), and APARCH (1, 1) with different error distributions (ED) using the mean absolute percentage error (MAPE), the root mean square error (RMSE), Theils inequality coefficient (TIC) and QLIKE. It was concluded that the proposed BSGARCH (1, 1) model outperforms the traditional GARCH-type models that were considered in the study based on the performance metrics, and thus, it can be used for estimating volatility of stock markets.

A simplified mathematical model of immune responds to Hepatitis B Virus (HBV) infection is presented. This focuses on the control of the infection by the interferons, the innate and adaptive immunity. The model was compartmentalized as appropriate and the resulting model equations were solved numerically. A mathematical analysis of the model shows that both disease-free and endemic equilibrium point exist and we derive conditions for their stability. We perform sensitivity analysis on the model parameters, to account for the variability and speed of adaptation.

We present a Caputo fractional order mathematical model that describes the cellular infection of the Hepatitis B virus and the immune response of the body with Holling type II functional response. We study the existence of unique positive solutions and the local and global stability of virus-free and endemic equilibria. Finally, we present numerical results using the Adam-type predictor–corrector iterative scheme.

Exchange rate prediction is a crucial aspect of international finance, impacting decisions by governments, investors, and businesses. Accurate prediction supports the development of sound monetary policies, effective risk management, and strategic international trade planning. According to literature, traditional econometric models like ARIMAX and VAR often struggle to capture the complex, non-linear dynamics of foreign exchange markets. In contrast, machine learning methods, particularly Extreme Gradient Boosting (XGBoost), have shown superior performance due to their ability to handle large datasets, model non-linear relationships, and resist overfitting. This study evaluates the efficacy of the Extreme Gradient Boosting (XGBoost) model by predicting the GHc/USD, GHc/GBP and GHc/EUR exchange rates. Four different types of XGBoost models were employed on the financial data to determine the best performed model. The four different XGBoost models include, the XGBoost all feature, the XGBoost difference feature, the XGBoost ratio feature and the XGBoost lagged feature. The data sourced from Bank of Ghana and World Bank websites spans from January 2015 to March 2025. Findings from the study reveals that the XGBoost lagged feature and XGBoost all feature models outperformed the other two models, with an average R²of 99%, RMSE of 0.05, and MAE of 0.01. Gold price was the biggest contributor to the GHc/USD exchange rate with the feature important score of 80% followed by monthly interest rate 60%, Government debt 25%, M2 20%, price of oil 20%, BCI 15%, and CCI 10%. This result provides valuable insight for financial analyst and policymakers seeking to forecast/predict exchange rates and develop policies aimed at addressing exchange rate menace in Ghana.

In this study, we explore the causal influence of export earnings of cocoa towards economic growth (i.e., gross domestic product (GDP)). The augmented Dickey–Fuller (ADF) and the Phillips–Perron (PP) unit root tests indicate that the two series are integrated of order one, I (1). The results of the trace and the maxi-eigenvalue cointegration test based on Johansen’s procedure indicate the existence of a cointegration between export earnings of cocoa and GDP. Thus, the two variables of the study have a long-run equilibrium relationship. The vector error correction model of order two, VECM (2), was considered as the “best” model after evaluating other competing models. It is observed that, in the long-run, previous year export earnings of cocoa is positively related to economic growth. In the short run, the results revealed that the previous GDP has positive effect on current GDP; and higher export earnings of cocoa have positive effect on GDP. Feedback causality is observed between economic growth (GDP) and export earnings of cocoa. This suggests a bi-directional causality from export earnings to economic growth (GDP). Keywords: Cointegration, VECM, Johansen, ADF, Causality.