This study investigates the impact of global financial crisis and the present COVID-19 pandemic on daily and weekly Crude oil futures using four variants of ARMA-GARCH models: ARMA-sGARCH, ARMA-eGARCH, ARMA-TGARCH and ARMA- aPARCH with dummy variables We also investigated the persistence, half-life and backtesting of the models. This study therefore seeks to contribute to the body of literature on the impact of global financial crisis and the present COVID-19 pandemic on crude oil futures market. This investigation of the impact of global financial crisis and the COVID-19 on crude oil futures has not been much studied at present. We obtained and analyzed the daily and weekly crude oil futures from secondary sources. Daily crude oil futures used in this study covers the period from the 4th January 2000 to 27th April 2020 while the weekly crude oil futures covered from 2ndJanuary 2000 to 26th April 2020 . The global financial crisis period covered from 2nd July 2007 to 31st March 2009 and the current COVID-19 pandemic covered from 1st January 2020 to 27th April, 2020. The study used both student t and skewed student t innovations with AIC, goodness-of-test fit and backtesting to select the best model. Most of the estimated ARMA-GARCH models are supported by skewed student t distribution while most of the ARMA-GARCH models exhibited high persistence values in the presence of global financial crisis and the COVID-19 pandemic. In the overall, the estimated ARMA(1,0)-eGARCH(2,1) and ARMA(1,0)-eGARCH(2,2) model for daily crude oil futures and weekly crude oil futures respectively have been significantly impacted by the global financial crisis and the Present COVID-19 pandemic while the preferred estimated models also passed the goodness-of-test fit and backtesting.This study recommends shareholders and investors should think outside the box as crude oil futures tend to be affected by global financial crisis and COVID-19 pandemic while countries also that depend mostly on crude oil are encouraged to diversify their economy in other to survive and be sustained during financial and health crisis.

This study compared the performance of five Family Generalized Auto-Regressive Conditional Heteroscedastic (fGARCH) models (sGARCH, gjrGARCH, iGARCH, TGARCH and NGARCH) in the presence of high positive autocorrelation. To achieve this, financial time series was simulated with autocorrelated coefficients as ρ = (0.8, 0.85, 0.9, 0.95, 0.99), at different time series lengths (as 250, 500, 750, 1000, 1250, 1500) and each trial was repeated 1000 times carried out in R environment using rugarch package. And the performance of the preferred model was judged using Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE). Results from the simulation revealed that these GARCH models performances varies with the different autocorrelation values and at different time series lengths. But in the overall, NGARCH model dominates with 62.5% and 59.3% using RMSE and MAE respectively. We therefore recommended that investors, financial analysts and researchers interested in stock prices and asset return should adapt NGARCH model when there is high positive autocorrelation in the financial time series data.

In financial time series modelling and forecasting, combining ARMA and GARCH models tend to produce superior and reliable models for volatility persistence, half-life volatility and backtesting (application of model in real life). In Nigeria, banking stocks are mostly traded because of its potential benefits to investors. This study modelled and forecasted the Guaranty Trust (GT) Bank daily stock returns from January 2^nd 2001 to May 8^th 2017 data set collected from a secondary source. The ARMA-GARCH models, persistence, half-life and backtesting were used to analysed the collected data using student t and skewed student t distributions, and the analyses are carried out R environment using rugard and performanceAnaytics Packages. The study revealed that using the lowest information criteria values only could be misleading rather we added the use of backtesing. The ARMA(1,1)-GARCH(1,1) models fitted exhibited high persistency in the daily stock returns while the days it takes for mean-reverting of the models ranges from 5 days to 100 days but unfortunately the models failed backtesting. The results further revealed ARMA(1,1)-eGARCH (2,2) model with student t distribution provides a suitable model for evaluating the GT bank stock returns among the competing models while it takes less than 30 days for the persistence volatility to return back to its average value of the stock returns. This study recommended that researchers should adopt backtesting approach while fitting GARCH models while GT bank stocks investor should be assured that no matter the fluctuations in the stock market, the GT bank stock returns has the ability to returns to its mean price return.

Total Nigeria Plc is a Marketing and Services subsidiary of Total; a multinational energy company operating in more than 130 countries and committed to providing sustainable products and services for its customers. For over 50 years, Total Nigeria Plc has remained the leader in the downstream sector of the Nigerian oil and gas industry. This study investigated the volatility of the stock price of Total Petroleum Nigeria plc using nine (9) GARCH models namely sGARCH, gjrGARCH, eGARCH, iGARCH, aPARCH, TGARCH, NGARCH, NAGARCH and AVGARCH. We also investigated the Value-at-Risk (VaR) and Backtesting of the Models. The aim actually of this study is to boost the confidence of the shareholders and investors of the Total Nigeria plc. To achieve this, daily stock price for Total petroleum Nigeria plc from secondary was collected from January 2^nd 2001 to May 8^th 2017. . The study used both normal and student t innovations, using Akaike Information Criterion (AIC) to select the best model, for normal innovations for log returns and cleansed log returns of Total plc, the eGARCH and sGARCH models performed best respectively, while NGARCH model performed best for student t innovation for both log returns and cleansed returns of Total plc. The persistence of the models are stable except in few cases where iGARCH, eGARCH where not stable. Also for student t innovation, the sGARCH and gjrGARCH fails to converge. The mean-reverting number of day for the returns of Total Nigeria plc differs from model to model. Evidence from the VaR Analysis revealed from the selected models revealed that the Risk of VaR losses is high at 99% confidence level, slightly high at 95% confidence level and better at 90% confidence level. Although The Duration-Based Tests of independence conducted revealed that the models are correctly specified since in all cases the null hypotheses were accepted. This means that the probability of an exception on any day did not depends on the outcome of the previous day. Finally, both the unconditional (Kupiec) and conditional (Christoffersen) coverage tests for the correct number of exceedances for both Total stock returns and cleansed returns. The tests revealed rejection of the models at 1% level of significance. This confirms that unconditional (Kupiec) and conditional (Christoffersen) coverage tests for the correct number of exceedances are reliable compared to the Duration-Based Tests of independence. Therefore we recommend that shareholders and investors in Total Nigeria plc are to remain and continue to investment in Total Nigeria plc because if there is any form of losses, the price of the stock has the potentials to improve in the future. Again, though the risk is high at 99% confidence level, this in line with the financial theory that states that an asset with high expected risk would pay higher return on the average.

It is a common practice to detect outliers in a financial time series in order to avoid the adverse effect of additive outliers. This paper investigated the performance of GARCH family models (sGARCH; gjrGARCH; iGARCH; TGARCH and NGARCH) in the presence of different sizes of outliers (small, medium and large) for different time series lengths (250, 500, 750, 1000, 1250 and 1500) using root mean square error (RMSE) and mean absolute error (MAE) to adjudge the models. In a simulation iteration of 1000 times in R environment using rugarch package, results revealed that for small size of outliers, irrespective of the length of time series, iGARCH dominated, for medium size of outliers, it was sGARCH and gjrGARCH that dominated irrespective of time series length, while for large size of outliers, irrespective of time series length, gjrGARCH dominated. The study further leveled that in the presence of additive outliers on time series analysis, both RMSE and MAE increased as the time series length increased.