Double-Edged Sword of Global Financial Crisis and COVID-19 Pandemic on Crude Oil Stock Returns

This study investigates the impact of global financ i l risis and the present COVID-19 pandemic on daily and weekly Crude oil futures using four va iants of ARMA-GARCH models: ARMAsGARCH, ARMA-eGARCH, ARMA-TGARCH and ARMAaPARCH w ith 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 th e impact of global financial crisis and the present COVID-19 pandemic on crude oil futures mark et. This investigation of the impact of global financial crisis and the COVID-19 on crude o il futures has not been much studied at present. We obtained and analyzed the daily and wee kly crude oil futures from secondary sources. Daily crude oil futures used in this study covers the period from the 4 th January 2000 to 27 April 2020 while the weekly crude oil futures cove red from 2January 2000 to 26 th April 2020 . The global financial crisis period covered f rom 2 July 2007 to 31 st March 2009 and the current COVID-19 pandemic covered from 1 st January 2020 to 27 th April, 2020. The study used both student t and skewed student t innovations wit h AIC, goodness-of-test fit and backtesting to select the best model. Most of the estimated ARMAG RCH 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 a nd the COVID-19 pandemic. In the overall, the estimated ARMA(1,0)-eGARCH(2,1) and ARMA(1,0)-eGARC H(2,2) model for daily crude oil futures and weekly crude oil futures respectively h ave 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.Thi s 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 t hat depend mostly on crude oil are encouraged to diversify their economy in other to s urvive and be sustained during financial and health crisis.


Introduction
Today, the world is battling COVID-19, also known as coronavirus pandemic. The outbreak of this virus was first noticed in Wuhan, China in December 2019. The World Health Organization (WHO) declared it a "public health emergency of international concern" on 30 January, 2020, and finally, on 11 March, 2020, WHO declared the coronavirus outbreak a pandemic (Wikipedia, (2020). The virus, which, according to WHO, is highly contagious and spreads very fast, has spread to over two hundred (200) countries, has infected over four million seven hundred and thirty-one thousand, four hundred and fifty-eight (4,731,458) people globally, has also resulted in over three hundred and sixteen thousand, one hundred and sixty-nine (316,169) fatalities globally (WHO, 2020). It therefore poses a very serious health challenge.
In addition to its serious implications for people's health, COVID-19 is also impacting businesses and economy significantly. This is supported by Adrian &Natalucci (2020) in opining that necessary measures taken to contain the spread of the virus have resulted in economic downturn, and with financial system being adversely impacted. But whether it will lead to financial crisis, (Kenton, reviewed by Scott, 2020) opines that though it is too early to tally up the total cost of the coronavirus pandemic, there is sign of worldwide supply interruption, heightened volatility and steep losses in financial markets, all of which are a warning for financial crisis.
Global financial crisis is financial crisis that affects many countries of the world at the same time. Financial crisis can be defined as any of the very many situations in which values of some financial assets suddenly nosedive, businesses and consumers become unable to pay their bills, and financial institutions experience liquidity shortages, (Kenton, reviewed by Scott, 2020). The current global financial crisis is being exacerbated by COVID-19 which necessitated lockdown of cities and businesses the world over, which impacts crude oil market with the price of oil steeping terribly.
Crude oil price is presently in steep decline, this is because crude oil, being a commodity, the price tends to fluctuate with larger fluctuations than more stable investments such as stocks and bonds (Lioudis, 2020). It listed some of the causes of crude oil price fluctuation to include OPEC decisions, laws of demand and supply, natural disaster, and political unrest affecting large oil producing nation(s). COVID-19 pandemic is a disaster which impact on crude oil prices is enormous and devastating.
Efforts to study the relationship between crude oil prices and the stock returns in the industry have yielded different results: for instance, whereas Sari &Soytas (2006) found that oil price shocks do not have a significant impact on real stock returns in the Istanbul Stock Exchange, Adekunle, et.al (2020) studying the nexus between crude oil prices and stock returns of nine major oil and gas companies currently listed on the Nigerian Stock Exchange, using data that span over the period of January 2014 to November 2019, found that oil price matters in the predictability of stock returns for some listed oil and gas companies in Nigeria. However, other factors that can affect stock prices, according to Hall (2019) include interest rate, saying that there is inverse relationship between interest rate and stock price; and Ramsharan (2019) submits that financial crisis affects stock value adversely.
This paper studies the double-edged sword effect of global financial crisis and covid-19 pandemic on crude oil stock returns, which in effect aims to study if crude oil prices have been impacted by financial crisis and health crisis. And to achieve this, volatility modeling of financial time series will be explored.
In the last few years, modeling and forecasting volatility of a financial time series has attracted a lot of interest of researchers; mainly because volatility, which is a statistical measure of dispersion of returns of a given security or market index, is taken as an important concept for many economic and financial applications such as portfolio and risk management and pricing of assets (Arum &Uche, 2017;Emenogu, et.al, 2020). Emenogu, et.al (2020) further states that volatility can be measured by either standard deviation or variance between returns of the same security or market index, and opined that the higher the volatility of an asset return, the riskier the security. Crude oil prices fluctuate heavily in global market and its stock prices or returns are also highly volatile (Ulusoy&Ozdurak, 2018). The high risk in investment in oil security and the high volatility of oil stock returns does not discourage investors from investing in the stock of oil companies; believing that the higher the risk, the higher the return.
In studying the volatility of oil stock returns occasioned by the fluctuations in crude oil prices resulting from the current global financial crisis and covid-19 pandemic, this study uses the combination of autoregressive moving average (ARMA) models, a popular and excellent model for modeling and forecasting univariate time series data (Emenogu, et.al, 2019) proposed by Box & Jenkins (1970), and generalized autoregressive conditional heteroscedasticity (GARCH) family models. The flexibility and simplicity of the ARMA model, combined with the capacity of the GARCH models to capture volatility in financial time series, to yield the ARMA-GARCH models (Emenogu, et.al, 2019). The advantage of this combination is its flexibility and tractability in allowing the model to capture both the mean and the variance components in the financial time series volatility (Lange, 2011;Panait&Slavescu, 2012), thus yielding a more reliable estimates for decision making.

Literature reviews
Crude oil futures stock prices and returns have attracted many researchers to study the effect of wars, financial crisis on the performance of the market in such periods. This section provides brief literature reviews in that regards. Bencivenga et al. (2012) investigated crude oil volatility and its relationships with Dollar/Euro exchange rate, US interest rate, the crude oil futures open interest, US oil imports and gold price over the period of 1993 to 2009. The study revealed one long run relationship equation and the authors found that exchange rate and gold price played dominant role in the estimated model. Zavadska et al. (2020) investigated Bent crude oil and futures prices using GARCH type models during first gulf war 1990/91, Asian financial crisis 1997/98, the US terrorist attack 2001 and global financial crisis 2008/2009. The authors found that higher level of volatility during crisis was associated with oil supply and demand disruptions while higher volatility persistence associated with financial and economic crises.
Singh and Singh (2017) examined the impact of financial crisis on the volatility of crude oil market in India. The study revealed no evidence of structural break was found except for CUSUM of squares test. Evidence from EGARCH model with dummy, the study found that there was no effect of global financial crisis on the volatility of crude spot market of India. Chen (2014) investigated the fear spillovers between for implied volatility indices which include MVX in Canada, VXJ in Japan, VDAX in Germany and VIX in the United States using a Copula-based bivariate Markov-switching model. They found that there exist linkages between implied volatility indices and are more evident when the indices rise. Yousef (2020) investigated the impact of Coronavirus on stock market volatility in United States using three stock market indexes (S&P500, Dov Jones and Nasdaq). The study employed GARCH, TGARCH and GJR-GARCH models with COVID-19 as dummy variable. The results revealed in the condiational variance equation that COVID-19 had a significant positive impact on all the three stock indexes, implying that COVID-19 had increased market volatility. Wei and Wei (2019) investigated the long-term connection between crude oil futures market and China stock market across financial crises with the use of non-linear threshold cointegration method. The study revealed exchange rate market as a key factor in transferring the impact of oil price on China stock market after the financial crisis. Junttila and Raatikainen (2018) used daily data from 1989 to 2016 to investigate the correlations between gold and oil market futures and equity returns in the aggregate US equities increased in crisis period while gold futures became negatively correlated with US equities around times of financial crises. Zhang and Wang (2013) investigated the price discovery and risk transfer functions in the crude oil and gasoline futures market. The authors found financial crisis has not significantly influenced the price discovery and risk transfer functions between crude oil and gasoline futures markets. Kang and Yoon (2017) examined the dynamic spillover effects among crude oil, precious metal, and agricultural commodity futures markets using multivariate DECO-GARCH model and spillover index. The study revealed a positive equicorrelation between commodity futures market return increased shaply during financial crises period. Jiang and Zhou (2014) investigated the weak-form efficient of the WTI crude oil futures market. They found that market is inefficient after turbulent and crises event occurred such as oil crash in 1985, the Gulf war and the oil price crash in 2008. Luo et al. (2020) introduced the Infinite Hidden Markov (IHM) model to forecast crude oil realized volatility in the presence of structural breaks such as change in policy, exogenous shock and other factors like financial and health crisis. The authors recommended that IHM-HAR models with exogenous factors are superior for short term forecast and provides economic gains. Adenomon et al. (2020) examined the effect of COVID-19 outbreak on the performance of the Nigeria stock exchange using historical data covering 2 nd March 2015 to 16 th April, 2020 sourced from a secondary source. The authors applied Quadratic GARCH (QGARCH) and Exponential GARCH (EGARCH) models with dummy variable were applied to the stock returns and the results shown that the COVID-19 has had negative effect on the stock returns in Nigeria.
From the foregoing, crude oil futures have not be study especially with the impact of the global financial crisis and the present COVID-19 pandemic. Hence the reason for this present study.

Methodology
This study uses GARCH and ARMA models in data analysis in order to select the model or models that give the best possible forecast. Particularly, the combination of the GARCH and ARMA models, the so called ARMA-GARCH models come very handy here due to their robustness for forecasting the volatility of financial time series data, and because of their superiority in modelling the conditional mean and conditional variance (volatility) of any financial time series (Ruppert, 2011).So we present in this section ARMA-GARCH models and some of their extensions.

The ARMA-GARCH Model
The ARMA (p,q)-GARCH (1,1) model can be specified as follows: (1) ) D(0, , where r t is the daily rate of return, θ is the AR(p) term in the mean equation in order to account for time dependence in returns, φ is the MA(q) term in the mean equation, t ε is the residual term in the mean equation, Z t is the standardized residual sequence of iid random variable with mean zero and variance as one while D represents distribution of the shock returns (Moshiri and Foroutan, 2006;Chen et al., 2013;Meitz and Saikkonen, 2011;Francq and Zakoian, 2004;Ling and McAleer, 2003;Koul and Ling, 2006;Zhu and Ling, 2011).

ARCH Model
Autoregressive Conditional Heteroskedasticity (ARCH) Family Model or Generalized Autoregressive Conditional Heteroskedasticity (GARCH) Family Model, according to Atoi (2014), requires two distinct specifications, namely: the mean and the variance equations. The mean equation for a conditional heteroskedasticity in a return series, t y is given by The mean equation in equation (2) also applies to other GARCH family models. (.) is the expected value conditional on information available at time t-1, while t ε is the error generated from the mean equation at time t and t φ is the sequence of independent and identically distributed random variables with zero mean and unit variance. The variance equation for an ARCH(p) model is given by The equation reveals that large values of the innovation of asset returns have bigger impact on the conditional variance because they are squared, meaning that a large shock tends to follow another large shock and that is the same way the clusters of the volatility behave. So the ARCH(p) model becomes: ε is assumed to follow the standard normal or a standardized student-t distribution or a generalized error distribution (Tsay 2005). Asymmetric Power ARCH Rossi (2004) asserts that the asymmetric power ARCH model below, which was proposed by Ding, Engle & Granger (1993) forms the basis for deriving the GARCH family models Given that: This model is known to impose a Box-Cox transformation of the conditional standard deviation process and the asymmetric absolute residuals. The leverage effect is the asymmetric response of volatility to positive and negative "shocks".

Standard GARCH(p, q) Model:
The mathematical model for the sGARCH(p,q) model is obtained from equation (5)   ( t r is the continuously compounded log return series), and t ε~N (0,1) iid , the parameter i α is the ARCH parameter and j β is the GARCH parameter, and (Rossi, 2004;Tsay, 2005 andJiang, 2012 To expatiate on the properties of GARCH models, the following representation is necessary: Let (4), the GARCH model can be rewritten as It can be seen that { t η } is a martingale difference series (i.e., E( t η )= 0 and A GARCH model can be regarded as an application of the ARMA idea to the squared series 2 t a . Using the unconditional mean of an ARMA model, results in this provided that the denominator of the prior fraction is positive. (Tsay, 2005) When p =1 and q =1, we have GARCH(1, 1) model given by:

EGARCH Model
The Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) Model proposed by Nelson (1991), Enocksson and Skoog(2012) pointed out, was to overcome some weaknesses of the GARCH model in handling financial time series, in particular, to allow for asymmetric effects between positive and negative asset returns, he considered the weighted whereθ and γ are real constants. Both t ε and The asymmetry of ) ( t g ε can easily be seen by An EGARCH(m, s) model, according to Tsay (2005), Dhamija and Bhalla (2010), Jiang (2012), Ali (2013) and Grek (2014), can be written as Which specifically results in EGARCH (1, 1) being written as

TGARCH(p, q) Model
TGARCH model stands for Threshold GARCH model, and it is another model used to handle leverage effects.The TGARCH(p, q) model is given as follows: and i α , i γ , and j β are nonnegative parameters satisfying conditions similar to those of GARCH models, (Tsay, 2005). When 1 , 1 = = q p , the TGARCH(1, 1) model becomes:

ARCH-LM
The Autoregressive Conditional Heteroscedasticity-Lagrange Multiplier (ARCH-LM) test proposed by Engle's (1982) for testing autoregressive conditional heteroscedasticity is the undisputed and most commonly applied standard test for detecting the presence of ARCH effect (Sjölander, 2011). The test circumvents the problem of high dimensionality in multivariate tests for ARCH in (VAR) models (Catani &Ahlgren, 2017), and Tsay (2005) says that it is one of two tests available for detecting ARCH effect in the residual of a mean equation, the other method being the usual Ljung-Box statistic, Q(m).
If we let = − be the residuals of the mean equation, the test is equivalent to the usual F statistic for testing the hypothesis that there is ARCH effect; H 0 : = 0, ( = 1, 2, ⋯ , ) in the linear regression: where, is the error term, is a prespecified positive integer, and is the sample size. (Tsay, 2005;Sjölander, 2011) Now, let where ! " = # ∑ # is the sample mean of , where % is the k regression.

Thus the test statistic is
which is asymptotically chi-squared distribution with p degrees of freedom ; that is , ( ) under H 0 , and the decision rule is to reject H 0 if & > , (.) , if the p-value of F is less than (Tsay, 2005).

Materials and Methods
The data used in this study were collected from www.investing.com. The study used daily and The returns were calculated using the following formula: where is return at time t; ln is the natural logarithm; 1 is the current stock price at time t, and 1 is the previous stock price at time − 1. ARCH LM-test: Chi-squared = 1220.1, df = 12, p-value < 2.2e-16

Descriptive Statistics
In bold significant Jarque-Bera (JB) test at 5% level of significance.
The table 1 presents the descriptive statistics of the daily crude oil futures price from the 4 th January 2000 to 27 th April 2020. the maximum stock price for Crude oil futures was $145.2900 that occurred during the global financial crisis and the minimum stock price for the crude oil futures was $10.01000 which occurred during the COVID-19 crisis. The price had high standard deviation with positive skewness with a moderate kurtosis but the stock price was not normally  pandemic. In addition, 0.319640 was maximum returns, -0.601680 was minimum returns with standard deviation of 0.107638, but the returns exhibited negative skewness and high kurtosis value of 13.93005 and finally, the returns series was not normally distributed and there was no evidence of arch effects in the returns series.  The bold denotes evidence of no unit root in the series.
The unit root testing was carried using ADF, DF-GLS and PP test on the daily crude oil stock prices and returns. In the table 3 above, the stock price of the crude oil for the full sample was not stationary but for the stock returns for the full sample was stationary.  The bold denotes evidence of no unit root in the series.
The unit root testing was carried using ADF, DF-GLS and PP test on the daily crude oil stock prices and returns during the global financial crisis and COVID-19 crisis. In the table 4 above, the stock returns of the crude oil for the Crisis periods (global financial crisis and COVID-19 crisis) was stationary.  In addition, high volatility occurred during 2008 and 2020.   The log stock prices and returns almost exhibited similar pattern during the global financial crisis as shown in fig. 9 above. crisis. This is similar to the pattern in fig. 11 above.  The bold denotes evidence of no unit root in the series.
The unit root testing was carried using ADF, DF-GLS and PP test on the weekly crude oil stock prices and returns. In the table 7 above, the stock price of the crude oil for the full sample was not stationary but for the stock returns for the full sample was stationary. The bold denotes evidence of no unit root in the series.
The unit root testing was carried using ADF, DF-GLS and PP test on the weekly crude oil stock prices and returns during the global financial crisis and COVID-19 crisis. In the table 8 above, the stock returns of the crude oil for the Crisis periods (global financial crisis and COVID-19 crisis) were stationary.

Results and Analysis
This section focused on univariate time series analysis and the univariate GARCH analysis of the daily and weekly stock prices and returns by including global financial crisis and COVID-19 crisis as dummy variables. The Auto.arima function in R software was used to implement the best ARIMA model for the log stock price and returns as presented in table 9 above. For the log stock prices ARIMAX(1,1,0) was optimal for the daily crude oil futures and the autoregressive coefficient was significant while the global financial crisis (gfc)and the impact of COVID-19 were positive but not significant. The model was not adequate while the residual is not normally distributed and exhibited Arch effects.
For the log returns ARIMAX(1,,0) was optimal for the daily crude oil futures returns and the autoregressive coefficient was significant while the impact of global financial crisis (gfc) was negative and the impact of COVID-19 was positive but not significant. The model was not adequate while the residual is not normally distributed and exhibited Arch effects. The table 10 above presents the selection criteria values for daily crude oil futures based on the student t and skewed student t distributions. The ARMA model favoured the ARMA(1,0) model while the competing GARCH models (that is sGARCH, eGARCH, TGARCH and apARCH) used a maximum lag of 2. The selection criteria seems to be lowest for skewed student t distribution compared to student t distribution except in few cases. This is so because the return series was negatively skewed. Among the competing ARMA-GARCH models, ARMA(1,0)-eGARCH(2,1) with skewed student t distribution had the least selection criteria values while ARMA(1,0)-eGARCH(2,2) with student t distribution. Hence ARMA(1,0)-eGARCH(2,1) with skewed student t distribution emerged as the superior model for daily crude oil futures with the effects of global financial crisis and the present COVID-19 crisis.  Table 11 presented the persistence and half-life values for the competing models with the student t and skewed student t distributions. The entire ARMA-GARCH model exhibited very high persistence value though less than 1 (one), this could be due to the impact of global financial crisis and the present COVID-19 pandemic, but the entire model exhibited stability. For the ARMA(1,0)-eGARCH(2,1), the persistence value is 0.9933146 while it take about 104 days for mean-reverting to take place.  The Auto.arima function in R software was used to implement the best ARIMA model for the log stock price and returns as presented in table 13 above. For the log stock prices ARIMAX(1,1,0) was optimal for the weekly crude oil futures and the autoregressive coefficient was not significant while the global financial crisis (gfc)and the impact of COVID-19 were negative but not significant. The model was not adequate while the residual is not normally distributed and exhibited Arch effects.
For the log returns ARIMAX(1,,0) was optimal for the weekly crude oil futures returns and the autoregressive coefficient was not significant while the impact of global financial crisis (gfc) was negative and the impact of COVID-19 was positive but only COVID-19 effect was significant.
The model was not adequate while the residual is not normally distributed and exhibited Arch effects. The table 14 above presents the selection criteria values for weekly crude oil futures based on the student t and skewed student t distributions. The ARMA model favoured the ARMA(1,0) model while the competing GARCH models (that is sGARCH, eGARCH, TGARCH and apARCH) used a maximum lag of 2. The selection criteria seems to be lowest for skewed student t distribution compared to student t distribution except in few cases. This is so because the return series was negatively skewed. Among the competing ARMA-GARCH models, ARMA(1,0)-eGARCH(2,2) with skewed student t distribution had the least selection criteria values while ARMA(1,0)-eGARCH(1,1) with student t distribution. Hence ARMA(1,0)-eGARCH(2,2) with skewed student t distribution emerged as the superior model for weekly crude oil futures with the effects of global financial crisis and the present COVID-19 crisis.  Table 15 presents the persistence and half-life values for the competing models with the student t and skewed student t distributions. The entire ARMA-GARCH model exhibited very high persistence value except for ARMA(1,0)-eGARCH(2,1) though less than 1 (one), this could be due to the impact of global financial crisis and the present COVID-19 pandemic, but the entire model exhibited stability. For the ARMA(1,0)-eGARCH(2,2), the persistence value is 0.9263762 while it take about 10 weeks for mean-reverting to take place. Unconditional and conditional coverage while 95% Value-at-Risk the model also passed using the unconditional coverage and passed using the conditional coverage. On the overall the estimated ARMA(1,0)-eGARCH(2,2) model is reliable.
The  (Chen, 2014;Adenomon, et al. 2020, Yousef, 2020. The pandemic. In addition, 0.319640 was maximum returns, -0.601680 was minimum returns with standard deviation of 0.107638, but the returns exhibited negative skewness and high kurtosis value of 13.93005 and finally, the returns series was not normally distributed and there was no evidence of arch effects in the returns series. This descriptive statistics is similar to (Chen, 2014;Adenomon, et al. 2020, Yousef, 2020. Evidences from the figures during crisis revealed that crude oil futures could be affected negatively leading to losses in value of stock returns (Junttila and Raatikainen , 2018).
The unit root testing was carried using ADF, DF-GLS and PP test on the daily crude oil stock prices and returns. In the table 3 above, the stock price of the crude oil for the full sample was not stationary but for the stock returns for the full sample was stationary.
The unit root testing was carried using ADF, DF-GLS and PP test on the daily crude oil stock prices and returns during the global financial crisis and COVID-19 crisis. In the table 4 above, the stock returns of the crude oil for the Crisis periods (global financial crisis and COVID-19 crisis) was stationary. These results regarding stationarity of returns are expected as returns are mostly stationary .
The  (Chen, 2014). The log stock prices and returns almost exhibited similar pattern during the global financial crisis as shown in fig. 9 above. Fig. 11 presents the time plot of the weekly crude oil futures stock prices from 5 th January 2020 to 26 th April 2020 during the COVID-19 Crisis, the stock price shows almost steady decline in the crude oil futures during this present COVID-19 crisis. Fig. 12 presents the time plot of the weekly crude oil futures stock prices and returns from 5 th January 2020 to 26 th April 2020 during the COVID-19 Crisis, the stock price shows almost steady decline in the crude oil futures during this present COVID-19 crisis. This is similar to the pattern in fig. 11 above. This shows that crude oil futures are more volatile during crisis Zavadska et al. (2020).
The unit root testing was carried using ADF, DF-GLS and PP test on the weekly crude oil stock prices and returns. In the table 7 above, the stock price of the crude oil for the full sample was not stationary but for the stock returns for the full sample was stationary. The unit root testing was carried using ADF, DF-GLS and PP test on the weekly crude oil stock prices and returns during the global financial crisis and COVID-19 crisis. In the table 8 above, the stock returns of the crude oil for the Crisis periods (global financial crisis and COVID-19 crisis) were stationary. This is expected as returns are mostly stationary ).
The univariate time series analysis and the univariate GARCH analysis of the daily and weekly stock prices and returns by including global financial crisis and COVID-19 crisis as dummy variables.The Auto.arima function in R software was used to implement the best ARIMA model for the log stock price and returns as presented in table 9 above. For the log stock prices ARIMAX(1,1,0) was optimal for the daily crude oil futures and the autoregressive coefficient was significant while the global financial crisis (gfc)and the impact of COVID-19 were positive but not significant. The model was not adequate while the residual is not normally distributed and exhibited Arch effects. For the log returns ARIMAX(1,,0) was optimal for the daily crude oil futures returns and the autoregressive coefficient was significant while the impact of global financial crisis (gfc) was negative and the impact of COVID-19 was positive but not significant.
The model was not adequate while the residual is not normally distributed and exhibited Arch effects.
The Auto.arima function in R software was used to implement the best ARIMA model for the log stock price and returns as presented in table 13 above. For the log stock prices ARIMAX(1,1,0) was optimal for the weekly crude oil futures and the autoregressive coefficient was not significant while the global financial crisis (gfc)and the impact of COVID-19 were negative but not significant. The model was not adequate while the residual is not normally distributed and exhibited Arch effects. For the log returns ARIMAX(1,,0) was optimal for the weekly crude oil futures returns and the autoregressive coefficient was not significant while the impact of global financial crisis (gfc) was negative and the impact of COVID-19 was positive but only COVID-19 effect was significant. The model was not adequate while the residual is not normally distributed and exhibited Arch effects.
The above results from ARMA models for daily and weekly crude oil futures helped us to identify the order of the ARMA model but failed certain assumptions. Thus the need to fit ARMA-GARCH Models.
The table 10 above presents the selection criteria values for daily crude oil futures based on the student t and skewed student t distributions. The ARMA model favoured the ARMA(1,0) model while the competing GARCH models (that is sGARCH, eGARCH, TGARCH and apARCH) used a maximum lag of 2. The selection criteria seems to be lowest for skewed student t distribution compared to student t distribution except in few cases. This is so because the return series was negatively skewed. Among the competing ARMA-GARCH models, ARMA(1,0)-eGARCH(2,1) with skewed student t distribution had the least selection criteria values while ARMA(1,0)-eGARCH(2,2) with student t distribution. Hence ARMA(1,0)-eGARCH(2,1) with skewed student t distribution emerged as the superior model for daily crude oil futures with the effects of global financial crisis and the present COVID-19 crisis. This is in line with Wilhelmsson (2006). Table 11 presents the persistence and half-life values for the competing models for daily crude oil futures with the student t and skewed student t distributions. The entire ARMA-GARCH model exhibited very high persistence value though less than 1 (one), this could be due to the impact of global financial crisis and the present COVID-19 pandemic Zavadska et al. (2020), but the entire model exhibited stability. For the ARMA(1,0)-eGARCH(2,1), the persistence value is 0.9933146 while it take about 104 days for mean-reverting to take place.
The used a maximum lag of 2. The selection criteria seems to be lowest for skewed student t distribution compared to student t distribution except in few cases. This is so because the return series was negatively skewed. Among the competing ARMA-GARCH models, ARMA(1,0)-eGARCH(2,2) with skewed student t distribution had the least selection criteria values while ARMA(1,0)-eGARCH(1,1) with student t distribution. Hence ARMA(1,0)-eGARCH(2,2) with skewed student t distribution emerged as the superior model for weekly crude oil futures with the effects of global financial crisis and the present COVID-19 crisis.This is in line with Wilhelmsson (2006). Table 15 presents the persistence and half-life values for the competing models with the student t and skewed student t distributions. The entire ARMA-GARCH model exhibited very high persistence value except for ARMA(1,0)-eGARCH(2,1) though less than 1 (one), this could be due to the impact of global financial crisis and the present COVID-19 pandemicZavadska et al.
The table 16 above presents backtesting test for ARMA(1,0)-eGARCH(2,2) model obtained for weekly crude oil futures. At 99% Value-at-Risk, the model passed the backtesting test using the Unconditional and conditional coverage while 95% Value-at-Risk the model also passed using the unconditional coverage and passed using the conditional coverage. On the overall the estimated ARMA(1,0)-eGARCH(2,2) model is reliable (Nieppola, 2009).
Lastly the estimated ARMA(1,0)-eGARCH(2,1) and ARMA(1,0)-eGARCH(2,2) 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 Zavadska et al. (2020) while the preferred estimated models also passed the goodness-of-test fit.

Conclusion and Recommendation
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 backtestingof 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. The study used both student t and skewed student t innovations with AIC, goodness-oftest 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 Zavadska et al. (2020) 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 tends to be affected by global financial crisis and COVID-19 pandemic (Junttila and Raatikainen, 2018) Countries also that depend mostly on crude oil are encouraged to diversify their economy in other to survive and sustain during financial and health crisis.