Modelling the Impact of the COVID-19 Pandemic on Some Nigerian Sectorial Stocks: Evidence from GARCH Models with Structural Breaks

: This study provides evidence of the impact of COVID-19 on ﬁve (5) Nigerian Stock Exchange (NSE) sectorial stocks (NSE Insurance, NSE Banking, NSE Oil and Gas, NSE Food and Beverages, and NSE Consumer Goods). To achieve the goal of this paper, daily stock prices were obtained from a secondary source ranging from 2 January 2020 to 25 March 2021. Because of the importance of incorporating structural breaks in modelling stock returns, the Zivot–Andrews unit root test revealed 20 January 2021, 26 March 2020, 27 July 2020, 23 March 2020 and 23 March 2020 as potential break points for NSE Insurance, NSE Food, Beverages and Tobacco, NSE Oil and Gas, NSE Banking, and NSE Consumer Goods, respectively. This study investigates the volatility in daily stock returns for the ﬁve (5) Nigerian Stock Exchange (NSE) sectorial stocks using nine versions of GARCH models (sGARCH, girGARCH, eGARCH, iGARCH, aPARCH, TGARCH, NGARCH, NAGARCH, and AVGARCH); in addition, the half-life and persistence values were obtained. The study used the Student t - and skewed Student t -distributions. The results from the GARCH models revealed a negative impact of COVID-19 on the NSE Insurance, NSE Food, Beverages and Tobacco, NSE Banking, and NSE Consumer Goods stock returns; however, the NSE Oil and Gas returns showed a positive correlation with the COVID-19 pandemic. This study recommends that the shareholders, investors, and policy players in the Nigerian Stock Exchange markets should be adequately prepared in the form of diversiﬁcation of investment in stocks that can withstand future possible crises in the market.


Introduction
The COVID-19 virus was first reported in Wuhan city of China in December 2019 [1], while in Africa, the first COVID-19 case was recorded in Egypt on 14 February 2020 [2]; however, for Nigeria, the first COVID-19 case was recorded on 27 February 2020 [3,4]. COVID-19 impacted both developed economies such as USA, Germany, UK, etc., and developing economies such as Nigeria, Egypt, Morocco, etc. [5]. The impact of COVID-19 on any economy cannot be overemphasized: world economies are expected to have a negative growth of 4.4% in 2020 [6], while developing economies would record a negative growth of 5.6% in the year 2020 [7].
Due to the impact of COVID-19, many economies were in lockdown, leading to a loss in Gross Domestic Product (GDP), of which the financial market was not left out.
In the past, world stock markets have suffered from global financial crises and now, the market is been plague by the COVID-19 pandemic of which developed, developing, and the equity markets. The study revealed that panic generated by the news outlets regarding COVID-19 is associated with increasing volatility in the equity markets. The study of [16] considered the financial markets under the impact of COVID-19 using daily data up to 27 March 2020 for certain countries. The study implemented the volatility analysis, correlation analysis, and minimum spanning tree. Their work confirmed that market risks have increased substantially in response to the pandemic: individual stock market reactions are linked to the severity of the outbreak of the pandemic. Ref. [17] investigated the impact of COVID-19 pandemic uncertainty on financial market volatility, with an interest in new infection cases and the fatality ratio reported at the global level and in the US. The study employed a three-month realized volatility index of S&P 500 as a proxy for the US financial markets' volatility, while the test of simple Ordinary Least Squares (OLS) and the stepwise procedure were implemented. The study concluded that the US financial markets were affected by the persistence of the COVID-19 crisis. The work of [18] investigated the impact of COVID-19 on the volatility of stock prices in India (Nifty and Sensex Daily closing prices of stock indices from 3 September 2019 to 10 July 2020) using the GARCH model. Further, the study made comparisons of the stock price return in the pre-COVID-19 and COVID-19 periods. The findings revealed that the stock market in India has experienced volatility during the pandemic period. In addition, the study found that the return on the indices is higher in the pre-COVID-19 period than in the period of COVID-19. Ref. [19] considered the impact of the COVID-19 pandemic and inflation on the All Share Index (ASI) in Nigeria using the GARCH and GJR-GARCH models. The study shows that the COVID-19 pandemic increased volatility and distorted the possible positive relationship between inflation and stock market returns in Nigeria. Ref. [19] considered ASI, whereas our study focuses on sectorial stock returns to see the effect on each sector of the Nigeria Stock Exchange (NSE).
Other previous studies include the following: Ref. [20] investigated the economic impact of government interventions during the COVID-19 pandemic in relation to international evidence from financial markets; Ref. [21] studied the effect of the COVID-19 pandemic on the transmission of monetary policy to financial markets. In Nigeria, other studies include [9,22].
From the foregoing, much of the work had not paid attention to the sectorial stocks of the Nigerian Stock Exchange. Hence, this study investigates the impact of COVID-19 on five (5) Nigerian Stock Exchange (NSE) sectorial stocks, namely, NSE Insurance, NSE Banking, NSE Oil and Gas, NSE Food and Beverages, and NSE Consumer Goods, using some family GARCH models (sGARCH, girGARCH, eGARCH, iGARCH, aPARCH, TGARCH, NGARCH, NAGARCH, and AVGARCH); in addition, the values of the half-life and persistence were obtained.

Variants of GARCH Models
The general form of the standard GARCH(p,q) model (also known as the GARCH model) is a combination of ARCH and GARCH parameters, and is given as where a t = r t − µ t (r t is obtained as the continuous compounding log return series), ε t ∼ N(0, 1) iid, α i is the ARCH parameter and β j is the GARCH parameter, and ω > 0, α i + β j < 1, which means that the GARCH model is stable and suitable for forecasting [23][24][25][26].
For instance, the condition on ARCH and GARCH parameters α i , β j suggests that the volatility (a i ) is finite in nature while the conditional standard deviation (σ i ) increases as well. We observe that if q = 0, then the GARCH model parameter β j becomes extinct and what is left is called the ARCH(p) model. Suppose p = 1 and q = 1, then the GARCH(1,1) model can be presented as The GARCH (1,1) model is popular in modelling financial returns such as in the works of [27,28].

The Asymmetric Power ARCH
The asymmetric power ARCH model is another interesting GARCH model that was proposed by Ding, Engle, and Granger in 1993 [23]. The asymmetric power ARCH model can be stated as where The asymmetric power ARCH model utilizes a Box-Cox transformation of the conditional standard deviation process and the asymmetric absolute residuals. The leverage effect in the asymmetric power ARCH is the asymmetric response of volatility relating to both positive and negative shocks.

GJR-GARCH(p,q) Model
The GARCH model that attempts to address volatility clustering in the innovation process is called the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model. The GJR-GARCH model is obtained by letting δ = 2.
When δ = 2 and 0 ≤ γ i < 1, That is, FinTech 2023, 2 5 The Equation (5) above is the GJR-GARCH model [23]. However, when −1 ≤ γ i < 0, recall Equation (4): The Equation (6) obtained above allows the positive shocks to have a stronger effect on volatility than the negative shocks [23]. Lastly, when p = q = 1, we can write the GJR-GARCH(1,1) model as follows: 2.1.4. Integrated GARCH(1,1) Model The unit-root GARCH models are referred to as the Integrated GARCH (IGARCH) models [26]. The IGARCH(1,1) model can be specified as where ε t ∼ N(0, 1) iid, and 0 < β 1 < 1. It can be noted that α i can be used to denote 1 − β i [29]. The model also used an exponential smoothing model for the series a 2 t . Then, the model can be rewritten as By repeated substitution, we obtain which, in the financial model literature, is a well-known exponential smoothing formation in which β 1 is regarded as the discounting factor [26]. The advantage of the threshold GARCH model is the capacity to handle leverage effects in financial time series. The TGARCH(1,1) model can be specified as follows: where N t−i refers to an indicator for negative a t−i , which can further be specified in detail as follows, where α i , γ i , and β j refer to nonnegative parameters that follow conditions similar to those of family GARCH models [26]. Suppose p = 1, q = 1, then the TGARCH(1,1) model will become as shown in Equation (12) below: 2.1.6. Nonlinear GARCH(p,q) Model The nonlinear GARCH model has been explored in various ways in the literature by the following scholars: [30][31][32][33]. The NGARCH(p,q) model can be shown as follows: where h t is known as the conditional variance, while ω, β, and α should satisfy ω > 0, β ≥ 0, and α ≥ 0. Finally, the NGARCH(p,q) can then be written as 2.1.7. EGARCH Model The exponential GARCH (EGARCH) model is an important model that was proposed around 1991 [34] for the purpose of overcoming some challenges inherent in the GARCH model when handling financial time series [35]. Particularly, the EGARCH model allows for asymmetric effects between positive and negative asset returns. Consider the weighted innovation as shown in Equation (15) below: where θ and γ can be seen as real constants. Both ε t and |ε t | − E(|ε t |) are zero-mean iid sequences that follow continuous distributions. Therefore, E[g(ε t )] = 0, then the asymmetry of g(ε t ) can be rewritten as An EGARCH(p,q) model, according to [24,26,29,36,37] can be written as The EGARCH(1,1) can then be written as where |a t−1 | − E(|a t−1 |) is an iid with zero mean. The EGARCH model with a Gaussian distribution error term, E(|ε t |) = √ 2/π, can be specified as: In the work of [29], an asymmetric GARCH (AGARCH) can be written as However, the absolute value GARCH (AVGARCH) model can be written as

Nonlinear Asymmetric GARCH Model
Another interesting GARCH model that plays a key role in option pricing with stochastic volatility is called the nonlinear asymmetric GARCH (NAGARCH). The NAGARCH can be written as Suppose that z t ∼ I IDN(0, 1), where z t is independent of σ 2 t . Then, σ 2 t is now only a function of an infinite number of past-squared returns, which can derive the long run and unconditional variance under the NGARCH model with the assumption of stationarity: where σ 2 = E σ 2 t and E σ 2 t = E σ 2 t+1 because of the condition of stationarity. Then which exists and is positive if, and only if, α 1 + δ 2 + β < 1. The following are the implications: (i) The persistence index of an NAGARCH(1,1) can be seen as α 1 + δ 2 + β and not simply α + β as is common with other GARCH models; and Further details on these implications can be seen in [34,[38][39][40][41].

Persistence and Half-Life Volatility
The study of persistence and half-life is very important in financial time series modelling. They help to determine if the estimated GARCH model is stable and how long it takes for the mean reversion. The persistence value of a GARCH model can be obtained as the sum of the GARCH (β 1 ) coefficient and the ARCH (α) coefficient (α + β 1 ). In empirical financial time series, persistence values are often very close to one (1) [28,42]. Persistence can assumed as the following: (i) When α + β 1 < 1, the GARCH model is stationary and has a positive conditional variance. (ii) When α + β 1 = 1, the model is strictly stationary. In addition, the GARCH model has an exponential decay model which makes the half-life value become infinity. Lastly, when α + β 1 > 1, the GARCH model is assumed to be unstable and nonstationary [27,28].
The half-life volatility of a GARCH model is a statistic that measures the mean reverting speed (known as average time) of a stock return under study. The half-life volatility can be written as When the value of α + β 1 is very close to one (1), we can expect that the volatility shocks of the half-life of the estimated GARCH model will be longer [28].

Distributions of GARCH Models
The distribution plays a significant role on the performance of the estimated GARCH model for any given financial time series data. This study used two innovations (namely the Student tand skewed Student t-distributions); this is because they have the potential to account for the excessive kurtosis and non-normality inherent in financial returns [27,43,44].
The Student t-distribution can be written as While, on the other hand, the Skewed Student t-distribution can be written as where ν is the shape parameter of the Skewed Student t-distribution with 2 < ν < ∞, while λ is the skewness parameter with −1 < λ < 1. a, b and c are constants given as In the Skewed Student t-distribution above, µ and σ are known as the mean and standard deviation, respectively.

Materials and Methods
Daily stock prices were obtained from a secondary source ranging from 2 January 2020 to 25 March 2021. The data covers this range due to the availability of data for all the sectors considered in this study; Nigerian Stock Exchange (NSE) sectorial stocks, namely, NSE Insurance, NSE Banking, NSE Oil and Gas, NSE Food and Beverages, and NSE Consumer Goods were collected from www.investing.com accessed on 25 March 2021. The structural break was coded as 0 for the period before the break and 1 for the period from the break onward.
We calculated the returns using the following formula in (29) below: In the formula in (29), R t represents the return at time t; the natural logarithm is represented as ln; P t represents the current daily stock price at time t; while P t−1 represents the previous daily stock price at time t − 1. As mentioned earlier, we employed the Student t-distribution and Skewed Student t-distribution in this study.

Results
The rugarch package in the R environment [45] was employed in the analyses of this study. Figure 1 shows some evidence of volatility; however, there is a sharp drop at data point 261, which is evidence of a break.                             Figure 5 shows some evidence of volatility at the midpoint of the return series; however, there is a sharp drop at data point 142, which is evidence of a break. Figure 5 shows some evidence of volatility at the midpoint of the return series; however, there is a sharp drop at data point 142, which is evidence of a break.       Figure 6 shows that the return series is stationary with a potential break point at 142 (27 July 2020). Figure 5 shows some evidence of volatility at the midpoint of the return series; however, there is a sharp drop at data point 142, which is evidence of a break.         Figure 7 shows some evidence of volatility at the beginning of the return series; however, there is a sharp drop at data point 50, which is evidence of a break.         Figure 7 shows some evidence of volatility at the beginning of the return series; however, there is a sharp drop at data point 50, which is evidence of a break.       Figure 9 shows some evidence of volatility at the beginning of the return series; however, there is a sharp drop at data point 57, which is evidence of a break. Figure 9 shows some evidence of volatility at the beginning of the return series; however, there is a sharp drop at data point 57, which is evidence of a break.         Figure 9 shows some evidence of volatility at the beginning of the return series; however, there is a sharp drop at data point 57, which is evidence of a break.     In Table 1 above, the sectorial stock return series are not normally distributed and all the series exhibited evidence of ARCH effects, which shows the appropriateness of the application of GARCH models. The Zivot-Andrews unit root test was applied to the sectorial stock returns, and the results of the unit root test revealed 20 January 2021, 26 March 2020, 27 July 2020, 23 March 2020, and 23 March 2020 as potential break points In Table 2 above, the TGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. The estimated TGARCH model is stable, while the mean reverting takes an average of four days. With the TGARCH(1,1), the effect of COVID-19 is positively correlated, while with EGARCH(1,1), the effect of COVID-19 is negatively related, though it is not significant in both models. For the NSE Insurance returns, all the estimated GARCH models revealed an absence of serial correlation using the weighted Ljung-Box Test. In addition, the ARCH LM test revealed an absence of ARCH effects in the residuals of the estimated GARCH models. In Table 3 above, the EGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. The estimated EGARCH model is stable, while the mean reverting takes an average of 12 days. The effect of COVID-19 is negatively correlated with the NSE Food, Beverages and Tobacco returns, and significant (p < 0.05) for the model with the Student t innovation and not significant with the Skewed Student t innovation. For the NSE Food, Beverages and Tobacco returns, all the estimated GARCH models revealed an absence of serial correlation using the weighted Ljung-Box Test, while the ARCH LM test revealed an absence of ARCH effects in the residuals of the estimated GARCH models.

Zivot and Andrews Unit Root Test
In Table 4 above, the EGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. The EGARCH model is stable, while the mean reverting takes an average of 20 days. The effect of COVID-19 is positively correlated with the returns and significant (p < 0.05). For the NSE Oil and Gas returns, all the estimated GARCH models revealed an absence of serial correlation using the weighted Ljung-Box Test, while the ARCH LM test revealed an absence of ARCH effects in the residuals of the estimated GARCH models.
In Table 5 above, the iGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. However, with EGARCH (1,1), the model is stable, while the mean reverting takes an average of 12 days. The effect of COVID-19 is positively correlated with the returns and not significant using the EGARCH (1,1) model. For the NSE Banking returns, all the estimated GARCH models revealed an absence of serial correlation using the weighted Ljung-Box Test, while the ARCH LM test revealed an absence of ARCH effects in the residuals in the estimated GARCH models.
In Table 6 above, the EGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. The EGARCH model is stable, while the mean reverting takes an average of 11 days. The effect of COVID-19 is negatively correlated with the NSE consumer returns and not significant using the EGARCH (1,1) model. For the NSE Consumer Goods returns, all the estimated GARCH models revealed an absence of serial correlation using the weighted Ljung-Box Test, while the ARCH LM test revealed an absence of ARCH effects in the residuals in the estimated GARCH models.

Discussion
In this study, the sectorial return series are not normally distributed, while the return series exhibited evidence of ARCH effects, which shows the appropriateness of the application of GARCH models. The Zivot-Andrews unit root test was applied to the sectorial stock returns and the results of the test revealed 20 January 2021, 26 March 2020, 27 July 2020, 23 March 2020, and 23 March 2020 as potential break points for NSE Insurance, NSE Food, Beverages and Tobacco, NSE Oil and Gas, NSE Banking, and NSE Consumer Goods, respectively. Each of the above dates is associated with some significant economic and financial event. In 20 January 2021, the National Bureau of Statistics (NBS) in Nigeria projected that the economy will grow by 3 percent in 2021, while the World Bank projected that it will grow by 1.7 percent. This was expected to improve the insurance sector in the year 2021 [46].
The Central Bank of Nigeria (CBN) on 16-18 March 2020 announced a set of policy measures to counter the impact of the fast-spreading coronavirus (such policies as interest rate cut, provision of credits to SMEs, injections of USD 3.25 billion into the manufacturing sector, etc.) [47]. This would have caused the structural breaks of 23 and 26 March 2020 in NSE Food, Beverage and Tobacco, NSE Banking, and NSE Consumer Goods, respectively.
Lastly, towards the end of July 2020, OPEC and other non-OPEC producers including Russia and Mexico agreed to extend its record oil production cuts [48]; this would have caused the structural break in NSE Oil and Gas at 27 July 2020 in Nigeria.
For the NSE Insurance returns, the TGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. The TGARCH model is stable, while the mean reverting takes an average of four days. With the TGARCH(1,1), the effect of COVID-19 is positively correlated with the returns, while with EGARCH(1,1), the effect of COVID-19 is negatively related with the returns, though it is not significant in both models. This finding is related to the study of [19], whose findings revealed that there are negative effects of COVID-19 on the NSE Insurance returns in Nigeria. This finding is also similar to the study of [18] that studied the Indian stock market.
For NSE Food, Beverages and Tobacco, the EGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. The EGARCH model is stable, while the mean reverting takes an average of 12 days. The effect of COVID-19 is negatively correlated with the returns, and significant (p < 0.05) for the model with the Student t innovation and not significant with the Skewed Student t innovation. This finding is related to the study of [19], whose findings revealed that there is a negative impact of COVID-19 on the NSE Food, Beverage and Tobacco returns in Nigeria. Our result is also related to the work of [5] which showed that Nigeria is a net recipient of United States risk spillovers.
For NSE Oil and Gas, the EGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. The EGARCH model is stable, while the mean reverting takes an average of 20 days. The effect of COVID-19 is positively correlated with the returns and significant (p < 0.05). This shows that the NSE Oil and Gas has higher volatility during the COVID-19 period. This is similar to the result of [18], which showed that the return on the indices is higher in the pre-COVID-19 period than in the period of COVID-19.
For NSE Banking, the iGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. However, with EGARCH (1,1), the model is stable, while the mean reverting takes an average of 12 days. The effect of COVID-19 is positively correlated with the returns and not significant using the EGARCH (1,1) model. This shows that the NSE Banking return has higher volatility during the COVID-19 period. This is similar to the result of [18], which showed that the return on the indices is higher in the pre-COVID-19 period than in the period of COVID-19.
For NSE Consumer Goods, the EGARCH(1,1) model has the least AIC value for both Student t and Skewed Student t innovations. The EGARCH model is stable, while the mean reverting takes an average of 11 days. The effect of COVID-19 is negatively correlated with the NSE Consumer Goods returns and not significant using the EGARCH (1,1) model. This finding is related to the study of [19] whose findings revealed that there is a negative impact of COVID-19 on the NSE Consumer Goods returns in Nigeria. Our result is also related to the work of [5], which showed that Nigeria is a net recipient of United States risk spillovers.

Conclusions
This study provides evidence of the impact of COVID-19 on five (5) Nigerian Stock Exchange (NSE) sectorial stocks (NSE Insurance, NSE Banking, NSE Oil and Gas, NSE Food and Beverages, and NSE Consumer Goods). In order to achieve the goal of this paper, daily stock prices were obtained from a secondary source ranging from 2 January 2020 to 25 March 2021. Because of the importance of incorporating structural breaks in modelling stock returns, the Zivot-Andrews unit root test revealed 20 January 2021, 26 March 2020,