COMPARING DIFFERENT SYMMETRIC AND ASYMMETRIC GARCH MODELS IN THREE ERROR DISTRIBUTIONS USING NIGERIAN CRUDE OIL EXPORT PRICES

By

Zainul-Abidina Salisu[1]; Hussaini Garba Dikko[2]; Sani Ibrahim Doguwa[3]

ABSTRACT

Volatility modelling has always been an important subject of inquiry and research in Financial Markets (such as crude oil markets). However, the review of relevant literatures showed that Autoregressive Conditional Heteroscedasticity (ARCH) model and its various extensions are being applied for modelling the volatility of Financial Time Series to capture the stylized facts incorporated in these series, but less attention has been given to the contribution of the error distribution assumptions while modelling the volatility. Accordingly, this study examined the monthly series of the Nigerian Crude Oil Price (NCOP) with a scope of January, 1982 to March, 2019 by employing simple GARCH, GARCH-M, EGARCH, TGARCH, GJR-GARCH, and APARCH models each in three main error distributions which are normal error, student’s t error and General Error Distribution (GED). The Log-likelihood function, Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) were used to compare the performances of the estimated models. The results indicated the existence of volatility clustering, leptokurtic behavior, leverage effect and high volatility persistence in the Nigerian crude oil price and that the asymmetric EGARCH model in Student’s-t distribution and GED has the best specifications for explaining Nigerian Crude Oil Price conditional volatility than the normal error.  It was concluded that the ARCH/GARCH models are suitable for dealing with volatility in oil price market and that, GARCH models with normal errors are not capable to fully capture the leptokurtic in empirical Time Series. Furthermore, the asymmetric effects are indeed present in empirical data and the asymmetric GARCH models perform better than the symmetric GARCH model in explaining conditional volatility. The work recommends that more credence may be given to asymmetric models for modelling oil price volatility, and thus investors to react to bad news than good news.

Keywords: Asymmetric effect, errors distribution, modelling, volatility, oil price, time series, volatility persistence.

INTRODUCTION

Crude oil is one of the greatest vital commodities in the world and despite the crusade for green energy and other sources of power, it is still an exclusive commodity in the international market. Crude oil applications are pervasive in daily life, regardless of being non-renewable, the world consumes crude oil every single minute as it is hard to find an alternative source that can parallel its performances. For the past two decades, oil has been experiencing some ups and downs. In early 1999, there was the Asian Financial Crisis, along with Iraq deciding to increase oil production, which caused oil prices to reach a bottom. But the market adjusted quickly and reached over U.S. $34 by late 2000.  The dotcom bubble in 2001 caused another round of economic panic, which caused it to drop until early 2002. Then, the global economy had been regaining momentum which resulted in a few years of bullish state (Lam, 2013). Accordingly, oil prices had been spiking up.

The limiting amount of global oil supply and hostile relationships between the U.S. and a number of oil producing countries caused oil price touched an all-time high of U.S. $147.30 in which the housing bubble in the U.S. started to burst, and an unprecedented credit crisis was followed. The dramatic decline in oil price that followed was difficult to model. Even though the price had, in general, exhibited a steady level of recovery after the financial crisis in 2008, it still posts a great challenge to find a model that performs consistently well, when being confronted with such unpredictable circumstances.

Nevertheless, crude oil price behaves like any other commodity with price swings in times of shortage and surplus. Such price swings have multiplier effect on our daily life ranging from diesel and gasoline to detergent, medicines and household appliances. Because of the multifaceted usefulness of crude oil, there is broad consensus that its price volatility can have significant impact on the financial market and economy. For instance, an increase in oil price induces higher cost of production and changes capacity utilization of firms. Such higher costs of production are usually passed on to consumers through soaring prices of consumer’s goods.

Furthermore, oil prices could lead to the implementation of policies. For example, interest in energy-related policy was revived with the high oil prices in the late 2000s, and the composition of governmental supports across sectors shifted tectonically with the American Recovery and Reinvestment Act, away from fossil fuels and toward unprecedented levels of support for renewable energy. Thus, modelling and forecasting crude oil prices have attracted the interest of business moguls’, energy researchers and policy makers. Therefore, an accurate forecast of oil prices is of great interest to investors, Energy Researchers and policymakers.

However, the review of relevant literatures shows that Autoregressive Conditional Heteroscedasticity model (ARCH model) and its various extensions are being applied for modelling volatility of financial Time Series data in order to captures the stylized facts incorporated in these series such as volatility clustering, persistence, heavy tail distribution, and leverage effects. But less attention has been given to the contribution of the error distribution assumptions while modelling volatility. And according to Deebom and Essi (2017), the wrong use of an appropriate error distribution in volatility model for financial Time Series may cause misspecification in volatility model, leptokurtic and autocorrelation behaviour of such series. Sequel to this, this study tends to model and forecast the volatility of Nigerian monthly crude oil Price series using two symmetric GARCH (GARCH and GARCH-M) models and four different asymmetric GARCH (EGARCH; TGARCH; GJR-GARCH and APARCH) models by considering or assuming three different error distributions which are the Normal Error Distribution, Student’s t Error Distribution and the Generalized Error Distribution. Thus, the aim of this study is to compare the performance of different symmetric and asymmetric GARCH models in three different distributional assumptions namely: normal error distribution; student’s t distribution and the Generalized Error Distribution (GED); and the objectives are:

1. To determine the volatility of the NCOP by estimating a series of symmetric and asymmetric GARCH models.
2. To select the best model and appropriate error distribution for explaining conditional volatility of the Nigerian Crude Oil Price based on information criterion, AIC and BIC.
3. To validate the homoscedasticity assumption of the best model selected using the LM-Test.

LITERATURE REVIEW

Historical examination of Financial Time Series data has revealed that volatility is not constant over time. It has been found that great returns are frequently followed by further large returns and vice versa termed as volatility clustering (Mandelbrot, 1963). The swing in volatility over time is due to the supposed market and unique risks not being constant, leading to risk at different periods (Brooks, 2014). Further on, Asset returns and commodities prices have leptokurtic unconditional distributions (Mandelbrot 1963, Fama 1965), which is related to the time varying volatility and are characterized by volatility clustering (Mandelbrot 1963, Fama 1965). Schwartz (1989) proved that Financial Time Series reveals high and low volatility episodes at any time, entailing that volatility in current time will lead to the probability of volatility many periods in future. Skewness can be associated to the fact that stock prices are liable to cluster; large (small) changes are followed by large (small) changes (Bollerslev 1986).

The first break-through in modelling Time Series that exhibit the above characteristics was championed by Engle (1982) which he termed as Autoregressive Conditional Heteroskedasticity (ARCH) model. Engle (1982) demonstrated that conditional heteroskedasticity can be modelled using an auto-regressive conditional variance of the disturbance term with linear combination of the square disturbance in the recent past second. Meanwhile Engle (1982) modelled the heteroskedasticity by relating the conditional variance of the errors term to the linear combination of the squared errors in the recent past.

However, when using the ARCH model in determining the optimal lag length of variables are very cumbersome (Deebom & Essi, 2017), therefore, often time users encounter problems of over parameterization. Sequel to this and many other lapses and challenges encountered in the ARCH model, Bollerslev (1986) proposed extension to ARCH model which was referred to as Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. Bollerslev (1986) generalized the ARCH model by modelling the conditional variance to depend on its lagged values as well as squared lagged values of disturbance. This was done with view to achieving parsimony just like the idea behind Autoregressive Moving Average (ARMA) model.

 Ever since the work of Engle (1982) and Bollerslev (1986), various variants of GARCH model have been developed to model volatility. Some of the models include IGARCH model originally proposed by Engle and Bollerslev (1986); GARCH in- Mean (GARCH-M) model introduced by Engle et al (1987); the standard deviation GARCH model introduced by Taylor (1986) and Schwert (1989); the EGARCH or Exponential GARCH model proposed by Nelson (1991); Threshold ARCH or TARCH and Threshold GARCH model introduced independently by Zakoïan (1994) and Glostenet al (1993), the Power ARCH model generalised by Ding et al (1993) among others (Tasi’u, et al., 2014).

Most of the studies which applied ARCH/GARCH models on Financial Time Series to measure or model volatility found the existence of non-normality, volatility clusters, negative skewness, leptokurtosis in their respective Time Series from different countries such as the work of Floros (2008) and Emenike (2010) in Nigeria, Su (2010) in China, Angabini and Wasiuzzaman (2011) in Malaysia, Abd el Aal (2011) and Ezzat (2012) in Egypt and Freedi et al.. (2012) in Saudi Arabia which all applied TGARCH, EGARCH, and GJR GARCH models in their studies and their findings revealed that EGARCH and GJR-GARCH are the best models for measuring volatility, detecting clustering effect, leptokurtosis and the leverage effect.

Similarly, Miron and Tudor (2010) compared several statistical models for daily stock return volatility in terms of sample fit and out-of-sample forecast ability. The focus is on U.S. and Romanian daily stock return data corresponding to the 2002-2010 time intervals. The work investigates the presence of leverage effects and estimate different asymmetric GARCH-family models (which are EGARCH, PGARCH and TGARCH) specifying successively a Normal, Student’s t and GED error distributions. They find that GARCH family models with normal errors are not capable to capture fully the leptokurtic in empirical Time Series, while GED and Student’s t error provide a better description for the conditional volatility. Additionally, they outline some stylized fact about volatility that are not captured by conventional ARCH or GARCH models but are consider by the asymmetric models and document their presence in empirical Time Series. Finally they report that volatility estimates given by the EGARCH model exhibit generally lower forecast errors and are therefore more accurate than the estimate given by the other asymmetric GARCH models.

In order to determine volatility modelling of exchange rate between US dollar (USD) and Nigerian Naira (NGN), Tasi’u et al. (2014) investigated the volatility of daily exchange rate of Naira vis-à-vis United State Dollar using GARCH, GJR-GARCH, TGARCH and TS-GARCH models by using data over the period June 2000 to July 2011. The result shows that the GJR-GARCH and TGARCH models show the existence of statistically significant asymmetric effect. The forecasting ability is subsequently assessed using the symmetric loss functions which are the Mean Absolute Error (MAE), Root Mean Absolute Error (RMAE), Mean Absolute Percentage Error (MAPE) and Theil-inequality coefficient. The results show that TGARCH model provide the most accurate forecasts. They conclude that this model (TGARCH model) captured all the necessary stylized facts (common features) of financial data, such as persistent, volatility clustering and asymmetric effects.

Hussaini et al. (2015) evaluated volatility forecasts of the Nigerian Stock Exchange rate obtained through asymmetric models. They make use of monthly data from January, 2000 to January, 2012 to evaluate the parameter of each model and produce volatility estimates. Their result shows that the coefficient of determinant of the presence of volatility clustering is statistically significant in the EGARCH model; this appears to show the presence of volatility clustering. The forecasting ability was subsequently assessed using the symmetric lost functions which are the Mean Absolute Error (MAE), Root Mean Absolute Error (RMAE), Mean Absolute Percentage Error (MAPE) and Theil-inequality coefficient. The results show that GJR-GARCH model provides best estimates for persistence, volatility clustering and leverage effects are absent or minimal, GARCH model provides the most accurate forecast of future volatility.

In the same vein, the volatility of crude oil prices has been of growing area of research. Accordingly, the studies of oil price volatility are covering a number of different areas and issues and examine the characteristics of these prices in various respects. Many empirical studies showed evidence that Time Series of crude oil prices, likewise other financial Time Series, are characterized by the common features of financial data called stylized facts such as heavy tail distribution, volatility clustering, asymmetry and leverage effects.

Narayem and Narayan (2007) use the Exponential Generalized Conditional Heteroscedasticity (EGARCH) model with a daily data for the period 1991-2006 with the intention of checking for evidence of asymmetry and persistence of shocks. In their work, volatility is characterized in various sub-samples to judge the robustness of their results. Across the various subsamples they show an inconsistent evidence of asymmetry and persistence of shocks and also across full sample period, evidence suggests that shocks have permanent effects and asymmetric effects on volatility. Thus Narayan and Narayan (2007) findings imply that behaviour of oil prices tends to change over short periods of time.

Olowe (2009) investigated weekly oil price volatility in Nigeria using EGARCH (1, 1) within January 3, 1997 – March 6, 2009. The result shows that the oil price return series has high persistence of volatility, volatility clustering and asymmetric characteristics.

 In like manner, Tatyana (2010) studied the dynamics of oil prices (Brent and WTI crude oil markets) and their volatilities by Linking four GARCH related models namely; GARCH (1,1), GJR – GARCH (1,1), EGARCH (1,1) and APARACH (1,1). The findings of this study showed that oil shocks have permanent impact and there exist asymmetric consequence on the volatility of the markets under consideration.

Alhassan and Kilishi (2016) provide analytical insight on modelling macroeconomic and oil price volatility in Nigeria. Mainly, they employed GARCH model and its variants (GARCH-M, EGARCH and TGARCH) with daily, monthly and quarterly data. The findings reveal that: all the macroeconomic variables considered (real gross domestic product, interest rate, exchange rate and oil price) are highly volatile; the asymmetric models (TGARCH and EGARCH) outperform the symmetric models (GARCH (1 1) and GARCH – M), and that, the asymmetric effects are important in modelling oil market in Nigeria. They also conclude that crude oil price is a major source of macroeconomic volatility in Nigeria.

The study of Deebom and Essi (2017) was targeted at modelling price volatility and the risk-return related to crude oil export in Nigerian crude oil market using the first order asymmetric and symmetric univariate Generalized Autoregressive Conditional heteroscedasticity (GARCH) family models. Three objectives with three research questions and two hypotheses were raised for the study. The results from the statistical analysis reveal that the markets were optimistic of their investment and other trade related activities. Sequel to that, there were high probabilities of gains than losses. Although, the variables use in the markets were extremely volatiles and shows evidence there exists positive risk first-rated meaning that investments or investors deserved rewards for holding risky assets. However, the selected models were subjected to several diagnostic test such as ARCH effect test, test for serial correlation and QQ-plot in order to validate their fitness which was confirmed to be appropriate. For investors or marketers in these markets, they were advice to be mindful in trading in a highly volatile period especially when there is evidence of high standard deviation in the descriptive statistic of the return series and in modelling volatility of price return of certain micro/ macro-economic variable the leverage effect of such variable should be properly estimated using asymmetric GARCH model.

Aigheyisi (2018) investigated the effect of oil price volatility on the business cycle (measured as fluctuations in real GDP) in Nigeria, while controlling for effects of other variables such as inflation, exchange rate, money supply, trade openness and foreign direct investment. Volatility in real GDP and oil price is generated through the EGARCH process. The paper recommends channelling of efforts by the government towards diversifying the productive base and exports of the country as measure to reduce volatility in the real GDP.

However, the review of relevant literatures showed that Autoregressive Conditional Heteroscedasticity (ARCH) model and its various extensions are being applied for modelling volatility of financial Time Series data in order to captures the stylized facts incorporated in these series such as volatility clustering, persistence, heavy tail, and leverage effects. But less attention has been given to the contribution of the error distribution assumptions while modelling volatility. According to Deebom and Essi (2017), the wrong use of an appropriate error distribution in volatility modelling for financial Time Series may cause misspecification in volatility model, leptokurtic and autocorrelation behaviour of such series. Sequel to this, this study tends to examine the volatility of Nigerian monthly crude oil Price series using two symmetric GARCH (GARCH and GARCH-M) models and four different asymmetric GARCH (EGARCH; TGARCH; GJR-GARCH and APARCH) models by considering or assuming three different error distributions which are the Normal Error Distribution, Student’s t Error Distribution and the Generalized Error Distribution. Thus, the study investigates best model for the Nigerian crude oil price with best error distribution.

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