Oil price volatility in the Philippines using generalized autoregressive conditional heteroscedasticity

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1 Oil price volatility in the Philippines using generalized autoregressive conditional heteroscedasticity Carl Ceasar F. Talungon University of Southern Mindanao, Cotabato Province, Philippines Date received: 26 April 2016 Date accepted: 3 September 2016 Date published: 16 December 2016 ABSTRACT This study was conducted to estimate the diesel oil price volatility of the three big oil companies (Caltex, Petron and Shell) in the Philippines using Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. The diesel price of the three big oil companies in the Philippines had exhibited wild fluctuations at various times which confirmed that the diesel prices are wildly volatile. The ARMA model was used to model the data series of the big three oil companies diesel prices based on the modified Box-Jenkins approach. The big three oil companies diesel prices fluctuated wildly indicating that the series are volatile. ADF test revealed that all series are stationary at level except Shell. Among the three diesel prices, only the ARMA model for Petron diesel price possessed residuals with white noise process and significant in ARCH LM test. The residuals of ARMA model for Caltex and Shell diesel prices failed to satisfy white noise process and ARCH LM test. Thus, only Petron diesel price proceeded to GARCH model identification for price volatility estimation. ARMA (2,2) model was chosen for Petron diesel price based on selected criteria while GARCH (1,1) model, came out to be the best model to estimate Petron diesel price volatility. Keywords: ARMA, Diesel, Oil Price Volatility, Univariate. INTRODUCTION Oil plays an important role in an economy because it is most influential to physical commodity in the world (Kuncoro, 2011). In modern industry, oil is the lifeblood in industrialization and urbanization process and oil s consumption dramatically increased in most of the emerging countries (Lu, 2011). Specifications of Diesel fuel differ for various fuel grades and in different countries ( Diesel fuel is an important input for many industries (transportation, construction, fisheries, agriculture). It is usually used because it is more efficient and the diesel engines are cheaper and easier to maintain (Bajjalieh, 2010). Oil price is one of the key factors in financial markets because it affects significantly the option pricing, portfolio management and risk measurement processes (Wei et al., 2010). Volatility is a measure of the degree of prices of commodity that fluctuates (Al-Fattah, 2013). Oil prices fluctuations have greater impacts on different economies. High volatility of oil prices will lead to economic instability both countries that export and import oil. In particular, economies that depend on oil are highly affected by uncertainty, because high volatility of exports increase government revenues (Gileva, 2010). In 1982, Robert Engle introduced the Autoregressive Conditional Heteroscedasticity (ARCH) model as a way to not just correct, but directly model heteroscedasticity in one time series data by augmenting a time series regression with a separate conditional variance equation. Bollerslev (1986) developed this model into GARCH (Generalized Autoregressive Conditional Heteroscedasticity), 10

2 which has become an invaluable tool for researchers using heteroscedasticity time series data in finance, macroeconomics and energy economics. The 1990 s and early 2000 s saw several empirical studies using univariate GARCH models with energy data. This study was conducted to provide empirical evidence in estimating oil price volatility in the Philippines showing patterns of trends of diesel oil prices of the three big oil companies (Caltex, Petron and Shell) in the Philippines from March 2005 to December Likewise, the generalized autoregressive conditional heteroscedasticity model was employed to determine volatility of oil. METHOD Time series analysis is concerned with the past behavior of a variance in order to predict future behavior, time series analysis is a method to understand the underlying theory on the sequence of observations ordered in time, or to forecast the identified pattern based on past events. This study used univariate time series analysis using the Bollerslev (1986) Generalized Autoregressive Conditional Heteroscedasticity (GARCH). This is the process of checking the residuals and adjusting the values of p and q continues until the resulting residuals contain no additional structure. Once a suitable model is selected, the model can proceed to ARCH/GARCH LM test, when the result of test is significant, the model can proceed to GARCH Modelling. The time series data of diesel oil price of the three big oil companies in the Philippines from March 2005 to December 2013 were obtained from were used. Testing for Stationarity Stationarity is a critical assumption of time series analysis. It is import to initially test the time series variables for stationarity before employed in any regression analysis, it is important condition because violations of some assumptions will cause to ruin (Gujarati, 1995). A stochastic time series ( ) is said to be stationary if it satisfies the following requirements: (a) E ( ) = ( has constant mean); (b) Var ( ) = = ( has constant variance); (c) Cov (, ) = for all (the covariance between any two of the terms of the series is a function only of the distance between them). The first and second requirement implies that the means and variances are the same over time. The third requirement implies that the covariance in observations of the series is a function of how far apart is the time and not the time at which they occur (Greene, 2001). In other words, stationarity appear in a time series when mean, variance and autocorrelation structure do not change over time ( If the series does not qualify one or more of the conditions, it would mean that the series is not stationary, and to proceed with regression analysis would lead to false results, where it is possible to obtain very high value for but insignificant estimates. Testing for unit Roots A unit root was conducted to determine stationarity. Stationarity refers to a condition wherein the series have constant variance. Augmented Dickey-Fuller (ADF) test is used in testing for the presence of unit root and is applied to the data series. The specification is: = where is a white noise error term. The error term is said to be independent and identically distributed. Failure of ADF test to reject the null hypothesis indicates there is a presence of a unit root. In case of non-stationarity, to have a stationary condition can be attained through differencing process. Differencing process is frequently employed to detrend the data and control autocorrelation by subtracting each datum in a series from its predecessor ( If ADF test results will reveal that the series are stationary in level, then the time series analysis can now proceed to 11

3 autoregressive moving average (ARMA) and generalized autoregressive conditional heteroscedasticity (GARCH) modelling. ARIMA Model Autoregressive moving average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the auto-regression and the second for the moving average. The general ARMA model was described by Whittle (1951) on his thesis, and it was popularized by the book of Box and Jenkins (1975). Given a time series of data X t, the ARMA model is a tool in understanding and predicting future values in this series. The model consists of two parts, the autoregressive (AR) part and a moving average (MA) part. The model is referred to as the ARMA (p,q) model where p is the order of the autoregressive part and q is the order of the moving average part AR or MA models will become complicated if one may need a high-order model with many parameters to adequately describe the dynamic structure of the data. To overcome this difficulty, the ARMA models are introduced by Whittle (1951). An extremely general model that encompasses equations AR and MA modelsor the Autoregressive moving average model is ARMA (p, q) model: = + + where: t = time subscript. = is the time series of diesel oil prices (Big 3 oil companies) in the Philippines. = constant. and = are the coefficients. p = the order of autoregressive part of the model. q= the order of the moving average portion of the model. = is a white noise error term of time t. In ARMA modeling it conclude the suitable model order (p, q). Box and Jenkins (1976) developed steps in building ARMA models, which has an impact on the time series analysis and forecasting applications. Box and Jenkins(1976) recommended to use the Partial Autocorrelation Function (PACF) and the Autocorrelation Function (ACF) of the sample data as the important tools to determine the order of the ARMA model. In the classification step, data transformation is necessary to create the time series stationary. Stationarity is an essential stage in creating an ARMA model applied for predicting. A stationary time series is described by statistical characteristics for instance the mean and the autocorrelation structure being stable ultimately. While the experimental time series shows heteroscedasticity and trend, power transformation and differencing are used to the data to eliminate the trend and to make the variance constant before can be fitted an ARMA model (Ahmed and Shabri, 2013) If white noise error term in model above has a constant variance and once the order of the ARMA (p, q) model for each series will be determined, this constant variance assumption will be dropped and the models allow for the autoregressive conditional heteroscedasticity (ARCH) effect will be estimated. Testing of ARCH/GARCH 1. Simple Test The simplest approach to test for GARCH effect is to test the squares of residuals. The processes are the following steps: Step 1. Fit a mean equation and obtain the residuals { }. Step 2. Compute the OLS regression of =ω Step 3. Test the joint significance of,,, 12

4 If these coefficients are significantly different from zero, the null hypothesis of conditionally homoscedastic disturbances is rejected in favor of the alternative hypothesis ARCH/GARCH disturbances. 2. Lagrange multiplier (LM) Test A Lagrange multiplier (LM) test is used to test for autoregressive conditional heteroscedasticity (ARCH) in the residuals. To examiine the null hypothesis that there will be no ARCH up to order in the residuals Ho: = = = 0 = ω where is a residual and is error term. This is a regression of the squared residuals on constant and lagged squared residuals up to order q. Engle s (1982) LM test statistic, computed the number of observations (n) times the R 2 from the test regression. The LM test statistic is asymptotically distributed x 2 (q) under quite general conditions, that is, n times R 2 ~ x 2 (q). ARCH will be accepted if the p-value for LM test is found to be smaller than the significance level. GARCH Model Generalized ARCH was developed by Bollerslev (1986). This model is commonly used in many branches of econometrics, specifically on energy and finaciall time series analysis. The underlying regression is the usual one as in the case of equation (4) above. Conditioned on information set at time 1, GARCH (p, q) process is defined as follow, = + + (4) With the constraints > 0, > 0 for all =1,2. and > 0 for all =1,2. With the following assumptions: 1. [ ] = 0, the mean is zero. where: 2. [ ] = 0, conditional heteroscedasticity is inevitable. Hence, the autocovariance of any pair of elements from the series is expected to be zero resulting in lack of serial correlation. For > 0, is not correlated with. 3. The unconditional variance = = a series with mean zero and has no serial correlation. It s treated as a white noise sequence for further manipulation. From = it is possible to write: = + = where: = [ + ] =max (, ), = 0 for > and =0 for > + In other words, is an ARMA (r,p) process with a white noise innovation. Using stationarity which implies [ ]=, the unconditional variance will be, [ ]= + [ ] [ + ] 13

5 Univ. of Min. Intl. Mult. Res. Jour. 2016, 1(2), Or [ ]= 1 RESULTS This study was conducted to estimate the diesel oil price volatility of the three big oil companies in the Philippines using Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. The diesel price of the oil companies exhibited wild fluctuations uations at various times and shows that the diesel prices are wildly volatile. The secondary data on Philippine Diesel prices of the three big oil companies were obtained from the website and its source is the Philippine Department of Energy. The ARMA model was used to model the oil companies diesel prices of the series. The modified Box-Jenkins approach was followed to identify the best fitting ARMA model. The trends of the diesel oil prices of the oil companies in the Philippines from the period of March 2005 to December 2013 which are plotted in Figures 1, 2 and 3.There are erratic behavior of the diesel oil prices of the three big oil companies. The three data series depict the same trend as peaks and troughs behave similarly but in different level among the three companies. As seen in the figures, prices of diesel oil reached its highest peak in the mid s of 2008, the time when there is a price increase in world oil market due to reduction on non-opec (organization of oil exporting countries) supply of oil. The world, not only the Philippines experienced this price shock. Figure 1, 2 and 3 show the diesel prices of the big three oil companies have been fluctuating wildly and significant at various times, indicating that each series is volatile. Figure 1. Diesel Prices of Caltex in the period of March 2005 to December [Source: 14

6 Figure 2. Diesel Prices of Petron in the period of March 2005 to December 2013 [Source: Figure 3. Diesel Prices of Shell in the period of March 2005 to December [Source: Testing for Stationarity The diesel oil prices of the oil companies are plotted using the autocorrelation function (ACF) and sample partial function (SPACF). The shape of correlograms of the autocorrelation function (ACF) and sample partial autocorrelation function (PACF) helps to distinguish the property of time series. The visual inspection on the ACF and PACF of diesel prices of the big three oil companies are shown that the series is stationary in process as characterized by a correlogram that decays rapidly. However, by merely looking at the correlograms alone, it is difficult to detect stationarity, in such case, the Augmented Dickey Fuller (ADF) test was employed to check the series property. Variable Caltex diesel oil Petron diesel oil Shell diesel oil Table 1.Augmented Dickey Fuller (ADF) test results Random Walk Random with Drift Walk(None) (Intercept) Mixed Process (Trend and intercept) ns ns * ns * ns ns ns ns ns *significant at 10% not significant Results of the unit root test are summarized in Table 1. Results reveal that all series are stationary at 10% significant level except shell diesel prices. The same table shows that Caltex yielded significant 15

7 result in mixed process indicating a stationary process. Petron yielded significant result in random walk with drift indicating a stationary process. While Shell yielded insignificant statistics in all process, thus the series is non-stationary and it must go through differencing process. A. Differencing Shell Diesel Prices The result on the ADF test reveal that the series in Shell exhibits a unit root. This would mean that Shell is non-stationary. The main premise for differencing shell is to smoothen the data series. Using first difference value, the shell was found to be stationary at first differencing. This implies that the variable attained stationarity after first differencing and the series is said to be integrated of order 1. The result of first differencing on EVIEWS output shows that the series is significant at all process. ARIMA Model Identification Following the Box-Jenkins framework, a number of ARIMA models were selected for the diesel price of the oil companies. The orders of ARIMA models were summarized and the statistics of the model are presented in Table 2, 3 and 4 for Caltex, Petron and Shell respectively. In the Tables on ARMA model identification (Table 2, 3 and 4), Akaike Information Criterion (AIC) and Schwarz Information Criterion (SIC) of each model is shown at the second and third columns of the table. The fourth column of the table shows the value of the Q-statistic at lag 12 (Q 12 ).This statistic was computed from the squares of the first twelve residual autocorrelations (with p-value enclosed) and the fifth column of the table shows the number of non-significant parameter estimates based on the p-values of ARMA models (Shown in Appendix 4). The model that minimizes both AIC and SIC, has the highest coefficient of determination or the R 2, has the least non-significant parameter estimates and non-significant Q 12 statistic and is the preferred model. If the p-value is associated in Q 12 is larger than 10% significant level, the hypothesis of the first twelve (12) autocorrelations are not significantly different from zero is accepted and therefore, indirectly accepting that the model is correct. Table 2. ARMA models used to represent the Caltex diesel prices Model SIC Nonsignificant R 2 AIC Q 12 Parameters AR (0.000) None AR (0.001) None AR (0.003) AR (0.001),, ARMA (1,1) ARMA (2,1) ARMA (1,2) ARMA (2,2) (0.000) None (0.011) None (0.001) None (0.004) p-value in parenthesis 16

8 Based on the result presented in Table 2, Caltex has five (5) candidate models for ARMA, these include AR (1), AR (2), ARMA (1,1), ARMA (2,1) and ARMA (1,2) based on the selection criteria. These models have all yielded high coefficient of determination, smaller values on both AIC and SIC and all the parameter estimates are significant at 10% significant level. However Q 12 is also significant at 10% alpha level. Since all the Ljung Box Q 12 is statistically significant, it shows that neither of the estimated ARMA model fits the Caltex diesel price reasonably well. That is because it violates the criteria on ARMA model identification. The non-significant Ljung-Box Q 12 statistic implies that all the estimated ARMA models possess residuals which do not follow the white noise process. Hence, none of them best fits the data series. Model AIC SIC Q Table 3. ARMA models used to represent the Petron diesel prices. significant R 2 Non- 12 Parameters AR (0.000) None AR (0.544) None AR (0.710) None AR (0.643) ARMA (1,1) (0.459) None ARMA (2,1) (0.651), ARMA (1,2) (0.716) None ARMA (2,2) (0.989) None p-value in parenthesis On the other hand, Petron (Table 3) has six (6) candidate models, these include AR (1), AR (2), AR (3), ARMA (1,1), ARMA (1,2) and ARMA (2,2). The candidate models of ARMA have all yielded high coefficient of determination and all the parameter estimates are significant and only AR (1) has significant Q 12 at 10% level of significance. Second-order autoregressive and second-order moving average or ARMA (2,2) model was chosen as the appropriate model for the Petron diesel prices because it satisfy all the criteria. ARMA (2,2) model registered an R 2 of which indicates that 85.53% of the variation in Petron diesel price is explained by the model. ARMA (2,2) model equation form: =

9 Table 4. ARIMA models used to represent the Shell diesel prices. Q 12 Model AIC SIC significant Parameters Non- R 2 AR (0.000) AR (0.034) AR (0.020),, AR (0.013),,, ARIMA (1,1, 1) (0.026) ARIMA (2,1,1) (0.036), ARIMA (1,1,2) (0.036), ARIMA (2,1,2) (0.021),, p-value in parenthesis Table 4 shows the first difference of Shell diesel price on ARIMA modeling. All models yielded very low coefficient of determination and all the models are significant on the Q 12 at 10% level of significance, it means that all the models cannot advance to ARMA modeling because neither of the estimated ARMA models fit the data series. That is because all the estimated ARMA models failed to satisfy the selection criteria. The plot of the sample autocorrelation function (SACF) and sample partial autocorrelation function (SPACF) of the squares of the residuals of the model of Petron diesel prices. The ACF and PACF of the squared of the residuals of Petron from the estimated model is a white noise process. It is crucial to obtain white noise residuals, since failure to capture all of the dependence in the conditional values for Akaike Information Criterion (AIC) and Schwarz Information Criterion (SIC) were used to compare the fitted models to those which allow for autoregressive conditional heteroscedasticity (ARCH) type dynamics in the conditional mean. Test for ARCH/GARCH Effects Based on the residuals from the mean equation it is possible to test for the existence of ARCH or GARCH effect. Table 5 shows the results of ARCH LM test of model of Petron diesel prices. The last column of Table 5 includes the p-values that indicate rejection of the null hypothesis that there is no ARCH effect at the first lag at 10% level of significance. The results indicate that the Petron series are volatile as indicated by the significant ARCH LM test. Hence, the residuals qualified to be modeled using ARCH or GARCH models. Table 5.ARCH LM Test Summary Statistics. Company Obs*R 2 t-stat Lag(p) P-value Petron % significant GARCH Model Identification Petron s GARCH model for the residuals is found to be the only significant in LM test among the three series. Likewise, only the residuals of Petron diesel prices proved non-significant Ljung-Box Q12 statistic. Hence only the Petron oil price was subjected to the analysis for GARCH model 18

10 identification. The Caltex and Shell diesel price volatility came out to be non-significant based on the result of ARCH LM test, thus, there is no evidence that supports to model its volatility using GARCH process. In practice, for ARCH modeling, a large lag q is often needed and this requires that estimating a large number of parameters. To reduce the computational burden, a GARCH model with low lags can help. This result in a more parsimonious representation of the conditional variance process.the modeling procedure of ARCH models can also be used to build a GARCH model. However, specifying the order of a GARCH model is not easy. Only lower order GARCH models are used in most applications, say, GARCH (1,1), GARCH (1,2), GARCH (2,1), and GARCH (2,2) models. Computing different statistics such Akaike Information Criterion (AIC) and Bayesian information or Schwarz Criterion (SIC) in order to choose the best model. Table 6.GARCH Models for Petron Diesel Prices. Model AIC SIC Nonsignificant Parameters GARCH (1,1) None GARCH (1,2) , GARCH (2,1) GARCH (2,2) None Table 6 presents the results of the models used in specifying the GARCH model for the Petron oil price series. The second and third columns are the AIC and SIC from each of the GARCH model. The fourth column of the table shows the parameters that have a non-significant parameter estimates at 10% level in the corresponding model. In GARCH identification, the model that has significant parameters and both parsimony in AIC and SIC is the preferred model for GARCH. GARCH (1,1) is the chosen model for diesel oil price because the model has the smallest value in both AIC and SIC and all parameter estimates are significant. Thus, the volatility of the Petron diesel price can be explained by the presence of first-order autoregressive GARCH term and a first-order moving average ARCH term. This means that the oil price volatility is influenced by the volatility from the previous period even if prices of diesel oil rises or fall. GARCH (1,1) model equation: σ^2 _t = _(t-1) _(t-1) The ACF and PACF in Appendix 8 show the correlograms and indicate that the residuals from the estimated model are white noise processes. This means that GARCH (1,1) is appropriate model for the Petron diesel oil price series. CONCLUSION The diesel prices series is volatile and appeared that prices fluctuated at different periods. Only the Petron diesel oil price possesses the white noise process and qualified for GARCH model identification. 19

11 REFERENCES Ahmed, R. A., & Shabri, A. B. (2013). Fitting GARCH models to crude oil spot price data. Life Science Journal, 10(4), Al-Fattah, S. M. (2013). Artificial Neural Network Models for Forecasting Global Oil Market Volatility. USAEE Working Paper No Bajjalieh, J. W. (2010). Forecasting diesel fuel prices (Master Thesis, University of Illinois at Urbana- Champaign). Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), Box, G. E., & Jenkins, G. M. (1976). Time series analysis: forecasting and control, revised ed. Holden-Day. 575 Danao, R., Introduction to Statistics and Econometrics. Quezon City: University of the Philippines Press. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 50(4), Gileva, T. (2010). Econometrics of crude oil markets (Master Thesis) University Paris I Sorbone. Greene, W. (2001). Estimating econometric models with fixed effects. NYU Leonard N. Stern School of Business Department of Economics Working Paper, (2001m10). Kuncoro, H. (2011). The volatility of world crude oil prices. Economic Journal of Emerging Markets, 3(1), Wei, Y., Wang, Y., & Huang, D. (2010). Forecasting crude oil market volatility: further using GARCH-class models. Energy Economics, 32(6), evidence Whittle, P. (1951). Hypothesis Testing in Time Series Analysis-Hafner. New York, p