Modeling and Volatility Analysis of Oil Companies in Pakistan

Modeling and Volatility Analysis of Oil Companies in Pakistan Firdos Khan Scientific officer, Climate Section, Global Change Impact Studies Centre (GCISC), National Centre for Physics (NCP) Complex, Quaid-i-Azam University Campus Islamabad, Pakistan. fkyousafzai@gmail.com Zahid Asghar Assistant Professor at Quaid-i-Azam University, Islamabad. g.zahid@gmail.com ABSTRACT We focus on modeling of conditional mean and conditional variance of the share prices of two stocks, i.e. Shell Gas (LPG) Pakistan and Attock Petroleum (AP) from Karachi Stock Exchange by using ARIMA and ARCH models. We have daily data for shell gas (LPG) Pakistan and Attock Petroleum (AP) from 3rd of August 1998 to 31st of January 2007. Firstly we have constructed ARIMA modeling. Unit root testing with different lag length criteria has been performed. We find that there are more than one potential models which fit to this data. The performances of these models are not different from each other. The best model for AP series is ARIMA(1,1,0) and for LPG ARIMA((1,2),1,0) is the best model among the competing models. Secondly we have measured the volatility of the stocks by ARCH family of models. Maximum likelihood method of estimation is used to estimate the unknown parameters of the models. Different lag length criteria like AIC, BIC etc are used to find the best possible model. The potential models are then analyzed on the basis of their forecast performance. GARCH model is the best model for AP series and EGARCH model is the best model for LPG series among the competing models. This study is an important application of all the modern time series econometric tools. Moreover effort is made to remove a misnomer (which our old econometricians and books show that true model is known) that there is only one model which fits the data. Keywords: ARIMA Models, ARCH Models, Unit Root test, AIC, BIC, ARCH-LM test, Volatility, MSE, RMSE, MAE, TIC, TGARCH Model, LM-Test, GARCH model, EGARCH model, JGR-GARCH model, P-Value, White noise, Stationarity. 1 Firdos Khan and Zahid Asghar 1. Introduction. Recent development in financial time series requires the use of models and techniques that are able to model the attitude of the investors not only toward expected returns, but also towards risk or uncertainty. Therefore the modeling of uncertainty is more important. To model the uncertainty of a series it requires the models to be capable of dealing with the uncertainty or variance of the series. The most suitable models that deal with such phenomena are the ARCH class of models. An important assumption of the conventional econometrics is that the disturbance terms appearing in a regression equation are homoskedastic i.e. the variance of the disturbance terms remains constant over time. But in practical analysis this assumption dose not holds. In case of heteroskedasticity the OLS estimators are unbiased, consistence but no longer efficient and therefore they are not BLUE. Before the introduction of the ARCH models the researchers thought that heteroskedasticity is the problem of cross-sectional data. However, some researchers tried to forecast financial time series like stocks prices, inflation rates, foreign exchange rates etc, they observed that their forecast variable varies from one time period to another time period. The forecast errors are small for some period and large for other period. This variability could be due to volatility in financial market. This would suggest that the variance of the forecast error is not constant, but varies from period to period. It means that there is some kind of autocorrelation in the variance of forecast error. Since the behavior of the forecast error can be assumed to depend on the behavior of the (regression) disturbance. It can be make a case of autocorrelation in the variance of (regression) disturbance. To capture this autocorrelation Engle (1982) introduced a model called ARCH model. Prior to the introduction of the ARCH model the researchers used different types of model, which are based on the sample standard deviation to model the current uncertainty and forecast the future uncertainty of the series. For example Fama (1965) used RW model for this purpose. According to random walk model the standard deviation of a series for today is the estimated forecast for tomorrow and so on. The other models, which were used by researchers, include the models like HISVOL models, moving average models, EWMA models etc. 2 Modeling and Volatility Analysis of Oil Companies in Pakistan There are several reasons that we may wish to model and forecast volatility of the data. First we may need to analyze the risk of holding an asset. Second forecast confidence interval may be time varying, so that more accurate interval can be obtained by modeling the variance of the error terms. The third reason is that more efficient estimates can be obtained if heteroskedasticity in the error terms is handle properly. ARCH models are especially designed to model and forecast conditional variance. The variance of the dependent variable is modeled as a function of the past value of the dependent variable and independent or exogenous variables. ARCH model introduced by Engle’s (1982) and generalized as GARCH by Bollerslev (1986) and Taylor (1986) independently, are popular in financial time series and are widely used in econometrics analysis. The ARCH and GARCH model perform very well but these models only consider the magnitude of the random shocks and do not utilize the direction of the shocks. The non-negativity constraints on the parameters create difficulties in estimation, and in some studies they are found negative. Another problem with the ARCH and GARCH models is, whether a shock persists to the conditional variance, if yes then how long it well last? Keeping in view the drawbacks of the ARCH and GARCH models, Nelson (1990) proposed a new model called EGARCH model. EGARCH model overcomes the drawbacks of ARCH and GARCH models. EGARCH model captures the asymmetry of the data. Asymmetry means positive and negative shocks of the same magnitude have not the same effect on the conditional variance. Glosten et al (1993) proposed a model for modeling the second moment of the financial time series, called GJR-GARCH model due to Glosten, Jagannathan, and Runkle. GJR-GARCH model is also called TARCH model. Engle and Ng (1993) introduced PNP model and made comparison of PNP and other ARCH class models like GARCH, EGARCH, and GJR-GARCH. Except the GARCH model the remaining models gives similar results. The Violation of constrain of nonnegativity of the parameters in ARCH and GARCH models were frequently found. To overcome this problem Geweke (1986) and Milhoj (1987 a) suggested log ARCH model. The log-ARCH models have no need of non-negativity constraint. Bera and 3 Firdos Khan and Zahid Asghar Higgins (1992) proposed non-linear ARCH model. This model includes linear ARCH model as a special case and log ARCH as a limiting case. A number of studies in the literature can be found in which the researchers made comparison among different volatility models. Akgiray (1989) have found evidence in favour of GARCH (1,1) model over other model like ARCH (2), EWAM and HISVOL model. He found that GJR-GARCH is the best model in all the employed models. Figlewski (1997) compared HISVOL model and GARCH (1,1) model using the data on three stocks and one exchange rate. For some past data set GARCH model gives best result and for some data set other models were superior. Lee (1991) compared different types of ARCH models by using weekly data on different five exchange rates. He found evidence in favour of non-linear ARCH models. McMillan et al (2000) used daily, weekly and monthly data for two stocks of UK and compared different Volatility models. They concluded that RW model is the best model if the crash of 1987 is included and all models gives similar results if 1987 crash is excluded. Brooks (1998) used daily data and made an interesting comparison among different volatility models. Brooks also concluded that the performance of all of the employed models were similar if the 1987 crash is not included in the analysis. The main objective of the study is to provide suitable model to the investors that can be used by investors to predict or forecast the future expected share prices and make investment accordingly. For this purpose we first do modeling of the chosen series by assuming homoscedasticity i.e. ARIMA modeling. We also relaxed the assumption of homoscedasticity and do ARCH modeling of the selected series. By using ARCH model the investors can forecast expected future return as well as the uncertainty of the series, so it is very helpful for investors to do investment or not. The rest of the paper is distributed as, in section two we present methodology adopted in the study as well as some models, which are employed in the study. In section three data analysis and results are given, and section four is reserved for conclusion. At the end a list of references is given. 4 Modeling and Volatility Analysis of Oil Companies in Pakistan 2. Methodology and models used in the study. This section consists on methodology and models used in the study. In the first step we performed unit root testing, because stationarity is one of the assumption of Box and Jenkins (1976) methodology of ARIMA modeling. After achieving stationarity we build ARIMA models to each series. ARIMA modeling consists of three steps. The first step is to identify the order of the model, ACF’s and PACF’s. The next step is the estimation of the unknown parameters of the identified models using OLS method if the model is linear, otherwise non-linear methods can be used to estimate the unknown parameters. Diagnostic checking of the modeling completes the ARIMA modeling. Various diagnostic checks are available in the literature to check the adequacy of the model. In current study we have applied two diagnostic checks, one is the serial correlation LM test, and the other is Ljung and Box Q-Statistics. For both series we have more than one models that perform similarly, therefore they are compared on the basis of statistics like AIC, BIC, TIC, ME, MAE, RMSE etc. In the second phase of the study we model the conditional mean as well as the conditional variance of the given series simultaneously, which is called ARCH modeling. The analysis is completely based on the residuals obtained from the fitted ARIMA model built in the first phase. ARCH modeling consists of four steps. Identification of the order of the model, testing for ARCH error, estimation of the model, and diagnostic checking of the model. For identification purpose we have used ACF’s to identify the order of the model. Tests are also available in the literature to test for ARCH error. We have used ARCH-LM test of Engle (1982) at various lag length to test for ARCH error. ML method is used to estimate the parameters of the model. It is assumed that the residuals from ARIMA models are normally distributed. We can use OLS method to estimate the parameters of the ARCH models, but the ML method is efficient than OLS method in case of ARCH modeling (Engle, 1982). In the last step diagnostic (ARCH-LM) test is used to test the adequacy of the estimated models. We also used Q-statistics to check the adequacy of the conditional mean model, and Q-statistics of square residuals to check the adequacy of the conditional variance model. Next we present the introduction of the models used in the study. There are a number of ARCH models in literature, but we confine ourselves to four main ARCH models, which are widely used in ARCH modeling 5 Firdos Khan and Zahid Asghar literature. In this Study the parameters of ARIMA model is represented by English letters and parameters of ARCH model is represented by Greek letters. 2.1 ARIMA Model ARIMA model is the combination of AR process and MA process when the given series is non stationary. Stationary is the main difference between ARMA model and ARIMA model. In the former the order of integration is equal to zero, while in the later it is greater than zero. Mathematically the general form of an ARIMA process can be represented by the following equation p q i =1 j =1 Δ d Yt = a0 + ∑ ai Δ d Yt −i + ∑ b j ε t − j + ε t (2.1.1) Where Δ d Yt show that original series is integrated of order d time. The Δ d Yt −i are the previous values of the series of order p, and ε t − j are the previous random shocks of order q. Where p is the order of AR terms and q is the order of MA process, and ai , for i = 1, 2,3,..., p are the AR parameters and b j for j = 1, 2,3,..., q are the MA parameters which are to be estimated. The error terms are assumed to be white noise. The estimation of the identified ARIMA model is not straightforward, because the ordinary least square method cannot be used to estimate an ARIMA model. Non-linear estimation algorithm can be used to estimate an ARIMA model. An ARIMA model of order of integration d, order of AR terms p, and order MA terms q, can be simply represented as ARIMA ( p,d,q ) 2.2 ARCH Models Engle (1982) introduced ARCH model. ARCH model suggests that the variance of residuals at time t depend on the square errors terms from the past. The ARCH model can be defined as follow. Let Yt is the return of the stock, then the following model is called ARCH (p) model. p q i =1 j =1 Y t = a0 + ∑ aiY t −i + ∑ b j ε t − j + ε t (2.2.1) ε t = υt ht (2.2.2) 6 Modeling and Volatility Analysis of Oil Companies in Pakistan s ht = α 0 + ∑ α iε 2t −i (2.2.3) i =1 Where p and q are the order of AR and MA process to yield an ARMA process. It is assumed that the error terms have mean zero and no autocorrelation at any lag. To specify an ARCH process it is assumed that the equation (2.2.2) can be decomposed in to two parts, υt which is homoskedastic, with mean zero and σ υ2 = 1 , and ht is heteroskedastic with variance given by equation (2.2.3) The generalization of ARCH model is called GARCH model. Bollerslov (986) generalized ARCH model as GARCH model by allowing the past conditional variances in the conditional variance equation. The generalization was done in the similar way as AR model was generalized to ARIMA model. The GARCH model can be defined as follow, Let Yt be the return of a particular stock, and suppose Yt follow an ARIMA process p q i =1 j =1 Y t = a0 + ∑ aiY t −i + ∑ b j ε t − j + ε t (2.2.4) ε t = υt ht (2.2.5) s r i =1 j =1 ht = α 0 + ∑ α iε 2t −i + ∑ β j ht − j (2.2.6) Where p and q are the order of AR and MA process to yield an ARIMA model. It is assumed that the error terms have mean zero and no autocorrelation. In equation (2.2.6) s is the order of ARCH terms and r is the order of GARCH process. Also α i of order s are the coefficients of ARCH terms and β j of order r are the coefficients of GARCH terms. The above Model is denoted by GARCH (s, r) model. It is important to remember that the GARCH model consist of two parts. One is an ARIMA equation and other is GARCH equation. If r = 0, the GARCH model reduced to ARCH model. Black (1976) has noted a negative correlation between current returns and future volatility that tends to rise in response to bad NEWS and fall in response to good NEWS. It means that the impact of good NEWS is not the same as that of bad NEWS. GARCH model is symmetric and gives same weight to good and bad NEWS of the same magnitude and ignore the sign of the NEWS. The GARCH model failed to interpret the 7 Firdos Khan and Zahid Asghar persistence of the shock to the conditional variance. In GARCH model some restrictions are imposed on the parameters to insure the positiveness of the conditional variance. Nelson (1991) introduced a model known as EGARCH model that can be defined as follow Let Yt be the return of a particular stock, and suppose Yt follow an ARIMA process p q i =1 j =1 Y t = a0 + ∑ aiY t −i + ∑ b j ε t − j + ε t (2.2.7) ε t = υt ht (2.2.8) s ln ( ht ) = α 0 + ∑ α i i =1 ε t −i ht −i w + ∑ λj j =1 εt− j ht − j r + ∑ β k ln ( ht − k ) (2.2.9) k =1 Where p and q are the order of AR and MA process respectively to yield an ARIMA (p,d,q) model. Where ai are AR parameters of order p and b j are MA parameters of order q which are to be estimated. It is assumed that the error terms have mean zero and no autocorrelation at any lag. To specify an ARCH process it is assumed that equation (2.2.8) can be decomposed in to two parts, υt which is homoskedastic, with mean zero and σ υ2 = 1 , and ht is heteroskedastic with variance given by equation (2.2.9). In equation (2.2.9) s, r, and w are the order of ARCH terms, GARCH terms and asymmetric terms respectively. Where a0 , ai b j , i = 1, 2,3,... p and j = 1, 2,3,...q are the parameters of the mean equation. and α 0 , α i , β j , and λk for i = 1, 2,3,...s , j = 1, 2,3,...w , and k = 1, 2,3,...r are the parameters for the variance equation which are to be estimated. The GJR-GARCH model is a member of ARCH family. This model is called GJR-GARCH due to the Lawrence R. Glosten, Ravi Jagannathan and David E. Rankle proposed it in (1993) is also celled TGARCH model. This model was independently introduced by Zakoin in (1990). The major drawback of ARCH and GARCH models is the fact that they are symmetric. These models only consider the magnitude of the error term and ignore the sign. Therefore in ARCH and GARCH models a big negative shock will have exactly the same effect on volatility of the series as a big positive shock of the same magnitude and its looking plausible due to their symmetry. Therefore, there was 8 Modeling and Volatility Analysis of Oil Companies in Pakistan need of models which take in to account the asymmetry of the time series data especially financial time series data. Because the sign play important role in volatility analysis. Despite the advantages EGARCH model the empirical estimation of the model is technically difficult as it involves highly non-linear algorithms. In contrast the GJRGARCH model is much simpler than EGARCH, though it not as elegant as EGARCH model. The general GJR-GARCH model is specified as follow p q i =1 j =1 Y t = a0 + ∑ aiY t −i + ∑ b j ε t − j + ε t (2.2.10) ε t = υt ht (2.2.11) s r w i =1 j =1 i =1 ht = α 0 + ∑ α iε 2t −i + ∑ β j ht − j + ∑ γ iδ iε 2t −i (2.2.12) ⎧1 if ε t −i < 0 ⎩0 if ε t −i ≥ 0 δt = ⎨ Where Yt the returns, p and q are the order of AR and MA process respectively to yield an ARIMA process. It is assumed that the error terms have average value equal to zero and no autocorrelation at any lag. To specify an ARCH process it is assumed that equation (2.2.11) can be decomposed in to two parts, υt which is homoskedastic, with zero mean and σ υ2 = 1 , and ht is heteroskedastic with variance given by equation (2.2.12). So, γ i catches the asymmetry in response of volatility to the shocks in a way that impose a prior belief that for a positive shock and a negative shock of the same magnitude, future volatility is always higher, or at least the same, when the sign of the shock is negative. 3. Data analysis and results The first step in time series analysis is to check the series for unit root. So in the first step we have used Augmented Dickey – Fuller (ADF) (1979) test to examine the possibility of unit root in the series. We run the ADF regression. Estimation of ADF equation would be quite simple by using standard OLS method. p Δ ln yt = ρ ln yt −1 + ∑ φ j Δ ln yt − j + υt (3.1) j =1 In ADF test our null hypothesis is “there is a unit root”. The null hypothesis is tested and the results are given in the following table for the selected series. 9 Firdos Khan and Zahid Asghar Table 3.1: Results of ADF test. Test statistic Variables 5% critical value First Levels difference First Levels difference AP 1.701370(37) -2.632981(37) -1.9403 -1.9403 LPG 0.241245(66) -4.353701(66) -1.9395 -1.9395 Note: Figures in parenthesis represent the number of lags that are included in ADF test. At level we accept the null hypothesis of unit root at 5% level of significance in both time series. At first difference we reject the null hypothesis of unit root at 5% level of significance, and conclude that the given series are stationary at first difference. The selected series are integrated of order one denoted as I(1). In further analysis we will use these integrated series. To identify the order of the ARIMA model we used ACF,s and PACF,s of the given time series. We already determined the order of d in ARIMA model, which is equal to one. Now we want to determine the order of AR terms and MA terms. From the correlogram we observed that the tentative ARIMA model will be any combination of AR (1) AR(2) AR(3) MA (1) MA (2) MA (3) MA (4) for AP series. However, the final three competing models are ARI (1), MA (1), and AR (1) AR (3) MA (3) MA (4) for AP series. For LPG series from the correlogram it is infered that the final model will be any combination of the AR(1) AR(2) MA(1) MA(2) terms. Four models i.e. AR(1) AR(2), AR(1) MA(1), AR(1) MA(2) and MA(1) MA(2) are the final competing models for LPG series. The final estimation results for competing models for both series are given in the following table. 10 Modeling and Volatility Analysis of Oil Companies in Pakistan Table: 3.2 Estimation results of ARIMA modelling for AP and LPG AP LPG S.No Model Coefficients Std. Error P-Values Coefficients Std. Error P-Values 0.2548 0.0453 0.0000 0.0801 0.0241 0.0009 AR(2) ------- ------- ------- 0.0241 0.0003 AR(1) 0.2897 0.0450 0.0000 0.0352 0.0000 MA(1) ------- ------- ------- 0.0396 0.0000 AR(1) 0.1882 0.0223 0.0000 0.0242 0.0010 -0.8671 0.0236 0.0000 ------- ------- ------- ------- ------- 0.0242 0.0002 0.8701 0.0210 0.0000 ------- ------- ------- 0.1604 0.0287 0.0000 ------- ------- ------- ------- ------- ------- 0.0777 0.0241 0.0013 ------- ------- ------- 0.0904 0.0241 0.0002 AR(1) 1 2 AR(3) 3 MA(2) MA(3) MA(4) MA(1) 4 MA(2) 0.0882 0.8462 -0.8288 0.0794 ------0.0888 From the table it is clear that all the parameters are significant at 5% and 1% significance level. Next we checked the adequacy of the model by using the diagnostic checking test like LM-test and Ljung-Box Q-statistics. All the models mentioned in the table qualify both tests. So, these all models i.e. four for LPG series and three for AP series are the suitable models. However, these competing models are compared to judge the performance of each of these competing models. The forecast performance is also evaluated in each case. The statistics used for this purpose are, AIC, BIC, SSR, LL, SER, ME, MAE, RMSE and TIC. These statistics are useful to choose the best possible model among the competing models that is widely used by researchers in time series analysis and econometrics analysis. The following table gives summary of the mentioned statistics for the competing models in each case. 11 Firdos Khan and Zahid Asghar Table: 3.3 Comparison and forecast evaluation of competing models for AP and LPG AP LPG Statistic AR (1) MA (1) AR (1) AR (3) MA (3) MA (4) AR(1) AR(2) AR(1) MA(1) AR(1) MA(2) MA(1) MA(2) AIC -4.5001 -4.4936 -4.5203* -5.1338 -5.1751* -5.1344 -5.1344 BIC -4.4911* -4.4845 -4.4836 -5.1275* -5.1687 -5.1280 -5.1280 SSR 0.2914 0.2939 0.2805* 0.5894 0.5659* 0.5894 0.5898 1021.20* 1014.314 1016.531 4432.554 4434.46* 4399.604 4402.174 SER 0.0255 0.0256 0.0251* 0.0187* 0.0187* 0.0187* 0.0187* ME 0.0006 0.0003* 0.00032 0.0002* 0.0003 0.00032 0.00046 RMSE 0.0263 0.0265 0.0291* 0.0053* 0.0053* 0.0053* 0.0053* MAE 0.0201* 0.0203 0.0216 16.4136 16.4040 16.4040 16.3944* TIC 0.5146* 0.8900 0.7518 0.8500* 0.8900 0.9100 0.8600 LL Note: “*” indicate that the statistic choose the model among the competing models. From the above table we can conclude that AIC choose AR(1) AR(3) MA(3) MA(4) model for AP series, but BIC which choose more parsimonious model than AIC chooses AR(1) model for AP series. Log likelihood also going in favour of AR(1) model. MAE and TIC also choose AR(1) model as the best model for AP series. Therefore, the final ARIMA model for AP series ARIMA(1,1,0). On the other hand for LPG series among 9 statistics in the above table 5 choose ARIMA((1,2),1,0) as the best model for LPG series. After ARIMA modeling we used ARCH model to model the conditional mean and conditional variance of the series simultaneously. ARCH-LM test is used to test for ARCH error at different lag length. The number of lags included in the ARCHLM test is the order of the ARCH model. It is necessary to apply the formal ARCH test to confirm the order of the ARCH process, for this purpose we have used the Lagrange multiplier test of Engle (1982), which is called ARCH-LM test, especially design to test for ARCH error. ARCH-LM test is distributed as χ 2 with p degree of freedom under the 12 Modeling and Volatility Analysis of Oil Companies in Pakistan null hypothesis to be true, where p is the number of lags, which are included in the test for testing ARCH error. Our null hypothesis is H 0 = There is no ARCH error in the series. Vs H A = There is ARCH error in the series. In our analysis the value of the ARCH-LM test is 58.27565; the lags included in the test are only 2. The corresponding P-Value is 0.00000, which is very low. So we have no difficulty to reject the null hypothesis of no ARCH error, and conclude that there is an ARCH error in the series. This confirms that the order of the ARCH error is two for AP series. We have also estimated other ARCH models i.e. GARCH model, EGARCH model and GJR-GARCH model for both series. The estimation results are given in the following table. Table: 3.4 Estimation Results of various ARCH models for AP. Coefficient ARCH GARCH EGARCH GJR-GARCH Mean Equation Constant a1 ______ ______ ______ ______ 0.210767 0.207782 0.227461 0.217172 (0.0000) (0.0000) (0.0000) (0.0000) Variance Equation α0 0.000339 0.000108 -1.979132 0.000110 (0.0000) (0.0075) (0.0009) (0.0057) α1 0.276070 0.222069 0.409465 0.178956 (0.0048) (0.0059) (0.0001) (0.0386) α2 0.212106 ______ ______ ______ (0.0067) ______ ______ 0.606761 ______ 0.777937 ______ 0.590732 ______ (0.0000) (0.0000) (0.0000) ______ ______ -0.064140 -0.134311 ______ ______ (0.3007) (0.2466) β1 λ1 Note: Numbers in the parenthesis are the P-Values. Table 3.4 is the estimation results of various ARCH models. AR(1) is the mean equation in all ARCH models. ARCH (1), GARCH(1,1), EGARCH(1,1) and GJR-GARCH(1,1) are the variance equation in the four estimated ARCH models. 13 Firdos Khan and Zahid Asghar Table: 3.5 Estimation Results of various ARCH models for LPG. Coefficient ARCH GARCH EGARCH GJR-GARCH Mean Equation Constant ______ ______ Trend a1 a2 a3 b1 b3 -0.004238 ______ ______ (-11.0970**) ______ ______ ______ 0.00000329 ______ ______ ______ (8.898950**) ______ ______ ______ 0.74339 ______ ______ ______ (16.84051**) 0.100110 (3.433833**) ______ ______ ______ 0.110905 ______ (3.319231**) -0.321423 ______ -0.349591 ______ -0.361832 (-2.256154*) (-1.925883) (-2.42033*) ______ ______ ______ -0.68886 ______ 0.098 (-15.07004**) ______ ______ ______ ______ 0.392585 (2.894435**) ______ ______ 0.413 ______ (2.884995**) (2.437575*) ______ 0.42416 (2.761010**) ______ b2 ______ ______ ______ Variance Equation α0 α1 α2 β1 λ1 0.000204 0.000115 -4.028927 0.00012 (188.2402**) (14.97888**) (-18.11205**) (13.60119**) 0.221432 0.257928 0.569548 0.424892 (7.230734**) (7.967976**) (15.32145**) (6.366758**) 0.224575 ______ ______ ______ (10.32825**) ______ ______ 0.423238 ______ 0.541754 ______ 0.04312 ______ (11.57114**) (21.23085**) (9.321583**) ______ ______ 0.235790 -0.30854 ______ ______ (7.696042**) (-4.682034**) Note: “**” and “* “denote that the parameters are significant at 1%and 5% significance level respectively. Table 3.5 is the estimation results of four employed ARCH model for LPG series. The situation is quite different here as compare to the estimation results of AP series. Mean equation in ARCH models is not same in all estimated ARCH models. AR(2) 14 Modeling and Volatility Analysis of Oil Companies in Pakistan AR(3), AR(3) MA(2) MA(3), AR(1) MA(1) and AR(2) AR(3) MA(3) are the mean equations for ARCH, GARCH, EGARCH and GJR-GARCH models respectively. The order of ARCH model is 2 for LPG. The asymmetric coefficient is significant in both asymmetric models, which show that the response of good and bad NEWS is not symmetric. After estimating ARCH models we have used ARCH-LM test to see whether there is any further ARCH error in the series? We have found that there is no evidence of any further ARCH in both AP and LPG series. The ARCH-LM values are 0.3358, 0.1736, 0.1980 and 0.1067 for ARCH, GARCH, EGARCH and GJR-GARCH models respectively and P-value is greater than 0.85 in each case for AP series. This proves that there is no ARCH error remains in AP series. Similar results are obtained for LPG series. As we have estimated four ARCH models, but we want to know that, which model performs better. To judge the performance of these models we compared these models and also evaluate their forecast performance by using some statistics. The comparison and forecast evaluation results are given in the following table for the estimated models. Table: 3.6 Comparison and forecast evaluation of various ARCH models for AP and LPG AP Statistic AIC BIC SSR LL SER ME RMSE MAE TIC LPG AR CH GAR CH EGARCH GJRGARCH AR CH GAR CH EGARCH GJRGARCH -4.58352 -4.59270 -4.58888 -4.59319* -5.436465 -5.438305 -5.459839* -5.446338 -4.54699 -4.55618* -4.54322 -4.54753 -5.417371 -5.419212 -5.434405* -5.424063 0.291984 0.292070 0.291606* 0.291818 0.556017 0.553432* 0.579262 0.553863 1035.293 1037.358 1037.500* 1037.469 4656.895 4658.470 4684.352* 4666.343 0.02558* 0.02559 0.02559 0.02560 0.018059 0.018016* 0.018432 0.018029 0.002299 0.002300 0.002295* 0.002298 0.004299 0.002500 0.002255* 0.002278 0.026277 0.026270* 0.026271 0.026275 0.018352 0.018440 0.018857 0.018348* 0.020144 0.020135* 0.020140 0.020142 0.005522 0.005224* 0.006812 0.005533 0.770943 0.731367* 0.768558 0.760030 0.937972 0.940851 0.904861* 0.929894 Note: “*” indicate that the statistic choose the model among the competing models. 15 Firdos Khan and Zahid Asghar Table 3.6 is the comparison and forecast evaluation results for four estimated ARCH models for AP and LPG series. From table we conclude that GARCH model perform better than any other model for AP. Four out of nine statistics choose GARCH model among four estimated ARCH models. For LPG, EGARCH model is the best model. It is obvious from the table that out of nine statistics five choose EGARCH model. The smaller values of AIC and BIC indicate best model among the competing models. These both statistics choose EGARCH model for LPG. Log likelihood is maximum for EGARCH model and show that it is the best model. ME and TIC are minimum for EGARCH model showing that EGARCH model is the best model among the four competing ARCH model used for LPG data set. 4. Summary and conclusion After achieving stationarity by using unit root testing, we have used ARIMA modeling to model the conditional mean of both AP and LPG series. Various models have been used for both series to choose the best model. Three different ARIMA models are the final competing models for AP series and four different ARIMA models are the competing models for LPG series. These all models are good and can be used for forecasting, however, we made comparison and evaluate the forecast performance of these models to choose a single model among the competing models. ARIMA(1,1,0) is the final model for AP series and ARIMA((1,2),1,0) is the final model for LPG series. After ARIMA modeling we used ARCH models to model the conditional mean as well as the conditional variance simultaneously. For this purpose first we used ARCH-LM test to test whether there is any ARCH error. The test results show that there is an ARCH error in both series. Four different ARCH models viz., ARCH, GARCH, EGARCH and GJRGARCH models are used for both series. After estimation of ARCH models we again use ARCH-LM test to test whether there is any further ARCH error in both series. The test results show that there is no further ARCH error in both series after estimation of ARCH models. We also compared the estimated ARCH models for both series and check which model is the best model among the competing models. After comparison we conclude that GARCH model is the best model for AP series and EGARCH model is the best model for LPG series. The asymmetric parameter is insignificant in AP series and significant in LPG series. It is also important to note that ARCH models performed better 16 Modeling and Volatility Analysis of Oil Companies in Pakistan than ARIMA model in each case. For ARIMA model in case of LPG series minimum value of AIC is -5.1751 and BIC is -5.1275, while for ARCH models minimum value of AIC is -5.4598 and BIC is -5.4344. These statistics clearly support ARCH models instead of ARIMA models. It is easy to look at table 3.3 and table 3.6 to judge the performance of ARCH and ARIMA models. Almost all statistics support ARCH models. Similar situation is observed for AP series. References Akgiray, V. (1989): “Conditional Heteroskedasticity in Time Series of Stock Returns. Evidence and Forecast,” Journal of Business. 69, 55-80. Bera, A. K. and Higgins, M. L. (1992): “A Test for Conditional Heteroskedasticity in Time Series Model,” Journal of Time Series Analysis, 13, 501-519. Black, F. (1976): “Studies in Stock Price Volatility Changes, Proceeding of the 1976 Business Meeting of the Business and Economic Section,” American Statistical Association, 177-181. Bollerslev, T. (1986): "Generalized Autoregressive Conditional Heteroskedasticity,” Journal of Econometrics, 31, 307-327. Box, G. E. P., and Jenkins, G. C. (1976): “Time series analysis, forecasting and control,” San Francisco: Holden day. Brooks, C. (1998): “Predicting Stock Market Volatility,” Journal of Forecasting 17: 1, 59-80. Dickey, D. A., and Fuller, A. W. (1979): “Distribution of the Estimates for Autoregressive Time Series With Unit Root,” Journal of the American Statistical Association 74, 427-431. 17 Firdos Khan and Zahid Asghar Engle, R. F. (1982): "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation," Econometrica, 50, 987-1008. Engle, R. F., and Ng, V. K. (1993): “Measuring and Testing the Impact of News on Volatility,” The Journal of Finance. Vol. 48, 5, 1749-1778. Fama, E. F., (1965): “The Behavior of Stock Market Prices,” Journal of Business 38, 34- 105. Figlewski, S. (1997): “Forecasting Volatility in Finan. Markets,” Inst. Instruments, NYU, Salomon Center, 6:1 1-88. Geweke, J. (1986): “Modelling the persistence of conditional variance. Comments,” Econometric Reviews, 5 57-61. Glosten, L. R., Jagannathan, R., and Runkle, D. E. (1993): “On the Relation Between the Expected Value and the Volatility of the Nominal excess Returns on Stocks,” Journal of Finance 48, 1779-1801. Lee, K. Y. (1991): “Are the GARCH Models Best in out-of-sample Performance,” Economic Letters 37:3, 305-308. McMillan, D., Speight, A H., and Owain, A. G. (2000): “Forecasting UK Stock Market Volatility,” Journal of Applied Econometrics, 10, 435-448. Milhoj, A. (1987 a): “A Multiplicative Parameterization of ARCH Model,” Research Report, No 101, Institute of Statistics, University of Copenhagen. Nelson, D. B. (1991): “Conditional Heteroskedasticity in Asset Returns. A new Approach,” Econometrica 59, 347-370. Taylor, S. J. (1986): “Modelling Financial Time Series,” (Wiley, New York, NY). 18 Modeling and Volatility Analysis of Oil Companies in Pakistan Zakoian, J-M. (1990): “Threshold Heteroskedastic Model,” Unpublished manuscript (INSEE, Paris). 19 Firdos Khan and Zahid Asghar GLOSSARY Asymmetric autoregressive conditional heteroskedastic AARCH Augmented Dickey-Fuller ADF AIC Akaike information criteria AR Autoregressive ARCH Autoregressive conditional heteroskedasticity ARCH-LM Autoregressive conditional heteroskedastic langrage multiplier (Test) ARCH-M Autoregressive conditional heteroskedastic in mean ARIMA Autoregressive integrated moving average ARMA Autoregressive moving average BIC Bayesian information criteria Best Linear Unbiased Estimate BLUE DF Dickey-Fuller DGP Data generating process EGARCH Exponential Generalize autoregressive conditional heteroskedastic ES Exponential smoothing EWMA Exponential weighted moving average FARIMA Fractional autoregressive integrated moving average FTSE Financial Times Stock Exchange GARCH Generalize autoregressive conditional heteroskedastic GARCH-M Generalize autoregressive conditional heteroskedastic in mean 20 Modeling and Volatility Analysis of Oil Companies in Pakistan GJR-GARCH Glosten, Jaghanathan and Runkle-Generalize autoregressive conditional heteroskedastic HISVOL Historical volatility model IGARCH Integrated generalize autoregressive conditional heteroskedastic ISD Implied standard deviation LL Log Likelihood LM Langrage multiplier MA Moving average MAE Mean absolute error MAPE Mean absolute percent error ME Mean error MSE Mean square error NARCH Non-linear autoregressive conditional heteroskedastic NEWS North East West South PNP Partially non-parametric QGARCH Quadratic generalize autoregressive conditional heteroskedastic RMSE Root mean square error RSS Residuals sum of square RW Random walk SER Standard Error of Regression SV Stochastic volatility TARCH Threshold autoregressive conditional heteroskedastic TIC Theil’s Inequality Coefficient 21