AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 015,Science Huβ, http://www.scihub.org/ajsir ISSN: 153-649X, doi:10.551/ajsir.015.6.6.131.140 ARCH modeling of the returns of first bank of Nigeria 1 Emmanuel Alphonsus Akpan and 1 Imoh Udo Moffat (Correspondent author, eubong44@gmail.com, +34803600343, 1 Department of Mathematics and Statistics University of Uyo, Nigeria, ABSTRACT This study considers daily closing share prices of First Bank of Nigeria plc from 4 January, 000 to 31 December, 013. The data were obtained from the Nigerian Stock Exchange. This study seeks to check if ARCH effect exists in the returns of First Bank of Nigeria and to establish the volatility structure for modeling the time-varying conditional variance in the returns of First Bank of Nigeria. The share price series was found to be non-stationary. The returns series which is the first difference of log of the share price series was found to be stationary. ARCH effect of the residuals of ARIMA(,0,) model was checked and found to exist using Lagrange Multiplier test and, Ljung and Box Q-statistic. ARCH(1) model provided the volatility structure that is appropriate for modeling the returns of First Bank of Nigeria. Keywords: ARCH modeling, conditional variance, First Bank of Nigeria. INTRODUCTION One of the assumptions of conventional time series is constant variance. But the fact that large absolute returns of stocks tend to be followed by large absolute returns is hardly compatible with the assumption of constant variance. Engle (198) introduced the ARCH (Autoregressive Conditional Heteroscedasticity) process that changed the assumption of constant variance. The ARCH process allowed the conditional variance to change over time as a function of past errors. The term heteroscedasticity refers to changing variance. But it is not the variance itself which changes with time according to an ARCH model, rather, it is the conditional variance which changes in a specific way, depending on the available data. In ARCH model, the key concept is the conditional variance, that is, the variance is conditional on the past information. Therefore, in this paper, we seek to check if ARCH effect, that is changing variance exists in the returns of First bank of Nigeria and to establish the heteroscedastic structure that is appropriate for modeling the returns of FBN. ARCH family models are good candidates for modeling and estimating varying variance in emerging markets in that neglecting the presence of ARCH (autoregressive conditional heteroscedasticity) effects in regression models results in inefficient ordinary least squares estimates (Yet, still being consistent). The covariance matrix of the parameters would be biased, with invalid t-statistics (Asteriou and Hall, 007). Besides the lack of asymptotic efficiency, it might also lead to over-parameterization of an (ARMA) model and to over-rejection of conventional tests, for example tests for serial correlation (Fan and Yao, 005). Setting up a model which explicitly accounts for the presence of Autoregressive Conditional Heteroscedasticity (ARCH) effects leads to an efficient estimator and will ensure the calculation of a valid covariance matrix. However, such a model is usually not estimated by an ordinary least squared estimator, but by the iterative solving of a nonlinear maximation problem, namely by using a maximum-likelihood procedure (Brunhart, 011). Engle (198) introduced the autoregressive conditional Heteroscedasticity (ARCH) to model volatility by relating the conditional variance of the disturbance term of the linear combination of the squared disturbances in the recent past. Bollerslev (1986) generalized the ARCH model by modeling the conditional variance to depend on its lagged values as well as squared lagged values of disturbance. In literature, studies like Campbell and Hentschel (199), Braun, Nelson and Sunier (1995) and LeBaron (000) provide evidence that stock return has timevarying volatility. While Bekaert and Harvey (1997) and Aggarwal, Inclan and Leal (1999) in their studies of emerging markets volatility, confirm the ability of asymmetric GARCH models in capturing asymmetry in stock return volatility. In Nigeria, the studies of Arowolo (013), Yaya (013), Emenike (010), Ogum, Beer and Nouyrigat (005), Mgbame and Ikhatua (014), Atoi (014), Onwukwe, Samson and Lipcsey (014) and Aliyu (009) revealed that stock returns exhibit time-varying conditional variance.
METHODOLOGY Returns: Most financial studies involve returns, instead of prices of assets. Campbell, Lo, and Mackindlay (1997) give two main reasons for using returns. First, for average investors, return of an asset is a complete and scale free summary of the investment opportunity. Second, returns series are easier to handle than price series because the former have more attractive statistical properties. Although volatility is not directly observable, it has some properties that are commonly seen in returns. These properties are volatility clustering, leptokurtosis and leverage effect. The returns can be defined as follows R t = log ( P t P t 1 ) = log P t logp t 1 (3.1) where P t is the share price at time t, and P t 1 is the share price at time t 1. The series {R t } is referred to as the returns series (Karlsson, 013). ARCH Model The first model that provides a systematic framework for modeling volatility is the ARCH model of Engle (198). Specifically, an ARCH (q) model assumes that, R t = μ t + a t, a t = σ t e t, σ t = α 0 + α 1 a t 1 + + α q a t q (3.) Table 1: Output of Augmented Dickey Fuller Test for Share Price Series Null Hypothesis: FBN has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic - based on SIC, maxlag=60) where [e t ] is a sequence of independent and identically distributed (i.i.d.) random variables with mean, zero, and variance, 1, α 0 > 0, and α 1,.., α q 0. The coefficients α i, for i > 0, must satisfy some regularity conditions to ensure that the unconditional variance of a t is finite. Data analysis and Discussion : Daily closing share prices of First Bank of Nigeria were obtained from the Nigerian Stock Exchange (NSE) for a period spanning from January 4, 000 to December 31, 013. This series consists of 3464 observations. In the preliminary analyses, we assessed the time series plots of the share price series and the returns series which is the first difference of the log of the share price series. The plot of share price series of FBN (Figure 1) appears to contain a trend component which suggests that the series is non-stationary while the returns series (Figure ) suggests that volatility clustering is quite evident and the series appears to be stationary. We also applied the Augmented Dickey- Fuller (ADF) test to the share price series and the returns series to test for the presence of unit root. The results from (Table 1) show that the test fails to reject the null hypothesis of a unit root for the share price series while the results from (Table ) show that the test rejects the null hypothesis of a unit root for the returns series. Thus, the results of ADF test show that the share price series is non-stationary while the returns series is stationary. t-statistic Prob.* Augmented Dickey-Fuller test statistic -3.10993 0.1057 Test critical values: 1% level -3.960691 5% level -3.411104 10% level -3.17375 *MacKinnon (1996) one-sided p-values. 13
Table : Output of Augmented Dickey Fuller Test for Return Series Null Hypothesis: D(FBN) has a unit root Exogenous: Constant, Linear Trend Lag Length: 4 (Automatic - based on SIC, maxlag=60) t-statistic Prob.* Augmented Dickey-Fuller test statistic -9.95644 0.0000 Test critical values: 1% level -3.960694 5% level -3.411105 10% level -3.17376 *MacKinnon (1996) one-sided p-values. Linear Model Identification and Estimation: The tentative ARIMA models for FBN returns series based on the autocorrelations and partial autocorrelations would be ARIMA(1,0,1), ARIMA(,0,) and ARIMA(3,0,3) with their outputs shown in Tables 3,4 and 5 respectively. ARIMA(,0,) is the model with the smallest information criteria with only the constant term not significant, hence, the selected model for FBN returns series is the ARIMA(,0,) without the constant term shown in Table 6. Table 3: Output of ARIMA(1,0,1) Model Model 1: ARMA, using observations -3464 (T = 3463) Dependent variable: ld_fbnh Standard errors based on Outer Products matrix Coefficient Std. Error Z p-value Const 8.5757e-05 0.000561633 0.1470 0.8831 phi_1 0.018439 0.14141 0.1304 0.8963 theta_1 0.10196 0.140709 0.744 0.4688 Mean dependent var 0.000084 S.D. dependent var 0.09658 Mean of innovations 1.54e-06 S.D. of innovations 0.09441 Log-likelihood 794.499 Akaike criterion 14581.00 Schwarz criterion 14556.40 Hannan-Quinn 1457.1 Table 4: Output of ARIMA(,0,) Model Model 4: ARMA, using observations -3464 (T = 3463) Dependent variable: ld_fbn Standard errors based on Hessian Coefficient Std. Error Z p-value Const 8.3538e-05 0.00046381 0.1801 0.8571 phi_1 1.0538 0.0945795 11.1365 <0.0001 *** phi_ 0.407489 0.0958636 4.507 <0.0001 *** theta_1 0.93683 0.0993007 9.4343 <0.0001 *** theta_ 0.66707 0.1019.6099 0.0091 *** Mean dependent var 0.000084 S.D. dependent var 0.09658 Mean of innovations 6.37e-07 S.D. of innovations 0.09303 Log-likelihood 7310.806 Akaike criterion 14609.61 Schwarz criterion 1457.71 Hannan-Quinn 14596.44 133
Table 5: Output of ARIMA(3,0,3) Model Model 5: ARMA, using observations -3464 (T = 3463) Dependent variable: ld_fbn Standard errors based on Hessian Coefficient Std. Error Z p-value Const 8.111e-05 0.00044837 0.1854 0.859 phi_1 1.375 0.64168.168 0.0334 ** phi_ 0.785051 0.68065 1.1534 0.487 phi_3 0.4898 0.75481 0.9038 0.3661 theta_1 1.1415 0.63667 1.9468 0.0516 * theta_ 0.66901 0.608776 1.098 0.3031 theta_3 0.756 0.196168 1.1585 0.467 Mean dependent var 0.000084 S.D. dependent var 0.09658 Mean of innovations.1e-06 S.D. of innovations 0.099 Log-likelihood 731.049 Akaike criterion 14608.10 Schwarz criterion 14558.90 Hannan-Quinn 14590.53 Table 6: Output of ARIMA(,0,) Model without a Constant Term. Model 7: ARMA, using observations -3464 (T = 3463) Dependent variable: ld_fbn Standard errors based on Hessian Coefficient Std. Error Z p-value phi_1 1.0531 0.0945018 11.1449 <0.0001 *** phi_ 0.40753 0.0954444 4.697 <0.0001 *** theta_1 0.936755 0.09911 9.440 <0.0001 *** theta_ 0.66759 0.101735.61 0.0087 *** Mean dependent var 0.000084 S.D. dependent var 0.09658 Mean of innovations 0.000090 S.D. of innovations 0.09303 Log-likelihood 7310.790 Akaike criterion 14611.58 Schwarz criterion 14580.83 Hannan-Quinn 14600.60 Fig. 1: Plot of Share Price Series of FBN 134
Fig. : Plot of Returns Series of FBN Fig. 3: ACF of Squares of Residuals of ARIMA(,0,) Fig. 4: PACF for Squares of Residuals of AR1MA(,0,) Model 135
Identification of ARCH Effect: For ease in notation, let a t = R t μ t be the residuals of the mean equation. The squares of series, a t is then used to check for conditional heteroscedasticity, which is also known as the ARCH effects. If at least one lag term in the squares of residual series is found to be statistically significant, this confirms the presence of ARCH effects (Khan and Azim, 013). To perform the test, the usual Ljung-box statistic, Q(m), is applied to the { a t } series (Mcleod and Li, 1983). The null hypothesis is that, the first m lags of ACF of the a t series are zero. This implies that ARCH effects do not exist in a t. The null hypothesis is rejected at 5% significance levels if the probability value corresponding to the Q-statistic is less than the level of significance. Another approach for testing the ARCH effects is to apply the Lagrange Multiplier (LM) test of ARCH(q) against the hypothesis of no ARCH effects to { a t } series. The LM test is carried out by computing, χ = TR in the regression of a t on a constant and q lagged values. T is the sample size and R is the coefficient of determination. Under the null hypothesis of no ARCH effects, the statistic has a Chi-square distribution with q degrees of freedom. If the LM test statistic is larger than the critical value, then, there is evidence of the presence of ARCH effects (Greene, 00). After taking the residual series of the estimated ARIMA(,0,) model, which was selected on the basis of the significance of all its parameters, we check if there exists any ARCH effect or not in the residuals. From (Figures 3 and 4) respectively, the ACF and the PACF of the squares of residuals exhibit significant spike at lag 1 while dying down to zero fast, indicating the presence of ARCH effect. In order to test statistically for the presence of ARCH effect, the Lagrange Multiplier (LM) test is applied to the squares of the residuals. According to Table 7, the hypothesis of no ARCH effects is rejected at 5% level of significance since the LM test statistic = 6.7313 at lag 1 > χ 0.05,1 = 3.841 with corresponding p-value 0.0095. Table 8: Q- statistics for Squares of Residuals of ARIMA(,0,) Model without a Constant Term. Autocorrelation function for usq LAG ACF PACF Q-stat. [p-value] 1 0.044 *** 0.044 *** 6.759 [0.009] 0.006 0.0006 6.7818 [0.034] 3 0.0001-0.0000 6.7819 [0.079] 4 0.0001 0.0001 6.7819 [0.148] 5 0.001 0.001 6.7971 [0.36] 6 0.0087 0.0085 7.0584 [0.315] 7 0.000-0.0006 7.0585 [0.43] 8 0.0019 0.0019 7.0706 [0.59] 9-0.0006-0.0007 7.0717 [0.630] 10 0.004 0.005 7.095 [0.717] 11 0.000 0.0017 7.106 [0.790] 1-0.0004-0.0007 7.1068 [0.850] 13 0.009 0.0030 7.136 [0.895] 14 0.0054 0.005 7.390 [0.95] 15 0.009 0.004 7.681 [0.950] 16 0.0066 0.0063 7.4188 [0.964] 17-0.0017-0.003 7.488 [0.977] 18-0.0011-0.0009 7.437 [0.986] 19-0.009-0.009 7.468 [0.991] 0-0.0010-0.0008 7.466 [0.995] 1-0.003-0.003 7.4854 [0.997] -0.008-0.007 7.5130 [0.998] 3-0.0015-0.001 7.509 [0.999] 4-0.0018-0.0017 7.5317 [0.999] 136
Table 9: Output of ARCH (1) Model 5 0.005 0.007 7.5539 [1.000] 6 0.004 0.00 7.5740 [1.000] 7 0.0001-0.0001 7.5740 [1.000] 8-0.0005-0.0005 7.5748 [1.000] 9 0.0014 0.0014 7.5817 [1.000] 30-0.008-0.009 7.6085 [1.000] 31-0.0016-0.0014 7.6178 [1.000] 3-0.0035-0.0033 7.6597 [1.000] 33-0.0031-0.007 7.699 [1.000] 34-0.0016-0.0013 7.7019 [1.000] 35-0.0013-0.0011 7.7074 [1.000] 36-0.001-0.0010 7.716 [1.000] 37-0.0038-0.0036 7.767 [1.000] 38-0.000 0.000 7.769 [1.000] 39-0.0007-0.0006 7.7644 [1.000] 40 0.0019 0.0019 7.7767 [1.000] 41-0.008-0.009 7.803 [1.000] 4-0.0036-0.0034 7.8489 [1.000] 43-0.0000 0.0004 7.8489 [1.000] 44-0.009-0.009 7.8784 [1.000] 45-0.0015-0.001 7.8868 [1.000] 46 0.000 0.0004 7.8869 [1.000] 47 0.005 0.0053 7.9814 [1.000] 48 0.0018 0.0015 7.999 [1.000] 49-0.0019-0.000 8.006 [1.000] 50-0.0035-0.003 8.0487 [1.000] 51-0.009-0.006 8.0788 [1.000] 5-0.0033-0.0031 8.1181 [1.000] 53-0.009-0.007 8.1484 [1.000] 54-0.0017-0.0015 8.1581 [1.000] 55-0.001-0.0019 8.174 [1.000] 56-0.005-0.00 8.1958 [1.000] 57-0.0013-0.0010 8.017 [1.000] 58-0.0017-0.0015 8.14 [1.000] 59-0.0019-0.0017 8.5 [1.000] 60 0.0034-0.003 8.668 [1.000] Model 6: GARCH, using observations -3464 (T = 3463) Dependent variable: uhat1 Standard errors based on Hessian Coefficient Std. Error z p-value Const 0.0019894 0.0003610 5.5077 <0.0001 *** alpha(0) 0.00040155 1.51344e-05 6.533 <0.0001 *** alpha(1) 0.850956 0.0586056 14.501 <0.0001 *** Mean dependent var 0.000090 S.D. dependent var 0.09308 Log-likelihood 7699.888 Akaike criterion 15391.78 Schwarz criterion 15367.18 Hannan-Quinn 1538.99 137
Table 10: Lagrange Multiplier Test for ARCH(1) Model Heteroscedasticity Test: ARCH F-statistic 0.1413 Prob. F(1,3458) 0.6435 Obs*R-squared 0.1434 Prob. Chi-Square(1) 0.6434 Table 11: Q-Statistic for Squares of residuals of ARCH(1) Model Date: 10/07/15 Time: :09 Sample: 1/07/000 4/1/013 Included observations: 3461 Q-statistic probabilities adjusted for ARMA term(s) AC PAC Q-Stat Prob 1-0.008-0.008 0.146-0.00-0.00 0.75 3 0.004 0.004 0.779 0.598 4 0.000 0.000 0.779 0.870 5 0.00 0.00 0.899 0.96 6 0.010 0.010 0.687 0.960 7 0.000 0.000 0.689 0.987 8 0.004 0.004 0.6930 0.995 9 0.003 0.003 0.704 0.998 10 0.006 0.006 0.8631 0.999 11 0.004 0.004 0.919 1.000 1-0.000-0.000 0.9193 1.000 13 0.007 0.007 1.0996 1.000 14-0.003-0.003 1.118 1.000 15-0.001-0.001 1.138 1.000 16 0.01 0.01 1.65 1.000 17-0.003-0.003 1.678 1.000 18 0.00 0.00 1.691 1.000 19-0.005-0.006 1.7864 1.000 0 0.005 0.005 1.8649 1.000 1-0.00-0.00 1.883 1.000-0.003-0.003 1.9169 1.000 3-0.001-0.001 1.9188 1.000 4-0.008-0.008.173 1.000 5 0.005 0.005.05 1.000 6 0.001 0.001.077 1.000 7 0.005 0.005.971 1.000 8-0.005-0.005.388 1.000 9-0.005-0.005.4809 1.000 30-0.003-0.003.5150 1.000 31-0.001-0.001.5186 1.000 138
3-0.008-0.008.7658 1.000 33-0.006-0.006.8858 1.000 34-0.004-0.004.9365 1.000 35-0.003-0.003.9685 1.000 36-0.005-0.005 3.0676 1.000 37-0.006-0.006 3.006 1.000 38 0.00 0.00 3.193 1.000 39 0.001 0.001 3.17 1.000 40 0.001 0.00 3.60 1.000 41-0.009-0.009 3.506 1.000 4-0.009-0.009 3.8149 1.000 43 0.005 0.005 3.9091 1.000 44-0.007-0.007 4.0966 1.000 45-0.00-0.00 4.1149 1.000 46-0.00-0.00 4.153 1.000 47 0.007 0.007 4.873 1.000 48 0.001 0.00 4.946 1.000 49-0.009-0.008 4.5535 1.000 50-0.006-0.006 4.6934 1.000 51-0.004-0.004 4.7367 1.000 5-0.005-0.005 4.8339 1.000 53-0.006-0.006 4.9675 1.000 54 0.004 0.004 5.00 1.000 55-0.006-0.005 5.1306 1.000 56-0.006-0.007 5.76 1.000 57 0.010 0.011 5.6443 1.000 58-0.006-0.006 5.7684 1.000 59-0.005-0.005 5.844 1.000 60-0.003-0.003 5.8756 1.000 Also, evidence from Ljung and Box Q-Statistic (Table 8) confirms that ARCH effects exist in the squares of the residuals at lag 1 since Q-statistic = 6.759 > χ 0.05,1 = 3.841 with corresponding p-value 0.009. Therefore, it is concluded that ARCH effects exist in the returns series and can be modeled using an ARCH(1) model as shown in equation (4.1) below R t = 1.9894e 3 + a t, s.e: (3.61 10 4 ) z-ratio: ( 5.5077) p-value: (0.0001) σ t = 4.0155e 4 + 0.850956a t 1 (4.1) s.e: (1.5134 10 5 ) (0.05861) z-ratio: (6.533) (14.501) p-value: (1 10 4 ) (1 10 4 ) [Excerpts from Table 9]. Under the diagnostic checking of the ARCH(1) model, the parameter of ARCH (1) model is statistically significant. The LM test indicates that there is no ARCH effect in the residuals (Table 10), since LM = 0.143 at lag 1 < χ 0.05,1 = 3.841 with corresponding probability value of 0.6434. Also, evidence from Q- statistic (Table 11) confirms that the model is adequate since Q-statistic = 5.8756 at lag 60 < χ 0.05,59 = 77.931 with corresponding probability value of 1.000. CONCLUSION This study provides evidence to show that ARCH effect exists in the returns series of FBN. Also, ARCH(1) model provides the volatility structure that is 139
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