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The study explored both Box and Jenkins, and iterative outlier detection procedures in determining the efficiency of ARIMA-GARCH-type models in the presence of outliers using the daily closing share price returns series of four prominent banks in Nigeria (Skye (Polaris) bank, Sterling bank, Unity bank and Zenith bank) from January 3, 2006 to November 24, 2016. The series consists of 2690 observations for each bank. The data were obtained from the Nigerian Stock Exchange. Unconditional variance and kurtosis coefficient were used as criteria for measuring the efficiency of ARIMA-GARCH-type models and our findings revealed that kurtosis is a better criterion (as it is a true measure of outliers) than the unconditional variance (as it can be depleted or amplified by outliers). Specifically, the strength of this study is in showing the applicability and relevance of iterative methods in time series modeling.

The generalized autoregressive conditional heteroscedastic (GARCH-type) models were introduced to account for heteroscedasticity (changing variance), a phenomenon which occurs as a result of violation of assumption of constant variance in time series. The GARCH-type models are further divided into symmetric and asymmetric. The symmetric GARCH models (for example ARCH and GARCH) rely on modeling the conditional variance as a linear function of squared past residuals. The strength of this specification is in allowing the conditional variance to depend only on the modulus of the past variables (past positive and negative innovations have the same effect on the current conditional variance). The most interesting feature not addressed by GARCH model is the leverage effect which occurs when an unexpected drop in price (bad news) increases predictable volatility more than an unexpected increase in price (good news) of similar magnitude (Engle and Ng [

Originally, the GARCH model was specified based on the normal distribution for the innovations yet could not capture the heavy-tailed characterizations. Similarly, the student-t distribution which is traditionally specified to remedy the weakness of the normal distribution in accommodating the heavy-tailed property, is found wanting in many applications to account for excess kurtosis and thus, the resulting estimates of GARCH models are not efficient (Moffat and Akpan [

Furthermore, previous studies have shown that the heavy-tailed property indicates the presence of excess kurtosis which in turn is a measure of outliers (Moffat and Akpan [

Hence, the aim of this study is to determine the efficiency of GARCH-type models with outliers taken into consideration using kurtosis coefficient which is at least approximately mesokurtic, and in particular seeks to improve on the work of Akpan, Lasisi and Adamu [

Moreover, the remaining part of this work is organized as follows; Section 2 takes care of materials and method then followed by results and discussion in Section 3, while the conclusion of overall results is handled in Section 4.

The return series R t can be obtained given that P t is the price of a unit share at time, t and P t − 1 is the share price at time t − 1 .

R t = ∇ ln P t = ( 1 − B ) ln P t = ln P t − ln P t − 1 (1)

The R t in Equation (1) is regarded as a transformed series of the share price, P t meant to attain stationarity, that is, both mean and variance of the series are stable (Akpan and Moffat [

Box, Jenkins and Reinsel [^{th} difference is a stationary ARMA model. Denoting the d^{th} difference of X t by

φ ( B ) = ϕ ( B ) ∇ d X t = θ ( B ) ε t (2)

where φ ( B ) is the nonstationary autoregressive operator such that d of the roots of φ ( B ) = 0 are unity and the remainder lie outside the unit circle. ϕ ( B ) is a stationary autoregressive operator.

The first model that provides a systematic framework for modeling heteroscedasticity is the ARCH model of Engle [

R t = μ t + a t , a t = σ t e t ,

σ t 2 = ω + α 1 a t − 1 2 + ⋯ + α q a t − q 2 . (3)

where [ e t ] is a sequence of independent and identically distributed (i.i.d.) random variables with mean zero, that is E ( e t ) = 0 and variance 1, that is E ( e t 2 ) = 1 , ω > 0 , and α 1 , ⋯ , α q ≥ 0 (Francq and Zakoian [

Although the ARCH model is simple, it often requires many parameters to adequately describe the volatility process of a share price return. Some alternative models must be sought. Bollerslev [

a t = σ t e t ,

σ t 2 = ω + ∑ i = 1 q α i a t − i 2 + ∑ j = 1 p β j σ t − j 2 . (4)

where again e t is a sequence of i.i.d. random variance with mean, 0, and variance, 1, ω > 0 , α i ≥ 0 , β j ≥ 0 and ∑ i = 1 max ( p , q ) ( α i + β i ) < 1 (Tsay [

Here, it is understood that α i = 0 , for i > p , and β i = 0 , for i > q . The latter constraint on α i + β i implies that the unconditional variance of a t is finite, whereas its conditional variance σ t 2 , evolves over time.

Model

The EGARCH model represents a major shift from ARCH and GARCH models (Nelson [

R t = μ t + a t , a t = σ t e t ,

ln σ t 2 = ω + ∑ i = 1 q α i | a t − i σ t − i 2 | + ∑ k = 1 r γ k ( a t − k σ t − k 2 ) + ∑ j = 1 p β j ln σ t − j 2 . (5)

where again, e t is a sequence of i.i.d. random variance with mean, 0, and variance, 1, and γ k is the asymmetric coefficient.

The GJR-GARCH (q, p) model proposed by Glosten, Jagannathan and Runkle [

a t = σ t e t

σ t 2 = ω + ∑ i = 1 q α i a t − i 2 + ∑ i = 1 p γ i I t − i a t − i 2 + ∑ j = 1 p β j σ t − j 2 . (6)

where I t − 1 is an indicator for negative a t − i , that is,

I t − 1 = { 0 if a t − i < 0 , 1 if a t − i ≥ 0 ,

and α i , γ i and β j are nonnegative parameters satisfying conditions similar to those of GARCH models. Also the introduction of indicator parameter of leverage effect, I t − 1 in the model accommodates the leverage effect, since it is supposed that the effect of a t − i 2 on the conditional variance σ t 2 is different accordingly to the sign of a t − i .

To successfully fit the processes describe by subsections 2.2 and 2.3, the approach is based on Box and Jenkins three iterative procedures which includes model identification, model estimation and diagnostic checking are summarized as follows.

Identification Stage: employs the plots of estimated acf (autocorrelation function) and pacf (partial autocorrelation function) as guides to selecting one or more ARIMA models that appears suitable. At this stage, models whose theoretical acf and pacf most closely resemble the estimated acf and pacf are selected tentatively.

Estimation Stage: The tentatively entertained model is fitted to data and its parameters estimated using maximum likelihood techniques.

Diagnostic checking Stage: Diagnostic checks are carried to help determine if an estimated model is statistically significant targeted at uncovering possible lack-of-fit. If no lack-of-fit is indicated, the model is ready to use. If any inadequacy is found, the iterative cycle of identification, estimation, and diagnostic checking is repeated until a suitable representation is found. (See Box, Jenkins and Reinsel [

Generally, a time series might contain several, say k outliers of different types and we have the following general outlier model;

Y t = ∑ j = 1 k τ j V j ( B ) I t ( T ) + X t , (7)

where X t = ( θ ( B ) ) / ( φ ( B ) ) a t , V j ( B ) = 1 for an AO, and V j ( B ) = θ ( B ) φ ( B ) for an IO at t = T j , V j ( B ) = ( 1 − B ) − 1 for a LS, V j ( B ) = ( 1 − δ B ) − 1 for an TC, and τ is the size of outlier. For more details on the types of outliers and estimation of the outliers effects (see Moffat and Akpan [

Moreover, in financial time series, the residual series, a t is assumed to be uncorrelated with its own past, so additive, innovative, temporary change and level shift outliers coincide, and where both the mean and variance equations evolves together, we have for example GARCH(1,1) model:

R t − μ t = a ˜ t + τ I t ( T ) . (8)

a ˜ t = σ t e t . (9)

σ t 2 = ω + α 1 a ˜ t − 1 2 + β 1 σ t − 1 2 . (10)

where a ˜ t is the outliers contaminated residuals.

One approach for correcting the series for outliers is using standard criteria and then estimates the conditional variance. This approach involves detecting and correcting of outliers before estimating the conditional variance (Carnero, Pena and Ruiz [

Step I: Given an ARIMA model fitted to the data, all the potential outliers are detected based on preliminary model parameter estimates.

Step II: Joint estimates of the model parameters and outliers effects are obtained using the accommodated outlier information of step I.

Step III: Outliers are identified and their effects estimated again based on the adjusted estimates of model parameters obtained in Step II.

Efficiency is a measure of quality of an estimator of a model. It is often expressed using variance or mean square error. For the purpose of this study which looks at a unified effect of outliers, unconditional variance and coefficient of kurtosis are considered as the measures of efficiency of estimator of heteroscedastic model. The application of coefficient of kurtosis in this case is to ensure that the existence of heavy-tailed is taken care of.

For ARCH(q) model which is equivalent to GARCH(q, 0) model, the unconditional variance is given as follows:

σ 2 = ω 1 − ∑ i = 1 q α i . (11)

For GARCH(q, p) model, the unconditional variance is expressed thus:

σ 2 = ω 1 − ∑ i = 1 q α i − ∑ j = 1 p β i . (12)

For EGARCH(q, p) model, the unconditional variance is expressed as follows:

σ 2 = exp ( ω 1 − ∑ j = 1 p β i ) , (13)

where exp represent natural exponential function.

For GJR-GARCH(q, p) model

σ 2 = ω 1 − ∑ i = 1 q α i − ∑ i = 1 q γ i 2 − ∑ j = 1 p β i . (14)

Kurtosis coefficient for a centered (that is, zero-mean) distribution is defined as the ratio of the fourth-order moment, which is assumed to exist, to the squared second-order moment (Francq and Zakoian [

a t = σ t e t , (15)

σ t 2 = ω + α 1 a t − 1 2 + β 1 σ t − 1 2 , (16)

Note that, E ( e t ) = 0 , Var ( e t ) = 1 , and E ( e t 4 ) = K e + 3 , where K e is the excess kurtosis of the innovation, e t .

Also,

Var ( a t ) = E ( σ t 2 ) = ω [ 1 − ( α 1 + β 1 ) ] . (17)

E ( a t 4 ) = ( K e + 3 ) E ( σ t 4 ) provided that E ( σ t 4 ) exists.

But,

E ( σ t 4 ) = ω 2 ( 1 + α 1 + β 1 ) [ 1 − ( α 1 + β 1 ) ] [ 1 − α 1 2 ( K e + 2 ) − ( α 1 + β 1 ) 2 ] , (18)

provided that 1 > α 1 + β 1 ≥ 0 and 1 − α 1 2 ( K e + 2 ) − ( α 1 + β 1 ) 2 > 0 . the excess kurtosis of a t , if it exists, is then

K a = E ( a t 4 ) [ E ( a t 2 ) ] 2 − 3 = ( K e + 3 ) [ 1 − ( α 1 + β 1 ) 2 ] 1 − 2 α 1 2 − ( α 1 + β 1 ) 2 − K e α 1 2 − 3. (19)

This excess kurtosis can be written in an informative expression. Considering the case where e t follows a normal distribution, K e = 0 ,

K a ( g ) = 6 α 1 2 1 − 2 α 1 2 − ( α 1 + β 1 ) 2 , (20)

where the superscript, g, is used to denote the Gaussian distribution. The same idea applies to other GARCH-type models (Tsay [

Data collection is based on secondary source as documented in the records of Nigerian Stock Exchange. The documented data on the daily closing share prices of the sampled banks (Skye bank, Sterling bank, Unity bank and Zenith bank) from January 3, 2006 to November 24, 2016 were purchased from the Nigerian Stock Exchange and delivered through contactcentre@nigerianstockexchange.com. Since the data were obtained from a credible and secured source therefore reliable.

The share prices of the four prominent Nigerian banks considered are found to be nonstationary given the random fluctuations away from the common mean (see Figures 1-4).

To achieve stationarity, Equation (1) was applied to the share price series and these transformed series were found to cluster round the common mean and thus indicated the presence of heteroscedasticity (see Figures 5-8).

Based on Box and Jenkins procedures, out of the several models identified tentatively, the following joint ARIMA-GARCH-type models with respect to both normal (norm) and student-t (std) distributions in (

Here, we examined the residuals series of the fitted ARIMA models for detection

Bank | Model | Parameter | Estimate | s.e | t-ratio | p-value | Information Criteria | ||
---|---|---|---|---|---|---|---|---|---|

AIC | BIC | HQIC | |||||||

SKYE | ARIMA(1,1,0)-GARCH(1,1)-norm | μ | −2.6e^{−5} | 1.1e^{−5} | −2.3973 | 0.0165 | −4.5293 | −4.5183 | −4.5253 |

φ 1 | 3.87e^{−4} | 0.0214 | 0.0181 | 0.9855 | |||||

ω | 2.0e^{−6} | 0.0000 | 73.3223 | 0.0000 | |||||

α 1 | 0.1305 | 7.996e^{−3} | 16.3261 | 0.0000 | |||||

β 1 | 0.8685 | 6.671e^{−3} | 130.1908 | 0.0000 | |||||

STERLING | ARIMA(2,1,0)-EGARCH(1,1)-norm | μ | −2.89e^{−4} | 1.57e^{−4} | −1.8335 | 0.06673 | −4.3888 | −4.3734 | −4.3832 |

φ 1 | −0.0190 | 0.0102 | −1.8642 | 0.0623 | |||||

φ 2 | 0.0298 | 5.205e^{−3} | 5.7258 | 0.0000 | |||||

ω | −0.0943 | 1.607e^{−3} | −58.7069 | 0.0000 | |||||

α 1 | −0.0734 | 4.024e^{−3} | −18.2286 | 0.0000 | |||||

β 1 | 0.9859 | 1.7e^{−5} | 5863.5496 | 0.0000 | |||||

γ 1 | 0.0975 | 3.588e^{−3} | 27.1638 | 0.0000 | |||||

UNITY | ARIMA(0,1,1)- GARCH (1,1)-norm | μ | 1.38e^{−4} | 1.9e^{−5} | 7.1606 | 0.0000 | −4.8711 | −4.8601 | −4.8671 |

θ 1 | 0.1009 | 0.0236 | 4.2767 | 1.9e^{−5} | |||||

ω | 4.0e^{−6} | 0.0000 | 115.7800 | 0.0000 | |||||

α 1 | 0.2368 | 0.0117 | 20.2352 | 0.0000 | |||||

β 1 | 0.7622 | 7.927e^{−3} | 96.1509 | 0.0000 | |||||

ZENITH | ARIMA(2,1,1)-EGARCH(1,1)-std | μ | 0.0000 | 0.0000 | 1.2065 | 0.2276 | −7.0189 | −6.9992 | −7.0118 |

φ 1 | −0.3086 | 0.0106 | −29.0801 | 0.0000 | |||||

φ 2 | 0.0495 | 0.0171 | 2.8972 | 0.0038 | |||||

θ 1 | 0.2912 | 0.0105 | 27.7875 | 0.0000 | |||||

ω | −0.0291 | 8.62e^{−4} | −33.7888 | 0.0000 | |||||

α 1 | −0.6979 | 9.8e^{−5} | −7150.4179 | 0.0000 | |||||

β 1 | 0.9996 | 6.3e^{−5} | 15825.7924 | 0.0000 | |||||

γ 1 | 0.6983 | 9.8e^{−5} | 7147.6333 | 0.0000 |

of possible potential outliers in the returns series of the banks under study. The iterative procedure of Chen and Liu [

Bank | Model | Standardized Residuals | Standardized Squared Residuals | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Lag | Weighted LB | p-value | Lag | Weighted LB | p-value | Lag | Weighted ARCH ?LM | p-value | ||

SKYE | ARIMA(1,1,0)-GARCH(1,1)-norm | 1 | 0.0132 | 0.9085 | 1 | 0.0004 | 0.9833 | 3 | 0.0005 | 0.9829 |

2 | 0.015 | 1.0000 | 5 | 0.0014 | 1.000 | 5 | 0.0011 | 1.0000 | ||

5 | 0.0237 | 1.0000 | 9 | 0.0023 | 1.000 | 7 | 0.0016 | 1.0000 | ||

STERLING | ARIMA(2,1,0)-EGARCH(1,1)-norm | 1 | 0.0944 | 0.7587 | 1 | 0.0003 | 0.9851 | 3 | 0.9837 | |

5 | 0.1192 | 1.0000 | 5 | 0.0012 | 1.0000 | 5 | 0.0010 | 1.0000 | ||

9 | 0.1588 | 1.0000 | 9 | 0.0021 | 1.0000 | 7 | 0.0015 | 1.0000 | ||

UNITY | ARIMA(0,1,1)- GARCH (1,1)-norm | 1 | 0.0031 | 0.9557 | 1 | 0.0008 | 0.9776 | 3 | 0.0008 | 0.9776 |

2 | 0.0031 | 1.0000 | 5 | 0.0024 | 1.0000 | 5 | 0.0019 | 1.0000 | ||

5 | 0.0035 | 1.0000 | 9 | 0.0039 | 1.0000 | 7 | 0.0028 | 1.0000 | ||

ZENITH | ARIMA(2,1,1)-EGARCH(1,1)-std | 1 | 0.0014 | 0.9697 | 1 | 0.0014 | 0.9704 | 3 | 0.0014 | 0.9704 |

8 | 0.0066 | 1.0000 | 5 | 0.0041 | 1.0000 | 5 | 0.0033 | 0.9999 | ||

14 | 0.0111 | 1.0000 | 9 | 0.0069 | 1.0000 | 7 | 0.0049 | 1.0000 |

LB = Ljung-Box, LM = Lagrange Multiplier.

About twenty six (26) different outliers were identified to have contaminated the residuals series of ARIMA(1,1,0) model using the critical value, C = 4; six (6) innovation outliers (IO), six (6) additive outliers (AO) and fourteen (14) temporary change (TC) as indicated in (

About seven (7) different outliers were identified to have contaminated the residual series of ARIMA(2,1,0) model using the critical value, C = 5 one (1) innovation outlier (IO), four (4) additive outliers (AO) and two (2) temporary change (TC) as shown in (

About thirty three (33) different outliers were identified to have contaminated the residual series of ARIMA(1,1,0) model using the critical value, C = 5; two (2) innovation outliers (IO), six (6) additive outliers, fifteen (15) temporary change (TC) and ten (10) level shift (LS) as indicated in (

About forty two (42) different outliers were identified to have contaminated the residual series of ARIMA(2,1,1) model using the critical value, C = 5; thirteen (13) innovation outliers (IO), nine (9) additive outliers and twenty (20) temporary change (TC) as shown in (

Type | Observation index | Location | Estimate | T-statistic |
---|---|---|---|---|

IO | 211 | 13/11/2006 | −0.20150630 | −8.487698 |

IO | 1841 | 21/06/2013 | −0.10241849 | −4.313995 |

IO | 1843 | 25/06/2013 | 0.09870872 | 4.157735 |

IO | 2178 | 31/10/2014 | 0.10295915 | 4.336768 |

IO | 2263 | 05/05/2015 | 0.09758512 | 4.110407 |

IO | 210 | 10/11/2006 | 0.81215294 | 34.804236 |

AO | 1726 | 04/01/2013 | 0.09492679 | 4.068020 |

AO | 1984 | 21/01/2014 | −0.10371058 | −4.444443 |

AO | 2281 | 31/05/2015 | 0.09978000 | 4.276001 |

AO | 2414 | 15/10/2015 | 0.10169110 | 4.357900 |

AO | 2456 | 14/12/2015 | −0.09871728 | −4.230459 |

AO | 209 | 09/11/2006 | 0.20948475 | 10.861708 |

TC | 740 | 09/01/2009 | −0.09161349 | −4.750126 |

TC | 742 | 12/01/2009 | −0.07866550 | −4.078778 |

TC | 827 | 18/05/2009 | 0.07862559 | 4.076708 |

TC | 1723 | 13/12/2012 | 0.08946532 | 4.638744 |

TC | 2311 | 15/05/2015 | 0.09068887 | 4.702185 |

TC | 2381 | 26/08/2015 | 0.09747559 | 5.054074 |

TC | 2468 | 05/01/2016 | −0.10240036 | −5.309421 |

TC | 2590 | 29/06/2016 | −0.08395679 | −4.353129 |

TC | 2592 | 01/07/2016 | −0.12692535 | −6.581033 |

TC | 2599 | 15/07/2016 | 0.10544854 | 5.467469 |

TC | 2314 | 20/05/2015 | −0.10163346 | −4.363007 |

TC | 212 | 14/11/2006 | −0.10846069 | −5.731470 |

TC | 741 | 12/01/2009 | 0.07648229 | 4.225631 |

TC | 2589 | 28/06/2016 | 0.07256176 | 4.009023 |

Type | Observation index | Location | Estimate | T-statistic |
---|---|---|---|---|

AO | 184 | 03/10/2006 | 0.6913146 | 27.668748 |

AO | 655 | 02/09/2008 | −0.1764822 | −7.063413 |

AO | 2371 | 12/08/2015 | −0.1398721 | −5.598155 |

TC | 183 | 29/09/2006 | 0.2415834 | 11.950355 |

TC | 1672 | 22/05/2012 | −0.1075351 | −5.319415 |

IO | 185 | 04/10/2006 | −0.1964258 | −7.900298 |

AO | 2372 | 12/08/2015 | 0.1407112 | 5.678009 |

Type | Observation index | Location | Estimate | T-statistic |
---|---|---|---|---|

IO | 2293 | 20/04/2015 | −0.180979695 | −7.444781 |

AO | 248 | 10/01/2007 | 1.098612289 | 45.331893 |

AO | 1906 | 24/09/2013 | −0.200532990 | −8.274566 |

AO | 2292 | 17/04/2015 | 2.302585093 | 95.011264 |

TC | 247 | 09/01/2007 | 0.365004553 | 19.903211 |

TC | 1736 | 18/01/2013 | 0.107790532 | 5.877674 |

TC | 1745 | 01/02/2013 | 0.112419182 | 6.130068 |

TC | 1753 | 13/02/2013 | −0.118923561 | −6.484743 |

TC | 1762 | 26/02/2013 | 0.091895380 | 5.010932 |

TC | 2291 | 16/04/2015 | 0.758297010 | 41.348923 |

TC | 2298 | 27/04/2015 | −0.142415961 | −7.765752 |

TC | 2304 | 06/04/2015 | −0.098876918 | −5.391626 |

TC | 2446 | 30/11/2015 | −0.093629262 | −5.105479 |

TC | 2458 | 16/12/2015 | 0.112419118 | 6.130064 |

TC | 2460 | 18/12/2015 | 0.104980801 | 5.724463 |

TC | 2467 | 04/01/2016 | −0.106045605 | −5.782525 |

TC | 2469 | 06/01/2016 | −0.119002493 | −6.489047 |

IO | 1905 | 23/09/2013 | 0.127354627 | 5.132279 |

AO | 1904 | 20/09/2013 | −0.163097022 | −6.592937 |

LS | 243 | 29/12/2006 | −0.003141767 | −5.772181 |

LS | 251 | 15/01/2007 | −0.002771753 | −5.084049 |

LS | 347 | 11/06/2007 | −0.002837928 | −5.102009 |

LS | 520 | 19/06/2008 | −0.003114010 | −5.387808 |

LS | 598 | 13/06/2008 | −0.003027395 | −5.143002 |

LS | 613 | 04/07/2008 | −0.003035068 | −5.137530 |

LS | 631 | 30/07/2008 | −0.002988789 | −5.037234 |

LS | 635 | 05/08/2008 | −0.003001055 | −5.052994 |

LS | 2286 | 09/04/2015 | −0.008202488 | −6.130741 |

TC | 1901 | 17/09/2013 | 0.097493034 | 5.208012 |

TC | 2477 | 18/01/2016 | 0.096324642 | 5.145597 |

LS | 607 | 26/06/2008 | 0.022239276 | 28.623733 |

AO | 1336 | 09/06/2011 | −0.128187865 | −5.110079 |

AO | 1872 | 05/08/2013 | −0.144642972 | −5.766045 |

Type | Observation index | Location | Estimate | T-statistic |
---|---|---|---|---|

IO | 396 | 17/08/2007 | −0.09816377 | −13.221339 |

IO | 840 | 05/06/2009 | 0.04253167 | 5.728444 |

IO | 2221 | 06/01/2015 | −0.03927918 | −5.29037 |

IO | 2263 | 05/03/2015 | 0.04378397 | 5.897112 |

IO | 2281 | 18/03/2013 | 0.03787680 | 5.101495 |

IO | 2473 | 12/01/2016 | −0.03936074 | −5.301362 |

IO | 2475 | 14/01/2016 | −0.04230834 | −5.698364 |

IO | 2525 | 24/03/2016 | −0.06416968 | −8.642792 |

IO | 2565 | 24/05/2016 | 0.04357221 | 5.868590 |

IO | 2568 | 27/05/2016 | −0.03892676 | −5.242911 |

AO | 839 | 04/06/2009 | −0.17023181 | −23.610345 |

AO | 1051 | 13/04/2010 | −0.10685231 | −14.819909 |

AO | 1971 | 31/12/2013 | −0.04930715 | −6.838668 |

AO | 2027 | 21/03/2014 | −0.04348949 | −6.031786 |

AO | 2223 | 08/01/2015 | 0.04365213 | 6.054343 |

AO | 2389 | 07/09/2015 | 0.03931499 | 5.452803 |

AO | 2453 | 09/12/2015 | 0.04052320 | 5.620376 |

AO | 2483 | 26/01/2016 | −0.03680867 | −5.105188 |

TC | 395 | 16/08/2007 | −0.04944665 | −8.279793 |

TC | 691 | 27/10/2008 | −0.03018987 | −5.055264 |

TC | 710 | 21/11/2008 | −0.03064893 | −5.132133 |

TC | 747 | 20/01/2009 | −0.03068035 | −5.137395 |

TC | 802 | 08/04/2009 | 0.02900192 | 5.029855 |

TC | 838 | 03/06/2009 | −0.05601975 | −9.380451 |

TC | 2477 | 18/01/2016 | 0.03332783 | 5.580712 |

IO | 1970 | 30/12/2013 | 0.03696756 | 5.089711 |

IO | 2269 | 13/03/2015 | −0.03667538 | −5.049484 |

TC | 698 | 05/11/2008 | 0.03138467 | 5.372151 |

TC | 754 | 29/01/2009 | 0.03144461 | 5.382412 |

TC | 802 | 08/04/2009 | 0.02900192 | 5.029855 |

IO | 2569 | 31/05/2016 | −0.04117313 | −5.684147 |

TC | 394 | 15/08/2007 | 0.03079012 | 5.941387 |

TC | 833 | 26/05/2009 | 0.02933330 | 5.660273 |

TC | 850 | 19/06/2009 | −0.02610763 | −5.037836 |

TC | 2212 | 18/12/2014 | 0.02755604 | 5.317326 |

AO | 2282 | 25/01/2016 | 0.03601640 | 5.029504 |
---|---|---|---|---|

TC | 2484 | 27/01/2016 | 0.02586343 | 5.048003 |

TC | 824 | 13/05/2009 | 0.02437288 | 5.000302 |

TC | 890 | 14/08/2009 | −0.02533852 | −5.198411 |

TC | 2217 | 29/12/2014 | −0.02467081 | −5.061424 |

TC | 2450 | 04/12/2015 | −0.02471591 | −5.070677 |

TC | 919 | 28/09/2009 | 0.02369936 | 5.067229 |

However, with the identified outliers being adjusted for, we obtained a new series (outlier adjusted return series). Again, Box and Jenkins iterative procedures were applied and those models in (

The ARIMA(1, 1, 0)-GARCH(1, 1)-norm model fitted to both outlier adjusted return series and outlier contaminated series with the same unconditional variance, 0.0016. Again, the value of kurtosis, 2.9465 captured by GARCH(1, 1)-norm model on outlier adjusted return series is nearly the value accommodated by the normal distribution while GARCH(1, 1)-norm model on outlier contaminated series seems inferior with excess kurtosis of 132.8707 (see

From (

From (

The ARIMA(2, 1, 1)-EGARCH(1, 1)-std model fitted to both outlier adjusted return series and outlier contaminated series converges to respective unconditional variances of 5.118125e^{−5} and 5.887684e^{−36} with corresponding kurtosis values of 3.5746 and 26.3794. Though, the unconditional variance of ARIMA(2, 1, 1)-EGARCH(1, 1)-std model fitted to the outlier contaminated series is smaller than that of the outlier adjusted return series, the model for the outlier adjusted return series is more efficient given that the kurtosis value is near three, the value occupied by normal distribution (

Bank | Model | Parameter | Estimate | s.e | t-ratio | p-value | Information Criteria | ||
---|---|---|---|---|---|---|---|---|---|

AIC | BIC | HQIC | |||||||

SKYE | ARIMA(1,1,0)-GARCH(1,1)-norm | μ | −1.36e^{−4} | 5.0e^{−6} | −25.9283 | 0.0000 | −4.6518 | −4.6409 | −4.6479 |

φ 1 | −0.0466 | 0.0211 | −2.2101 | 0.0271 | |||||

ω | 2.0e^{−6} | 0.0000 | 67.0558 | 0.0000 | |||||

α 1 | 0.1228 | 7.675e^{−3} | 15.9933 | 0.0000 | |||||

β 1 | 0.8762 | 6.665e^{−3} | 131.4656 | 0.0000 | |||||

STERLING | ARIMA (2,1,2)-EGARCH(1,1)-std | μ | 0.0000 | 0.0000 | −1.51 | 0.1310 | −5.7621 | −5.7401 | −5.7541 |

φ 1 | 6.085e^{−3} | 1.0e^{−6} | 5673.10 | 0.0000 | |||||

φ 2 | 0.9386 | 1.58e^{−4} | 5935.97 | 0.0000 | |||||

θ 1 | 0.2025 | 3.2e^{−5} | 6283.61 | 0.0000 | |||||

θ 2 | −0.7371 | 1.2e^{−4} | −6163.17 | 0.0000 | |||||

ω | −0.0929 | 2.74e^{−4} | −338.38 | 0.0000 | |||||

α 1 | 0.5597 | 6.7e^{−5} | 8413.54 | 0.0000 | |||||

β 1 | 0.9925 | 1.42e^{−4} | 6970.63 | 0.0000 | |||||

γ 1 | 0.5599 | 6.5e^{−5} | 8667.79 | 0.0000 | |||||

UNITY | ARIMA(1,1,0) -GJR-GARCH(1,0)-norm | μ | −2.284e^{−3} | 1.0e^{−5} | −220.0970 | 0.0000 | −4.2296 | −4.2406 | −4.2336 |

φ 1 | 0.0559 | 2.66e^{−4} | 209.9780 | 0.0000 | |||||

ω | 1.0e^{−6} | 0.0000 | 3.4940 | 0.0005 | |||||

α 1 | 0.8566 | 3.888e^{−3} | 220.3020 | 0.0000 | |||||

γ 1 | 0.2175 | 0.0114 | 19.0470 | 0.0000 | |||||

ZENITH | ARIMA(2,1,1)-EGARCH(1,1)-std | μ | 0.0000 | 0.0000 | 0.3698 | 0.7116 | −7.0635 | −7.0438 | −7.0564 |

φ 1 | 0.6094 | 0.0092 | 66.5553 | 0.0000 | |||||

φ 2 | 0.0852 | 0.0045 | 18.7897 | 0.0000 | |||||

θ 1 | −0.6012 | 0.0038 | −159.8213 | 0.0000 | |||||

ω | −0.0960 | 2.3e^{−4} | −418.1551 | 0.0000 | |||||

α 1 | −0.8581 | 1.05e^{−4} | −8158.1158 | 0.0000 | |||||

β 1 | 0.9903 | 8.6e^{−5} | 11523.0226 | 0.0000 | |||||

γ 1 | 0.8591 | 1.05e^{−4} | 8183.2788 | 0.0000 |

Our study has shown that the use of minimum unconditional variance as a measure of efficiency of heteroscedastic models in the presence of outliers is not heuristic as outliers are capable of inflating or reducing the unconditional variance. To this end, the use of kurtosis coefficient as a measure of heteroscedastic

Bank | Model | Standardized Residuals | Standardized Squared Residuals | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Lag | Weighted LB | p-value | Lag | Weighted LB | p-value | Lag | Weighted ARCH-LM | p-value | ||

SKYE | ARIMA(1,1,0) -GARCH(1,1)-norm | 1 | 1.323 | 0.2500 | 1 | 2.227 | 0.1356 | 3 | 1.113 | 0.2914 |

2 | 2.710 | 0.0617 | 5 | 3.072 | 0.3941 | 5 | 1.375 | 0.6257 | ||

5 | 4.284 | 0.1838 | 9 | 3.533 | 0.6687 | 7 | 1.472 | 0.8269 | ||

STERLING | ARIMA(2,1,2) -EGARCH(1,1)-std | 1 | 0.0014 | 0.9707 | 1 | 0.0013 | 0.9716 | 3 | 0.0013 | 0.9716 |

11 | 0.0087 | 1.0000 | 5 | 0.0038 | 1.0000 | 5 | 0.0030 | 0.9999 | ||

19 | 0.0145 | 1.0000 | 9 | 0.0063 | 1.0000 | 7 | 0.0045 | 1.0000 | ||

UNITY | ARIMA(1,1,0) -GJR-GARCH(1,0) -norm | 1 | 0.0081 | 0.9283 | 1 | 0.3091 | 0.5782 | 2 | 0.0990 | 0.7530 |

2 | 0.0158 | 1.0000 | 2 | 0.3587 | 0.7639 | 4 | 2.5450 | 0.3332 | ||

5 | 4.3656 | 0.1730 | 5 | 2.3489 | 0.5386 | 6 | 2.8354 | 0.5386 | ||

ZENITH | ARIMA(2,1,1) -EGARCH(1,1)-std | 1 | 0.0010 | 0.9752 | 1 | 0.0009 | 0.9756 | 3 | 0.0009 | 0.9756 |

8 | 0.0058 | 1.0000 | 5 | 0.0028 | 1.0000 | 5 | 0.0022 | 0.9999 | ||

14 | 0.0100 | 1.0000 | 9 | 0.0047 | 1.0000 | 7 | 0.0033 | 1.0000 |

LB = Ljung-Box, LM = Lagrange Multiplier.

Efficiency Measurement Criteria | ARIMA(1,1,0)-GARCH(1,1)?norm Model fitted to Returns Series of Skye Bank | ARIMA(1,1,0)-GARCH(1,1)-norm Model fitted to Outlier Adjusted Return Series of Skye Bank |
---|---|---|

Unconditional Variance | 0.0016 | 0.0016 |

Kurtosis Coefficient | 132.8707 | 2.9465 |

Efficiency Measurement Criteria | ARIMA(2,1,0)-EGARCH(1,1)-norm Model fitted to Returns Series of Sterling Bank | ARIMA(2,1,2)-EGARCH(1,1)-std Model fitted to Outlier Adjusted Return Series of Sterling Bank |
---|---|---|

Unconditional Variance | 0.0012288 | 4.234715e^{−6} |

Kurtosis Coefficient | 80.0303 | 3.6829 |

Efficiency Measurement Criteria | ARIMA(0,1,1)-GARCH (1,1)-norm Model fitted to Returns Series of Unity Bank | ARIMA(1,1,0) -GJR-GARCH(1,0)-norm Model fitted to Outlier Adjusted Return Series of Unity Bank |
---|---|---|

Unconditional Variance | 0.0037 | 2.104825e^{−5} |

Kurtosis Coefficient | 888.5032 | 3.2678 |

Efficiency Measurement Criteria | ARIMA(2,1,1)-EGARCH(1,1)-std Model fitted to Returns Series of Zenith Bank | ARIMA(2,1,1)-EGARCH(1,1)-std Model fitted to Outlier Adjusted Return Series of Zenith Bank |
---|---|---|

Unconditional Variance | 5.887684e^{−36} | 5.118125e^{−5} |

Kurtosis Coefficient | 26.3794 | 3.5746 |

models in the presence of outliers becomes more tractable irrespective of the choice of distribution of the innovations. In addition, this study highlights that the applicability of iterative methods in time series modeling and in gauging model efficiency yet failed to consider the application of iterative methods in forecasting. It is recommended that further studies be extended to focus mainly on the application of iterative methods in maximum likelihood estimation of GARCH parameters.

The authors declare no conflicts of interest regarding the publication of this paper.

Akpan, E.A., Lasisi, K.E., Adamu, A. and Rann, H.B. (2019) Application of Iterative Approaches in Modeling the Efficiency of ARIMA-GARCH Processes in the Presence of Outliers. Applied Mathematics, 10, 138-158. https://doi.org/10.4236/am.2019.103012