International Journal of Analysis and Applications 

Volume 17, Number 4 (2019), 530-547 

URL: https://doi.org/10.28924/2291-8639 

DOI: 10.28924/2291-8639-17-2019-530 

  

 

Received 2019-05-01; accepted 2019-06-10; published 2019-07-01. 

2010 Mathematics Subject Classification. 82B31. 

Key words and phrases. ARIMA; outliers; return series; time series. 

©2019 Authors retain the copyrights 

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 

530 

 

MODELING THE EFFECTS OF OUTLIERS ON THE ESTIMATION OF LINEAR 

STOCHASTIC TIME SERIES MODEL 

 

EMMANUEL ALPHONSUS AKPAN1,* AND IMOH UDO MOFFAT2 

 

1Department of Mathematical Science, Abubakar Tafawa Balewa University, Bauchi, Nigeria 

2Department of Mathematics and Statistics, University of Uyo, Nigeria 

*Corresponding author: eubong44@gmail.com 

 

ABSTRACT. This study investigates the effects of outliers on the estimates of ARIMA model 

parameters with particular attention given to the performance of two outlier detection and 

modeling methods targeted at achieving more accurate estimates of the parameters. The two 

methods considered are: an iterative outlier detection aimed at obtaining the joint estimates of 

model parameters and outlier effects, and an iterative outlier detection with the effects of outliers 

removed to obtain an outlier free series, after which a successful ARIMA model is entertained. We 

explored the daily closing share price returns of Fidelity bank, Union bank of Nigeria, and Unity 

bank from 03/01/2006 to 24/11/2016, with each series consisting of 2690 observations from the 

Nigerian Stock Exchange. ARIMA (1, 1, 0) models were selected based on the minimum values of 

Akaike information criteria which fitted well to the outlier contaminated series of the respective 

banks. Our findings revealed that ARIMA (1, 1, 0) models which fitted adequately to the outlier 

free series outperformed those of the parameter-outlier effects joint- estimated model. Furthermore, 

we discovered that outliers biased the estimates of the model parameters by reducing the estimated 

values of the parameters. The implication is that, in order to achieve more accurate estimates of 

ARIMA model parameters, it is needful to account for the presence of significant outliers and 

preference should be given to the approach of cleaning the series of outliers before subsequent 

entertainment of adequate linear time series models.   

 

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-530


Int. J. Anal. Appl. 17 (4) (2019) 531 

 

1. INTRODUCTION 

 Outliers are common characterizations of every time series. In general, outliers are 

extreme observations that deviate from the overall pattern of the sample. Statistically, outliers 

are those observations whose standard deviations are greater than 3 in absolute value, which is 

the value of kurtosis occupied by the normal distribution. However, the effects of outliers on 

the linear time series models cannot be overemphasized; such effects range from false inference, 

introduction of biases in the model parameters, model misspecification and misleading 

confidence interval ([1], [2], [3], [4]).  

 By efficiency, we mean the goodness of an estimator of a model which can be measured 

by variance, that is, a model with the smallest variance is considered to be superior as regarding 

efficiency. To reiterate the need for efficiency of the estimates of model parameters by 

considering the presence of outliers, this study applied two outlier identification and modeling 

methods. The first is the modified iterative method proposed by [5], which involves the joint 

estimation of the model parameters and the magnitude of outlier effects. The second is the 

modified iterative method proposed by [6], which involves identification of outliers 

sequentially by searching for most relevant anomaly, estimating its effect and removing it from 

the data. The estimation of the model parameters is again done on the outlier corrected series, 

and further iteration of the process is carried out until no significant perturbation is found. 

 Actually, the motivation for this study is derived from the fact that previous studies 

such as [7], [8], [9], [10] failed to consider outliers while modeling returns series in Nigeria. 

Thus, this gap in knowledge is fully addressed in our work. 

 This work is further organized as follows: section 2 takes care of materials and methods; 

section 3 handles the results and discussion while section 4 treats the conclusion. 

2. MATERIALS AND METHODS 

2.1 Return Series 

The returns series (𝑅𝑡) can be obtained given that 𝑃𝑡 is the price of a unit shares at time t 

and 𝑃𝑡−1  is the price of shares at time t−1. Thus 

𝑅𝑡 = ∇𝑙𝑛𝑃𝑡  = (1 − 𝐵)𝑙𝑛𝑃𝑡   = 𝑙𝑛 𝑃𝑡  − 𝑙𝑛 𝑃𝑡−1                                                      (1) 

In equation (1), 𝑅𝑡 is regarded as a transformed series of the price (𝑃𝑡) of shares meant to attain 

stationarity such that both the mean and the variance of the series are stable [11] while 𝐵 is the 

backshift operator. 

 



Int. J. Anal. Appl. 17 (4) (2019) 532 

 

2.2 Autoregressive Integrated Moving Average (ARIMA) Model 

[3] considered the extension of ARMA model to deal with homogenous non-stationary 

time series in which 𝑋𝑡 , is non-stationary but its 𝑑
𝑡ℎ difference is a stationary ARMA model. 

Denoting the 𝑑𝑡ℎ difference of 𝑋𝑡 by   

𝜑(𝐵) = 𝜙(𝐵)∇𝑑 𝑋𝑡 = 𝜃(𝐵) 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 while 𝜙(𝐵) is a stationary autoregressive 

operator. It should be noted that in equation (2), the presence of outliers is not taken into 

consideration. 

2.3 Joint Model of ARIMA and Outlier-effects 

𝑅𝑡 =  ∑ φjRt−j
p
j=1 +  ∑ θi

𝑞
𝑖=1 𝑎t−i + 𝑎𝑡 + ∑ 𝜔𝑗

𝑘
𝑗=1 𝑉𝑗(B)𝐼𝑡

(𝑇)
,                                        (3) 

where  𝑉𝑗(B) = 1 for an AO, and 𝑉𝑗(B) =   
𝜃(𝐵)

𝜑(𝐵)
 for an IO at t = 𝑇𝑗, 𝑉𝑗(B) = (1 –  𝐵)

−1 for a LS,   𝑉𝑗(B) 

= (1 – 𝛿 𝐵)−1 for an TC, and 𝜔 is the size of the outlier. For more details on the types of outliers 

and estimation of their effects, see [1], [12], [3], [4], [5], [13].  

2.4 ARIMA Model for Outlier-Adjusted Return Series 

𝑅𝑡  −  ∑ φjRt−j
p
j=1 −  ∑ θi

𝑞
𝑖=1 𝑎t−i − ∑ 𝜔𝑗

𝑘
𝑗=1 𝑉𝑗(B)𝐼𝑡

(𝑇)
=  𝑎𝑡,                                              (4) 

where 𝑎𝑡  is the outlier free series. Meanwhile, equations (3) and (4) represent major 

modifications on equation (2) to account for the presence of outliers. 

2.5      Outliers in Time Series 

Generally, in time series, four types of outliers are identified and they are as follows: 

additive outlier, innovative outlier, level shift outlier and temporary outlier [12]. 

2.5.1 Additive Outlier (AO)  

 A time series  𝑌1, …, 𝑌𝑇  affected by the presence of an additive outlier at t = T is given by    

𝑌𝑡 =   {
𝑋𝑡  ,             𝑡 ≠ 𝑇   
𝑋𝑡 +  𝜔,   𝑡 = 𝑇 

= 𝑋𝑡 +  𝜔𝐼𝑡
(𝑇)

   =   
𝜃(𝐵)

𝜑(𝐵)
𝑎𝑡  +  𝜔𝐼𝑡

(𝑇)
                                            (5) 

for t = 1, …,T, where 𝐼𝑡
(𝑇)

= {
1 ,             𝑡 = 𝑇,
0,              𝑡 ≠ 𝑇,

  is the indicator variable representing the presence 

or absence of an outlier at time T, 𝑋𝑡 follows an ARIMA model, 𝜔  is an outlier size. Hence, an 

additive outlier affects only a single observation (see also [1], [12], [3], [4]). 

2.5.2 Innovative Outlier (IO)  

 A time series𝑌1, …, 𝑌𝑇  affected by the presence of an innovative outlier at t = T is given 

by 



Int. J. Anal. Appl. 17 (4) (2019) 533 

 

𝑌𝑡 = 𝑋𝑡 +  
𝜃(𝐵)

𝜑(𝐵)
𝜔𝐼𝑡

(𝑇)
=   

𝜃(𝐵)

𝜑(𝐵)
(𝑎𝑡  +  𝜔𝐼𝑡

(𝑇)
)                                                                        (6) 

hence, an innovative outlier affects all observations 𝑌𝑡 , 𝑌𝑡+1 ,…, beyond time T through the 

memory of the system described by 𝜓(B) = 
𝜃(𝐵)

𝜑(𝐵)
, such that 𝑌𝑡 = 𝑋𝑡 +  ψ(B)𝜔𝐼𝑡

(𝑇)
. 

Meanwhile, according to [12], the innovation of a time series  𝑌1, …, 𝑌𝑇  is affected by 

𝑌𝑡 = 𝑒𝑡 +  𝜔𝐼𝑡
(𝑇)

                                                                                                                    (5) 

where 𝑒𝑡are the innovations of the uncontaminated series  𝑋𝑡. 

2.6.3 Level Shift (LS)  

A time series  𝑌1, …, 𝑌𝑇  affected by the presence of a level shift at t = T is given by 

𝑌𝑡 = 𝑋𝑡 +  𝜔𝑆𝑡
(𝑇)

                                                                                                                   (6) 

where 𝑆𝑡
(𝑇)

= (1 − 𝐵)−1𝐼𝑡
(𝑇)

. Note that level shift affects all the observation of the series after t = 

T. Hence, according to [12], level shift serially affects the innovations as follows: 

𝑎𝑡 = 𝑒𝑡 +  π(B)𝜔𝑆𝑡
(𝑇)

                                                                                                            (7) 

where 𝜋(𝐵)   =   (1 −  𝜋1𝐵 −  𝜋2𝐵
2  − ⋯ ) 

2.7.4 Temporary Change (TC) 

A time series 𝑌1, …, 𝑌𝑇  affected by the presence of a temporary change at t = T is given 

by 

𝑌𝑡 = 𝑋𝑡 +
1

1−𝛿𝐵
 𝜔𝐼𝑡

(𝑇)
                                                                                                             (8) 

where 𝛿 is an exponential decay parameter such that  0 < 𝛿 < 1. If 𝛿 tends to 0, the temporary 

change reduces to an additive outlier, whereas if 𝛿 tends to 1, the temporary change reduces to 

a level shift. The temporary change affects the innovations as follows: 

𝑎𝑡 = 𝑒𝑡 +
𝜋(𝐵)

1−𝛿𝐵
 𝜔𝐼𝑡

(𝑇)
                                                                                                            (9) 

If 𝜋(𝐵) is close to 1 − 𝛿𝐵, the effect of temporary change on the innovations is very close to the 

effect of an innovative outlier. Otherwise, the temporary change can affect several observations 

with a decreasing effect after t = T [12].  

3. RESULTS AND DISCUSSION 

3.1  Time Plots 

 Inspecting the plots in Figures 1-3, it is obvious that they are characterized by upward 

and downward movements away from the common mean, which clearly indicates the existence 

of nonstationarity.   

 



Int. J. Anal. Appl. 17 (4) (2019) 534 

 

 

Figure 1: Price Series of Fidelity Bank shares  

 

 

Figure 2: Price Series of Union Bank shares  

 

 

Figure 3: Price Series of Unity Bank shares 

Also, the plots in Figures 4 - 6 indicate that the returns series cluster around the mean 

which implies that the series are stationary.  

 

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Int. J. Anal. Appl. 17 (4) (2019) 535 

 

 

Figure 4: Return Series of Fidelity Bank 

 

 

 

Figure 5: Return Series of Union Bank 

 

Figure 6: Return Series of Unity Bank 

 

 

 

 

-0.3

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Int. J. Anal. Appl. 17 (4) (2019) 536 

 

3.2 Linear Time Series Modeling Return Series of Fidelity Bank  

      From Figures 7 and 8, both ACF and PACF indicate that mixed model is possible. The 

following models, ARIMA(1, 1, 0), ARIMA(0, 1, 1), ARIMA(1, 1, 1), ARIMA(1, 1, 2) and ARIMA 

(2, 1, 1) are entertained tentatively. 

 

Figure 7: ACF of Return Series of Fidelity Bank  

 

Figure 8: PACF of Return Series of Fidelity Bank  

From Table I, ARIMA(1, 1, 1) model has the smallest AIC  but one of  its parameters is 

not significant. Secondly, ARIMA(1, 1, 2) model has the second smallest AIC  yet its parameters 

are not significant. Hence, ARIMA(1, 1, 0) model is selected based on the ground that its 

parameter is significant and has the nearest minimum AIC. 

 

Table I: ARIMA Models for Return Series of Fidelity Bank  

Model 

Parameter Akaike 

Information 

Criteria (AIC) Log likelihood 𝝋𝟏 𝝋𝟐 𝛉𝟏 𝛉𝟐 

ARIMA(1,1,0) 0.1606∗∗∗    −11562.17 5783.09 

ARIMA(0,1,1)   0.1494∗∗∗  −11559.28 5780.64 

ARIMA(1,1,1) 0.2569∗∗∗  − 0.0986  −11563.16 5783.58 

ARIMA(1,1,2) −0.0498  0.2071 0.0628 −11562.88 5784.44 

ARIMA(2,1,1) −0.0721 0.0619 0.2288  −11561.98 5783.99 

                                                                                                         *** significance at 5% level 



Int. J. Anal. Appl. 17 (4) (2019) 537 

 

Furthermore, evidence from Ljung-Box Q-statistics shows that  ARIMA(1, 1, 0) model is 

adequate at 5% level of significance given the Q-statistics at lags 1, 4, 8, and 24 given by Q(1) = 

0.0376, Q(4) = 5.4261, Q(8) = 9.8001, and Q(24) = 23.379 with the respective p-values of 0.8462, 

0.2463, 0.2793, and 0.4975. 

3.3 Identification of Outliers in the Residual Series of ARIMA(1, 1, 0) Model fitted to the 

Return Series of Fidelity Bank  

Considering the critical value, C = 4, and based on the condition that n ≥ 450,  we identified 

sixteen (16) different outliers that have contaminated the residual series of ARIMA(1, 1, 0) 

model, as indicated in Table II. They are: two (2) innovation outliers (IO), five (5) additive 

outliers, and nine (9) temporary change outliers.  

 

Table II: Types of Outliers Identified in the Residual Series of ARIMA(1, 1, 0) Model  

                 fitted to the Return Series of Fidelity Bank 

Type Observation index Location Estimate T-statistic 

IO 1555 26/04/2012 -0.09798041 -4.198390 

AO 1789 08/04/2013 -0.10865950 -4.715609 

AO 1841 21/06/2013 -0.10673597 -4.632131 

AO 2539 15/04/2016 -0.17301613 -7.508560 

AO 2042 11/04/2014 -0.30477209 -13.226510 

TC 827 18/05/2009 0.07540548 4.049288 

TC 847 16/06/2009 -0.07692527 -4.130901 

TC 859 02/07/2009 -0.07537282 -4.047534 

TC 1665 04/10/2012 0.08360953 4.489847 

TC 1724 01/02/2013 0.07510564 4.033187 

TC 2263 05/03/2015 0.07816849 4.197662 

TC 2280 30/03/2015 0.09555288 5.131207 

IO 2292 17/04/2015 -0.09220965 -4.046644 

AO 2043 14/04/2014 0.24193892 10.598998 

TC 691 27/10/2008 -0.06641433 -4.025161 

TC 950 11/11/2009 0.06557061 4.004060 

 

                                                                     



Int. J. Anal. Appl. 17 (4) (2019) 538 

 

To account for the effect of outliers, the method of joint estimation of the parameter of 

ARIMA (1, 1, 0) model with outliers identified in Table II is performed as indicated in Table III. 

Comparing the values of AIC = −11922.67 and log likelihood = 5979.34 of the joint model of 

ARIMA(1, 1, 0) with outliers effects with that of ARIMA (1, 1, 0) model having AIC = −11562.17 

and log likelihood = 5783.09, it is obvious that the joint model  of ARIMA (1, 1, 0) with outliers 

effects  has a lower AIC and a higher log likelihood value, thus making it a better model than 

the ARIMA (1, 1, 0) model where the influence of outliers is not taken into consideration. 

 

Table III: Joint Model of ARIMA (1, 1, 0) and Outlier-effects fitted to Return Series 

                  Fidelity Bank  

 Estimate Std. Error z value Pr(>|z|) 

ar1           0.171530 0.019142 8.9607 < 2.2e−16 ∗∗∗ 

IO1555      -0.084464 0.024209 -3.4890 0.0004849∗∗∗ 

AO1789      -0.109211 0.025841 -4.2262 2.376e−05 ∗∗∗ 

AO1841      -0.107286 0.025841 -4.1518 3.299e−05 ∗∗∗ 

AO2042    -0.273962 0.026198 -10.4573 < 2.2e−16 ∗∗∗ 

AO2539     -0.173178 0.025830 -6.7045 2.022e−11 ∗∗∗ 

TC827        0.075179 0.021072 3.5677 0.0003601∗∗∗ 

TC847       -0.076153 0.021068 -3.6147 0.0003007 ∗∗∗ 

TC859        -0.074623 0.021069 -3.5418 0.0003974 ∗∗∗ 

TC1665         0.083147 0.021082 3.9439 8.016e−05 ∗∗∗ 

TC1724        0.074547 0.021090 3.5348 0.0004081∗∗∗ 

TC2263         0.078614 0.021097 3.7264 0.0001943∗∗∗ 

TC2280         0.095246 0.021087 4.5168 6.277e-06∗∗∗ 

IO2292      -0.071450 0.024232 -2.9486 0.0031921∗∗ 

AO2043       0.197831 0.026196 7.5520 4.286e-14∗∗∗ 

TC691       -0.070928 0.021078 -3.3651 0.0007653∗∗∗ 

TC950          0.071171 0.021068 3.3782 0.0007295∗∗∗ 

 

3.4 Building ARIMA(1, 1, 0) Model for Outlier-Adjusted Return Series of Fidelity Bank 

            Here, the second method is applied which is the removal of the outliers effects to obtain 

an outlier-adjusted series. Then, ARIMA(1, 1, 0) model  fitted well to the outlier-adjusted series 

with its parameter significant at 5% level [see Table IV] and is found to be adequate given the 



Int. J. Anal. Appl. 17 (4) (2019) 539 

 

Q-statistics at lags 1, 4, 8, and 24 given by Q(1) = 0.0003, Q(4) = 4.2007, Q(8) = 13.92, and Q(24) = 

29.649 with the corresponding p-values of 0.99, 0.38, 0.09, and 0.20. 

 

Table IV: ARIMA (1, 1,  0) Model for Outlier-Adjusted Return Series of Fidelity 

                 Bank 

Model Parameter (𝝋) 

Akaike Information 

Criteria Log likelihood 

ARIMA (1, 1, 0)  0.1715∗∗∗ −11954.67 5979.34 

                                                                                        *** significance at 5% level 

ARIMA (1, 1, 0) model with the least AIC = −11954.67 appears to be better than that of the joint 

model of ARIMA (1, 1, 0) with outliers effects. 

 On comparing the estimates of ARIMA(1, 1, 0) model fitted to the outlier contaminated 

series with the ARIMA(1, 1, 0) model when adjusted for outliers using the two proposed 

methods, it is found that the estimates of both the joint ARIMA(1, 1, 0) model with outliers 

effects and the ARIMA(1, 1, 0) model fitted to the outlier adjusted series are the same. However, 

the later tends to outperform the former on the basis of smallest information criteria. Of 

paramount interest is the discovery that outliers introduced substantial bias in the estimate of 

ARIMA (1, 1, 0) model by 0.0109 as shown in Table V.  Again, the modified iterative method 

produced a model with smallest variance as indicated in Table V, hence, adjudged the most 

efficient method.  

 

Table V: Effect of Outliers on Estimate of ARIMA(1, 1, 0) Model for Return Series 

                of Fidelity Bank  

Model 

ARIMA (1,1,0) 

(For outlier- 

contaminated) 

Joint ARIMA 

(1,1,0) with 

Outliers Effects 

ARIMA (1,1,0) 

(For outlier- 

adjusted) 

Bias 

Introduced 

Parameter (𝝋𝟏) 0.1606 0.1715 0.1715 −0.0109 

AIC −11562.18 -11922.67    −11954.67 

 

 

 

Standard error 0.0190 0.0191 0.0189 

Variance 0.000795 0.000691 0.000687 

Log likelihood 5783.09 5979.34 5979.34 

 

 



Int. J. Anal. Appl. 17 (4) (2019) 540 

 

3.5 Linear Time Series Modeling of Return Series of Union Bank  

From Figures 9 and 10, both ACF and PACF indicate that the following mixed model could 

be entertained tentatively: ARIMA(1, 1, 0), ARIMA(0, 1, 1) and ARIMA(1, 1, 1). 

 

Figure 9: ACF of Return Series of Union Bank 

 

Figure 10: PACF of Return Series of Union Bank 

From Table VI, ARIMA (1, 1, 0) model is selected based on the ground that its parameter 

is significant and has the minimum AIC. 

Table VI: ARIMA Models for Return Series of Union Bank 

Model 

Parameter Akaike Information 

Criteria (AIC)  Log likelihood 𝝋𝟏 𝛉𝟏 

ARIMA (1,1,0) 0.1014∗∗∗  −9132.26 4567.13 

ARIMA (0,1,1)  0.0963∗∗∗ −9130.87 4566.43 

ARIMA (1,1,1) 0.2455 − 0.1453 −9131.12 4567.56 

                                                                                                        *** significance at 5% level      

Furthermore, evidence from Ljung-Box Q-statistics shows that ARIMA(1, 1, 0) model is 

adequate at 5% level of significance given the Q-statistics at lags 1, 4, 8 and 24 given by  Q(1) = 



Int. J. Anal. Appl. 17 (4) (2019) 541 

 

0.0133, Q(4) = 2.3753, Q(8) = 4.318 and Q(24) = 7.9309 with the corresponding p-values of 0.9082, 

0.6671, 0.8274, and 0.9991. 

3.6 Identification of Outliers in the Residual Series of ARIMA(1, 1, 0) Model fitted to the 

Return Series of Union Bank 

Here, we consider the critical value, C = 5 given that C = 4 was not sufficient for computing 

weights of outliers and about nineteen (19) different outliers are identified to have 

contaminated the residual series of ARIMA(1, 1, 0) model, four (4) innovation outliers (IO), 

eight (8) additive outliers and seven (7) temporary change outliers, as shown in Table VII.  

Table VII: Types of Outliers identified in the Residual Series of ARIMA(1, 1, 0) Model      

                    fitted to the Return Series of Union Bank 

Type Observation index Location Estimate T-statistic 

IO 458 16/11/2007 -0.20259320 -9.867965 

IO 1472 23/12/2011 -0.22031597 -10.731210 

IO 1831 07/06/2013 0.10533493 5.130683 

IO 1843 25/06/2013 0.10590627 5.158511 

AO 150 15/08/2006 -0.13856541 -6.783874 

AO 705 14/11/2008 -0.20086454 -9.833910 

AO 1471 22/12/2011 1.67935140 82.217553 

AO 1830 06/06/2013 -0.11483241 -5.621956 

AO 1842 24/06/2013 -0.10581300 -5.180384 

AO 1984 21/01/2014 -0.10648119 -5.213098 

AO 1994 04/02/2014 0.16239480 7.950512 

TC 691 27/10/2008 -0.08071046 -5.129738 

TC 901 31/08/2009 -0.08274861 -5.259278 

TC 1470 22/12/2011 0.53378545 33.925958 

TC 1523 09/03/2012 -0.08218825 -5.223663 

TC 1541 04/04/2012 0.07869209 5.001456 

TC 1824 28/05/2013 0.11353246 7.215815 

TC 2534 08/04/2016 -0.08059290 -5.122266 

AO 1748 06/02/2013 -0.11923771 -5.160464 

 

  



Int. J. Anal. Appl. 17 (4) (2019) 542 

 

Again, applying the first method as indicated in Table VIII, it is found that the values of 

AIC = −11560.27 and log likelihood = 5800.13 for the joint model of ARIMA(1, 1, 0) with outliers 

effects when compared to that of ARIMA (1, 1, 0) model with  AIC = −9132.26 and log 

likelihood = 4567.13 are respectively smaller and higher, making the former a better  model than 

the later. 

Table VIII: Joint Model of ARIMA (1, 1, 0) and Outliers Effects fitted to Return Series of 

Union Bank 

                      Estimate Std. Error z value Pr(>|z|) 

ar1           0.265411 0.018828 14.0965 < 2.2e−16 ∗∗∗ 

IO458     -0.176690 0.024975 -7.0747 1.497e−12 ∗∗∗ 

IO1472    -0.045126 0.026449 -1.7061 0.0879825 . 

IO1831          0.049676 0.025686 1.9340 0.0531185 . 

IO1843       0.049638 0.025664 1.9341 0.0530983 . 

AO150      -0.152926 0.027115 -5.6399 1.701e−08 ∗∗∗ 

AO705     -0.209666 0.027091 -7.7393 9.999e−15 ∗∗∗ 

AO1471      1.676966 0.029599 56.6554 < 2.2e−16 ∗∗∗ 

AO1830    -0.122852 0.027860 -4.4096 1.036e−05 ∗∗∗ 

AO1842    -0.094486 0.027815 -3.3969 0.0006816 ∗∗∗ 

AO1984    -0.118687 0.027104 -4.3790 1.192e−05 ∗∗∗ 

AO1994        0.169260 0.027084 6.2495 4.117e−10 ∗∗∗ 

TC691     -0.072536 0.023951 -3.0285 0.0024576 ** 

TC901      -0.076155 0.023946 -3.1803 0.0014712 ** 

TC1470    -0.004099 0.025804 -0.1589 0.8737844 

TC1523    -0.075499 0.023947 -3.1528 0.0016173 ** 

TC1541        0.079253 0.023929 3.3120 0.0009264 *** 

TC1824        0.106075 0.024035 4.4134 1.018e−05 ∗∗∗ 

TC2534    -0.084954 0.023936 -3.5492 0.0003865 *** 

AO1748    -0.110790 0.027141 -4.0821 4.463e−05 ∗∗∗ 

 

3.7 Building ARIMA (1, 1, 0) Model for Outlier-Adjusted Return Series of Union Bank 

              Using the second method, which is removing the effects of the outliers and afterward, 

ARIMA(1, 1, 0) model is fitted to the outlier-adjusted series with its parameter significant at 5% 

level [Table IX], it is found to be adequate at 5% level of significance given the Q-statistics at 



Int. J. Anal. Appl. 17 (4) (2019) 543 

 

lags 1, 14, 18, and 24 having Q(1) = 0.0030, Q(14) = 19.228, Q(18) = 24.611 and Q(24) = 27.717 

with the corresponding p-values of 0.956, 0.1564, 0.136, and 0.2722. 

Table IX: ARIMA (1,1,0) Model for Outlier Adjusted Return Series of Union Bank 

Model Parameter (𝝋) Akaike Information Criteria Log likelihood 

ARIMA (1,1,0)  0.2654∗∗∗ −11598.27 5800.13 

                                                                                   *** significance at 5% level 

ARIMA (1, 1, 0) model fitted to the outlier adjusted series with least AIC = −11598.27 is found 

to be a better model than that of the joint estimation of ARIMA (1, 1, 0) with outliers effect, and 

that of ARIMA (1, 1, 0) model without outliers effect. 

Again, the effects of outliers on the estimate of ARIMA(1, 1, 0) model fitted to the  return 

series of Union bank is similar to that of the Fidelity bank although the estimate of the model is 

reduced by 0.164 and the modified iterative method is also adjudged superior in term of 

efficiency given that it produced a model with minimum variance as shown in Table X. 

Table X: Effect of Outliers on Estimate of ARIMA (1, 1, 0) Model for Return Series of Union 

Bank 

Model 

ARIMA (1,1,0)  

(For outlier 

contaminated) 

Joint ARIMA (1,1,0) and 

Outlier Effect 

ARIMA (1,1,0) 

(For outlier 

adjusted) 

Bias 

Introduced 

Parameter 0.1014 0.2654 0.2654 −0.164 

AIC -9130.26 -11560.27 -11598.27 

 

Standard error 0.0192 0.0188 0.0186 

Variance 0.001963 0.000785 0.000784 

Log-likelihood 4567.13 5800.13 5800.13 

 

3.8 Linear Time Series Modeling of Return Series of Unity Bank 

 Again, using the same procedures as in the first two banks, ARIMA(1, 1, 0) model is 

found to be adequate for the return series of the Unity bank. However, about thirty three (33) 

different outliers are identified to have contaminated the residuals series of ARIMA(1,1,0) 

model, two (2) innovation outliers (IO), six (6) additive outliers, fifteen (15) temporary change 

and ten (10) level shift at C = 5 as shown in Table XI and the joint estimation of the parameter of 

ARIMA(1, 1, 0) model and outliers effects is shown in Table XII.  

 



Int. J. Anal. Appl. 17 (4) (2019) 544 

 

Table XI: Types of Outliers identified in the Residual Series of ARIMA (1, 1, 0) Model                    

fitted to the Return Series of Unity Bank 

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 

 



Int. J. Anal. Appl. 17 (4) (2019) 545 

 

Table XII: Joint Model of ARIMA (1, 1, 0) and Outliers Effect fitted to Return Series of                    

Unity Bank  

 Estimate Std. Error z value Pr(>|z|) 

ar1 0.22870458 0.01895846 12.0635 < 2.2e−16 ∗∗∗ 

IO2293 0.00020692 0.02847462 0.0073 0.9942018 

AO248 1.09943444 0.03077300 35.7272 < 2.2e−16 ∗∗∗ 

AO1906 -0.19318151 0.03242140 -5.9585 2.546e−09 ∗∗∗ 

AO2292 2.30319017 0.03189738 72.2062 < 2.2e−16 ∗∗∗ 

TC247 -0.00386092 0.03081394 -0.1253 0.9002878 

TC1736 0.09934947 0.02489870 3.9901 6.603e−05 ∗∗∗ 

TC1745 0.11091559 0.02491948 4.4510 8.549e−06 ∗∗∗ 

TC1753 -0.12798775 0.02489883 -5.1403 2.743e−07 ∗∗∗ 

TC1762 0.10385821 0.02488517 4.1735 3.000e−05 ∗∗∗ 

TC2291 0.00423615 0.02733850 0.1550 0.8768592 

TC2298 -0.12316459 0.02511656 -4.9037 9.404e−07 ∗∗∗ 

TC2304 -0.07679364 0.02506410 -3.0639 0.0021848∗∗ 

TC2446 -0.08214679 0.02500377 -3.2854 0.0010185∗∗ 

TC2458 0.09014905 0.02690186 3.3510 0.0008051∗∗∗ 

TC2460 0.07420557 0.02694465 2.7540 0.0058872∗∗ 

TC2467 -0.07462796 0.02693742 -2.7704 0.0055984∗∗ 

TC2469 -0.08905130 0.02691343 -3.3088 0.0009370∗∗∗ 

IO1905 0.08527383 0.03084516 2.7646 0.0056997∗∗ 

AO1904 -0.19483838 0.03033146 -6.4236 1.331e−10∗∗∗ 

LS243 0.00112671 0.01580103 0.0713 0.9431541 

LS251 0.00098712 0.01609793 0.0613 0.9511048 

LS347 -0.00156994 0.00489397 -0.3208 0.7483691 

LS520 -0.00603173 0.00523178 -1.1529 0.2489502 

LS598 -0.02950821 0.01304279 -2.2624 0.0236717∗ 

LS613 -0.05489807 0.01706044 -3.2179 0.0012915∗∗ 

LS631 0.00951673 0.01947408 0.4887 0.6250634 

LS635 -0.00408405 0.01769983 -0.2307 0.8175170 

LS2286 -0.00206496 0.00221715 -0.9314 0.3516679 

TC1901 0.11851208 0.02564415 4.6214 3.811e−06 ∗∗∗ 

TC2477 0.10312584 0.02499954 4.1251 3.706e−05 ∗∗∗ 

LS607 0.08270945 0.01887366 4.3823 1.174e−05 ∗∗∗ 

AO1336 -0.12237989 0.02901086 -4.2184 2.460e−05 ∗∗∗ 

AO1872 -0.12246391 0.02910831 -4.2072 2.586e−05 ∗∗∗ 



Int. J. Anal. Appl. 17 (4) (2019) 546 

 

The effects of outliers on the estimate of ARIMA (1, 1, 0) model fitted to the return series 

of Unity bank is similar to those of the first two banks only that  the estimate of the model is 

reduced by 0.1501, as shown in Table XIII. 

Table XIII: Effect of Outliers on Estimate of ARIMA (1, 1, 0) Model for Return Series Unity 

Bank  

Model 

ARIMA (1,1,0) 

(For outlier 

contaminated) 

Joint ARIMA (1,1,0) and 

Outliers Effects 

ARIMA (1,1,0) 

(For outlier 

adjusted) 

Bias 

Introduced 

Parameter  0.0786 0.2287 0.2287 −0.1501 

AIC −7588.08 −11206.23 −11272.23 

Standard error 0.0192 0.0190 0.0188 

Variance 0.00348 0.000885 0.000884 

Log likelihood 3795.04 5638.12 5638.12 

 

4. CONCLUSION 

 In all, it is discovered that outliers introduced substantial biases in the estimates of the 

ARIMA models of the returns series considered and the two methods employed are sufficient 

and adequate in handling outliers in such time series. Meanwhile, to ensure efficiency of the 

estimated parameters of linear models, it is needful and commendable to account for the 

presence of outliers with preference given to modified iterative method. Furthermore, the fact 

that volatility clustering exist in the return series calls for entertainment and modeling of 

heteroscedasticity in future studies. 

 

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