10 
 

 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) 
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

© Global Society of Scientific Research and Researchers  

http://asrjetsjournal.org/  
 

On the Efficiency of Outlier Generating Mechanisms in 

Multivariate Time Series 

Olusesan Oluyomi Olufolaboa*, Olanrewaju Ismail Shittub, Kazeem Adesola 

Adepojuc 

aDepartment of Statistics, Yaba College of Technology, Lagos, Nigeria 

b,cDepartment of Statistics, University of Ibadan, Ibadan, Nigeria 
aEmail: olufolabosesan@yahoo.com 

bEmail: oi.shittu@hotmail.com 
cEmail: ka.adepoju@.ui.edu.ng 

 

 

Abstract 

In this paper, two new outlier generating mechanisms for the detection of outliers in multivariate time series 

setting were derived. This is achieved by specifying two-variable vector autoregressive models and assuming 

additive and convolution effect of outliers on time series data. The magnitude and variance of outlier were 

derived for the generating models by method of least squares. Also a modified test statistics were developed to 

detect single outliers both in the response and explanatory variables. In order to establish the validity and 

efficiency of the derived models, the models were applied to both simulated and existing data. The results from 

the analysed data were also compared to some existing models and the result showed that the convolution model 

is best in terms of the number of outliers detected and the residual variance. This result confirms the finding in 

previous studies of outlier detection in univariate time series. 

Keywords: Additive outlier; Convolution outlier; Innovative outlier; Multiplicative outlier; Vector auto 

regressive.  

1.  Introduction 

The problem of outlier detection in time series has gained much attention in recent times and various methods of 

detection are available, but in most cases, it is limited to univariate time series.  

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

11 
 

It is a known fact, that in time series analysis, outliers can cause biases in parameter estimation as well as 

inappropriate predictions, resulting in misleading conclusion [28]. The essence of outlier detection is to discover 

the unusual data; whose behaviour is very exceptional when compared to the rest of the data set. Examining the 

extraordinary behaviour of outliers  

will surely help to uncover the valuable knowledge hidden behind them and to help the decision makers to 

improve on the quality of data.  

Generally, detection methods are divided into two parts: univariate and multivariate methods. In univariate 

methods, observations are examined individually while in multivariate methods, associations between variables 

in the same dataset are taken into account.  

Several outlier detection methods have been proposed for univariate time series including [11,8,9,26,1,14 and 

32]. All the listed works were based on time domain and almost all make use of iterative procedure in the outlier 

detection process. However, [25] in their work considered the identification of outliers in frequency domain 

using the spectral method.  

On the detection of outlier in multivariate time series, [12] made use of projection pursuit technique while [3] 

proposed the Independent Component Analysis (ICA) as a tool capable of identifying the locations of multiple 

outliers in multivariate time series. The authors [10] used meta-heuristic methods to detect additive outliers in 

multivariate time series. The work of [13] introduced the coefficient of vector autocorrelation, obtained its 

influence function together with its distribution and used it to test the hypothesis of presence of outliers. [31] in 

his paper used an efficient two-phase algorithm for detecting outlying samples in multivariate time series 

datasets. The Bounded Coordinate System metric was used to measure the similarity between two multivariate 

time series samples, and the outlierness of a sample is measured by average distance to its k nearest neighbours. 

Then a heuristic and two pruning rules were utilized to quickly remove multivariate time series samples that are 

not possible outlier candidates, reducing significantly the distance computation among objects. 

As a result of outlier masking effect of both Additive and Innovative on the estimates of parameters and the 

multiplicative effect on parameters estimated, [26] introduced two other types of outliers which are Convolution 

Outlier (CO) and Multiplicative Outlier (MO) for univariate time series. The work of [26] was extended to 

multivariate time series by [19] whereby two generating mechanisms; Innovative and Multiplicative were 

considered. It was concluded that Multiplicative outlier model was more sensitive to outlier with minimum 

standard error of the estimate. 

 For this paper, two outliers generating mechanisms; Convolution and Additive will be extended to multivariate 

time series and their performances in terms of outlier detection will be compared to the existing ones. 

2. Methodology 

In this section, by assuming that outliers have either Additive or Convolution effect on a series for bivariate time 

series and specifying two-variable Vector Autoregressive (VAR) models, the estimate of the parameter for the 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

12 
 

two models will be derived and their corresponding test statistics developed. 

2.1   Derivation of Outlier Generating Mechanisms for Additive Outlier (AO) Model 

In this subsection outlier generating mechanism for additive outlier model will be derived. 

Generally, an additive outlier represents an unexpected change in the value of one of the observations. It can 

appear as a result of a recording or measurement error or other single effect. 

         The additive outlier is defined as  

)(T
ttt ZX ξω+=         (1) 

where   tX = (𝑥𝑥1𝑡𝑡,𝑥𝑥2𝑡𝑡, … … … . . 𝑥𝑥𝑘𝑘𝑡𝑡 ) is a k-dimensional time series, Zt is an outlier free time series that is 

assumed to follow the Autoregressive Moving Average of Order (p,q) i.e. ARMA (p,q),
)(T

tξ is a time indictor 

such that  1)( =T
tξ  for all 0)( == T

tandTt ξ  otherwise, and  𝜔𝜔 = �ω 1, … , ω k�
′
 is the size or the 

magnitude of outlier. 

Now, given vector models 1tX and 2tX  such that 1tX contains outlier and 2tX is outlier free, the magnitude of 

such outlier and its corresponding variance can be obtained by specifying the two variable VAR(2) as: 

1 11 1 1 12 2 1   1

2 21 2 1 22 1 1 2

                                                                                                          (2) 

                                      

t t t t

t t t t

X X X a

X X X a

ϕ ϕ

ϕ ϕ

− −

− −

= + +

= + +                                                                      (3)  

where, 1tX and 1 1tX − are the current and lag values of the response variable respectively, 2tX and 2 1tX −  are 

current and lag values of the explanatory variable respectively. 

Now considering equation (3) 

𝑋𝑋2𝑡𝑡 = ∅21𝑋𝑋2𝑡𝑡−1 + ∅22𝑋𝑋1𝑡𝑡−1 + 𝑎𝑎1𝑡𝑡        

When 𝑋𝑋2𝑡𝑡−1 is contaminated and assumed additive model, we have 

𝑋𝑋2𝑡𝑡 = ∅21�𝑍𝑍𝑡𝑡−1 + 𝜔𝜔𝜉𝜉𝑡𝑡−1
(𝑇𝑇) � + ∅22𝑋𝑋1𝑡𝑡−1 + 𝑎𝑎1𝑡𝑡       (4) 

 = ∅21 𝑍𝑍𝑡𝑡−1 + 𝜙𝜙21 𝜔𝜔𝜉𝜉𝑡𝑡−1
(𝑇𝑇) + ∅22𝑋𝑋1𝑡𝑡−1 + 𝑎𝑎1𝑡𝑡  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

13 
 

𝑋𝑋2𝑡𝑡 = ∅22 𝑋𝑋1𝑡𝑡−1 + 𝜙𝜙21 𝑍𝑍𝑡𝑡−1 + 𝜙𝜙21 𝜔𝜔𝜉𝜉𝑡𝑡−1
(𝑇𝑇)  

𝑋𝑋2𝑡𝑡 = ∅22 𝑋𝑋1𝑡𝑡−1 + 𝜙𝜙21 𝜑𝜑(𝛽𝛽)𝜉𝜉𝑡𝑡−1 + 𝜙𝜙21 𝜔𝜔𝜉𝜉𝑡𝑡−1 

𝑋𝑋2𝑡𝑡 = ∅22 𝑋𝑋1𝑡𝑡−1 + 𝜙𝜙21 𝜉𝜉𝑡𝑡−1�𝜔𝜔 + 𝜑𝜑(𝛽𝛽)�          

Therefore, the general the additive model is given as 

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴:𝑋𝑋𝐴𝐴𝐴𝐴𝑡𝑡 = 𝜙𝜙𝐴𝐴𝐴𝐴𝑋𝑋𝑗𝑗𝑡𝑡−1 + ∅𝐴𝐴𝑗𝑗𝜖𝜖𝑡𝑡−1 �𝜔𝜔 + 𝜑𝜑(𝛽𝛽)�                  (5) 

2.1.1 Derivation of the Magnitude of Outlier for AO 

With 1 11 1 1 12 2 1 1  as defined in equation (2)t t t tX X X aϕ ϕ− −= + +  

Then, 

( )

( )
( )

( )
( ) ( )

( ) ( )
11 12 2 1 1

( ) ( )
11 12 1

                                                                     (6)

 

                                              

T T
t t t t t t

T T
t t t t t

Z Z X a

a a

ωξ ϕ ωξ ϕ

θ β θ β
ωξ ϕ ωξ ϕ φ β

ϕ β ϕ β

−+ = + + +

 
+ = + + + 

 
l      

 

( )
( )

( )
( ) ( ) )(

12
)(

111 T
tt

T
tta ωξβϕφω

βφ
βθφ

βφ
βθ

−+







∈+=








− 

                        
 

( )
( ) ( )

( )
( ) 1

)(
12

)(
11

−

−+







+

=

βθ
βφ

ωξβϕφωξ
βθ
βφφ T

tt
T

t

ta


                (7)

 

 Summing the square of equation (1.9) over n we have  

( )
( ) ( )

( )
( )

∑ ∑
= =



















−

−+







+

=
n

t

n

t

T
tt

T
t

ta
1 1

)(
12

)(
11

2

1
βθ
βφ

ωξβϕφωξ
βθ
βφφ 

                     (8)

 

Differentiating equation (1.7) with respect to ω  and setting to zero, we obtain the magnitude of outlier in the 

model as  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

14 
 

( )
( )

( )
( ) ( ) 0

1

2
ˆ

)(
12

)(

1
112

2

=−+







+









−

=
∂
∂ ∑

=

T
tt

T
t

n

tA

ta
ωξβϕφωξ

βθ
βφφ

βθ
βφω

l

          (9)

 

( )
( ) ( )∑ ∑∑

= ==

=−+







+

n

t

n

t
t

n

t 1 1
12

1
11 0ωβϕφω

βθ
βφφ 

  
 

( )
( ) ( )

( ) ( )
( ) ( ) t

t





βϕφ
βφ
βθφφω

ωβϕφωφ
βθ
βφφ

121111

121111

1

0

+=−

=−++








 

( )
( ) ( )

11

1211

1
ˆ

φ

βϕφ
βφ
βθφ

ω
−

+
=

t

A



      
 

( )( )
11

12

1
1ˆ
φ
φβϕω

−
+

= t
A



                                                                                            (10)
 

Therefore, the estimate of the variance is  

 ( )
( )

2
2 2
12

2
11

ˆ
1

a

AV
φ β ϕ σ

ω
ϕ

 
 
 =

−
 

2 2
212

111 a
ϕ φ β σ
ϕ

  
=   −                          (11)

 

With the estimates of mean and variance of the magnitude of AO derived, the test statistic for testing the 

presence of outlier for additive model is constructed as follows: 

( )

12 11

11 12

12

12

1 1.                                                            
1

1
                                                                                               (12) 

t
i

a

t
i

a

ϕ ϕλ
ϕ ϕ σ

ϕ
λ

ϕ σ

+ −
=

−

+
=

l

l

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

15 
 

2.2 Derivation of Outlier Generating Mechanisms for Convolution Outlier (MO) Model 

The outlier effect on a given series may be either additive or innovative and the effect may be a combination of 

the two [26]. 

By this, we propose the convolution of the additive and innovative outliers for the multivariate setting as 

follows: 

The innovative and additive models are defined respectively as follows: 

 

( )

             model additivefor            

         model innovativefor           

)(T
tttA

T
tttI

ZX

ZX

ωξ

ωξβϕ

+=

+=
 

The convolution involved adding both innovative and additive models [26]. This gives 

( )( )βϕωξ ++= 12 T
tttC ZX        (13) 

For the general case of CO, now considering, 

𝑋𝑋2𝑡𝑡 = ∅21𝑋𝑋2𝑡𝑡−1 + ∅22𝑋𝑋1𝑡𝑡−1 + 𝑎𝑎1𝑡𝑡 as defined in equation (2) 

Assuming 𝑋𝑋2𝑡𝑡−1 is contaminated, we have 

𝑋𝑋2𝑡𝑡 = ∅21 �2𝑍𝑍𝑡𝑡−1 + 𝜔𝜔𝜉𝜉𝑡𝑡
(𝑇𝑇)(1 +𝜑𝜑(𝛽𝛽))� + ∅22𝑋𝑋1𝑡𝑡−1 + 𝑎𝑎1𝑡𝑡                                                     (14) 

𝑋𝑋2𝑡𝑡 = 2∅21𝑍𝑍𝑡𝑡−1 + ∅21𝜔𝜔𝜉𝜉𝑡𝑡
(𝑇𝑇)(1 +𝜑𝜑(𝛽𝛽)) + ∅22𝑋𝑋1𝑡𝑡−1 + 𝑎𝑎1𝑡𝑡 

where 𝑍𝑍𝑡𝑡 = 𝜑𝜑(𝛽𝛽)𝜉𝜉𝑡𝑡
(𝑇𝑇)and 𝑍𝑍𝑡𝑡−1 = 𝜑𝜑(𝛽𝛽)𝜉𝜉𝑡𝑡

(𝑇𝑇) 

we then have 

𝑋𝑋2𝑡𝑡 = 2∅21𝜑𝜑(𝛽𝛽)𝜉𝜉𝑡𝑡
(𝑇𝑇) + ∅21𝜔𝜔𝜉𝜉𝑡𝑡

(𝑇𝑇)(1 +𝜑𝜑(𝛽𝛽)) + ∅22𝑋𝑋1𝑡𝑡−1 + 𝑎𝑎1𝑡𝑡                  (15) 

𝑋𝑋2𝑡𝑡 = 2∅21𝜑𝜑(𝛽𝛽)𝜉𝜉𝑡𝑡
(𝑇𝑇) + ∅21𝜔𝜔𝜉𝜉𝑡𝑡−1 + ∅21𝜔𝜔𝜉𝜉𝑡𝑡

(𝑇𝑇)𝜑𝜑(𝛽𝛽) + ∅22𝑋𝑋1𝑡𝑡−1 + 𝑎𝑎1𝑡𝑡 

𝑋𝑋2𝑡𝑡 = ∅22𝑋𝑋1 𝑡𝑡−1 + 2∅21𝜑𝜑(𝛽𝛽)𝜉𝜉𝑡𝑡
(𝑇𝑇) + ∅21𝜔𝜔𝜉𝜉𝑡𝑡

(𝑇𝑇) + ∅21𝜔𝜔𝜉𝜉𝑡𝑡
(𝑇𝑇)𝜑𝜑(𝛽𝛽) 

𝑋𝑋2𝑡𝑡 = ∅22𝑋𝑋1 𝑡𝑡−1 + 2∅21𝜑𝜑(𝛽𝛽)𝜉𝜉𝑡𝑡
(𝑇𝑇) + ∅21𝜔𝜔𝜉𝜉𝑡𝑡

(𝑇𝑇)�1 + 𝜑𝜑(𝛽𝛽)� 

𝑋𝑋2𝑡𝑡 = ∅22𝑋𝑋1 𝑡𝑡−1 + ∅21𝜉𝜉𝑡𝑡
(𝑇𝑇)[2𝜑𝜑(𝛽𝛽) + 𝜔𝜔(1 + 𝛽𝛽)]                                                   (16) 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

16 
 

Therefore, in general, the CO generating mechanism is 

𝐶𝐶𝐴𝐴𝐶𝐶𝐴𝐴𝐴𝐴𝐴𝐴𝐶𝐶𝐴𝐴𝐴𝐴𝐴𝐴𝐶𝐶𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴:𝑋𝑋𝐶𝐶𝐴𝐴𝑡𝑡 = 𝜙𝜙𝐴𝐴𝐴𝐴𝑋𝑋𝑗𝑗𝑡𝑡−1 + ∅𝐴𝐴𝑗𝑗𝜉𝜉𝑡𝑡−1𝑇𝑇 [2𝜑𝜑(𝛽𝛽) + 𝜔𝜔(1 + 𝛽𝛽)]                           (17) 

2.1.2 Derivation of Magnitude of Outlier for CO 

Now, specifying  tttt aXXX 1121211111 ++= −− φφ
                         (18)

 

and substituting  tCX  in equation (18) gives 

( )( ) ( )( )( ) tt
T

tt
T

tt aXZZ 11212
)(

111
)( 1212 ++++=++ −− φβϕωξφβϕωξ

                     (19) 

( ) ( )( ) ( ) ( )( )[ ] ( ) tt
T
tt

T
tt aaa 1112111

)( 1212 ++++=++ −− βϕφβϕωξβϕφβϕωξβϕ  

( ) ( ) ( ) ( )( ) ( )( )βϕωξφβϕωξβϕφβϕφβϕ +++−−− −=− 1122 )(
11

)(
1121111

T
t

T
ttttt aaa   

( ) ( )[ ] ( ) ( )( ) ( )( )βϕωξφβϕωξβϕφφβϕβϕ +++−=−− − 11122 )(
11

)(
11211

T
t

T
ttta       (20) 

Summing and squaring equation (20) gives 

( ) ( )( ) ( )( )[ ]
( ) ( )[ ]∑

∑
=

=
−

−−

−++−
=

n

t

n

t

T
t

T
tt

ta
1

2
11

1

2
11112

2

122

11

φβϕβϕ

βϕωξφβϕωξβϕφ 

                        (21)
 

 

 Differentiating equation (21) with respect to ω and equating to 0 we have  

( )( ) ( )( )
( ) ( )[ ]

0 
122

112
2

11

11
2

=
−−

+++
=

∂
∑∂

φβϕβϕ
βϕφβϕ

ω
ta

`                        (22)
 

( ) ( )( ) ( )( )[ ] 011
1

11112 =+++−∑
=

−

n

t
t βϕωφβϕωβϕφ 

    
 

( ) ( )( ) ( )( ) 011 11112 =+++−− βϕωφβϕωβϕφ t       

( )( ) ( )( ) ( ) 11211 11 −=+−+ tβϕφβϕωωφβϕω       



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

17 
 

( )( ) ( )( )[ ] ( ) 11211 11 −=+−+ tβϕφβϕφβϕω                   

( )
( ) ( )( )βϕφ

βϕφ
ω

−−
= −

11
ˆ

11

112 t
C



                              (23)
 

The corresponding variance is  

( )
( ) ( )( )

2 2
12

22
11

ˆV( )             
1 1

a
C

ϕ φ β σ
ω

ϕ φ β
=

− −                            (24)
 

Therefore, the test statistic is 

 ( )C

C
i eS ω

ωλ
ˆ.

ˆ
=

         

( ) ( ) ( )( )

( ) ( )( )22
11

2
2

2
12

11111

11

11/

βϕφσβϕφ

βϕφβϕφ
λ

+−







+−
= −

a

t
i


                                     (25)

 
( )

( ) ( )( )
( ) ( )( )

( ) a

t

σβϕφ
βϕφ

βϕφ
βϕφ

12

11

11

112 11*
11

+−
+−

= −  

a

t
i σ

λ 1       −=


                           (26) 

Table 1:  Summary of Estimates and Test Statistic for the two models when 1tX contains outlier 

MODELS 

 

MAGNITUDE VARIANCE TEST 

STATISTIC 

Additive ( ) ( )12

11

1
1

tϕ β φ λ
φ
+

−
 2

2
2

11

12

1 aσβϕ
φ

φ
















−

 
( )

a

t

σφ
φ

12

121 +
 

Convolution ( )
( ) ( )( )βϕφ

βϕφ
−−

−

11 11

112 t  

( ) ( )( )22
11

2
2

12

11 βϕφ

σβϕφ

−−









a

 a

t

σ
1−  

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

18 
 

3. Application 

In this section, analysis of both simulated and real data sets will be used to test the validity and efficiency of the 

derived outliers generating mechanisms. In other to compare the performance of the two newly derived models 

with the existing ones, Data on Nigerian Bank Deposits and Loans from Annual Statistical Bulletin of the 

Central Bank of Nigeria, 2011 were made used of.  

From the derived outlier generating mechanisms in section 2 and with the estimation of the magnitudes of 

outliers and their variances, the test statistics constructed will be used to detect the existence of outliers in both 

the generated series and real data. 

For the simulated data, a uniform distribution is assumed with contaminated observation with varying sizes of 

10, 50, and 100. The data were analysed with the R-package of version 3.0.1.  

3.1 Analysis of Simulated Data when X1t Contains an Outlier 

The results of the models on simulated data assuming a uniform distribution in terms of their outlier detection 

performance are tabulated below.  

The sample sizes considered are 10, 50 and 100.    

Table2: Summary of Result on Detection Rate of the Models on Simulated Data when 1tX  contains outlier 

            N=10                N=50         N=100 

 

Model Type No of 

outliers 

injected 

No of 

outliers 

detected 

% of 

outliers 

detected 

No of 

outliers 

injected 

No of 

outliers 

detected 

% of 

outliers 

detected 

No of 

outliers 

injected 

No of 

outliers 

detected 

% of 

outliers 

detected 

Additive 2 1 50 5 4 80 8 6 75 

Convolution 2 2 100 5 5 100 8 8 100 

Innovative* 2 0 0 5 2 40 8 2 25 

Multiplicative* 2 2 100 5 4 80 8 5 80 

 Source* [19] 

The Convolution model from the summary in Table2 had 100% outlier detection compared to Additive model as 

the sample size increases. 

 When compared with existing models, the Convolution model is most sensitive to outlying observations. 

 3.2 Detection of Outlier in Real Data 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

19 
 

In order to investigate the performance of the proposed models, a pair of data on Deposit and Loan was used. 

The data was extracted from the Annual Statistical Bulletin of the Central Bank of Nigeria, 2011. 

3.2.1. Assumed Model of Deposits and Loans 

Here two cases are considered. The first case is when loan is contaminated.  

The vector autoregressive model is given as 

𝑋𝑋1𝑡𝑡 =  ∅11𝑋𝑋1𝑡𝑡−1 + ∅12𝑋𝑋2𝑡𝑡−1 + ℓ𝑡𝑡         (27) 

where  𝑋𝑋1𝑡𝑡  is the current value of deposit, 𝑋𝑋1𝑡𝑡−1  is the immediate past value of deposit, and 𝑋𝑋2𝑡𝑡−1  is the 

immediate past value of loan. 

The estimated VAR model via the use of statistical package R is as follows 

X1t = 0.4826 X1t-1 –– 0.1579 X2t-1         (28) 

s.e (0.1836) (0.1561) 

t  (2.628)  (–1.012) 

P-value (0.0142) (0.3210) 

When deposit is contaminated, the vector autoregressive model is: 

𝑋𝑋2𝑡𝑡 =  ∅21𝑋𝑋2𝑡𝑡−1 + ∅22𝑋𝑋1𝑡𝑡−1 + ℓ𝑡𝑡                          (29) 

where 𝑋𝑋2𝑡𝑡 is the current value of loan,𝑋𝑋2𝑡𝑡−1 is the immediate past value of loan and 𝑋𝑋1𝑡𝑡−1 is the immediate past 

value of deposit. 

The estimated VAR model via the use of statistical package R is as follows 

X2t = 0.9605 X2t-1 –– 0.3339 X1t-1        (30) 

S.e (0.1712) (0.2015) 

t  (5.610)  (–1.657) 

P  (6.78e.06) (0.1095) 

The detection performance of both Additive and Convolution models on the real data are shown on tables 3 and 

4 below.  



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

20 
 

Table 3: Detection Performance of Additive Model on Deposit and Loan Data 

Deposit  Loan  (Wa) T Remarks 

111.7 

131.2 

276.6 

311.4 

873.5 

1229.2 

1378.4 

5722.0 

8360.1 

10580.7 

4612.2 

19542.2 

4855.2 

8807.1 

12442.0 

19047.6 

18513.8 

15860.5 

20640.9 

16875.9 

14861.6 

20551.8 

64490.0 

18461.9 

3118.6 

3082.3 

13411.8 

3296.2 

3953.1 

35.9 

44.2 

58.2 

114.9 

373.6 

492.8 

659.9 

3721.1 

4730.8 

5962.1 

1895.3 

10910.4 

1602.2 

8659.3 

4411.2 

11158.6 

11852.7 

7498.1 

11150.3 

12341.0 

8942.2 

11251.9 

34118.5 

16105.5 

24274.6 

27263.5 

46521.5 

15590.5 

63769.4 

-7425.1230   

-7287.8225  

 -7320.9776  

 -6766.7184  

 -6641.4233   

-6645.0523  

 -2347.0674  

 -1321.7090   

 -214.7493   

-7060.4273 

10107.6624  

-10360.6635    

-790.9809   

2051.1317   

6231.8871   

3575.8049   

1289.6953   

6662.9285   

1167.7164   

1158.2959 

7283.8830  

48840.8554 

 -14779.9643  

-10755.6646   

-2097.8978   

8721.0511   

-3338.5137   

-2683.9759    

747.8194 

-3.5562723  

-3.4905121  

-3.5063918  

-3.2409286  

-3.1809184  

-3.1826565  

-1.1241310  

-0.6330342  

-0.1028544  

-3.3816009  

4.8410781 

-4.9622533  

-0.3788413  

0.9823922  

2.9847704  

1.7126364  

0.6177012  

3.1912183  

0.5592793  

0.5547673  

3.4886253 

23.3923914 

-7.0788832  

-5.1514396 

 -1.0047909  

4.1769588 

-1.5989855  

-1.2854938  

0.3581691 

ND* 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

D** 

D 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

D 

D 

D 

ND 

D 

ND 

ND 

ND 

D** = Outlier detected 

ND* = No outlier detected 

The critical value (c) = 4  

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

21 
 

Table 4: Detection Performance of Convolution Model on Deposit and Loan Data 

Deposit  Loan  (Wc) T Remarks 

111.7 

131.2 

276.6 

311.4 

873.5 

1229.2 

1378.4 

5722.0 

8360.1 

10580.7 

4612.2 

19542.2 

4855.2 

8807.1 

12442.0 

19047.6 

18513.8 

15860.5 

20640.9 

16875.9 

14861.6 

20551.8 

64490.0 

18461.9 

3118.6 

3082.3 

13411.8 

3296.2 

3953.1 

35.9 

44.2 

58.2 

114.9 

373.6 

492.8 

659.9 

3721.1 

4730.8 

5962.1 

1895.3 

10910.4 

1602.2 

8659.3 

4411.2 

11158.6 

11852.7 

7498.1 

11150.3 

12341.0 

8942.2 

11251.9 

34118.5 

16105.5 

24274.6 

27263.5 

46521.5 

15590.5 

63769.4 

 

 543.83931   

533.78300   

536.21138   

495.61569   

486.43869   

486.70449   

171.90659    

96.80611    

15.72891   

517.12786 

-740.31692   

758.84751    

57.93392  

 -150.23132   

-456.44297   

-261.90317   

 -94.46133  

 -488.01380    

-85.52721    

-84.83722 

-533.49445  

-3577.25755  

1082.53098   

787.77864   

153.65662   

-638.75715   

244.52322   

196.58282    

-54.77264 

3.4944194   

3.4298029   

3.4454064   

3.1845603   

3.1255938   

3.1273017   

1.1045794   

0.6220241   

0.1010655   

3.3227859 

-4.7568790   

4.8759466   

0.3722523   

-0.9653058   

-2.9328574   

-1.6828492   

-0.6069578   

-3.1357146  

 -0.5495519   

-0.5451185 

-3.4279489  

-22.9855362   

6.9557628   

5.0618425   

0.9873149   

-4.1043105   

1.5711749   

1.2631357 

-0.3519396 

ND* 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

D** 

D 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

ND 

D 

D 

D 

ND 

D 

ND 

ND 

ND 

D** = Outlier detected 

ND* = No outlier detected 

The critical value (c) = 4  

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

22 
 

Table 5: Summary of Outlier Detection of the Two Models on Deposits and Loan Data 

Model No of outliers detected  

Convolution  6 

Additive  6 

**Innovation 5 

**Multiplicative Nil 

**Source: [19]  

 

4. Discussion of Results 

From the analyzed simulated data with varying sample sizes of 10, 50, and100, the average percentage rates of 

outlier detection for AO and CO are 68% and 100% respectively of the injected outliers. From the result, CO 

was consistent in outlier detection as the sample size increases. Comparing the performance of these two newly 

derived models with the existing models, the CO outperformed both Multiplicative and Innovative models that 

have average detection rate of 86.7% and 21.7% respectively for the simulated data.    

For the real data set of Deposit and Loan, 6 outliers were equally detected by the two models when we consider 

the case of deposit depending on loan. The two derived outlier-generating mechanisms were able to detect 

potential outlier independently in multivariate time series. However, comparing the performance of these 

models with the existing ones, AO and CO detected 6 outliers while Innovative model was able to detect 5 but 

Multiplicative model detected no outlier as a result of non-multiplicative nature of data. [19].    

In summary, CO was found to be most sensitive to outliers for the simulated data sets as the sample increases 

and also for the real data. When compared also with the existing models, CO has been found to be most efficient 

with minimum standard error of the estimate and is therefore recommended for outlier detection in multivariate 

time series data. 

5. Conclusion 

This work was undertaken to develop test statistic for detecting outliers assuming two different outliers 

generating mechanisms in multivariate time series models. In line with the main objective of this paper, the test 

statistics were derived for each generating mechanism namely; the Additive and Convolution models. The 

model with greatest detective power in terms of their sensitivity to the number of outliers detected by applying 

the models to both simulated and a pair of real data were determined. All these were achieved using theoretical 

and analytical methods. The convolution model was found to be most sensitive to outlier detection when 

compared with existing models, it is therefore recommended for outlier detection in multivariate time series.    

 



American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 22, No  1, pp 10-25 

 

23 
 

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