@1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

Time Series Forecasting by Using Box-Jenkins Models 
 

Hazim  M. Gorgess 
Raghad  Ibrahim 

Dept. of Mathematics/College of Education for Pure Science(Ibn AL-Haitham) / 
University of Baghdad 

 
Received in:10 October  2012  ,  Accepted in:9  December  2012 

 
Abstract 
        In this paper we introduce a brief review about Box-Jenkins models. The acronym 
ARIMA stands for “autoregressive integrated moving average”. It is a good method to 
forecast for stationary and non stationary time series. According to the data which obtained 
from Baghdad Water Authority, we are modelling two series, the first one about pure water 
consumption and the second about the number of participants. Then we determine an optimal 
model by depending on choosing minimum MSE as criterion. 
 
Key Words :  Forecasting, Box-Jenkins, autoregressive integrated moving average 
(ARIMA), Autoregressive (AR), moving average (MA), autocorrelation function (ACF), 
partial autocorrelation function (PACF) 
 

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

Introduction  
        Time series forecasting is one of the most common forecasting techniques used in 
operation management due to its simplicity and ease of usage. Many time series are non 
stationary so, we cannot use AR, MA or ARMA processes directly. One possible way to 
make them stationary is to apply differencing. The first differences namely (Xt – Xt – 1) = (1 – 
B)Xt, may themselves be differenced to give second differences and so on. The d th 
differences may be written as (1 – B)dXt. If original data series is differenced  d  times before 
fitting an ARMA(p,q) process, then the model for the original undifferenced series is said to 
be an ARIMA(p,d,q) process where the letter I refers to integrated and  d  denotes the 
number of differences taken. 

 
Autoregressive (AR) Processes 
        A time series {Xt} is said to be an autoregressive process of order p (abbreviated AR(p)) 
if it is a weighted linear sum of the past  p  values plus a random shock so that: 
Xt = φ1Xt – 1 + φ2Xt – 2 + … + φpXt – p + at                                                                                           
…(1) 
where {at} denotes a purely random process with zero mean and constant variance 2aσ . Using 
the backward shift operator “B” such that BjXt = Xt – j, the AR(p) model may be written in the 
compact form  
φ(B)Xt = at                                                                                                                         …(2) 
where φ(B) = 1 – φ1B – φ2B2 – … – φpBp is a polynomial in B of order p.  
An AR process is stationary provided that the roots of φ(B) = 0 lie outside the unit circle, [1]. 
The simplest case of an AR process is the first order case AR(1) process given by: 
Xt = φ1Xt – 1 + at                                                                                                                  …(3) 
or 
(1 – φ1B)Xt = at                                                                                                                                   
…(4) 
In order to be stationary the root of φ(B) = (1 – φ1B) = 0 must be outside of the unit circle. 
That is for stationary process, we have φ1 < 1. The AR(1) process is some times called the 
Markov process because the value of xt is completely determined by the knowledge of Xt – 1. 
The autocovariances can be obtained as follows: 
γk = E(Xt – kXt) = E(φ1Xt – 1Xt – k) + E(Xt – k at) 
    = φγk – 1,  k ≥ 1                                                                                                                  …(5) 
and hence the autocorrelation function becomes  
ρ k = φρ k – 1 =

k
1φ  , k ≥ 1                                                                                                      …(6) 

where we use the fact that ρ 0 = 1. 
 
      The magnitude of these autocorrelations decrease exponentially. For higher order 
stationary AR processes, the ACF will typically be a mixture of terms which decrease 
exponentially or of damped sine or cosine waves. For an AR(1) process, the PACF from (6) 
is: 

1 1
kk

for k 1
0 for k 2
ρ = φ =

φ = 
≥

                                                                                               …(7) 

 
        Hence, the PACF of AR(1) process cuts off at lag 1. A useful property of an AR(p) 
process is that it can be shown that the PACF is zero at all lags greater than p. This means that 
the sample PACF can be used to help determine the order of an AR process (assuming the 
order is unknown as it is usually the case) by looking for the lag value at which the sample 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

PACF cuts off (meaning that it should be approximately zero, or at least not significantly 
different from zero, for higher lags). 
 
Moving Average (MA) Processes 
        A time series {Xt} is said to be a moving average process of order q ( abbreviated 
MA(q)) if it is a weighted linear sum of the last q random shocks so that: 
Xt = at – θ1at – 1 –… – θqat – q                                                                                             …(8) 
or 
Xt = θ(B) at                                                                                                                       …(9) 
where θ(B) = (1 – θ1B – … – θqBq) is a polynomial in B of order q. 
It can be shown that a finite order MA process is stationary for all parameter values. 
However, it is customary to impose a condition on the parameter values of an MA model 
called the “invertibility condition”. This moving average process is invertible if the roots of 
θ(B) = 0 lie outside of the unit circle, [2]. Moving average processes are useful in describing 
phenomena in which events produce an immediate effect that only lasts for short periods of 
time. To discuss further properties of the MA(q) process, let us consider the following simpler 
case when θ(B) = 1 – θ1B we have the first order moving average MA(1) process 
Xt = at – θ1at – 1 = (1 – θ1B) at                                                                                          …(10) 
where {at} is a zero mean white noise process with constant variance 2aσ . It can be shown that 
the autocovariances of the process are: 

t t kk X X

(1 k 0

E k 1
0 k 1

a

a

σ

σ
−

2 2
1

2
1

 + θ ) =


γ = = − θ =
 >

                                                                        …(11) 

and hence the autocorrelation function becomes: 

k

1 k 0

k 1
1

0 k 1

ρ 1
2
1

 =


−θ
= =

+ θ
 >

                                                                                              …(12) 

which cuts off after lag 1. 
Using (11) and (12) the PACF of an MA(1) process can be easily seen to be: 

11 1
(1 )

1 1
ρ

2
1 1 1

2 4
1 1

− θ − θ − θ
φ = = =

+ θ − θ
 

2
1

22 2
1

(1 )
1 1 1

ρ
ρ

2 2 2
1 1 1

2 4 6
1 1 1

− θ − θ − θ
φ = − = =

− + θ + θ − θ
 

In general: 
k

kk 2(k +1)

(1 )
1

2
1 1

1

− θ − θ
φ =

− θ
, for k ≥ 1                                                                                           …(13) 

The PACF of an MA(1) model tails off exponentially in one of two forms depending on the 
sign of θ1 (hence on the sign of ρ1). If alternating in sign, it begins with positive value, 
otherwise, it decays on the negative sign. 
For the general qth order moving average process the variance is: 

q

0 j
j=0

aσ 2 2∑γ = θ                                                                                                                   …(14) 

where θ0 = 1, and the other covariances are 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

k 1 k 1 q k q
k

( k 1, 2,..., q

0 k q
aσ
2

− − −θ + θ θ + ... + θ θ ) =γ = 
>

                                          …(15) 

Hence the ACF becomes 
k 1 k 1 q k q

qk

k 1, 2,..., q
1 ...

0 k q
ρ

− −
2 2
1

−θ + θ θ + ... + θ θ
= + θ + + θ= 

 >

                                                     …(16) 

Thus the ACF cuts off at lag q. This property may be used to try to assess the order of the 
process (i.e. the value of q) by looking for the lag beyond which the sample ACF is not 
significantly different from zero. The PACF of the general MA(q) process tails off as a 
mixture of exponential decays and / or damped sine waves depending on the nature of the 
roots of θ(B) = 0. The PACF will contain damped sine waves if some of the roots are 
complete. The dual relationship between an AR(p) and an MA(q) processes exists and may be 
described as follows [3]. A finite order stationary AR(p) process corresponds to an infinite 
order MA process and a finite order invertible MA(q) process corresponds to an infinite order 
AR process, this relationship also exists in the ACF and PACF. The AR(p) process has its 
autocorrelations tailing off and partial autocorrelations cutting off while MA(q) process has 
its autocorrelations cutting off and partial autocorrelations tailing off. 
Table (1) represent the quantity of water consumption of Baghdad City for the period (2000-
2008) and table (2) represent the number of participants for the period (2000-2008), where 
figure (1) represent series water consumption and figure (2) represents the series of number of 
participants which are non stationary. It is clear that the first is stationary. 
 
 
Modeling by Box-Jenkins Method  
        Box-Jenkins is a famous method which has a high qualified in time series analysis that 
reflect behaviour of series while if it was seasonal or non seasonal. In this research ,we 
modeled two series. We depend on SPSS and Minitab programmes to determine tables and 
figures. Table (3) represents the ARIMA model for water series consumption and table (4) 
represents ARIMA models for series number of participants. 
 
Conclusion 
        It is clear that by applying Box-Jenkins methods depending on minimum MSE as 
criterion to choose an optimal model, the best method for water series consumption is (2,0,1) 
which has no differences and means the series stationary, and the best model for series 
number of participants is (2,1,2). 
 
References 
1. Chris Chatfield (2000) Time Series Forecasting, Univ. of Bath, Dep. Of Math. Sciences. 
2. Wei,W.W.S. (1989) Time Series Analysis, Addision Wesley Publishing Company Inc. 
3. Margherita G. (2010) ARIMA and SARIMA Models. 
 
 
 
 
 
 
 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

Table (1):Quantity of Water Consumption of Baghdad City for the period (2000-2008) 
period year Water Consumption 

31/12/1999 – 30/4/2000 
1/5/2000 – 31/8/2000 
1/9/2000 – 31/12/2000 

2000 
150199 058 
117836291 
110120423 

31/12/2000 – 30/4/2001 
1/5/2001 – 31/8/2001 
1/9/2001 – 30/12/2001 

2001 
192430197 
260990152 
248181835 

31/12/2001 – 30/4/2002 
1/5/2002 – 31/8/2002 
1/9/2002 – 30/12/2002 

2002 
192892689 
198365209 
195628949 

31/12/2002 – 30/4/2003 
1/5/2003 – 31/8/2003 
1/9/2003 – 30/12/2003 

2003 
112386361 
110389561 
111387961 

31/12/2003 – 30/4/2004 
1/5/2004 – 31/8/2004 
1/9/2004 – 30/12/2004 

2004 
106390698 
168258446 
119950103 

31/12/2004 – 30/4/2005 
1/5/2005 – 31/8/2005 
1/9/2005 – 30/12/2005 

2005 
100917650 
134876155 
140109023 

31/12/2005 – 30/4/2006 
1/5/2006 – 31/8/2006 
1/9/2006 – 30/12/2006 

2006 
112130164 
168329582 
113161258 

31/12/2006 – 30/4/2007 
1/5/2007 – 31/8/2007 
1/9/2009 – 30/12/2007 

2007 
164912005 
112033870 
148864908 

31/12/2007 – 30/4/2008 
1/5/2008 – 31/8/2008 
1/9/2008 – 30/12/2008 

2008 
202032553 
223789575 
129457565 

 
Table (2):Number of Participants for the period (2000-2008) 

period year Number of Participants 
31/12/1999 – 30/4/2000 
1/5/2000 – 31/8/2000 
1/9/2000 – 30/12/2000 

2000 
552825 
564314 
556327 

31/12/2000 – 30/4/2001 
1/5/2001 – 31/8/2001 
1/9/2001 – 30/12/2001 

2001 
571845 
571731 
555949 

31/12/2001 – 30/4/2002 
1/5/2002 – 31/8/2002 
1/9/2002 – 30/12/2002 

2002 
557170 
557109 
557140 

31/12/2002 – 30/4/2003 
1/5/2003 – 31/8/2003 
1/9/2003 – 30/12/2003 

2003 
557102 
555611 
556357 

31/12/2003 – 30/4/2004 
1/5/2004 – 31/8/2004 
1/9/2004 – 30/12/2004 

2004 
558600 
559519 
559837 

31/12/2004 – 30/4/2005 
1/5/2005 – 31/8/2005 
1/9/2005 – 30/12/2005 

2005 
560903 
561551 
563386 

31/12/2005 – 30/4/2006 
1/5/2006 – 31/8/2006 
1/9/2006 – 30/12/2006 

2006 
566547 
568128 
571265 

31/12/2006 – 30/4/2007 
1/5/2007 – 31/8/2007 
1/9/2009 – 30/12/2007 

2007 
573615 
575894 
576591 

31/12/2007 – 30/4/2008 
1/5/2008 – 31/8/2008 
1/9/2008 – 30/12/2008 

2008 
578429 
580834 
582955 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

Table (3):ARIMA Model for Water Series Consumption 
ARIMA Parameters MSE 
(1,0,0) p(1)   0.2218 0.07942 

(0,1,0) no autoreg. or movi. average – 

(0,0,1) q(1)   – 0.1980 0.07977 
(1,1,0) p(1)   – 0.4183 0.11435 

(1,0,1) p(1)      0.2044 q(1)   – 0.0185 0.08303 

(0,1,1) q(1)   0.9142 0.09364 

(1,1,1) p(1)   0.2026 q(1)   0.9161 0.09405 

(2,0,0) p(1)      0.2245 p(2)   – 0.0108 0.08302 

(0,2,0) no autoreg. or movi. average – 

(0,0,2) q(1)   – 0.2502 q(2)   – 0.0993 0.08250 

(2,2,0) p(1)   – 0.8795 p(2)   – 0.3389 0.21035 

(2,0,2) 

p(1)   – 0.8198 
p(2)   – 0.4803 
q(1)   – 1.1620 
q(2)   – 0.8325 

0.08240 

(0,2,2) q(1)      1.4100 q(2)   – 0.4599 0.12312 

(2,2,2) 

p(1)   – 0.9689 
p(2)      0.0336 
q(1)      0.0079 
q(2)      0.8890 

0.12964 

(1,0,2) 
p(1)   0.7532 
q(1)   0.6492 
q(2)   0.2550 

0.08358 

(1,2,0) p(1)   – 0.6493 0.22316 

(2,0,1) 
p(1)   – 0.8216 
p(2)      0.1759 
q(1)   – 0.9322 

0.07903* 

(2,1,0) p(1)   – 0.5041 p(2)   – 0.2087 0.11492 

(2,1,1) 
p(1)   – 1.1045 
p(2)   – 0.1028 
q(1)   – 0.9768 

0.10921 

(1,2,1) p(1)  – 0.1770 q(1)     1.1350 0.11346 

(1,1,2) 
p(1)   0.2120 
q(1)   0.9152 
q(2)   0.0229 

0.09728 

(2,1,2) 

p(1)     0.7569 
p(2)   – 0.2899 
q(1)     1.2970 
q(2)   – 0.2021 

0.09458 

 
 
 
 
 
 
 
 
 
 
 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

Table (4):ARIMA Models for Series Number of Participants 
ARIMA Parameters MSE 

(0,1,0) no autoregr. or movin. average – 

(0,0,1) q(1)   – 0.9241 0.00010437 
(1,1,0) p(1)   – 0.3621 0.00008927 

(1,0,1) cannot be estimated with these data – 

(0,1,1) q(1)   0.4067 0.00008747 

(1,1,1) p(1)   – 0.0440 q(1)      0.3684 0.00009159 

(2,0,0) p(1)    0.7202 p(2)    0.2791 0.00009805 

(0,2,0) no autoregr. or movin. average – 

(0,0,2) q(1)   – 0.5492 q(2)   – 0.5263 0.00013817 

(2,2,0) p(1)   – 0.9207 p(2)   – 0.7206 0.00009393 

(2,0,2) 

p(1)      1.3055 
p(2)   – 0.3062 
q(1)      0.6447 
q(2)   – 0.2022 

0.00010666 

(0,2,2) cannot be estimated with these data – 

(2,2,2) 

p(1)   – 0.4740 
p(2)   – 0.5481 
q(1)      0.4746 
q(2)   – 0.3658 

0.00012311 

(1,0,2) cannot be estimated with these data – 

(1,2,0) p(1)   – 0.6501 0.00016255 

(2,0,1) 
p(1)   0.8022 
p(2)   0.1971 
q(1)   0.1456 

0.00010206 

(2,1,0) p(1)   – 0.4047 p(2)   – 0.2017 0.00009108 

(2,1,1) 
p(1)   0.5321 
p(2)   0.2363 
q(1)   1.1833 

0.00006983 

(1,2,1) cannot be estimated with these data – 

(1,1,2) 
p(1)   – 0.5343 
q(1)   – 0.4204 
q(2)      0.5306 

0.00007867 

(2,1,2) 

p(1)    – 0.9599 
p(2)   – 0.4592 
q(1)   – 0.8780 
q(2)      0.0874 

0.00006792* 

 
 
 
 
 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

 

Fig (1):Series Water Consumption 
 

Fig (2):Series Number of Participants 

 

 

 

 

 

 

 

 

 

 
 
 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                           Vol. 26 (1) 2013 

 التنبؤ بالسالسل الزمنیة بإستخدام اسالیب بوكس جینكیز
 

 حازم منصور كوركیس
 رغد إبراھیم

 جامعة بغداد / )ابن الھیثم( للعلوم الصرفة كلیة التربیة /الریاضیات  علوم قسم
 

 2012االول  كانون  9: قبل البحث في ، 2012تشرین االول    10: استلم البحث في
 

 الخالصة
اختصاراً  باالنحدار الذاتي واالوساط  ARIMAجینكیز تمثل -في ھذا البحث قدمنا نبذة مختصرة عن نماذج البوكس        

المتحركة وھو اسلوب جید للتنبؤ بالسالسل الزمنیة المستقرة وغیر المستقرة. ووفقا ً للبیانات التي حصلنا علیھا من دائرة 
نموذج السلسلة اعداد المشتركین ومن ثم تحدید ا االخرىولى سلسلة استھالك الماء وماء بغداد قمنا بنمذجة سلسلتین: اال

 نموذج.الالختیار ا اً االفضل باالعتماد على اختیار أقل متوسط مربعات الخطأ معیار
 

المشترك، دالة التنبؤ، بوكس جینكیز، النماذج المختلطة، انحدار ذاتي، اوساط متحركة، دالة االرتباط  :  الكلمات المفتاحیة
 االرتباط المشترك الجزئي الذاتي.

 
 
 

345 | Mathematics 


	قسم علوم الرياضيات / كلية التربية للعلوم الصرفة (ابن الهيثم) / جامعة بغداد
	في هذا البحث قدمنا نبذة مختصرة عن نماذج البوكس-جينكيز تمثل ARIMA اختصارا ً بالانحدار الذاتي والاوساط المتحركة وهو اسلوب جيد للتنبؤ بالسلاسل الزمنية المستقرة وغير المستقرة. ووفقا ً للبيانات التي حصلنا عليها من دائرة ماء بغداد قمنا بنمذجة سلسلتي...