Applied Mathematics, 2011, 2, 633-645
doi:10.4236/am.2011.25084 Published Online May 2011 (http://www.SciRP.org/journal/am)
Uses of the Buys-Ballot Table in Time Series Analysis
Iheanyi S. Iwueze1, Eleazar C. Nwogu1, Ohakwe Johnson2, Jude C. Ajaraogu3
1Department of Statistics, Federal University of Technology, Owerri, Nigeria
2Department of Mathematics and Statistics, Anambra State University, Uli, Nigeria
3Department of Mathematics and Statistics, Federal Polytechnic, Owerri, Nigeria
Received December 14, 2010; revised March 30, 2011; accepted April 3, 2011
Abstract
Uses of the Buys-Ballot table for choice of appropriate transformation (using the Bartlett technique), assess-
ment of trend and seasonal components and choice of model for time series decomposition are discussed in
this paper. Uses discussed are illustrated with numerical examples when trend curve is linear, quadratic and
exponential.
Keywords: Buys-Ballot Table, Trend Assessment, Assessment of Seasonality, Periodic Averages, Seasonal
Averages, Data Transformation, Choice of Model
1. Introduction
A time series is a collection of observations made se-
quentially in time. Examples occur in a variety of fields,
ranging from economics to engineering and methods of
analyzing time series constitute an important area of sta-
tistics [1]. Time series analysis comprises methods that
attempt to understand such time series, often either to un-
derstand the underlying context of the data points (Where
did they come from? What generated them?), or to make
forecasts. Time series forecasting is the use of a model to
forecast or predict future events based on known past
events.
Methods for time series analyses are often divided into
three classes: descriptive methods, time domain methods
and frequency domain methods. Frequency domain me-
thods centre on spectral analysis and recently wavelet
analysis [2,3], and can be regarded as model-free ana-
lyses. Time domain methods [4,5] have a distribution-
free subset consisting of the examination of autocorrela-
tion and cross-correlation analysis.
Descriptive methods [1,6] invo lve the separation of an
observed time series into components representing trend
(long term direction), the seasonal (systematic, calendar
related movements), cyclical (long term oscillations or
swings about th e trend) and irregular (uns ystematic, s hor t
term fluctuations) components. The descriptive method
is known as time series decomposition. If short period of
time are involved, the cyclical component is superim-
posed into the trend [1] and the observed time series

,
t
X
can be decomposed into the trend-cycle
component

1, 2,,tn


t
M
, seasonal component and the ir-
regular/residual component .

t
S
t

t
e
tt
Decomposition models are typically additive or multi-
plicative, but can also take other forms such as pseudo-
additive/mixed (combining the elements of both the ad-
ditive and multiplicative models).
Additive Model: t
X
MSe

 (1)
Multiplicative Model: tttt
X
MS e
tt
(2)
Pseudo-Additive/Mixed Model; tt
X
MSe (3)
The pseudo-additive model is used when the original
time series contains very small or zero values. For this
reason, this paper will discuss only th e additive and mul-
tiplicative models.
As far as the traditional method of decomposition is
concerned (to be referred to as the Least Squares Method
(LSE)), the first step will usually be to estimate and eli-
minate t
M
for each time period from the actual data
either by subtraction for Equation (1) or division for
Equation (2). The de-trended series is obtained as t
X
ˆt
M

for Equation (1) or ˆ
tt
X
M for Equation (2). In
the second step, the seasonal effect is obtained by esti-
mating the average of the de-trended series at each sea-
son. The de-trended, de-seasonalized series is obtained as
t
ˆ ˆ
tt
X
MS

 for Equation (1) or

ˆ ˆ
ttt
X
MS for Equa-
tion (2). This gives the residual or irregular component.
Having fitted a model to a time series, one often wants to
see if the residuals are purely random. For detailed dis-

I. S. IWUEZE ET AL. 634
t
s

cussion of residual analysis, see [4,7].
It is always assumed that the seasonal effect, when it
exists, has periods. That is, it repeats after s time per iods.
,forall
ts t
SS
 (4)
For Equation (1), it is convenient to make the further
assumption that the sum of the seasonal components over
a complete period is zero.
10
s
tj
j
S


 (5)
Similarly, for Equations (2) and (3), the convenient
variant assumption is that the sum of the seasonal com-
ponents over a complete period is s.
1
s
tj
j
S



 (6)
It is also assumed that the irregular component t is
the Gaussian e

2
1
0,N

white noise for Equation (1),
while for Equation (2), is the Gaussian
t
e


2
2
N1,

white noise.
This paper discusses the uses of the Buys-Ballot table
for 1) choice of appropriate transformations (using the
Bartlett technique) 2) assessment of trend and seasonal
components and 3) choice of model for time series de-
composition. We describe in great detail the Buys-Ballot
table in Section 2 for better understanding of the methods
used to achieve these objectives.
When any of the assumptions underlying the time se-
ries analysis is violated, one of the options available to
an analyst is to transform the study series. The choice of
appropriate transformation for a study series using Buys-
Ballot table is described in Section 3. The presence and
nature of trend and seasonal component of a study series
can be inferred from the plot and values of the periodic
/annual and seasonal averages. Assessment of trend and
seasonal component of the actual series from the Buys-
Ballot table was discussed in Sections 4 and 5 respec-
tively. A major problem in the use of the descrip tiv e time
series analysis is the choice of appropriate model for
time series decomposition. This problem was addressed
using Buys-Ballot table in Section 6. Numerical exam-
ples are also given to illustrate these uses.
2. Buys-Ballot Table
A Buys-Ballot table summarizes data to highlight sea-
sonal variations (Table 1). Normally, each line is one pe-
riod (usually a year) and each column is a season of the
period/year (4 quarters, 12 months, etc), A cell,

,i

j,
of this table contains the mean value for all observations
made during the period i at the season j. To analyse the
data, it is helpful to includ e the period and seasonal to tals


..
and
i
TT
j
, period and seasonal averages

.i
X
and

.
j
X
, period and seasonal standard deviations

.
ˆi

and

.
ˆ
j

, as part of the Buys-Ballot table. Also included
for purposes of analysis are the grand total

, grand
mean

..
T


..
X
and pooled standard deviation

..
ˆ

.[8] cred-
its these arrangements of the table to [9], hence the table
has been called the Buys-Ballot table in the literature.
For easy understanding of Table 1, we define the row
and column totals, averages and standard deviations as
follows:


.1
1
.1
1
,1,2,,
,1,2,,
s
iisj
j
m
jisj
i
TX i
TX j










,
,
m
s
Table 1. Buys-Ballot Table.
Season (j)
Period (i) 1 2 …j … s i.
T i. X .
ˆi

1 1
X 2 X …j X …
s
X 1. T 1. X 1.
ˆ

2 1
s
X 2s X …
s
j
X

… 2
s
X 2. T 2. X 2.
ˆ

3 21
s
X 22s X …2
s
j
X

… 3
s
X 3. T 3. X 3.
ˆ

… … … …… … … … … …
i

11is
X



12is
X …

1isj
X

 …

1iss
X

 .i
T .i X . ˆi

… … … …… … … … … …
m

11ms
X



12ms
X …

1msj
X

 … ms
X .m T .m X . ˆm

.j
T .1 T .2 T ….j T … .
s
T .. T - -
.j
X .1 X .2 X ….j X … .
s
X - .. X -
.
ˆj

.1 ˆ

.2 ˆ

…. ˆj

… . ˆ
s

- - .. ˆ


I. S. IWUEZE ET AL.635
formation can be a complex issue and the usual statistical
technique used is to estimate both the transformation and






.
.... .
11
... ..
...
2
.. 1
1
2
..
1
1
2
..
1
11
,,1,2,,,
,1,2,,, ,
1
ˆ,1,2,,
1
1
ˆ,1,2,,
1
1
ˆ1
ms
i
iji
ij
j
j
s
ii isj
j
m
jj
isj
i
ms
isj
ij
T
TTTX im
s
TTT ,
X
jsX n
mmsms
XXi m
s
ms
X
Xj s
m
XX
n










 
 














where ,1,2,,
t
X
tn
m is the number ois the observed value of the se- ries, f periods/years,
s
is the perio-
le
d seasonal indices from the chosen de-
sc
dicity, he total number of observations
/samp size.
Finally, Buys-Ballot table is used to estimate the trend
component an
and nms  is t
riptive time series model. This method, called Buys-
Ballot estimation procedure uses the periodic means

., i
X
i

1, 2,,m and the overall mean

..
X
to es-
timate the trend component. Seasonal means

.,
j
X
the overall mean are usedstimate
the seasonal indices. The advantages of the Buys-t
cedure are that 1) it computes trend easily,
2) gets over the problem of de-trending a series before
computing the estimates of the seasonal effects and 3)
estimates the error variance without necessarily decom-
posing the series. For further details on the Buys-Ballot
estimation procedure, see [10-14].
3. Choice of Appropriate Tr

2, ,jm and1, to eBallo
estimation pro
ansformation
es
e measurement scale of a variable. Reasons for trans-
Transformation is a mathematical operation that chang
th
formation include stabilizing variance, normalizing, re-
ducing the effect of outliers, making a measurement
scale more meaningful, and to linearize a relationship.
For further details on reasons for transformation, see
[3,14]. Many time series analyst assume no rmality and it
is well known that variance stabilization implies normal-
ity of the series. The most popular and common are the
powers of transformations such as log ,
et
X
log ,
et
X
1,
t
X
1,
t
X
2,
t
X
2 1t
X
. Selecting the best trans-
required model for the transformed t
X
at th e sa me t i me
[15].
[16] have shown how to apply Bartlett transformation
technique [17] to time series data using the Buys-Ballot
ta
hms of the group
sta
ble and without considering the time series model struc-
ture. The relation between variance and mean over sev-
eral groups is what is needed. If we take random sam-
ples from a population, the means and standard devia-
tions of these samples will be independent (and thus un-
correlated) if the population has a normal distribution
[18]. Furthermore, if the mean and standard deviation are
independent, the distribution is normal.
[16] showed that Bartlett’s transformation for time se-
ries data is to regress the natural logarit
ndard deviations


.
ˆ,1,2,,
iim
  against the natu-
ral logarithms of the group means

.,1,2,, i
X
im
and determine the slope,

, of the relationship.
..
ˆ
loglog error
eie i
X   ) (7
For non-seasonal data that require transf
split the observed time series ormation, we ,,1,2,
t
X
tn chrono-
logically into m fairly equal different parts and compute


.,1,2,, i
X
im and

.
ˆi
 for the
parts. For seasonal data with the length of the periodic
itions the
observed data into m periods or rows for easy application.
[16] showed that Bartlett’s transformation may also be
regarded as the power transformation

1
log ,1
et
t
X
Y


m
interval, s, the Buys-Ballot table naturally part
,1 ,2,,i
,1
t
X









 (8)
Summary of transformations for various values of

is given in Table 2. However, [16] concluded that it is
better to use the estimated value of the slope,

,
rectly in the power transformation (Equation (8)) than to
approximate to the known and popular logarithmic, uare
root, inverse, inverse of the square root, squares and in-
verse of the squares transformations.
An example that requires logarithmic transformation is
the Nigerian Stock Exchange (NSE) All
di-
sq
Shares Index
(1985 – 2005) that is listed as Appendix A [19]. Sum-
mary of the regression analysis of


ˆ
logeI

on


log ,
ei
X
1, 2,, 21i

 is given in Table 3.
mation for some values of β.
S/No 1
Table 2. Bartlett’s transfor
2 3 4 5 6 7
β 0 1/2 1 3/2 2 3 –1
Transformation No transformation t X loget
X 1t
X 1t
X 2 1t X 2
t
X

I. S. IWUEZE ET AL. 636
Table 3. Ression analysis of on egr ˆ gei. σlo log ,
.
X1,2,L,
ei i for various transformations of the Nigerian Stock Ex-
change (NSE) All Shares Index (1985-2005).
21
testtfor ˆ1.0  
Transformations
2
Regression equation
R
df valuet
:Original
t
X ˆ log2.5797 1.0260log
eie i
X    0.94 19 0.44
log
te
YX
tˆ
log2.8857 0.2490log
ei ei
Y    0.02 19 N/A

ˆ
1ˆ
, 1.0260
tt
WX



 ˆ
log6.2186 0.0794log
eie i
W    0.00 19 N/A
It is clear from Table 3 that we can approximate the
value ofand the suitable trans-
rm . Regression
an
ˆˆ 1.0260 to1.0  
ation using Table 2 is
t
fo logYX
tet
alysis summaries for loget YX and


ˆ
1,
tt
WX
 

ˆ1.0260
 able 3. The transforma-
tion

ˆ
1ˆ
, 1.0260
tt
WX
 
 srela-
tionships between the stanions an
/yearlyroupings.
4. Assessment of Trend
are also given in T
u
dard
rely rem
deviat
oves the
d the means
for the row g
he time plot of a time series, according to [1] reveals
describe the pattern in
es, the time plot of the pe-
d additive model)
ulation of 100 values from the ad-
T
the nature of the trend which can
e series. Among other featurth
riodic means follows the same pattern as the plot of the
entire series with respect to the trend. Therefore, instead
of looking at the plot of the entire series, one may look at
only the plot of the period/annual means in order to
choose the appropriate trend. We use the following ex-
amples to illustrate this.
4.1. Linear Trend
he data of Table 4 (linear trend anT
and Figure 1 is a sim
itive model d
ttt
X
abtS e  (9)
with 5.0,a 0.2,b 11.5,S 22.5,S 33.5,S

44.5S an
. The dat d t
e b
a of Tabl
eing Gau
e 4 (linear
ssian
tr

0,1N white
end and multiplica-noise
tive mulat
from ultiplicative model

ttt
odel) and Figure 2 is a simion of 100 values
the m
X
abtS e 
(10)
with 5.0,a 0.2,b 10.6,S 
21.1,S30.9,S

S and
4
noise1.4 . Listed t
e being
in Table 4
Gaussian
are the

1.0,0.01 white
eriodic/row means
N
p

., i1,2,,
X
im
of
a

plottted data, while the s of thectual series and perians are
Figures 12
he simula
timeodic/row me
shown in and , respectively. It is clear that
the tim plot ofe .i
X
for a with linear trend curve
mimics the time plot of the etire series arow av-
erages
dat
nnd the


.i
X
can therefore, be used to estimate trend.
tire series can be d
n p
The estimate of the parameters of the trend of the en-
etermined from the estimate of the
trend of the periodic means by recognizing that periodic
averages are centred at the midpoints of the periodic in-
tervals. Thus, whilesuccessive values in the actual series
are one unit of time apart starting from 1t, successive
values ieriodic average series

.i
X
are s units of
tim e apart starting from

12ts . Thus, periodic
averages are derived from the original series by transla-
tion of the original series by a factor


12s and
dilation by a factor s. That is, periodic averages,
.,1,2,, i
X
im may be looked at as th e values of the
original series at times ,1,2,,
i
ti m.t is, Tha
.i iti.
X
Xabt (11)
where

211
13 15 1
,,,,
22 2
i
ms
sss
t


1, 2, 3,, im
2, for

, Hence,
.1) 1 i
s
X (2
1()
2
''
i
t
i
Xab
s
ab bsi
abi







 



(12)
where
2
1
'2
s
aab





, 'bbs or 'bbs

,
1
'2
s
aab





.
As an illustration, we observe from Figure 1 that
4,s

0.8003,b


4.6971.a


Hence, '0.8003bbs 4
0.2001,









'aa 14.69711 4 1 2bs 2 0.200
4.9972

.
The estimate of the parameters of the trend of the en-
tire series can be determined from the estimate of the
trend of the periodic means in quadratic [12] and expo-
nential [13] trend curves as shown below.

I. S. IWUEZE ET AL.637
able 4. Periodic means of
Linear Equations (9) and (10) Quadraq(16) and ( 17)
T
tic E
the simulated series.
uations (13) and (14) Exponential Equations
Period/Year

i Add.

i.
X Mult.

i.
X Add.


i.
X Mult.


i.
X Add.

i.
X Mult.


i.
X
1 5.631 10.647 11.260 5.950 56.900 8.180
2 6.106 6.396 67.800 19.830 11.197 11.580
3 6.200 760 187 710
4 8.260 9.288 124.300 90.800 13.728 15.680
168. 101.
10.224. 190.
10.292. 264.
10.
11 13.
13 14.
1232. 1319.
1580. 1720.
947 6.147 90.33.12.10.
5 8.320 7.340 400 000 14.101 12.210
6 9.568 060 800 000 15.755 16.580
7 10.359 911 000 000 17.053 18.020
8 11.267 11.881 370.500 347.200 18.576 19.670
9 11.519 417 459.600 365.000 19.562 17.450
10 13.019 14.206 561.100 577.000 21.922 24.230
396 14.125 672.600 667.000 23.299 24.700
12 13.937 12.031 795.600 629.000 24.989 21.380
520 13.255 929.800 785.000 26.883 24.280
14 16.377 18.476 1076.400 1203.000 30.227 34.800
15 16.880 18.418 900 000 32.408 35.700
16 17.583 17.168 1400.800 1324.000 34.995 34.100
17 18.712 20.478 400 000 38.232 42.460
18 19.977 24.094 1771.200 2168.000 41.850 51.580
19 20.258 20.132 1972.300 1940.000 44.740 44.800
20 19.843 14.640 2183.900 1517.000 47.222 34.110
21 21.522 23.094 2408.800 2558.000 52.110 56.400
22 21.978 20.834 2643.600
2448.000 56.098 53.200
23 22.310 16.033 2889.600 1958.000 60.330 42.130
24 24.741 30.636 3148.800 3983.000 67.052 85.100
25 24.467 23.045 3416.500 3156.000 71.492 67.200
..
X 15.100 15.162 1221.600 1174.900 32.266 32.373
Note: Add = Additive; Mult =plicative Multi
(a) Actual series (b) Periodic means
Figure 1. Time plot of actual series and periodic means of simulated series using Equation (9).

I. S. IWUEZE ET AL. 638
(a) Actual series (b) Periodic means
Figure simulated seri (10).
.2. Quadratic Trend
The data of Table 4 (quadratic trend and additive model)
and Figure 3 is a simulation of 100 values from the ad-
ditive model
t
2. Time plot of actual series and periodic means ofes using Equation
4

2
tt
X
abtctSe  (13)
with 4,s 5.0,a 0.35,b 0.35,c 1=50, S

2= 3S0, 3
S80, S460 and


0, 1. The
plicative mo-
t
eN
ultidata of Table 4 (quadratic trend and m
del) and Figure 4 is a simulation of 100 values from the
multiplicative model
t

2
tt
X
abtct Se  (1 ) 4
with 4,
s 5.0,a 0.35,b 0.35,c 10.6,S

21.1, SS 30.9, 41.4S and
le 4 are th
2
1,0.333)
ow means
(
t
eN
e periodic/rwhite noise. Listed in Tab

.i,1,2,,
X
im of the simulated data, while the
time plots of the actual series and periodic/row means are
shown in Figure 3 and Figure 4, respectively.
Figures 3 and 4 show that the graphs of the periodic
means follow the same pattern as the plot of the actual
series with respect to quadratic trend in b
ultiplicative m, as in the
ayook at only the plo periodic me
means by recognizing that periodic
averages are centred at the midpoints of
tervals. That is, periodic averages,
oth the additive
and models. Thuslinear trend,
one m lt of theans in
order to choose the appropriate trend. As in linear trend
also, the estimate of the parameters of the trend of the
entire series can be determined from the estimate of the
trend of the periodic the periodic in-
.,1,2,,
i
X
i m
e expssed in terms of the original series at times
as
may bre
,1,2,,
i
tm
2
.1it i

2
2
22
2
(21) 1(21) 1
22
11
22
(1) ()
'' '
isis
ab c
ss
ab c
bscssic si
abici
 

 




 


 
 
(15)
where 2
11
',
22
ss
aabc


 





'1bbscss 
, the length of the pe-
and
Figure 4
riodic interval s = 4, the parameters of the trend of the
periodic averag

2
'ccs
As an illustration, from
es are'6.65a
i
X
Xabtct
41,

'5.5799b and 'c
5.5992

. Hence, the parameters of the actual series are:

2
5.5992c
cs

2
0.3500,
4
1bcss
bs

2
2
5.57990.350044 10.3451
4
11
22
41 41
6.6541 0.34510.3500
22
5.3490
ss
aab c

The data of T (exponential trend and additive mo-
del) and Figure 5 is a simulation of 100 v
additive model



 




 








4.3. Exponential Trend
able 4alues from the

I. S. IWUEZE ET AL.639
(a) Actual series (
Figure 3. Plot of actual series and periodic means of simulated series using Equation (13).
b) Periodic means
(a) Actual series (b) Periodic means
eans of simulated series using Equation (14). Figure 4. Plot of actual series and periodic m
(a) Actual series (b) Periodic means
Figure 5. Actual series and periodic means of simulated series from Equation (16).

I. S. IWUEZE ET AL. 640
t
ct
tt
X
beS e (16)
with 4,s 10,b 0.02,c 11.5,S 22.5,S

e data of
odel) and
3
S
Ta 3.5, S ble 4 (expon
44.5
ential tr
and t
eN
end and m

0, 1. Th
ltiplicative mu
Figure 6 is a simulation of 100 values from the multi-
plicative model
t
ct
tt
X
beSe (17)
with 4, s 10,b 0.02,c 10.6,S 21.1,S

3
S
noise. 0.9, 4 S
Listed in Ta1.4 and ble 4 a

2
,0.333
odic/row
1
t
eN
re the peri white means

., i
X

1,
of the act
2, ,im
ual se
of the sim
ries and pe
ulate
riodic/row m
d data, while the time
eans are shown in
plots
Figures 5 and 6, respectively.
The corresponding graphs of the actual series and th
periodic means, sho 5 and 6, indicat
clearly that the pattern in thehe periodic means
is similar to that
nd multiplicative models.
As in linear and quadratic trend curves, the es timate of
the parameters of the exponential trend of the entire se-
ries can be determined from the estimate of the trend of
the periodic means. That is, periodic averages,
e
e wn in Figures
plot of t
of the actual series in both the additive
a
., i
X
i

inal 1, 2,, m
series at tim
may be expressed in terms of the orig
es as ,1,2,,
i
ti m
(2 1)1
2
.
1()
2
i
i
is
c
ct
it
s
ccsic i
XXbebe
beebe




 
 

 





(18)
where
1
2
'
s
c
bbe




 and 'ccs

.
As an illustration, from Figure 6 the length of the pe-
riodic interval s = 4, the parameters of the trend of the
periodic averages are and 0.0780c


9.9323,b


.
Hence, the parameters of the actual series are:
141
0.0195
22
0.0780 0.0195,
4
9.9323 10.2273
s
c
c
cs
bbe e
  
 
 

 


5. Assessment of Seasonal Component
The seasonal component consists of effects that are rea-
sonably stable with respect to timing, direction and mag-
nitude. Seasonality in a time series can be identified from
the time plot of the entire series by regularly spaced
peaks and troughs which have a consistent direction and
approximately the same magnitude every period/year, r
lative to the trend.
onal effect, the
overall average
e-
For time series which contain a seas


..
X
and the seasonal average

.,
j
X

1, 2,,js
the effects either as a
of the Buys-Ballot table are used to assess
difference

...j
X
X or as a ratio


...j
X
X
sonal averages a
. That is, the deviations of the differences sea-
nd the overall average (additive model)
from zero or the ratios of the seasonal averages to the
overall average from unity (multiplicative model) is used
to assess the presence of seasonal effect. The wider the
deviations, the greater the seasonal effect.
This is illustrated below with stimulated time series
data for the additive and multiplicative models when
trend-cycle components are assumed 1) linear (Equations
(9) and (10), respectively) 2) quadratic (Equations (13)
and (14), respectively) and 3) exponential (Equation s (16)
and (17), respectively). The assessed values of the se a-
(a) Actual series
(b) Periodic means
ns of simulated series from Equation (17). Figure 6. Actual series and periodic mea

I. S. IWUEZE ET AL.641
sonal effects from these series are given in Table 5 while
rresponding in Figures 7, 8 and their cographs enare giv
respectively. As Table 5 and Figures 7, 8 and 9 show, 9
the patterns of the deviations ...j
X
X (additive model)
of the seasonal averages (.
j
X
) from the overae
(ll averag
..
X
) and ratios ..j.
X
X (multiplicative model) mim-
ics/follow those of the actual seasonal indices
j
S used
in the simulation in all series. Thus, an analyst interested
in studying the seasonal effect in any study series only
eeds to look at either n..j.
X
X or ..j.
X
X to deter-
ive test
mine if there is seasonal effect or not.
However, this should not be used as a conclus
for the presence of or otherwise of seasonal effect in a
study series. The use of seasonal averages to measure
seasonal effect is most appropriate in a series with no
trend. When trend dominates other components in any
series, the true seasonal effect may be visible from
...j
X
X or ...j
X
X. Therefore, this assessment pro-
cedure should be used with gr eat caution.
6. Choice of Appropriate Model
te
multiplicative ular varia
change as
hen the origi-
na
parameters of the Buys-Ballot table? The relationship
between the seasonal means
Traditionally, the ime plot of the entire series is usd to
make the appropriate choice between the additive and
models. In some time series, the amplitude
of both the seasonal and irregtions do not
the level of the trend rises or falls. In such
cases, an additive model is appropriate. In many time
series, the amplitude of both the seasonal and irregular
variations increases as the level of the trend rises. In this
situation, a multiplicative model is usually appropriate.
The multiplicative model cannot be used w
l time series contains very small or zero values. This is
because it is not possible to divide a number by zero. In
these cases, a pseudo-additive model combining the ele-
ments of both the additive and multiplicative models is
used.
How can an appropriate model be obtained from the

.,1,2,, j
X
js and the
Table 5. Actual values of seasonal effects

j
S
, deviations of seasonal averages from the overall averages


.
j..
X
X and
ratios of seasonal averages to the overall averages


.
j..
X
X for the simulated series.
Linear Quadratic Exponential
Additive
Equation (9) Multiplicative Equation (10) Additive Equation (13) Multiplicative Equation (14) Additive Equation (16) Multiplicative Equation (17)
Season j
j
S ..j XX
.j
S ..j XX
.j
S..j XX.j S..j XX.
j
S ..j.
X
X j
S ..j XX.
1 –1.5 –1.71 0.61 0.62 –50–103.0 0.60.60 –0.6–1.47 0.6 0.61
2 2.5 2.24 1.08 1.01 30 12.00 1.10.99 1.1 0.62 1.1 1.01
3 3.5 3.47 0.93 0.90 80 97.00 0.90.90 0.9 1.09 0.9 0.89
4 –4.5 –4.01 1.38 1.47 –60–7.00 1.41.51 –1.4–0.24 1.4 1.50
(b) Multiplicative model
effects when trend is linear.
(a) Additive model
Figure 7. Assessment of seaso nal

I. S. IWUEZE ET AL. 642
(a) Ad ditidel ve mo( Mcatil
re 8ssment oason wn tr qdratic.
b)ultiplive mode
Figu. Assef seal effectheend isua
(a) Additive model (b) Multiplicative model
Fi l.
seasonal standard deviations
gure 9. Assessment of seasonal effect when trend is exponentia


.
ˆ,1,2,,
jjs
 
odel. An additive m
l standard deviations
ase relative to any increase
eans. On the other ha
appropriate when th
ow appreciable increa
se/decrease in the seasonal
strated in Table 6 for
odels. Time plots of
gives
an indication of the desired model is
appropriate when the seasona show
no appreciable increase/decre
or decrease in the seasonal mnd, a
multiplicative model is usuallye sea-
sonal standard deviations shse/de-
crease relative to any increa
means. This is vividly demonaddi-
tive and multiplicative m.
j
X
and
.
ˆ,(1,2, ,)
jjs


for values of Table 6
are given in Figures 10 th 12
.
As Tab les 6 an d Figures 10 through 12 show, the o
servations are true furves and for both
additive and multiplicative models. For the multiplicative
model, the plot of standard deviation,
rough
b-
or all trending c
(.
ˆ
j

) clearly show
appreciable increase or decrease as the seasonal averages
) change. For the additive model, the plot of (.
ˆ
j

)
of sea-
(.
j
X
show no appreciable change relative to the plot
sonal ave rages (.
j
X
) in all the series. However, as
earlier, when trend dominates other components, the sea
sonal standard deviations may not follow the observed
pattern and therefore, may not be used effectively for
choice of appropriate model for decomposition. Therefore,
the use of the plot of the seasonal averages and standard
deviations as basis for the choice of appropriate model
should be done wi
noted
-
th great care.

I. S. IWUEZE ET AL.643
Tab ies.
Linear Quadratic Exponential
le 6. Seasonal means and standard deviations for the simulated ser
Additive
Equation (9) Multiplicative Equation (10) Additive Equation (13)Multiplicative Equation (14) Additive Equation (16) Multiplicative Equation (17)
Season j
.j
X ˆj

.j
X ˆj

.j
Xˆj

.j
Xˆj

.j
X ˆj

.j
X ˆj

1 13.40 6.05 9.41 5.13 1119103470870730.8 17.95 19.61 13.28
2 17.34 5.94 15.25 7.89 123410541161106832.8818.14 32.56 20.08
3 18.57 6.07 13.7 7.83 1319107410581113c 18.72 28.68 21.11
4 11.09 5.84 22.29 10.55 121510941772174032.0318.88 48.64 31.8
(a) Additive model
b) Multiplicative model (
Figure 10. Line plot of
.j
X
and ˆ
.j
σ for simula ted data when trend is linear (Equations 9 and 10).
(a) Additive m odel (b) Multiplicativeodel
Figure 11. Seasonal means
m


.j
X


ˆ
j
σ and standarviations d deof simulated series from quadre
14).
atic trnd (Equations 13 and

I. S. IWUEZE ET AL. 644
(a) Additive model
(b) Multiplicative model
Figure 12. Seasonal means


.j
X
and standard deviations


ˆ
j
σ of simulated series from exponential trend.
7. Conclusions
This paper has examined four uses of the Buys-Ballot
table. Uses examined in detail include 1) data transfor-
mation 2) assessment of trend 3) assessment of seasonal-
ity and 4) choice of model for decomposition. Use of
Buys-Ballot table for the estimation of trend and compu-
tation of seasonal indices was not discussed in details.
For data transformation, the relationship between period
/annual averages and standard deviations was used. As-
sessment of trend is based on the period/annual averages
while the assessment of the seasonal effect is based on
the seasonal and overall averages. The choice of appro-
priate model for decomposition is based on the seasonal
averages and standard deviations.
8. References
lysis o
tion,” Chapman and Hall/CRC Press, Boca Raton, 2004.
[2] D. B. Percival and A. T. Walden, “Wavelet Methods for
Time Series Analysis,” Cambridge University Press,
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[3] M. B. Priestley, “Spectral Analysis and Time Series
Analysis,” Academic Press, London, Vols. 1-2, 1981.
[4] G. E. P. Box, G. M. Jenkins and G. C. Reinsel, “Time
Series Analysis, Forecasting and Control,” 3rd Edition,
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[5] W. W. S. Wei, “Time Series Analysis: Univariate and
Multivariate Methods,” Addison-Wesley, Redwood City,
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[6] M. G. Kendal and J. K. Ord, “Time Series,” 3rd Edition,
Charles Griffin, London, 1990.
[7] G. M. Ljung and G. E. P. Box, “On a Measure of Lack of
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[8] H. Wold, “A Study in the Analysis of Stationary Time
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1938.
[9] C. H. D. Buys-Ballot, “Leo Claemert Periodiques de
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[10] I. S. Iwueze and A. C. Akpanta, “Effect of the Logarith-
mic Transformation on the Trend-Cycle Component,”
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2414-2422.
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[15] G. E. P. Box and D. R. Cox, “An Analysis of Transfor-
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I. S. IWUEZE ET AL.645
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[19] Central Bank of Nigeria, The Statistical Bulletin, Vol. 18,
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Appendix All Shares Index of the Nigerian Stock Exchange (1985-2005)
Month
.i
X .
ˆi

Year Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
1985 111.3 112.2 113.4 115.6 116.5 116.3117.2117.0116.9119.1124.6 127.3 117.34.7
1986 134.6 139.7 140.8 146.2 144.2 147.4150.9151.0155.0160.9163.3 163.8 149.89.5
1987 166.9 166.2 154.2 196.1193.4193.0194.9176.917.9
1988 .1 615.9
256.9 257.5 257.1 259.2269.
0 349.3 356.0 362.0 382.3 417.4445.63.1
6.2 6298.5 6113.9 6033.9 5892.15817.
5494.8 5376.5 5456.2 5315.7 5315.7 5977.
3 20128.9 15560.02502.0
161.7 157.5 154.8193.4 190.9
190.8 191.4 195.5 200199.2 206.0211.5217.224.1228.5231.4 233.6 210.8
1989 239.7 251.0 2281.0279.9298.4311.2 325.3 273.926.1
4463.6468.2480.3502.6 513.8 423.71990 343.
1991 528.7 557.0 601.0 625.0 649.0 651.868
1992 794.0 810.7 839.1 844.0 860.5 870.8879
1993 1113.4 1119.9 1130.5 1147.3 1186.9 1187.5118
1994 1666.3 1715.3 1792.8 1845.6 1875.5 1919.1192
1995 2285.3 2379.8 2551.1 2785.5 3100.8 3586.5431
1996 5135.1 5180.4 5266.2 5412.4 5704.1 5798.759
1997 7268.3 7699.3 8561.4 8729.8 8592.3 8459.38148.
8.0712.1737.3757.5769.0 783.0 671.683.7
.7969.31022.01076.51098.0 1107.6 931.0117.0
0.81195.51217.31310.91414.5 1543.8 1229.0131.1
6.31914.11956.02023.42119.3 2205.0 1913.2154.5
4.34664.64858.15068.05095.2 5092.2 3815.01149.0
19.46141.06501.96634.86775.6 6992.1 5955.0652.0
87682.07130.86554.86395.8 6440.5 7639.0876.0
05795.75697.75671.05688.2 5672.7 5961.9293.61998 6435.6 642
1999 94964.44946.24890.85032.55133.2 5266.4 5264.2304.3
77394.17298.97415.37164.4 8111.0 6701.0778.02000 5752.9 5955.7 5966.2 5892.8 6095.4 6466.76900.
2001 8794.2 9180.5 9159.8 9591.6 10153.8 10937.310576.410329.010274.211091.411169.6 10963.1 10185.0825.0
8.212327.911811.611451.511622.7 12137.7 11632.0637.0
62.0 15426.016500.518743.5 19319.
2002 10650.0 10581.9 11214.4 11399.1 11486.7 12440.71245
2003 13298.8 13668.8 13531.1 13488.0 14086.3 14565.5139
2004 22712.9 24797.4 22896.4 25793.0 27730.8 28887.427062.123774.322739.723354.823270.5 23844.5 24739.02131.0
2005 23078.3 21953.5 20682.4 21961.7 21482.1 21564.821911.022935.424635.925873.824635.9 24085.8 22877.01563.0
.j
X 5533.1 5648.2 5579.6 5818.3 5981.3 6216.66099.86077.66129.16357.26329.4 6472.8 6020.3
.
ˆj

6955.2 7116.6 6743.2 7281.1 7539.4 7782.97503.37246.97369.47761.47609.5 7723.7 7235.3
Source: Statistical Bulletin of the Central Bank of Nigeria (CBN, 2007)