J. Service Science & Management, 2010, 3 : 91 -97
doi:10.4236/jssm.2010.31011 Published Online March 2010 (http://www.SciRP.org/journal/jssm)
91
Padé Approximation Modelling of an
Advertising-Sales Relationship*
M. C. Gil-Fariña, C. Gonzalez-Concepcion, C. Pestano-Gabino
Department of Applied Economics, Faculty of Economics and Business Administration, University of La Laguna (ULL),
Campus of Guajara, Tenerife, Spain.
Received December 18th, 2009; revised January 21st, 2010; accepted February 20th, 2010.
ABSTRACT
Forecasting reliable estimates on the future evolution of relevant variables is a main concern if decision makers in a
variety of fields are to act with greater assurances. This paper considers a time series modelling method to predict
relevant variables taking VARMA and Transfer Function models as its starting point. We make use of the rational
Padé-Laurent Approximation, a relevant type of rational approximation in function theory that allows the decision
maker to take part in the building of estimates b y providing the available info rmation and expectations for the decision
variables. This method enhances the study of the dynamic relationship between variables in non-causal terms and al-
lows for an ex ante sensibility analysis, an interesting matter in applied studies. The alternative proposed, however,
must adhere to a type of model whose properties are of an asymptotic nature, meaning large chronological data series
are required for its efficient application. The method is illustrated through the well-known data series on advertising
and sales for the Lydia Pinkham Medicine Company, which has been used by various authors to illustrate their own
proposals.
Keywords: Time Series Modelling, Expectations, Economics, Numerical Methods, Padé Approximation
1. Introduction
The possibility of forecasting reliable estimates for the
future behaviour of relevant variables is main concern to
decision making in numerous fields (including business,
industry, energy, environmental, government agencies
and medical and social network fields). Consequently,
from a scientific standpoint, it is necessary to investigate
alternative methods that can prov ide estimates while also
introducing them into a technological system that allows
the decision maker himself to participate in the composi-
tion of said predictions. Many researchers have at-
tempted to satisfy this requirement from different per-
spectives, such that the prediction problem is always
present in any generic data-mining task. And yet, the
technique selected fo r use depend s on the availab ility an d
type of data in relation to the hypotheses of the methods
that sustain the desired technique and which yield dif-
ferent degrees of accuracy, time horizons and different
computational and social costs. The use of a relevant
technique within the scope of rational approximation,
namely the Padé Approximation (PA), has had a gratify-
ingly enriching and stimulating effect on the study of the
dynamic relationship structure between variables, espe-
cially within the context of identifying univariate and
multivariate ARMA models (see, among others, [1–5]).
In particular, within the scope of rational model for-
mulation in causal terms, this technique acquires a spe-
cial relevance for characterizing simple models at a
computational level with which to specify the determi-
nistic part in certain time series models. At the same time,
it provides reliable initial estimates for obtaining a de-
finitive model by using more efficient, iterative methods
once the random component has been ide nti fi ed.
The choice of a particular VARMA model, namely the
Transfer Function [6] model which constitutes one of the
most widely used representations in the input-output
context of dynamic stochastic systems through the use of
polynomial rational expressions, serves to highlight the
influence that the expectations or forecasts for an ex-
planatory variable (input) exerts on the model under con-
sideration.
In this sense, the formulation of the time series analy-
sis instruments that encompass the available sampling
*This research was partially funded by Plan Nacional Projectno.
MTM2008-06671, MTM2006-14961-C05-03 and the Centro de
I
nvestigación Matemátic a d e Ca n a r ias (CIMAC).

Padé Approximation Modelling of an Advertising-Sales Relationship 92
information and the establishment of non-causal models
into which the forecasts are incorporated, that is, the
knowledge th at by way of ex ante or ex post expectations
is provided by economic theory or by the empirical evi-
dence, if any, about the model’s explanatory variable(s),
provide a basic framework for tackling the study of the
rational identification of time series models from a more
general context.
The method that allows for a sustained study of the
time series identification from an evolutionary, but not
necessarily causal, standpoint of the variables involved is
based on a generalization of the PA concept to the study
of formal Laurent series [7]. The use of this approach
allows for the study of new dynamic identification pro-
cedures by means of a single model that simultaneously
approximates both time directions in a formal Laurent
series while encompassing the classical case as a specific
one when the expectations are not included (see, for ex-
ample, [8,9 ]).
The consideration of this broader dynamic framework,
however, into which future information on the model’s
variables is incorporated, favours a continuous feedback
process through which the formulated expectations are
replaced period by period by updated information as the em-
pirical evidence modifi es or confirm s the predictions made.
It is worth noting that the models are in no way insen-
sitive to changes in the way the expectations are made;
on the contrary, these changes lead to different dynamic
behaviours insofar as it is the precise nature of the ex-
pectations that determines the explicit dynamics of the
forecasted variables.
In this sense, the possibility of offering different mod-
els for a single data series by simply modifying the ex
ante expectations made by economic agents allows us to
obtain future knowledge about their influence on the
model and therefore to contrast and compare different
dynamic specifications so as to yield an optimum model.
In short, the performance of a sensibility analysis will
permit for a contrast of the extent to which the forecasts
of a variable’s future behaviour affect the mode’s predictions
and, as a consequence , its adequacy to the empirical data.
So as to illustrate the rational Padé Approximation in
modelling time series, we develop an application for the
study of data from the advertising-sales series involving
the activity of the Lydia E. Pinkham Medicine Company
for the period 1907–1960 [10]. This series, which has
been analyzed by numerous authors to illustrate their
proposed methods, presents, as noted in [11,12], various
characteristics which make it the ideal example for
studying the relationship between both variables. Some
of the reasons that ju stify the pro minent role of this series
in studies conducted to date are, among others, the im-
portance of advertising as the company’s sole marketing
instrument, as well as the elevated advertising costs to
sales ratio (above 40%) which, in some instances, even
exceeded 80%.
This paper is structured into three sections. The first
two present certain theoretical foundations for the prop-
erties of the PA that allow for the identification of the
orders in VARMA models and TF models with expecta-
tions. We note the last section, which is devoted to the
empirical results of the study on the aforementioned ad-
vertising-sales series. We conclude the work with the
more relevant conclusions and the main references.
2. VARMA Models
A non-deterministic, k-dimensional centred process can
be expressed as a vector autoregressive moving average
model (VARMA(p, q)) if () ()
p
tq t
A
BXB Ba, where
1
() ...
p
pp
A
BIAB AB 
and
1
( )...q
qq
BBI BBBB 
are kxk polynomial matrices in the lag operator B, th at is,
the coefficients i
A
and (1, ...,;1, ...,)
j
Bipj q are
kxk matrices.
The k vector t a is a series of i.i.d. random variables
with a zero mean multivariate norm and covariance ma-
trix .
The series t
X
is said to be stationary when the zeros
of )
P
A
B（ are outside the unit circle and invertible
when the same can be said for those of )
q
BB（ .
This type of model is useful for understanding the dy-
namic relationships between the components of the series
t
X
. In this sense, one series can cause another or there
may be a feedback relationship or they may be contem-
poraneously related.
In the case of the Lydia Pinkham advertising-sales se-
ries, the joint modelling of these effects by means of the
procedures described in [13] allows the type of dynamic
relationship existing between both variables to be deter-
mined [14].
A consideration of the PA matrix method yields the
following theorem, which can be used in the first steps of
the VARMA model identification.
Theorem 1 [15].- Let t
X
be a second-order station-
ary k-dimensional process. Let - ()( )
tth
RhCovX X be
the covariance matrices of the process. Let
1
,0
1( ,)(((1))j
kh
MijRijk h

 .
Then,
~
t
XVARMA(,)pqrank 1( ,)
M
ij=
rank 1(1,1) ,/,
M
ij ijiqjp

 

Padé Approximation Modelling of an Advertising-Sales Relationship 93
3. Transfer Function Models with
Expectations
Based on the PA definitions for a formal power series
[16] and its extension to the study of formal Laurent se-
ries [7], we can provide a characterization for the identi-
fication of a TF model with expectations by means of the
Toeplitz determinants
,,1
()( )
g
fgif kj kj
Tc detc
 


Given two stationary time series ,
tt
yx, let us assume
the existence of a unidirectional causal dynamic rela-
tionship tt
x
y given by the combination of simulta-
neous and shifted effects of the input variables (including
the presence of expected values that may or may not fol-
low the same distribution as the data), and let us consider
a generalized TF model of the form:
(); ()i
ttt i
i
yvBxNvB vB


 

in which the ti
x
 refers to the present and past of the
input series (data) if 0i and to the expected values of
the input series (expectation s) if 0i, and such that the
exogenous variables represented by xt satisfy a VARMA
model.
A finite-order representation for the Impulse Response
Function (IRF), namely, () i
i
i
vB vB


 that simulta-
neously approximates both directions in time and enables
the estimation of a finite number of observations will be
characterized by the following result:
Theorem 2 [17].- Given the series () i
i
i
vB vB


,
the following conditions are equivalent:
a)
11
()
(1)(1 )
Ki
i
iH
NU
ii
ii
ii
aB
vB
qB qB








b)
,,,
()0, ()0; ()0 <H>N;
HN iKU iJM i
Tv Tv TvJM
,
()0
JM i
Tv JKMU
Due to the properties of the lag operator, the following
equivalency holds:
,,
,
() ()
() ()
HK IL
ttttt
NU M
AB AB
yxNxN
QB QB


where
,
, , ,() y ()
IL M
I
HNLKNM NUABQB 
are polynomials of the form:
,0
() ()
LM
ii
ILiMi
iI i
A
BaBQBqB



Assuming the non-existence of common roots and the
conditions for ensuring the stability of the model hold,
the problem will consist of determining, in keeping with
the sample information available and by means of a di-
rect estimation of the IRF, the best orders ,,
I
LM that
describe ()vB.
With this in mind, the steps to follow in studying the
dynamic identification for a generalized formulation of
the TF model are:
1) Obtain the estimates ˆi
v for the weights i v of the
IRF ()vB, approximating ()vB by a finite number of
terms, that is, 'ˆ
() ki
i
ik
vB vB

with ,'kk Z, normally
0, '0kk

.
2) Calculate the Toeplitz determinants associated with
the series of estimated relative weights ˆ
ˆˆ
max
i
i
i
kik
v
v



and arrange them in a tabular form (T-table) as a gener-
alization of the C-Table for the classic case.
3) Estimate the model parameters.
4. Empirical Results
The first model for the Lydia Pinkham advertising-sales
for the period 1907–1960 involving the TF models
method was proposed by [10], which assumed a unidi-
rectional causal dynamic relationship from advertising to
sales. The possible existence of a relationship in the other
direction has, however, been mentioned by other authors.
Various subsequent papers, including [14,18,19], have
illustrated the application of this method using the ana-
lyzed series as reference. This assumption of unidirec-
tional causality has been questioned on several occasions,
however, as some maintain there is a feedback relation-
ship [11,14,19,20]. That is why, in what follows, we
analyze two cases, one involving the identification of a
VARMA model, and another in which, assuming the
existence of a unidirectional causal dynamic relationship
from advertising to sales, we present various TF models
and analyze the influence on the sales trends of different
behaviour schemes or ex ante forecasts for the variable
input, which in this case is advertising.
In any case, and given that both series exhibit non-
stationarity, we present the models by taking first differences.
4.1 VARMA Models
Once the ranks for 1( ,)
M
ij, 05, 05ij  are
obtained, applying Theorem 1 to the aforementioned data

Padé Approximation Modelling of an Advertising-Sales Relationship 94
provides the results shown in Tables 1 and 2, depending
on the sample size used in the estimates. This indicates,
as per the above theorem, that the differentiated data fol-
low a VARMA (0, 1) model exchangeable with (1, 0),
according to Table 1, and a VARMA (0, 2) interchange-
able with (2, 0), according to Table 2.
Note how in Table 1, even though square (1, 1) does
not have the same value as squares (i, i), i2 due to
rounding errors in the calculations, using theoretical Padé
Matrix Approximation results we know that these values
have to coincide.
Once the model orders are calculated, the maximiza-
tion of the likelihood function allows for a determination
of efficient estimators. Using the Time Series Processor
(TSP) software package [21], the results for the estimated
VARMA (1, 0) and VARMA (2, 0) models yield:
1
0.20 0.40
0.05 0.45
44132
23893 46790
ttt
XX















and
12
0.250.550.490.04
0.09 0.510.320.08
33326
18249 45672
tt tt
XX X





 









Table 1. Orders of VARMA models. Alternative 1
0 1 2 3 4 5
0 0 0 0 0 0 0
1 2 1 0 0 0 0
2 4 2 2 0 0 0
3 5 4 2 2 0 0
4 6 5 4 2 2 0
5 7 6 5 4 2 2
Table 2. Orders of VARMA models. Alternative 2
0 1 2 3 4 5
0 0 0 0 0 0 0
1 2 2 1 0 0 0
2 3 3 3 1 0 0
3 4 3 3 3 1 0
4 5 4 3 3 3 1
5 6 5 4 3 3 3
where 1,2 ()
t
ttt
XXX and 1t
X
and 2t
X
are the first
differences of the advertising and sales data, respectively.
As for the relationship between variables, we can con-
clude that a) the value of element (1, 2) in coefficient 1
A
in both models, namely 0.40 and 0.55, indicates the exis-
tence of a relationship between sales and future advertis-
ing in a period, b) the negligible value of element (2,1) in
coefficient 1
A
, namely -0.05 and -0.09, indicates a weak
relationship between advertising and future sales, and c)
the estimated correlation between residuals 1
ˆt

and 2
ˆt

indicates that advertising and sales are contemporane-
ously related.
Figures 1 and 2 show both models.
4.2 FT Models with Expectations
Taking into account the relationship between sales and
future advertising noted in the above VARMA models,
we now build transfer function models associated with
three specific cases, depending on whether the advertis-
ing expectations follow a increasing or decreasing trend
or whether they respect the predictions of the ARMA
model for the input series (advertising), that is,
24
(1 0.2690.325)tt BBxN .
ADV ERTIS ING. F IRS T DIFFERENCES
-800
-600
-400
-200
0
200
400
600
1 6
11 16 21 26 31 36 41 46 51
Adv Dif
VARMA (1,0)
VARMA (2,0)
Figure 1. Advertising. First differences
SALES. FIRST DIFFERENCES
-800
-600
-400
-200
0
200
400
600
800
1 9
17 25 33 41 49
Sales Dif
VARMA (1,0)
VARMA (2,0)
Figure 2. Sales. First differences

Padé Approximation Modelling of an Advertising-Sales Relationship 95
Starting from a generalized TF model for the advertis-
ing (t
x
) and sales ()
t
y variables, and keeping in mind
that the series are first-order integrable, we formulate the
model
()
ttt
yvBxa
in which ()vB is approximated by a finite number of
terms from k to k’, that is, 'ˆ
() ki
i
ik
vB vB

, normally
0, '0kk.
Following the steps suggested by the method proposed
yields the following results:
Case a: The expectations follow an increasing trend
After estimating the weights ˆi v, the resulting T-table for
the series of estimated relative weights is as shown in
Table 3.
The behaviour of this table suggests a model of the
form 3,0 2,1
1,1 0,2
ˆ
() ()
ˆ
() ()
A
BAB
QB QB



Specifically, once the parameters are estimated and the
noise process is analyzed using the SCA software [22],
the resulting model is:
1
2
.4334 .3267
1 1.5010.6518
ttt
BB
yxa
BB

 

Table 3. T-Table. Orders of TF models with expectations in
Case a
1 2 3 4 5 6 7
-12 -.29 .09 -.02 .01 .00 .00 .00
-11 -.11 .21 -.05 .05 -.02 .01 .00
-10 .68 .45 .32 .24 .14 .09 .05
-9 -.11 -.07 .12 .24 .07 .04 .08
-8 .12 .04 .10 .21 -.03 -.05 .09
-7 .23 .14 .01 .19 .16 .14 .06
-6 -.76 .47 -.28 .16 .04 -.18 .25
-5 .46 .05 .16 .20 .20 .17 .45
-4 -.22 -.15 -.06 .06 .15 .34 .27
-3 .43 .23 .08 .06 .01 .44 .56
-2 .21 -.13 .013 .05 -.17 .57 .12
-1 .40 -.05 .31 .44 .51 .78 .82
0 1.00 .93 .87 .75 .51 .35 -.01
1 .16 -.16 .16 .28 -.01 .17 .02
2 .18 .00 .08 .11 -.09 .08 -.02
3 .22 .09 .04 .07 .01 .03 .03
4 -.26 .08 -.06 .04 .02 .00 -.03
5 -.05 -.09 .01 .04 .01 .02 .02
Case b: The expectations follow a decreasing trend
In this case, the method yields the following model:
1
2
.4331 .3285
1 1.5016.6514
ttt
BB
y
xa
BB




as suggested by Table 4.
Note the analogy between the models for cases a) and b).
Case c: The exp ectatio ns are g enerat ed by th e ARMA
model of the input series
In this case, the T-table obtained by the series of esti-
mated relative weights is shown in Table 5 and suggests
a model of the form 3,0 1,2
2,1 0,3
ˆ
() ()
ˆ
() ()
A
BAB
QB QB


 which, once
estimated, yields
1
2
.5413 .1775
1 1.3754.6612
ttt
B
yxa
BB




whose parameters differ significantly from those ob-
tained in the first two cases.
The results for the three cases considered are shown in
Figures 3, 4 and 5.
As we can see, the incorporation in the model of the ex
ante advertising forecasts as well as the various forma-
tive mechanisms that not only include the information
contained in the available historical data but also that
derived from the company’s desires and the strategies
and outlooks devised by the economic agents in the deci-
sion-making process, facilitate obtaining valid dynamic
formulations from a data fitting perspective. In addition,
Table 4. T-Table. Orders of TF models with expectations in
Case b
1 2 3 4 5 6 7
-12 -.21 .04 -.01 .00 .00 .00 .00
-11 -.14 .19 -.06 .04 -.02 .01 .00
-10 .83 .68 .58 .51 .42 .34 .28
-9 -.14 -.13 .20 .47 .10 .01 .25
-8 .18 .07 .18 .39 .01 -.07 .21
-7 .25 .21 .02 .32 .28 .25 .14
-6 -.82 .56 -.38 .25 .04 -.30 .51
-5 .46 .02 .16 .24 .27 .27 .76
-4 -.23 -.13 -.05 .06 .15 .44 .40
-3 .39 .20 .07 .05 -.02 .50 .69
-2 .22 -.11 .10 .06 -.16 .60 .18
-1 .41 -.05 .26 .41 .50 .78 .83
0 1.00 .91 .83 .71 .47 .33 -.02
1 .21 -.13 .16 .26 -.03 .15 .01
2 .18 -.02 .07 .10 -.08 .07 -.01
3 .23 .10 .04 .06 .01 .02 .03
4 -.23 .08 -.06 .03 .02 .00 -.03
5 -.10 -.07 .01 .03 .01 .02 .03

Padé Approximation Modelling of an Advertising-Sales Relationship 96
Table 5. T-Table. Orders of TF models with expectations in
Case c
1 2 3 4 5 6 7
-12 -.25 .06 -.02 .00 .00 .00 .00
-11 -.11 .21 -.05 .05 -.02 .01 -.01
-10 .79 .61 .50 .43 .33 .26 .20
-9 -.12 -.12 .19 .40 .12 .03 .22
-8 .17 .06 .17 .32 .01 -.10 .22
-7 .27 .21 .04 .25 .26 .27 .15
-6 -.79 .50 -.29 .15 .12 -.34 .53
-5 .49 .08 .18 .23 .26 .20 .69
-4 -.20 -.16 -.04 .06 .15 .39 .38
-3 .41 .21 .07 .04 -.01 .47 .66
-2 .23 -.12 .11 .04 -.14 .58 .15
-1 .41 -.06 .27 .40 .49 .77 .83
0 1.00 .92 .83 .70 .46 .31 -.04
1 .21 -.14 .17 .27 -.01 .15 .02
2 .18 -.02 .08 .11 -.09 .07 -.02
3 .25 .10 .05 .07 .01 .02 .03
4 -.24 .08 -.06 .04 .02 .00 -.03
5 -.07 -.08 .01 .04 .02 . 02 .02
ADV ERTISING. ARM A M O DEL
0
500
1000
1500
2000
2500
1 7
13 19 25 31 37 43 49 55 61 67
ADVERTISING
DATA
ARMA MODEL
Figure 3. Advertising. ARMA model
ADVERTISING. DATA AND EXPECTATIONS
0
500
1000
1500
2000
2500
1 8
15 22 29 36 43 50 57 64
DECREAS ING
EXP.
INCREASING
EXP.
DATA AND
AR MA MODEL
EXP.
Figure 4. Advertising. Data and expectations
SALES . TRANSFER FUNCTION MODEL W I TH
ADVERTISING EXPECTATIONS
0
500
1000
1500
2000
2500
3000
3500
4000
1 8
15 22 29 36 43 50
SALES
TF MODEL
DECR EXP
TF MODEL
INCR E XP
TF MODEL
ARMA EXP
Figure 5. Sales. Transfer function model with advertising
expectations
to the extent that a knowledge can be had of the future
influence of said forecasts on the relationship under con-
sideration, it is possible to evaluate its sensitivity to al-
ternative dynamic specifications and, as a consequence,
determine the optimum model.
5. Conclusions
In this paper we show the usefulness of the Padé-Laurent
Approximation to the study of the deterministic part in
the dynamic relationship between various variables, since
it allows for the introduction of expectations or expected
future values for certain variables. We illustrate the tech-
nique described by modelling the dynamic relationship
between advertising and sales for the available data in the
references consulted for the Lydia E. Pinkham Company.
Also included is a sensitivity analysis of the estimates as
a function of the expected future values or expectations
of the decision maker for the advertising variable.
It would be interesting to combine this approach with
the use of hybrid models that include a proper combina-
tion of linear and/or non-linear models of the type stud-
ied in [23].
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Padé Approximation Modelling of an Advertising-Sales Relationship 97
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