Date post: | 02-Jun-2018 |
Category: |
Documents |
Upload: | miguel-costa |
View: | 216 times |
Download: | 0 times |
of 11
8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
1/11
8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
2/11
8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
3/11
during
maturity,
with elasticities
increasing slightly
through
the saturation and
decline
stages
of
the
product
life
cycle.
Similarly,
he
postulated
time
varying
elasticities for
four other
marketing
instru-
ments:
price,
service,
product
quality,
and
packag-
ing.
Kotler
(1971, p. 63)
indicated that firms
are
increasingly
interested
in
measuring
the
time-
dependent
elasticities
of their
marketing
nstruments
and reported that one packaged goods company
has
been
measuring
advertising elasticity
for
a
wide
range
of
its
products
at different
stages
of
their
life
cycle
and
has found a
general pattern
of
falling
advertising
elasticities
as the
products pass
through
their
life
cycles.
These and other
findings
clearly
suggest
the need
for the
development
of
approaches
which can
assist
marketing managers
in
evaluating
the
time effectiveness of
their
decision
variables.
It
is
important
to
understand how
the
different
marketing
nstruments
relate to the
market
response
over time
so that
relative
allocations of
the
market-
ing budget to different marketinginstruments can
be
improved.
In
view
of
the
above,
it is
clear
that
in
the
development
of
a
market
response
model,
the task
of the
model-builder
is to
develop
a
model
which
adapts
to
the
variability
of
marketing
conditions
(Little
1966).
The
objective
of this
paper
is
to
demonstrate
the use
of
adaptive
approaches
to
estimate
time-varying
coefficients
of
managerial
decision
variables
in
the
market
response
model.
The
marketing
iterature n
this
area
has
been
sparse.
Econometric
(such
as
least
squares)
and
other
time
series
analysis
approaches
(such
as
Box-Jenkins)
utilizing
long
series of historical data to
develop
the
market
response
model
implicitly
assume
that
the
coefficients
of the
controllable
marketing
vari-
ables
and
uncontrollable
environmental
variables
remain
stable
over
theentire
time
interval
of
analysis
(Box
and
Jenkins
1976;
Geurts
and
Ibrahim
1975;
Helmer and
Johansson
1977; Weiss,
Houston,
and
Windal
1978).
However,
the
longer
the
time
interval
of
analysis,
the
more
tenuous
this
assumption
is
likely
to
be.
If
the
structural
changes
in
market
response
occur
at known
points
in
time,
then the
changes
in
the
coefficients of the
relevant variables
can be represented by dummy variables (Palda 1964;
Parsons and
Schultz
1976)
or
by
estimating separate
regressions
on
selected
subsets of
observations
(i.e.,
moving
window
regression,
see
Wildt
1976).
The
major
problem
with
these
approaches
is the
diffi-
culty
of
defining
a
priori
time
segments
since the
timing
of
structural
changes
is
rarely
known. In
recent
studies,
Beckwith
(1972),
Erickson
(1977),
and
Parsons
(1975)
assumed that the
coefficients
of the
decision-variables could
be
expressed
as a
function
of
observed
variables.
Studying
the
time-
varying
effectiveness
of
advertising,
Beckwith
(1972)
and Erickson
(1977) represented
the
variation
in
coefficients
as a
polynomial
function
of
time
and Parsons
(1975)
employed
an
exponential
form.
The
major
problem
with these
approaches,
referred
to as the
systematic
parameter
variation
methods,
is that
they
assume
a
priori
the time
path
of
coefficients. Other suggested approaches include
the random coefficient
models
and the
sequential
variation models.
In the former
models,
the
random
parameters
are assumed
to constitute
a
sample
from
a common
multivariate
distribution with an
estimat-
ed
given
mean and
variance-covariance
structure
(see
Swamy
1974).
The
sequential
variation
models,
on the
other
hand,
assume
that time variation
of
coefficients
is the realization
of
a
stochastic
process
(e.g.,
First-order
Markov
process)
and there
is
a
determinate form
to the time
variations of coeffi-
cients
(Cooley
and Prescott
1973;
Little
1966;
Winer
1978).'
This
paper presents adaptive
approaches
to
the
estimation of
coefficients of decision
variables in
the market
response
model.
These
approaches
use
the
concept
of
feedback
from
the decision
making
environment.
They require
no
a
priori
assumptions
about
the time
path
of coefficients
or
knowledge
about
the nature or causes
for time variations
in
the coefficients.
The use
of such
approaches
pro-
vides
self-adaptive
coefficients
of decision
variables
which
can
adjust
automatically
to
changing
data
patterns
in
the
postulated
market
response
model.
The
Feedback
Framework
In the
introduction to
model
building
approaches
to
marketing
decision
making,
Kotler
(1971,
pp.
14-15) emphasizes
the
point
that the
marketing
decision
system
is not static.
In the
models
of
marketing
decision,
provision
must
be
made
for
continuous
revision and reevaluation
n
light
of
new
data and
insights
about
the
decision
making
en-
vironment.
Describing
the
marketing
planning
and
control
system,
he
suggests
that
before
allocating
the marketing budget across different marketing
instruments
(and
different
territories)
at time
t,
the
'The
October 1973
issue of the
Annals
of
Economic
and
Social
Measurement
contains
a
collection of
papers
dealing
with
various
types
of
time-varying
estimation
schemes
and
provides
a
useful
status
report
of
current research
on
the
problem
of
estimating
time-varying
parameter
structures.
For
an
exhaustive
survey
of
the
literature on
this
problem,
see
Rosenberg
(1973).
For
a
brief
summary
of
relevant
issues,
see
Parsons
and
Schultz
(1976,
pp.
155-164)
and
Wildt
and
Winer
(1978).
72
/
Journal of
Marketing,
Winter
1980
This content downloaded from 193.136.144.3 on Sun, 19 Oct 2014 10:04:06 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
4/11
manager
irst
considers the
previous
period's
market
response
(sales,
market
share,
etc.)
and the
market-
ing
expenditures
of the
company
and its
competi-
tors.
The
manager
compares
these
results
to
pre-
vious
predictions
and
adjusts
the
mathematical
response
model where
required.
The
manager
then
forecasts
the
future
environment
and
competitors'
strategies.
These
become
input
to
a
mathematical
decision rule that produces recommended alloca-
tions
of
budget
along
with
predicted
market
re-
sponse.
The whole
process
is
repeated
in
each
period.
The
key
element in
the
marketing
planning
and
control
process
is
the notion
of
feedback
from
the
decision
making
environment.
Figure
1
depicts
this
feedback
framework
for the
development
of
the
adaptive
market
response
model. At
every
time
period
t,
the
predicted
market
response
made
in
period
(t
-
1)
is
compared
with
the
actual
response
and the
error is
fed
back
in
order
to
adjust
the
coefficients or parametersof the postulated market
response
model.
The
revised
model is
then
used
to
predict
the
market
response
for
the
period
(t
+
1).
In
order to
examine
the
adaptive
approaches
to
update
or
adjust
the
coefficients of
decision
variables,
consider
the
following
terminology:
y,
=
actual
market
response at
period
t
f,
=
predicted
market
response
for
period
t
13,
=
coefficient
of
the
i-th
decision
variable
at
time t
e,
=
error at
time
t,
which
is
equal
to
(y,
-
f
,)
x,
=
i-th
decision
variable
at
time
t
I3,
=
estimate
of
the
coefficient
of the i-th
decision
variable
at
time
t
f
=
postulated
form of the
function
relating
deci-
sion
variables to
the
market
response
p
=
number of
decision
variables
considered
Also,
let
,
=f(xl,,
x2t,
.
-X,
;
Plt,,
9
2t,9
..
pt)
(1)
The
whole
idea
behind
the
adaptive
approaches
to
the
estimation
of
coefficients is
to
use
the
informa-
tion provided by the error between the actual and
predicted
market
response,
e,,
to
update
the
esti-
mates
of
coefficients,
1,,.
That
is,
i,,,
=
,,
+
A,
(e,) (2)
The
reevaluation or
reestimation
of the
coefficients
involves
specification
of the
feedback
filter or
adapter
A
(e,).
Note
it is
the
feedback
filter
A,
e,)
that
produces
time
variations in
[,,
and
makes
the
postulated
market
response model,
equation
(1),
FIGURE
Adaptive
Market
Response
System
Marketing
Decision
Dynamic
Marketing
Actual
Market
Variables
Decision
System
Response
xt-I'
Yt-
I
Yt
Model
of the
Forecast
Market
+
Marketing
Decision
O
Response
System
it
Values for
the
Adjustment
of
the
Estimation
Parameters
Parameter
Values
Error
adaptive
to
changing
data
patterns.2
Gelb
(1974)
provides
a
review
of the different
possible mathe-
matical
formulations
for the feedback
filter.
Two
of the feedback
filters that
recently
have
been
developed
are
the ones
suggested
by
Widrow
and
Glover
(1975),
Widrow
and McCool
(1976)
and
Carbone
and
Longini
(1977).
Based
on the
steepest
descent
method
of
optimization
to
minimize
mean
squared
error,
Widrow
et al.
derived
the
following
formulation
for the
adapter:
A
i
(e,)
=
2K
x
it
e,
(3)
where
K is
a
learning
factor
between
zero
and
one,
and
determines
the
speed
of
adaptation.
Carbone
and Longini (1977)proposed the following formula-
tion
for
the
feedback
filter:
e
xi
A,
(et)=
It
^t
.
-
-K
(4)
yt
xit
where
K is the
learning
factor between
zero
and
one, and determines
the
speed
of
adaptation.
i,,,
in
equation
(4),
is
an
updated
average
for the
i-th
decision
variable.
An
exponential
smoothing
scheme
is used
to calculate
this
average,
i.e.,
Xi,
=
w
xit
+
(1
-
w)
i,t,_1
(5)
where
w is
between
zero
and
one,
and
depends
upon
the
forgetting
rate
of
past
observations
on
the market
response
process.
Equation
(4)
is
based
upon
concepts
from
electrical
engineering
and
feed-
back
systems.
In contrast
to
equation
(3),
this
feedback
filter
was
developed
in
a
heuristic
manner
2Note,
the model
is
adaptive,
but
it is
intended
as
only
an
approximation
of the
underlying
system
and
we
are not
assuming
that
the
system
is
adaptive
in
the same
way
as
the
model.
Modeling
Structural
Shifts in
Market
Response
/
73
This content downloaded from 193.136.144.3 on Sun, 19 Oct 2014 10:04:06 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
5/11
through
experimentation
and
logical
considerations
rather
than via
deductive
reasoning
(Carbone
and
Longini
1977).
Note,
the
objective
of
equations
(3)
and
(4)
is
to
determine the
magnitude
by
which the
previous
estimates of
the
coefficients should
be
adjusted
(see
equation
(2)). Equation
(3) specifies
thatthis
magni-
tude
for
time
(t
+
1)
is
determined
by
the
error
between the actual and
predicted
market
response
at time
t,
the
value of
the
decision
variable at
time
t,
and a
constant which
determines
the
contribution
of
the
above
two
factors to
the new
value of
the
coefficient.
The
adapter
in
equation
(3)
is
an
ap-
proximation
for the
gradient
decent
direction
at
time
t
which
minimizes the
expected
square
error
at
time
t
(see
Widrow
and
McCool
1976).
Similarly,
equation
(4) specifies
the
change
in
the
coefficient
value
to
be
determined
by
a
percentage
error,
the
present
value of
the decision
variable
scaled
by
its
smoothed
mean,
and
the
learning
factor. If
the error
between
the actual and predicted market response, e,, is
zero
at
time
t,
then
A,
(e,)
=
0
and
the
value
of
coefficients
at
time
t
+
1
and
t
are
identical.
Furthermore,
a
negative
value
of
error at
time
t
results in
a
negative
adjustment
to
the
coefficient
value at
time
t
to
time t
+
1
and a
positive
value
results in
a
positive
adjustment.
That
is,
the
feed-
back
approaches
follow
the
data
patterns
and
using
the
filters,
such
as
equation
(3)
or
(4),
determine
the
contribution
of
data
pattern
changes
to
the
previous
estimates of
coefficients.
The
application
of
the
Widrow
et
al.,
adapter
for
univariate
time-series forecastinghas been dem-
onstrated
by
Makridakis
and
Wheelwright
(1977).
Carbone
and
Longini
(1977)
have
demonstrated
the
use of
their
adapter
for
real
estate
assessment.
Studies
are
currently
underway
to
assess
the
relative
efficiency
of
these
two
adapters. However,
because
of
its
availability
and
some
indications
of
its
competitive
performance
(Bretschneider,
Carbone,
and
Longini
1979),
the
adapter
suggested
by
Carbone
and
Longini
will
be
used
in
the
next
section
to
illustrate
its
application
to
the
marketing
data.
Note
that
given
some
initial
values
for
the
coefficients,
the
adaptive
estimation
approaches
are
designed to
automatically
capture
the
types
of
processes
governing
the
change
in
coefficient
val-
ues.
This
aspect
is
crucial
since it
is
generally
impossible
to
assume
a
priori
knowledge
of
the
processes
governing
structural
shifts
in
market
re-
sponse.
Furthermore,
these
adaptive
approaches
do
not
impose
any
restriction on
the
type
of
coefficient
variation
that
may
arise.
They
are
truly
self-adaptive
and
can
adjust
automatically
to
changing
data
pat-
terns
(Makridakis
and
Wheelwright
1978,
p.
287).
FIGURE
Sales
and
Advertising
of
LydiaE.
Pinkham
Medicine
Company
(1907-1960)
3,600
3,000
2,400
1,800
05
1 2 0 0
Advertising
600
1907
'10 '15
'20 '25
'30 '35 '40
'45 '50
'55
'60
An
Example
The use of adaptiveapproachesto obtaintime-vary-
ing
coefficients
of
decision
variables
in
the
market
response
model
can
be
illustrated
with
the
help
of
an
example
drawn
from
a
distributed
lag model
of
advertising carryover
effect.
The
general
distrib-
uted
lag
model
can
be written
as
(FitzRoy
1976,
pp.
172-173):
y,
=
[ox,
+
Pix,_,
+
P2
xt-2
+
...
+
e, (6)
where
y,=
the sales
in
period
t
x,
= the
advertising
in
period
t
e = the error at time t
This model
is
completely specified
once
the
relative
magnitude
of the
weighting
coefficients
has
been
determined,
reflecting
the form
of
distributed
lag.
These
weights
are
generally
assumed
to
decline
monotonically,
i.e.,
P,
>
P,+,,
although
other
pat-
terns have
been
suggested
(Parsons
and
Schultz
1976,
pp.
167-188).
A
number
of methods
have
been
suggested
to
replace
the
infinite
sum of
equation
(6)
with
a
single
term.
The best
known
is
that
attributed
to
Koyck,
in which
it is
assumed
that
the
effect of
advertising
decays
geometrically
over
time, i.e.,
S,
=
3X'
(7)
where
P
and
X are constants
and the value
of
X
is between
zero and
one. Substitution
of
equation
(7)
into
equation
(6)
and further
simplification
yields
the
simplest
version
of the cumulative
effects
model
(FitzRoy
1976,
p.
173):
y,
=
3x,
+
hy,_1
(8)
In
equation
(8),
the
coefficient
p
reflects
the
current
74
/
Journal
of
Marketing,
Winter
1980
This content downloaded from 193.136.144.3 on Sun, 19 Oct 2014 10:04:06 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
6/11
impact
of
advertising
while X
represents
the
car-
ryover
from
past
advertising
or the
retention
coeffi-
cient. The
firm
analyzed
is the
oft-studied
Lydia
E.
Pinkham
Medicine
Company
and its
product,
the
Lydia
Pinkham
vegetable
compound,
originally
examined
by
Palda
(1964). Figure
2
depicts
the
annual data
from 1906-1960.
There
are
several
unique
features of this
product
that
have
made
this
datapopularfor studying sales-advertisingrelation-
ships.
The firm
spent
almost all its
promotion
budget
on
advertising.
Price of
the
product
varied
little
over
the
years
of
analysis.
The firm
did
not
have
a clear
cut
competitor.
One
easily
identifiable
factor
that could
have
caused
a
structural
shift in
the
market
response
is
the
advertisingcopy.
In
all,
four
periods
of
copy
can
be identified:
1907-1914,
1915-
1925,
1926-1940,
and
1941-1960.
A
number
of
econometric
models
have
been
tested on
this
data.
Weiss,
Houston,
and
Windal
(1978) provide
a
review of
such
efforts.
Helmer
and
Johansson
(1977) have also used this data to
illustrate
the
use of
Box-Jenkins
transfer
function
analysis
to
marketing
data.
As
indicated
earlier,
the
major
problem
with
the
use of
the
econometric
and
Box-Jenkins
approaches
is
that
they
assume
the
stability
of
coefficients
of
decision
variables
for the
entire
period
of
analysis.
Other
nvestigations
on
this
data
base
include
the
study
by
Caines,
Sethi,
and
Brotherton
(1977)
to
establish
the
causality
relationship
between
sales
and
advertising
and
stud-
ies
by
Winer
(1978)
and
Beckwith
(1972)
to
illustrate
the
use
of
sequential
parameter
variation
and
sys-
tematic
parameter
variation
methods,
respectively.
It should be noted here that our
objective
of
using
this
data
and
particularly
the
distributed
lag
model,
equation
(8),
is
to
illustrate
the
use
of
adaptive
approaches
to
obtain
estimates
of
time-
varying
coefficients
of
decision
variables in
the
market
response
model.
The
attainmentof
a
model
that
best
describes
this
data is
a
secondary
objective.
Data
Analysis
Before
estimating
time-varying
coefficients
of
the
distributed
ag
model,
equation
(8),
for the
Pinkham
data,
ordinary
east-squares
estimates
were
obtained
for the different
advertising
copy
era. Table
1
reports
these results.
It should
be noted
in
Table
1
that there
is
a
significant
difference
in the
values
of the advertising coefficient P and the retention
coefficient
X,
across
the different
time
periods.
Furthermore,
note that
the distributed
lag model
does
not describe
the
process
for
the
years
1915-
1925
(i.e.,
X >
1).
This
initial data
analysis
suggests
the
following:
*
Since
the
coefficients
vary
across
the
dif-
ferent
periods
of
advertising
copy,
it is
quite
possible
that
they
may
vary
within
each
advertising copy
era because
of the
factors
that cannot
be
identified
or
specified
a
priori.
*
The
strategy
of
segmenting
the
data
according
to known structuralshifts,
(e.g.,
advertising
copy)
may
result
in small
subsets
of
data
points,
causing
questionable
confidence
in
the
estimates
of
the
coefficients.
Next, the
time-varying
estimates
of the
advertis-
ing
and retentioncoefficients
were obtained
by
using
the
moving
window
regression
procedure
(Wildt
1976)
and
the
feedback
filter
suggested
by
Carbone
and
Longini
(1977).
As
mentioned
earlier,
the
mov-
ing
window
regression
procedure
involves
estimat-
ing
separate
regressions
on selected
subsets
of
observations.
The
underlying
idea
is
to
obtain
moving
estimates
of the coefficients
by
substi-
tuting
the most recent
for the oldest
observation.
Three
different
moving
time-spans
(windows)
of
10,
15,
and
20
years
were
considered.
Table
2
presents
the results
for
the
20-year
moving
time-
span.
An
examination
of the
table
indicates
that
when
applying
the
moving
window
regression
TABLE
1
Ordinary
Least-Squares
Results for
Different
Time
Periods,
y,
=
3x,
+
Xy,_
Advertising Retention
Time
Period
Number
of
Coefficient
Coefficient
Covered
Observations
A R
1908-1914
7
.995
.485
.39
1915-1925
11
.067
1.0543
.93
1926-1940
15
.503
.6321
*
.85
1908-1946
39
.274
.860
.87
*significant
at a
=
.05.
Modeling
Structural
Shifts
in
Market
Response
/
75
This content downloaded from 193.136.144.3 on Sun, 19 Oct 2014 10:04:06 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
7/11
TABLE
Time-varying
Coefficients
of the
Sales
Response
Model
Using Moving
Window
Regression
Analysis (first
40
years)
Time
Advertising
Retention
Period
Coefficient Coefficient**
Covered
p
h
R2
1908-1927 .1895 .9224 .9044
1909-1928
.0602 .9835
.8973
1910-1929
-.0294
1.0287
.8892
1911-1930
-.0916
1.0591 .8790
1912-1931 -.0260 1.0176
.8566
1913-1932 -.0568 1.0298 .8431
1914-1933 .0605 .9684
.8287
1915-1934 .0248 .9878 .8089
1916-1935 .0553 .9669
.7824
1917-1936 .1787
.8940
.7712
1918-1937
.1610 .9036 .7740
1919-1938
.2167
.8675
.8362
1920-1939 .2636 .8357
.8580
1921-1940
.3141
.8102
.8488
1922-1941
.3949 .7632
.8478
1923-1942
.3728
.7745 .8421
1924-1943 .3492
.7816
.8417
1925-1944 .3561
.7702
.8184
1926-1945
.3160 .7921 .7476
1927-1946
.4012* .7572
.7283
*significant
at
a
=
.05.
**All
the
X
values
are
significant
at a
=
.05.
procedure
the
distributed
lag
model,
equation
(8),
does not describe the process for the years 1910-
1929,
1911-1930,
1912-1931,
and
1913-1932
(i.e.,
negative
values of
p
and
h
>
1).
Similar results
were obtained for
the
10-
and
15-year
moving
time-spans.
Because
of its
inability
to
describe the
process,
the
forecasting
efficiency
of the
distributed
lag
model,
calibratedvia the
moving
window
regres-
sion
procedure,
was
not
considered.
Recalling
equations
(4)
and
(5),
the
time-varying
coefficients via the
feedback
filter were
obtained
by
using
the
following:
i,t+l
=
it
+
it
yt t)
K
(9)
where
2it
=
wx,,
+
(1
-
w)
Xi,tl1
The use of
equation
(9)
first
involves
the selection
of
the
values of
the
learning
factor
K
and
the
forgetting
rate
factor w. In
order to
determine
these
adaptive
model
parameters,
the
first 40
data
points,
i.e.,
1906-1946,
are
used. The
following
steps
are
undertaken to
determine
these
values
(details
of
the
procedure
are
given
in
the
Appendix):
TABLE3
Time-varying
Coefficients
of the
Sales
Response
Model
Using
Adaptive
Estimation
Procedure
(first
40
years)
Advertising
Retention
Coefficient Coefficient
Year
3
h
1908 .4768 .7960
1910 .4734
.7911
1912
.4735
.7892
1914
.4716
.7874
1916
.4992
.7882
1918
.5000
.8392
1920
.5016
.8398
1922
.6002
.8440
1924
.4515
.8408
1926
.4331
.7639
1928
.4196
.7275
1930
.4131
.7107
1932
.4095
.6993
1934
.4063
.6951
1936
.4291
.6853
1938 .4469 .7227
1940 .4700
.7494
1942
.4752
.7866
1944
.4785
.7949
1946
.4653
.7709
*
Initial estimates
for the
coefficients
Pit,
w,
and
K are selected.
*
The set of 40
observations is then
iterated
several times
through
the
filter
equation
(9)
while
adjusting
the value of
K,
if
necessary,
until
convergence
to
a
pattern
of
change
in
the values of the coefficients occurs.
For
the 40
data
points
used,
Table 3
gives
the
estimated
time-varying
coefficients for
the
even
years.
The
estimated
values
of K and
w
are
.16
and
.01
respectively.
Once
the
values
of
K,
w,
and
estimates of
3,t
for
period
t are
known,
equation
(9)
can
be
used
to
forecast
the
market
response
for
period
(t
+
1)
and
so on.
This
was
done for
the
remainder
of
14
data
points,
i.e.,
1947-1960.
The
mean
absolute
forecast
error
for
the 14
ob-
servations is
74.90
and
mean
square
forecast
error
is
9224.
Some
important
comments on
these
results
are
warranted:
*
The
value of
the
advertising
coefficient
P
shows
a
maximum
variation
of
about 25%
(see
Table
3).
Furthermore,
the
value
of
the
coefficient
increases
up
to
1922,
declines
from
1923 to
1934,
and
then
increases
again.
The
maximum
value
of
the
coefficient is
for
the
year
1922
and
minimum
for
the
year
1934.
76
/
Journal
of
Marketing,
Winter
1980
This content downloaded from 193.136.144.3 on Sun, 19 Oct 2014 10:04:06 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
8/11
*
The value of the
retention
coefficient
shows
a
similar
trend. It
generally
increases
up
to
1922
and
declines from 1923
to 1936 before
increasing again.
The maximum
value
of
the
coefficient
is .8440
in
the
year
1922
and
minimum is
.6853
in
the
year
1936.
It is
very
clear from
the above
trends
that
factors
other than the
advertising
copy
could
have been
operating
under the market re-
3
sponse
process.
*
The mean
square
forecast
error for
the
years
1947-1960,
which
was not
included in
the
estimation
of the
learning
parameter
K
and
the
forgetting
rate
parameter
w
is
9224.
This
mean
square
forecast
error is
compara-
ble to
the
mean
square
errors of
9155
and
8912
reported
by
Helmer
and
Johansson
(1977)
for
two
different
Box-Jenkins
transfer
function
models
of
Pinkham
data.
However,
the
adaptive
model of
the
Pinkham
data
out
performs
all other econometric
models
sum-
marized
by
Helmer
and Johansson
(1977).
Figures
3 and
4
summarize
the estimates
of
the
advertising
coefficient
P
and the retention
coeffi-
cient
X,
obtained
by ordinary
least
squares, seg-
mented
regression,
moving
window
regression,
and
the
adaptive procedure
over the
period
1908to
1946
(see also Tables 1, 2, and 3). In examining these
results,
several
strengths
and weaknesses
of
each
procedure
should
be
kept
in mind.
Though
ordinary
least
squares
and
segmented
regression
provide
static
estimates
of the
parameters
(ordinary
least
squares
for the entire
time horizon
and
segmented
regression
for
different
time
segments),
they
do
allow
one to
perform
formal tests
of
hypothesis.
Unlike
segmented
regression,
moving
window
re-
gression
and
adaptive
estimation
procedures
require
no a
priori
knowledge
of structural
shifts.
However,
the
major
drawback
of these two
procedures
is
that
their statistical
propertiesare
not well
defined.
That
is,
hypothesis
tests
on the
significance
of
parameter
paths
and
changes
in
parameter
values
do
not
presently
exist
for these two
procedures.
In
the
case
of the
moving
window
procedure,
a
basic
application
consideration is
the
length
of
the win-
dow.
Furthermore,
this
procedure
results in
time-
varying
estimates
for
all
model
coefficients. In
the
case
of
adaptive
procedures,
it
is
possible
to
con-
sider
a
selective
set of
model
coefficients
to
vary
3It
should
be
noted
here that
the
feedback
approaches
are not
explicitly
concerned
with
the
identification
of
the
factors
that
cause
structural
shifts in
the market
response.
In
fact,
the
causes and
the timing
of
structural
changes
are
rarely
known
a
priori
(Parsons
and
Schultz
1976,
p. 155).
The
biggest
advantage
of
these
approaches
is
that
they
can
capture
the
time-varying
effect
without the knowl-
edge
of
the
causes
of
change.
However,
these
models do
provide
information
that
can
be
used
to
study
the
why
question
by
discussing
the
changes
in the
parameters
with the
management.
Unfortunately,
lack of
secondary
information
does
not
permit
this
analysis
for
the
Pinkham
data.
FIGURE
Time-Varying
Estimates of
Advertising
Coefficient
e-o
Adaptive
Estimation
Procedure
-1.0
Moving
Window
Regression
-
Segmented
Ordinary
Least-Squares
-
.8
-
---
Ordinary
Least-Squares
C
.4-,
.6
0
o
.4
-
~
.0
1
1 1
1
I I
I
I I I I I
I I I I I
I I I
1
1 1 1
1
1
11
I I I
I I I I
i I
I
I
I
I
I
1908
1913
1918
1923
1928
1933
1938
1943
1946
-
Yea s
Modeling
Structural
Shifts in
Market
Response /
77
This content downloaded from 193.136.144.3 on Sun, 19 Oct 2014 10:04:06 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
9/11
FIGURE
Time-Varying
Estimates
of Retention
Coefficient
*-.
Adaptive
Estimation
o-o
Moving
Window
Regression
Segmented
Ordinary
Least-
Squares
Ordinary
Least-
Squares
1.08
.96
S.84
0
.72
-
o
c
.60
48
I I I
I I
I I I
I I
I I I
I I I
I I
I I I
I I
I 1
I
I I I I I
I I I
I
I
I I
I
1908
1913
1918
1923
1928
1933
1938
1943
1946
-* Years
over
time
by
specifying
different
values
of
K for
different
coefficients in
equation
(4).
For
example,
value
of
K,
=
0 will
resultin
a
time
invariant
estimate
for coefficient 3,i.
Conclusion
and
Summary
The
basic
objective
of this
paper
has
been
to
demonstrate
the
use of
feedback
approaches
to
develop
self-adaptive
market
response
models.
Such
approaches
provide
time-varying
coefficients
of
the
postulated
market
response
models.
Popular
approaches
of
parameter
estimation such
as
least
squares, Box-Jenkins,
and
other
econometric
ap-
proaches
assume
the
effectiveness
of
the
controlla-
ble
marketing
decision
variables
and the
uncon-
trollableenvironmentalvariablesremainstable over
the
entire
time
interval of
analysis.
The
use
of
such
approaches,
in
the
long
run,
may
lead to
the
deve-
lopment
of a
market
response
model
which is
insensitive to
the
reality
of
marketing
conditions
resulting
in
a
nonoptimal
allocation
of
marketing
resources. In
the
presence
of
a
decision
making
environment
which
is
relatively
stable
over
time,
the
feedback
approaches
provide
a
means
to
diag-
nose
the
stability.
The feedback
approaches
to
modeling
structural
shifts
in
market
response
look
promising
and
should
be
considered
as
an
alternative
to
existing
ap-
proaches to handle structural shifts. These ap-
proaches
are
especially
useful when
the
timing
of
structural
changes
is not known.
The
adaptive
approaches
offer
a
deep
insight
into the
structure
of
the
decision
variable
space
and
the
sensitivity
and
effectiveness
of
decision
variables
over
time.
In the model
building
context,
the
objective
of
the
model
builder
is
to
develop
a model which
adapts
to the
variability
of
marketing
conditions.
The
feedback
approaches
can
help
the
analyst
to
assess
the
time
effectiveness
of
managerial
decision
vari-
ables.
If the
effectiveness
of decision
variables
is
stable
over
time,
these
approaches
would
provide
constant
coefficients
of the variables
in the
model,
thus
indicating
that
the
popular
nonadaptive
ap-
proaches
provide
a
good
set
of
coefficients.
The
feedback
approaches
provide
a better
comprehen-
sion
of the decision
variables
and
help
in
developing
a
model
which
is
self-adaptive
and can
adjust
to
changing
data
patterns
from
the
environment.
With
growing
interest
in the
development
and
use of models
for
marketing
decision
making,
it
is
imperative
that models
be
developed
that
capture
78
/
Journal of
Marketing,
Winter
1980
This content downloaded from 193.136.144.3 on Sun, 19 Oct 2014 10:04:06 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
10/11
the
dynamic
nature
of the
managerial
decision
variables.
The
feedback
approaches
offer
an
avenue
to
develop time-varying parameter
structures
of
market
response
models. These
approaches
have
found
their
dissemination in the model
building
literature
n the
last few
years
(Carbone
and
Longini
1977;
Makridakisand
Wheelwright 1977).
Although
they
are still at the
development
and
testing
stage
and lack statistical
properties
of the coefficient
estimates,
they
are
intuitively appealing,
theoreti-
cally
strong,
and
extremely
practical.
This
inves-
tigation
has
focused
upon
the
application
of these
procedures
to
study
time
effectiveness of
marketing
instruments in
sales
response
models.
Future
ap-
plications
of these
procedures may
include
studying
time-varyingaspects
in
other models such
as
product
life
cycle
and
product growth
models,
market share
models,
brand
switching
models,
time-series
fore-
casting,
and sales
territory
models.
Appendix
Consider the
following
model
p
yt=
,ix,i
+e,
and
t=
1,2,...
T
(Al)
i=
1
where
y,
represents
the
actual
market
response
at
time
t,
x,1
is the
value
of
the
i-th
decision
variable
at
time
t,
p
is
the
number
of decision
variables,
P,,
is the coefficient
of the
i-th decision
variable
at
time
t,
and
e,
is
a
random
disturbance.
The
time-varying
estimates
of the coefficients,
it,,
are
obtained
by using
the
following:
P i t +
I y,
- ,
xP p,
itl=
:
+
|
_i'_
i
y
) K)
(A2)
and
it
=
wx,,
+
(1
-
w)
xi,,t-
p
and ,= , (A3)
i=1
The
use of
equation
(A2)
involves
the
selection
of the values
of K
(0
-
K
1),
w
(0
:
w
:
1)
and initial
value for
each
coefficient
io.
For a
particular
problem,
the values
of
K,
w,
and
Pf1o
an be obtained
by
using
certain
estimation
criteria
such
as the minimization
of
the sum
of
squares
of
errors, i.e.,
T
Minimize
Z
=
(y,
-
)Y,)2
(A4)
t=l
where
y^,
is
given by
equations
(A2)
and
(A3).
However, when equations (A2) and
(A3)
are substi-
tuted
in
equation
(A4),
it is
analytically
not
possible
to solve
equation
(A4)
for
w,
K,
and
P3
;
a
nonlinear
programming
solution
algorithm
is
needed
(Him-
melblau
1972).
An
algorithm
is
currently
available
which
when
given
some
initial values
of the
coeffi-
cients,
K and w
goes through
a series
of
iterations
until
convergence
in Z
(and/or
other
desirable
fit
statistics)
is obtained
yielding
the
appropriate
values
of
K,
w,
and
time-varying parameter
estimates
over
the
sample.
REFERENCES
Beckwith,
N.
E.
(1972),
Regression
Estimation of
the
Time-Varying
Effectiveness of
Advertising,
unpub-
lished
paper.
Box,
G. E.
P. and
G. M.
Jenkins
(1976),
Time
Series
Analysis:
Forecasting
and
Control,
San
Francisco:
Holden-Day.
Bretschneider, S.,
R.
Carbone,
and R.
L.
Longini
(1979),
An
Adaptive
Approach
to
Time
Series
Forecasting,
Decision Sciences, 10 (April), 232-244.
Caines,
P.
E.,
S.
P.
Sethi,
and T.
W.
Brotherton
(1977),
Impulse
Response
Identification and
Causality
Detec-
tion for
the
Lydia-Pinkham
Data,
Annals
of
Economic
and Social
Measurement,
6,
147-163.
Calantone,
R. J.
and A.
G.
Sawyer
(1978),
The
Stability
of
Benefit
Segments,
Journal
of
Marketing
Research,
15
(August),
395-404.
Carbone,
R.
and R. L.
Longini
(1977),
A
Feedback
Model
for
Automated
Real
Estate
Assessment,
Management
Science,
24
(November),
241-248.
Cooley,
T.
F.
and E.
C.
Prescott
(1973),
Varying
Parameter
Regression:
A
Theory
and Some
Applications,
Annals
of
Economic
and Social
Measurement,
2
(October),
463-473.
Erickson,
G. M.
(1977),
The
Time-Varying
Effectiveness
of
Advertising,
in
Educators'
Proceedings,
B.
A.
Green-
berg
and
D. N.
Bellenger,
eds.,
Chicago:
American
Marketing
Association,
125-128.
FitzRoy, P. T. (1976), Analytical Methods for
Marketing
Management,
Maidenhead,
England:
McGraw-Hill.
Gelb,
A.
(1974),
Applied
Optimal
Estimation,
Cambridge,
MA:
The
M.I.T.
Press.
Geurts,
M. D.
and
I.
B.
Ibrahim
(1975),
Comparing
the
Box-Jenkins
Approach
with
the
Exponentially
Smoothed
Forecasting
Model
with an
Application
to
Hawaii
Tourists,
Journal
of
Marketing
Research,
12
(May),
182-187.
Helmer,
R. M.
and J.
K.
Johansson
(1977),
An
Exposition
of
the
Box-Jenkins
Transfer
Function
Analysis
with an
Application
to
the
Advertising-Sales
Relationship,
Jour-
Modeling
Structural
Shifts in
Market
Response /
79
This content downloaded from 193.136.144.3 on Sun, 19 Oct 2014 10:04:06 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response
11/11
nal
of
Marketing
Research,
14
(May),
227-239.
Himmelblau,
D. M.
(1972),
Applied
Nonlinear
Programming,
New
York: McGraw-Hill.
Houston,
F.
S. and
D.
L. Weiss
(1975),
Cumulative
Advertising
Effects: The Role
of Serial
Correlation,
Decision
Sciences,
6
(July),
471-481.
Kotler,
P.
(1971),
Marketing
Decision
Making:
A
Model
Building
Approach,
New York:
Holt,
Rinehart,
and
Winston.
Little, John D. C. (1966), A Model of Adaptive Control
of Promotional
Spending,
Operations
Research,
14
(No-
vember-December),
1075-1097.
Makridakis,
S. and
S. C.
Wheelwright
(1977),
Adaptive
Filtering:
An
Integrated Autoregressive
/Moving Average
Filter
for Time Series
Forecasting, Operations
Research
Quarterly,
28
(October),
425-437.
-
,
and
(1978),
Forecasting,
Methods and
Applications,
New
York:
John
Wiley
and
Sons.
Mickwitz,
G.
(1959),
Marketing
and
Competition,
Helsing-
fors,
Finland:
Centraltryckeriet.
Moinpur,
R.,
J. M.
McCullough,
and
D. MacLachlan
(1976),
Time
Changes
in
Perception:
A
Longitudinal Application
of Multidimensional
Scaling,
Journal
of Marketing
Re-
search,
13
(August),
245-253.
Monroe, K. B. and J. P. Guiltinan (1975), Path-Analytic
Exploration
of Retail
Patronage
Influences,
Journal
of
Consumer
Research,
2
(June),
19-28.
Morrison,
D. G.
(1966),
Interpurchase
Time
and Brand
Loyalty,
Journal
of
Marketing
Research,
3
(August),
289-291.
Myers,
John
G.
(1971),
The
Sensitivity
of
Time-Path
Typologies,
Journal
of Marketing
Research,
8
(No-
vember),
472-479.
--
,
and Francesco
M.
Nicosia
(1970),
Time Path
Types:
From
Static to
Dynamic Typologies, Manage-
ment
Science,
16
(June),
584-596.
Palda,
K.
(1964),
The Measurement
of
Cumulative
Advertis-
ing
Effects, Englewood
Cliffs,
NJ:
Prentice-Hall.
Parsons,
L.
J.
(1975),
The
Product
Life
Cycle
and Time-
Varying
Advertising
Elasticities,
Journal
of Marketing
Research,
12
(November),
476-480.
--
,
and R. L. Schultz
(1976),
Marketing
Models and
Econometric
Research,
New
York:
North-Holland
Pub-
lishing
Company.
Rosenberg,
B.
(1973),
A
Survey
of
Stochastic Parameter
Regression,
Annals
of
Economic
and Social Measure-
ment,
2
(October),
381-397.
Swamy,
P. A.
V.
B.
(1974),
Linear
Models
with
Random
Coefficients,
in
Frontiers in
Econometrics,
P. Zaremb-
ka, ed.,
New York:
Academic
Press,
143-168.
Weiss,
D.
L.,
F. S.
Houston,
and
P.
Windal
(1978),
The
Periodic
Pain
of
Lydia
E.
Pinkham,
Journal
of
Business,
51
(January),
91-101.
Wichern,
D. W. and
R.
H.
Jones
(1977), Assessing
the
Impact
of
Market Disturbances
Using
Intervention Anal-
ysis, Management
Science,
24
(November),
329-337.
Widrow,
B. P. and J. R. Glover
(1975),
Adaptive
Noise
Cancelling: Principles
and
Applications,
Proceedings
of
IEEE,
63
(December),
1692-1716.
-,
and J. M.
McCool
(1976), Stationary
and Non-
stationary Learning
Characteristics
of
the LMS
Adaptive
Filter, Proceedings of the IEEE, 64 (August), 1151-1162.
Wildt,
A. R.
(1976),
The
Empirical Investigation
of
Time
Dependent
Parameter Variation in
Marketing
Models,
in
Educators'Proceedings,
K. L.
Bernhardt, ed.,
Chicago:
American
Marketing
Association,
466-472.
,,
and R. S.
Winer
(1978), Modeling
Structural
Shifts
in
Market
Response:
An
Overview,
in
Educators'
Proceedings,
S.
C.
Jain, ed.,
Chicago:
American Market-
ing
Association,
96-101.
Winer,
R.
S.
(1978),
An
Analysis
of the
Time-Varying
Effects of
Advertising,
Research
Paper
No.
144A,
New
York: Graduate School
of
Business,
Columbia
Universi-
ty.
80
/
Journal
of
Marketing,
Winter
1980
hi d l d d f 6 6
http://www.jstor.org/page/info/about/policies/terms.jsp