Feedback Approaches to Modeling Structural Shifts in Market Response

8/10/2019 Feedback Approaches to Modeling Structural Shifts in Market Response

1/11


2/11


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

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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



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



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



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



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



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



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



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

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