J BUSN RES 247 1985:13:247-257

Transfer Function Analysis of the Relationship Between Advertising and Sales: A Synthesis of Prior Research

Mark M. Moriarty

The University of’ IOM~I

Prior research into the relationship between advertising and sales has generally been based on methodologies involving either direct lag or distributed lag models. The current study demonstrates that these classes of models are nested within a general model which can easily be represented and estimated with transfer function methods. The general model is used to identify empirically models of the sales-advertising relationship for a data set that is aggregated into data intervals of varying length. Results of the aggregation process illustrate the effect of data interval bias on the estimate of the duration of the advertising effect.

Econometric models have been used in marketing to describe and sometimes to forecast the outputs of a market system based on input factors such as advertising and price. Occasionally, such models are used to develop normative rules for optimally controlling the inputs. A generic feature of most econometric models is that they are part stochastic and part systematic [ 111 . The systematic components of such models are understandably interesting to marketers, since they provide the wealth of managerial implications which have come from this research tradition, particularly with respect to the cumulative effects of advertising. On the other hand, the specification of the stochastic structure of such models has typically relied heavily on a priori knowledge of the process generating the disturbance terms of such models. The recent introduction of Box-Jenkins procedures in marketing [6, 7, 91 , particularly transfer function analysis, has heightened interest in empirical analysis of the stochastic structure of time-series models. Transfer function models are well suited to analysis of the joint influence of structural marketing influences and complex disturbance term structures because of the flexibility of their specification.

The purpose of this paper is to show that all of the currently used cumulative advertis- ing effects models can be expressed as transfer function model forms, and that all of these alternative forms are special cases of a single form. This general transfer function form is important because it can be used to specify which of several “nested” model forms is appropriate for a given data set. To document the importance of this property of transfer functions, the paper first demonstrates that prior advertising effects models are special

Address correspondence to Mark M. Moriarty, The University of Iowa, Iowa City, IA 52242.

Journal of Business Research 13, 247-257 (1985) 0 Elsevier Science Publishing Co., Inc. 1985 52 Vanderbilt Ave., New York, NY 10017

0148-2963/85/.$3.30

248 J BUSN RES 1985:13:247-257

M. M. Moriarty

cases of a single transfer function form. Then, we use the transfer function modeling process to point out the advantages of model specification using a general transfer func- tion. Finally, the implications arising from the application of this process are examined for modeling cumulative advertising effects, with particular attention given to data

interval bias [4] . Data interval bias is an aggregation bias which results from grouping data into time intervals which are too long, and often leads to an overestimate of the

duration of the advertising effect. To accomplish these objectives, the literature pertaining to cumulative advertising

effects models is reviewed to determine the alternative model forms that have appeared. These forms suggest a general model form from which each can be derived based on either simplifying assumptions or empirical model tests. The general model nests four com- monly occurring models and can be used as a general test equation. After the general model is derived, the frequently cited Lydia Pinkham data [lo] are used to illustrate its application to modeling advertising effects. The context of this application allows the problem of data interval bias to be illustrated [4].

Transfer Function Analysis and Models of Advertising Effectiveness

Box and Jenkins [3] postulate that many dynamic time series relationships can be de- scribed by a combined transfer function-noise model. As applied to the advertising-sales relationship, the model has the following form:

s, = A-‘(L)b(L)A,_i + ivr (1)

where S, is sales at time t, At-i is advertising expenditures at time t-i, h-‘(L)b(L) is a compact notation describing the transfer function relationship between S, and A,_i (see

Box and Jenkins [l, p. 370]), and Nt = $-‘(L)0(L)(l - L)dat [G(L) is the autoregres-

sive process of order p, e(L) is the moving average process of order 4, and (1 - L)d is the difference operator of order d] , L is a backshift operator, and ut - N(O,uO’) and is known as white noise. Readers unfamiliar with the backshift operator should consult McCleary and Hay [8] for a complete discussion of this notation. The expression LA, means that L operates on A, to shift it backward one period (i.e., A,_ 1), and L2At would shift A, back two periods, such that L2At = A,_,.

Equation (1) is a flexible model of time-series relationships which accommodates many alternative specifications. In the following development, it is shown that the cur- rently used models of advertising effect can be developed from (1) through appropriate specification of h-‘(L)B(L) andNr.

In the marketing literature, two general classes of advertising effectiveness models have been used [4], namely, direct lag models and distributed lag models. The direct lag model postulates a relationship between sales and advertising as follows:

where S,, A,_i, and ar are defined as in equation (1) and, n is the number of direct lagged advertising terms. The distributed lag model includes lagged values of the dependent variable in addition to the advertising terms appearing in the direct lag model. The general form of this model is:

St = g XjS,_j + g bi A,_i + a, j=l i=O

Transfer Function Analysis J BUSN RES 1985:13:247-259

249

where S,_i is sales at time t-j, m is the number of lagged sales terms, and the other variables are defined as in equation (1). Even more sophisticated models can be derived by specifying a more complicated structure for the disturbance term. That is, we could specify that the disturbance term follows a general ARMA process [7] such that the model in equation (3) would be:

St = S! AjS,_j + 2 bi A,_i + Nt

JU w Transfer function Noise function

component component

(4)

where Nt is defined as in equation (1). The model represented by equation (4) can be specified for transfer function analysis [7], because the transfer function component of (4) is one specification of ~I-‘(L)~(L)A,_~. Thus, all of the common cumulative adver- tising effects models can be expressed as transfer functions. The transfer function component of (4) can be written as v(L)A,, where v(L) is the polynomial operator representing the transfer function relating sales to advertising. The polynomial v(L) has a general form which can be written:

v(,r) = b Lb X(L)

where b(L) is the s*-order polynomial operator, h(L) is the r*-order poly- nomial operator, and L b is the bth-order “dead time” operator representing the number of periods before any effect is measurable.

Specification of (r s b) involves specification of integers for these three values, which will be greater than or equal to zero. A full specification of the transfer function-noise model (r s b) (p d q) would include both the above notation for the transfer function component and the noise function component (p d 4) defined in equation (1). In Table 1, commonly used models and their literature references are summarized. In addition, the corresponding transfer function model forms using the (r s b) (p d q)

convention of Box and Jenkins [7] are specified. In some cases, particularly with the partial adjustment models, special conditions must hold for the model to be considered appropriate for a data set.

Several observations can be made about the models in Table 1. First, the maximum order of decay (I) evident in the literature is 3. While Bass and Clarke [l] considered model forms with three lagged sales terms, they rejected the necessity of three lagged sales terms in their empirical tests. In fact, in the majority of cases, a single lagged sales term was sufficient for modeling the cumulative effects of advertising. Second, the order

of the (s) specification has a maximum order of 1 in the models reviewed, and the speci-

fication is 1 only for the geometric lag models and the direct lag models. Of course, this does not preclude the possibility of additional lags for the direct lag model (s = 2,3,. . .) or for time delays in the geometric lag model before the advertising decay occurs (s = 2,3,. . .). Third, the maximum autoregressive order for the disturbance term (p) is 2. The other three terms-(b), (d), and (q)-do not exceed 0, although logic does not exclude these possibilities. If (b) is 1, for instance, there is one period of time when no

advertising effect occurs, and the first effect would be felt after one period. Such a circumstance is likely to occur in situations in which the time intervals of data collection are short (e.g., one day, one week). If data are collected at longer intervals, the dead time between advertising impulse and sales response is unlikely to occur (i.e., for data col-

Tab

le

1.

Com

mon

ly

Occ

urri

ng

Adv

ertis

ing

Eff

ectiv

enes

s M

odel

s &

0

Tra

nsfe

r Fu

nctio

n M

odel

For

m

Spec

ial

Nam

e M

odel

For

m

(rsb

) @

dcl)

C

ondi

tions

R

efer

ence

w

u =w

!$

C

urre

nt

effe

cts’

B

asic

mod

el

Aut

oreg

ress

ive

Geo

met

ric

lag’

Im

med

iate

de

cay

b Fi

rst

orde

r

St

= b

,At

+at

(0

00)

(000

)

(000

) (1

00)

Cla

rke

[4]

Cla

rke

[4]

b St

=

AA

t+at

l-

h,L

(1

00)

(000

)

(200

) (0

00)

(300

) (0

00)

Bas

s an

d C

lark

e [ 1

]

Seco

nd o

rder

Thi

rd o

rder

Dec

ay a

fter

on

e pe

riod

b

Firs

t or

der

Seco

nd o

rder

Thi

rd o

rder

Aut

oreg

ress

ive

dist

urba

nce

term

Firs

t or

der

St

=

bo

1 -

h,L

-

h,L

’ A

t +

at

St

=

be

1 -

h,L

-

h,L

” -

h,L

3 A

t +

at

s

t = (b

, +

b,L

) At+

at

l-&

L

St =

(b

, +

b,L

)

1 -

h,L

-

h,L

’ A

t+at

St =

(b

, +

b,L

)

1 -

h,L

-

h,L

Z

- h,

L”

At

+at

&=

b,

At+

?-

1 -h

,L

1 -(

YL

(110

) (0

00)

(210

) (0

00)

(310

) (0

00)

(100

) (1

00)

Bas

s an

d C

lark

e [ l

]

Bas

s an

d C

lark

e [ 1

I

Bas

s an

d C

lark

e [ 1

1

Bas

s an

d C

lark

e [ 1

]

Bas

s an

d C

lark

e [ 1

1

Wei

ss a

nd W

inda

l [ 1

21

Seco

nd o

rder

aut

oreg

ress

ive

dist

urba

nce

term

Fi

rst

orde

r

Part

ial

adju

stm

ent’

B

asic

mod

el

Aut

oreg

ress

ive

b St

= z

Ar+

at

(1

00)

(200

) 1

-h,L

l-

&L

-P

LZ

be

St

= ~

A

r+L

(1

00)

(100

) l-

&L

1-

h,L

b St

=

0 A

t+

at

(100

) (2

00)

1-h,

L

(1 -

P,L

J (1

-&

L)

Wei

ss a

nd W

inda

l [ 1

21

A,

=a

Wei

ss a

nd W

inda

l [ 1

21

a=p,

+9

P=

-P*h

,

Wei

ss a

nd W

inda

l [ 1

21

or

b,

St =

-

At+

at

(s

ee s

peci

al c

ondi

tions

) 1-

h,L

l-

oL-P

L’

Dir

ect

lag-

one

lag

term

’ B

asic

mod

el

Sr =

b,A

t +

b,A

t_l

+at

(0

10)

(000

) C

lark

e [4

]

Bas

ic m

odel

-aut

oreg

ress

ive

St =

b,A

t +

b,

At_

, +

-f!

--

(010

) (1

00)

See

Hel

mer

and

Joh

anss

on

I7 1

1-

CX

L

for

spec

ial

case

s

Bas

ic m

odel

-sec

ond-

St

=b,

At+

b,A

t_,

+

at

(010

) (2

00)

See

Hel

mer

and

Joh

anss

on

[ 7 I

orde

r au

tore

gres

sive

1

-d-P

L2

for

spec

ial

case

s

i T

he i

nter

cept

s fo

r th

ese

mod

els

are

dele

ted

for

clar

ity.

The

tra

nsfe

r fu

ncti

on

anal

ysis

al

low

s fo

r th

em

if n

eede

d.

The

six

m

odel

s re

pres

ente

d by

the

se

two

clas

ses

of m

odel

s ar

e fr

om

mas

s an

d C

lark

e [ I

]. T

hey

have

be

en

rew

ritt

en

in t

his

form

to

sho

w

thei

r re

lati

onsh

ip

to

tran

s-

fer

func

tion

an

alys

is.

252 J BUSN RES 19a5:13:247-257

M. M. Moriarty

lected monthly, quarterly, yearly, etc.). The author is aware of only one case where (d) is specified as 1 (Helmer and Johansson [7]) and no cases where (4) is 1. The Helmer and Johansson transfer function model [(OlO)( 1 lo)] , h owever, is a special case of the [(0 10) (200)] model where CY = Z + p1 and p1 is the estimate of the autoregressive parameter associated with the specification r = 1 (see Table 1).

These observations suggest that a reasonable general advertising effects model based on the currently available literature would be the [(3 10)(200)] transfer function model, which is written in explicit form below for clarity:

s, = ( b, •t blL

1 - X,L - X,L2 - h3L3 ) At •t (I-a:-PV)

All the cited models are nested within this model and can be specified from it by making assumptions or testing assumptions about its parameters. For instance, if the decay in effect is immediate, b, = 0, and if a simple geometric lag is appropriate, h2 and h3 = 0. Further, if the disturbance term is not autoregressive, (Y and p = 0. Finally, if a direct lag model is appropriate, hr , h2, and h3 = 0, and if a period of dead time is expected,dt can be replaced with A,_j (i = 1,2,. . .) depending on the number of dead time periods.

Transfer Function Analysis and Model Specification

Tests for discriminating among different models of advertising effectiveness have received recent research interest [ 1,5, 11, 121 . The model specified in equation (5) can be used as a reasonable starting point for developing equations to be applied to data to determine which of a number of nested models is appropriate. The Lydia Pinkham data base [lo] provides a convenient application to illustrate the specification testing process based on (5). The deseasonalized monthly Pinkham data are available for the period from January 1954 to June 1960, providing 78 observations for monthly sales and advertising. The data will be analyzed monthly, quarterly, and in six-months intervals based on aggregations into the latter two time intervals. Thus, there are 26 quarters, and 13 six-month observa- tions for study. The purpose of analyzing these three time intervals is to illustrate the development of different model forms from equation (5) and to show the effects of data interval bias.

The Monthly Model

The results of estimation of the (310) (200) transfer function model based on the monthly data appear in Table 2. The results suggest that h2, X3, and br are not statistical- ly significant. Thus, it would appear that the lagged sales term structure is unnecessarily complicated. It also appears that the sales response to advertising decays immediately because br is 0. These parameters are deleted from the model, and analysis is conducted on a (100) (200) model, i.e., one involving only one lagged sales term, hr , and a second- order autoregressive disturbance process. Estimation of this model suggests that the inclusion of a second-order autoregressive parameter (/3) is unnecessary (see Table 2).

Finally, given the nonsignificance of the second-order autoregressive disturbance parameter, a (100) (100) model is tested. This model meets the constraints imposed for single lagged sales term models (0 < ir < 1,O < 6, < 1; see Bass and Clarke [l , p. 301]), and has statistically significant coefficients. Examination of the residual autocorrelation function suggests the need for three MA terms (4 = 3) in the noise function at lags of 3,

Transfer Function Analysis J BUSN RES 253 1985:13:247-257

Table 2. Estimation of the Monthly Model

Model Parameter Estimates Standard Errors f- Ratio

(310) (200) 51 = 1.384 h=I =- 1.160 h ..I = 0.316 b ..o = 0.134 b = ,I 0.054

3 = = 0.487 0.384

(100) (200)

(100) (103) (100) (103)

$ = 0.773 b = -0 0.275 d = = 0.161 0.336

Constant =- 165.69

41 = 0.758 b = -0 0.304 (Y = 0.314 ^ 03 =- 0.528 0‘ =- 0.263 0, = 0.282 Constant = - 182.65

0.75 1 1.84 0.728 - 1.59 0.539 0.59 0.057 2.35 0.114 0.47 0.120 4.06 0.122 3.15

0.100 7.73 0.079 3.48 0.119 2.82 0.124 1.30

46.944 - 3.53

0.087 8.71 0.056 5.43 0.166 2.71 0.095 - 5.56 0.096 - 2.74 0.096 2.94

33.857 - 5.39

5, and 7. Thus, the (100) (103) model is considered appropriate for the monthly data. This model implies that there is a geometric lag structure for the advertising decay process and that the decay commences without delay. The estimate of the total advertising effect based on the geometric hypothesis (ha/(1 - ir)) is equal to 0.304/(1 - 0.758) = 1.26; and the 80% implied duration interval estimate of the time length of the advertising effect [log,c(l - 0.8)/log10 x1] is 5.8 months (see Clarke [4] for derivations of these estimates). Thus, virtually all of the advertising effect dissipates within six months of an advertising impulse.

Based on Bass and Clarke’s [l] analysis, the probability of purchase in period i in response to an advertising input at time i = 0 is $i = ii (1 - i\) (i = 0,1,2,. . .). Given the parameter estimate for hr = 0.758, fi,, = 0.242, r;r = 0.183,& = 0.139,& = 0.105, and

3 C = 0.669.

i=o

Thus, at the end of the fourth month, two-thirds of the advertising effect has dissipated. This result suggests that the quarterly model is likely to lack a substantial lag structure and that most of the advertising effect is hkely to be in the current effect, be, Q, of the quarterly model.

The Quarterly Model

An elaborate structure for the general transfer function is not suggested for the quarterly data based on the previous analysis of the monthly data. Hence, the (100) (100) model structure is used as a starting point for analysis of the quarterly data because this is the

254 J BUSN RES 1985:13:247-257

M. M. Moriarty

Table 3. Estimation of the Quarterly Model

Model Parameter Estimates Standard Errors t- Ratio

(100) (101) &I = - 0.062 0.311 - 0.20 b = ..o 0.440 0.152 2.89 01 = 0.651 0.203 3.24 . 0 = - 0.391 0.224 - r&ant

1.75 = - 879.78 297.714 - 2.96

(000) (101) Ho = 0.441 0.114 3.87 (Y = 0.666 0.170 3.92 ^ % = - 0.389 0.214 - 1.82 Constant = - 859.98 251.898 - 3.41

(010) (101) 80 = 0.444 0.154 2.88 b = ..I 0.024 0.141 0.17 OL = 0.653 0.206 3.17 ^ 0, = - 0.388 0.225 - 1.72 Constant = - 831.27 475.117 - 1.75

highest order of structure for r and p, which were found to be significant based on the monthly results. Estimation of this model suggests the need for an MA term (4) of order 1 at lag = 3 based on the residual autocorrelation function. The (100) (101) model results are shown in Table 3; they imply no carryover effect beyond the current quarter because hr is not significant. The value ie is significant, which indicates a current quarterly effect of advertising. Thus, the model conforms to the expectations derived from the monthly model. The quarterly model is insufficiently sensitive to specify a carryover effect in the second quarter. The estimate of the total advertising effect (he/l - ir)) is 0.440/(1 - (- 0.062)) = 0.414, which seriously underestimates the total effect, assuming the monthly results (= 1.26) to be correct.

Because the kr term is not significant, it was decided to test further the possibility of a carryover effect using a direct lag model. Estimation results of the (000) (101) and (010) (101) direct lag quarterly models also appear in Table 3. The direct lag coefficient, br , at lag 1 for the (010) (101) model is not significant, thus supporting the previous result that advertising carryover does not occur. The other observation is that 6, is very stable regardless of which of the quarterly models in Table 3 is examined. Thus, we conclude, on the basis of these results, that there is no carryover effect of advertising beyond one quarter and that the monthly and quarterly results disagree.

The Six-Month Model

Given the results for the quarterly data, the structure of the six-month model should be relatively simple. We start with the (100) (100) model, whose estimates appear in Table 4.

The lagged sales term (h, ) is not significant and 6, is significant. Therefore, a current effects model appears to be appropriate. Estimates of the direct lag models, (000) (100) and (0 10) (loo), confirm the results based on the (100) (100) model. Only the current term ie is significant in the (010) (100) model; hence, the final model is the (000) (100) model. The total advertising effect (&/(l - hr) = Lo) is 0.877 based on this model. If the (100) (100) model estimates are used, the total advertising effect (he/(1 --?I,)) would be 1.035/(1 - 0.213) = 1.32. This estimate is approximately equal to the estimate for the total advertising effect obtained from the monthly data.

Transfer Function Analysis J BUSN RES 255 1985:13:241-257

Table 4. Estimation of the Six-Month Model

Model Parameter Estimates Standard Errors r- Ratio

(100) (100) i = 0.213 0.182 40 = 1.035 0.240 OL = 0.351 0.466 Constant = - 3805.0 836.276

(000) (100) 40 = 0.877 01 = 0.711 Constant = - 3424.5

(010) (100) 80 = 0.850 b = -1 0.155 (Y = 0.919 Constant = - 3357.6

0.147 5.97 0.204 3.49

501.888 - 6.82

0.141 6.03 0.129 1.20 0.196 4.69

2201.63 - 1.53

1.17 4.31 0.75

- 4.55

Data Interval Bias

Close examination of the results in Tables 2, 3, and 4 suggests that as the data are aggre- gated from months to quarters to six-month data intervals, the estimate of bc increases steadily (0.304 to 0.441 to 0.877), and the estimate of Xi first declines and then rises (0.773 to - 0.062 to 0.213). Bass and Leone [2] have examined the estimation of the coefficient of advertising and the coefficient of lagged sales for data grouped at different levels of temporul aggregation. Their research provides a theoretical explanation for the observation that the parameter estimates mentioned above can vary with the length of the data interval for the same model and thus imply different duration intervals of advertising effect. Starting with a micro sales-advertising equation involving lagged sales and current advertising as independent variables, they show that increasing the data interval results in the convergence of the advertising coefficient (be in the present research) to the total cumulative effect of a unit impulse of advertising assuming the geometric lag hypothesis. Further, the coefficient on the lag of sales (which measures the duration of the advertis- ing effect) will approach zero (hi in this case). However, they show empirically that the implied duration interval for advertising increases with the data interval as a result of data interval bias. The reason for the data interval bias is that empirically the rate of decrease in the lag coefficient does not compensate sufficiently for the rate of increase in the data interval, thus causing the implied duration interval to increase with the length of the data interval. The evidence presented here is consistent with their conclusions. Thus, they suggest that when interest focuses upon the duration interval, an attempt should be made to capture the coefficient of lagged sales (hi) for the appropriate data interval.

Our analysis for the monthly, quarterly, and six-month data suggests that the advertis- ing duration interval is at most six months. Based on the monthly model, the 80% duration interval is 5.8 months. For the quarterly model, a measurable advertising effect was not observed past the current quarter, and for the six-month data not measurable effect is observed past the first six months. However, if data are aggregated into longer intervals, the potential for observing data interval bias is high. In particular, it is likely that analysis of the yearly Pinkham data will show the effects of data interval bias and a concomitantly high estimate of the advertising duration interval.

The Yearly Model

The yearly F’inkham data [lo] are available for advertising and sales for the years 1907- 1960. These data will be analyzed to determine whether the advertising duration interval

256 J BUSN RES 1985:13:247-257

M. M. Moriarty

Table 5. Estimation of the Yearly Model

Model Parameter Estimates Standard Errors t- Ratio

(300) (200) -1 = 0.359 h ,a =- 1.04 A ,s = 0.090 b -0 = 0.478 z = = 0.345 1.202

Constant = - 284.190

(100) (200) II = 0.379 b -0 = 0.554

; = = - 0.271 1.130

Constant = - 481.20

0.349 1.03 0.333 - 1.31 0.339 0.27

0.172 2.78

0.169 7.11

0.170 - 2.03 244.199 - 1.57

0.253 1.50 0.146 3.79

0.164 6.89

0.155 - 1.75

186.286 - 2.58

estimate based on them is longer than the estimates developed above based on the shorter data intervals. If the duration interval is longer, the yearly model results would be con- sistent with Bass and Leone’s [2] conclusions and would demonstrate data interval bias.

Table 5 presents the yearly results of estimating the (300) (200) and the (100) (200) models. The (3 10) (200) general model was not estimated here, since no delay in response is reasonable given the nature of yearly data. The (300) (200) model results show that the ha and As terms are unnecessary given their small t ratios. The (100) (200) model appears adequate because all coefficients are significant at OL = 0.10 or less. The estimate of Xi now clearly appears to be biased upward, and is larger than either the quarterly or the six- month estimate of hr. Based on the yearly estimate of Xi, the implied 80% duration interval for advertising effect is [logia(l - 0.8)/logn, i,) = 1.66 years. Obviously, this estimate exceeds the previous estimates based on the shorter data intervals. It is clear from Tables 2-5 that consistent estimates of the duration interval of advertising effect can be obtained only from analyses conducted at levels of aggregation of one month, one quarter, or at most six months. Higher levels of aggregation are suspect, including a one- year level of aggregation.

Discussion

In the marketing literature, the two general classes of advertising effectiveness models, namely, direct lag and distributed lag models, are shown to be nested versions of the (310) (200) transfer function model. Thus, it is possible to employ this general model form whether the researcher’s purpose is to identify a model form or to test competing model forms. For identification purposes, the researcher is assured that all reasonable model forms are embedded in the (3 10) (200) model and can be derived from it, depend- ing on the significance of various parameter estimates obtained from a given data set. This relieves the researcher of a prior? specification of either a direct lag model or a dis- tributed lag model.

For hypothesis testing, alternative model forms can be tested against the general model form. Implications of alternative models for the significance of the parameter estimates of the general model can be derived, and the significance of the estimates of the test equation can be used as a basis for rejecting one model in favor of another. For instance, if the researcher wants to test the first-order geometric lag model, with no delay in adver- tising decay against the direct lag model with one lag term, the implications for the

Transfer Function Analysis J BUSN RES 257 1985:13:247-257

general model parameters would be that hr > 0, hz = h3 = 0, bo > 0, bl = 0 in the former case and X1 = hz = hs = 0, b. > 0, and bI > 0 in the latter case. Thus, there are a variety of uses of the model which should benefit researchers interested in the cumu- lative effects of advertising.

An application to the issue of data interval bias is illustrated in the paper. The general model is used to demonstrate that the coefficients of current advertising and lagged sales for varying lengths of data intervals behave in a manner consistent with the theoretical results recently published by Bass and Leone [2]. Results suggest a pattern of upward bias in the coefficient of lagged sales, which in turn leads to an upward bias in the dura- tion interval estimate of the advertising effect.

Subsequent research on the transfer function approach could focus on refining the testing process and determining its sensitivity in detecting model differences among rival hypotheses of the advertising effect. For applications involving model identification, the convergence of the method presented here with traditional methods of specification of dynamic relationships should be compared to determine which approach is preferred.

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