Advertising Themes

8/2/2019 Advertising Themes

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Vol. 26, No. 2, MarchApril 2007, pp. 179195

issn 0732-2399 eissn 1526-548X 07 2602 0179

informs

doi10.1287/mksc.1060.0208 2007 INFORMS

Wearout Effects of Different Advertising

Themes: A Dynamic Bayesian Model of theAdvertising-Sales Relationship

Frank M. Bass, Norris Bruce, Sumit Majumdar, B. P. S. MurthiSchool of Management, The University of Texas at Dallas, Richardson, Texas 75083-0688

{[email protected], [email protected], [email protected], [email protected]}

Models of advertising response implicitly assume that the entire advertising budget is spent on dis-seminating one message. In practice, managers use different themes of advertising (for example, priceadvertisements versus product advertisements) and within each theme they employ different versions of anadvertisement. In this study, we evaluate the dynamic effects of different themes of advertising that have beenemployed in a campaign. We develop a model that jointly considers the effects of wearout as well as that offorgetting in the context of an advertising campaign that employs five different advertising themes. We quantify

the differential wearout effects across the different themes of advertising and examine the interaction effectsbetween the different themes using a Bayesian dynamic linear model (DLM). Such a response model can helpmanagers decide on the optimal allocation of resources across the portfolio of ads as well as better managetheir scheduling. We develop a model to show how our response model parameters can be used to improve theeffectiveness of advertising budget allocation across different themes. We find that a reallocation of resourcesacross different themes according to our model results in a significant improvement in demand.

Key words : Bayesian dynamic linear models; Gibbs sampling aggregate advertising models; wearout effects;forgetting effects; copy effects; scheduling of ad copy

History : This paper was received June 30, 2005, and was with the authors 4 months for 1 revision; processedby Greg Allenby.

1. IntroductionManagers, policy makers, and researchers are inter-ested in understanding the effect of advertising ondemand. A large number of response models have

been proposed in the literature linking advertisingexpenditures to sales or market shares. These studieshave focused on the shape of the response function(linear or S-shaped), the dynamic effects of advertis-ing (carry-over effects or wearout), and interactioneffects with other marketing mix variables, consistentwith the desirable properties of advertising responsemodels in Little (1979).

In these response models, advertising expendituresare aggregated and there is an implicit assumptionthat advertising expenditure is spent on propagatingone message or theme. In practice, firms concurrentlyrun several themes in their advertising campaigns(for example, price and product advertisements). Eachtheme may have multiple versions, or executions,which are rotated over time. In effect, there is a portfo-lio of ads that is run in each week. Managers need tounderstand the individual effects as well as the inter-action effects among the different versions of advertis-ing on overall demand. Such an understanding then

Deceased.

would allow managers to allocate their budgets overmultiple themes more effectively. Most of the pub-

lished research on the impact of advertising themesis categorized under copy research and these stud-ies have employed experiments. They have focusedon the effects of different copy on individual con-sumer attributes such as brand awareness and atti-tudes toward the brand. The literature that links thecopy effects to sales or market share is limited to ex-perimental studies and does not consider wearout ef-fects (Aaker and Carmen 1982, Eastlack and Rao 1989).

The substantial research issues that we address inthis study focus on understanding the effects of dif-ferent themes of advertising on demand and on how

to allocate a firms resources across different themesto improve sales performance. The specific researchquestions pertinent to the study of the effectivenessof multiple themes in advertising are: How differ-ent are the wearout effects for the different themesof advertising? How can researchers accurately assessthe magnitude of such wearout effects? What is thenature of interaction between the different themes ofads? For a given advertising budget, what is a moreeffective way to allocate resources across differentadvertising themes?

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Bass, Bruce, Majumdar, and Murthi: Wearout Effects of Different Advertising Themes180 Marketing Science 26(2), pp. 179195, 2007 INFORMS

Our response model extends the model by Naiket al. (1998) to multiple themes of advertising. Theyestimate a dynamic model of the effect of advertis-ing on consumer awareness, rather than on demand,using Kalman filtering methods. Their model cap-tured decay effects in the presence and absence of

advertising and found that a gap in advertising hasthe effect of restoring ad quality. They also sepa-rate out the effects of two different types of wearout:copy wearout and repetition wearout. However, intheir paper, they study the effectiveness of a singlead campaign with a single message and do not con-sider multiple themes of advertising which may bescheduled by the manager for the same product cat-egory. Our study generalizes their advertising modelto account for multiple thematic executions of adver-tising. In addition, there are a few other notable dif-ferences between the two models. Naik et al. (1998)use brand awareness as the dependent variable while

we use a measure of demand which is more man-agerially useful. We employ a different methodology,Bayesian estimation of a dynamic linear model (DLM)as in West and Harrison (1997).

Our paper makes three contributions. First, weextend earlier research on advertising by developinga generalized model of multiple themes of advertisingwhich are aired concurrently in a given week. Sucha model can allow managers to assess the wearouteffects of different themes of advertising as well asconsider the interaction effects between these differ-ent themes. Second, we capture the dynamic effects of

different advertising themes as a function of wearoutand forgetting. In other words, the parameters ofthe model are time varying and we are able to cap-ture the dynamic path of these parameters. To dothis, we employ Gibbs sampling in estimating thestate space model of West and Harrison (1997). Tworecent papers in marketing have also employed Gibbssampling in a DLM framework (Neelamegham andChintagunta 2004, Van Heerde et al. 2004). Leichtyet al. (2005) employed Gibbs sampling to study thedynamic development of consumer preferences in aconjoint application. Finally, we develop a model tohelp managers conduct a what-if scenario analysis so

that they can allocate their resources over differentadvertising themes more effectively.

Our data are obtained from a major telecom com-pany that is a monopolist in its category. We have dataon demand in terms of calling time, prices per minuteof calling, and advertising spending for each of thethemes that are defined by the company. A uniqueaspect of our data is that advertising is measured ingross rating points (GRPs) and not in dollars. OneGRP represents advertising exposure to one percentof the population that owns television sets as defined

by the ACNielsen TV ratings. There are two advan-tages of using GRPs instead of advertising dollars.First, GRPs provide a more accurate picture of adver-tising input than advertising expenditures since itis not clear how much advertising exposure can bepurchased for a given budget. Second, most media

buying is done in terms of GRPs and managers eval-uate the effectiveness of their campaigns in terms ofdemand generated per GRP. The company classifiesits advertising into five advertising themes: call stimu-lation ads, product offer ads, price offer ads, reconnectads, and reassurance ads. We restrict our attention tothese five themes.

The remainder of the paper is organized as follows.In 2, we provide a brief review of the relevant liter-ature on estimation of advertising response with par-ticular emphasis on wearout and dynamic effects. In3, we present the details of our econometric model.In 4, we present the data and discuss the results of

our estimation. Finally, we conclude with an overviewof findings, the managerial implications, and the lim-itations of the study.

2. Literature ReviewWe present a brief overview of response models andreview the literature on wearout effects and dynamiceffects in greater detail. We also present argumentsfor studying the differences in wearout effects acrossmultiple themes.

2.1. Response ModelsIn a seminal paper, Little (1979) stated that aggregate

advertising response models should have the follow-ing desirable characteristics: (a) the effect of adver-tising should be nonlinear, (b) the models shouldcapture the dynamic effects of wearout and forgetting,(c) models should consider the effect of competitiveadvertising, and (d) the ad effects could change overtime due to changes in media and copy. Most of themodels in the literature have been developed consis-tent with some, if not all, of these principles.

Early aggregate advertising response models linkedadvertising expenditures to sales or market share(Bass and Clarke 1972, Blattberg and Jeuland 1981,Hanssens et al. 1990) and considered the carry-over effects of advertising (Bass and Leone 1983,Broadbent 1984, Clarke 1976, Srinivasan and Weir1988). These models used distributed lag models tocapture the carry-over effect. Clarke (1976) showedthat the magnitude of the effect of advertising andthe duration of carry-over effects depended on thedata interval used. The response models were thenused to appropriate advertising dollars to maximizeprofits in both monopoly and oligopoly contexts(Simon 1965, Nerlove and Arrow 1962, Telser 1964,Palda 1964). These models used aggregate advertising


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Bass, Bruce, Majumdar, and Murthi: Wearout Effects of Different Advertising ThemesMarketing Science 26(2), pp. 179195, 2007 INFORMS 181

expenditures and did not consider the effect of mul-tiple themes. A good recent review of the advertisingliterature is in Vakratsas and Ambler (1999).

Another stream of research discussed the shapeof the advertising response function: whether it isconcave or S-shaped. Wittink (1977), Rao and Miller

(1975), and recently Vakratsas et al. (2005) found evi-dence of an S-shaped function while Simon (1969)found no evidence for the S-shape. This was animportant question because the theoretical modelsshowed that the phenomenon of pulsing in advertis-ing was related to the S-shape of the response func-tion (Simon 1982, Mahajan and Muller 1986, Feinberg1992). In a recent paper, Naik et al. (1998) show thatpulsing can occur due to ad copy wearout, whileBronnenberg (1998) shows that pulsing can also occurin the context of a monopolist facing a Markoviansales response function.

2.2. Wearin and WearoutWearin refers to the positive effect on consumers whoare exposed to an ad (Pechmann and Stewart 1990).The term wearout refers to the decay in advertis-ing quality of an ad over time (Grass and Wallace1969, Strong 1972, Calder and Sternthal 1980, Simon1982). An ad is worn out if it either does not haveany significant effect on consumers or has a negativeeffect. Both wearin and wearout effects depend onfactors such as whether the ad was based on an emo-tional appeal or a rational appeal, whether the per-suasion in the message was strong or weak, whetherconsumers were motivated or not to process the ad,and whether the level of competitive ads was highor low (Pechmann and Stewart 1990). The wearouteffects may also depend on the change in ad copy.This is based on research that suggests that varia-tions in copy improve the effectiveness or, specifically,recall of ads (Grass and Wallace 1969).

In a series of experiments to study of the effects ofrepetition of ads, Ray and Sawyer (1971a, b) foundthat the response functions for repetition varied acrossdifferent measures (e.g., recall or intention), segments,

brands, and type of advertising. They also studied theeffect of different messages on repetition functions.Relevant to our study, they found that grabber adswere less effective over repetitions (i.e., had higherwearout) than nongrabber ads. MacInnis et al. (2002)find evidence of a significant positive relationship

between ad repetition and sales when emotional adsare employed, but not for rational ads. They arguethat one possible explanation of the above finding isthat emotional ads have less rapid wearout.

Naik et al. (1998) model two sources of wearoutrepetition wearout and copy wearout. When a customeris exposed to ads repeatedly, she can become bored,irritated, or simply lose interest as the benefits of pro-cessing the ad are perceived to be worthless (Berlyne

1970, Greyser 1973, Weilbacher 1970). This leads torepetition wearout, which depends on the amountof advertising that is done. Copy wearout, on theother hand, is the decay in advertising effectivenessdue to the passage of time, which is independent ofthe amount of advertising. Such a decay may be the

result of a change in consumers conditions such asincreased knowledge about product attributes overtime (Calantone and Sawyer 1978). Other reasons forcopy wearout include the imitation of an ad strategy

by competing firms or by firms in other product cat-egories (Axelrod 1980) and an increase in ad clutter(Corkindale and Newall 1978). A good discussion ofthese two effects is in Naik et al. (1998).

2.3. ForgettingAnother factor that affects quality dynamics is theeffect of forgetting. Consumers tend to forget an adwhen it is not being aired in a given period. Forget-

ting has a negative effect by reducing aggregate brandawareness (Mahajan et al. 1984). On the other hand,the literature also suggests that there is a rejuvenatingeffect of advertising when an ad is taken off the media(Grass and Wallace 1969, Greenberg and Suttoni 1973,Corkindale and Newall 1978, Naik et al. 1998). Grassand Wallace (1969) conclude that a period of no adver-tising enhances consumers attention to ads. Simi-larly, Calder and Sternthal (1980) have found that theamount of cognitive responses increases when thereis a break in advertising. The argument for improve-ment in ad quality, when it is not aired for a period oftime, is that consumers may forget the particulars of

a given ad and may feel that the ad is fresh when itis reintroduced. This suggests that as the period thatan ad is pulled off the media is increased, there is acorresponding increase in forgetting and a consequentincrease in the restoration of ad quality (Corkindaleand Newall 1978).

2.4. Differential Effects of Wearout Across ThemesAs stated earlier, wearout is affected by several fac-tors such as the type of appeal, level of competitiveadvertising, and strength of persuasion. Experimen-tal evidence shows that emotional ads wearout moreslowly than ads based on nonemotional or rational

appeals (Hitchon et al. 1988). In this laboratory study,the dependent variables were attitude toward the ad,

brand attitudes, and purchase intentions. The authorsfound that unemotional ads exhibited faster wearoutin all of the three dependent variables when com-pared to emotional ads. This could be due to the factthat ads with emotional images elicit imagery process-ing while verbal arguments elicit cognitive processing(MacInnis and Price 1987). Silk and Vavra (1974) andRay and Sawyer (1971b) also suggest that soft sellads, which use emotional images, wear out slower


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than hard sell ads, which are based on verbal argu-ments. These arguments highlight the need for study-ing the differential wearout effects of different themesof advertising and their effect on demand.

2.5. Need for Time-Varying CoefficientsAdvertising effects accumulate over time, hence it is

reasonable to expect that the response coefficients willalso vary over time. Researchers have modeled thephenomenon of time-varying parameters using datafrom different time periods and estimating a sep-arate coefficient for each time period (Winer 1979,Bronnenberg et al. 2000, Mela et al. 1997). Since onlya part of the data is used in each estimation, theseestimates are likely to be inefficient.

Other researchers have employed a random coeffi-cients approach (Jedidi et al. 1999) where the param-eter is assumed to be distributed according to aprobability distribution over time. In these models,the variance of the distribution is estimated; thus

researchers are able to control for the time-varyingnature of the response parameters but are unable torecover the parameter paths over time. The modelused in our research, the dynamic linear model(DLM), is able to estimate the dynamic path of theresponse coefficient over time and the estimates areefficient.

In a recent paper, Dub and Manchanda (2005)study the differences in dynamics of marketing mix(price and advertising) across several markets. Theyconclude that a firms current and past advertisinghas a larger effect on its own demand, especially inlarger markets. They also conclude that competitors

advertising has a much smaller effect on a firmsdemand than the firms own advertising. This worksuggests that ignoring competitive advertising whilemodeling a firms advertising-to-sales relationship isunlikely to cause serious error.

2.6. Need for InteractionsThe interaction effects between advertising and othermarketing mix variables, especially price, have beenwell-documented (Eskin and Baron 1977, Wittink1977, Krishnamurthi and Raj 1985, Winer and Moore1989). Therefore, it is necessary to model the inter-action between the different themes of advertising as

well. A recent paper by Naik et al. (2005) makes astrong case for modeling interactions between adver-tising and promotions in developing demand mod-els in a competitive environment. In sum, our paperextends the literature on advertising response mod-els by considering wearout effects, interaction effects,and dynamic effects of different themes of advertisingwhile being consistent with prior research guidelines.

Montgomery et al. (2005) use a survey to provide evidence thatmanagers, when making decisions, place considerably less weighton competitive reactions and strategic competitive reasoning.

3. ModelWe develop a model of advertising that links demandto the advertising expenditure that is incurred in air-ing multiple themes of advertising in a given week.Since advertising effects last longer than a week, wecan assume that multiple themes are being aired con-

currently. The model should consider the dynamiceffects of wearout and forgetting as described earlier.We begin by generalizing the model proposed

by Naik et al. (1998), who estimate a model thatlinks awareness to the advertising expenditure data.Though they estimated a number of advertising mod-els that have been proposed in the marketing lit-erature such as Nerlove and Arrow (1962), Vidaleand Wolfe (1957), Little (1975), Blattberg and Golanty(1978), and Blackburn and Clancy (1982), they findthat the Nerlove-Arrow model provides them the bestfit. Given this information, we focus our attention inthis paper on the Nerlove-Arrow model.

In the Nerlove-Arrow model, the rate of change ingoodwill Gt is a function of the advertising expen-diture per week At that can be measured eitherin terms of gross rating points (GRP) or in dollars.Specifically,

dGt

dt= qAt Gt (1)

whereq = effectiveness of ad spending (assumed con-

stant, set equal to one in Nerlove-Arrow). = rate of decay of goodwill due to forgetting.

At = advertising spending in each time period t.

Gt = goodwill in each time period.More generally, one can use a function of advertis-

ing expenditure fAt, which need not be linear asin the Nerlove-Arrow model.

Naik et al. (1998) extend the above model by mak-ing the advertising effectiveness parameter q to betime dependent. In other words, q is not assumed to

be constant over time, but is a function of time as wellas advertising expenditure. The rationale for such anapproach is that it allows the capture of differenttypes of wearout effectscopy wearout and repetitionwearout. Copy wearout refers to the decay in adver-tising effectiveness over time due to the message

becoming less effective, and this effect is assumed tobe independent of the amount of advertising. Naiket al. (1998) point out that copy wearout can occurdue to several reasons: (1) the consumer becomesmore knowledgeable about the product attributes,(2) the amount of competitive advertising in responseto this firms advertising can reduce attention, or(3) other firms imitate the advertising style. Repetitionwearout, on the other hand, depends on the amountof advertising expenditure and occurs because con-sumers get bored of seeing the same ad or perceive


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the value of processing the ad a second time as beingsmall. The greater the amount of advertising, thegreater is the repetition wearout effect.

To describe the evolution of the effectiveness param-eter q over time, Naik et al. (1998) propose a differen-tial equation as follows:

dqdt

= aAq + 1 IA1 q (2)

where

aA = c + wAt (3)

c is the copy wearout parameter and w is the repeti-tion wearout parameter.

In the above equations, there are two types ofeffects. When the advertising is on in a given timeperiod, the indicator function IA takes the value = 1and the second term on the RHS of Equation (2)

becomes zero. The effect of advertising is captured

by copy wearout parameter c and repetition wearoutparameter w in Equation (3). Note that repetitionwearout depends on advertising level At. Whenadvertising is off, there is a rejuvenating effect ofadvertising that is captured by the second term on theRHS of Equation (2).

We now generalize the Nerlove-Arrow and Naikmodels to accommodate multiple ad themes. A dif-ferent combination of ad themes is aired in eachweek. We model the ad spending rate in Equation (1)as an additive function of ad spending on individ-ual themes. The modified equation of the rate ofchange in goodwill is given by a generalization of the

Nerlove-Arrow model for advertising themes:

dG

dt=

mi=1

qi

gAi + i

mj=1j=i

hAi Aj

G (4)

where qi is the effectiveness of ad theme i, gAi is afunction of the advertising expenditure for theme i,m is the number of ad themes, G is the goodwill,and Ai is the advertising expenditure for each adtheme. Note that both goodwill and advertising varyover time. Because of availability of data on severalad themes, we are able to understand the interactioneffects among the different themes. The term

i

mj=1j=i

hAi Aj

is an interaction effect for the ith theme. We estimatea separate interaction effects parameter i for eachtheme, thus allowing for possible asymmetric effects.It is an overall measure of how the ith theme interactswith all other concurrent advertising themes at time t.We suppress the notation for time for ease of exposi-tion. One can potentially estimate interaction param-

eters for all pairwise combinations of ad themes, butwe estimate only one coefficient for each theme for thesake of parsimony. Furthermore, we have no theoryto explain the nature of pairwise interactions, whichcould depend on the nature of ad themes and theirexecution.

In the estimation, we use gAi = ln1 + Ai andhAi Aj = ln1 + Ai ln1 + Aj. The justification foruse of the semilog specification has been presentedin Doyle and Saunders (1990). They rule out lin-ear and exponential functional forms as they do notexhibit diminishing returns, which is a characteristicof advertising response models (Simon 1970). Poly-nomial functions exhibit supersaturation and do notfare well in optimization (Doyle and Saunders 1990).The semilog specification has also been favored byLambin (1969) and Carroll et al. (1979). Jagpal et al.(1979) also recommend using a log-log specificationand consider the interaction term as the product of

two lnAi terms as we have done.The change in ad effectiveness qi is given by thefollowing equation:

dqidt

= aAiqi + 1 IAi1 qi

i = 1 2 m (5)

whereaAi = ci + wiAit (6)

Note that there are five different equations forEquation (5), one for each advertising theme. Fur-ther, for tractability we assume that the rate of changein ad quality for the ad themes dq

i

/dt are inde-pendent of each other. There is some support forthe assumption of independence in the study byBlair and Rabuck (1998). Based on an analysis ofover 500 case studies they conclude that commercialswithin a campaign wear out independently of theirpool partners. Even though there may be similarities

between some ad themes which could be modeled,such a dependency would complicate the estimationof the dynamic parameters in our model. We thereforeleave the modeling of such dependencies between adthemes for future research.

Thus, our model allows for the estimation of dy-namic effects of advertising themes in the presence ofwearout effects. We control for cross-sectional hetero-geneity between the ad themes by estimating separatedynamic parameters for each ad theme. We employthe Gibbs sampler for estimation of our parameters,as in West and Harrison (1997). There are two advan-tages of using a DLM model. First, we are ableto provide an understanding of how the responseparameters themselves change over time. To under-stand the differences in advertising effectiveness oftwo different ad themes, it is not enough to observedifferences in mean parameter values because even


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if the average effect of a particular response coeffi-cient may be the same across two advertising themes,their parameter paths may be substantially differentover time. Other researchers have employed a ran-dom coefficients specification to model time-varyingparameters (Jedidi et al. 1999). Such models can con-

trol for the time-varying nature of parameters but donot provide an estimate of the parameter at a givenpoint in time. DLM also handles missing observationsin a trivial manner as no updating takes place.

Second, advertising response may be nonstation-ary. In time-series analysis, researchers filter the data(say, by taking first- or higher-order differences) inorder to make the series stationary. West and Harrison(1997) point out that these filtering methods affect theinterpretation of the model by confounding differentmodel components. Further sources of nonstationar-ity that deviate from the process implied by the filterare not captured. They suggest that DLM presents a

better method for handling nonstationarity.

4. Data and Estimation

4.1. DataWe obtained data for a major telecommunicationsservices company in which the demand for residen-tial telephone services has been measured in two

Figure 1 Plot of GRPs for Different Themes Over Time

0 20 40 60 80 100 1200

200

400

600

Time in weeks Time in weeks



Call stim ads

0 20 40 60 80 100 1200

100

200

300

400

Price off ads

0 20 40 60 80 100 1200

100

200

300

400

500

Product off ads

0 20 40 60 80 100 1200

50

100

150

200

250

Reconnect ads

0 20 40 60 80 100 1200

50

100

150

200

Reassure ads

0 20 40 60 80 100 1200

500

1,000

1,500

Competitive ads

GRP

GRP

GRP

GRP

GRP

GRP

ways: minutes of call time and number of calls. Wehave used the total call time in millions of hoursas our dependent variable. We aggregated demandfor three call typeslocal, regional, and nationaleach of which are classified as weekday and week-end calls. We have not considered international calls.

The covariates that are available to model demand arethe average price per minute of a call, the numberof land lines available, and competitive advertising.We compute the weighted average price per minuteover the different categories that we considered sincethe rates are different for these categories. In addi-tion, we have data on the number of lines available toaccount for the increase in capacity over the 114 weeksunder study. Note that the telecommunications com-pany is a monopoly in the land line telecommuni-cations business and competition exists only in thewireless markets. We group all competitive advertis-ing expenditures into one category.

The company classifies its advertising into fivethemescall stimulation ads, product offer ads, priceoffer ads, reconnect ads, and reassurance ads. Adver-tising expenditure for each theme of advertising ismeasured in terms of gross rating points (GRP).Advertising GRPs of different themes of advertisingare available for a period of 114 weeks between 1995and 1997. In Figure 1, we provide plots of GRPs for


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Figure 2 Plot of Total GRP Over Time

0 20 40 60 80 100 1200

100

200

300

400

500

600

Time in weeks

Aggregate firm ads

GRP

different advertising themes as well as for total GRPof competitive ads over the 114-week period. The totalamount of GRP across all themes for the firm is givenin Figure 2. Descriptive statistics are given in Tables 1and 2.

4.2. Model EstimationWe model the demand for residential telephone ser-vices yt at time t as a function of goodwill Gt, avector Xt with variables price, number of lines, andcompetitive advertising, and mean-zero normally dis-tributed error t :

yt = Gt +Xt + t where t N 0 2 (7)

This unobservable error t could be related to de-mand factors such as the growth in mobile telephoneservice or the availability of the Internet. We assumethese omitted demand factors are perceived by thefirm and can influence price, suggesting a potential

endogeneity problem (Villas-Boas and Winer 1999).Therefore, we use instruments W to address thisproblem. We used retail price index, number of house-holds, consumer sentiment, and household spendingas instruments in addition to competitive advertisingand number of lines:

pt = ptW + t (8)

Another source of endogeneity bias could be dueto the assumption of advertising GRP as an exoge-nous variable. If firms choose the level of advertisingstrategically (say, in response to competitors advertis-

ing), then these variables are endogenous. This poten-tial endogeneity bias could be mitigated if we hadgood instruments in our data. We leave the study ofendoeneity bias as an issue for future research. Weexpect that the potential bias in estimates could affectthe solution to our allocation model.

To estimate the parameters of our model, we con-vert the above system of Equations (4)(6) to discrete

We are grateful to an anonymous reviewer for bringing this issueto our attention.

Table 1 Descriptive Statistics of Main Variables

Mean gross Standard

rating points deviation

Call stimulation ads 10096 12859

Price ads 3168 7321

Product ads 6387 10777

Reconnect ads 4562 6133Reassurance ads 2125 4401

Competition ads 50582

Mean call volume (hours) 16.42 million

Mean weighted price per minute 246

Mean line capacity (number of lines) 22.63 million

time and rewrite the generalized model in more for-mal, state space notation:

yt = 1 0 0 Ft

Gt

q1t

qmt

t

+Xt + t (9)

Gt

q1t

qmt

t

=

1 gA1t gAmt

01aA1t

1IA1t 0

0 0 1aAmt

1IAmt

Ht

Gt1

q1t1

qmt1

t1

+0

1 IAit

1 IAmt

ut

+w0tw

1t wmt

wt

(10)

where

gAit = gAit + i

mj=1j=i

hAit Ajt i = 1 m

Note that, as stated earlier, in the estimation weuse gAit = ln1 + Ait and hAit Ajt = ln1 + Ait ln1 + Ajt .

As shown above, t is the state vector whosefirst element is goodwill and remaining elements arem = 5 advertising qualities, one for each theme dur-ing time t. The m + 1 m + 1 transition matrix Htcaptures the time-varying effects of ad spending,wearout, and forgetting on the portfolio of advertisingthemes and goodwill across time. The constant vectorFt reflects the impact of goodwill on telephone usage.Error terms wt and t are assumed to be mean-zeroindependent normals. Thus, we can rewrite Equations(9) and (10) in a more compact notation to obtain the


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Table 2 Correlation Between GRPs of Different Themes

Reconnect Reassurance Competit ive

Price ads Product ads ads ads ads

Call stimulation ads 0137 (ns) 0242 0269 0014 ns 002 ns

Price ads 0038 ns 0037 ns 0157 035

Product ads 0029 ns 0175 0024 ns

Reconnect ads 00001 ns 0186

Reassurance ads 0119 ns

Significant at p < 010.Significant at p < 001; ns indicates nonsignificant.

standard Bayesian dynamic linear model (DLM) ofWest and Harrison (1997):

yt = Ftt +Xt + t (11)

t = Htt1 + ut + wt

wheret N 0

2 wt N 0 W (12)

Gathering the parameters of the model togetherin a tuple constitutes the DLM specification Ft Ht2 W. The model is then completed with specifi-cations of: (i) prior distributions for the unobservedelements of the transition matrix Ht , , ci wi i

mi=1,

(ii) prior distributions for , the coefficient of otherindependent measures that affect the demand for tele-phone services, (iii) prior distributions for unobservedobservation and system variances 2 and W, and(iv) initial guess for the state vector of containinggoodwill and advertising quality 0 at time t = 0

based on initial information denoted by D0. We willassume standard conjugate forms for all priors in ouranalysis. Prior specifications are chosen to allow thedata to dominate the results.

While many state space methods (Harvey 1994) relyon maximum likelihood estimation (MLE) for estima-tion, the DLM is based on Bayesian estimation. Westand Harrison (1997) suggest the use of traditionalMonte Carlo Markov chain (MCMC) techniques suchas Gibbs sampling. There are several reasons why theDLM is particularly suited to studying time-varyingadvertising effects. Relative to the static Bayesianmodels seen in the marketing literature (Allenby andRossi 2003), the DLM offers improved estimation viaadaptation and Bayesian learning. Moreover, withsuitable informative priors, relatively accurate fore-casts can be produced from series too short for purelydata driven (frequentist) analyses.

We estimate the above system of discrete Equations(9) and (10) by specifying priors for the model param-eters and using MCMC simulation of the full poste-rior based on the entire data series t = 1 2 T . LetT = 012 T and y

T = y1 y2 yTbe the state parameters and telephone usage over theentire data set. Let = ci wi i

mi=1 be a vec-

tor of forgetting, wearout, and interaction parameters

defined in Ht, and the coefficient for other explana-tory variables that influence telephone usage. Nowassume that the prior on 2 is inverse gamma and theprior on W is independent inverse Wishart. Then, byusing a direct Gibbs sampling approach (Gelfand andSmith 1990), we can compute the complete joint pos-terior pT W 2 y

T, iteratively resamplingconditional posteriors pT yT 2

W andpW 2 t y

T (see Appendix A for details).

4.3. Model IdentificationTo show identification in our DLM, we need to inves-tigate whether enough prior structure exists in thesystem of Equations (11) and (12). First, we need toconstruct an observationally equivalent system M1 =F1t H1t

2 W1, where observability (Crassidis and

Junkins 2004) relates to our ability to recover all statevectors T = 012 T given observationsyT = y1 y2 yT. An equivalent system is achieved

by substituting the linear transformation 1t = Ltinto Equations (11) and (12), where Lt is a m + 1 m + 1 nonsingular matrix. We obtain F1t = FtL

1,H1t = LHtL

1, u1t = Lut , and W1 = LWL. Under this

transformation, the model M1 is indistinguishablefrom our system described in Equations (11) and (12).Thus, for identification it is sufficient to show thatenough prior structure exists in Ft and Ht such that alltransformations are precluded except L = I (the iden-tity matrix). It turns out that L = I for the Naik et al.(1998) and our model (see Appendix B for a proof).

Finally, for the incorporation of price endogeneity,we take a two-step approach. In the first step, we esti-mate a Bayesian regression model with all the exoge-

nous variables and instruments as independent vari-ables. In the second step, we replace prices in the

For comparison, we also implemented a fully Bayesian instrumen-tal variable procedure given in Rossi et al. (2005). Their approachis different from that in Lancaster (2004). An important distinction

between these two treatments is the specification of the prior onthe covariance matrix of the reduced form errors. This matrix isa function of the coefficient of the endogenous variable. Lancaster(2004) assumes that the prior is independent of the coefficient ofthe endogenous variable while Rossi et al. (2005) do not, whichwe believe is more reasonable. We find that our substantive resultswere unchanged.


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DLM observation equation with their respective pre-dicted values obtained from the first stage estima-tion. These predicted values are obtained from theposterior predictive distribution, consistent with ourBayesian framework.

5. ResultsWe report the results of our estimation in Tables 3to 9. In Table 7, we provide values of the log Bayesfactors for comparing the model fits across alternatemodels. In Tables 4 and 5, we provide estimates of ourcomplete model with interactions between differentthemes. We also estimate a model in which there areno interaction effects between the different ad themes(Table 7). To compare the results, we estimate anothermodel in which we aggregate the advertising expen-ditures of different themes and treat the demand as afunction of total advertising GRPs (Table 8).

To compare model fit across two models that differin parameters, we employ Bayes factors or log Bayesfactors (West and Harrison 1997, Congdon 2001). Alog Bayes factor of one or greater indicates evidencein favor of our model. From Table 3(a), we see thata model with no advertising interactions is rejectedin favor of a model with advertising interactions.Similarly, a model that considers aggregate advertis-ing GRPs across the different themes as a depen-dent variable is also rejected in favor of our proposedmodel; this highlights the value of modeling differentresponses for different advertising themes. We alsoreject the linear and square root functional forms for

gAit in favor of the lnAit specification.We also compare the predictive performance of

alternate models and compute the mean absolute de-viation (MAD) and mean square error (MSE) betweenpredictions of demand from the model and the actualdemand. In Tables 3(b) and 3(c), we compute MADand MSE using data from all 114 weeks and use theparameters to predict demand for 1 to 114 weeksand 51 to 114 weeks. The MAD and MSE values in

both sections are the lowest for the full model withinteractions. We also provide out-of-sample forecastperformance of alternate models in Table 3(d). Weestimate Equations (9) and (10) using two sample

timeframes (t = 160, and t = 1100) and generate out-of-sample (step ahead) forecasts for 10 time periodsk = 1 2 3 10. We report MAPE (mean absolutepercentage error) across the k = 10 time periods forall model forecasts. The evidence clearly supports thepredictive performance of our proposed model.

In Table 7, we present the estimates of our modeland we see that the effect of price per minute ondemand is negative. To obtain an understanding ofthe significance level, we report the 95% HPDI (high-est probability density interval). We note that price

Table 3 Comparison of Current Specification with Alternate

Specifications

(a) Log Bayes Factors

Specification Description Log Bayes factor

Model 2 No advertising interactions 98.767

Model 3 Aggregate GRPs 87.712Model 4 Square root GRP 21.565

Model 5 Linear GRP 360.19

Note. T = 1: 116 weeks.

(b) Predictive Performance

Specification Description MAD MSE

Model 1 Full model 04194 03499

Model 2 No advertising interactions 09837 14942

Model 3 Aggregate GRPs 09072 12871

Model 4 Square Root GRP 07538 09082

Model 5 Linear GRP 14564 30890

(c) Predictive Performance

Specification Description MAD MSE




Model 4 Square Root GRP 08975 12051


Note. T = 50: 116 weeks.

(d) Model Forecast Performance

Specification Description MAPE1 MAPE2




Model 4 Square root GRP 1186 2753


1Sample t= 160.2Sample t= 1100.

per minute has a significant effect on demand at 95%confidence level. Further, capacity as measured by thenumber of lines also has a significant positive effecton demand, while competitive advertising has a neg-ative but nonsignificant effect on demand. The lack ofan effect due to competition may be due to the factthe firm is dominant in its category. These effects are

consistent with our expectations.The forgetting rate is 0.034, significant and consis-

tent with values obtained in earlier studies (Naik et al.1998). This means that when an ad is taken off con-sumers tend to forget the message and this could leadto an enhancement in quality of the ad when it is airedagain after some time. The copy wearout effects c areall positive and significant. The coefficients are differ-ent for different themes and range from a low valueof 0.1625 for reassurance ads to 0.5725 for price offerads. This suggests that price offer ads have a higher


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Table 4 Estimates by Message Type (with Interactions)

Standard

Parameters Estimate deviation 95% HPDI

Price per minute 119171 63939 226110 14258

Competitive advertising 00467 01434 02860 01844

Capacity (number of lines) 11192 05879 01347 20848

Forgetting rate 00344 00089 00236 00469

Initial goodwill G0 14326 09774 12833 16014

Observation variance 01280 00287 00870 01796

System variance 00796 00103 00642 00979

(goodwill) 1

Call stimulation

Copy wearout c 02494 00893 01421 03649

Repetition wearout w 00329 00222 00612 00053

Initial quality q10 01214 00627 00252 02316

System variance 00020 2.5599e4 00016 00024

(ad quality) 2

Price offer

Copy wearout c 05725 01947 03264 08363



System variance 1.984e4 2.5863e5 1.6008e4 2.4405e4

(ad quality) 3

Product offer

Copy wearout c 04041 01279 02420 05725



S ystem vari ance 7.9286e4 1.0349e4 6.3891e4 9.7657e4

(ad quality) 4

Reconnection

Copy wearout c 02275 00644 01577 03087



S ystem vari ance 7.9796e5 1.0335e5 6.4505e5 9.7978e5

(ad quality) 5

Reassurance

Copy wearout c 01625 00606 01064 02315



S ystem vari ance 1.5915e4 2.0762e5 1.2836e4 1.947e4(ad quality) 6

rate of copy wearout than that of reassurance ads. Weprovide a discussion of these parameters later.

We find that the repetition wearout effects are nega-tive and significantly different from zero. For instance,the repetition wearout effect for call stimulation adis 00329. This suggests that there is no repetitionwearout, but in fact there is wearin. It is important tonote that Naik et al. (1998) obtain positive and signif-icant effects for repetition wearout while we get sig-

nificant negative coefficients. This may be explained by the fact that the firm employed different themesand multiple executions within each theme. This rota-tion of ads across different themes could have theeffect of keeping the ads fresh and may explain thewearin effect of advertising. This is consistent withLodish et al. (1995), in which they conducted a meta-analysis of a large number of split cable televisionexperiments and concluded that an increase in salesdue to an increase in media weight is more likelywhen copy strategy is changed. So while repeating the

Table 5 Analysis of Wearout Parameters

No. of Copy wearout Repetition wearout

executions parameter parameter

Rational appeal

Price offer ads 60 05725 01084

Product ads 22 04041 00666

Average 41 04883 00875

Emotional appeal

Call stimulation ads 43 02494 00329

Reassurance ads 17 01625 00432

Reconnect ads 16 02275 00488

Average 25.3 02131 00416

same ad might cause wearout, having different execu-tions of the same message may contribute to a wearin.Research in laboratory experiments also indicates thatrepeated ads using multiple executions lead to bet-ter recall (Unnava and Burnkrant 1991). Our resultson wearout differ from Naik et al. (1998) and demon-strate the importance of considering the effect of dif-ferent themes in obtaining correct wearout effects.

To understand the reason for differences in copyeffects between different themes, we draw on the find-ings in the review by Pechmann and Stewart (1990).They suggest that emotional ads and ads with greateremotional imagery wear out slower than ads withoutemotional persuasion. We conduct an informal anal-ysis of this argument as we do not have informationabout the actual content. We speculate that price andproduct offer ads have a more rational appeal and wesuspect that call stimulation, reassurance, and recon-

nect ads have a greater emotional appeal. Based onthis classification, we see in Table 5 that rational adshave larger copy wearout parameters than emotionalads. These findings are consistent with the research byHitchon et al. (1988) and the verbal arguments in Silkand Vavra (1974) and Ray and Sawyer (1971) whichsuggest that hard sell ads wear out faster than softsell ads.

Further, the literature review posits that wearoutcan be mitigated by using a number of different copyexecutions. We show the number of executions foreach theme and the repetition wearout parameters inTable 5. Within the category of rational ads, we see

that as the number of executions increases, the rep-etition wearout decreases. We do not find the same

Table 6 Advertising Interactions

Standard

Copy interactions Estimate deviation 95% HPDI

Call stimulation 00773 00268 01194 00314

Price offer 00162 00548 00957 00818

Product offer 00384 00362 00949 00228

Reconnection 00846 00273 01217 00372

Reassurance 00969 00228 01277 00595


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pattern within the category of emotional appeal ads;however, if we take the average number of execu-tions across rational and emotional ads, there is alink between the number of copy executions andwearin. As the number of copy executions increases,the wearin is higher (i.e., wearout is lower). While

these conclusions are not definitive evidence, they areconsistent with what researchers have observed inlaboratory settings and provide validity to our modeland estimation. Note that the above analysis is basedon our ad hoc classification of ads as either rationalor emotional.

In Table 6, we present the interaction effects ofthe various themes. We find that all the interactioncoefficients are negative, and the interactions of callstimulation, reconnection, and reassurance ads arestatistically significantly different from zero. Note thatin our model we assume that the interaction of, say,a call stimulation ad with all other ads has the same

coefficient. The negative parameters suggest that theinteraction between the different themes mitigates thegoodwill generated by the ad campaign. There isconsiderable evidence for negative interaction effectseven among noncompeting advertisements. Calderand Sternthal (1980) show that when subjects arerepeatedly exposed to an ad embedded in a collectionof commercials over a period of time, they have neg-ative evaluations of the ad and the associated prod-uct. Attention to a specific message theme is theorizedto diminish if the message is dominated by a clutterof other message themes. Moreover, there is evidence

that the persuasive impact of advertising is much lessreliable when a target ad is presented in an environ-ment that includes other ads (Belch 1982, Burke andSrull 1988, Ray and Sawyer 1971, Rethans et al. 1986,Malaviya et al. 1999). It is interesting to see that bothprice ads and product offer ads with rational appealdo not have significant interaction effects. This may

be suggesting that attention to rational price offer orproduct offer messages is less diminished by messageclutter. Emotional ads appear to have negative inter-action effects on goodwill even though they reducewearout.

In Table 7, we present the estimates of our model

without interaction effects. We find that this modelis rejected by log Bayes factor criterion. The copywearout and repetition wearout effects are consistentwith those in the earlier model. We find that rationalads do have faster copy wearout but slower repetitionwearin. The forgetting parameter is 0.0784, significantand higher than that obtained earlier. In fact, withoutthe interaction terms all the wearout effects are largerin magnitude, thus suggesting bias if one does notcapture these interaction effects. The magnitude of the

bias averages 58% with a range from 21% to 146%.

Table 7 Estimates by Message Type (No Interactions)

Standard


Price per minute 238591 15290 56999 31266



Forgetting rate 00784 00345 00419 01301Initial goodwill G0 165864 26498 14219 23106


System variance 00815 00107 00658 01005

(goodwill) 1

Call stimulation

Copy wearout c 05119 01297 03446 06758



System variance 00020 2.5920e4 00016 00024

(ad quality) 2

Price offer

Copy wearout c 07097 01775 04388 09267



Syst em va ri anc e 1 .98 91 e4 2.5992e5 1.6072e4 2.4427e4

(ad quality) 3

Product offer

Copy wearout c 06196 01225 04663 07762



Syst em va ri anc e 7 .94 34 e4 1.0497e4 6.3979e4 9.809 e4

(ad quality) 4

Reconnection

Copy wearout c 02749 00586 01850 03344



Syst em va ri anc e 8 .07 00 e5 1.0522e5 6.4914e5 9.499 e5

(ad quality) 5

Reassurance

Copy wearout c 02807 00559 02028 03424


Initial quality q50 02203 01034 01020 04722Syst em va ri anc e 1 .59 89 e4 2.1034e5 1.2847e4 1.9690e4

(ad quality) 6

In Table 8, we present estimates of our model usingaggregate advertising GRP instead of considering theGRP of the different themes separately. Using logBayes factor (87.712), we can reject this model in favorof our proposed model. We find that the effect ofprice, competitive advertising, and capacity are con-sistent with earlier results. In the aggregate model, wefind that the forgetting rate is 0.93, which is very highcompared to estimates obtained in the above mod-els as well as the estimates obtained in earlier stud-ies. Note that the forgetting parameter is identified byperiods when there is no advertising. When we addup the GRPs of all themes of ads (Figure 2), thereare very few weeks in which there is no advertising.Thus, the aggregate model will provide a biased esti-mate. We find the copy wearout parameter is nega-tive and the repetition wearout parameter is positive.These results suggest that when there are multiplethemes being advertised, an aggregate model mightprovide misleading interpretation.


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Table 8 Estimates of Aggregate Model

Standard


Price per minute 104908 43619 17823 32900



Forgetting rate 09322 00523 08594 09911Initial goodwill G0 15253 03622 09251 21201


System variance 00293 00039 00234 00362

(goodwill) 1

Aggregated

Copy wearout c 00293 00164 00494 00093

Repetition wearout w 2.8089e4 6.4801e5 1.9014e4 3.7902e4


S ystem variance 7.40 72e4 9.6713e5 5.959e4 9.1178e4

(ad quality) 2

Figure 3 shows the evolution of the goodwillparameter Gt and the five quality parameters qit .

To provide a measure of the models predictive per-formance, we use the parameters from our proposedmodel in Table 4 and predict the number of hours oftalk time. The plot of predicted versus actual hoursof talk time is given in Figure 4. The plot indicates avery good fit of the model with the data across 114weeks. The predicted values track the variations inactual data quite closely.

5.1. Advertising Policy ImplicationsGiven our estimates in Table 4, we can reconsiderthe advertisers budget allocation decisions across thefive advertising themes. For example, we estimated

that over the 114-week period of our sample, priceand product ads had the two highest copy wearoutestimates but the lowest repetition wearouts. On theother hand, reconnection and reassurance advertisingthemes had the lowest copy wearouts and call stim-ulation and reassurance had the highest repetitionwearouts. What are the implications of these resultson budget allocation in each of the 114 weeks? Moreimportantly, could the advertising have been moreefficient in generating greater demand for telephoneservices?

To answer the above questions, we develop a modelto reallocate the total advertising GRPs bt in each

period across the five m = 5 ad themes to maxi-mize total expected telephone service demand over114 weeks. We solve the following large-scale nonlin-ear optimization problem P1:

maxA11A1mAT1ATm

Tt=1

Eyt Dt1

s.t.m

i=1

Ait bt Ait 0 t = 1 T

Varyt Dt1 2t t = 1 T

where Eyt Dt1 and Varyt Dt1 are the expec-tation and variance of the one-step-ahead forecastdistribution (see Appendix A, Equation A3) and Aitthe GRP for an ad theme at time t. Thus, we solve aprogram that involves 570 5 114 variables and 114(114 weeks) linear constraints.

The problem (P1) is solved in SNOPT, a set ofFortran routines developed by Stanford OptimizationLaboratory. These routines are called within MAT-LAB. SNOPT is a general purpose system for solvingoptimization problems involving many variables andconstraints, and so is particularly suited for our prob-lem. It uses a sequential quadratic programming algo-rithm that obtains search directions from a sequenceof quadratic programming subproblems (Gill et al.2002). Given the size of the problem it solves, SNOPTfinds solutions that are locally optimal and, ideally,any nonlinear objective functions should be smooth.We use multiple start values and choose solutionsthat give allocations that represent an improvementin advertising efficiency. We caution the readers thatthe solution may not be optimal but provides a betterallocation.

One such allocation is reported in Table 9 anddepicted in Figure 5. The horizontal bar graph com-pares the improved and the actual allocation ofresources (measured in total GRPs) across the dif-ferent themes. Our model suggests increasing adver-tising expenditure on reconnect and reassurance adswhile decreasing on the other three themes. The per-centage changes in allocation of GRPs are quite largeas seen in Table 9. They range from a low of 50% toa high of 276% change.

A reallocation of advertising along the suggestedlines would generate an additional 35.82 millionhours of calling time which represents a 2% increaseover the current level of demand. We thus demon-strate how our model can be used to reallocate re-sources over different themes to improve the desiredoutcome, whether it is awareness as in Naik et al.(1998) or demand as in our case.

6. ConclusionWe have developed a model of demand that consid-ers the dynamic effects of multiple themes of tele-

vision advertising. We believe that this is a firstattempt to help managers allocate their advertisingresources across different themes of advertising. Themodel considers wearout effectsboth copy wearoutand repetition wearoutas well as forgetting. Adver-tising is assumed to affect goodwill which in turnaffects demand for a product or a service. A modifica-tion of the Nerlove-Arrow specification is employed

because it has been shown to perform the best (Naiket al. 1998). Our model extends the specification inNaik et al. (1998) by considering the evolution ofgoodwill and advertising effectiveness over time for


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Figure 3 Plot of the Dynamic Effects of the Coefficients with 95% HPDI

0 20 40 60 80 100 12012

14

16

18

20

Time in weeks



Time in weeks

Goodwill

95% HPDI

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

Call stim ads

95% HPDI

0 20 40 60 80 100 1200

0.05

0.10

0.15

0.20

0

0.05

0.10

0.15

0.20

Price off ads

95% HPDI

0 20 40 60 80 100 120

Product off ads

95% HPDI

0 20 40 60 80 100 1200

0.05

0.10

0.15

0.20

0.25

Reconnect ads

95% HPDI

0 20 40 60 80 100 1200

Reassure ads

95% HPDI

different ad themes. Further, we employ dynamiclinear Bayesian estimation techniques (West and Har-rison 1997), a relatively new method, to estimate ourmodel parameters. We also develop a model to showhow reallocation of resources can be done using the

parameters of our proposed model.We find that modeling different advertising efforts

on different themes yields better insights and unbi-ased estimates relative to a model that aggregates theadvertising effort. We find that copy wearout effectsof different themes are positive and systematically dif-ferent. Based on our ad hoc classification of ad themesas either emotional or rational ads, we find some sup-port for the premise proposed in behavioral literature(Pechmann and Stewart 1990) that emotional ads tendto wear out faster than nonemotional ads. Repetition

wearout parameters are negative, suggesting wearineffects for this data. We find some evidence that rep-etition wearout is lower if the number of copy execu-tions is higher.

Thus, changing the execution of the message ap-

pears to refresh the message and mitigates wearout.There is evidence of positive forgetting effects whenthere is no advertising. Thus, our model is consistentwith earlier findings on the effects of advertising withrepetition. The main value of our model comes fromthe fact that we have linked demand to the wearoutand forgetting effects and so have provided a wayfor managers to use these parameters and insights inmaking allocation decisions. Our policy experimentsuggests that a reallocation of advertising effort acrossthe different themes could result in a higher payoff.


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Figure 4 Plot of Predicted and Actual Hours of Talk Time Over

114 Weeks

0 20 40 60 80 100 12013

14

15

16

17

18

19

20

Generalized model

Totalhours(M)

Time in weeks

Predicted

Actual

There are a few limitations of our model. In theproposed model, we assume that advertising is an

exogenous variable. Since managers allocate advertis-ing in a strategic manner, one can make a case thatit should be treated as an endogenous variable as wehave done for price. This control for the endogene-ity of advertising can be modeled if good instrumentswere available in the data. We have not consideredthe dynamic effects of competitive advertising, whichcould be modeled analogous to the different themes.Given data on competitors demand, this extensioncan be attempted in future research. In our data, thefirm is a monopolist in its business and so we havenot modeled competition explicitly. In other productcategories, this would be an area for future research.

Our model can be extended to account for changesin copy and a model can be developed for allo-cation of resources across different copy. This is achallenging problem since the number of executionseven for a given theme can be large and one needs

Figure 5 Actual vs. Model-Based Allocation of GRPs for Each

Advertising Theme

0 2,000 4,000 6,000 8,000 10,000 12,000

1

2

3

4

5Reassurance

Reconnect

Product offer

Budgetbyadtheme

Policy implications

Price offer

Call stimulation

Total GRPs

ActualOptimal

Table 9 Comparison of Actual and Model-Based GRPs

Model-based

Theme of advert ising Actual GRPs GRPs % Change (%)

Call stimulat ion 11509 5815 495

Price offer 3611 914 747

Product offer 7281 3218 558

Reconnect 5201 10962 +1107Reassurance 2422 9115 +2763

to also find a way to incorporate content of advertis-ing into the model. Other avenues for future researchare to extend our model to develop a media allo-cation model and to also consider the interaction

between different themes and different media. Theseare important issues and our paper presents a possi-

ble starting point to address them.

AcknowledgmentThe authors acknowledge first author Frank M. Bass, whodied December 1, 2006, for his lifetime of contributions tomarketing science.

Appendix AThis appendix provides an overview of the posterior sam-pling algorithm. The sampling scheme shown in A.1is direct application of DLM theory (West and Harrison1997, Chapter 4) and Gibbs sampling for state space mod-els developed by Carter and Kohn (1994) and Fruhwirth-Schnatter (1994). Our model belongs to the class of linearstate space models in which components of the transi-tion matrix Ht and the variance components

2 W

are treated as parameters to be estimated along with thesequence of state vectors t over time. We begin the for-ward filtering step with the most recent values of m0, Co ,

W, 2 , , and :

A.1. Sampling from pt Dt

Forward Filtering. We use the standard DLM framework(Equations A1 through A5) to infer the posterior distribu-tion t Dt over time, where Dt = yt Dt1 includes allinformation available to the researcher at time t. The poste-rior t Dt is then determined using standard mutivariatenormal theory:

Posterior distribution for t1,

t1 Dt1 Nmt1 Ct1 (A1)

Prior distribution for t : t Dt1 Nat Rt , where

at = Ht mt1 + ut and Rt = HtCt1Ht + W (A2)

Prior one-step-ahead forecast distribution: yt Dt1 N ft Pt,

ft = Ft at +Xt Pt = FtRtFt +

2 (A3)

t yt Dt1 N

at

ft

Rt RtFt

RtFt Pt

with

Covt yt = Covt Ftt +Xt + t

= Vart Dt1Ft = Rt Ft

(A4)


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Posterior distribution for t : t yt Dt N mt Ct, mak-ing use of the marginal properties of the normal in (A4),

mt = at + At yt ft At = RtFtP1t

where Ct = RtI FtAt (A5)

Backward Smoothing. We derive the backward smooth-

ing algorithm by using (A1) through (A2) to write downthe joint distribution of the parameters at t and t 1, giveninformation Dt1 to obtain Equation (A6):

t t1 Dt1 N

at

mt1

Rt GtCt1

GtCt1 Ct1

(A6)

Then, using standard multivariate normal theory, we obtainthe conditionals:

t1 tDt1 Nmt1 +Bt t at Ct1 BtRtBt (A7)

where Bt = GtCt1R1t and t is a random draw from the

posterior Nt Dt .We can now do a retrospective analysis using results

derived in Equations (A6) and (A7). That is, we take expec-tations and variance over all possible draws of the posteriorof t given information at time t, Dt . In other words, at anytime (say, t 1) in the analysis the researcher can updatehis posterior belief about the effects of advertising spend-ing given his new information at time t. The result is our

backward smoothing algorithm:

Et1 Dt = EEt1 t Dt1 Dt

= mt1 + Bt mt at (A8)

and

Vart1 Dt = EVart1 t Dt1 Dt

+ VarEt1 t Dt1 Dt

= Ct1 BtRt CtBt (A9)

We use the above (Equations A1 through A9) to samplethe state vector of goodwill and advertising copy quality asfollows:

Simulation for T.Step 1. For t = 1 T , compute the moments mt Ct

for the multivariate normal pt Dt 2 W by applying

the sequential updating procedure described in the aboveforward filter section (Equations A2, A3, and A5).

Step 2. At the end of the series t = T , sampleT from the posterior distribution: pT DT

2

W =NT mT CT.

Step 3. For t = T 1, sample pt1 t 2 W con-

ditional on the latest draw t .The results are the draws T = 012 T

from the full conditional posterior.

A.2. Sampling from pW 2 T yT

Conditional on all the states and the data T yT, ourDLM Equations (9) and (10) simplify to a linear multivari-ate system with unknowns, parameters , and variancecomponents W 2 . Consequently, the Gibbs sampler stepto estimate the joint posterior of the nonstate parameters

conditional on the data and all states pW 2 T yT

is very straightforward. We refer the reader to Gelfand andSmith (1990) for an overview. For the interested reader, wenote several DLM features that highlight the simplicity ofthis step:

(1) The observation Equation (9) and system Equa-tion (10) errors are mutually independent (see West and

Harrison 1997). Thus, conditional on , we can sampleW 2 independently.

(2) Similarly, conditional on T yT, the states and thedata are independent and are thus sampled sepa-rately.

Appendix BTo show identification in our DLM, we need to investi-gate whether enough prior structure exists in the systemof Equations (11) and (12). First, we need to construct anobservationally equivalent system M1 = F1t H1t

2 W1,

where observability (Crassidis and Junkins 2004) relatesto our ability to recover all state vectors T = 012 T, given observations y

T = y1 y2 yT. An

equivalent system is achieved by substituting the lineartransformation 1t = Lt into Equations (11) and (12),where Lt is a m + 1 m + 1 nonsingular matrix (West andHarrison 1997, Chapter 4). Thus, we have:

F1t = Ft L1 (B1)

H1t = LHtL1 (B2)

W1 = LWL (B3)

u1t = Lut (B4)

We illustrate identification for a model with a single mes-sage theme (i.e., m = 1; Naik et al. 1998). We will show

that enough prior structure exists in Ft and Ht such thatall transformations are precluded except L = I. The resultsgeneralize to the multitheme case.

Define the nonsingular matrix

L =

l11 l12

l21 l22

and recall and note constraints in Ft Equations (11).From Equation (B1) we have

1 0

=

1 0 l11 l12

l21 l22

Thus, l11 = 1, l12 = 0, and l22 = 0 (L is nonsingular).Now, recall and note the constraints in the system matrix

Ht from Equation (12), when advertising is off IA = 0.From Equation (B2), we have:

h11 0

0 h22

H1t

1 0

l21 l22

=

1 0

l21 l22

1 0

0 1c

Ht

By inspection, H1t = Ht and l21 = 0.Given our specification of the quality evolution (5) and

H1t = Ht , we must have u1t = Lut and as a result, l22 = 1.


16/17


That is, enough prior structure exists in Ft and Gt such thatall transformations are precluded except L = I. QED

Note that Naik et al. (1998) is a model of pulsing. Wenow consider identification if this model was applied tosituations in which there is no pulsing IA = 1:

H1t= 1 0l21 l22

1 gA

0 1 aA Ht

1 0l21 l22 1

=1

l22

al22 bl21 b

al21l22 bl221 cl21l22 bl21 + cl22

where a = 1 , b = gA, and c = 1 aA.Given the structure of Ht , al21l22 bl

221 cl21l22 = 0. Thus,

l21 = 0 or l21 = a c/bl22. Note that if we take l21 =a c/bl22, H1t 1 1 = c and H1t2 2 = a, which is incon-sistent with evolution of quality or goodwill (Equations 4and 5). Thus, we take l21 = 0. If there is no pulsing, Equa-tion (B4) provides no information since u1t = ut = 0. Con-sider Equation (B3):

W1 =

1 0

0 l22

w11 w12

w21 w22

W

1 0

0 l22

=

w11 l22w12

l22w21 l222w22

Identification can be achieved if we restrict the system vari-ance diagonal to be equal, giving l22 = 1 or l22 = 1. Wereject l22 = 1 since this implies H1t1 1 < 0, which isimpossible given the constraint in Ht1 1 > 0. Thus, wehave L = I. QED

Note that imposing the above constraint leads to model

identification.

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