Predicting Purchase Behavior from Stated Intentions: A Unified Model

Baohong Sun and Vicki G. Morwitz1

1 Baohong Sun is Assistant Professor of Marketing at Kenan-Flagler Business School, University of North Carolina. Tel: 919-962-9579, Fax: 919-962-7186, Email: sunb@bschool.unc.edu. Vicki G. Morwitz is Associate Professor of Marketing at Leonard N. Stern School of Business, New York University. Tel: 212-998-0618, Fax: 212-995-4006, Email: vmorwitz@stern.nyu.edu. We would like to thank seminar participants at the University of Chicago and Marketing Science Conference for valuable comments.

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Predicting Purchase Behavior from Stated Intentions: A Unified Model

Abstract

Stated intentions are imperfect measures of actual purchasing. Intentions data can contain systematic biases, intentions can change over time and the correlation between intentions and actual purchase can be imperfect. Ignoring any of the discrepancies between intentions and purchasing can result in biased estimates of variable coefficients and biased forecasts of future demand. Previous models are either aggregate level models or only take into account a subset of these discrepancies. Thus, these models are limited in their ability to aid managers in targeting the right consumers with the right marketing offer. In this paper, we develop a unified model of the relationship between intentions and purchasing that (1) takes into account all possible sources of discrepancies between intentions and purchasing; (2) forecasts purchasing probability at the individual level by linking explanatory variables (e.g., socio-demographics, product attributes and promotion variables) and intentions with actual purchasing; (3) considers multiple levels of purchase decisions rather than the simple purchase / no-purchase decision. We empirically demonstrate that this model provides more accurate individual level purchase predictions and is therefore more useful for guiding targeting efforts.

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INTRODUCTION Self-reported purchase intentions, because of their flexibility, ease-of-use, and

inexpensiveness, are one of the most widely used proxy measures of purchasing in sales

forecasts (Chandon, Morwitz and Reinartz, in press; Infosino 1986; Wittink 2001; Verhoefl and

Hans Franses 2002; Whitlark, Geurts and Swenson 1993) and new product tests (Jamieson and

Bass 1989; Silk and Urban 1978; Urban and Hauser 1980; Urban and Katz 1983). Intentions are

also used to segment markets and evaluate the effectiveness of promotions for different

individuals (Sewall 1978).

The widespread use of intentions to forecast actual purchasing relies on the strong

assumption that intentions are good indicators of individuals’ purchase behavior. A key question

then is whether self-reported intentions are reliable indicators of individuals’ subsequent

purchasing? If not, how should marketers combine stated intentions measures with other

available data to forecast respondents’ probability of purchase? Past research has shown that the

predictive validity of intentions is questionable (e.g. Belk 1985; Clawson 1971). Individual-level

purchase behavior differs from stated intentions and those individual-level discrepancies do not

cancel in the aggregate. This results in a discrepancy between the overall mean stated purchase

intention and the subsequent proportion of buyers. Moreover, models ignoring this discrepancy

will not only lead to inaccurate forecasts, but also provide biased estimates of the relationship

between correlates of intentions (e.g., socio-demographics, product attributes and promotional

variables) and purchasing (Hsiao and Sun 1999; Young, Morwitz and DeSarbo). Thus,

segmentation and promotion strategies made based on these models can be sub-optimal.

Previous research in marketing and psychology has identified three main reasons why

stated intentions and actual purchasing differ: (i) systematic intentions biases (systematic biases

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in the reporting of stated intentions at the time of the survey, e.g., social desirability bias)

(Granois and Summers 1975; Hsiao and Sun 1999; Kalwani and Silk 1982; Morrison 1979); (ii)

changes in true intentions over time (changes in true intentions between the time of the survey

and the time of the purchasing associated with changes in explanatory variables, e.g., price

increase) (Manski 1990; Morrison 1979); (iii) an imperfect correlation between true intentions

and purchasing (e.g, the psychological difference between intentions and behavior) (Bagozzi and

Dholakia 1999; Gollwitzer 1999a,b).

[Insert Table 1 here]

Table 1 summarizes previous research on the intentions-behavior relationship and

highlights our contribution to this research stream. In an influential paper, Morrison (1979)

develops a modified version of the beta-binomial model that allows for the impact of exogenous

events on true intentions. Kalwani and Silk (1982) further analyze and apply Morrison's model to

different product categories. Bemmaor (1995) extends Morrison's (1979) model to allow for

heterogeneous switching probabilities. Infosino (1986) interprets intentions as a monotonic

transformation of a latent value (willingness-to-pay minus price) and examines the effect of

promotions on the probability of subsequent purchase. The purpose of most of these studies was

to identify rather than explain discrepancies between stated intentions and actual purchasing. In

addition, since all of these are aggregate-level models, they forecast the same purchase

probabilities for all respondents with the same stated level of intentions, and therefore they are

only useful for forecasting aggregate sales and not for identifying which individuals are more

likely to buy.

In recent years, a stream of models has emerged that examines the relationship between

intentions and behavior at the respondent or segment level. These models incorporate some of

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the possible sources of discrepancies between these two variables. In a descriptive study,

Morwitz and Schmittlein (1992) segment respondents using demographic information, and

develop separate forecasts for each segment. Using binary and multinomial choice models,

Young et al (1998) and Hsiao and Sun (1999) demonstrate how to recover true intentions from

stated intentions when systematic intentions biases are present. Both models do not take into

account the possibility that true intentions may change over time nor the imperfect correlation

between true intentions and purchasing. The way we model systematic intentions biases is an

extension of Young et. al. (1998) and Hsiao and Sun (1999). However, there are substantive

differences between our model and theirs. First, their studies focus on modeling systematic

response biases and still rely on the assumption that true intentions equal purchasing. Second,

their models are only based on intentions survey data and the derived intention-purchase

relationship is not validated using purchase data. Third, their models assume intentions without

bias are equivalent to subsequent purchasing and therefore the purchase model is the same as the

intentions model.

Moreover, most existing aggregate and disaggregate models only consider the binary

purchase/no purchase decision. However, more and more companies are offering multiple levels

of products and services and managers are interested in predicting purchases for these different

levels, in addition to predicting simple purchase/ no purchase decisions. For example, in the

cellular phone market, an intentions survey data might ask: “Which usage plan to intend to sign

up for?”, with response options, “high usage plan,” “medium usage plan,” “low usage plan,” and

“no plan.” Alternatively, we observe marketers asking consumers about their purchase timing

decisions, with questions such as: “When do you intend to purchase the product?“ with response

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options, “within six months,” “7-12 months,” or “not within a year.”2 These data cannot be

handled by conventional intentions models such as Morrison’s (1979) which were designed to

only consider purchase / no-purchase decisions.

In summary, to better use self-reported intentions for sales forecasting, new product

testing, promotion evaluation and targeting, there is a need for a unified individual level model

that (1) takes into account all possible sources of discrepancies between intentions and

purchasing; (2) forecasts purchasing probability at the individual level by linking explanatory

variables (e.g., socio-demographics, product attributes and promotion variables) and intentions

with actual purchasing; (3) considers multiple levels of purchase decisions rather than being

restricted to binary purchase / no-purchase decisions.

In this paper, we develop a conceptual framework of the intentions - purchase

relationship, and propose a unified individual-level intentions model that corrects for systematic

intentions biases, adjusts for changes in true intentions over time associated with changes in

related explanatory variables, allows for an imperfect correlation between intentions and

purchasing, and allows for multiple levels of purchase choices. We then develop a forecasting

method to predict individual purchase probabilities using intentions survey data and demonstrate

that this model predicts purchase probabilities at the individual level that are more accurate and

useful for guiding targeting effort than other models currently used in marketing. Thus, our

paper is the first paper that proposes a unified statistical framework to take into account the three

possible causes of discrepancies between stated intentions and purchasing and allows individual-

level correlates and intentions to explain purchasing. It unifies stated intentions and purchasing

by allowing them to be both directly and indirectly connected.

2 Morwitz and Schmittlein (1992) treat these alternatives as ordered choices. This is a reasonable assumption since consumers who state they intend to purchase within 6 months are also more likely to purchase than those who state intentions to purchase within 7-12 months and so on.

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In the next section, we develop a conceptual framework linking explanatory variables,

stated intentions and actual purchasing. We then propose a unified statistical model to model the

discrepancies between stated intentions and actual purchasing. We next estimate the unified

purchase-forecasting model using intentions survey data related to automobiles and personal

computers. Finally we discuss managerial implications and opportunities for future research.

CONCEPTUAL FRAMEWORK

Systematic Intentions Bias

It is well known that stated intentions may not perfectly reveal respondents' true

intentions. Respondents may not reveal their true intentions because they (1) do not understand

the nature of their own preferences (Balasubramanian and Kamakura 1989; Kahneman and Snell

1992); (2) want to impress the interviewer; (3) try to guess the correct answer (e.g., what the

sponsor or interviewer would prefer them to say); or (4) simply misunderstand the question

(Andersen 1988, chapter 9). These effects can lead to a difference between an individual’s stated

and true intentions. The random component of measurement error in individual reports cancels

out but the systematic component persists at the aggregate level (Kalwani and Silk 1982). We

use the term systematic intentions bias, to refer to systematic differences between reported and

true intentions at the time of the survey.

The direction of the systematic intentions bias can be two-sided or one-sided. For

instance, when asked their intentions to buy a newly developed high-tech product, respondents

may not have sufficient knowledge and may construct impromptu answers in order to avoid the

embarrassment of not having an answer for the question. Thus, they may provide erroneous

answers that could be biased in either direction and the resultant stated intentions are

contaminated with two sided intentions bias. Alternatively, there could also be a “social

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desirability bias” (Bagozzi 1994; Bagozzi, Yi and Lynch 1999) and the respondent may over-

report “desirable” behaviors and under-report “undesirable” ones (Sherman 1980). Thus, the

stated intentions are always skewed downward or upward. In such cases, the resultant stated

intentions are contaminated with one-sided intentions bias. 3

Change in true intentions over time

Even if respondents state their true intentions, it is very likely that, at the time of the

survey, they may not possess all the relevant information related to an intended future purchase.

Many unexpected events can occur between the time of the survey and the time of the purchase,

leading true intentions to change over time (Manski 1990; Morrison 1979). For example,

respondents’ true intentions might shift with an unanticipated change of price or improvement of

the product. Infosino (1986) suggests that events such as promotions shift the entire intentions

distribution. The unexpected promotion leads to increased purchasing because more of the

distribution is above the threshold required for purchase.

Imperfect correlation between true intentions and purchase

Even if there are no systematic intentions biases and true intentions do not change over

time, respondents' true intentions at the time of purchase may still differ from their actual

purchasing behavior. We attribute this difference to all unobservable factors that cause true

intentions to be an imperfect representation of actual purchasing. For example, this may be

related to the notion in psychology that there is a fundamental psychological difference between

forming an intention to perform a goal and achieving a goal (Bagozzi and Dholakia 1999;

Gollwitzer 1999a, b). In a purchase context, this may be due to unobservable factors; e.g. the

3 Of course, there might exist other forms of intentions bias, for example, mixtures of two-sided and one-sided biases where respondents randomly select an answer that is close to their true intention. In this paper, we only consider two-sided and one-sided intentions biases. We do not aim to model all forms of intentions biases.

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intended product is out of stock (Bagozzi and Dholakia 1999) or the under-estimated cost of

searching for the right product.

STATISTICAL MODEL

In this section we propose a unified model to analyze categorical purchase intentions

data.4 Since it has become fairly common to collect intentions data measured in a graded order,

we assume that a respondent is confronted with a set of ordered levels j=0, …,M. The set of

respondents is denoted by n=1, 2, …, N.

Intentions Model

The basic framework we shall follow is the conventional random utility model (e.g.

McFadden 1974). Let Un be the (indirect) utility for the nth respondent. We assume that it can be

decomposed into two components.

nn

nnn

XU

ξαξµ+=

+= (2)

where µn is the deterministic part of the latent utility that determines true intentions, and Xn

denotes some socio-demographics, product attributes and marketing variables, e.g. product

properties, price, age, gender, present job category, etc. The coefficients of these variables are

defined by α. ξn represents all other unobservable random factors and is assumed to follow an

i.i.d. normal distribution.

Let the observed stated intention level yjn be defined as: 4 Note that conventionally intentions are measured on 5-point likelihood scales or 11-point probability scales and purchases are measured as binary (i.e., the product is purchased or not). We adopt a multinomial ordered probit model of intentions and purchasing for the following reasons: First, it is becoming more common for marketers to measure intentions with multiple levels or responses (e.g., usage categories, timing levels, etc.) Second, we want to develop a more general model that can incorporate situations with more than two intentions and purchase levels. The binary buy / no buy decision model is nested within our general model. Third, we can distinguish two-sided intentions bias and one-sided intentions bias models in the multinomial case while, in binary case, these two types of response bias models collapse to the same model. Our model, thus, is applicable to general (ordered) discrete choices of multi levels.

(1)

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=.otherwise,0

,respondent by thechosen isintention level theif,1 nthjthy jn (3)

For notational convenience, we let *jny be an indicator denoting latent true intentions.

≤<

=.otherwise,0

if,1 ,1* jnj-jn

lUly (4)

where jl is the threshold for intention level j. Then we have the ordered probit model for true

intention, which is based on utility maximization.

)()()1( 1*

njnjjn llyP µµ −Φ−−Φ== − , (5)

for j=1, .., M-1 . Φ is the normal distribution function derived from the density function of ξn.

)()1( 0*0 nn lyP µ−Φ== and )(1)1( 1

*nMMn lyP µ−Φ−== − . We define )1( * == jnjn yPF , which is

the probability of having j as the true purchase intention.

The first model we consider corresponds to the case when respondents report their stated

intentions based on their true preference as defined by equation (2). That is,

jnjnjn FyPyP ==== )1()1( * (6)

We call the model in the above equation the random utility model. It corresponds to the case

when stated intentions are perfect indicators of true intentions.

In the second model, we allow for the existence of two-sided intentions bias. We assume

that with probability πj for j=0, …, M the nth respondent chooses the jth intention level

irrespective of his or her underlying preference, and that with probability 1-∑ =

M

j j0π , the

respondent provides his or her true intention, consistent with utility maximization. Therefore, the

probability of observing the nth individual choosing the jth level is given by

(7) ,)1()1(0∑ =

−+==M

j jnjjjn FyP ππ

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for j=0, …, M. We call this second model the two-sided intentions bias model.5 Intuitively, with

the two-sided intentions bias model, the probability of observing stated intention j is the sum of

the probability of randomly choosing the jth intention and the joint probability of not giving a

random answer and choosing the jth intention based on the utility function.6

In the third model, we consider the case in which the intentions bias can be either positive

or negative. In the analysis to follow, we assume that respondents' stated intentions always over-

report their true intentions, however under-reporting can be modeled similarly. A respondent has

a probability πj to state random intentions j for j=0 …, M. The respondent always chooses the jth

intention level if the jth intention level is the higher of his or her random intention and true

intention.

∑∑==

−+==M

jijni

j

iinjjn FFyP ]1[][)1(

0ππ , (8)

for j=1, … M and

∑=

=−==M

jjnn yPyP

10 ).1(1)1( (9)

We call this the one-sided intentions bias model. Intuitively, respondents always report the

higher of their random intention and their true intention, and thus their reported intention can be

higher than their true intention. In other words, this model allows respondents to exaggerate their

stated intentions.

5 Hsiao and Sun (1999) provides a more detailed explanation on how to model intentions bias. Murthi and Srinivasan (1999) also develop a similar model to address different issues. 6 Note our focus is not to provide a process-level explanation on how respondents offer biased answers. Rather we adopt a statistical approach to take into consideration intentions bias. While equations (6), (7) and (8-9) represent

statistical assumptions we made in the model, we do not claim they represent the process by which respondents provide biased intentions.

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Note that when πj is zero for all j, both the two-sided intentions bias model and the one-

sided intentions bias model become the random utility model as defined in equation (6). Thus the

random utility model is nested in both the two-sided and the one-sided intentions bias models.

This implies that, given an intentions survey, we can always assume the data are contaminated

with (constant) intentions biases and start with the two intentions bias models. If intentions

biases do not exist in the data, the estimates of πj for all j will not be significantly different from

zero. The usual model selection criteria such as AIC and BIC can be used to choose between the

two-sided and the one-sided intentions bias models.

Instead of assuming every respondent has the same probability πj of giving biased

intentions,, we allow πj to vary across respondents by making πj a function of Wn, a vector of

observable explanatory variables. We define

)( njjn Wγπ Φ= . (10)

)( njWγΦ takes a multinomial probit form for all j in π. Wn include variables representing the

types of respondents that are associated with providing biased responses to the intentions

question (e.g., product knowledge, education, gender). γj measures the effect of Wn on the

probability of stating biased intentions.

Using models (7) and (8-9), the underlying true intentions can be derived by

disentangling the systematic intentions bias from stated intentions. We next consider the portion

of the unified model that describes the relationship between intentions and actual purchase. We

note that this component was ignored in Young et. al (1998) and Hsiao and Sun (1999).

Purchasing Model

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We assume that the purchase decision also follows a conventional random utility model.

Let 'nU be the (indirect) utility level for the nth respondent. Louviere et. al. (1999) note that the

fundamental latent constructs of preference and utility are quite stable and that, after controlling

for context effects, intentions can be used to predict actual choices. Hsiao, Sun and Morwitz

(2002) show that purchasing is a function of intentions. Based on these empirical findings, we

assume that the purchase decision depends on both latent true intentions at the time of the survey

Un and changes in explanatory variables Xn between the time of survey and the time of purchase

( nX∆ ). Thus, we have the following equations,

'

'

'

)'('

''

nnn

nnnn

nnn

nnn

XX

eXXeXU

eU

ξλα

βξλβαλβ

µ

+∆+=

++∆+=+∆+=

+=

where 'nµ is the deterministic part of latent purchase utility and '

ne denotes all the unobservable

factors that affect the purchase decision. We assume 'ne is normally distributed. The inclusion of

nX∆ captures changes in the explanatory variables which lead to a shift in the purchase utility

and hence can affect the subsequent purchase decision. Replacing Un by its expression as in

equation (2), give us equation (13). Letting βαα =' and nnn e'' += βξξ , we obtain Equation (14).

With appropriate normalization, we normalize the variance of 'nξ to one so that '

nξ is standard

normal. Since α can be estimated from the intentions model, we only need to estimate

parameters β and λ. Coefficient β measures the direct effect of latent true intentions and λ

measures the effect of changing explanatory variables on purchasing.

(11)

(12)

(13)

(14)

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Let zjn be an indicator variable to denote that the nth respondent purchase the jth

alternative:

≤≤

=.otherwise,0

''' if,1 ,1 jnj-jn

lUlz (15)

where jl ' is the threshold for purchasing j. Then the probability of purchasing the jth

alternative is:

)''()'()1( 1'

njnjjn llzP µµ −Φ−−Φ== − (16)

for j=1,..,M-1 and Φ is the normal distribution function derived from the density function of 'nξ .

)'()1( '00 nn lzP µ−Φ== and )''(1)1( 1 nMMn lzP µ−Φ−== − . We define )1(' == jnjn zPF , which

is the probability of purchasing choice j.

There are unforeseeable events or unobservable factors that can lead to an imperfect

relationship between true intentions and purchasing. This is captured by the correlation between

nξ and 'nξ , which we define as ρ. We assume that

''2' 1 nnn ξρξρξ −+= (17)

where nξ and ''nξ are i.i.d. nξ captures unobserved events that occur at the time of survey. '

nξ

captures the new events that affect consumers' actual purchasing. ρ measures the persistence of

nξ between the time of the survey and the time of purchase. In one extreme case ρ=0,

0),cov( ' =nn ξξ which implies such events at the time of survey and at the time of purchase are

independent. In another extreme case ρ=1, nn ξξ =' , which implies events at the time of survey

and at the time of purchase are the same. In our model, we consider the general case without

putting any constraints on ρ .

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Assuming that there are no systematic intentions biases, the joint probability that true

intentions equal j and actual purchasing equals j' at the time of the purchase is given by

njjnjjn FjjGjjGjjGjjGzyP '''''

'* '),1()1,()1,1(),()1,1( =−−−−−−+=== (18)

The function G(j,j') denotes the joint probability of njn l µξ −< and njn l ''' µξ −< .

Unified Model

The above equation gives the joint probability that true intentions equals j and actual

purchasing equals j' at the time of purchase. However, to complete our unified model, we also

need to consider the possibility of systematic intentions biases at the time of the survey. The joint

probabilities of stating intention j and purchasing 'j when two-sided biases are present are given

by:

∑ =−+===

M

j njjjnnjjnnjjn FFzyP0

''

''' )1()1,1( ππ (19)

Similarly, the joint probabilities of stating intention j and purchasing 'j when one-sided biases

are present are given by:

∑∑=

=−+===

M

jinjjin

j

i njjjnnjjn FFzyP ''0

''' ]1[][)1,1( ππ (20)

where ''njF and '

'njjF are given by equations (16) and (18). We call (19) and (20) the unified

models. We call these unified models since they integrate systematic intentions biases, changes

in true intentions over time and the imperfect correlation between true intentions and actual

purchasing in the same model. They also relate intentions with purchasing both directly and

indirectly.

The log-likelihood function is given by

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∑ ∑ ===n j njjnnjjn zyPzyL

j

)]1,1(log[logmax '',,,, ρλγβα (21)

Note that at the heart of the unified model is a bi-variate probit model. For identification

purposes, we follow the convention and normalize one of the constants in both the intentions

and purchase utility functions to be zero, that is 00 =l and 0'0 =l . We also normalize the

variance of nξ and 'nξ to be 1.

PURCHASE PREDICTION USING INTENTIONS DATA

In the above model development, we assumed both stated intentions and purchase data

are available. However, firms are more interested in using intentions data to forecast individual

purchase probabilities in situations when actual purchase data are not yet available (e.g., prior to

a new product launch or before a change in the marketing program is made). Specifically, it is 'α

(or α given β) and forecasts of actual purchasing, after adjusting for discrepancies between

intentions and purchasing, that managers can use to make marketing and targeting decisions

before purchasing data are available. In practice, we assume we have a historical sample for

which both intentions and purchase data are available and apply our unified model to the

historical data to obtain estimates of α, β, γj, λ, and ρ. Once we have reliable estimates of β, λ

and ρ from the historical data that characterize the relationship between true intentions and

purchasing, we use them together with the new intentions survey to obtain new estimates of α

and γj to forecast future purchasing. To determine the probability that respondent n purchases at

level j' conditional on stated intention j, we use Bayes Law:

)1Pr()1,1Pr(

)1|1Pr( ''' =

======

jn

njjnjnnjnjj y

zyyzA (22)

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where njjA ' denotes the conditional probability that the nth respondent purchases the jth

alternative conditional on his or her stated intention j. The term )1,1Pr( ' == njjn zy can be

obtained by calculating the joint probability of purchasing the jth alternative and stating the jth

intention alternative (equation 19 or 20 using known values of β, λ and ρ and the prediction

intentions data). The term )1Pr( =jny can be obtained by calculating the probability of stating

the jth intention (using equation 7 or 8-9 with prediction intentions data). Thus, the probability

that the nth respondent purchases the jth alternative ( njA ' ) is given by

)]1Pr([0

'' == ∑=

jn

M

jnjjnj yAA (23)

In general, the unified model can be applied to intentions survey data to forecast

purchasing by following a four-step procedure. (1) First, we estimate various systematic

intentions bias models (equations 7, 8-9) to determine whether any systematic intentions biases

exist in the stated intentions and, if so, in which direction. (2) Once we find out the direction and

magnitude of any systematic intentions biases, we then apply the corresponding unified model

(equation 19 or 20) to a historical sample of intentions and purchase to obtain estimates of β, λ

and ρ . (3) Next, we input β, λ and ρ together with the prediction intentions data and any other

available information such as demographic variables to the unified model again to obtain more

accurate estimates of α and γj. (4) Finally, we predict the individual-level purchase probabilities

using equation (23) and obtain more accurate estimates of 'α .

Note, in the historical sample, we have Xn measured both at the time of survey and at the

time of purchase. However, in the prediction sample, Xn is only measured at the time of the

survey. Fortunately, in practice, many variables that may cause true intentions to systematically

change over time are product and promotion related and may be known or even planned by

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management (Infosino 1988). For example, management may plan to introduce a 30 percent

price cut for a personal computer a few months after the intentions survey. The price drop will

not affect the true intention measured at the time of the survey but will increase the demand at

the time of purchase. If management is interested in predicting sales one year after the survey,

they should take into account the price drop, or nX∆ , to obtain a more accurate forecast.

Since the same β, λ and ρ from the historical sample are used for prediction, we are

assuming the intentions-purchasing relationship we observed in the historical sample carries over

to the prediction sample. Thus the choice of the historical sample has to be consistent with the

above assumption. If historical data are not available, there are other ways to obtain a calibration

sample. For example, a two-stage study can be done. In the first stage, a pilot study can be

conducted where intentions and purchasing data are collected from a small number of

respondents. We then estimate β, λ and ρ from the pilot sample. In the second stage, we collect

intentions from a large sample of respondents and then use the prediction intentions data

combined with β, λ and ρ from the pilot study to forecast sales.

EMPIRICAL APPLICATION

Data Description

To demonstrate the predictive power of our unified model, we apply the model to two

different intentions survey datasets involving automobiles and personal computers. Multiple

waves of surveys were conducted to measure intentions to buy an automobile or a home personal

computer using two different but similar U.S. consumer mail panels. For the focal prediction

samples, we used stated intentions collected from a randomly selected 2000 households in the

fourth quarter of 1989 for the automobile data , and those from 3315 households collected in

January 1987 for the personal computer data. For the historical samples for which actual

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purchase information is also available, we used stated intentions collected from 1000 households

in the fourth quarter of 1988 for the automobile data, and from 1105 households collected in

January 1986 for the personal computer data.

For the automobile data the intentions question asked: “When will the next new (not

used) CAR (not truck or van) be purchased by someone in your household?” with response

alternatives: “6 months or less, 7-12 months, 13-24 months, 25-36 months, Over 36 months, and

Never.” For the personal computer data, the intentions question asked: “Do you or does anyone

in your household plan to acquire a (another) personal computer in the future for use at home?”

with response alternatives: “Yes, in the next 6 months, Yes, in the next 7 to 12 months, Yes, in

the next 13 to 24 months, Yes, sometime, but not within 24 months, and No, will not acquire

one.” Whether an automobile was purchased was directly asked in each survey wave. For the

personal computer data, we used respondents' reports of product ownership to infer whether a

household purchased a computer in a given time period. We assume a household bought a

personal computer if they switched from being a non-owner to an owner from one wave to the

next.8

For the automobile data, we define the purchase intentions j (purchase 'j ) as 1 if the nth

consumer intends to purchase (actually purchases) an automobile within 12 months and 0 if he or

she does not intend to purchase ( does not actually purchase) within 12 months. For the personal

computer data, purchase intentions j and purchase 'j are constructed as follows: 9 10

8 For the personal computer data, we have excluded repeat purchasing since we cannot determine when a repeat purchase occurred with these data. Thus, we only include respondents who intend to purchase their first personal computer. 9 Note the major purpose of our study is to establish that our unified model, which is developed for general ordered discrete choices, provides more accurate purchase predictions at the individual level than existing forecasting models. The empirical applications are only for demonstration purposes. Our goal is not to provide a detailed accounting of consumers automobile and personal computer purchase timing decisions, but rather is to describe the unified model, and illustrate how it can be used. We treat timed purchase intentions as ordered purchase intentions.

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20

=

.months 12 within occurs)(actually intended is purchase no if,0months 12-7ithin computer w personal a purchase (actually) tointendsconsumer nth theif,1

months 6within computer personal a purchase (actually) tointendsconsumer nth theif,2)'( jj

In addition to the intentions and ownership questions, extensive demographic information

was also collected including the size of household, annual household income, age of the head of

household, marital status, home ownership, household stage of life, occupation, education of the

head of household, race, number of cars owned, regional dummy variables, ownership of a

cellular phone, etc.

In the automobile data, we only have a single measurement of the demographic variables

and most of these demographic variables are unlikely to change over time. Thus, we assume no

change in the demographic variables. Fortunately, for the personal computer data, we do have

demographic information collected at each survey wave. In order to demonstrate how changes in

explanatory variables can lead true intentions to change over time, we analyze the impact of

changes in one particular demographic variable, the number of cars owned.

[Insert Table 2 Here]

Table 2 lists all the variables used in the estimation and provides some descriptive

statistics. Comparing the percent of respondents who stated an intention to buy with the

corresponding percent that actually purchased, we see that for both the historical and prediction

samples, intentions overstate purchasing. For example, in the personal computer prediction

This follows Morwitz and Schmittlein (1992) who use the same data set and treat the alternatives as ordered choices. As long as the intentions or purchases are discrete and ordered, our model should be applicable. 10 The reason we aggregate the multiple intention levels to a binary choice for the automobile data is that the data only allow us to construct binary purchase information (buy within a year or otherwise). Similarly, we aggregate intentions and purchase categories to 3 levels for PC data because the ownership information collected at each survey wave only allow us to construct 3 levels discrete purchase information. It is important to note that the categorization is only due to data limitation. It is not required for the application of our unified model. How the categorization affects the magnitude of our parameter estimates is another research topic, which we leave for future research.

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21

sample, 15.26 percent of the respondents stated they would purchase a computer within 6-12

months but only 5.67 percent actually bought. This demonstrates that respondents exaggerated

their demand to buy in the future, suggesting that direct use of stated intentions as a proxy for

purchasing would lead to inaccurate sales forecasts.

RESULTS

Before we use the unified model to describe the relationship between intentions and

purchasing, we need first to determine whether any systematic intentions biases exist in the

reported intentions, and if so, in which direction. We estimate three models, namely, the random

utility model, the two-sided intentions bias model and the one-sided intentions bias model using

the historical data assuming constant πj. Loglikelihood ratio tests reject the hypothesis that πj =0.

We then compare log-likelihood values, AIC and BIC between the two competing intentions bias

models. We find that the one-sided intentions bias model provides a better fit indicating that

respondents were overstating their intentions of purchasing an automobile as well as a personal

computer at the time of the surveys. This is consistent with the existing literature that shows that

respondents often exaggerate their future demand for socially desirable and high-tech new

products. For the remainder of the paper, we will assume the existence of one-sided intentions

bias.

To examine whether admitting individual level explanatory variables, adding the

purchase model and considering the various discrepancies indeed improve model fit and

predictive accuracy, we estimate seven models. The first model is Morrison’s (1979) model

estimated using the MLE method proposed by Kalwani and Silk (1982). The second model is

Bemmaor’s (1995) model which represents the most recent development in aggregate intentions

models. Since these two aggregate models can only predict purchase / no-purchase decisions, we

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modify our unified model by defining buying within 12 months as a purchase and all other cases

as a non-purchase within a year. The third model is our unified model ignoring purchase

information (λ=0 and ρ=1). It is similar to the most recent development in disaggregate models,

i.e., Hsiao and Sun (1999) for the multinomial case and Young et. al. (1999) for the binary case.

These models only focus on modeling response biases. Their objective is to derive *jny which is

defined by equation (2). The fourth model is our unified model ignoring systematic intentions

biases (πj=0 for all j). It still assumes intentions are directly and indirectly related to purchasing

and forecasts purchase based on Equation (23). The fifth is our unified model without changes in

true intensions over time (λ=0). The sixth model is our unified model assuming perfect

correlations between true intentions and purchasing (ρ=1). The seventh model is our unified

model, which nests models 3 to 6. The comparison of our unified model with these four nested

models shows the relative importance of each component in explaining the data. Note, for the

automobile data, since we do not have information on changes in explanatory variables to

explain changes in true intentions over time, we do not estimate model 5.

[Insert Table 3 here]

In Table 3 we report the model fit statistics of the competing models estimated using the

historical data. For discrete choices, research has shown that percentage correctly predicted (CP)

and Efron’s R2 are more powerful criteria for selecting the best fitting models than are AIC and

BIC (Amemiya 1985). 11 We report CP, Efron’s R2 and the simulated numbers of purchases for

the total sample (equation 23) as well as for each level of stated intentions (equation 22). We

11 Efron’s R-Square is calculated as ∑ ∑∑ ∑

= =

= =

−

−− N

n

M

j jnjn

N

n

M

j jnjn

YY

FY

1 02

1 02

)(

)ˆ(1 . It conveys the proportion of the variance of the

dependent variable explained by the independent variable and is a more reliable model selection criterion for discrete choice models (Amemiya 1985). Hence, the higher Efron’s R2, the more desirable the model.

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23

compare them with the observed numbers from the sample. It is important to note that i) Models

1 and 2 are not directly applicable to the automobile data which has binary intentions;12 ii)

Models 1 and 2 only offer probabilities of binary purchase / no-purchase decisions at each

intentions level. To obtain the number of simulated purchases, we use the total number of

respondents times the simulated percentages. This is different from what we do for the

disaggregate models, for which we simply count the number of consumers who are simulated to

purchase.

We first compare Models 1 and 2 with the unified model (Model 7). Both CP and Efron’s

R2 indicate that the unified model outperforms Models 1 and 2. This suggests that model fit can

be improved by further disaggregating the data from the intentions level to the individual level.

We note that aggregate models can also be quite accurate in predicting aggregate sales. This is

because these models are designed to fit aggregate sales instead of modeling individual behavior.

The advantage of disaggregate models is that they predict purchase probability at the individual

level which is more useful for guiding marketing and targeting efforts.

Model 3 represents existing disaggregate models (it is the same as Young et. al. for the

automobile data and the same as Hsiao and Sun for the personal computer data). CP, Efron’s R2

and the predicted total purchases indicate that our unified model improves existing disaggregate

models. The unified model outperforms Models 4 to 6 suggesting that it is important to consider

all three discrepancies when characterizing the relationship between intentions and purchasing.

Comparisons of the improvement of model selection criteria for Models 3, 4, 5, and 6 with

Model 7 suggest that adding the purchase model and considering systematic intentions bias are

the most important elements for improving data fit, followed by incorporating changes in true

12Bemmaor’s model is applicable to data with more than 2 intentions levels. For binary intentions, Bemmaor’s model can help specify upper and lower bounds for aggregate proportion of purchasers.

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intentions over time and incorporating an imperfect intentions-purchasing correlations.13 Since

Model 7 is the best fitting model, we focus just on Model 7 when we discuss the estimation and

prediction results in the following discussion.

[Insert Table 4 Here]

In Table 4 we report the estimation results using the historical data for all the competing

models. Assuming one-sided intentions bias, education has a negative impact on the probability

of stating biased intentions indicating that higher educated respondents state more accurate

intentions. Perhaps more educated respondents can better understand the question and therefore

can more accurately formulate their answers. We calculated the average probability of giving

biased intentions. For example, for the personal computer data, π1 and π2 are calculated to be 3.1

and 11.9, respectively, indicating that there is a 3.1 percent probability that respondents state an

intention to buy a computer within 6 months while their true intention is to either to buy within 7

to 12 months or not buy within a year. There is an 11.9 percent chance that respondents state an

intention to buy within 7 to 12 months while their true intention is not to buy within a year.

For the personal computer data, in order to demonstrate the impact of changes in

explanatory variables over time, we looked at how changes in the number of cars owned affected

the probability of purchasing a personal computer. Assuming these changes are foreseeable, we

found that buying additional cars in the recent past decreases the probability of acquiring a

personal computer in the near future (λ=-0.021). We think this happens because purchasing a car

reduces a household’s disposable income, which in turn is likely to lower their propensity to buy

an expensive product like a personal computer. Ideally, one would look at how changes in

product and promotion related variables systematically alter respondents’ true intentions over

13 We caution against generalizing the results on the relative importance of including these components in improving data fit because they are likely to be product specific.

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25

time. However, in our data sets we only had demographic variables available and the number of

cars owned was the only variable that changed over a year and had a significant impact on

purchasing. We believe that the inclusion of product and promotion related variables would

enhance the results.

ρ is estimated to be 0.732 (t=3.12) for the automobile data and 0.548 (t=22.42) for the

personal computer data. The log likelihood ratio tests reject the hypothesis that ρ=1, indicating

that true intentions and actual purchasing are positively but not perfectly correlated even after we

take into account systematic intentions biases and changes in true intentions over time.14 This

suggests that unobserved shocks in the environment affect both true intentions and actual

purchasing in the same direction but not identically in magnitude.

In the intentions model, all the demographic variables are significant and have the

expected signs. Variables such as marital status, occupation, home ownership, number of cars,

family size, education level, gender of head of household, social status and income are all

significantly related to purchase intentions for both products.

In the purchase model, parameter β is estimated to be 1.109 (t=5.44) for the automobile

and 1.497 (t=4.01) for the personal computer data, implying that latent true intentions

significantly and positively affect purchasing. This confirms Hsiao, Sun and Morwitz (2002) that

intentions can be used to predict purchase. It is important to note that 'α , which is equal to α

multiplied by a scalar β, denotes the vector of parameters representing the effect of descriptor

variables on purchase (i.e., in our case socio-demographic variables, but in general this could

include product and promotion related variables as well). It is the more accurate estimates of

14 The significance level is under-stated because ρ was estimated as a bounded variable.

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these variables that can help managers target the right consumers more efficiently with a more

effective marketing strategy.

[Insert Table 5 Here]

After calibrating the unified model using historical data and obtaining estimates of

β, λ and ρ, we then re-apply the unified model to the focal prediction sample (where only

intentions data are available) to obtain individual sales forecasts and more accurate estimates of

'α . In Table 5, we report CP, Efron’s R2 and the predicted number of purchases. The comparison

results of prediction power across the seven competing models are similar to those described

earlier for model fit. All the prediction criteria show that the unified model predicts actual

purchasing more accurately than the other models. The relative importance of including the

various components in our model in improving prediction power as we go from Model 3 to 7

follows the same trend as we saw when we discussed data fit. Thus, by recognizing the different

kinds of discrepancies between intentions and purchasing and by allowing intentions and

purchasing to be directly and indirectly connected, the unified model is able to provide more

accurate purchase predictions.

Since our random utility framework allows purchasing to depend on individual

characteristics, we can examine what type of consumers are more likely to purchase and

calculate individual probabilities of purchase using equation (23). To concisely present the

prediction results, we do not report purchase probabilities for each respondent but instead report

gains charts for the disaggregate models (Models 3 to 7) in Figure 1.15 To obtain the gains charts,

we begin by selecting the 10 percent of the respondents from the calibration sample who,

according to the predicted purchase probabilities from the model, are most likely to make a

15 Those interested in reading more about the gains chart methodology are referred to Drozdenko and Drake (2002).

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purchase. Next we compute the number of accurately predicted purchases from this group,

relative to the total number of purchases in the entire sample. This percentage is the gain due to

using the model and represents the value of sorting the predicted purchase probability based on

the model for more precise targeting. We repeat this for each model. Analogous values are also

computed for each percentile of the holdout sample (top 20%, top 30% etc.). The “No Model”

line depicts a situation in which customers are grouped randomly. So, if we have a randomly

drawn group that constitutes 10% of the sample it will on average contain 10% of the overall

number of purchases etc. The greater the difference between the gains curve and the baseline

model, the better the model in guiding targeting efforts.

The results of this analysis show that the accuracy of predicting individual purchases for

our unified model is superior to that of the baseline case and the competing disaggregate models.

The results are robust at each percentile. Models 1 and 2 are aggregate models that cannot give

individual purchase probabilities. We therefore cannot include them in the gains chart to show

their value in guiding targeting efforts. While aggregate models can be useful for forecasting

aggregate sales and helping to make go / no-go new product launch decisions, disaggregate

models are useful in guiding marketing and targeting efforts in addition to offering more accurate

overall sales forecasts.

GENERAL DISCUSSION

“For any curious human being, asking questions is easy. But for a professional

researcher, it can be a daunting challenge fraught with innumerable chances to destroy a study's

validity (Anderson 1988).” Marketing decisions made based on raw, unadjusted self-reported

intentions can be sub-optimal. In this paper, we propose a unified model that simultaneously

takes into account systematic intentions biases, changes in true intentions over time, and the

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imperfect correlation between true intentions and actual purchasing. It also unifies stated

intentions and purchasing by allowing them to be both directly and indirectly connected.

Specifically, the unified model allows us to (1) recover underlying true intentions from biased

stated intentions, (2) obtain more accurate estimates of marketing and socio-demographic

coefficients, which characterize which consumers are more likely to purchase and how

marketing strategies affect their purchase decisions, and (3) derive individual multi-level

forecasts of actual purchase probabilities. The proposed model outperforms models existing in

the literature that only consider a subset of the intentions-purchase discrepancies discussed in

this paper. This model helps to make purchase intentions a better-understood and more useful

tool for forecasting actual purchasing and targeting the individual consumers.

Our results offer implications for managers who rely on intentions to forecast sales. First,

managers should be aware that intentions surveys are usually contaminated with systematic

intentions biases, especially for new products for which consumers do not have enough

knowledge, durable products for which consumers do not have enough experience, or socially

sensitive products for which consumers may want to hide their true intentions to comply with

social norms. To minimize the likelihood that respondents report biased intentions, care should

be taken in survey design to provide as much information about the product or privacy to

respondents as possible. Second, managers should know that the longer the period between the

intentions survey and time period of interest for measuring behavior, the less accurately stated

intentions will reflect subsequent purchasing. Managers can either reduce the time between these

two events or closely track the possible factors that may lead true intentions to change over time.

Third, managers should consider factors that affect the correlation between true intentions and

purchasing. For example, if it is significantly harder for consumers to actually pay for and buy a

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product than it is to hold a true intention to buy, managers should develop marketing programs

that ease the burden of purchasing and facilitate the transaction, e.g., no-payment-no-interest-for-

a-year programs for expensive products and education programs for new and technically

complex products. Fourth, given limited resources, managers should focus their marketing

efforts on the individual customers or segments of consumers who are predicted to be most likely

to purchase the product.

Our model is limited by several simplifying assumptions, which in turn, offer avenues for

future research. First, more general model can be developed to nest both one-sided and two-

sided intentions bias model. Future extensions can also be made to consider unordered choices.

We should note, however, that the one-sided intentions bias model is no longer meaningful if the

choices are not ordered. Second, we can modify the model to accommodate brand choices. Third,

we can incorporate the finding that asking consumers purchase intent questions has an impact on

their actual purchase incidence (Chandon, Reinartz and Morwitz; Fitzsimons and Morwitz 1996;

Morwitz, Johnson and Schmittlein 1993; Louviere et. al. 1999; Sherman 1980). Fourth, we can

better take into account consumer heterogeneity by allowing the model parameters to be

heterogeneous across consumers (Hutchinson, Kamakura and Lynch 2000, Jedidi, Jagpal and

DeSarbo 1997). Fifth, the purchase timing contained in the data sets can also be formally

modeled. Sixth, if one has multiple measurements of consumers’ latent true intentions, latent trait

or itemized response models can be used to study the relationship between intentions and

purchasing (Bagozzi, Yi and Nassen 1999). Finally, it would be interesting to apply the unified

model to multiple product categories and study how the nature of the systematic intentions bias,

changes in true intentions over time and the imperfect correlation between true intentions and

purchasing varies across different types of product categories.

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Verhoef1, Peter C. and Philip Hans Franses (2002), “On combining revealed and stated preferences to forecast customer behavior: three case studies,” working paper, Erasmus University Rotterdam. Wittink D.R. (2001), Forecasting with conjoint analysis, Principles of Forecasting: A Handbook for Researchers and Practitioners (ed. Armstrong J.S.), Boston: Kluwer, 147-167 Whitlark, David B. Michael D. Geurts and Michael J. Swenson (1993), “New Product Forecasting with a Purchase Intention Survey,” Journal of Business Forecasting, 12, 3(Fall), 18-21. Young, Martin, Wayne S. DeSarbo, and Vicki G. Morwitz (1998), “The Stochastic Modeling of Purchase Intentions and Behavior,” Management Science, 44, 2 (February), 188-202.

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Table 1. Comparison of Our Model to Past Intentions Models

Sources of Discrepancy Level of

Modeling Systematic intentions

bias

Changes in true

intentions over time

Imperfect correlation between

true intentions

and purchasing

Modeling Approach

Allow for Explanatory

Variables

Data Application

Results

Morrison (79) Aggregate at intentions level

Y

Y Y Beta-binomial

N (aggregate)

Auto Linear relation between stated intentions and true intentions

Kalwani and Silk (82)

Aggregate at intentions level

Y Y Y Beta-binomial

N (aggregate)

Durable and packaged goods

Heterogeneous piecewise linear relation across product categories

Bemmaor (95) Aggregate at intentions level

Y Y Y Beta-binomial

N (aggregate)

Durable goods, services, and

other activities

Heterogeneous switching probabilities; Upper and lower bounds

Infosino (86) Aggregate at intentions level

Y N Y Probit N (aggregate)

Service Systematic promotion effects

Morwitz and Schmittlein (92)

Aggregate at intentions level

N N N Segment Y (individual)

Durable goods Forecast sales based on segmentation

Young et. al. (98) Individual Y N N Bayesian Y (individual)

Durable goods

Improved estimates αa and better purchase prediction

Hsiao and Sun (99) Individual Y N N Logit Y (individual)

Telecomm Improved estimates of α and better purchase prediction

This Paper Individual Y Y Y Multivariate Ordered Probit

Y (individual)

Automobile Personal

Computer

More accurate estimates of α’ and more accurate forecasts of individual purchase probabilities

a. α refers to the estimated coefficients of explanatory variables, e.g. socio-demographics, product attributes and marketing variables.

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Table 2: Descriptive Statistics

Automobile

Mean or Percentage

Variable Name

Variable Descriptions Historical Sample

Prediction Sample

Number of observations 2000 1000 INTENTION 1 0 PURCHASE 1a 0

intend to purchase automobile within 12 months do not intend to purchase automobile within a year purchase automobile within12 months after survey do not purchase automobile within a year after survey

9.87% 90.13% 5.01%

94.99%

9.54% 90.46% 5.13%b 94.87%

MARITAL1 FEMPL OCC1 OCC5 OCC6 EDU LIVE3 LIVE4 OWN1 INCOME YEAR NUMCARS

Married couple Female is full-time or part-time employed Household head education – manager, professional Household head education – craft and repair Household head education – operator and laborer Education level of household head (0=less than grade school; 1=Grade school; 2=Grad grade school; …; 7=post graduate university) Type of residence – mobile home Type of residence – condominium Own home Income (1=<10000; 2=10000-19999; 3=20000-29999; 4=30000-44900) Years of car currently in household Number of cars currently owned by the household

57.95% 31.1%

23.01% 7.0%

8.22% 4.11

6.13% 2.14% 7.58% 2.86

2.84 2.02

55.32% 29.4%

25.14% 6.88% 8.53% 4.27

5.99% 2.06% 7.36% 2.71

2.99 1.86

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Personal Computer Mean or Percentage

Variable Name

Variable Descriptions

Historical Sample Prediction Sample

Number of observations 3315 1105 INTENTION 2 1 0 PURCHASE 2a 1 0

intend to purchase PC within 6 months intend to purchase PC within 7 to12 months do not intend to purchase PC within a year purchase PC within within 6 months after survey purchase PC within 7 to12 months after survey do not purchase PC within a year after survey

7.52% 13.97% 78.51% 6.69% 5.51%

87.80%

7.38% 15.26% 77.36% 6.42%b 5.67%

87.91%

CARS BABY YOUNG EDUCATION LGSIZE NEW-HOUSEHOLD UPSCALE MIDAGE-NO KIDS PROFESSIONAL CLERICAL WORKING-HOURS MALE-HEAD WHITE-COLLAR INCOME

number of cars family has children under 6 year-old age <= 30 householder education(0=less than grade school; 1=Grade school; 2=Grad grade school;… ;7=post graduate university) large size family with more than 6 members household's life cycle - new household household's life cycle - upscale families household's life cycle - mid aged without children householder occupation -- professional householder occupation -- clerical number of working hours of householder household head is male household head is white collar household income

1.62 16.95% 31.78%

4.19b

0.093% 14.85% 22.80% 23.52% 24.80% 28.20%

2.50 79.44% 34.65% $38,868

1.67 16.54% 32.61%

4.25

0.117% 14.37% 21.46% 22.65% 25.37% 29.61%

2.61 76.79% 35.42% $37,932

a. For the personal computer data, purchase is inferred from a change in self reported ownership. For the automobile data, purchase is directly measured. b. Note the purchase information for the prediction sample is for validating predictive accuracy. In applying the unified model to the focal prediction sample, we

assume this information is not available.

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Table 3. Comparison of Data Fittinga Automobileb

(1) Morrison’s

Model

(2) Bemmaor’s

Model

(3) Ignore

Purchase Information

(4) Ignore

Intentions Bias

(6) Ignore

Imperfect Correlation

(7) Unified Model

Model Selection -Log-L value AIC BIC

1731.0 1735.0 1746.2

NA

895.8 908.8 945.2

1722.5 1735.5 1771.9

1684.3 1697.3 1750.5

1664.5 1677.5 1734.5

CP Intenders Nonintenders Effron’s R2

Intenders Nonintenders Derived Purchases Purchases 1 0 Intentions 1 197 87 110 0 1803 13 1790

0.870 0.715 0.887

0.086 0.077 0.087

NA NA

0.900 0.818 0.909

0.086 0.068 0.088

NA NA

0.871 0.824 0.875

0.081 0.044 0.085

93 106 16 1785

0.863 0.841 0.862

0.079 0.042 0.083

94 106

16 1787

0.912 0.864 0.915

0.086 0.049 0.090

91 106

15 1788

0.963 0.887 0.968

0.092 0.064 0.095

89 108

14 1789 Personal Computer

a. The results in Tables 3 and 4 are obtained using historical samples in which both intentions and purchase data are available. b. There are 2000 households in the historical sample for the automobile data. There are 3315 households in the historical sample for the personal computer data. c. 151 is the first entry of the 3x3 matrix. It indicates that out of the 249 respondents stating an intention of purchasing within 6 months (intention 2), 151 actually made a purchase within 6 months. d. Since Model 1 and 2 can only predict purchase and no purchase, we treat purchase within 12 months as purchase. In addition, these models can only predict percentage of purchase without

knowing who are more likely to purchase. We obtain 164 using 249x66%. 164 indicate that out of the 249 respondents stating intention 2, 66% or 164 actually made a purchase within 12 months. This number should be compared with 151+28 in the sample, 154+30 in the unified model, and so on.

(1) Morrison’s

Model

(2) Bemmaor’s

Model

(3) Ignore

Purchase Information

(4) Ignore

Intentions Bias

(5) Ignore true

intention shift

(6) Ignore

Imperfect Correlation

(7) Unified Model

Model Selection -Log-L value AIC BIC

2244.5 2248.5 2260.7

NA

1045.9 1062.9 1114.8

2238.6 2254.6 2303.4

2166.5 2186.5 2247.6

2150.0 2169.0 2227.0

2131.1 2150.1 2208.1

CP Intenders 1 Intenders 2 Nonintenders Effron’s R2

Intenders 1 Intenders 2 Nonintenders Derived Purchases Purchases 2, 1 0 Intentions 2 249 151c 28 70 1 463 28 145 290 0 2603 43 10 2550

0.890 0.710 0.841 0.916

0.082 0.060 0.077 0.085

66% (164d) 35% (162)

3% (78)

0.910 0.788 0.852 0.932

0.084 0.072 0.079 0.086

69% c(172) 37% (171) 5% (130)

0.88 0.846 0.842 0.890

0.075 0.070 0.072 0.076

159 34 56 34 149 280 38 13 2552

0.87 0.841 0.801 0.885

0.073

0.0644 0.072 0.074

156 33 60 33 148 282 40 13 2550

0.91 0.863 0.851 0.925

0.082 0.072 0.082 0.083

156 31 62 33 147 283 41 11 2551

0.92 0.871 0.862 0.935

0.084 0.0754 0.083 0.085

155 30 64 31 148 284 40 12 2551

0.95 0.897 0.894 0.965

0.088 0.0776 0.088 0.089

154 30 65 30 146 287 41 11 2551

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Table 4. Maximum Likelihood Estimates of Competing Models

Automobile

Variables

(3) Ignore Purchase

Information

(4) Ignore Intentions

Bias

(6) Ignore Imperfect

Correlation

(7) Unified Model

INTENTION MODEL α THRESHOLD1 l1 MARITAL1 FEMPL OCC1 OCC5 OCC6 LIVE3 LIVE4 OWN1 INCOME YEAR π constant EDU PURCHASE MODEL β ρ

0.505(2.27)a -1.601 (2.52) 0.129(2.23) 0.305(2.40) -0.480(1.97) -0.288(2.65) 0.731(2.43) 0.952(2.79) -0.194(1.99) 0.166(2.06) 0.118(2.14)

0.0514

-2.206(2.42) -0.183(2.06)

0.426(2.21) -1.612 (2.43) 0.133(3.04) 0.309(2.42) -0.492(2.38) -0.296(2.88) 0.711(2.74) 0.988(3.05) -0.174(1.91) 0.166(2.00) 0.111(2.43)

0.834(4.17)

0.541(2.11)

0.427(3.45) -1.627 (2.72) 0.140(3.50) 0.311(2.60) -0.480(2.45) -0.301(2.90) 0.710(3.24) 1.061(3.53) -0.160(2.37) 0.165(2.21) 0.111(2.97)

0.0465

-2.320(2.64) -0.188(2.41)

1.057(4.09)

0.817(4.03) -1.732 (3.02) 0.139(3.48) 0.295(3.01) -0.477(2.55) -0.310(2.88) 0.712(3.22) 1.055(3.15) -0.159(2.38) 0.177(2.30) 0.101(2.99)

0.0423

-2.331(2.01) -0.192(2.35)

1.109(5.44) 0.732(3.12)

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Personal Computer

a. T-statistics are reported in the parenthesis.

Variables

(3) Ignore Purchase

Information

(4) Ignore Intentions

Bias

(5) Ignore changes in

intentions

(6) Ignore Imperfect

Correlation

(7) Unified Model

INTENTION MODEL α THRESHOLD1 l2 THRESHOLD1 l1 CARS BABY LGSIZE EDUCATION NEW-HOUSEH. PROFESSIONAL WORKING-HOUR MALE-HEAD WHITE-COLLAR INCOME CELLULAR π1

CONSTANT EDUCATION π2

CONSTANT EDUCATION PURCHASE MODEL β λ CHANGE OF CARS ρ

0.937(15.21) 1.238(20.73) 0.059(1.49) -0.280(3.92) 0.288(1.29) 0.030(1.68) -0.213(2.64) 0.091(3.22) -0.029(0.57) 0.230(0.38) 0.150(2.61) 0.230(2.38) 0.309(2.55)

0.027 -2.765(1.63) -0.401(2.43)

0.101 -2.022(2.09) -0.371(3.07)

0.624(10.72) 0.996(8.83) 0.063(1.30)b -0.262(3.10) 0.279(1.39) 0.030(2.02) -0.203(2.70) 0.098(2.69) -0.024(0.85) 0.236(1.31) 0.098(2.55) 0.229(2.41) 0.328(2.23)

0.852(2.09)

-0.011(1.90)

0.301(20.55)

1.325(17.45) 1.756(52.91) 0.053(15.44) -0.304(20.17) 0.297(2.86) 0.031(1.97) -0.232(3.85) 0.103(2.65) -0.030(2.15) 0.241(2.74) 0.075(2.91) 0.212(3.03) 0.328(2.95)

0.029 -2.752(1.63) -0.401(2.43)

0.115 -2.022(2.09) -0.371(3.07)

1.337(3.49)

0.479(20.33)

1.338(19.18) 1.892(53.92) 0.058(18.76) -0.293(32.99) 0.310(2.73) 0.028(2.00) -0.237(3.94) 0.097(2.71) -0.026(2.20) 0.238(2.72) 0.077(2.86) 0.215(3.21) 0.330(2.74)

0.029 -2.749(1.85) -0.341(2.75)

0.117 -2.064(2.31) -0.369(3.12)

1.328(3.54)

-0.024(1.93)

1.450(19.52) 1.889(63.86) 0.054(2.78) -0.294(4.36) 0.319(2.69) 0.028(2.11) -0.221(4.12) 0.109(3.03) -0.029(1.99) 0.238(2.36) 0.075(2.79) 0.218(3.30) 0.307(2.73)

0.031 -2.725(1.92) -0.353(2.81)

0.119 -2.059(2.41) -0.358(3.23)

1.497(4.01)

-0.021(2.03)

0.548(22.42)

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Table 5. Comparison of Predictive Accuracya Automobileb

(1) Morrison’s

Model

(2) Bemmaor’s

Model

(3) Ignore Purchase

Information

(4) Ignore

Intentions Bias

(6) Ignore

Imperfect Correlation

(7) Unified Model

CP Intenders Nonintenders

Effron’s R2

Intenders Nonintenders Predicted Total Purchases Purchases 1 0 Intentions 1 95 42 53 0 905 9 896

0.860 0.708 0.876

0.084 0.074 0.085

NA NA

0.87 0.756 0.882

0.086 0.077 0.087

NA NA

0.86 0.841 0.862

0.070 0.0416 0.073

46 49 13 882

0.85 0.803 0.855

0.070

0.0416 0.073

44 51 11 884

0.88 0.823 0.886

0.084 0.0461 0.088

44 51 10 885

0.92 0.863 0.926

0.089

0.0501 0.093

43 52

9 886

Personal Computer

(1) Morrison’s

Model

(2) Bemmaor’s

Model

(3) Ignore

Purchase Information

(4) Ignore

Intentions Bias

(5) Ignore true

intention shift

(6) Ignore

Imperfect Correlation

(7) Unified Model

CP Intenders 1 Intenders 2 Nonintenders Effron’s R2

Intenders 1 Intenders 2 Nonintenders Predicted Total Purchasesc Purchases 2 1 0 Intentions 1 82 50 8 23c 2 168 11 53 104 0 858 10 3 842

0.820 0.642 0.739 0.853

0.068 0.040 0.063 0.072

61% (50) 35% (59) 3.4% (29)

0.840 0.695 0.743 0.873

0.070 0.049 0.065 0.073

64% (52) 37% (62) 5.3% (45)

0.84 0.745 0.774 0.862

0.066

0.0617 0.0669 0.0635

58 7 17 16 59 93 14 5 839

0.82 0.726 0.839 0.769

0.060

0.0593 0.0598 0.0601

56 8 18 14 58 96 13 5 840

0.85 0.767 0.809 0.866

0.073

0.0612 0.0721 0.0743

53 9 20 13 56 99 12 4 842

0.87 0.861 0.849 0.875

0.077

0.0678 0.0769 0.0779

53 9 20 12 56 100 102 4 852

0.900 0.887 0.901 0.901

0.079 0.0712 0.0772 0.0801

52 9 22 10 55 103 12 3 843

a. The results in Tables 5 are obtained using the prediction intentions sample in which only stated intentions are available. b. There are 1000 households in the prediction sample for the automobile data. There are 1105 households for the personal computer data. c. This is the actual number purchasing from the prediction intentions data. This information is not used for estimation and prediction. It is only used as a benchmark for examining the

predictive accuracy of the different models.

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Figure 1. Gain’s Chart

Persaonal Computer

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Percentage of Consumers

Cum

ulat

ive

Gai

ns

no-modelModel 3Model 4Model 5Model 6Model 7

Automobile

0%

20%

40%

60%

80%

100%

0% 20%

40%

60%

80%

100%

Percentage of Consumers

Cum

ulat

ive

Gai

ns no-modelModel 3Model 4Model 6Model 7

MARK-05-04