1. Joint estimates of purchase timing and brand switch tendency: results from a scanner panel
data set of frequently purchased products
F. F. Gönül, P. T. L. Popkowski Leszczyc, and T. Sugawara
Carnegie Mellon University, University of Alberta, and
Carnegie Mellon University
ABSTRACT. In the marketing literature, purchase timing and brand choice behavior of households have
generally been treated independently. The subsumed independence of the two events can be restrictive. In
this paper we relax and test the independence assumption. We study purchase behavior in two stages: In
the first stage we model the timing of purchases. The second stage focuses on the conditional probability of
switching given that a purchase took place. We use longitudinal data on household purchases for two
different product categories; diapers and ketchup. The joint formulation exploits the information in the
scanner panel data more fully than has been possible before.
I. INTRODUCTION
Our purpose in this paper is to measure the correlation between purchase timing and brand switch behavior
of households. More specifically, we want to study (i) whether households switch brands more often for
the products they purchase more frequently or for the products they purchase less frequently. (ii) We want
to identify the demographic variables that influence purchase timing; and to identify the marketing variables
that influence the brand switching decisions. These research issues are of utmost importance to brand
managers of consumer products who are interested in encouraging purchase acceleration and loyalty to their
brand. Managers can implement our model in identifying market segments that buy more often but switch
less often, the heavy users of a brand or segments with growth potential.
Our data is from the scanner panel purchase data collected in two cities by the A.C. Nielsen Company for
a period of approximately one year. We aggregate the purchases to weekly units and select two categories.
We test our model and find that while for purchases of diapers there is no evidence of a significant
correlation between purchase timing and brand switching, for ketchup purchases, there is a significant
positive relationship between the timing of purchases and brand switching.
II. ECONOMETRIC MODEL
The likelihood function for a purchase occasion i that constitutes a switch from the previous purchase that

2. took place t periods ago is,
Li = Pr(j → k,t)
= Pr(j → k  t) ⋅ f(t)
where j denotes the origin brand, k denotes the destination brand, and t stands for the interpurchase time.
Conversely, the likelihood function for a repeat purchase occasion i (a nonswitch event) after t periods is
given by, Li = (1-Pr (j → k  t)) • f(t). We assume the switch from j to k is a discrete choice governed by a
random utility model. That is,
Ujk = _jk + ujk,
where ujk is normally distributed ( u jk _ N(0, σ u 2)). 1 We assume brand switching imposess switching
costs and hence we assume different utilities for switching and staying with the same brand. A switch
occurs if the utility of switching is greater than the utility of not swtching. (We arbitrarily normalize the
utility of not switching to zero). We further assume that the interpurchase time (t) is lognormal, and is
correlated with the random utility component. Then Li has a closed-form expression given by a bivariate
normal. More explicitly,
Pr(j → k) = Pr(_jk + ujk > 0), and
Pr(j → k  t) = Pr(_jk + ujk > 0  ln(t)) J,
where J is the Jacobian of the transformation from t to ln(t). Then, the likelihood for a switch occasion
is,
Li = Pr(_jk + ujk > 0  ln(t)) J f(t).
Since the conditional distribution, g(ujk  ln(t)), is a normal distribution, the likelihood function is the
product of two normal distributions and is easy to evaluate. Specifically,
σ u (ln(t) - ))] 1 _ φ ( ln(t) - µ t )
Li = [1 - Φ(- _ jk - ρ µt 2
σt ln(t) σ t σt
2
where ln( t) _ N( µ t , σ ), φ ( _ ) 3 stands for the standard normal p . d . f ., Φ (•) 4 denotes the standard
t
normal c.d.f., and the correlation coefficient, ρ = Corr(ujk, ln(t)), measures the correlation between purchase

3. timing and the probability of a switch. A positive correlation indicates that when interpurchase time
2
increases the likelihood of a switch increases. We impose the restriction that σ (1 - ρ ) = 1 5 in order to
2
u
identify the bivariate normal model.
III. ESTIMATION
We estimate our model on two data sets, with diaper purchases and ketchup purchases. The diaper data
consists of 152 households who regularly bought diapers from among 3 national brands during the course
of a year. There are a total of 2675 purchases. The ketchup data consists of 200 households who regularly
bought ketchup from among 2 national brands and store brands (combined as one brand, since the purchase
data is across stores) over the course of one year. There are a total of 1567 ketchup purchases. The average
interpurchase time for disposable diapers is about 2 weeks, and for ketchup about 10 weeks. Our findings
are presented in Table 1.
--Insert Tabel 1 here--
Diapers
Households that have higher incomes are likely to buy diapers more frequently, ceteris paribus, and
households that are better educated are likely to buy diapers less frequently, ceteris paribus. Coupons rather
than price are the determining factor in influencing brand switches in the diaper market. There is no
significant correlation between interpurchase time and the propensity to switch.
Ketchup
Households that are larger in size are likely to buy ketchup more often, ceteris paribus. The remaining
demographic variables have no influence on purchase timing. Price, significantly and negatively influences
the probability of a switch. In-store displays do not play a role in encouraging consumers to switch brands.
The correlation coefficient is positive and significant. Households that buy less often may "forget" the
brand and tend to switch more often than other households.

4. IV. CONCLUSION
Our empirical analysis shows that it is possible to identify the characteristics of households that buy a
certain product more often than others. This facilitates target marketing and enables managers to better
target advertising and promotional campaigns to different consumer segments. For example, a managerial
policy could be to target reminder advertising to heavy users and target price incentives to switchers. We
also show that brand switching is caused by different characteristics depending on the product category.
For diapers, price is not a significant factor but coupon is. In a category like ketchup, where coupons are
negligible, store display does not induce households to switch, while price does.
Different from the prior literature, in this work, we measure the correlation between the timing of
purchases and brand switches. We find that when the purchase interval gets longer, consumers are more
likely to switch. Hence, frequent purchasers are more likely to be brand loyal. For ketchup purchases we
observed a significant correlation between interpurchase time and the propensity to switch. Thus, we are
led to conclude that if the product is essential and bought regularly like diapers, loyalty may continue.
However, if the product is not essential like ketchup, where there are many substitutes, then switching may
be more prevalent. These results are consistent with findings in the promotional literature. Price
promotions, reflected in the price of ketchup, tend to lead to purchase acceleration, while coupons (mostly
used for diaper purchases) generally do not lead to purchase acceleration.

5. _____________________________________________________________________________________
Table 1
Results of Analysis of Purchase Timing and Brand Switching
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DIAPER KETCHUP
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Interpurchase Time
Constant 0.5941** 1.7766**
(0.0704) (0.1179)
Education 0.0316** 0.0135
(0.0092) (0.0130)
Income -0.0172** 0.0016
(0.0058) (0.0089)
Household Size ---- -0.0808**
(0.0173)
Switch Propensity
Constant -0.9121** 0.1074
(0.3360) (0.2579)
Price -0.0131 -0.3526**
(0.0230) (0.0661)
Coupon 0.4387** ----
(0.0584)
Display ---- -0.1427
(0.0960)
Correlation between 0.0251 0.0871*
Interpurchase Time and Switch (0.0278) (0.0440)
Propensity
____________________________________________________________________________________
Notes: Standard errors are in parentheses. Significance at 1% is denoted by (**) and at 5% by (*).
Household size is not included in the diaper model, since there is not much variation in size
among the purchasing households. Coupon is not included in the ketchup model, since coupon
activity is negligible in the category. Likewise, display activity is virtually nonexistent in the
diaper category. We exclude these variables without loss of significant information.
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