A Dynamic Model of Purchase Timing with Application to Direct Marketing

Article excerpt


Successful direct marketing initiatives require firms to predict the behavior of specific individuals. At major investment brokerage firms, such as Merrill Lynch, Fidelity, and Schwab, 5-10% of customers switch firms each year, taking with them billions of dollars in assets and close to $100 million in revenue. A critical problem faced by these investment firms, and for many direct marketing companies, is predicting when specific individuals are likely to stop being regular customers and enter periods of relative inactivity. With such prediction, firms can intervene aggressively when they feel a customer is likely to move from a frequent to a less frequent level of purchase behavior due to the fear that they are taking their business to a competitor.

The field of direct marketing has grown considerably, largely due to the reduced costs associated with collecting and storing customer records and the availability of distribution channels (e.g. the internet, fax machines, and delivery firms such as Federal Express) that provide direct access to the customer. However, despite the ready availability of customer records, direct marketers currently operate in an environment characterized by datasets with severe limitations. For example, although information might be available to a firm about the date and price paid for an item, typically no information is available on the availability of goods from other manufacturers or whether a consumer had considered these other goods when making their purchase. Therefore, it is often not possible to estimate constructs such as a consumer's utility function, which is assumed to drive the purchase decision. Instead, firms have been forced to use statistical models of purchase data that by necessity ignore competitive effects and unobserved customer behavior (e.g., purchases of competitive products).

In this article we develop a dynamic model of purchase timing that provides an early indication of when specific customers are likely to move from an active state to a less active state. The model has three distinct features:

1. It uses a generalized gamma distribution that we find useful for modeling interpurchase times.

2. Random effects are introduced through a hierarchical Bayes structure that yields individual-level estimates of key model parameters and functions of parameters.

3. Temporal variation in an individual's expected interpurchase time is permitted by introducing covariates that change through time.

From a direct marketing perspective, this model has two major advantages. First, the hierarchical Bayes random-effects specification, coupled with the temporal component of the model, allows for a customer-specific baseline rate of interaction with the firm that can fluctuate over time. Second, and more important, the covariates permit prediction of when this rate is likely to change.

In the next section we discuss the data and provide descriptive statistics. In Section 3 we introduce the model and discuss the estimation algorithm. We give results in Section 4, comparing model predictions to simpler methods of generating individual-level forecasts of future interpurchase times. We present concluding remarks in Section 5.


The data are provided by a major investment brokerage firm that buys and sells financial instruments such as stocks, bonds, and mutual funds for their customers. Revenues are generated as a percentage of the trade amount. In consultation with senior managers and analysts from the firm, we decided to measure interpurchase times (in this case, intertrading times) in terms of calendar weeks. There were three reasons for this. First, managers at the firm felt that for the purpose of decision making, it was not necessary to react to changes in interpurchase times in units less than a week. They did not see any advantage in a model of dally (versus weekly) purchase timing. …