## Specifying and Estimating a Sales Distance Model

The preceding sections of this chapter demonstrated the theoretical reasons for using a social gravity model as the basis for modeling the sales-distance relationship. In most cases, the classic gravity model equation is the simplest and best choice. In other cases, a different specification may be selected that is better-suited for the data. We describe the estimation of the parameters of such models in this section; we also discuss an alternative specification, a polynomial model.

Data collection note: The data required for both specifications are the same. "Data Collection and Formatting" on page 351 describes the type of data needed and how to place the data into bins.

Step-by-Step Method

We recommend that the following steps be done after the acquisition of the data:

1. Initial exploratory data analysis: A good approach to this and other problems is to first explore the data. Exploratory Data Analysis (EDA) is a fruitful source of analytical techniques for this purpose. We describe techniques of exploring the data in "Geographic Analysis of Sales" on page 340. Before attempting to specify and estimate a sales-distance model, explore the spatial data patterns.

A second, simple EDA technique is to simply plot the distance-sales relationship on an X-Y chart, with dots indicating each (sale and distance) pair. With the data already analyzed in the manner described above, a histogram of the data, broken down by distance, would provide excellent summary information.

2. Identifying analog stores: A second step that should be taken in many cases is to look for existing retail outlets with similar characteristics and examine their sales-distance relationships. This is sometimes called the analog method after William Applebaum popularized it in the 1960s.346

3. Primary model specification—gravity model: With the data prepared in bins as described above, specify and estimate the distancesales relationship. We provide examples below.

Example I: Monte Carlo Data on Retail Sales We first provide an example of this technique using Monte Carlo data:

1. To create this test data and then estimate it, we used the retail_test function which is part of the Business Economics Toolbox. It first takes in assumptions about the underlying relationship and generates the random numbers to include in the data. Second, using the assumptions and the random numbers, it creates test data. Third, using the special functions we have developed for modeling retail sales, it estimates those parameters.

2. Figure 13-7, "Underlying sales-distance relationship: gravity model," shows the test data generated to test our ability to properly identify a gravity model. The underlying relationship is shown in the top pane. The data are based on this relationship, plus some random noise, and are shown in the bottom pane.

3. Using only the test data, we estimated the relationship as shown in Figure 13-8, "Estimated sales-distance relationship." Note that the actual parameters were somewhat different from our estimated parameters, indicating that (as should be expected in all such cases) the random noise in the data obscured the precise relationship.

4. The command line output from the function is shown in the following code fragment.

346 W.Applebaum, "Methods for determining store trade areas, market penetration and potential sales", J. Mark. Res., 3,1966. Cited in, e.g., A.Ghosh, Retail Management (Fort Worth, TX:

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