Question on aggregate logit estimation using revenue shares

[Taha Ferhan]

Dear Professor Bierlaire,

I am currently working on a retail site location problem and would greatly appreciate your perspective on a methodological issue.

For each geographic market (defined by an isochrone), I observe annual revenues for my stores and competing stores, as well as market population. However, I do not observe individual customer choices or transaction counts.

I am considering estimating a logit model using revenue shares as proxies for market shares. My understanding is that, under random utility maximization and homogeneous preferences, aggregate shares can be used for estimation. However, I have two concerns:

- Since revenue equals number of customers times average basket size, under what conditions can revenue shares be interpreted as consistent proxies for choice probabilities?

-Given that store locations and formats are endogenously chosen based on expected demand, to what extent can the estimated parameters be given a causal interpretation?

I would be very grateful for any guidance you may have on the identification and interpretation issues in this setting or any research papers in this area.

Thank you for your time and consideration.

Best regards,

Taha FERHAN

[Michel Bierlaire]

This is a very interesting question, but also a complex one that cannot really be resolved in a short email.

In brief, estimating a logit model from aggregate shares is standard when individual choices are unobserved, but the theory applies to unit market shares (customers/transactions), not directly to revenues. Revenue shares can proxy choice probabilities only under fairly restrictive conditions (e.g., similar basket sizes across stores or a model that explicitly accounts for spending). Otherwise, the estimates confound store choice with basket-size differences and their interpretation becomes unclear.

A second key issue is endogeneity: store locations, formats, and prices are typically chosen in anticipation of demand. Without instruments, panel variation, or a structural model of entry/location, estimated parameters should be interpreted primarily as descriptive rather than causal.

You may want to have a look at these papers:
https://www.nber.org/system/files/working_papers/w6481/w6481.pdf
https://faculty.wharton.upenn.edu/wp-content/uploads/2012/04/21_Bell-Ho-Tang-%281998%29-JMR.pdf
https://doi.org/10.1509/jmkr.46.2.260

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Michel Bierlaire
Transport and Mobility Laboratory
School of Architecture, Civil and Environmental Engineering
EPFL - Ecole Polytechnique Fédérale de Lausanne
http://transp-or.epfl.ch
http://people.epfl.ch/michel.bierlaire