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Additive models with random scaling factors: applications to modeling price response functions


  • Peter Wechselberger


  • Stefan Lang


  • Winfried J. Steiner



We discuss inference for additive models with random scaling factors. The additive effects are of the form (1+g)f(z) where f is a nonlinear function of the continuous covariate z modeled by P(enalized)-splines and 1+g is a random scaling factor. Additionally, monotonicity constraints on the nonlinear functions are possible. Our work is motivated by the situation of a retailer analyzing the impact of price changes on a brand's sales in its orange juice product category. Relating sales to a brand's own price as well as to the prices of competing brands in the category, we estimate own- and cross-item price response functions flexibly to represent nonlinearities and irregular pricing effects in sales response. Monotonicity constraints are imposed so that a brand's own price is inversely related and the prices of competing brands are directly related to the number of items sold, as suggested by economic theory. Unobserved store-specific heterogeneity is accounted for by allowing the price response curves to vary between different stores.

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  • Peter Wechselberger & Stefan Lang & Winfried J. Steiner, 2007. "Additive models with random scaling factors: applications to modeling price response functions," Working Papers 2007-27, Faculty of Economics and Statistics, University of Innsbruck.
  • Handle: RePEc:inn:wpaper:2007-27

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    P-splines; Monotonicity constraints; multiplicative random effects; price response; own- and cross-item price effects;

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