Additive Models with Random Scaling Factors: Applications to Modeling Price Response Functions
We discuss inference for additive models with random scaling factors. The additive effects are of the form (1+γ)f(z) where f is a nonlinear function of the continuous covariate z modeled by P(enalized)-splines and 1 + γ 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.
Belitz, C., und Lang, S. (2007). Simultaneous selection of variables and smoothing parameters in structured additive regression models. Technical report, submitted for publication.
Blattberg, R., und Wisniewski, K. (1989). Price-induced patterns of competition. Marketing Science, 8, 291-309.
Bollaerts, K., Eilers, P. H. C., und Mechelen, I. van. (2006). Simple and multiple P-splines regression with shape constraints. British Journal of Mathematical and Statistical Psychology, 59, 451-469.
Brezger, A., und Lang, S. (2006). Generalized structured additive regression based on Bayesian P-splines. Computational Statistics and Data Analysis, 50, 947-991.
Brezger, A., und Steiner, W. (2007). Monotonic regression based on Bayesian P-splines: An application to estimating price response functions from store-level scanner data. Journal of Business and Economic Statistics. (In press)
De Boor, C. (2001). A Practical Guide to Splines. Springer, New York.
Eilers, P. H. C., und Marx, B. D. (1996). Flexible smoothing using B-splines and penalized likelihood (with comments and rejoinder). Statistical Science, 11, 89-121.
Fahrmeir, L., Kneib, T., und Lang, S. (2007). Regression. Modelle, Methoden und Anwendungen. Berlin: Springer.
Gelman, A., und Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. New York: Cambridge University Press.
Hanssens, D., Parsons, L., und Schultz, R. (2001). Market Response Models - Econometric and Time Series Analysis. London: Chapman and Hall.
Hastie, T., und Tibshirani, R. (1990). Generalized Additive Models. London: Chapman and Hall.
Hastie, T., und Tibshirani, R. (1993). Varying–Coefficient Models. Journal of the Royal Statistical Society B, 55, 757-796.
Heerde, H. van, Leeflang, P., undWittink, D. (2001). Semiparametric analysis to estimate the deal effect curve. Journal of Marketing Research, 38, 197-215.
Heerde, H. van, Leeflang, P., und Wittink, D. (2002). How promotions work: SCAN*PRO-based evolutionary model building. Schmalenbach Business Review, 54, 198-220.
Kalyanam, K., und Shively, T. (1998). Estimating irregular pricing effects: a stochastic spline regression approach. Journal of Marketing Research, 35, 19-29.
Lang, S., und Brezger, A. (2004). Bayesian P-splines. Journal of Computational and Graphical Statistics, 13, 183-212.
Montgomery, A. (1997). Creating micro-marketing pricing strategies using supermarket scanner data. Marketing Science, 16, 345-337.
Ramsay, J., und Silverman, B. (2002). Applied Functional Data Analysis. Berlin: Springer.
Ramsay, J., und Silverman, B. (2005). Functional Data Analysis. Berlin: Springer.
Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Chapman and Hall.
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