Monotonic regression based on Bayesian P-splines: an application to estimating price response functions from store-level scanner data

Generalized additive models have become a widely used instrument
for flexible regression analysis. In many practical situations,
however, it is desirable to restrict the flexibility of
nonparametric estimation in order to accommodate a presumed
monotonic relationship between a covariate and the response
variable. For example, consumers usually will buy less of a brand
if its price increases, and therefore one expects a brand's unit
sales to be a decreasing function in own price. We follow a
Bayesian approach using penalized B-splines and incorporate the
assumption of monotonicity in a natural way by an appropriate
specification of the respective prior distributions. We illustrate
the methodology in an empirical application modeling demand for a
brand of orange juice and show that imposing monotonicity
constraints for own- and cross-item price effects improves the
predictive validity of the estimated sales response function
considerably.