Estimating Nonlinear Parameters in the Multinomial Logit Model

Multinomial logit models, especially those calibrated on scanner data, often use explanatory variables that are nonlinear functions of the parameters to be estimated. A common example is the smoothing constant in an exponentially weighted brand loyalty variable. Such parameters cannot be estimated directly using commercially available logit packages. We provide a simple iterative method for estimating nonlinear parameters at the same time as the usual linear coefficients. The procedure uses standard multinomial logit software and, in experience to date, converges rapidly. We prove that, under suitable conditions, the resulting parameter values are maximum likelihood estimates and show how to calculate asymptotic standard errors from normal computer output. Three applications illustrate the method in practice.