Identification of inferential parameters in the covariate-normalized linear conditional logit model

TL;DR

This paper develops an analytical method to recover original parameters in covariate-normalized linear conditional logit models, addressing issues of numerical instability and inference perturbation caused by normalization.

Contribution

It introduces formulas to recover true model parameters from normalized data for two common normalization schemes, enhancing inference accuracy.

Findings

Analytical recovery of original parameters from normalized data

Asymptotic distributions of recovered parameters derived

Numerical validation confirms effectiveness of the approach

Abstract

The conditional logit model is a standard workhorse approach to estimating customers' product feature preferences using choice data. Using these models at scale, however, can result in numerical imprecision and optimization failure due to a combination of large-valued covariates and the softmax probability function. Standard machine learning approaches alleviate these concerns by applying a normalization scheme to the matrix of covariates, scaling all values to sit within some interval (such as the unit simplex). While this type of normalization is innocuous when using models for prediction, it has the side effect of perturbing the estimated coefficients, which are necessary for researchers interested in inference. This paper shows that, for two common classes of normalizers, designated scaling and centered scaling, the data-generating non-scaled model parameters can be analytically recovered along with their asymptotic distributions. The paper also shows the numerical performance of the analytical results using an example of a scaling normalizer.