The Rescaling Problem in Nonlinear Probability Models, Dr. Christopher Achen and Dr. Won-Ho Park, October 27|28, 2022

Researchers using nonlinear probability models, such as logit and probit, often study how the original coefficients change when additional covariates are added or subtracted from the model. Such comparisons are illegitimate. The problem occurs because the estimated parameters in such models are only identified “up to a scale,” which means that the estimated coefficients are scaled by the standard deviation of the unobserved disturbance term. Hence, adding additional covariates decreases the residual variance, which then inflates all the estimated coefficients even if their true values have not changed.

In the paper, we study alternative estimators that address the rescaling problem, including y-scale standardization, AMEs (average marginal effects), and the prominent KHB method developed by Karlson, Holm, and Breen (2012). A series of Monte Carlo simulations show that AME outperforms other alternatives in nearly all cases with a wide range of error distributions for the disturbances, while y-scale standardization is biased when the disturbances are not normally distributed, and KHB typically has proportional sampling errors averaging about half again as large as those for the AMEs. Thus, we recommend the use of AMEs to cope with the scaling issue in nonlinear probability models.