Identification Robust Confidence Sets Methods for Inference on Parameter Ratios and their Application to Estimating Value-of-Time

The problem of constructing confidence set estimates for parameter ratios arises in a variety of econometrics contexts; these include value-of-time estimation in transportation research and inference on elasticities given several model specifications. Even when the model under consideration is identifiable, parameter ratios involve a possibly discontinuous parameter transformation that becomes ill-behaved as the denominator parameter approaches zero. More precisely, the parameter ratio is not identified over the whole parameter space: it is locally almost unidentified or (equivalently) weakly identified over a subset of the parameter space. It is well known that such situations can strongly affect the distributions of estimators and test statistics, leading to the failure of standard asymptotic approximations, as shown by Dufour. Here, we provide explicit solutions for projection-based simultaneous confidence sets for ratios of parameters when the joint confidence set is obtained through a generalized Fieller approach. A simulation study for a ratio of slope parameters in a simple binary probit model shows that the coverage rate of the Fieller's confidence interval is immune to weak identification whereas the confidence interval based on the delta-method performs poorly, even when the sample size is large. The procedures are examined in illustrative empirical models, with a focus on choice models