A control-function approach to correct for endogeneity in discrete choice models estimated on SP-off-RP data and contrasts with an earlier FIML approach by Train & Wilson

It is common practice to build Stated Preference (SP) attributes and alternatives from observed Revealed Preference (RP) choices with a view to increasing realism. While many surveys pivot all alternatives around an observed choice, others use more adaptive approaches in which changes are made depending on what alternative was chosen in the RP setting. For example, in SP-off-RP data, the alternative chosen in the RP setting is worsened in the SP setting and other alternatives are improved to induce a change in behaviour. This facilitates the creation of meaningful trade-offs or tipping points but introduces endogeneity. This source of endogeneity was largely ignored until Train and Wilson (T&W) proposed a full information maximum likelihood (FIML) solution that can be implemented with simulation. In this article, we propose a limited information maximum likelihood (LIML) approach to address the SP-off-RP problem using a method which does not need simulation, can be applied with standard software and uses data that is already available for the stated problem. The proposed method is an application of the control-function (CF) method to correct for endogeneity in discrete choice models, using the RP attributes as instrumental variables. We discuss the theoretical and practical advantages and disadvantages of the CF and T&W methods and illustrate them using Monte Carlo and real data. Results show that, while the T&W method may be more efficient in theory, it may however fail to retrieve consistent estimators when it does not account properly for the data generation process if, e.g., an exogenous source of correlation among the SP choice tasks exists. On the other hand, the CF is more robust, i.e. less sensitive, to the data generation process assumptions, and is considerably easier to apply with standard software and does not require simulation, facilitating its adoption and the more extensive use of SP-off-RP data.