Maximum Score Methods

In a seminal paper, Manski (1975) introduces the Maximum Score Estimator (MSE) of the structural parameters of a multinomial choice model and proves consistency without assuming knowledge of the distribution of the error terms in the model. As such, the MSE is the rst in-stance of a semiparametric estimator of a limited dependent variable model in the econometrics literature. Maximum score estimation of the parameters of a binary choice model has received the most attention in the literature. Manski (1975) covers this model, but Manski (1985) focuses on it. The key assumption that Manski (1985) makes is that the latent variable underlying the observed binary data satises a linear -quantile regression specication. (He focuses on the linear median regression case, where = 0:5.) This is perhaps an under-appreciated fact about maximum score estimation in the binary choice setting. If the latent variable were observed, then classical quantile regression estimation (Koenker and Bassett, 1978), using the latent data, would estimate, albeit more eciently, the same regression parameters that would be estimated by maximum score estimation using the binary data. In short, the estimands would be the same for these two estimation procedures