The equivalence of three latent class models and ML estimators

•We establish the equivalence of three latent class models.•They are the binary Roy model, the probit model with a misclassified dependent variable and a trivariate probit model with partial observability.•The probit model with measurement error is an enhanced version of existing models.•A researcher working on one of these estimators may benefit from the literature and software related to others.

The purpose of this letter is to show the equivalence of three latent class models; the switching regression model with endogenous switching and a latent outcome (the binary Roy model), the probit model with a systematically misclassified dependent variable, and a trivariate probit model with partial observability. The probit model with measurement error is an enhanced version of existing models which allows for the potential correlation between error terms. Establishing this connection, we hope, will help a researcher working on one of these classes of estimators to benefit from the literature and software related to other families.