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We propose a general approach to detect measurement non-invariance in latent Markov models for longitudinal data. We define different notions of differential item functioning in the context of panel data. We then present a model selection approach based on the Bayesian information criterion (BIC) to choose both the number of latent states and the measurement structure. We show the practical relevance by means of an extensive simulation study, and illustrate its use on two real–data examples from the social sciences. Our results indicate that BIC is able to select the correct measurement equivalence structure more than 95% of times. [ABSTRACT FROM AUTHOR]

While latent class (LC) models with distal outcomes are becoming popular in literature as a consequence of the increasing use of stepwise estimators, these models still suffer from severe shortcomings. Namely, using the currently available stepwise estimators the direct effects between the distal outcome and the indicators of the LC membership cannot be easily modeled. At the same time using the traditional Full Information Maximum Likelihood (FIML) approach the LC solution can become dominated by the distal outcome, especially when model misspecifications occur, and the relationship between the distal outcome and LC is strong. In this paper, we consider a more general formulation, typical in cluster-weighted models, which embeds both the latent class regression and the distal outcome models. This allows us to test simultaneously both whether the distribution of the distal outcome differs across classes, and whether there are significant direct effects of the distal outcome on the indicators, by including most of the information about the distal outcome – latent variable relationship. We propose a two-step estimator for these models that makes it possible to separate the estimation of the measurement and structural model, that is much desired for distal outcome models, while keeping the possibility of modeling direct effects open. We show the advantages of the proposed modeling approach through a simulation study and an empirical application on assets ownership of Italian households. [ABSTRACT FROM AUTHOR]