Latent Trait Models with Indicators of Mixed Measurement Level

In this chapter we deal with the formulation and estimation of simul­ taneous equation models in metric latent endogenous variables that are connected to observed variables of any measurement level. The literature on this topic has focused on simultaneous equation models with metric (cf. J6reskog & S6rbom, 1984) and ordinal indicators (Muth€n, 1984). The use of ordinal indicators is based on an normal theory threshold concept implying an ordinal probit model. Concepts based on normal distribution theory are given up when qualitative variables are used as indicators for a latent metric variable (cf. the multinomial logit latent trait model of Bock, 1972). The proposed model structure consists of the following parts: First, causal relations in metric latent variables are modeled as an ordinary simultaneous equation system including dependence on exo­ genous variables. Second, each observed variable is connected to a possibly vector-valued underlying variable-from now on denoted as the latent indicator-via threshold or random utility maximization sub­ models. Third, in the single indicator case the latent indicator is equal to an endogenous variable in the structural equation system. This may happen only if the latent indicator is not vector valued. Hence,