in Equation (2). However, in some cases, it may indeed be appropriate to allow between the latent variables. Section 3.1 discusses the identification considerations in this case. Note also that our model formulation and estimation technique are readily applicable to this case of inter-related latent constructs too as long as the identification considerations in Section 3.1 are met. Latent Variable Measurement Equation Model Components We will consider a combination of continuous, ordinal, count, and nominal outcomes (indicators) of the underlying latent variable vector . However, these outcomes may be a function of a set of exogenous variables too. Let there be continuous outcomes with an associated index . Let in the usual linear H h regression fashion, where is an vector of exogenous variables (including a constant) as well as possibly the observed values of other endogenous continuous variables, other endogenous ordinal variables, other endogenous count variables, and other endogenous nominal variables (introduced as dummy variables). is a corresponding compatible coefficient vector. is an vector of latent variable loadings on the continuous outcome, and is a normally distributed hth measurement error term. Stack the continuous outcomes into an vector and the error terms H y, H into another vector . Also, let be the covariance matrix of , which is restricted to be diagonal. This helps identification because there is already an unobserved latent variable vector that serves as a vehicle to generate covariance between the outcome variables (as we discuss in the next section). Define the matrix and the matrix of latent variable loadings Then, one may write, in matrix form, the following measurement equation for the continuous outcomes: . (3) Next, consider ordinal outcomes (indicator variables) for the individual, and let be N n the index for the ordinal outcomes . Also, let be the number of categories for the ordinal nth outcome and let the corresponding index be. Let be the latent underlying variable whose horizontal partitioning leads to the observed outcome for the ordinal variable. Assume that nth the individual under consideration chooses the ordinal category. Then, in the usual ordered response formulation, for the individual, we may write: (4)
in Equation (2). However, in some cases, it may indeed be appropriate to allow between the latent variables. Section 3.1 discusses the identification considerations in this case. Note also that our model formulation and estimation technique are readily applicable to this case of inter-related latent constructs too as long as the identification considerations in Section 3.1 are met. Latent Variable Measurement Equation Model Components We will consider a combination of continuous, ordinal, count, and nominal outcomes (indicators) of the underlying latent variable vector . However, these outcomes may be a function of a set of exogenous variables too. Let there be H continuous outcomes with an associated index h . Let in the usual linear regression fashion, where is an vector of exogenous variables (including a constant) as well as possibly the observed values of other endogenous continuous variables, other endogenous ordinal variables, other endogenous count variables, and other endogenous nominal variables (introduced as dummy variables). is a corresponding compatible coefficient vector. is an vector of latent variable loadings on the hth continuous outcome, and is a normally distributed measurement error term. Stack the H continuous outcomes into an vector y, and the H error terms into another vector . Also, let be the covariance matrix of , which is restricted to be diagonal. This helps identification because there is already an unobserved latent variable vector that serves as a vehicle to generate covariance between the outcome variables (as we discuss in the next section). Define the matrix and the matrix of latent variable loadings Then, one may write, in matrix form, the following measurement equation for the continuous outcomes: . (3) Next, consider N ordinal outcomes (indicator variables) for the individual, and let n be the index for the ordinal outcomes . Also, let be the number of categories for the nth ordinal outcome and let the corresponding index be. Let be the latent underlying variable whose horizontal partitioning leads to the observed outcome for the nth ordinal variable. Assume that the individual under consideration chooses the ordinal category. Then, in the usual ordered response formulation, for the individual, we may write: (4)
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