into an vector and the upper thresholds into another vector . Then, in matrix form, the measurement equation for the ordinal outcomes (indicators) for the decision-maker may be written as: . (5) For the count variable, let the index be for the count categories and let be the actual g r observed count value for the household. Then, a generalized version of the negative binomial count model may be written as (see Castro, Paleti, and Bhat, or CPB, 2012 and Bhat et al., 2013): (6) , , and . (7) In the above equation, is a latent continuous stochastic propensity variable that maps into the observed count through the vector (which is a vertically stacked column vector of thresholds r is an vector of latent variable loadings on the count outcome, and is a standard normal random error term. is a column vector corresponding to the vector (including a constant) of exogenous observable covariates and endogenous outcomes. in the threshold function of Equation (7) is the inverse function of the univariate cumulative standard normal. is a parameter that provides flexibility to the count formulation, and is related to the dispersion parameter in a traditional negative binomial model if the general negative binomial structure collapses to a general Poisson structure). is the traditional gamma function; . The threshold terms in the vector satisfy the ordering condition (i.e., as long as The presence of the terms in the thresholds provides substantial flexibility to accommodate high or low probability masses for specific count outcomes (see CPB, 2012 for a detailed discussion). For identification, set and . In addition, we identify a count value above which is held fixed at ; that is, if where the value of can be based on empirical testing. Doing so is the key to allowing the count model to predict beyond the count range available in the estimation sample. For later use, let ( vector) (assuming . Next, consider the nominal (unordered-response) outcome for the individual, and let be i the corresponding ( = 1, 2, 3,…, Let the individual under consideration choose the I). alternative . Also, assume the usual random utility structure for each alternative i. (8) where is the same fixed vector of exogenous variables as earlier, is an column vector of corresponding coefficients, and is a normal random error term. is a matrix of variables
into an vector and the upper thresholds into another vector . Then, in matrix form, the measurement equation for the ordinal outcomes (indicators) for the decision-maker may be written as: . (5) For the count variable, let the index be g for the count categories and let r be the actual observed count value for the household. Then, a generalized version of the negative binomial count model may be written as (see Castro, Paleti, and Bhat, or CPB, 2012 and Bhat et al., 2013): (6) , , and . (7) In the above equation, is a latent continuous stochastic propensity variable that maps into the observed count r through the vector (which is a vertically stacked column vector of thresholds is an vector of latent variable loadings on the count outcome, and is a standard normal random error term. is a column vector corresponding to the vector (including a constant) of exogenous observable covariates and endogenous outcomes. in the threshold function of Equation (7) is the inverse function of the univariate cumulative standard normal. is a parameter that provides flexibility to the count formulation, and is related to the dispersion parameter in a traditional negative binomial model if the general negative binomial structure collapses to a general Poisson structure). is the traditional gamma function; . The threshold terms in the vector satisfy the ordering condition (i.e., as long as The presence of the terms in the thresholds provides substantial flexibility to accommodate high or low probability masses for specific count outcomes (see CPB, 2012 for a detailed discussion). For identification, set and . In addition, we identify a count value above which is held fixed at ; that is, if where the value of can be based on empirical testing. Doing so is the key to allowing the count model to predict beyond the count range available in the estimation sample. For later use, let ( vector) (assuming . Next, consider the nominal (unordered-response) outcome for the individual, and let i be the corresponding ( = 1, 2, 3,…, I). Let the individual under consideration choose the alternative . Also, assume the usual random utility structure for each alternative i. (8) where is the same fixed vector of exogenous variables as earlier, is an column vector of corresponding coefficients, and is a normal random error term. is a matrix of variables