Bayesian estimation of multivariate-normal models when dimensions are absent

Multivariate economic and business data frequently suffer from a missing data phenomenon that has not been sufficiently explored in the literature: both the independent and dependent variables for one or more dimensions are absent for some of the observational units. For example, in choice based conjoint studies, not all brands are available for consideration on every choice task. In this case, the analyst lacks information on both the response and predictor variables because the underlying stimuli, the excluded brands, are absent. This situation differs from the usual missing data problem where some of the independent variables or dependent variables are missing at random or by a known mechanism, and the holes in the data-set can be imputed from the joint distribution of the data. When dimensions are absent, data imputation may not be a well-poised question, especially in designed experiments. One consequence of absent dimensions is that the standard Bayesian analysis of the multidimensional covariances structure becomes difficult because of the absent dimensions. This paper proposes a simple error augmentation scheme that simplifies the analysis and facilitates the estimation of the full covariance structure. An application to a choice-based conjoint experiment illustrates the methodology and demonstrates that naive approaches to circumvent absent dimensions lead to substantially distorted and misleading inferences.