Seemingly unrelated clusterwise linear regression

Abstract

Linear regression models based on finite Gaussian mixtures represent a flexible tool for the analysis of linear dependencies in multivariate data. They are suitable for dealing with correlated response variables when data come from a heterogeneous population composed of two or more sub-populations, each of which is characterised by a different linear regression model. Several types of finite mixtures of linear regression models have been specified by changing the assumptions on the parameters that differentiate the sub-populations and/or the vectors of regressors that affect the response variables. They are made more flexible in the class of models defined by mixtures of seemingly unrelated Gaussian linear regressions illustrated in this paper. With these models, the researcher is enabled to use a different vector of regressors for each dependent variable. The proposed class includes parsimonious models obtained by imposing suitable constraints on the variances and covariances of the response variables in the sub-populations. Details about the model identification and maximum likelihood estimation are given. The usefulness of these models is shown through the analysis of a real dataset. Regularity conditions for the model class are illustrated and a proof is provided that, when these conditions are met, the consistency of the maximum likelihood estimator under the examined models is ensured. In addition, the behaviour of this estimator in the presence of finite samples is numerically evaluated through the analysis of simulated datasets.