SUMMARYWe propose a simple, fully Bayesian approach for multivariate receptor modeling that allows for flexible and consistent incorporation of a priori information. The model uses a generalization of the Dirichlet distribution as the prior distribution on source profiles that allows great flexibility in the specification of prior information. Heavy-tailed lognormal distributions are used as priors on source contributions to match the nature of particulate concentrations. A simulation study based on the Washington, DC airshed shows that the model compares favorably to Positive Matrix Factorization, a standard analysis approach used for pollution source apportionment. A significant advantage of the proposed approach compared to most popularly used methods is that the Bayesian framework yields complete distributional results for each parameter of interest (including distributions for each element of the source profile and source contribution matrices). These distributions offer a great deal of power and versatility when addressing complex questions of interest to the researcher.

Jeff Lingwall's work was supported in part by a grant from the Kauffman Foundation. The authors declare that they have no relevant or material financial interests that relate to the research described in this paper. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

This paper examines the effects of introducing compulsory attendance laws on the schooling of U.S. children for three overlapping time periods:

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SUMMARYMany approaches for pollution source apportionment have been considered in the literature, most of which are based on the chemical mass balance equations. The simplest approaches for identifying the pollution source contributions require that the pollution source profiles are known. When little or nothing is known about the nature of the pollution sources, exploratory factor analysis, confirmatory factor analysis, and other multivariate approaches have been employed. In recent years, there has been increased interest in more flexible approaches, which assume little knowledge about the nature of the pollution source profiles, but are still able to produce nonnegative and physically realistic estimates of pollution source contributions. Confirmatory factor analysis can yield a physically interpretable and uniquely estimable solution, but requires that at least some of the rows of the source profile matrix be known. In the present discussion, we discuss the iterated confirmatory factor analysis (ICFA) approach. ICFA can take on aspects of chemical mass balance analysis, exploratory factor analysis, and confirmatory factor analysis by assigning varying degrees of constraint to the elements of the source profile matrix when iteratively adapting the hypothesized profiles to conform to the data. ICFA is illustrated using PM 2:5 data from Washington D.C., and a simulation study illustrates the relative strengths of ICFA, chemical mass balance approaches, and positive matrix factorization (PMF).