Partial Trace Correlations
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原文摘要
THIS PAPER is a sequel to [5] in which a generalized correlation coefficient (the trace correlation) for multi-equation models was presented. This paper is concerned with the development of partial trace correlations. The relation between partial trace correlations and the trace correlation is analogous to the relation between partial correlation coefficients and the multiple correlation coefficient in a single-equation model.2 In the latter the multiple correlation coefficient measures the extent to which the regression relationship on all of the independent variables accounts for the observed variation in the dependent variable. In addition there are partial correlation coefficients which measure the extent to which the regression on a particular independent variable explains the observed variation in the dependent variable, holding the influence of the other independent variables constant. Similarly in multi-equation models there is the trace correlation3 which is a measure of the extent to which the regression relationship explains the variation in the set of jointly dependent variables. The purpose of this paper is to develop the analogue of partial correlation coefficients for multiequation models, i.e., to develop a measure of the degree to which the regression relationship on a subset of the independent variables accounts for the variation in the jointly dependent variables, while holding the influence of the other independent variables constant. As in the case of the trace correlation the basic concepts are developed through the use of canonical correlation theory. This is done in Section 2, and in Section 3 partial trace correlations are defined. In Section 4 the asymptotic sampling variances are given.