Minimum Variance Properties of Principal Component Regression

In the context of the usual regression model, , this article shows that the coefficient obtained when Y is regressed on the first principal component of the independent variables may be interpreted as an estimator of that , which can be estimated with least variance. Analogous results hold for succeeding components, and the property can be expressed in terms of volumes of confidence ellipsoids. To estimate using r principal components, choice of those components associated with the r largest characteristic roots minimizes the trace of a variance matrix. The results are illustrated with data on French imports.