Joint Likelihood-based Principal Components Regression
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We propose a method for estimating principal components regressions by maximizing a multivariate normal joint likelihood for responses and predictors. In contrast to classical principal components regression, our method uses information in both responses and predictors to select useful linear combinations of the predictors. We show our estimators are consistent when responses and predictors have sub-Gaussian distributions and the number of observations tends to infinity faster than the number of predictors. Simulations indicate our method is substantially more accurate than classical principal components regression in estimation and prediction, and that it often compares favorably to competing methods such as partial least squares and predictor envelopes. We corroborate the simulation results and illustrate the practical usefulness of our method with a data example with cross-sectional prediction of stock returns.
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