Envelopes and reduced-rank regression

We incorporate the nascent idea of envelopes (Cook et al., Statist. Sinica 20, 927-1010) into reduced-rank regression by proposing a reduced-rank envelope model, which is a hybrid of reduced-rank and envelope regressions. The proposed model has total number of parameters no more than either of reduced-rank regression or envelope regression. The resulting estimator is at least as efficient as both existing estimators. The methodology of this paper can be adapted to other envelope models, such as partial envelopes (Su & Cook, Biometrika 98, 133-46) and envelopes in predictor space (Cook et al., J. R. Statist. Soc. B 75, 851-77).