Bruce E. Hansen

Stein Combination Shrinkage in Vector Autoregressions

June 2016


This paper introduces Stein combination shrinkage for vector autoregressions (VARs). The proposed methods shrink unrestricted least-squares VAR estimates towards multiple user-specified linear constraints, including lag exclusion and autoregressive models. We propose weighted combination estimators, where the weights minimize an estimate of the mean-squared error (MSE) of a vector-valued parameter of interest. Particular attention is given to impulse response estimation and multi-period point forecasting. The combination estimators are similar to Stein shrinkage estimators. Our proposed weights are specific to the horizon, which allows the degree of shrinkage to adapt across horizons. The proposed methods are evaluated in a careful simulation experiment. The simulation evidence shows that the Stein combination methods have much lower MSE than conventional OLS and BVAR methods. We illustrate the methods with an application to a standard seven-variable system of U.S. macroeconomic aggregates.

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Some of the above material is based upon work supported by the National Science Foundation under Grants No. SES-9022176, SES-9120576, SBR-9412339, and SBR-9807111. Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the NSF.