Stein-Rule Estimation and Generalized Shrinkage Methods for Forecasting Using Many Predictors

Abstract

We examine the Stein-rule shrinkage estimator for possible improvements in estimation and forecasting when there are many predictors in a linear time series model. We consider the Stein-rule estimator of Hill and Judge (1987) that shrinks the unrestricted unbiased ordinary least squares (OLS) estimator toward a restricted biased principal component (PC) estimator. Since the Stein-rule estimator combines the OLS and PC estimators, it is a model-averaging estimator and produces a combined forecast. The conditions under which the improvement can be achieved depend on several unknown parameters that determine the degree of the Stein-rule shrinkage. We conduct Monte Carlo simulations to examine these parameter regions. The overall picture that emerges is that the Stein-rule shrinkage estimator can dominate both OLS and principal components estimators within an intermediate range of the signal-to-noise ratio. If the signal-to-noise ratio is low, the PC estimator is superior. If the signal-to-noise ratio is high, the OLS estimator is superior. In out-of-sample forecasting with AR(1) predictors, the Stein-rule shrinkage estimator can dominate both OLS and PC estimators when the predictors exhibit low persistence.

Citation

Hillebrand, E. and Lee, T.-H. (2012), "Stein-Rule Estimation and Generalized Shrinkage Methods for Forecasting Using Many Predictors", Terrell, D. and Millimet, D. (Ed.) 30th Anniversary Edition (Advances in Econometrics, Vol. 30), Emerald Group Publishing Limited, Leeds, pp. 171-196. https://doi.org/10.1108/S0731-9053(2012)0000030011