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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.

This chapter examines the asymptotic properties of the Stein-type shrinkage combined (averaging) estimation of panel data models. We introduce a combined estimation when the fixed effects (FE) estimator is inconsistent due to endogeneity arising from the correlated common effects in the regression error and regressors. In this case, the FE estimator and the CCEP estimator of Pesaran (2006) are combined. This can be viewed as the panel data model version of the shrinkage to combine the OLS and 2SLS estimators as the CCEP estimator is a 2SLS or control function estimator that controls for the endogeneity arising from the correlated common effects. The asymptotic theory, Monte Carlo simulation, and empirical applications are presented. According to our calculation of the asymptotic risk, the Stein-like shrinkage estimator is more efficient estimation than the CCEP estimator.

This chapter uses an application to explore the utility of Bayesian quantile regression (BQR) methods in producing density nowcasts. Our quantile regression modeling strategy is designed to reflect important nowcasting features, namely the use of mixed-frequency data, the ragged-edge, and large numbers of indicators (big data). An unrestricted mixed data sampling strategy within a BQR is used to accommodate a large mixed-frequency data set when nowcasting; the authors consider various shrinkage priors to avoid parameter proliferation. In an application to euro area GDP growth, using over 100 mixed-frequency indicators, the authors find that the quantile regression approach produces accurate density nowcasts including over recessionary periods when global-local shrinkage priors are used.

Vector autoregressions (VAR) combined with Minnesota-type priors are widely used for macroeconomic forecasting. The fact that strong but sensible priors can substantially improve forecast performance implies VAR forecasts are sensitive to prior hyperparameters. But the nature of this sensitivity is seldom investigated. We develop a general method based on Automatic Differentiation to systematically compute the sensitivities of forecasts – both points and intervals – with respect to any prior hyperparameters. In a forecasting exercise using US data, we find that forecasts are relatively sensitive to the strength of shrinkage for the VAR coefficients, but they are not much affected by the prior mean of the error covariance matrix or the strength of shrinkage for the intercepts.

These moments of the asymptotic distribution of the least-squares estimator of the local-to-unity autoregressive model are computed using computationally simple integration. These calculations show that conventional simulation estimation of moments can be substantially inaccurate unless the simulation sample size is very large. We also explore the minimax efficiency of autoregressive coefficient estimation, and numerically show that a simple Stein shrinkage estimator has minimax risk which is uniformly better than least squares, even though the estimation dimension is just one.