Beyond Optimal Forecasting

While the conditional mean is known to provide the minimum mean square error (MSE) forecast – and hence is optimal under a squared-error loss function – it must often in practice be replaced by a noisy estimate when model parameters are estimated over a small sample. Here two results are obtained, both of which motivate the use of forecasts biased toward zero (shrinkage forecasts) in such settings. First, the noisy forecast with minimum MSE is shown to be a shrinkage forecast. Second, a condition is derived under which a shrinkage forecast stochastically dominates the unbiased forecast over the class of loss functions monotonic in the forecast error magnitude. The appropriate amount of shrinkage from either perspective depends on a noisiness parameter which must be estimated, however, so the actual reduction in expected losses from shrinkage forecasting is an empirical issue. Simulation results over forecasts from a large variety of multiple regression models indicate that feasible shrinkage forecasts typically do provide modest improvements in forecast MSE when the noise in the estimate of the conditional mean is substantial.