Formalizing a Postprocessing Procedure for Linear–Convex Combination Forecasts

Abstract

We investigate mean squared forecast error (MSE) accuracy improvements for linear–convex combination forecasts, whose components are pretreated by a postprocessing procedure called “vector autoregressive forecast error modeling” (VAFEM). Assuming that the forecast error series of the individual forecasts are governed by a stable VAR process under classic conditions, we obtain the following results: (i) VAFEM treatment bias corrects all individual and linear–convex combination forecasts. (ii) Any VAFEM-treated combination has a smaller theoretical MSE than its untreated analog, if the VAR parameters are known. (iii) In empirical applications, VAFEM gains depend on (1) in-sample sizes, (2) out-of-sample forecast horizons, and (3) the biasedness of the untreated forecast combination. We demonstrate the VAFEM capacity in simulations and for realized-volatility forecasting, using S&P 500 data.