Nonnegativity Restricted Least Squares combinations

In combining reasonably efficient forecasts of a time series, nonnegative weights are intuitively meaningful and their constrained least squares estimators are known to have desirable theoretical properties. It is analytically shown that whether such inequality (or equality) constraints will result in gains in efficiency depends on the number of observations in the fit sample and the mean level of the fit set. Next, the performances of the combination forecasts of a macroeconomic time series obtained using Nonnegativity Restricted Least Squares (NRLS) and other combination methods are exhaustively compared. It is shown that NRLS combination models are more efficient, more robust, and less sensitive to sample size than Ordinary Least Squares, Equality Restricted Least Squares, minimum-variance, and outperformance models. They are more efficient than historical weighting and equal weighting methods only for large samples. Two heuristic NRLS algorithms are shown to yield equally accurate and robust combined forecasts.