An Optimal Property of Least Squares Weights in Prediction Models
Authors: Gross Alan L. 
Published: January 2006

In predicting\documentclass[12pt]{minimal}
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$$\tilde y$$
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$$(\tilde x)$$
\end{document} in a sample of size ñ, the optimal strategy (minimum expected loss), under certain assumptions, is shown to be based upon the least squares regression weights\documentclass[12pt]{minimal}
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$$(\hat \beta )$$
\end{document} computed from a previous sample. Letting\documentclass[12pt]{minimal}
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$$\tilde r(\hat \beta )$$
\end{document} represent the correlation between\documentclass[12pt]{minimal}
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$$\tilde y$$
\end{document} and the predicted values\documentclass[12pt]{minimal}
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$$(\hat \beta '\tilde x)$$
\end{document}, and letting\documentclass[12pt]{minimal}
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$$\tilde r(w)$$
\end{document} represent the correlation between\documentclass[12pt]{minimal}
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$$\tilde y$$
\end{document} and a different set of predicted values\documentclass[12pt]{minimal}
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$$(w'\tilde x)$$
\end{document}, where w is any weighting system which is not a function of\documentclass[12pt]{minimal}
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$$\tilde y$$
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$$\tilde r(\hat \beta )$$
\end{document} being less than\documentclass[12pt]{minimal}
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$$\tilde r(w)$$
\end{document} cannot exceed .50. The relationship of this result to previous research and practical implications are discussed.