Many researchers have demonstrated that fixed, exogenously chosen weights can be useful alternatives to Ordinary Least Squares (OLS) estimation within the linear model (e.g., Dawes, Am. Psychol. 34:571-582, 1979; Einhorn & Hogarth, Org. Behav. Human Perform. 13:171-192, 1975; Wainer, Psychol. Bull. 83:213-217, 1976). Generalizing the approach of Davis-Stober, Dana, and Budescu (Psychometrika 75:521-541, 2010b), I present an analytic method to determine when a choice of fixed weights will incur less mean squared error than OLS as a function of sample size, error variance, and model predictability. Geometrically, I solve for the region of population [beta] that favors a choice of fixed weights over OLS. I derive closed-form upper and lower bounds on the volume of this region, giving tight bounds on the proportion of population [beta] favoring a choice of fixed weights. I illustrate this methodology with several examples and provide a MATLAB[c] (The MathWorks, Matlab software, version 2009b, 2010) programming implementation of the major results. (Contains 2 footnotes and 4 figures.)