A Better Method of Applying OLS to the CAPM, Prediction, and Forecasting

Abstract

This paper formulates a weighting function from conventional least squares (LS) and combines it with estimation theory to provide the statistical estimate, expected value, and variance of any point on the polynomial constructed for fitting a set of existing data. This solves the problem of the missing variance at arbitrary points on the polynomial from LS derived by Gauss. The method includes three LS tricks: (a) Reframing LS from creating a polynomial for fitting existing data to estimating an already existing polynomial corrupted with statistically described sampling errors. (b) Restructuring LS processing from approximating polynomial coefficients to creating a weighting function for estimating the independent variable at any point on the LS polynomial. (c) Averaging the statistical deviations from the estimated LS polynomial to estimate the variance of sampling errors. The method is based on two cold hard CAPM data sets: (a) samples of the asset corrupted by statistically described errors and (b) deterministic samples of the corresponding market. Hand-waving arguments about market forces and investor behavior apply only after LS processing, not before or during. An example of the technique applied to fictitious sales as a function of GNP shows the technique applies to virtually any problem addressed by polynomial LS.