Ordinary Least Squares Revolutionized: Establishing the Vital Missing Empirically Determined Statistical Prediction Variance

Abstract

Although ordinary least squares (OLS) is generally regarded as the foundation of econometrics, the vital concept of the empirically determined statistical variance in an OLS prediction is misunderstood and missing. The fundamental deficiency is the failure to appropriately model the deterministic OLS processing of statistical errors. The variance in the OLS prediction is typically described in terms of standard properties of variance. However, those properties fail to correctly model the statistical OLS output. This paper fills that gap. Based on experience with well proven and long established engineering and math modeling techniques, OLS is modeled with methods of estimation theory analogous to the modeling of a signal processor or target tracker. In other words, OLS is a treated as a "black box" with "gosintas" and "gosoutas". A novel derivation of the OLS based on the orthogonality principle establishes the empirically determined statistical OLS output in the form of the prediction, variance, and expected value. Also included is the empirically determined statistical variance of future observations, which are independent of the statistical prediction. Thus, the statistical OLS prediction variance and the statistical variance of any new sample would both contribute to the uncertainty of any future observation. Both statistical variances are empirically estimated by OLS processing in advance.