Multi - dimensional Error Measures for Ex Post Facto Forecast and Estimation Evaluations

This paper deals with an issue that appears to be unexplored by demographers. The issue Is based on the question: "if one was not only doing ex post facto evaluations of estimates (or projections) of the total population but by characteristics such as race and geography, how would one summarize the summary measures of error?" The paper examines this issue in two dimensions using a standard summary measure of ex post facto error “Mean Algebraic Percent Error” (MALPE). It offers hypothetical, but empirical, examples of the issue using MALPE. It is aimed at answering the underlying question, which is “what is the nature of the relationship, if any, of summary errors of measure taken across different dimensions in conducting an ex post facto evaluation of the accuracy of a set of estimates or projections. The paper finds that relationships do exist and offers proof of these relationships both mathematically and empirically. It concludes by noting work to be done includes generalizing these findings to more than two dimensions and extending the findings to other commonly used summary measures of ex post facto error, such as Mean Absolute Percent Error (MAPE) and Root Mean Square Error (RMSE).