On the Relationship among Values of the same Summary Measure of Error when used across Multiple Characteristics at the same point in time: An Examination of MALPE and MAPE

Review of Economics & Finance Submitted on 15/04/2015 Article ID: 1923-7529-2015-03-01-14 David A. Swanson On the Relationship among Values of the Same Summary Measure of Error when it is used across Multiple Characteristics at the Same Point in Time: An Examination of MALPE and MAPE 1 Dr. David A. Swanson Department of Sociology, University of California Riverside Riverside, California 92521, U.S.A. Tel.: +1-951-827-4373 E-mail: dswanson@ucr.edu A bstract: This paper deals with an issue that appears to be unexplored by demographers and others who conduct cross-sectional forecasts of populations with multiple characteristics. The issue is based on the question: if one was conducting an ex post facto evaluation of a forecast or estimate that includes more than one characteristic (e.g., age, gender, race, and geography), how can one explain the fact that there are differences among the summary measures of error for all of the characteristics? Using a hypothetical demographic forecast as an illustration, the paper examines this issue for two characteristics (race and geography) using a standard summary measure of forecast errors for each of two error dimensions, bias and precision. For the bias dimension, we examine the “Mean Algebraic Percent Error” (MALPE) and “Mean Absolute Percent Error (MAPE) for the precision dimension. The paper finds that hitherto unknown relationships across characteristics exist for each of the two summary error measures. For example, in evaluating a forecast that includes more than one characteristic, one can express MAPE taken for one characteristic (e.g., race) in terms of each of the other characteristics (e.g., age and geography). This finding allows one to explain the reason for differences in the summary error measures used to evaluate a forecast or estimate done for more than one characteristic. The findings are informative and suggestive and are likely to be generalizable. While the discussion is not formally rigorous, a formal general proof is presented of the relationship between values of MALPE across multiple characteristics as well as the values of MAPE. Keywords: Forecasting; Errors; Summary measure reconciliation JEL Classifications: B4, C8, and Z1 1. The Problem Summary measures of error are part of the toolkit for those who evaluate forecasts and estimates. In the field of demography, the evaluations are often in the form of ex post facto assessments against census data (Chi and Voss, 2011; Chu, 1974; Coleman and Swanson, 2007; Long, 1995; Rayer, 2008; Shahidullah and Flotow, 2005; Smith and Sincich, 1992; Smith, Tayman, and Swanson, 2013: 324-328; Swanson, Tayman, and Barr, 2000; Tayman, 1996; Tayman and Swanson, 1999; Tayman, Swanson and Barr, 1999). The most commonly used summary measures of error are Mean Algebraic Percent Error – MALPE - and Mean Absolute Percent Error – MAPE This research was prompted by Dr. George Hough, the first person known by the author to ask how the differences among summary measures of forecast error taken across different characteristics of the same population could be explained.