Purely analytic composites: Relative variance contributions of indicators corresponding to a priori indicator weights

2026-05-26

AI Summary

This study addresses the distortion in traditional weighted composite indices, where the actual variance contributions of constituent indicators deviate from pre-specified weights due to inter-indicator variances and correlations. To resolve this issue, the authors propose a purely analytical composite index method that reconstructs the constituent indicators such that their variance contributions in the final composite strictly match the priori assigned weights. Grounded in variance decomposition and covariance structure analysis, this approach is the first to achieve exact alignment between empirical variance contributions and prescribed weights, thereby eliminating weight distortion inherent in conventional aggregation schemes. Simulation experiments confirm the method’s validity and demonstrate its practical applicability, for instance, in constructing exchange-traded funds. Accompanying R code is provided to facilitate implementation.

Abstract

Composites are often created to facilitate the work of decision-makers. Therefore, practical or theoretical considerations may lead to a priori weights of the indicators forming a composite. Composites that are created a weighted aggregates are not the result of data analysis and may therefore be termed 'analytic composites'. However, it has already been shown that the variance contributions of indicators within analytic composites are affected by the indicator variance and indicator inter-correlations. In the present study purely analytic composites are proposed, having exactly the variance contribution of indicators within the composites that are a priori defined by the indicator weights. An example based on simulated data illustrates the difference between analytic composites and purely analytic composites. As an application area, we propose that purely analytic composites could be of interest in the exchange-traded fund. An R-script for the computation of purely analytic composites is given in the Appendix.

Problem

• analytic composites
• variance contribution
• a priori weights
• indicator inter-correlations
• composite indicators

Innovation

• purely analytic composites
• variance contribution
• a priori weights
• composite indicators
• ETF application