Construction of an Index by Maximization of the Sum of its Absolute Correlation Coefficients with the Constituent Variables

7 Pages Posted: 25 May 2007

Date Written: May 26, 2007

Abstract

On many occasions we need to construct an index that represents a number of variables. Cost of living index, general price index, human development index, index of level of development, etc are some of the indices that are constructed by a weighted (linear) aggregation of a host of variables. The criterion on which importance of a variable (vis-à-vis other variables) is determined may be varied. In constructing a cost of living index, for instance, importance of a commodity is determined by the proportion of consumption expenditure allocated to it, and in constructing the human development index the variables such as literacy, life expectancy or income are weighted according to the importance assigned to them in accordance with their perceived roles in determining human development status.

In absence of any preferred means or logic to determine the relative importance of different variables, weights are assigned mathematically. One of the methods to determine such mathematical weights is the Principal Components (PC) analysis where weights are determined such that the sum of the squared correlation coefficients of the index with the constituent variables (used to construct the index) is maximized.

Although the PC analysis has excellent mathematical properties, one may face some difficulties in using it to construct a single index of poorly correlated variables. The method has a tendency to pick up the subset of highly correlated variables to make the first component, assign marginal weights to relatively poorly correlated subset of variables and/or relegate the latter subset to construction of the subsequent principal components. Consequently, the index obtained by PC analysis is elitist in nature that has a preference to the highly correlated subset over the poorly correlated subset of variables. Further, since there is no dependable method available to obtain a composite index by merging two or more principal components, the deferred set of variables may never find its representation in the index.

In this paper we propose to construct indices either (i) by maximization of the minimal correlation (I-M) or (ii) by maximization of the sum of absolute correlation (I-1) between the index and its constituent variables. Maximization has been done by the Differential Evolution method of global optimization. We find that I-1 is more inclusive, and has a tendency to represent even the poorly correlated variables. The I-M indices are egalitarian in nature. It would depend on the analyst whether he is interested in egalitarian, inclusive or elitist method of constructing indices when the constituent variables are not very highly correlated among themselves. This paper has opened up the option to choose the method of constructing a desired type of index.

Keywords: Index, construction, weighted linear aggregation, principal components, elitist, egalitarian, inclusive, maximin correlation index, sum of absolute correlation coefficients, Differential evolution

JEL Classification: C43, C61

Suggested Citation

Mishra, Sudhanshu K., Construction of an Index by Maximization of the Sum of its Absolute Correlation Coefficients with the Constituent Variables (May 26, 2007). Available at SSRN: https://ssrn.com/abstract=989088 or http://dx.doi.org/10.2139/ssrn.989088

Sudhanshu K. Mishra (Contact Author)

North-Eastern Hill University (NEHU) ( email )

NEHU Campus
Shillong, 793022
India
03642550102 (Phone)

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