The Importance of Variables in Composite Indices: A Contribution to the Methodology and Application to Development Indices

Abstract

The paper examines the issue of weights and importance in composite indices of development. Building a composite index involves several steps, one of them being the weighting of variables. The nominal weight assigned to a variable often differs from the degree to which the variable affects the scores of the overall index. The newly suggested notion of importance is based on the idea that an important indicator, if omitted from the index, causes large changes in countries’ results. We propose a method of measuring the importance and apply it to inequality variables in composite indices of development. The results show a low importance for most inequality variables, and for some of them, a large discrepancy between the nominal weights and the importance. We argue that the importance of variables should be considered in the process of index construction. This may imply a modification of the index when there is a large discrepancy between the nominal weights and importance and when the importance of some variables is extremely low. Whether any such modification is justified must be decided within the context of the particular index.

Appendix

See Figs. 1, 2, 3, 4 and 5.

1.1 Derivation of Formula (3)

Let us show that the correlation coefficient $r_{k}$ used in the definition of measure of absolute importance (MAI) of the k-th indicator can be expressed in form (3).

$$ \begin{aligned} r_{k} & = corr\left( {OCI,MCI_{k} } \right) = corr\left( {OCI,OCI - w_{k} x_{k} } \right) \\ & = \frac{{cov\left( {OCI,OCI - w_{k} x_{k} } \right)}}{{\sqrt {var\left( {OCI} \right)} \sqrt {var\left( {OCI - w_{k} x_{k} } \right)} }} = \frac{{var\left( {OCI} \right) - w_{k} cov\left( {OCI,x_{k} } \right)}}{{\sqrt {var\left( {OCI} \right)} \sqrt {var\left( {OCI} \right) - 2w_{k} cov\left( {OCI,x_{k} } \right) + w_{k}^{2} var\left( {x_{k} } \right)} }}. \\ \end{aligned} $$

The equalities above are based on the definition of correlation coefficient and properties of variance and covariance of a sum.

Now, we express $cov\left( {OCI,x_{k} } \right)$ by means of correlation coefficient $\rho_{k} = corr\left( {x_{k} ,OCI} \right)$.

Since $\rho_{k} = \frac{{cov\left( {OCI,x_{k} } \right)}}{{SD\left( {OCI} \right)SD\left( {x_{k} } \right)}}$, it holds $cov\left( {OCI,x_{k} } \right) = \rho_{k} SD\left( {OCI} \right)SD\left( {x_{k} } \right)$.

Using this expression, the formula for $r_{k}$ can be rewritten in the following form:

$$ \begin{aligned} r_{k} & = \frac{{var\left( {OCI} \right) - w_{k} \rho_{k} SD\left( {OCI} \right)SD\left( {x_{k} } \right)}}{{\sqrt {var\left( {OCI} \right)} \sqrt {var\left( {OCI} \right) - 2w_{k} \rho_{k} SD\left( {OCI} \right)SD\left( {x_{k} } \right) + w_{k}^{2} var\left( {x_{k} } \right)} }} \\ & = \frac{{var\left( {OCI} \right) - w_{k} \rho_{k} SD\left( {OCI} \right)SD\left( {x_{k} } \right)}}{{\sqrt {var\left( {OCI} \right)} \sqrt {var\left( {OCI} \right)\left[ {1 - 2w_{k} \rho_{k} SD\left( {x_{k} } \right)/SD\left( {OCI} \right) + w_{k}^{2} var\left( {x_{k} } \right)/var\left( {OCI} \right)} \right]} }} \\ \end{aligned} $$

The final step consists in dividing both nominator and denominator by $var\left( {OCI} \right) = SD\left( {OCI} \right)^{2}$. Denoting $RV_{k} = SD\left( {x_{k} } \right)/SD\left( {OCI} \right)$, it immediately follows that

$$ r_{k} = \frac{{1 - w_{k} \rho_{k} RV_{k} }}{{\sqrt {1 - 2w_{k} \rho_{k} RV_{k} + (w_{k} RV_{k} )^{2} } }}. $$

1.2 Derivation of Formula (4)

In the case of uncorrelated indicators $x_{1} , \ldots ,x_{p}$, it holds

$$ var\left( {\mathop \sum \limits_{i} w_{i} x_{i} } \right) = \mathop \sum \limits_{i} w_{i}^{2} var(x_{i} ),\quad cov\left( {x_{k} ,\mathop \sum \limits_{i} w_{i} x_{i} } \right) = w_{k} var\left( {x_{k} } \right), $$

and therefore

$$ \rho_{k} = \frac{{cov\left( {\mathop \sum \nolimits_{i} w_{i} x_{i} ,x_{k} } \right)}}{{SD\left( {OCI} \right)SD\left( {x_{k} } \right)}} = w_{k} \frac{{SD\left( {x_{k} } \right)}}{{SD\left( {OCI} \right)}}, $$

By substituting this term into expression (3), one obtain the following equality

$$ r_{k} = \frac{{1 - \frac{{\left( {w_{k} SD\left( {x_{k} } \right)} \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{p} \left( {w_{i} SD\left( {x_{i} } \right)} \right)^{2} }}}}{{\sqrt {1 - \frac{{\left( {w_{k} SD\left( {x_{k} } \right)} \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{p} \left( {w_{i} SD\left( {x_{i} } \right)} \right)^{2} }}} }} = \sqrt {1 - \frac{{\left( {w_{k} SD\left( {x_{k} } \right)} \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{p} \left( {w_{i} SD\left( {x_{i} } \right)} \right)^{2} }}} $$