On Construction of Robust Composite Indices by Linear Aggregation

weights to different indices, there are two approaches: the first in which the weights are determined on the basis of some information or considerations exogenous to the data on index variables ( ), X xkj ∈ and the second in which the weights are endogenously determined such that ). (X f w = The most robust composite index is the one that is exogenously determined since in that case w is used as a parameter and, therefore, ). | ( w X I φ = However, when weights are endogenously determined, we have )). ( , ( X f X I φ = In this latter case, the composite index, , I depends not only on the index variables, , X but also on the specification of the function, (.), f that obtains weights, , w from . X To make this point clearer, let kj x be perturbed such that kj kj kj x x ∂ + ⇐ where . 0 ≠ ∂ kj If weights are not derived from , X then kj j k k w I I ∂ + ⇐ and . i k i k I I ≠ ≠ ⇐ That is, the perturbation affects k I only. However, if weights are derived from , X a perturbation of one of the values of kj x would in most cases alter the values of w as well as the values of . , 1 n k I k = ∀ A perturbation of kj x will pervade throughout even though all of : ip x i k p j ≠ ∧ ≠ have remained unchanged. The extent of pervasiveness, which is not a desirable property of the composite index, would depend on the specification of ). (X f w = There is an additional point to be noted. When , ) | ( Xw w X I = = φ the weight, j w , which may be viewed as j x I ∂ ∂ is constant and hence I is indeed a linear combination of . X However, when )), ( , ( X f X I φ = the weight, , j w in general , is not constant and, therefore, I is not a linear combination of . X In that case, these weights, which may also be viewed as the rate of substitution among different constituent indices, lose interpretability in any simple manner and hence go far off the desirable property of easy comprehensibility.