The Differential Approach to Superlative Index Number Theory

Article excerpt

Diewert's "superlative" index numbers, defined to be exact for second-order aggregator functions, unify index number theory with aggregation theory but have been difficult to identify. We present a new approach to finding elements of this class. This new approach, related to that advocated by Henri Theil, transforms candidate index numbers into growth rate form and explores convergence rates to the Divisia index. Because the Divisia index in continuous time is exact for any aggregator function, any discrete time index number that converges to the Divisia index and that has a third-order remainder term is superlative.

Key Words: Divisia, index numbers, superlative indexes

According to Theil (1960, p. 464), "The subject of index numbers is one of the oldest in statistics and also, as regards the more specialized subject of cost of living index numbers, an old one in economics." Although an old subject, economists have long struggled to identify useful index numbers. The most influential selection criterion is that the index number be exact for an aggregator function that can produce a second order approximation to any twice continuously differentiable linearly homogeneous function. Diewert (1976) defined such index numbers to be "superlative." Superlative index numbers thus have known tracking ability relative to the exact aggregator functions of economic aggregation theory.

The class of superlative index numbers contains an infinite number of index numbers, since an infinite number of second-order aggregator functions exist, but only a small number of index numbers in the superlative class have so far been found. The search process has previously involved finding an index number that is exact for a known second-order algebraic aggregator function or searching for a second-order aggregator function for which a known index number is exact. No simple procedure has been found for either direction. For example, the minflex Laurent aggregator function, originated by Barnett and Lee over 15 years ago, is known to be second order, but no one has succeeded in finding the index number that can track it exactly. In the other direction, Fisher proposed many index numbers in his famous book, but to the present day, the aggregator functions tracked exactly by them remain unknown for most of those index numbers.

The Divisia continuous time index holds a prominent place in the literature because Newman and Ville, Hulten, Samuelson and Swamy, and Barnett and Serletis (p. 101-2) have shown that the Divisia line integral produces the unique exact index number formula for any neoclassical aggregator function. Similarly, the Divisia price index is the unique exact index number formula in continuous time for the neoclassical aggregator function's dual unit cost function. These results imply that the Divisia integral index is the prototype economic index number. For general use, however, the Divisia continuous time index must be adapted to apply to discrete data. A log-change form index is usual for all well-known discrete time approximations to the Divisia index. This observation alone is not sufficient to determine the weight function. Thus, large numbers of potential finite change approximations to the Divisia index have been published. Each is in log change form, and they are differentiated by their weights.

We show that Theil's differential approach, which he used to support the Tornqvist index (Theil 1973), can be used systematically to determine which finite change approximations to the Divisia index are superlative. Because the Divisia line integral in continuous time is exact for any aggregator function, any superlative discrete time index number must (1) converge to the Divisia index as the time intervals narrow and (2) have a third-order remainder term for finite-change time intervals. This is true regardless of whether or not we are capable of finding the second-order aggregator function for which the discrete time index number is exact in discrete time. …