Wednesday, October 20, 2010

Index Numbers and Consistency in Aggregation: Part II

This post gets even more technical than the last. I'm just blogging about what I'm reading in the course of my research. I read a whole bunch more papers on index numbers, which got more and more technical. The bottom line is that for most applications the chain Fisher index is an appropriate index to use.

An index is superlative if it is exact for an aggregator function (e.g. a production function) that can provide a second order differential approximation to an arbitrary twice differentiable linear homogenous function. A second order differential approximation is one where the level, first derivative, and second derivatives of the two functions are equal at the point of approximation.

Diewert (1978) shows that the Vartia I index differentially approximates a superlative index as long as both prices and quantities are strictly non-zero and there is actually no price and quantity change between the periods. What this means is that for relatively small changes in prices and quantities the Vartia I index will give very similar results to superlative indices like the Törnquis index and the Fisher index.

The nature of superlative indices themselves means that “chained” indices are always preferable to non-chained indices. A chain index is one where the index is computed for each year (or whatever is the smallest available gap between datapoints) and the product of those annual indices is used as the time series of the index over time even if we only want the change over a much longer period.

Diewert (1978) goes on to show that chained superlative indices will yield close to consistent aggregation for relatively small changes in prices and quantities. Diewert (1978) also shows in an empirical appendix that chained Vartia, Törnquist, and Fisher indices produce almost identical results and that two stage aggregation produces almost the same results as one step aggregation for Törnquist and Fisher.

An additional advantage of the Fisher index over logarithmic indices such the discrete Divisia index also known as the Törnquist index is that it can easily handle the introduction of new goods, as zero values pose no problem for it. One way to deal with this for the Törnquist or Vartia I indices is to compute the price index, assuming that the price of a new input was the same before its introduction as in the year of its introduction and then find the quantity index as the ratio of total value to the price index.

Diewert, W. E. (1978) Superlative index numbers and consistency in aggregation, Econometrica 46(4): 883-900.

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