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Some Observations on Price Index Estimators

Alan H. Dorfman, Sylvia G. Leaver, and Janice Lent


Price indexes can be divided into two broad classes, the superlative indexes, and the non-superlative indexes. Superlative indexes in theory approximate a true cost-of-living index (which it is impossible to directly calculate), and tend in practice to be very close to each other, so that anyone of them can be taken as representing the class. Non-superlative indexes (which are the ones actually used in practice, because of the timeliness with which their sample based estimates can be produced) deviate in theory from the cost of living index, and in practice from the superlative indexes. Particular forms of-non-superlative indexes tend to lie above the superlative indexes, others below. The (positive or negative) gap between a non-superlative index and the cost of living index [has been characterized as] is its substitution bias or substitution effect. Implicit is the notion that the indexes are population indexes, representing the totality of transactions of a given sort in a given economy. It seems natural to estimate the magnitude of the substitution effect of a given index by retrospectively measuring the distance between it and a superlative index. However, such measurements are necessarily made on sample estimates of the corresponding indexes. We show that the relationships among sample-based indexes, and between them and population indexes, are not necessarily a straightforward reflection of the relation between population indexes. Thus estimates of the direction and magnitude of the substitution effect are more imprecise than has previously been supposed.