Transitivity of Price Indices

Transitivity of price indices is investigated and recommended, in particular in case of dynamic populations of goods. The general form of a transitive price index is presented. Various transitive price indices are discussed. Methods to adjust intransitive price indices to transitive ones are dealt with, in particular the cycle method.
The concept of transitivity of price indices is explored, in particular in case of a dynamic population of goods. A transitive price index is (by definition) free of chain drift. Many classical (bilateral) price indices, such as those named after Laspeyres, Paasche, Fisher, Törnqvist, are not transitive, however. Other bilateral indices are transitive only if the population of goods is static, which is a rather academic situation. Intransitive price indices can be made transitive, which is called transitivization. Some of these transitivization methods are discussed in more detail in the paper: GEKS, MST and the cycle method. This later method can be shown to be a generalization of the GEKS method, and is of more recent origin. The cycle method is discussed in some detail and various applications of the method are presented. Contrary to GEKS, the cycle method has parameters to control the transitivization of an intransitive price index. The cycle method uses constrained weighted linear regression to compute the transitivized version of the input index. It also allows to quantify the extent to which an index is intransitive ('degree of intransitivity'). The cycle method is best applied to numerical examples. Symbolic expressions can be computed, but tend to be fairly complicated. In matrix form, however, these symbolic computations are straightforward and insightful. The same is true for GEKS, and probably for any other multilateral price index method. The cycle method can also by applied in an incremental approach, where every month the set of known indices is augmented with the newest results, which preserve the transitivity of the price index involved.