Aggregate Properties of Two-Staged Price Indices
JensÂ Mehrhoff
Statistics Department and Research Centre, Deutsche Bundesbank, Frankfurt am Main, Germany

It is customary in official statistics, although often neglected in theoretical papers, for most price indices to be calculated in (at least) two stages. At the first stage (lower level), elementary indices are calculated on the basis of prices or their relatives, without having information on quantities or expenditures. At the second stage (upper level), the aggregate index is calculated on the basis of the elementary indices from the first stage, using aggregate expenditure share weights.

While the Laspeyres and Paasche price indices are exactly consistent in aggregation, superlative indices, such as the Fisher price index, are approximately consistent in aggregation. This means that the result of a two-staged index calculation coincides (approximately) with that of a calculation in a single stage. However, when statistical offices cannot use a quantity or expenditure-weighted formula at the first stage of the aggregation process, owing to the unavailability of this information, they have to rely on an unweighted index. Such an index might not reflect the characteristics of the index formula at the upper level. This elementary index bias is equally applicable to the Laspeyres and Paasche price indices as well as to the Fisher price index, irrespective of which unweighted index is used. A two-staged index with a non-according formula at the lower level can lead to a different conclusion than the true price index. This is due to the fact that the elementary indices may not even be close to the desired target index.

The present paper contributes to the literature by looking at the empirical evidence of calculations of Laspeyres price indices formed from different elementary indices. For three German statistics - index of producer prices for industrial products (domestic sales), index of import prices and index of export prices - disaggregate official data are analysed. Numerous elementary indices, in particular a continuous variation over a range of generalised means of price relatives, are systematically calculated and plugged into the aggregate formula. The generalised mean represents a whole class of unweighted elementary indices, such as the Carli and Jevons indices. The basic idea behind this approach is that different elementary indices implicitly weight price relatives differently, although they do not imply an explicit expenditure structure.

The results point to widely different estimates between Laspeyres price indices based on the alternative elementary indices. Thus, the importance of the lower level and the elementary index cannot be emphasised enough. Biases of these indices at this level are probably more severe than the pros and cons of the methodology at the upper level. The two-staged index can never be better than its building blocks. Hence, more attention should be paid to the characteristics of two-staged price indices.

Keywords: Consistency in Aggregation; Laspeyres Price Index; Elementary Indices; Generalised Mean

Biography: Dr Jens Mehrhoff is a researcher in the Statistics Department of the Deutsche Bundesbank working on price statistics and seasonal adjustment. His current position which he has held for more than four years is a research post in official statistics. He gave talks at several high-profile international conferences as well as to major international organisations. Further research of his includes the development of econometric and statistical methods.