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U.S. DEPARTMENT OF LABOR Bureau of Labor Statistics OFFICE OF PRICES AND LIVING CONDITIONS Random Errors and Superlative Indexes

This paper demonstrates that random sampling error in CPI price and expenditure data at the item-area level can have a distorting effect on empirical cost-of-living indexes computed using those data. In particular, the expected values of the Fisher Ideal and Tornqvist indexes can be distorted downward by random error in basic index relatives. This, in turn, can cause the estimated Laspeyres substitution bias in the CPI to be overestimated. The issue is illustrated empirically using CPI data for the period 1987 through 1995. Estimated substitution bias is sharply higher in each year when smaller CPI “replicate” samples are used to compute indexes than when the full samples—which are subject to less sampling error—are employed. To address this problem, the paper derives and applies a composite-estimation approach, in which CPI item-area indexes are replaced by a weighted average of those indexes and the U.S.-level item indexes. This approach causes the estimated superlative index values to be higher, and the estimated substitution bias consequently lower. For example, in a comparison of annual chain Laspeyres indexes to chain Tornqvist indexes the adjusted estimate of substitution bias is about 0.08 percentage point rather than the roughly 0.12 percentage point previously estimated.