Mr. Levell also writes: "While the Carli's failure to satisfy time reversibility is indeed a problem, it is important to realise that the index numbers into which these elementary aggregates eventually feed (the RPI and CPI) are themselves not time-reversible, and nor would they be even if the elementary aggregates were time-reversible." This is true but surely doesn't mean we should not try to apply SATIRE formulas (formulas which satisfy the time reversal property) at the elementary aggregate level, where we can do so without violating the convention against revision of official consumer price indexes. Most countries use a chain Lowe formula to calculate official consumer price series at the basic aggregation level and up, because it is compatible with a non-revision policy, even though the chain Lowe formula does not satisfy the time reversal policy. Nevertheless, the practice is not universal. Statistics Sweden calculates its official CPI as a chain Walsh index since 2005. The US has been publishing a chain Törnqvist index, the Chained Consumer Price Index for All Urban Consumers (C-CPI-U) since August 2002. Both formulas are SATIRE formulas, and both indexes have a 24-month revision period. Of course, it is an open question whether the benefits from eliminating the upper level substitution bias imposed by using a chain Lowe formula outweigh the costs of giving up the longstanding no-revision policy. But surely there is no similar argument against eliminating the lower level bias imposed by using a non-SATIRE formula like the Carli formula by replacing it with something more suitable, like the Jevons formula. If the convention of a non-revision policy prevents a national statistical institute from removing one source of bias, why should that prevent it from removing other sources of bias?
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