Other Practical Measurement Issues: Aggregation at the Lower and Upper Level

1. Other Practical Measurement Issues: Aggregation at the Lower and Upper Level Joint SEACEN-IFC-BIS workshop on Inflation measurement: central bank perspectives 12-15 October 2009. Mick Silver The views expressed herein are those of the author and should not be attributed to the IMF, its Executive Board, or its management.

2. Aggregation • Formulas: • elementary level and • higher level • Revision of baskets and chaining

3. Choice of elementary aggregate formula • At basic level: building block of CPI, unweighted across (and within) outlets of well-specified variety. • Three main alternatives used in practice • Carli: Average of Price Relatives • Dutot: Ratio of Average Prices • Jevons: Geometric Average

4. DutotIndex • The ratio of average (arithmetic) prices • for a set of varieties in the current period • to the average price of the same (matching) set of transactions in the base period.

5. CarliIndex • The arithmetic average of price relatives • Unweighted average of the long-term price relatives (current /base period price) • For the same (matching) set of transactions.

6. JevonsIndex • The geometric average of price relatives • Unweighted average of the long-term price relatives (current /base period price) • For the same (matching) set of varieties. • = ratio of geometric avg. prices in current period to geo. Avg. prices in base period.

7. Example:

8. Carli • Carli fails time reversal test—it has an upward bias. • Carli can produce absurd results: If in example 10 to 12 and 12 to 10; Carli fall of 1.67 per cent. • Carli is not recommended for unweighted indexes. • Chained Carli can drift badly

9. Dutot • Dutot implicitly weights each price relative proportionally to its base period price • High weight to expensive varieties’ price changes even if they represent only a low share of total base year expenditures. • Only for homogeneous items: fails commensurability test—Different results arise if an item’s price is in kilos rather than pounds.

10. Jevons • Jevons passes all reasonable tests. • More difficult to explain. • If elasticities of substitution are unity and sampling is with probability proportionate to reference period’s expenditure then included substitution effect • Lower than Carli. • approved by the Statistical Office of the European Communities (Eurostat) for use in those countries’ Harmonized Indexes of Consumer Prices (HICP); used by 20 of 30 countries as a primary formula for computing the elementary indexes in their HICP.

11. Arithmetic Mean: Dutot Vs. Carli • Dutot and Carli are equal if and only if(1) the base prices of all varieties are equal, or (2) the price relatives for all varieties within an item are equal (prices of all varieties have changed in the same proportion). • Assuming that changes in the varieties’ price are equal is absurd as the objective is precisely to measure these changes. • On the other hand, assuming that the base prices of the different varieties are all equal requires the selection of perfectly homogenous items (very low dispersion of base prices).

12. Dutot and Jevons • There is an established relationship between Dutot and Jevons. Jevons is equal to Dutot times the (exponent of the) difference between the variance of (log) prices in the current period and the reference period. • If the variance of prices does not change they will be the same

13. United States • Change to Geometric mean January 1999 for 61 percent of expenditure. Shelter, utilities, medical remain arithmetic mean. • Aimed at addressing Boskin Commission concerns about substitution bias, along with increased frequency of reweighting. • Experimental indexes show that the geometric mean led to an overall decrease in CPI growth of about 0.28 percentage point per year over the period from December 1999 to December 2004.

14. Higher level formulas • Laspeyres • Paasche • Fisher • Törnqvist • Young (geometric Young) • Lowe • Chaining

15. Price index number formulas: Laspeyres price indexas a weighted arithmetic mean of price relatives where

16. Paasche price index as a weighted relative harmonic mean where

17. Fisher and some other symmetric index number formulas • Fisher ideal price index - geometric average of the Laspeyres and Paasche indices. passes factor reversal and time reversal. • Törnqvist price index - geometric average of the price relatives weighted by the average expenditure shares in the two periods. passes time reversal fails factor reversal. where

18. Laspeyres, Paasche, Fisher or Törnqvist? • Fixed basket: Laspeyres and Paasche hold quantities constant in a fixed period – both are equally justifiable, but generally yield different results. Fisher best average. • Axiomatic tests: both Laspeyres and Paasche fail the: • Time Reversal Test - the same result should be obtained whether the change is measured forwards in time, i.e., from 0 to t, or backwards in time from t to 0. • Factor Reversal Test - the product of the quantity index and the price index should be identical with the change in the value of the aggregate in question.However, Laspeyres and Paasche satisfy the additivity test. Fisher passes both. • Economic theory: both Laspeyres and Paasche assume no substitution behaviour – superlative formulas best.

19. Which formula? • But superlative formulas using symmetric price and quantity information are not feasible in real time. • So often “Laspeyres” used.

20. When is a Laspeyres not a Laspeyres? • Laspeyres requires the price reference period to be the same as the weight reference period • In practice such weights are not available. For example, for a CPI the expenditure survey for households may be conducted over June-July 2004, but it may take over 6 months to compile the weights for use in the index. Thus it will be in January 2005 that weights are first used even for an annually chained CPI. The Laspeyres price in the current period may be for May 2005, but the base period price would be on average January 2004. • The old weights may be price updated: a Lowe index, or may not be price updated, a Young index.

21. Laspeyres vs. Young Index • If we did not update the weights for price change during the rebasing, we would have a Young index • The Young index formula is: • Geometric Young

22. Laspeyres vs. Lowe Index • The Lowe index formula is:

23. Chaining • So far the discussion has been of direct comparisons – between periods 0 and t. We can compare periods 0 and t by comparing period 0 with 1, then 1 with 2,…..,then t-1 with t. The resulting indices or links between each period and its successive one can then multiplied to form a chain index. Any index number formula can be used for the links. • When a fixed base Laspeyres is used over a long period, the weights become progressively out of date and irrelevant. The base period has to be updated and the new index linked to the old. Chaining is simply the limiting case in which the weights are updated each period.

24. Chaining • A chain index between periods 0 and t is “path dependant”- it depends not just on the prices and quantities in 0 and t, but also on the prices and quantities in all the intervening places. • If the path is fairly smooth then the additional price and quantity information will tend to lead to a better measure of the overall change. It will tend to reduce spread between Laspeyres and Paasche. If not smooth, bouncing leads to drift. If the prices in 0 are the same as those in t, a chain Laspeyres index may exceed 100. • Rebasing is but a constrained form of chaining. Also opportunity to revise sample and methods.

25. CPI-U – United States • Aggregated from item-area indexes: 8,018 basic indexes weighted using Consumer Expenditure Survey data • Most basic indexes calculated using weighted geometric mean formula • Upper level uses Lowe index form • Biennial weight revisions since 2002 • 2-year weight reference periods • Prior to 2002, the expenditure base period was updated approximately every 10 years. • Base period has been 2005-2006 since January 2008

26. C-CPI-U • Uses same basic indexes as CPI-U • Series introduced in July 2002; published series go back to January 2000 • Monthly-chained Törnqvist index subject to two revisions • Preliminary indexes use geometric mean • Data are final through December 2007 • 2009 data become final in February 2011

27. Greenlees and Willams, 2009 undertook extensive simulations on alternative formulas • Used US’s continuous household budget survey and retrospective data. • Substitution effect: difference between CPI-U and C-CPI-U and annual chained Fisher: e.g. for 2007 annual change: 2.81, 2.50, and 2.53 respectively. • Estimates for annual rates for 2000 and 2001 using old weights from 1993-4 were: 3.33 and 2.81 compared with chained Laspeyres, 3.02 and 2.41 and chained Fisher: 2.45 and 2.29.

28. Greenlees and Williams • Price up-dating has little effect: difference between Lowe (CPI-U) (annual percentage change 2.50) and Young (2.42) for 2001-07. • Using chained annual Lowe vs biannual CPI-U (Lowe) effect is small – 2.50 and 2.42 for 2001-07. • Lowe 3 month weights with a 3 month lag compared with 6 months weight with 6 month lag – approach C-CPI-U as in Figure.