Workshop on Price Statistics Compilation Issues February 23-27, 2015 Compilation of Elementary Indices Gefinor Rotana Hotel, Beirut, Lebanon

Lecture Outline Overview Introduction Average of relatives versus relative of averages Arithmetic mean versus geometric mean Homogeneity of items Recommendations Particular circumstances

Introduction: Idealized World Laspeyres formula is equivalent to a weighted arithmetic average of price relatives (ratios) The weights are the base-period expenditure shares The prices, quantities and expenditure shares are for clearly-defined goods and services

Real World Huge number of transactions Must select a small subset No Transaction-level weights in CPI (only higher-level weights) Laspeyres concept: only at the higher level Unweighted averages: Within item cat’s Aggregating individual prices within item cat’s (The first step of index compilation) Without weights: an approximation to Laspeyres

Unweighted Index Formulas Carli:Average of Price Relatives (AR) Dutot:Ratio of Average Prices (RA) Jevons: Geometric Average (GA) All use “Matched Model” : same item varieties in 2 periods

Dutot Index(RA) Ratio of averages Arithmetic averages of the same set of varieties In period t, the current period In base period o, the base period

Carli Index (AR) Average of price relatives Unweighted arithmetic average of Long-term price relatives price in current period ( t, ) / price in base period ( o ) For the same (matching) set of items

Jevons Index (GA) Geometric average of price relatives Unweighted geometric average of the long-term price relatives price in current period ( t, ) / price in base period ( o ) For the same (matching) set of varieties. Note : geometric average of price relatives = = ratio of geometric averages of prices

Dutot, Carli, or Jevons Index Differ due to: the types of average Avg. prices vs. price relatives arithmetic vs. geometric the price dispersion the more heterogeneous the price changes within an item, the greater are the differences between the different types of formulas.

Elementary indices for an Item containing two varieties

Arithmetic Mean: Dutot vs. Carli Dutot 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. Carli weights each price relative equally different varieties’ price changes are equally representative of price trends of the item and gives each the same weight.

Dutot vs. Carli Dutot Carli each price relative weighted proportionally to its base period price each price relative weighted equally

Arithmetic Mean: Dutot vs. Carli Dutot and Carli are equal only if all base-period prices are equal, or all price relatives are equal (prices of all varieties have changed in the same proportion). If all price relatives are equal, every formula gives the same answer If the base prices of the different varieties are all equal the items may be perfectly homogenous What about different sizes? Example:Prices of Orange juice, 2 liter bottles, ½ liter bottles

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.

Desirable Properties for Index Formulas Axioms Proportionality X(P t, P 0 ) =  X(P t, P 0 ) Change in Units X(P t, P 0 ) =  X(A  P t, A  P 0 ) Time Reversal X(P t, P 0 ) = 1/ X(P 0, P t ) Transitivity X(P t, P t-2 ) = X(P t, P t-1 )* X(P t-1, P t-2 )

Time Reversal Test X(P t, P 0 ) = 1/ X(P 0, P t ) Carli fails—it has an upward bias. Multi-period Carli can produce absurd results Carli is not recommended

Units of Measurement Test Dutot fails: Different results if price is in kilos rather pounds. The weight given to a price relative is proportional to its price in the base period. QA’s to the base-period price affect the weights. Dutot is only recommended for tightly specified items whose base prices are similar.

Geometric Mean (Jevons) Index Average of relatives = ratio of averages Circular (multi-period transitivity) X(P t, P t-2 ) = X(P t, P t-1 ) * X(P t-1, P t-2 ) Incorporates substitution effects if sampling is probability proportionate to base period expenditure unity elasticity Sensitive to extreme price changes

Arguments against Geometric Mean not easily interpretable in economic terms (particularly for the producer price index) not as familiar as the arithmetic mean relatively complicated Not as transparent Inconsistent: use for elementary aggregates with use of the arithmetic mean at higher levels of aggregation (product groups and total index) But does not fail critical tests consistent with geometric Young

Homogeneity of Items An item is homogeneous if its transactions: (1) have the same characteristics and fulfill similar functions, and (2) have similar prices (or change in prices)

Homogeneity of Items How can homogeneity be achieved in practice? Define items at a very detailed level: could lead to lack of flexibility in the index classification and lead to items which would not have reliable aggregation weights. Reduce the number of varieties within items selecting a fewer number of varieties (select tennis balls as representative of sport items) Difficulty with customs data and unit values as surrogates for price relatives

Unit Values Indices Are they price indices? Common with electronic (point of sale) data (“scanner data”)

Example The unit value index is 4.6/1.7=2.71. Is this right?

Recommendations Select homogeneous items/products To reduce discrepancies between elementary level compilation methods. Don’t use Carli Use Dutot to calculate indices at the elementary aggregate level only for homogeneous products. Use Jevons to compile elementary indices. If data on weights are available, use them.

Thank you