Workshop on Price Statistics Compilation Issues

February 23-27, 2015

Compilation of Elementary Indices

Gefinor Rotana Hotel, Beirut, Lebanon

Lecture Outline

OverviewIntroductionAverage of relatives versus relative of averagesArithmetic mean versus geometric meanHomogeneity of itemsRecommendationsParticular circumstances

Introduction: Idealized World

Laspeyres formula is equivalent toa weighted arithmetic average of price relatives (ratios)

The weights are the base-period expenditure sharesThe prices, quantities and expenditure shares are for clearly-defined goods and services

0 0 0 00

0 0 0 0 0 0 0

0 00

0 0 0 0

ti it t ti i i i

Las i ii ij j j j j j ij j j

t ti i i i

ii ij j i ij

p q q q pI p p

p q p q p q p

p q p ps

p q p p

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 levelUnweighted 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

000

tii

tit i

Dutotjj jj

p

n pI

pp

n

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

00

1 t:t i

Carli ii

pI

n p

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

11

010 0

nn tti:t ii

Jevons nii ii

ppI

p p

Dutot, Carli, or Jevons Index

Differ due to:the types of average

Avg. prices vs. price relativesarithmetic vs. geometric

the price dispersionthe 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

Ratio of Average Prices Average of Price Relatives

t = 0 t = 1 Price Relative

p1 10.00 12.00 1.20

p2 20.00 30.00 1.50

==== ====

Arithmetic mean 15.00 21.00 Geometric mean 14.14 18.97

==== ======

Dutot (Arithmetic mean) 140.00 Carli (Arithmetic mean) 135.00 Jevons (Geometric mean) 134.16 Jevons (Geometric mean) 134.16

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 equallydifferent varieties’ price changes are equally representative of price trends of the item and gives each the same weight.

0 00

0 0 0 0 0 0

1 1t tit t ti i i i

Dutot i ii i ij j j i j ij j j j

p p p pI p p

p p p p p p

Dutot vs. Carli

Dutot Carli

00

0 0 0

t tit i i i

Dutot ij j ij j

p p pI

p p p

0

0

1 t:t i

Ci

pI

n p

each price relative weighted proportionally to its base period price

each price relative weighted equally

Arithmetic Mean: Dutot vs. CarliDutot and Carli are equal only if

all base-period prices are equal, orall 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

AxiomsProportionality

X(Pt, lP0) = l X(Pt, P0)

Change in Units X(Pt, P0) = X(AhPt, AhP0)

Time ReversalX(Pt, P0) = 1/ X(P0, Pt)

TransitivityX(Pt, Pt-2) = X(Pt, Pt-1)* X(Pt-1, Pt-2)

Time Reversal Test

X(Pt, P0) = 1/ X(P0, Pt)

Carli fails—it has an upward bias.Multi-period Carli can produce absurd resultsCarli is not recommended

Price A Price B Carli 1/Inverse CarliPeriod 0 1 1 1 1Period t 2 1 1.5 1.333333333relative 2 1

Inverse Relative 0.5 1

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 averagesCircular (multi-period transitivity)

X(Pt, Pt-2) = X(Pt, Pt-1) * X(Pt-1, Pt-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 meanrelatively 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 testsconsistent 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”)

1 1 0 0

1 1

1 0

1 1

M M

m m m mm m

M M

m mm m

p q p q

q q

Example

Size of refrigerator

Small Medium Large All sizes

Period q p v q p v q p v q p v

Now 2 2 4 3 4 12 5 6 30 10 4.6 46

Then 5 1 5 3 2 6 2 3 6 10 1.7 17

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

Recommendations

Select homogeneous items/productsTo 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 youThank you