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.