Transcript Carli - METAC

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)
0 t
I Las
t 0
p
 i i qi
0 
 q0

 q0


p
t
t
i
i
i

 i 
pi    i 
pi  0  
0 0
0 0
0 0
  j p j q j 
  j p j q j  pi  
 j pjqj
t
 p0 q0  pt 



p
0
i i
i
i
   i  si   0  
 i 
0 0 
0 
  j p j q j  pi  
 pi  

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
 i p 


t
n
p



i i


0
0
  j pj   j pj


 n 


t
i
I
0 t
Dutot
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
I
0:t
Carli
p 
1
 i  
n
p 
t
i
0
i
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
1n
I
0:t
Jevons
 pit 
 i  0 
 pi 
p



 p 
i
t 1n
i
i
0 1n
i
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.
I
0t
Dutot
t
p
i i
0 
 1

 1
 p0  pt 
t
t  pi 
i
i


 i 
pi    i 
pi  0     i 
0
0
0
0 
0 
  j p j 
  j p j  pi  
  j p j  pi  
 j pj
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
I
0t
Dutot
Carli
 p 0  pt  
p

i
i
i






0
0
0 
i
  j p j  pi  
 j pj
t
i
each price relative
weighted proportionally to
its base period price
I
0:t
C
1 p 
  
n p 
each price relative
weighted equally
t
i
0
i
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
t
0
t
0
X(P , lP ) = l X(P , P )
Change in Units
t
0
t
0
X(P , P ) = X(AhP , AhP )
Time Reversal
t
0
0
t
X(P , P ) = 1/ X(P , P )
Transitivity
t
t-2
t
t-1
t-1
t-2
X(P , P ) = X(P , P )* X(P , P )
Time Reversal Test
t
0
0
t
X(P , P ) = 1/ X(P , P )
Carli fails—it has an upward bias.
Multi-period Carli can produce absurd results
Carli is not recommended
Period 0
Period t
relative
Inverse Relative
Price A
1
2
2
0.5
Price B
1
1
1
1
Carli
1
1.5
1/Inverse Carli
1
1.333333333
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)
t
t-2
t
t-1
t-1
t-2
X(P , P ) = X(P , P ) * X(P , P )
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”)

1 1
  pm qm
 m 1M

1
  qm
 m 1
M







0 0 
  pm qm 
 m 1M



0
  qm 
 m 1

M
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