Workshop on Price Index Compilation
Issues
February 23-27, 2015
Compilation of Upper-Level
Indexes
Gefinor Rotana Hotel, Beirut, Lebanon
Calculation of higher level indices
The decision on how to calculate the higher level
indices
Purpose(s)
Ideal/target index
Actual index
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Calculation of higher level indices
The purpose(s) of the index; inflation
measurement, indexation or escalation
The ideal/Target index – the best estimate of
what the index ideally should measure:
• Economic indices: Fisher, Walsh or Törnqvist
• Basket indices; like Laspeyres or Lowe
The actual index – the formula to use for the
ongoing calculation of the PPI, with constraints
on data, time and resources
3
The index number problem
Problem of decomposing a change in values into
a change in quantities and prices.
Need to define the value aggregate.
n
n
i 1
i 1
V 0   pi 0 qi 0 ; V 1   pi 1 qi 1 .
Decompose
V 1 /V 0  V 0,1  P 0,1  Q 0,1
4
Alternative formulae
Laspeyres
Paasche
Fisher
Mashall-Edgeworth
Lowe
Walsh
Young
Törnqvist
5
How do we know which is best?
Some approaches:
Fixed basket
Divisia
Axiomatic
Stochastic
Economic
6
The fixed basket approach and the usual
suspects
Laspeyres
n
n
PL 

i 1
n

i 1
p 1i q i0

1
0
0 0
(
p
/
p
)
p
 i i i qi
i 1
n
0 0
p
 i qi
p i0 q i0
n
  ( pi1 / pi0 ) wi0 ;
i 1
i 1
Paasche
n
PP 
 p1i q 1i
i 1
n

i 1


  p 1i p i0
p i0 q 1i  i 1
n

1 1 
si 

1
.
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But…
Laspeyres and Paasche are equally plausible and can give
different answers;
Need a single ‘best’ measure – a symmetric (basket) average of
the two;
But which average: arithmetic mean (Drobisch) or geometric
mean (Fisher)?
Tests:
Time reversal test – an index number formula should work both
ways.
Laspeyres and Paasche do not, neither does Drobisch, but Fisher’s
is the only homogenous symmetric mean of Laspeyres and
Paasche to satisfy the time reversal test.
8
Furthermore…..
The product test – the product of a price and
quantity index should be a value index:
0 ,1
0 ,1
V
0,1
 PP Q L
V
0,1
 PL QP
V
0,1
0,1
0,1
0 ,1
0 ,1
 PF Q F
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Symmetric averages:
The Three Target Indexes if Weights were Available
Fisher
Walsh index
 n t 0  n t t
  pi qi    pi qi 
   i=1

PF   i=1
n
n
 p0 q0   pt q0 

i i 
i i 
 

i=1
i=1
 

PW 

pit
qi0 qit
i
p
0
i
qi0 qit
i
 p 
PT   

p
i 1 

N
Tornqvist
t
i
t 1
i
 w  w  /2
0
i
t
i
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Annual weights from previous years
In practice we do not have current period
weights available.
Also, it takes time to compile weights so
weight reference period (base year) differs
from price reference period (month).
We end up with a formula that resembles a
Laspeyres index but it is not exactly
Laspeyres.
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Calculation of higher level indices
What is the problem? The weights refer to an earlier period
than the price reference period
b
Weight ref.
0
Price ref.
t
current period
A true Laspeyres index can not be calculated on monthly
basis!
There are two options:
• Price-update the weights – calculate a Lowe index
• Do not price-update the weights – calculate a Young
index
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The Lowe Index
The quantity weights are from a
previous period in the past and the
price base is a more current month
n
t b
p
 i qi
t
b b
0


p
p
q
p
b

0
b

0
PLo  in1
  wi  i0  , where wi  n i i  ib
pi
pi 
b b
0 b
i

p
q
 k k
 pi qi
i 1
n
k 1
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The Young index
The Lowe index is based on representative
quantities, Young is based on representative
revenue shares:
n
PY   w
i 1
b
i
p
t
i
p .
0
i
where
wib 
pib qib
n
b b
p
 k qk
k 1
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Calculation of higher level indices
Theory
I0Young
:t
Practical calculation
j
j
I 0:Young

w

I
 b 0:t
t
i


p
i
t
  w b  i 
 p0 
i
i
p

q
b
b
w ib 
i
i
p

q
 b b

j
I 0:GeoYoung

I
  0:t 
t

I 0Lowe
:t
i i
i


p
q
p

i
t b
t

  w b ( 0)  i 
i i
 p 0q b
 p0 
w ib ( 0)
w ib pi0 pib

 w ib pi0 pib



wbj
j
j
I0Lowe

w

I
 b(0) 0:t
:t

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Calculation of higher level indices
Lowe is a fixed basket index – and easy to
interpret
Young is a fixed weight index – and perhaps not
so easy to interpret
The focus in Lowe and Young are different
A better alternative could be the Geometric
Young
In practice, the difference is whether the weights
are price-updated from b to 0, or not
Are Wib or Wib(0) the better estimate of the
average revenue shares from 0 to t?
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Calculation of higher level indices
An example ….
Assume Walsh fixed basket index is the target index
0:t
IWalsh

t
0
t
p
q

q
 i i i
0
p
 i
wiW

t


p
W
i
  wi   0 
0
t
qi  qi
 pi 

( wi0  wit ) / pit pi0



( wi0  wit ) / pit pi0

Whether a Lowe or a Young index is the best estimate
depends on whether Wib or Wib(0) is the best estimate of
Wi W
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Calculation of higher level indices
Törnqvist as the target index
I 0:Tornqvist
t
I
GeoYoung
0:t
 pit 
  0 
 pi 
 w  w  /2
 pit 
  0 
 pi 
0
i
t
i
wib
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Calculation of higher level indices
Calculation of a chain linked index
• Update the weights and chain link at lest every 5
years; more frequent if there are rapid changes
in production patterns
• Introduction of new elementary aggregates and
new higher level indices
• Partial re-weighting
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Calculation of higher level indices
Alternatives to fixed weight indices
• Annual chaining
• Long-term and short-term links
• Lloyd-Moulton formula using elasticity of substitution
• Calculate retrospective superlative indices as weights
become available
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Thank You
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