[PPT] - Retrospective Price Indices and Substitution Bias Retrospective PowerPoint Presentation

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Retrospective Price Indices and Substitution Bias Retrospective Price Indices and Substitution Bias

W. Erwin Diewert

U i i f B i i h C l bi V University of British Columbia, Vancouver

Marco Huwiler

Clariden Leu, Zurich

Ulrich Kohli

Swiss National Bank, Zurich

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Outline Outline

Introduction A retrospective measure of the price level A retrospective measure of the price level Comparison with Hansen Application to Swiss data C

l i

Conclusion

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1. Introduction

Most countries today still compute their consumer price indices Most countries today still compute their consumer price indices

(CPI) as direct (i.e. fixed-weighted) Laspeyres indices

Th

i ht d t d t di t i t l ft fi

The weights are updated at discrete intervals, often every five or

ten years, at which time the old and new series are spliced together together

It is well known that, in the consumer context, the Laspeyres

functional form tends to overestimate the price level due to a functional form tends to overestimate the price level due to a substitution bias

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1. Introduction (continued)

( )

Ideally, one should use chained, superlative indices

y,

, p

However, the necessary data (especially the quantity data) are

ften not available
ften not available

Nonetheless, sooner or later the baskets will need to be updated The question therefore arises at the time when the new

information becomes available whether it can be used to assess the importance of the substitution bias, and whether it can be exploited to improve the measure of past price level behavior

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1. Introduction (continued)

( )

The purpose of this paper is to propose a simple way to use the The purpose of this paper is to propose a simple way to use the

new information made available at the time of updating in order to get a superlative measure of the price change over the to get a superlative measure of the price change over the corresponding period

Retroactively computed price indices then make it possible to Retroactively computed price indices then make it possible to

assess the size of the substitution bias

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2. A retrospective measure of the price level

p p

Consider the following two runs of fixed-basket indices:

∑

i i i q

p

1

∑

i i i q

p

2

∑

− i i T i

q p

1

∑

i i T i q

p

, , ... , , 1, (1)

∑

i i i q

p

∑

i i i q

p

∑

i i i q

p

∑

i i i q

p

∑

T i i q

p 0

∑

T i i q

p1

∑

T i i q

p 2

∑

− T i T i

q p

1

, , ... , , 1, (1)

N t ti

t

d

t d

t th i d th tit f d

∑ ∑

i T i T i i i i

q p q p

∑ ∑

i T i T i i i i

q p q p

∑ ∑

i T i T i i i i

q p q p

∑ ∑

i T i T i i i i

q p q p

, , , ... , , 1 , (2)

Notation: pi

t and qi t denote the price and the quantity of good

i at time t, respectively; the initial period is denoted by 0, and the terminal one by T and the terminal one by T.

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2. A retrospective measure of the price level (continued)

p p ( )

Next, normalize run (2) by dividing all its elements by the first one:

∑

T

∑

T 1

∑

T 2

∑

T T 1

∑ ∑

i T i T i i T i i

q p q p 0

∑ ∑

i T i T i i T i i

q p q p1

∑ ∑

i T i T i i T i i

q p q p 2

∑ ∑

− i T i T i i T i T i

q p q p

1

1 , ... , , , , , (2)

∑ ∑

T i i i T i i

q p q p

1

∑ ∑

T i i i T i i

q p q p

2

∑ ∑

− T i i i T i T i

q p q p

1

∑ ∑

T i i i T i T i

q p q p

1, ... , , , , . (3)

∑

i i i q

p

∑

i i i q

p

∑

i i i q

p

∑

i i i q

p

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2. A retrospective measure of the price level (continued)

p p ( )

Finally, take the geometric means of the corresponding l f ( ) d ( ) h f ll i f elements of (1) and (3) to get the following sequence of pseudo Fisher indices:

∑ ∑

i i i i i i

q p q p

1

∑ ∑

i i i i i i

q p q p

2

∑ ∑

− i i i i i T i

q p q p

1

∑ ∑

i i i i i T i

q p q p

, , , ... , 1, (1)

∑ ∑

i T i i i T i i

q p q p

1

∑ ∑

i T i i i T i i

q p q p

2

∑ ∑

− i T i i i T i T i

q p q p

1

∑ ∑

i T i i i T i T i

q p q p

, , , . ... , 1, (3)

i i i i

∑ ∑ ∑ ∑

T i T i i i i i

q p q p

1 1

∑ ∑ ∑ ∑

− − T i T i T i i i T i

q p q p

1 1

∑ ∑ ∑ ∑

T i T i T i i i T i

q p q p

, . ... , , 1, (4)

∑ ∑

i T i i i i i

q p q p

∑ ∑

i T i i i i i

q p q p

∑ ∑

i T i i i i i

q p q p

, ( )

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3. Comparison with Hansen

p

HANSEN (2007) recently proposed the following Fisher-like HANSEN (2007) recently proposed the following Fisher like price index:

2 1 :

⎥ ⎥ ⎥ ⎥ ⎤ ⎢ ⎢ ⎢ ⎢ ⎡ =

∑ ∑

i T i i t i i t H

p s p p s P

(5) , t = 0, 1, ..., T,

⎥ ⎥ ⎦ ⎢ ⎢ ⎣∑

i t i i i

p s

where:

∑

=

i i i i i i

q p q p s

∑

=

i T i T i T i T i T i

q p q p s

, .

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3. Comparison with Hansen (continued):

p ( )

For comparison purposes, rewrite formulae (1) and (2) in For comparison purposes, rewrite formulae (1) and (2) in terms of price relatives and expenditure share weights:

i t i i i i i i i t i

p p s q p q p

∑ ∑ ∑

=

, t = 0, 1, ..., T (1’)

i

∑

t i T i T i t i

p q p

∑ ∑

( ’)

T i i i T i i T i T i i

p p s q p

∑ ∑

=

, t = 0, 1, ..., T. (2’)

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3. Comparison with Hansen (continued):

p ( )

Using (2’), it can be seen that the period t index in (3) can Using (2 ), it can be seen that the period t index in (3) can be written as follows:

T i t i i T i T i T i i T i t i

p p s q p q p

∑ ∑ ∑

=

, t = 0, 1, ..., T . (2’)

i i i i i

p q p

∑

T t i T i T i t i

p p s q p

∑ ∑

T i i i T i i i i T i i i

p p s p q p

∑ ∑ ∑

=

, t = 0, 1, ..., T . (3’)

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3. Comparison with Hansen (continued):

p ( )

Finally, using (1’) and (3’), it can be seen that the period-t d Fi h i i d (P 0 t ) i h d fi d b pseudo Fisher price index (PF

0:t ) in the sequence defined by

(4) can be written as follows:

i t i i i i i i i i t i

p p s q p q p

∑ ∑ ∑

=

, t = 0, 1, ..., T (1’)

T i T i T i t i i T i i T i i i T i t i

p s p p s q p q p

∑ ∑ ∑ ∑

=

, t = 0, 1, ..., T . (3’)

T i i i

p

∑

2 1 2 1 :

⎥ ⎥ ⎥ ⎤ ⎢ ⎢ ⎢ ⎡ ⋅ = ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ =

∑ ∑ ∑ ∑ ∑ ∑

T i t i i T i i t i i i T i T i t i i i t i t F

p p s p p s q p q p P

, t = 0, 1, ..., T . (4’)

⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎣ ⎥ ⎥ ⎦ ⎢ ⎢ ⎣

∑ ∑ ∑

T i i i T i i T i i i i i F

p p s q p q p

( )

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3. Comparison with Hansen (continued):

p ( )

2 1 :

⎥ ⎥ ⎥ ⎥ ⎤ ⎢ ⎢ ⎢ ⎢ ⎡ =

∑ ∑

i T i i i t i i t H

p s p p s P

, t = 0, 1, ..., T , (5)

⎥ ⎥ ⎦ ⎢ ⎢ ⎣∑

i t i i

p s

2 1 1

⎤ ⎡ ⎤ ⎡

t t

p p

2 2 :

⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ =

∑ ∑ ∑ ∑ ∑ ∑ ∑

T i i i T i T i i i T i i i i i i T i i i T i t i i i i i i t i t F

p p s p p s p p s q p q p q p q p P

, t = 0, 1, ..., T . (4’) Note that PH

0:T = PF 0:T.

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3. Comparison with Hansen (continued):

p ( )

“Laspeyres element”:

t L F i i t i i t L H

P p p s P

: ) ( : ) (

= = ∑

, t = 0, 1, ..., T .

t T i t i i T i t

p p s

: :

1

∑

“Paasche element”:

t 0 1 T

t P F T i i i T i i i i t i i T i t P H

P p p s p p p s P

: ) ( : ) (

1 = ≠ =

∑ ∑

, t = 0, 1, ..., T .

PH(P)

0:t can be interpreted as a harmonic period-T weighted

Young index It does not have the Paasche form though Young index. It does not have the Paasche form, though.

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4. Application to Swiss data

pp

The application is for the Swiss CPI data for the period 1993 to The application is for the Swiss CPI data for the period 1993 to

2000

At the time the Swiss CPI was computed as a direct Laspeyres At the time the Swiss CPI was computed as a direct Laspeyres

quantity index, with fixed weights using May 1993 as the reference period p

The weights were eventually revised in 2000 The number of categories common to both surveys amounts to 192

ut of 201

In value terms, these categories represented over 99% of the CPI

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Figure 1

Alternative consumer price indices, Switzerland, 1993-2000

106 107 106 107 Official CPI (1993-weights, 201 items) Lowe index (1993-weights, 192 items) Lowe index (2000-weights, 192 items) 104 105 104 105 Lowe index (2000 weights, 192 items) Geometric mean (192 items) 102 103 102 103 100 101 100 101 99 May-93 May-94 May-95 May-96 May-97 May-98 May-99 May-00 99

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5. Conclusion

Between 1993 and 2000 the Swiss price level (in terms of CPI) increased by

about 5.2%, rather than the 6.1% that the official data suggest

In terms of yearly averages, this implies an inflation rate of about 0.73%,

th th 0 85% rather than 0.85%

This suggests a yearly substitution bias of 0.13% on average This estimate gives considerable empirical support to BRACHINGER, SCHIPS and

STIER (1999) who contended that the (upper level) substitution bias probably does not exceed 0 15 percentage points per year does not exceed 0.15 percentage points per year

The approach that we have used for the CPI could obviously also be applied to

ther indices, including quantity indices
ther indices, including quantity indices