Unit Value Bias (Indices) Reconsidered Price- and Unit-Value-Indices - - PowerPoint PPT Presentation

*This paper represents the author's personal opinion and does not necessarily reflect the view of the Deutsche Bundes-

bank or its staff.

Unit Value Bias (Indices) Reconsidered

Price- and Unit-Value-Indices in Germany

Peter von der Lippe, Universität Duisburg-Essen Jens Mehrhoff*, Deutsche Bundesbank 11th Ottawa Group Meeting (Neuchâtel May 28th 2009)

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Agenda

• 1. Introduction and Motivation
• 2. Unit value index (UVI) and Drobisch's Index (PUD)
• 3. Price and unit value indices in German foreign trade

statistics (Tests of hypotheses)

• 4. Properties and axioms (uv, UVI, PUD)
• 5. Decomposition of the Unit Value Bias

(PUP/PL L- and S-effect)

• 6. Interpretation of the S-effect in terms of covariances

(using a generalized theorem of Bortkiewicz)

• 7. Conclusions

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• 1. Introduction and Motivation

❙ Export and Import Price Index Manual (XMPI Man. IMF, 2008) ❙ Unit Value Indices (UVIs) are used in

Prices of trade (export/import), land, air freight and certain services (consultancy, lawyers etc)

❙ Literature (UVIs cannot replace price indices)

Balk 1994, 1995 (1998), 2005 Diewert 1995 (NBER paper), 2004 etc. von der Lippe 2006 GER http://mpra.ub.uni-muenchen.de/5525/1/MPRA _paper_5525.pdf Silver (2007), Do Unit Value Export, Import, and Terms of Trade Indices Represent or Misrepresent Price Indices, IMF Working Paper WP/07/121

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• 1. Introduction and Motivation

2000 Jan – 2007 Dec

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• 2. UVI and Drobisch's index

Definitions and Formulas – 1 –

• 1. Unit value for the kth commodity number (CN)

k = 1, …, K Unit values are not defined over all CNs

Examples for CNs

∑ ∑ ∑ ∑

= =

j kj kj j k kj kj kj kj kj k

m p Q q p = q q p p ~

HS (Harmonized System) Germany (Warenverzeichnis)

19 05 90 Other Bakers' Wares,

Communion Wafers, Empty Capsules, Sealing Wafers

19 05 90 45 Cakes and similar small baker's wares (8 digits)

23 09 10 Dog or Cat Food, Put up for Retail Sale 23 09 10 11 to 23 09 10 90 twelve (!!) CNs for dog or cat food

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• 2. UVI and Drobisch's index

Definitions and Formulas – 2 –

• 2. German Unit Value Index (UVI) of exports/imports

the usual Paasche index (unit values instead of prices)

• 3. The Unit value index (UVI) should be kept distinct from

Drobisch's index (1871)

∑ ∑ ∑∑ ∑ ∑

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = =

K k m j k kj kj kt K k m j kjt kjt k kt k k kt kt P t

k k

Q q p Q q p Q p ~ Q p ~ PU

∑ ∑∑ ∑ ∑∑ ∑∑ ∑∑ ∑∑ ∑∑

= =

k k k j jk jk k kt k j jkt jkt k j jk k j jk jk k j jkt k j jkt jkt DR t

Q q p Q q p q q p q q p P

Aggregation in two stages; k = 1, …, K , j = 1, …, nK commodities in the kth CN; Σnk = n (all commodities)

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• 2. UVI and Drobisch's index

Definitions and Formulas – 3 –

Drobisch's index

However, Drobisch is better known for

( )

P t L t 2 1

P P +

It does not make sense to consider absolute unit values ("Euro per kilogram")

t t t t t DR t

Q Q Q ~ , Q ~ V p ~ p ~ P = = =

no information about quantities available

information about quantities

the same commodity in different outlets "normal" usage of the term "low level"

different goods

grouped by a classification

situation of a UVI (Σq needed for unit value)

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8 Absolute Unit Values in Austrian Statistics (publication of the Austrian National Bank OeNB 2006)

Austrian Import prices rose from ≈ 20 € per kilogram in 1995 to 25 € … in 2005

Glatzer et al "Globalisierung…" http://www.oenb.at/de/img/gewi_2006_3_tcm14-46922.pdf

"Because we use weights as units an increasing import price index could be ex- plained by either rising prices or reduced weights due to quality improvement"

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• 2. UVI and price indices (PI): System of possible indices

Price-indices Quantity-indices p

Laspeyres

PL PUL QL QUL

Paasche

PP PUP QP QUP

uv p uv

V = Σptqt/Σp0q0 = PPQL = PUP QUL

24 = 16 indices:

type of index (price vs quantity) Prices (p) vs unit values (uv) Laspeyres vs Paasche Export vs import

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• 3. Indices in Germany

(1) Data source, conceptual differences

Price index Unit value index Data Survey based (monthly), sample; more demanding (weights!) Customs based (by-product), census, Intrastat: survey Formula Laspeyres Paasche

Quality ad- justment

Yes No (feasible?) Prices,

aggregates

Prices of specific goods at time

• f contracting

Average value of CNs; time of crossing border New / dis- appearing goods

Included only when a new base period is defined; vanishing goods replaced by similar ones constant selection of goods * Immediately included; price quotation of disappearing goods is simply discontinued variable universe of goods

Merits Reflect pure price movement

(ideally the same products over time)

"Representativity" inclusion of all

products; data readily available Published in Fachserie 17, Reihe 11 Fachserie 7, Reihe 1 CN = commodity numbers * All price determining characteristics kept constant

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• 3. Indices in Germany

(2) Overview of Hypothesis

Hypothesis Argument

• 1. U < P, growing

discrepancy Laspeyres (P) > Paasche (U)

Formula of L. v. Bortkiewicz

• 2. Volatility U > P

U no pure price comparison

(U is reflecting changes in product mix [structural changes])

• 3. Seasonality U > P

U no adjustment for seasonally non-availability

• 4. U suffers from

heterogeneity Variable vs. constant selection of goods, CN less homogeneous than specific goods

• 5. Lead of P

Prices refer to the earlier moment of contracting

(contract-delivery lag; exchange rates)

• 6. Smoothing (due to

quality adjustment)

Quality adjustment in P results in smoother series

Price index (P) Unit value index (U)

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• 4. Properties and axioms: 4.1. unit values: one CN, two commodities

λ > 1

λ > 1 and μ < 1 → Δ < 0

less of the more expensive good 2 unit value declining

λ > 1 and μ > 1 → Δ > 0

more of the more expensive good 2 unit value rising λ < 1

λ < 1 and μ < 1 → Δ > 0

less of the cheaper good 2 unit value rising

λ < 1 and μ > 1 → Δ < 0

more of the cheaper good 2 unit value declining μ < 1 μ > 1

p10 = p1t = p p20 = p2t = λp μ = m2t/0.5 m10 = m20 = 0.5

( )( )

μ − λ − = − = Δ 1 1 2 p p ~ p ~

k kt "… 'unit value' indices … may therefore be affected by changes in the mix of items as well as by changes in their prices. Unit value indices cannot therefore be expected to provide good measures of average price change over time" (SNA 93, § 16.13)

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• 4. Properties and axioms: 4.2. ratios of unit values

k t ) k ( L t ) k ( L t kt k

Q ~ Q Q Q Q = ⋅

∑ ∑

=

k kt k kt k k k kt P t

Q p ~ Q p ~ p ~ p ~ PU

∑

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

j kt k kjt kj kj kjt k kt

Q p ~ q p p p p ~ p ~

1) UVI mean of uv-ratios 2) Ratio of unit values ≠ mean of price relatives 3) Proportionality (identity)

the weights do not add up to unity, but to

Contribution

• f k to S-effect

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• 4. Properties and axioms: 4.3. UVI and Drobisch's index

Axiom

Drobisch* German PUP

Proportionality U(p0, λp0, q0, qt) = λ (identity = 1) no no

Commensurability U( Λp0, Λpt, Λ-1q0, Λ-1qt) = U(p0, pt, q0, qt)

no no Linear homogen. U(p0, λpt, q0, qt) = λ U(p0, pt, q0, qt) yes yes Additivity** (in

current period prices)

U(p0, p*t, q0, qt) = U(p0, pt, q0, qt) + U(p0, p+

t, q0, qt) for p*t = pt + p+ t,

yes yes Additivity** (in

base period prices)

[U(p*0, pt, q0, qt)]-1 = [U(p0, pt, q0, qt)]-1 + [U(p+

0, p+ t, q0, qt)]-1 for p*0 = p0 + p+

yes yes Product test Implicit quantity index of PUD or PUP Σqt/Σq0 QUL Time re- versibility

U(pt, p0, qt, q0,) = U← = [U(p0, pt, q0, qt)]-1 = [U→]-1

yes (PUP←) = 1/(PUL→) Transitivity

U(p0, p2, q0, q2) = U(p0, p1, q0, q1). U(p1, p2, q1, q2) yes

no

* Balk1995, Silver 2007, IMF Manual; applies also to subindex ** Inclusive of (strict) monotonicity

k kt p

~ p ~

Axioms Drobisch's (price) index and the German UVI (= PUP)

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• 5. Decomposition of the discrepancy D

( )

L t P t L t L t L t t

P P Q P Q V − = − =

L t P t L t P t L t P t

P S PU QU S Q Q Q L ⋅ = ⋅ = =

Value index

∑ ∑

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

i i i i L t i it i L t i it

q p q p Q q q P p p C

S L P PU P P QU Q 1 P Q C P PU D

P t P t L t P t L t L t L t L t L t P t

⋅ = ⋅ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = =

L t P t P t L t t

QU PU QU PU V = =

Bortkiewicz Formula Discrepancy (uv-bias)

Ladislaus von Bortkiewicz (1923)

L t P t L t P t L t L t

P L PU QU L Q QU Q S ⋅ = ⋅ = =

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• 5. The two effects L and S - 1 -

L

P

P

P

L

PU

P

PU

L S

Quadrant I same Direction D > 1 IV opposite direction D indeterminate II opposite direction D indeterminate III same direction D < 1

L > 1 L < 1 S > 1

In I and III we can combine two inequalities S < 1 S = 1 S > 1 L < 1 PUP < PP < PL PUP < PL indefinite L = 1 PUP < PL = PP PUP = PP = PL PUP > PL = PP L > 1 indefinite PUP > PL PUP > PP > PL

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• 5. The two effects L and S - 2 -

Deflator X and M respectively taken for PP

S and L independent ?

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• 5. The two effects L and S - 3 -

Time path of S-L- pairs (left → right)

exports imports

Normal reaction: L and S negative

more likely in the case

• f imports

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• 6. Interpretation of S component (contributions to L as the model)

Y X sXY ⋅

∑ ∑

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − =

i i i i i L L i 1 i L L i 1 i L L P

q p q p Q Q q q P P p p P P P R

Interpretation L-Effect: contributions to the covariance (Szulc) R a "centred" covariance L = R + 1

• A. Chaffe, M. Lequain, G. O'Donnell, Assessing the Reliability of the CPI Basket Update

in Canada Using the Bortkiewicz Decomposition, Statistics Canada, September 2007

No L-effect (L = 1) if No S-effect (S = QL/QUL = 1) if

• 1. all p1/p0 equal (PL)
• r = 1
• 2. all q1/q0 = QL or = 1
• 3. covariance = 0
• 1. no CNs, only individual goods

(or: each nk = 1, perfectly homogeneous CNs)

• 2. all q1/q0 equal (or = 1) 3. all mkjt = mkj0

∀j, k 4. all prices in 0 equal pkj0 =

prices in t are irrelevant

k

p ~

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• 6. Contribution of a CN (k) to S as ratio of two linear indices

∑ ∑

⋅ = =

k kt k kt k k k t ) k ( L t L t L t

Q p ~ Q p ~ Q ~ Q QU Q S ∑ ∑

=

t t t t

y x y x X

∑ ∑

=

t

y x y x X

• 2. Generalized theorem of Bortkiewicz

for two linear indices Xt and X0

The "usual" theorem (page 15) is a special case →

Y X s 1 X X

xy t

⋅ + =

Y X y x y x w Y y y X x x s

t t t t xy

⋅ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

∑ ∑ ∑

∑

= y x y x w

t

X X w x x = =

∑

1.

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• 6. Generalized Theorem of Bortkiewicz

x0 = p0 y0 = q0 Xt = PP xt = pt yt = qt X0 = PL X0 =

yt = p0 xt = qt

Xt =

y0 = 1

x0 = q0 X0 =

yt = 1 xt = qt

Xt =

y0 = p0 x0 = q0

Theorem for the L-effect

• 1. for S
• 2. for 1/S

Y X s 1 X X

xy t

⋅ + =

∑ ∑

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

i i i i L t i it i L t i it

q p q p Q q q P p p C

( )∑

∑

− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −

kj kj k kj k t kj kjt

q q p ~ p Q ~ q q

∑ ∑

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −

kj kj kj kj k kj ) k ( L t kj kjt

q p q p p ~ 1 p 1 Q q q

k t

Q ~

k t

Q ~

) k ( L t

Q

) k ( L t

Q

L t L t

QU Q S =

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• 6. Two commodities example with both, S and L effect (example of page 12)

S-effect L-effect

π = η = 1

= 1 = 1 λ > 1 Δ < 0 → S < 1 Δ > 0 → S > 1 λ < 1 Δ > 0 → S > 1 Δ < 0 → S < 1 μ < 1 μ > 1

( )( )

t L t

p ~ 1 1 1 1 1 Q ~ Q S Δ + = λ + μ − λ − + = =

p10 = p1t = p p20 = p2t = λp μ =m2t/0.5 m10 = m20 = 0.5

π = p1t /p10 p2t / p20 = ηπ

( )

Δ = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

∑ ∑

t j j j j t j jt ) 1 ( xy

Q ~ q q p ~ p Q ~ q q s

( ) ( )

2 2 t ) 2 ( xy

p ~ 1 p ) 1 )( 1 ( Q ~ 2 s Δ − = λ + μ − − λ =

λ + ηλ + π = 1 ) 1 ( PL

t

( )

λμ + μ − ηλμ + μ − π = 2 2 PP

t

λμ + μ − λ + ⋅ ηλ + ηλμ + μ − == = 2 1 1 2 P P L

L t P t

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• 7. Conclusions

if π = η = 1

Δ* = Δ C = 0

( ) ( )

[ ]

)) 1 ( 1 2 2 p p ~ p ~ *

t

λ + − ηλ − μ − π = − = Δ

( )

2 t ) L ( xy

1 ) 1 )( 1 ( Q ~ 2 s C λ + μ − η − λ = = ( ) ( )

λ − − λη − π + λ + + π = − = Δ 1 1 Q ~ 2 ) 1 ( s Q ~ s p ~ p ~ *

t 2 L xy t 1 xy t

• 7. What remains to be done
• Analysis of the time series of UVIs and PIs on various levels of disaggre-

gation, cointegration and Granger-Causality

• Microeconomic interpretation of S-effect (in terms of utility maximizing

behaviour)

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No structural change between CNs (that is Qk0 = Qkt) yields This is, however, not sufficient for the S-effect to vanish

1 QU QU and PU PU V

P t L t L t P t t

= = = =

Discussion 1

1 Q S

L t 0 ≠

=

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

∑∑ ∑ ∑

* kjt kj kjt kj k j kj kjt P

q p q p p p PU

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

∑∑ ∑ ∑

kjt kj kjt kj k j kj kjt P

q p q p p p P

k kt k * kjt

Q Q q q

j

=

∑∑ ∑ ∑

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

j

k kj kt k kjt kj k j kj kjt L

q p Q Q q p p p PU

The same applies to Laspeyres

No mean value property of PUP

a fictitious quantity in t

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∑ ∑ ∑ ∑ ∑ ∑

= =

k j kj kj kt j kjt kj kt k P ) k ( k kt k k kt kt P

m p Q m p Q P Q p ~ Q p ~ PU

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

∑ ∑ ∑ ∑

n n n n n m 1 m m 1 m 1 m U

q q p q q p P

UVI in XMPI Manual

§ 2.14

Drobisch's formula

∑ ∑ ∑ ∑ ∑∑ ∑∑

= =

k j kjt kj kt j kjt kj kt k P ) k ( k j kjt kj k j kjt kjt P

m p Q m p Q P q p q p P

The relation S =PUP/PP instead of S = QL/QUL is not interesting

Sum of weights!