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)
*This paper represents the author's personal opinion and does not necessarily reflect the view of the Deutsche Bundesbank or its staff.
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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Unit Value Bias Reconsidered
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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)
~
pk 0 =
∑p q
∑q
kj0
kj0
= ∑ j p kj0
kj0
q kj0
Qk0
= ∑ j p kj0 m kj0
k = 1, …, K Unit values are not defined over all CNs
Examples for CNs
HS (Harmonized System)
Germany (Warenverzeichnis)
19 05 90 Other Bakers' Wares,
19 05 90 45 Cakes and similar
small baker's wares (8 digits)
Communion Wafers, Empty Capsules,
Sealing Wafers
23 09 10 Dog or Cat Food, Put up for
Retail Sale
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23 09 10 11 to 23 09 10 90
twelve (!!) CNs for dog or cat food
Unit Value Bias Reconsidered
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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)
k
~
∑ pk 0Q kt
k
nk
k
j
∑∑ p kjt q kjt
~
∑ pkt Q kt
PU 0Pt =
K
=
Aggregation in two stages;
k = 1, …, K ,
j = 1, …, nK commodities
in the kth CN; Σnk = n (all
commodities)
 n k p kj0 q kj0 
∑k Q kt  ∑j Q 
k0


K
3. The Unit value index (UVI) should be kept distinct from
Drobisch's index (1871)
P0DR
t
∑∑ p
=
∑∑ p
k
jkt
j
jk 0
k
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q jkt
j
∑∑ q
∑∑ q
k
q jk 0
k
jkt
j
j
∑∑ p
=
∑∑ p
k
jk 0
jkt
j
jk 0
k
Unit Value Bias Reconsidered
q jkt
j
∑Q
∑Q
kt
k
q jk 0
k0
k
6
2. UVI and Drobisch's index
Drobisch's index
Definitions and Formulas – 3 –
~
p t V0 t ~
Qt
DR
P0 t = ~ = ~ , Q 0 t =
p0 Q 0 t
Q0
However, Drobisch is better known for
no information about
quantities available
the same commodity in
different outlets
different goods
grouped by a classification
1
2
(P
L
0t
P
0t
+P
)
information about
quantities
"normal" usage of the
term "low level"
situation of a UVI
(Σq needed for unit value)
It does not make sense to consider absolute unit values ("Euro per kilogram")
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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 explained 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
24 = 16 indices:
type of index (price vs quantity)
Prices (p) vs unit values (uv)
Laspeyres vs Paasche
Export vs import
V = Σptqt/Σp0q0 = PPQL = PUP QUL
Price-indices
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Quantity-indices
p
uv
p
uv
Laspeyres
PL
PUL
QL
QUL
Paasche
PP
PUP
QP
QUP
Unit Value Bias Reconsidered
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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 adjustment
Yes
No (feasible?)
Prices of specific goods at time
aggregates of contracting
Prices,
New / disappearing
goods
Merits
Published in
Average value of CNs; time of
crossing border
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
Reflect pure price movement
(ideally the same products over time)
"Representativity" inclusion of all
products; data readily available
Fachserie 17, Reihe 11
Fachserie 7, Reihe 1
CN = commodity numbers
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* All price determining characteristics kept constant
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3. Indices in Germany
(2) Overview of Hypothesis
Price index (P) Unit value index (U)
Hypothesis
Argument
1. U < P, growing
discrepancy
Laspeyres (P) > Paasche (U)
2. Volatility U > P
U no pure price comparison
Formula of L. v. Bortkiewicz
(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
5. Lead of P
6. Smoothing (due to
quality adjustment)
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Variable vs. constant selection of goods,
CN less homogeneous than specific goods
Prices refer to the earlier moment of contracting
(contract-delivery lag; exchange rates)
Quality adjustment in P results in smoother series
Unit Value Bias Reconsidered
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4. Properties and axioms: 4.1. unit values: one CN, two commodities
p10 = p1t = p
p20 = p2t = λp
µ = m2t/0.5
m10 = m20 = 0.5
p
~
~
∆ = pkt − pk 0 = (1 − λ )(1 − µ )
2
λ>1
λ<1
λ > 1 and µ < 1 → ∆ < 0
λ > 1 and µ > 1 → ∆ > 0
less of the more expensive good 2
unit value declining
more of the more expensive good 2
unit value rising
λ < 1 and µ < 1 → ∆ > 0
λ < 1 and µ > 1 → ∆ < 0
less of the cheaper good 2
unit value rising
more of the cheaper good 2
unit value declining
µ<1
µ >1
"… '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
1) UVI mean of uv-ratios
~
pkt
P
PU 0 t = ∑ ~
p
k
k0
~
pk 0 Q kt
~
∑k pk 0Q kt
2) Ratio of unit values ≠ mean of price relatives
~
p kjt  p kj0 q kjt 
pkt


=
∑
~
j
pk 0
p kj0  ~
pk 0 Q kt 
Q k 0 L ( k ) Q 0Lt( k )
⋅ Q0 t = ~ k
the weights do not add up to unity, but to
Q kt
Q0t
3) Proportionality (identity)
Contribution
of k to S-effect
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4. Properties and axioms: 4.3. UVI and Drobisch's index
Axioms Drobisch's (price) index and the German UVI (= PUP)
Definition
Axiom
Proportionality
U(p0, λp0, q0, qt) = λ
(identity = 1)
Commensurability U( Λp0, Λpt, Λ-1q0, Λ-1qt) = U(p0, pt, q0, qt)
Drobisch*
German PUP
no
no
no
no
Linear homogen.
U(p0, λpt, q0, qt) = λ U(p0, pt, q0, qt)
yes
yes
Additivity** (in
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
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+0
yes
yes
Product test
Implicit quantity index of PUD or PUP
Σqt/Σ
Σq 0
QUL
Time re-
U(pt, p0, qt, q0,) = U←
= [U(p0, pt, q0, qt)]-1 = [U→]-1
yes
←) =
(PUP←
→)
1/(PUL→
current period prices)
Additivity** (in
versibility
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 pkt pk 0
** Inclusive of (strict) monotonicity
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5. Decomposition of the discrepancy D
Value index V0 t = PU 0Lt QU 0Pt = PU 0Pt QU 0Lt
Bortkiewicz Formula
 p it
L  q it
L  pi 0q i 0
− P0 t 
− Q 0 t 
C = ∑ 
i  pi 0
 q i 0
 ∑ pi 0q i0
= V0 t − Q 0Lt P0Lt = Q 0Lt (P0Pt − P0Lt )
Discrepancy (uv-bias)
 Q0Lt  P0Pt PU0Pt
PU0Pt  C
 = L ⋅ P = L⋅S
D = L =  L L + 1
L 
P0t
 Q0t P0t  QU0t  P0t P0t
Q 0Pt
Q 0Pt
PU 0Pt
L= L =
=
L
Q 0 t S ⋅ QU 0 t S ⋅ P0Lt
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Ladislaus von Bortkiewicz (1923)
Q 0Lt
Q 0Pt
PU 0Pt
S=
=
=
L
L
QU 0 t L ⋅ QU 0 t L ⋅ P0Lt
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5. The two effects L and S - 1 -
S>1
II opposite direction
D indeterminate
L<1
III same direction
D<1
P
L
L
P
P
Quadrant I same
Direction D > 1
S
L>1
IV opposite direction
D indeterminate
PU L
PU
P
In I and III we can combine two inequalities
S<1
S=1
S>1
L>1
II. indefinite
PUP > PL
I. PUP > PP > PL
L=1
PUP < PL = PP
PUP = PP = PL
PUP > PL = PP
L<1
III. PUP < PP < PL PUP < PL
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IV. indefinite
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5. The two effects L and S - 2 -
Deflator X and M respectively taken for PP
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Unit Value Bias Reconsidered
S and L independent ?
17
5. The two effects L and S - 3 -
Time path of S-L- pairs (left → right)
Normal reaction:
L and S negative
more likely in the case
of imports
exports
imports
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6. Interpretation of S component (contributions to L as the model)
Interpretation L-Effect: contributions to the covariance (Szulc)
 p1 p 0 − P   q1 q 0 − Q   p 0 q 0 
PP − PL
L
L
 ⋅  i i
 ⋅  i 0i 0 
R=
= ∑  i i
 ∑ p q 
PL
P
Q
i 
L
L
i i 






R a "centred" covariance s XY
L=R+1
X⋅Y
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
1. all p1/p0 equal (PL)
or = 1
2. all q1/q0 = QL or = 1
3. covariance = 0
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No S-effect (S = QL/QUL = 1) if
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 5. all quantities in 0 are
equal prices in t are irrelevant
Unit Value Bias Reconsidered
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6.
Contribution of a CN (k) to S as ratio of two linear indices
~
Q 0Lt
Q 0Lt( k )
pk 0 Q kt
1. S =
= ∑k ~ k ⋅
L
QU 0 t
pk 0Q kt
Q 0 t ∑k ~
2. Generalized theorem of Bortkiewicz
for two linear indices Xt and X0
x t yt
∑
Xt =
∑ x0yt
s xy
Xt
=1+
X0
X⋅Y
s xy
w 0 = x 0 y0
X0
∑ x 0 y0
 xt
 y t

= ∑  − X  − Y  w 0 =
 x0
 y 0

∑x y
∑x y
xy
∑
=
∑x y
t
0
0
0
xt
∑ x w 0 = X = X0
0
t
t
0
0
− X⋅Y
The "usual" theorem (slide 15) is a special case →
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6. Generalized Theorem of Bortkiewicz
s xy
Xt
=1+
X0
X⋅Y
Theorem for the L-effect
x0 = p0
y0 = q0
X t = PP
xt = pt
yt = qt
X 0 = PL
p
 q
 p q
C = ∑  it − P0Lt  it − Q 0Lt  i 0 i 0
i  pi0
 q i 0
 ∑ pi0q i0
S = Q 0Lt QU 0Lt
1. for S
x0 = q0
L( k )
Q
y0 = 1 Xt = 0 t
xt = qt
yt = p0 X = ~ k
Q0t
0
 q kjt ~ k 
q kj0
~
∑  q − Q 0 t (p kj0 − pk 0 ) q
∑ kj0

 kj0
2. for 1/S
~
x0 = q0 y0 = p0 Xt = Q 0k t
xt = qt
yt = 1
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L( k )
X0 =Q 0 t
 q kjt
 1
 p kj0 q kj0
1
L
(
k
)
∑  q − Q0 t  p − ~p  p q
k 0  ∑ kj0 kj0
 kj0
 kj0
Unit Value Bias Reconsidered
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6. Two commodities example with both, S and L effect (example of slide 12)
p10 = p1t = p
p20 = p2t = λp
µ =m2t/0.5
m10 = m20 = 0.5
π = p1t /p10
p2t / p20 = ηπ
λ>1
∆<0→S<1
∆>0→S>1
λ<1
∆>0→S>1
∆<0→S<1
µ<1
µ>1
S-effect
L-effect
π=η=1
Q 0Lt
(
1 − λ )(1 − µ )
∆
π(1 + ηλ )
L
S = ~ = 1+
= 1 + ~ P0 t =
1+ λ
1+ λ
p0
Q0t
=1
 q jt ~ 
q j0
~
π(2 − µ + ηλµ )
~


s = ∑j
− Q0t (p j0 − p0 )
= Q0t ∆ P0Pt =
q

∑q j0
j
0
2 − µ + λµ


=1
(1)
xy
s
( 2)
xy
~
2Q 0 t (λ − 1)(1 − µ)
∆
=
=− ~ 2
2
( p0 )
p(1 + λ )
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P0Pt
2 − µ + ηλµ
1+ λ
L = L ==
⋅
P0 t
1 + ηλ
2 − µ + λµ
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7. Conclusions
if π = η = 1
p
~
~
∆* = p t − p0 = [π(2 − µ(1 − ηλ )) − (1 + λ))]
2
~
2Q 0 t λ(1 − η)(1 − µ)
(L)
C = s xy =
2
(1 + λ )
∆* = ∆
C=0
1
L
2
s
s
(
1
+
λ
)
xy
xy
∆* = ~
pt − ~
p0 = π ~ +
+ π(1 − λη ) − (1 − λ )
~
Q0t
2Q 0 t
7. Future work
• Analysis of the time series of UVIs and PIs on various levels of disaggregation, cointegration and Granger-Causality
• Microeconomic interpretation of S-effect (in terms of utility maximizing
behaviour)
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Discussion 1
No structural change between CNs (that is Qk0 = Qkt) yields
P
0t
V0 t = PU = PU
L
0t
L
0t
P
0t
and QU = QU = 1
This is, however, not sufficient for
the S-effect to vanish
S = Q 0Lt ≠ 1
No mean value property of PUP
p kjt  p kj0 q kjt

PU = ∑k ∑ j
p kj0  ∑∑ p kj0q *kjt
P
q
*
kjt
= q k j0
Q kt
Qk0




p kjt  p kj0 q kjt 


P = ∑k ∑ j

p kj0  ∑∑ p kj0 q kjt 
P
The same applies to Laspeyres
PU = ∑k ∑ j
L
a fictitious quantity in t
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Unit Value Bias Reconsidered
p kjt
p kj0

Q 
p kj0  q kjt k 0 
Q kt 

∑∑ p kj0q k j0
24
Discussion 2
The relation S =PUP/PP instead of S = QL/QUL is not interesting
PU P
∑ ~p
=
∑ ~p
kt
Q kt
k
k0
Q kt
= ∑ P(Pk )
k
Q kt ∑ p kj0 m kjt
j
∑Q ∑p
kt
k
k
kj0
m kj0
j
PP
∑∑ p
=
∑∑ p
k
k
kjt
q kjt
j
kj0
j
q kjt
= ∑ P(Pk )
k
Q kt ∑ p kj0 m kjt
j
∑Q ∑p
kt
k
kj0
m kjt
j
Sum of weights!
UVI in XMPI
Manual
§ 2.14
Drobisch's formula
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 ∑m p1m q1m 

PU = 
 ∑ q1m 
m


Unit Value Bias Reconsidered
 ∑n p 0n q 0n

 ∑ q 0n
n





25