Unit Value Bias (Indices) Reconsidered Price- and Unit-Value-Indices in Germany Peter von der Lippe, Universität Duisburg-Essen Jens Mehrhoff*, Deutsche Bundesbank 11 th Ottawa Group Meeting (Neuchâtel May 28 th 2009) * This paper represents the author's personal opinion and does not necessarily reflect the view of the Deutsche Bundes- bank or its staff.
Agenda 1. Introduction and Motivation 2. Unit value index (UVI) and Drobisch's Index (P UD ) 3. Price and unit value indices in German foreign trade statistics (Tests of hypotheses) 4. Properties and axioms (uv, UVI, P UD ) 5. Decomposition of the Unit Value Bias (PU P /P L L- and S-effect) 6. Interpretation of the S-effect in terms of covariances (using a generalized theorem of Bortkiewicz) 7. Conclusions Ottawa Group 2009 Unit Value Bias Reconsidered 2
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 Ottawa Group 2009 Unit Value Bias Reconsidered 3
4 2000 Jan – 2007 Dec Unit Value Bias Reconsidered 1. Introduction and Motivation Ottawa Group 2009
2. UVI and Drobisch's index Definitions and Formulas – 1 – 1. Unit value for the k th commodity number (CN) ∑ p q q ∑ ∑ ~ = = kj 0 kj 0 kj 0 p = p p m ∑ k 0 kj 0 kj 0 kj 0 j j q Q kj 0 k 0 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 Communion Wafers, Empty Capsules, small baker's wares (8 digits) Sealing Wafers 23 09 10 Dog or Cat Food, Put up for 23 09 10 11 to 23 09 10 90 Retail Sale twelve (!!) CNs for dog or cat food Ottawa Group 2009 Unit Value Bias Reconsidered 5
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) m K ∑∑ k ∑ Aggregation in two stages; ~ p q p Q kjt kjt kt kt k = 1, …, K , = = k j P k PU ∑ j = 1, …, n K commodities ~ ⎛ ⎞ 0 t p Q m K p q ∑ ∑ in the k th CN; Σ n k = n (all k ⎜ ⎟ k 0 kt kj 0 kj 0 Q ⎜ ⎟ k kt commodities) Q ⎝ ⎠ k j k 0 3. The Unit value index (UVI) should be kept distinct from Drobisch's index (1871) ∑∑ ∑∑ ∑∑ ∑ p q q p q Q jkt jkt jkt jkt jkt kt = = k j k j k j k DR P ∑∑ ∑∑ ∑∑ ∑ 0 t p q q p q Q jk 0 jk 0 jk 0 jk 0 jk 0 k 0 k j k j k j k Ottawa Group 2009 Unit Value Bias Reconsidered 6
2. UVI and Drobisch's index Definitions and Formulas – 3 – ~ Drobisch's index p V ~ Q = = = DR t 0 t t P , Q ~ ~ 0 t 0 t p Q Q 0 0 0 t ( ) P + L P 1 P However, Drobisch is better known for 0 t 0 t 2 no information about information about quantities available quantities the same commodity in "normal" usage of the different outlets term "low level" different goods situation of a UVI ( Σ q needed for unit value) grouped by a classification It does not make sense to consider absolute unit values ("Euro per kilogram") Ottawa Group 2009 Unit Value Bias Reconsidered 7
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" Ottawa Group 2009 Unit Value Bias Reconsidered 8
2. UVI and price indices (PI): System of possible indices 2 4 = 16 indices: type of index (price vs quantity) Prices (p) vs unit values (uv) V = Σ p t q t / Σ p 0 q 0 = P P Q L = PU P QU L Laspeyres vs Paasche Export vs import Price-indices Quantity-indices p uv p uv P L PU L Q L QU L Laspeyres P P PU P Q P QU P Paasche Ottawa Group 2009 Unit Value Bias Reconsidered 9
3. Indices in Germany (1) Data source, conceptual differences Price index Unit value index Survey based (monthly), Customs based (by-product), Data sample ; more demanding (weights!) census , Intrastat: survey Formula Laspeyres Paasche Quality ad- Yes No (feasible?) justment Prices of specific goods at time Average value of CNs; time of Prices, of contracting crossing border aggregates Included only when a new base Immediately included; price New / dis- period is defined; vanishing quotation of disappearing goods appearing goods replaced by similar ones is simply discontinued goods constant selection of goods * variable universe of goods Reflect pure price movement " Representativity " inclusion of all Merits products ; data readily available (ideally the same products over time) Published in Fachserie 17, Reihe 11 Fachserie 7, Reihe 1 CN = commodity numbers * All price determining characteristics kept constant Ottawa Group 2009 Unit Value Bias Reconsidered 10
3. Indices in Germany (2) Overview of Hypothesis Price index (P) Unit value index (U) Hypothesis Argument 1. U < P , growing Laspeyres (P) > Paasche (U) discrepancy Formula of L. v. Bortkiewicz U no pure price comparison 2. Volatility U > P (U is reflecting changes in product mix [structural changes]) 3. Seasonality U > P U no adjustment for seasonally non-availability 4. U suffers from Variable vs. constant selection of goods, heterogeneity CN less homogeneous than specific goods Prices refer to the earlier moment of contracting 5. Lead of P (contract-delivery lag; exchange rates) 6. Smoothing (due to Quality adjustment in P results in smoother series quality adjustment) Ottawa Group 2009 Unit Value Bias Reconsidered 11
4. Properties and axioms: 4.1. unit values: one CN, two commodities p 10 = p 1t = p p ( )( ) ~ ~ Δ = − = − λ − μ p 20 = p 2t = λ p p p 1 1 kt k 0 μ = m 2t /0.5 2 m 10 = m 20 = 0.5 λ > 1 and μ < 1 → Δ < 0 λ > 1 and μ > 1 → Δ > 0 λ > 1 less of the more expensive good 2 more of the more expensive good 2 unit value declining unit value rising λ < 1 and μ < 1 → Δ > 0 λ < 1 and μ > 1 → Δ < 0 λ < 1 less of the cheaper good 2 more of the cheaper good 2 unit value rising 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) Ottawa Group 2009 Unit Value Bias Reconsidered 12
4. Properties and axioms: 4.2. ratios of unit values ~ ~ 1) UVI mean of uv-ratios p p Q ∑ = P kt k 0 kt PU ∑ ~ ~ 0 t p p Q k k 0 k 0 kt k 2) Ratio of unit values ≠ mean of price relatives ~ ⎛ ⎞ p p q p ∑ ⎜ ⎟ = kjt kj 0 kjt kt ⎜ ⎟ ~ ~ j ⎝ ⎠ p p p Q k 0 kj 0 k 0 kt L ( k ) Q Q ⋅ = L ( k ) k 0 0 t Q ~ the weights do not add up to unity, but to 0 t k Q Q kt 0 t 3) Proportionality (identity) Contribution of k to S-effect Ottawa Group 2009 Unit Value Bias Reconsidered 13
4. Properties and axioms: 4.3. UVI and Drobisch's index Axioms Drobisch's (price) index and the German UVI (= PU P ) German PU P Drobisch* Axiom U( p 0 , λ p 0 , q 0 , q t ) = λ no no Proportionality (identity = 1) U( Λ p 0 , Λ p t , Λ -1 q 0 , Λ -1 q t ) = U( p 0 , p t , q 0 , q t ) Commensurability no no U( p 0 , λ p t , q 0 , q t ) = λ U( p 0 , p t , q 0 , q t ) Linear homogen. yes yes U( p 0 , p* t , q 0 , q t ) = U( p 0 , p t , q 0 , q t ) + Additivity** (in yes yes U( p 0 , p + t , q 0 , q t ) for p* t = p t + p + t , current period prices) [U( p* 0 , p t , q 0 , q t )] -1 = [U( p 0 , p t , q 0 , q t )] -1 Additivity** ( in yes yes t , q 0 , q t )] -1 for p* 0 = p 0 + p + + [U( p + 0 , p + base period prices) 0 Σ q t / Σ q 0 Implicit quantity index of P UD or PU P QU L Product test U( p t , p 0 , q t , q 0 ,) = U ← (PU P ← ) = Time re- = [U( p 0 , p t , q 0 , q t )] -1 = [U → ] -1 yes 1/(PU L → ) versibility U( p 0 , p 2 , q 0 , q 2 ) = U( p 0 , p 1 , q 0 , q 1 ) . U( p 1 , p 2 , q 1 , q 2 ) yes no Transitivity ~ ~ p kt p * Balk1995, Silver 2007, IMF Manual; applies also to subindex k 0 ** Inclusive of (strict) monotonicity Ottawa Group 2009 Unit Value Bias Reconsidered 14
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