An Index Number Formula Problem: the Aggregation of Broadly Comparable Items

Transcription

1 An Index Nuber Forula Proble: the Aggregation of Broadly Coparable Ites Mick Silver* International Monetary Fund Presentation to the (Ottawa) International Working Group on Price Indices (May 27 29, 2009) Swiss Federal Statistical Office (FSO), Neuchâtel, Switzerland. The proble with unit value indices is that we hear a lot about their use for trade price indices and little about their use elsewhere, when in fact we should be hearing uch less about their use for trade price indices and a lot ore elsewhere. *The views expressed herein are those of the author and should not be attributed to the IMF, its Executive Board, or its anageent.

2 On forula and product heterogeneity If data on atched prices and uantities are available: a superlative price index nuber forula such as Fisher is best to aggregate heterogeneous ites, and a unit value index is biased, and a unit value index is best to aggregate hoogeneous ones, and a superlative index is biased. SNA 993 and 2008; CPI and PPI Manuals; Balk, Diewert and others. What about broadly coparable ones?

3 An illustration period 0 period price uantity value price uantityvalue A B total Unit value 0.97 Laspeyres/Fisher/Paasche

4 When to use unit value indexes and when Fisher? potential areas of application CPIs: sae ite different outlet: Reinsdorf; Hausan and Leibtag (2008). Health treatents: Aizcorbe and Nestoriak (2008). PPI: sae output (input) different custoers (suppliers). XMPIs: sae output (input) different country custoers (suppliers). SNA 2008 notes exception of institutionalized price discriination to using unit values for hoogeneous case. Asyetry in theory for iports and exports.

5 Unit value and Fisher price index P = UV M N t t 0 0 p p n n = n= M N t 0 n = n= P n n t t t 0 p i i p i i i= i= F n n 0 t 0 0 p i i p i i i= i=

6 Unit value indices for heterogeneous ites: The test approach (Balk and Diewert) The unit value index fails the Proportionality Test: that is, if all prices are ultiplied by the positive nuber λ, then the new price index is λ. The unit value index fails the Invariance to Changes in the Units of Measureent (coensurability) Test: that is, the price index does not change if the units of easureent for each product are changed. Inappropriate for XMPIs using custos data (Silver) Econoic theoretic approach Bradley (2005) copares the bias that results fro using unit values as plug-ins for prices for a COLI. Only in the uninteresting case of no price dispersion in either the current or reference period will the unit value (plug-in) index be unbiased against the COLI.

7 Unit value indices for hoogeneous ites: Passes the tie aggregation proble. If unit value index is used to deflate a corresponding value change, the result is a change in total uantity which is intuitively appropriate, M M M M t t t t 0 0 t p p p = = = = = N M M NM 0 0 t 0 0 p n n n= = = = Tests not designed for hoogeneous ites.

8 When to use unit value indexes and when Fisher? Balk: if the splitting into hoogeneous and heterogeneous ites was not feasible, he advised a price index. Diewert: if detailed data on strictly hoogeneous goods unavailable (specified ite by outlet for CPI), then unit value over outlets. Dálen argued for uality-adjusted unit value indexes that reove the effect on prices of product heterogeneity and de Haan ipleented this in a hedonic setting.

9 Nuerical relationship: unit value and Fisher price index Balk (998) and Parniczky (974) seinal work: decoposition in ters of uantity-weighted covariances. P P P P P P P P P P UV UV L UV L = = F L F L P 2 P P P P P P 2 ρxy, cv ( x) cv ( y) s0 s0 s0 = = = + F P P L F L 2

10 Nuerical relationship: unit value and Fisher price index P P UV P = 0 t ˆ t β ˆ 0 β P P UV L 0 ˆ t ( s ) β 0 = ρxy, cv( x) cv( y) +. t ˆ 0 β 0 ˆ t PUV PUV P L ( s ) 0 s 0 s β = == 0 2 ρxy, cv ( x) cv ( y) + t ˆ 0 PF PL PF β

11 t The difference between a Fisher price and a unit value index The first ter is the substitution effect. The second ter is the levels effect, that is the effect, for negatively sloping deand, of uantities shifting to prices at a lower level. The easure is based on the ratio of the slope coefficients fro the regressions of t on p 0 and of 0 also on p 0. As the slope of the period t line say increases, above average prices have lower uantities and below average prices have higher uantities the larger the increase, the greater the shift. The change in slopes capture a shift in levels.

12 Figure, Depiction of levels effect t* t, 0 t 0 0 t* 0 t 0 0 p p

13 Decoposition helps Identifies role of substitution effect; the unit value bias will be eual to zero if: all base period price OR uantity changes are eual to each other OR there is no (weighted) correlation between the base period price and uantity changes; AND all base period prices OR base and current period uantities are eual to each other OR there is no (unweighted) correlation between the base period prices and base and current period uantities; relative position of UV, Las, Pas, and Fisher; identifies levels effect; product heterogeneity via CVs.

14 When to use unit value indexes and when Fisher? Balk: if the splitting into hoogeneous and heterogeneous ites was not feasible, he advised a price index. Diewert: if detailed data on strictly hoogeneous goods unavailable (specified ite by outlet for CPI), then unit value over outlets. Dálen argued for uality-adjusted unit value indexes that reove the effect on prices of product heterogeneity and de Haan ipleented this in a hedonic setting.

15 If unit value indices are right for hoogeneous ites, what about broadly coparable goods and services? What about a uality-stripped unit value index? Dálen (200); De Haan (2004 and 2007)? Consider a regression of price on k uality characteristics: K τ τ τ τ p = β + 0 βk zk + k = K τ τ = τ τ pˆ p βk zk k = K ~ τ τ = + τ τ p β0 βk zk k = u τ ( τ ) z k P M M t t 0 0 pˆ ˆ pnn * = = M M t 0 n = = U =

16 But for broadly coparable goods soe of the price change is a unit value shift in levels and soe not... Weighted average of Fisher and ualityadjusted Fisher Weights based on: elasticity of substitution proportion of price variation due to heterogeneity: RSS M M M M t t t t t t 0 p ˆ p % p % * * = = = = U + F U = M M U + M M U t 0 0 p ˆ p % p % = = = = Pw P( w ) w ( w ) U

17 An appropriate index.. should have the property that if all price variation is explained by the hedonic regression, the index is a Fisher index; if none of the price variation is explained by the hedonic regression, the index is a unit value index; as the percentage of price variation explained by the hedonic regression increases, so too will the weight given to the Fisher coponent.

18 Epirical work Useful work by de Haan on unit values and ualityadjusted ones: TVs, washing achines, refrigerators and PCs for atched and unatched, OLS and WLS. For all products, except PCs, QAUV was less than weighted TDI, and for PCs pretty close. This work in progress is on television sets for atched odels sipler it is clear that hedonic is about product heterogeneity only. The decoposition is to explain why forulas differ in ρ ters of signs on rho and agnitude of CVs and to exaine different forulas.

19 Epirical work Figure, Unit value and price indices for 4in TVs Index January = Laspeyres Paasche Fisher Unit value

20 Epirical work Figure 2, Quality adjusted unit value and Fisher price indices. Index january= QA unit value Wgtd Avg QAUV and Fisher Fisher Unit value

21 Suary For the aggregation of hoogeneous ites, the unit value index is the best index and superlative index nubers biased, and for the aggregation of heterogeneous ites, superlative index nubers are best index and unit value index nubers biased. Institutionalized price discriination is exception for buyer of hoogeneous ites, but not seller. The factors deterining the difference between unit value indices and Laspeyres, Paasche and Fisher price indices were established. Quality adjustents to the prices to itigate price dispersion due to the slight product heterogeneity would be appropriate for unit value indices. For broadly coparable goods and services, it is ore coplex.