## _wp06174

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### Introduction
- Choice of lower-level CPI aggregation formula materially affects measured inflation: a change to a weighted geometric mean in January 1999 was estimated to reduce the annual rate of increase in the U.S. CPI by approximately 0.2 percentage points, implying a cumulative additional national debt from over-indexing of more than $200 billion over the 12-year period up to the mid-1990s (estimates based on Boskin et al., 1996 and 1998).
- This paper analyzes differences between two unweighted lower-level index formulas — the ratio of unweighted arithmetic means (Dutot) and the ratio of unweighted geometric means (Jevons) — in terms of price dispersion and product heterogeneity, using detailed scanner data.
- Empirical dataset summary:
  - 31,352 observations over 24 months.
  - 4.6 million transactions worth £1.7 billion.
  - Matched-sample observations for matched model comparisons: 20,574 observations.
  - Example observation: Toshiba 2173DB 21 inch model in January 1998 — unit value £275.80, volume 5,410 transactions.

### Elementary index formulas, usage, and axiomatic properties
- Main lower-level formulas for m = 1,..,M matched items:
  - Carli price index (arithmetic mean of price relatives), PC.
  - Dutot price index (ratio of arithmetic means), PD.
  - Jevons price index (geometric mean of price relatives, equal to ratio of geometric means), PJ.
- Usage in a sample of 37 countries:
  - 13 countries used the Dutot index.
  - 14 countries used the Jevons index.
  - 4 countries used the Carli index.
  - 6 countries primarily used Jevons but with Dutot/Carli for specific product groups.
- Axiomatic performance:
  - Carli fails the time reversal test and is upwards-biased.
  - Jevons satisfies all main tests.
  - Dutot satisfies main tests except the commensurability test (index may change if units change); Dutot can be used for homogeneous items as advised in the CPI Manual (ILO et al., 2004).
- Theoretical economic justifications:
  - Jevons is supported under substitution behavior (Cobb-Douglas, unitary cross-elasticities) and sampling proportional to base-period expenditure shares.
  - Dutot corresponds to Leontief preferences (no substitution) and sampling proportional to base-period quantities.

### Analytical results: relationship between Dutot and Jevons
- The difference between Dutot and Jevons depends on changes over time in price dispersion; an exact analytical expression links population Dutot and Jevons indexes to the difference in variances of log-prices between periods (equation (10) in the source).
- Statistical properties of sample indexes:
  - Sample Carli index PC: consistent and unbiased estimator of population Carli.
  - Sample Dutot PD: consistent but not unbiased estimator of the ratio of population means (population Dutot).
  - Sample Jevons PJ: consistent estimator of the population Jevons index but subject to small-sample bias.
- Hedonic (heterogeneity-controlled) framework:
  - Hedonic regression form: ln pmτ = β0 + β1 Dt + ∑k=2..K βk zkmτ + u mτ with u mτ ~ Normal(δτ, 2 τξ).
  - Hedonic-estimated Jevons index: P̂J* = exp(β̂1).
  - Consistent heterogeneity-adjusted Dutot estimator:
    - P̂D* = P̂J* . exp(.5(ξ̂0^2 − ξ̂t^2))
  - Difference between Jevons and Dutot hedonic price index is directly related to the change in variance of residuals (ξ̂t^2 − ξ̂0^2).
  - As product heterogeneity and price dispersion decline (2̂τξ → 0 for 0,tτ=), P̂D* → P̂J*.

### Empirical findings (U.K. TV scanner data, Jan-1998 to Dec-1999)
- Hedonic regressions:
  - Monthly regressions included 37 brand dummies, 3 outlet-type dummies, 19 screen size dummies, 6 tube-type dummies, and 23 further characteristics.
  - Regression performance: average R^2 = 0.91; maximum R^2 = 0.93; minimum R^2 = 0.89.
- Price-index movements (January 1998 to December 1999):
  - Dutot (raw) price fall: 26.4 percent.
  - Jevons price fall: 28.4 percent.
  - Heterogeneity-controlled Dutot (hedonic-adjusted) price fall: 27.2 percent.
- Variance and index relationships:
  - The Dutot index was 2.7 percent higher than Jevons in December 1999.
  - The variance-change adjustment from the analytical relationship (equation (10)) accounts very closely for the Dutot–Jevons difference; residual divergence is small and attributable to sample rather than population indexes and to deviations from distributional assumptions.
  - Hedonic adjustment reduced residual variances and brought Dutot closer to Jevons across periods (see Table 1 in source).

### Implications for CPI compilation
- The Dutot–Jevons difference is explainable by changes in price dispersion and by product heterogeneity; both analytical relation (variance-change term) and hedonic controls confirm this.
- Practical implications and sampling assumptions:
  - Jevons can be regarded as a sample estimator of a Törnqvist index under:
    1) sampling proportionate to base-period expenditure shares, and
    2) unity elasticity of substitution (equal base and current period expenditure shares).
  - Dutot can be shown to be a sample estimator of:
    - Laspeyres under sampling proportionate to quantity shares in the base period.
    - Paasche under sampling proportionate to quantity shares in the current period.
    - Fisher if base and current period quantity shares are equal.
  - Assumptions required for Jevons are generally less restrictive than those for Dutot, but their validity in practice is often questionable.
- Matched vs. unmatched samples:
  - Analysis used matched samples following statistical office practice.
  - For high-turnover product areas (domestic appliances, personal computers), unmatched new and disappearing models likely have different residual distributions; increased dispersion from unmatched items would enlarge the Dutot–Jevons difference relative to matched-sample results.
- Practical considerations:
  - Jevons does not fail the commensurability test and is unaffected by use of residuals from matched samples.
  - Geometric mean (Jevons) is more robust to outliers but less easy to explain to users than the arithmetic mean.

### Summary and policy guidance for statistical offices
- Analytical contributions:
  - Provides an exact expression for the Dutot–Jevons difference and frames sample indexes as estimators of population counterparts.
  - Decomposes the difference into components due to product heterogeneity (explained via hedonic residual variances) and due to the indexes being different types of averages.
- Practical recommendations:
  - (i) Jevons preferred where Dutot’s failure of the commensurability test and product heterogeneity are material — in the empirical TV example, heterogeneity accounted for over half the Dutot–Jevons difference.
  - (ii) Where heterogeneity explains only a small share of differences, choose formula based on sample design assumptions (see sampling-to-index mappings above).
  - (iii) For product areas with large product turnover, the case for a geometric formulation (Jevons) is stronger.
  - (iv) Much of the analysis carries over to producer price indexes, except where consumer cost-minimizing assumptions differ from producer revenue-maximizing behavior.
  - Note: the geometric mean is more robust to outliers but less straightforward to communicate than the arithmetic mean.

### Data annex — variable set and sample characteristics (selected)
- Observation-level variables: price (unit value across transactions in month/outlet-type), volume (number of transactions).
- Brand variables: 38 brands — 37 dummies benchmarked on Sony (excluded).
- Characteristics included (dummy variables unless stated): about 19 screen-size dummies; Nicam stereo sound; wide screen; teletext/fastext/top fastext; 6 reception-system types; continental monitor style; Dolby variants; tube types; s-vhs socket; satellite tuner analogue/digital; digital; DVD playback/recording; rear speakers; PC-internet flags; real flat tube; 100 hertz refresh; vintage; DIST (percentage of outlets where model sold).
- Outlet-types: multiple chains, mass merchandisers (department stores), independents, and catalogue stores.

*Source: Excerpt from _wp06174 (selected sections I–IV) as provided in the supplied content.*

### References..............................................................................................................

### _wp06174 - References..............................................................................................................

### Introduction
- Choice of formula for lower-level CPI aggregation materially affects measured inflation: change to a weighted geometric mean in January 1999 was estimated to reduce the annual rate of increase in the U.S. CPI by approximately 0.2 percentage points, implying a cumulative additional national debt from over-indexing of more than $200 billion over the 12-year period up to the mid-1990s (estimates based on Boskin et al., 1996 and 1998).
- Lower-level aggregation formulas are applied to finely defined goods (e.g., varieties of apples) and typically use unweighted averages of price observations; these lower-level indexes are building blocks of a CPI.
- This paper analyzes differences between two unweighted lower-level index formulas — the ratio of unweighted arithmetic means (Dutot) and the ratio of unweighted geometric means (Jevons) — in terms of price dispersion and product heterogeneity, using detailed scanner data.
- Empirical dataset: highly detailed scanner data from retailers’ barcode readers amounting to about 31,000 observations over 24 months on prices, characteristics, and brands of models of television sets sold in different outlet types.

### Elementary index number formulas, their use and justification
- Main lower-level formulas for m = 1,..,M matched items (prices pmτ and quantities qmτ for periods 0,tτ=) include:
  - Carli price index (arithmetic mean of price relatives), PC.
  - Dutot price index (ratio of arithmetic means), PD — which can be seen as a base-period price-share weighted Carli index.
  - Jevons price index (geometric mean of price relatives, equal to ratio of geometric means), PJ.
- Usage across CPIs (sample of 37 countries): 13 countries used the Dutot index, 14 the Jevons index, 4 the Carli index, and 6 other countries primarily used Jevons but with Dutot/Carli for specific product groups.
- Axiomatic performance:
  - Carli fails the time reversal test and is upwards-biased.
  - Jevons satisfies all main tests.
  - Dutot satisfies main tests except the commensurability test (if units change, the index may change); Dutot can be used for homogeneous items as advised in the CPI Manual (ILO et al., 2004).
- Economic (theoretical) justification:
  - Jevons: weak support from economic approach; justified when consumers substitute away from items with above-average price increases and cross-elasticities of demand are unitary (Cobb-Douglas preferences); sampling with probability proportional to base-period expenditure shares and constant expenditure shares makes Jevons acquire properties of symmetric superlative index.
  - Dutot: economic justification relies on Leontief preferences (no substitution); corresponds to sampling homogeneous items with probabilities proportional to base-period quantities.

### Differences between the Jevons and Dutot formulas
- Analytical focus: difference between Dutot and Jevons depends on changes over time in price dispersion; paper derives an analytical framework distinguishing calculated sample indexes as estimators of population counterparts.
- Statistical properties of sample indexes:
  - Sample Carli index PC is a consistent and unbiased estimator of the population Carli index.
  - Sample Dutot PD is a consistent but not unbiased estimator of the ratio of population means (population Dutot index).
  - Sample Jevons PJ is a consistent estimator of the population Jevons index but subject to small-sample bias as noted by McClelland and Reinsdorf (1999).
- Jensen’s inequality implies difficulty in determining dominance between Dutot and Jevons without distributional assumptions.
- Under the lognormal distribution assumption for log prices (logpτ ~ Normal(μτ, 2 τε) for 0,tτ=), a relationship is derived linking population Dutot and Jevons indexes to the difference in variances of log-prices between periods 0 and t (equation (10) in the source).
- Hedonic (heterogeneity-controlled) approach:
  - Use hedonic regression ln pmτ = β0 + β1 Dt + ∑k=2..K βk zkmτ + u mτ with u mτ ~ Normal(δτ, 2 τξ).
  - Hedonic estimated Jevons index: P̂J* = exp(β̂1).
  - Consistent estimator of heterogeneity-adjusted Dutot index:
    - P̂D* = P̂J* . exp(.5(ξ̂0^2 − ξ̂t^2))
    - Equivalently presented in the source as equation (13): ()() *22*22 100 ˆ ˆ exp.exp/ 2.exp/ 2 DtJt IPβξξξξ ⎡⎤⎡⎤ =−=− ⎣⎦⎣⎦ .
  - Difference between Jevons and Dutot hedonic price index is related to the change in the variance of residuals (ξ̂t^2 − ξ̂0^2).
  - As product heterogeneity and price dispersion decrease, the difference between the two indexes decreases; when 2̂τξ → 0 for 0,tτ=, P̂D* → P̂J*.

### Empirical work
- Data and variables:
  - Monthly scanner data on U.K. TV prices from January 1998 to December 1999; supplemented by price collectors for outlets without barcode readers (negligible).
  - Example observation: Toshiba 2173DB 21 inch model in January 1998 — unit value (price) £275.80, volume 5,410 transactions.
  - For the 24 months there were 31,352 observations covering 4.6 million transactions worth £1.7 billion.
  - Price indexes calculated over matched identical models; 20,574 observations used for matched price comparisons for January 1998 to December 1999.
  - Matched-sample practice: matched models vary month-to-month; sample renewed in January 1999 and linked to January 1998 = 100 reference period.
  - Hedonic regressions included 37 brand dummies, 3 outlet-type dummies, 19 screen size dummies, 6 tube-type dummies, and 23 further characteristics; estimated monthly for January 1998–December 1999.
  - Hedonic regression performance: average R^2 = 0.91 with maximum 0.93 and minimum 0.89; estimated coefficients almost invariably statistically significant with expected signs.
- Dutot and Jevons empirical findings:
  - Over the two-year period prices fall by 26.4 and 28.4 per cent respectively for the Dutot and Jevons indexes.
  - Columns 6 and 7 in Table 1 compare the ratio of Dutot to Jevons with and without the adjustment from equation (10) for calculated variances. In December 1999 the Dutot index is 2.7 percent higher than Jevons.
  - The variance-change adjustment from equation (10) accounts very closely for the Dutot–Jevons difference (column 7 in Table 1 close to unity), with only very slight divergence due to sample (rather than population) indexes and distributional assumption deviations.
- Heterogeneity-controlled Dutot indexes:
  - Using the same matched sample, the heterogeneity-controlled Dutot index shows a price fall of 27.2 percent, compared with 26.4 percent for the raw Dutot and 28.4 percent for Jevons (all figures for January 1998 to December 1999).

*Source: Excerpt from _wp06174 (selected sections I–IV) as provided in the supplied content.*

### 28.4 percent for the Jevons index.

### _wp06174 - 28.4 percent for the Jevons index.

### Implications for CPI compilation
- The difference between the Dutot and Jevons indexes can be explained in terms of the change in the dispersion of prices (equation (10)), as confirmed by the empirical work in section IV.
- Hedonic regressions can be used to control for price dispersion arising from product heterogeneity and to further explain the difference between the Jevons and Dutot indexes (equation (13)).
- In the empirical illustration:
  - Product heterogeneity was shown to sizably affect the Dutot index.
  - The reduction in price dispersion via hedonic regression served to bring the heterogeneity-controlled Dutot index closer to the Jevons index, accounting for over half of the disparity between the two raw price indexes in Columns 2 and 3.
  - The Jevons index does not fail the commensurability test and is thus unaffected by the use of residuals from matched samples.10
- Choice between Jevons and (heterogeneity-controlled) Dutot can be considered in terms of which approximates a superlative index (Fisher, Törnqvist, Walsh), but:
  - Superlative formulas use base and current period quantities, which Dutot and Jevons do not (they are unweighted).
  - Quantity/expenditure weights can be implicitly introduced if sampling is with probability proportionate to size (where “size” is the quantity/expenditure share).
- Sampling assumptions and implications:
  - Under assumptions of 1) sampling of prices with probability proportionate to base period expenditure shares, and 2) unity elasticity of substitution (thus equal base and current period expenditure shares), the Jevons index is a sample estimator of a Törnqvist index.
  - Under sampling with probability proportionate to quantity shares in the base period (current period) the Dutot index can be shown to be a sample estimator of a population Laspeyres (Paasche) index, and with a further assumption of equal base and current period quantity shares, the Dutot index is a sample estimator of a Fisher index.
  - The assumptions required for the Jevons index are not as limiting as those for the Dutot index, though their veracity in real CPI practice will often be questionable.
- Matched vs. unmatched samples:
  - Analysis was conducted for matched samples, following practice largely used by statistical offices.
  - For product areas with large product turnover (e.g., domestic appliances, personal computers), “new” unmatched models introduced in period t and “old” unmatched models disappearing from the sample are unlikely to have residuals of the same magnitude as matched models.
  - Silver and Heravi (2005) found for cameras, dishwashers, television sets, vacuum cleaners, and washing machines that the mean residual for unmatched new models was above the corresponding mean for unmatched old ones; the resulting increase in dispersion would produce a larger difference between Dutot and Jevons than reported for conventional CPI matched-model measurement.

10 The Jevons index does not fail the commensurability test and is thus unaffected by the use of residuals from matched samples.

### Summary and policy guidance for statistical offices
- Analytical contribution:
  - The paper provides an exact expression for the difference between Dutot and Jevons indexes (improving on prior Taylor-approximation–based work) and expresses indexes as sample estimators of population counterparts.
  - The difference is decomposed into that due to product heterogeneity and that due to the indexes being different types of averages.
- Practical advice:
  (i) Jevons can be said to be better because Dutot fails the commensurability (units of measurement) test and Jevons does not; in the empirical example product heterogeneity accounted for over half the difference between the two formulas, which argues for Jevons.
  (ii) If product heterogeneity explains only a small proportion of the difference, choice between formulas should be based on sample design assumptions:
    - Jevons as sample estimator of Törnqvist under: 1) sampling proportionate to base-period expenditure shares, and 2) unity elasticity of substitution.
    - Dutot as sample estimator of Laspeyres (Paasche) under sampling proportionate to quantity shares; and as Fisher under equal base and current period quantity shares.
    - Statistical offices might use different formulas for different product areas depending on the likely validity of these assumptions.
  (iii) For product areas with large product turnover the case for a geometric formulation (Jevons) is stronger.
  (iv) Much of the analysis carries over to producer price indexes, except where the economic assumption (cost-minimizing consumer behavior) differs from revenue-maximizing producer behavior (see Diewert, 2004).
- Additional note:
  - The geometric mean is more robust to outliers but less easy to explain than the arithmetic mean.

### Data annex — variable set and sample characteristics
- Variables on each observation included:
  - price, unit value of a model across all transactions in a month/outlet-type, volume (number of transactions during the period in the outlet type).
  - 38 brands — 37 dummy variables benchmarked on Sony (the excluded brand in the regression).
  - Characteristics (dummy variables unless stated):
    (i) size of screen — dummy variables for about 19 screen sizes;
    (ii) possession of Nicam stereo sound;
    (iii) wide screen;
    (iv) on-screen text retrieval news and information panels: teletext, fastext and top fastext — 3 dummy variables;
    (v) 6 types of reception systems — 5 dummy variables;
    (vi) continental monitor style;
    (vii) Dolby Pro, Dolby SUR/DPL, Dolby Digital sound — 3 dummy variables;
    (viii) Flat & Square, Super-Planar tubes — 2 dummy variables;
    (ix) s-vhs socket;
    (x) satellite tuner, analogue/digital — 2 dummy variables;
    (xi) digital;
    (xii) DVD playback or DVD recording — 2 dummy variables;
    (xiii) rear speakers;
    (xiv) without PC-internet/PC+internet;
    (xv) real flat tube;
    (xvi) 100 hertz, doubles refresh rate of picture image;
    (xvii) vintage;
    (xviii) DIST — the percentage of outlets in which the model was sold.
- Outlet-types: multiple chains, mass merchandisers (department stores), independents and catalogue stores.

### Table 1 — noted sample-index benchmarks and variances (selected)
- January 1998 = 100 for both Dutot and Jevons sample indexes (Jan-98 100.00, 100.00).
- The heterogeneity-controlled Dutot sample index is reported alongside sample variances in Table 1; hedonic adjustment reduced variances (last two columns) relative to raw variances (columns 4 and 5), bringing Dutot closer to Jevons across periods (examples include Feb-98, Mar-98, Apr-98, etc., as shown in Table 1).
- The empirical table reports monthly values for Dutot (D), Jevons (J), sample variances avarlogpτ2, var m uτ2 ξ, ratios D/J, and heterogeneity-controlled Dutot *DˆIJan ξ, for the period Jan-98 through Dec-99 (selected months and values are presented in Table 1 of the source).

*Source: _wp06174 - 28.4 percent for the Jevons index.*

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