## _wp12105 — Section IIB continues by considering the nature of and case for the geometric equivalents of

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---

### Geometric counterparts considered
- Geometric Young (Cobb‑Douglas) and Geometric Lowe indexes are presented as geometric equivalents of the arithmetic Young and Lowe indexes.
- Characteristics:
  - Use the same weights and price relatives as the Young and Lowe indexes and "can be readily adopted by statistical offices."
  - Share the practical advantage of being computable in real time.
- Formal definitions:
  - Geometric Young price index is given by equation (4).
  - Geometric Lowe price index is given by equation (5).
  - Törnqvist (superlative) index is given by equation (6).
  - Under constant expenditure shares over periods 0 and t (unitary elasticity of substitution), Geometric Young (4) equals Törnqvist (6).

### Formal and axiomatic observations
- Laspeyres:
  - Advantage: upper bound to a theoretical cost-of-living index (COLI).
- Lowe:
  - "Likely to fall above Laspeyres."
  - As a fixed quantity basket index it is "easy to explain; biased but easily explained."
  - Substitution bias: Balk and Diewert (2003) establish a Lowe CPI substitution bias; Lowe likely has an upward substitution bias relative to Laspeyres, and Laspeyres has an upward substitution bias relative to a superlative index.
  - Lowe has good axiomatic properties (ILO et al., 2004, chapter 16) despite substitution bias concerns.
- Young:
  - Fails the circularity and time reversal tests.
  - The Young index between periods 0 and t will exceed its time antithesis (its inverse between period t and 0) by a term equal to the Young index times the weighted variance of deviations of price relatives and their mean; since the variance must be positive, Young must exceed the inverse of its time antithesis except when there is no price change dispersion.
  - Analytical shortcomings: uncertainty a priori about the extent and nature of its deviations from Laspeyres and relatively poor axiomatic properties.
- Geometric Young and Geometric Lowe:
  - Geometric Young is consistent with unitary elasticity of substitution (geometric/Cobb‑Douglas preferences) and, under that condition, lies within the Laspeyres‑Paasche interval.
  - Geometric Young is less susceptible to substitution bias than the arithmetic Lowe (Balk, 2010).
  - Geometric Lowe has little standing in practice; the price‑updating rationale supporting the arithmetic Lowe has no analogous economic rationale for Geometric Lowe.

### Empirical evidence and findings (survey and literature)
- Hansen (2007) — UNECE/ILO survey:
  - Of 47 respondents as at September 2007, 32 national statistical offices used the (price‑updated) Lowe index and 15 used the original (survey period) Young weights.
  - Using Danish CPI data for 1996 to 2003, increases were: Young 17.49 percent, Lowe 18.01 percent, Törnqvist 17.08 percent. Geometric Young was below Törnqvist in Hansen’s results.
- U.S. CPI empirical results (section IV summary):
  - Geometric Young (unitary elasticity) has a downward bias.
  - U.S. data over the period studied demonstrate inelastic substitution (Greenlees, 2011).
  - Averaging can substantially offset Geometric Young’s downward bias; averaging variants have formal justification from Lent and Dorfman (2009).
  - Lowe price index (used in the U.S. and many countries) has a bias (against superlative indexes) several times that of some averaging variants; all averaging variants can be computed in real time using the same database as the Lowe index.
- Practical guidance:
  - CPI Practical Guide supports consideration of Geometric Young formulas (UNECE et al., 2009, page 160, ff. 50).
  - CPI Manual regards Geometric Young as a "serious practical possibility" but notes it is unlikely to gain general acceptance because "it cannot be interpreted as a fixed quantity basket index."

### Practical implications and operational recommendations
- Rationale for considering geometric counterparts:
  - Geometric Young can be computed in real time using existing weights and price relatives.
  - Geometric formulations maintain consistency with Jevons elementary‑level geometric practice and help preserve aggregation consistency.
  - Geometric Young exhibits lower substitution bias relative to arithmetic Lowe and can approximate superlative indexes under unitary elasticity of substitution.
- Operational note:
  - Statistical offices could compute Geometric Young and Geometric Lowe using the same real‑time data currently used for Young and Lowe to evaluate proximity to superlative indexes (e.g., Törnqvist) and to assess biases empirically.

*Source — _wp12105 (Section IIB and surrounding sections) from the provided PDF content.*

---

### _wp12105 — 16.51 percent. The differences between Young and Lowe are not always trivial.

### Summary findings (selected empirical comparisons)
- Paper documents substantive, non‑trivial differences between Lowe and Young aggregation formulas, and between their geometric counterparts and a superlative Törnqvist index.
- Selected empirical comparisons:
  - Annual inflation for 2004/5 and 2005/6: Young increased from 1.80 to 1.88 percent; Lowe decreased from 1.99 to 1.90 percent.
  - Greenlees and Williams (2010), US CPI, Dec 1990–Dec 2008: Lowe 18.88 percent, Young 18.24 percent, chained Törnqvist 16.78 percent; Geometric Young 15.84 percent.
  - Pike et al. (2009), New Zealand, June 2006–June 2008: Lowe 6.26 percent, Young 5.60 percent, Geometric Young 4.83 percent, retrospective Fisher 5.73 percent.
- Authors’ US dataset (Dec 1997–Dec 2010):
  - Fisher index increased in 2010 compared with 1998 at a compound annual average rate of 2.27 percent, compared with Lowe price index increase of 2.44 percent.
- General pattern:
  - Geometric Young generally falls below, and sometimes is closer to, the superlative Törnqvist index.
  - Lowe and Young may produce similar results, but their differences from Törnqvist are marked.

### Analytical decomposition: why Geometric Young and Geometric Lowe differ
- Log decompositions (equations (7)–(10)) identify determinants of the sign and magnitude of 0t GLo I / 0t GY I (equation (10)):
  - Period‑b weighted correlation between price relatives and logged price relatives, denoted ,b i s xy ρ —its sign dictates whether Geometric Lowe exceeds Geometric Young (positive) or vice versa (negative).
  - Dispersion of price changes: b i s x cv (coefficient of variation for x) and b i s y σ (standard deviation for y). Higher dispersion increases the magnitude of the difference; b i s y σ is the most potent driver because it is not corrected for changes in the mean.
  - Multiplicative interaction of correlation and dispersion terms — reducing any one term (e.g., minimizing weight‑update lags) reduces the difference between formulas.
- Conceptual clarification: difference is not characterized as substitution bias but stems from correlation across price‑change periods (price relatives vs logged price relatives).

### Comparisons with a superlative (Törnqvist) price index and elasticity implications
- Ratios of Geometric Lowe and Geometric Young to Törnqvist derived (equations (11)–(12)).
- Equality condition: Geometric Young equals Törnqvist if period‑b shares equal the average of period‑0 and period‑t shares (b i s = (0 i s + t i s)/2).
- Lloyd‑Moulton (CES) index (equation (13)): parameter η is elasticity of substitution.
  - Greenlees (2011) estimates η for US data: η varies between 0 and 1 for most years (inelastic substitution), with η = 0.521 to 0.655 for 1999/2000 to 2005/6, η = 0.981 for 2006/7, and η = 0.192 for 2007/8.
  - Inelastic substitution (η < 1) implies items with relatively higher price trends receive less importance in 0t GY I than in 0t T I, i.e., 0t GY I < 0t T I.
- Lent and Dorfman (2009) propose AG Mean (weighted average of arithmetic Laspeyres and Geometric Laspeyres) to approximate a superlative index when 0 ≤ η ≤ 1 (equations (14)–(16)); averaging arithmetic and geometric versions may better approximate superlative targets.

### Empirical results and data (authors’ US CPI analysis)
- Data and scope:
  - Elementary aggregate indexes and weights for U.S. Urban CPI, Dec 1997–Dec 2010, from U.S. BLS.
  - Elementary aggregates correspond to about 211 item strata.
  - CPI compilation: 211 product‑group strata × 38 area strata = 8,018 cells.
  - BLS uses weighted geometric means for most strata and Lowe for area aggregation; authors analyze using only the 211 product‑group weights.
- Weighting periods (mean‑annual expenditures → basis of weights):
  - 1993-1995 → Jan98-Dec01
  - 1999-2000 → Jan02-Dec03
  - 2001-2002 → Jan04-Dec05
  - 2003-2004 → Jan06-Dec07
  - 2005-2006 → Jan08-Dec09
  - 2007-2008 → Jan10-Dec11
  - 2008-2009 → Jan11-Dec12
- Performance highlights:
  - Authors’ chained Törnqvist closely matches BLS’ chained Törnqvist (C‑CPI‑U): correlations 0.99978 (levels) and 0.98855 (monthly annual changes) over Dec 1999–Dec 2010.
  - Figure descriptions:
    - Arithmetic Lowe is above Laspeyres; Young is closer to Laspeyres.
    - Geometric‑Lowe, Geometric Young, and Törnqvist track closely early in series; divergence after Dec 2003 with Geometric Lowe closer to Törnqvist than Geometric Young for Dec 2003–Dec 2007 in this dataset.
  - Geometric Young showed a marked downward bias relative to Törnqvist in authors’ US data, while Lowe tended to overstate Törnqvist — suggesting averaging the two formulas may better approximate the superlative target.

### Interpretation and operational implications
- Relative positioning of Geometric Lowe vs Geometric Young depends on long‑run persistence in price trends between the survey/reference period (b→0) and comparison period (0→t).
  - Empirical sign changes in ,b i s xy ρ (positive Jan 2008–Oct 2008, negative from Nov 2008 onward) illustrate long‑run trend reversals can change which geometric formula grows faster.
- Lags in weight updating (longer b→0 and 0→t spans), higher inflation, or higher relative‑price dispersion amplify differences between geometric formulas.
- Because correlation, coefficient of variation, and standard deviation are multiplicative, reducing any one (e.g., reducing weight‑update lags) reduces divergence.
- Practical consequence: with infrequent weight updates, choice among Lowe, Young, Geometric Lowe, and Geometric Young materially affects measured inflation relative to superlative indexes; averaging approaches (Lent‑Dorfman / AG Mean) can produce measures closer to superlative targets when η lies between 0 and 1.

*Source — _wp12105 - 16.51 percent. The differences between Young and Lowe are not always trivial.*

---

### _wp12105 — 2010. Lowe has the largest bias of about 1.7 percent from a superlative index and Young

### Key empirical findings (2010 summary)
- Lowe has the largest bias of about 1.7 percent from a superlative index; Young reduces the bias to about 0.5 percent.
- Geometric formulas relative to Törnqvist:
  - Geometric Lowe is, on average, 0.2 percent above Törnqvist.
  - Geometric Young is about 1.0 percent below Törnqvist.
- Fisher and Törnqvist track each other closely: Fisher annual growth 2.49033 and Törnqvist 2.49316 over the period December 198 to December 2010.
- In real time using the same database as Lowe, Lowe’s bias can be cut from 1.7 percent to 0.3 percent using Lent‑Dorfman (L‑D) approximations.
- Recommendation: Geometric Young:Lowe formulation is recommended as an adjustment, via Geometric Young, of the existing Lowe formula.
- Operational note: Geometric Young and Geometric Lowe are readily available compilers’ formulas; Geometric Young has a clear interpretation as weighted geometric average using survey‑period shares, while Geometric Lowe "has no meaningful interpretation."
- Averaging:
  - L‑D approximations (average of Lowe:Geometric Young and average of Young:Geometric Young) track Törnqvist and alleviate downward bias in Geometric Young and upward bias in Lowe/Young.
  - Simple geometric means of arithmetic and geometric formulas are described as “highly successful” and “vast improvements on other standard arithmetic and geometric formulas.”

### Methodology and formula adjustments
- Approximations to Laspeyres and Geometric Laspeyres (equation (14)) use Lowe/Young to approximate Laspeyres and Geometric Lowe/Geometric Young to approximate Geometric Laspeyres.
- Variants include geometric average of Young and Geometric Young, and geometric average of Lowe and Geometric Young.
- Tailoring of estimates of τη in (15) is based on a selected superlative benchmark (e.g., Törnqvist). Because Törnqvist cannot be estimated in real time, the most recently available estimates are used for real‑time computation.
- Lent‑Dorfman (L‑D) allows τη to change monthly but constrains changes to index rebasing points and holds τη constant until the next rebasing.
- Estimates of τη using equation (16) may be negative depending on relative positions of Geometric Young and Young to Fisher; adaptation using absolute values (equation (17)) ensures non‑negative weights.

### Performance metrics (average monthly percentage differences vs. target indexes, January 2004–December 2010)
- Targets: Fisher and Törnqvist.
- Arithmetic formulas:
  - Lowe: 1.712 (Fisher), 1.689 (Törnqvist)
  - Young: 0.466 (Fisher), 0.443 (Törnqvist)
- Geometric formulas:
  - Geometric Lowe (GLowe): 0.265 (Fisher), 0.242 (Törnqvist)
  - Geometric Young: -0.959 (Fisher), -0.981 (Törnqvist)
- Geometric means / hybrids:
  - GYoung‑Young: -0.249 (Fisher), -0.272 (Törnqvist)
  - GYoung‑Lowe: 0.368 (Fisher), 0.345 (Törnqvist)
  - GLowe:Young: 0.366 (Fisher), 0.343 (Törnqvist)
  - GLowe:Lowe: 0.986 (Fisher), 0.963 (Törnqvist)
- Lent‑Dorfman (η using lag):
  - GYoung:Young: -0.339 (Fisher), -0.375 (Törnqvist)
  - GYoung:Lowe: -0.196 (Fisher), -0.219 (Törnqvist)
  - GLowe:Young: 0.339 (Fisher), 0.316 (Törnqvist)
  - GLowe:Lowe: 0.581 (Fisher), 0.558 (Törnqvist)

### Conceptual and practical implications
- Arithmetic Laspeyres‑type aggregation formulas at higher weighted levels for CPIs (Young and Lowe) have limited theoretical justification and show upward drift against Laspeyres; Lowe drifts more.
- Lowe (and Laspeyres) advantage: ease of interpretation as fixed quantity basket but yields a "well‑defined, but biased, result."
- L‑D index: conceptual basis as a Taylor approximation to Törnqvist (or Fisher) and can be calculated in real time; picks up changes in buying patterns without iterative numerical procedures and is easily automated.
- Geometric Lowe vs Geometric Young difference is driven by long‑run unidirectional price changes (equations (10) and (11)), making Geometric Lowe unreliable for a predictable relationship with Geometric Young/Törnqvist.

### Policy recommendations and operational suggestions
- Debate on moving beyond Laspeyres‑type indexes is warranted; compile retrospective indexes to assess substitution bias magnitude.
- CPI users would be better served by a real‑time measure that more closely tracks a superlative index; public acceptance may follow demonstrated superior performance.
- Statistical agencies can compile L‑D approximations and simple geometric‑means adjustments in real time using existing price databases to substantially reduce Lowe’s bias.
- Geometric Young:Lowe and related L‑D adjustments are practical candidate formulas for immediate improvement, preserving real‑time computability and operational simplicity.

*Source — Excerpted content from _wp12105 - 2010.  Lowe has the largest bias of about 1.7 percent from a superlative index and Young*

### Section IIB continues by considering the nature of and case for the geometric equivalents of

### _wp12105 - Section IIB continues by considering the nature of and case for the geometric equivalents of

### Geometric counterparts considered
- Focus on the Geometric Young (sometimes referred to as the Cobb-Douglas) and Geometric Lowe indexes as geometric equivalents of the arithmetic Young and Lowe indexes.
- These geometric indexes:
  - Use the same weights and price relatives as the Young and Lowe indexes and "can be readily adopted by statistical offices."
  - Share the practical advantage of their arithmetic counterparts of being computable in real time.
- The Geometric Young price index is given by equation (4) and the Geometric Lowe by equation (5) in the source.
- The Törnqvist (superlative) index is given by equation (6). For constant expenditure shares over periods 0 and t (consistent with unitary elasticity of substitution), the Geometric Young index (4) equals the Törnqvist index (6).

### Formal and axiomatic observations
- Laspeyres:
  - Has the advantage of being an upper bound to a theoretical cost-of-living index (COLI).
- Lowe:
  - The widely used Lowe index is "likely to fall above Laspeyres."
  - As a fixed quantity basket index it is "easy to explain; biased but easily explained."
  - Balk and Diewert (2003) establish the substitution bias of a Lowe CPI; Lowe is likely to have an upward substitution bias against a Laspeyres index, and Laspeyres has an upward substitution bias against a superlative index.
  - The Lowe index has good axiomatic properties (ILO et al., 2004, chapter 16) despite substitution bias concerns.
- Young:
  - The Young index fails the circularity and time reversal tests.
  - The Young index between periods 0 and t will exceed its time antithesis (its inverse between period t and 0) by a term equal to the Young index times the weighted variance of deviations of price relatives (between periods 0 and t) and their mean; since the variance must be positive, Young must exceed the inverse of its time antithesis except when there is no price change dispersion.
  - Analytical shortcomings include uncertainty a priori about the extent and nature of its deviations from Laspeyres and relatively poor axiomatic properties.
- Geometric Young and Geometric Lowe:
  - Geometric Young is consistent with unitary elasticity of substitution (geometric/ Cobb-Douglas preferences) and, under that condition, lies within the Laspeyres-Paasche interval.
  - The Geometric Young is less susceptible to substitution bias than the arithmetic Lowe (Balk, 2010).
  - The Geometric Lowe has little standing in practice; the price-updating rationale that supports the arithmetic Lowe has no analogous economic rationale for the Geometric Lowe.

### Empirical evidence and findings
- Hansen (2007) — UNECE/ILO survey context:
  - In the joint UNECE/ILO survey on the CPI Manual of the 47 respondents as at September 2007, 32 national statistical offices used the (price-updated) Lowe index and 15 the original (presumably survey period) Young weights.
  - Using Danish CPI data for 1996 to 2003, Hansen (2007) found increases for the Young and Lowe indices of 17.49 and 18.01 percent, respectively, compared with an increase in the Törnqvist index of 17.08 percent. The Geometric Young index was below Törnqvist in Hansen’s results (source truncated).
- U.S. CPI empirical results (section IV summary):
  - The Geometric Young index, which is consistent with unitary elasticity of substitution, has a downward bias.
  - The U.S. data over the period studied demonstrate inelastic substitution (Greenlees, 2011).
  - Averaging can substantially offset the Geometric Young’s downward bias; averaging variants have a formal justification from Lent and Dorfman (2009).
  - The Lowe price index, as used in the U.S. and many other countries, is found to have a bias (against superlative indexes) several times that of some averaging variants, all of which can be computed in real time using the same database as the Lowe index.
- Practical guide / manual stance:
  - The CPI Practical Guide supports consideration of the Geometric Young formulas (UNECE et al., 2009, page 160, ff. 50).
  - The CPI Manual regards the Geometric Young as a "serious practical possibility" for CPI compilation but cites concern that it is unlikely to gain general acceptance because "it cannot be interpreted as a fixed quantity basket index."

### Practical implications and recommendations
- There is a prima facie case for formal and empirical analysis of the geometric counterpart to the arithmetic Lowe given:
  - The Geometric Young can be computed in real time using existing weights and price relatives.
  - Geometric formulations maintain consistency with the Jevons elementary-level geometric practice and help preserve aggregation consistency.
  - Geometric Young exhibits lower substitution bias relative to the widely used arithmetic Lowe and can approximate superlative indexes under unitary elasticity of substitution.
- Operational note:
  - Statistical offices could compute geometric counterparts (Geometric Young and Geometric Lowe) using the same real-time data currently used for Young and Lowe indexes to evaluate proximity to superlative indexes (e.g., Törnqvist) and to assess biases empirically.

*Italic: Source — _wp12105 (Section IIB and surrounding sections) from the provided PDF content.*

### 16.51 percent. The differences between Young and Lowe are not always trivial.

### _wp12105 - 16.51 percent. The differences between Young and Lowe are not always trivial.

### Summary findings
- The paper documents substantive, non-trivial differences between Lowe and Young CPI aggregation formulas, and between their geometric counterparts and a superlative Törnqvist index.
- Selected empirical comparisons cited:
  - Annual inflation for 2004/5 and 2005/6: increased from 1.80 to 1.88 percent using Young but decreased from 1.99 to 1.90 percent using Lowe.
  - Greenlees and Williams (2010) (US CPI, Dec 1990–Dec 2008): Lowe increase 18.88 percent, Young increase 18.24 percent, chained Törnqvist 16.78 percent; Geometric Young 15.84 percent.
  - Pike et al. (2009) (New Zealand, June 2006–June 2008): Lowe 6.26 percent, Young 5.60 percent, Geometric Young 4.83 percent, retrospective Fisher 5.73 percent.
- For the US data set analyzed by the authors (Dec 1997–Dec 2010), the Fisher index increased in 2010 compared with 1998 at a (compound) annual average rate of 2.27 percent, compared with a Lowe price index increase of 2.44 percent.
- The authors find that the Geometric Young generally falls below, and sometimes is closer to, the superlative Törnqvist index; Lowe and Young may produce similar results, but their differences from Törnqvist are marked.

### Analytical decomposition: why Geometric Young and Geometric Lowe differ
- Definitions and log decompositions are provided showing the difference between logarithms of Geometric Lowe and Geometric Young indexes (equations (7)–(10) in source).
- Key determinants of the sign and magnitude of the difference 0t GLo I / 0t GY I (equation (10)):
  - The period-b weighted correlation coefficient between price relatives and logged price relatives, denoted ,b i s xy ρ —its sign dictates whether Geometric Lowe exceeds Geometric Young (positive) or vice versa (negative).
  - The dispersion of price changes: b i s x cv (coefficient of variation for x over period b→0) and b i s y σ (standard deviation for y over period b→0). Higher dispersion increases the magnitude of the difference; b i s y σ is noted as the most potent driver because it is not corrected for changes in the mean.
  - The multiplicative interaction of the correlation and dispersion terms—reducing any one term (for example by minimizing weight-update lags) reduces the difference between formulas.
- Conceptual clarification: the authors do not characterize the Geometric Young vs Geometric Lowe difference as substitution bias; rather the difference stems from correlation across price-change periods (price relatives vs logged price relatives), not from price–quantity substitution.

### Comparisons with a superlative (Törnqvist) price index
- Ratios of Geometric Lowe and Geometric Young to Törnqvist are derived (equations (11)–(12)).
- The Geometric Young and Törnqvist are equal if period-b shares equal the average of period-0 and period-t shares (b i s = (0 i s + t i s)/2).
- Elasticity of substitution implications:
  - The Lloyd-Moulton (constant elasticity of substitution—CES) index (equation (13)) is introduced; parameter η is elasticity of substitution.
  - Greenlees (2011) estimates η for US data and finds η values between 0 and 1 for most years (inelastic substitution), with estimated η varying between 0.521 and 0.655 for 1999/2000 to 2005/6, 0.981 for 2006/7, and 0.192 for 2007/8.
  - Inelastic substitution (η < 1) implies items with relatively higher price trends receive less importance in 0t GY I than in 0t T I, i.e., 0t GY I < 0t T I.
- Lent and Dorfman (2009) propose an AG Mean (weighted average of arithmetic Laspeyres and Geometric Laspeyres) that can approximate a superlative index when 0 ≤ η ≤ 1 (equations (14)–(16)). The authors indicate averaging arithmetic and geometric versions (Lowe/Young or their geometric counterparts) may better approximate superlative targets.

### Empirical results and data (authors’ US CPI analysis)
- Data:
  - Elementary aggregate indexes and weights for the U.S. Urban CPI, Dec 1997–Dec 2010, from U.S. BLS.
  - Elementary aggregates correspond to about 211 item strata (product groups).
  - CPI compilation: 211 product-group strata × 38 area strata = 8,018 cells.
  - BLS uses weighted geometric means for most item/area strata and Lowe formula for aggregation across areas; authors analyze using only the 211 product-group weights (reflecting many countries’ data availability).
  - Dates of weights used (mean-annual expenditures → basis of weights):
    - 1993-1995 → Jan98-Dec01
    - 1999-2000 → Jan02-Dec03
    - 2001-2002 → Jan04-Dec05
    - 2003-2004 → Jan06-Dec07
    - 2005-2006 → Jan08-Dec09
    - 2007-2008 → Jan10-Dec11
    - 2008-2009 → Jan11-Dec12
- Empirical comparisons and performance:
  - The authors’ calculated chained Törnqvist index closely matches the BLS’ chained Törnqvist (C-CPI-U), with correlations of 0.99978 for levels and 0.98855 for monthly annual changes over Dec 1999–Dec 2010.
  - Figure 1 (described): arithmetic Lowe is above Laspeyres; Young is closer to Laspeyres (thus nearer the Laspeyres–Paasche interval). Differences are not large but are economically meaningful given CPI uses for escalations.
  - Figure 2 (described): Geometric-Lowe, Geometric Young, and Törnqvist track closely early in the series; after Dec 2003 Geometric Lowe and Geometric Young diverge. For Dec 2003–Dec 2007 Geometric Lowe is closer to Törnqvist than Geometric Young for this dataset.
  - The Geometric Young showed a marked downward bias relative to Törnqvist in the authors’ US data, while Lowe tended to overstate Törnqvist—suggesting an average of the two formulas may better approximate the superlative target.

### Interpretation and implications from the analysis
- The relative positioning of Geometric Lowe vs Geometric Young depends on long-run persistence in price trends between the survey/reference period (b→0) and the current comparison period (0→t).
  - Empirical sign changes in ,b i s xy ρ (positive Jan 2008–Oct 2008, negative from Nov 2008 onward) illustrate that long-run trends can reverse and change which geometric formula grows faster.
- Lags in weight updating (longer b→0 and 0→t spans) and higher inflation or higher relative-price dispersion amplify differences between geometric formulas.
- Because the three determining factors (correlation, coefficient of variation, standard deviation of logged relatives) are multiplicative, reducing any one (for example by reducing weight-update lags) reduces divergence between formulas.
- Practical consequence: for real-time CPI compilation with infrequent weight updates, choice among Lowe, Young, Geometric Lowe, and Geometric Young materially affects measured inflation relative to superlative indexes; averaging approaches (Lent-Dorfman / AG Mean ideas) can help produce measures closer to superlative targets when η lies between 0 and 1.

*Source: _wp12105 - 16.51 percent. The differences between Young and Lowe are not always trivial.*

### 2010.  Lowe has the largest bias of about 1.7 percent from a superlative index and Young

### _wp12105 - 2010.  Lowe has the largest bias of about 1.7 percent from a superlative index and Young

### Key empirical findings
- Lowe has the largest bias of about 1.7 percent from a superlative index; Young reduces the bias to about 0.5 percent.
- The Geometric Lowe is, on average, 0.2 percent above the Törnqvist while the Geometric Young is about 1.0 percent below.
- Fisher and Törnqvist are used as target superlative indexes, tracking each other very closely: Fisher has an annual growth of 2.49033 and Törnqvist 2.49316 over the period December 198 and to December 2010.
- In real time using the same database as used by Lowe, Lowe’s bias can be cut from 1.7 percent to 0.3 percent using L-D approximations.
- The Geometric Young:Lowe formulation is recommended as an adjustment, via the Geometric Young index, of the existing Lowe formula.
- The Geometric Young and Geometric Lowe price indexes are the two geometric formulations most readily available for compilers; Geometric Young has a clear interpretation as a weighted geometric average using survey-period expenditure shares, while Geometric Lowe “has no meaningful interpretation.”
- L-D approximations (average of Lowe:Geometric Young and average of Young:Geometric Young) successfully track the Törnqvist index and alleviate the downward bias in the Geometric Young and the upward bias in the Lowe or Young indexes.
- Simple geometric means of arithmetic and geometric formulas are “highly successful” and are “vast improvements on other standard arithmetic and geometric formulas.”

### Methodology and formula adjustments
- Approximations to Laspeyres and Geometric Laspeyres in equation (14) are constructed using Lowe and Young formulas to approximate Laspeyres and Geometric Lowe and Geometric Young formulas to approximate Geometric Laspeyres.
- Variants include geometric average of Young and Geometric Young, and geometric average of Lowe and Geometric Young.
- Tailoring of estimates of τη in (15) is done based on a selected superlative benchmark (e.g., Törnqvist). Because the Törnqvist index cannot be estimated in real time, the most recently available estimates are used to enable real-time computation (for example, for January 08 to December 2009 using most recent estimates available starting January 2008).
- The average τη over January 2004 to December 2005 are used in equation (15) for January 08 to December 2009, and similarly over other periods.
- Lent-Dorfman (L-D) formulation allows τη to change monthly but is constrained to change only at index rebasing and then held constant until next rebasing, aligning weight changes with rebasing timing to avoid CPI changes from weight shifts.
- Estimates of τη using equation (16) may be negative depending on relative positioning of Geometric Young and Young indexes to the Fisher index and bounds 0 ≤ τη ≤ 1. An adaptation using absolute values is used instead (equation (17) in source), ensuring non-negative weights.

### Performance metrics (Table 2: Average Monthly Percentage Differences vs. target indexes)
- Target indexes: Fisher and Törnqvist.
- Arithmetic formulas:
  - Lowe: 1.712 (Fisher), 1.689 (Törnqvist)
  - Young: 0.466 (Fisher), 0.443 (Törnqvist)
- Geometric formulas:
  - Geometric Lowe (GLowe): 0.265 (Fisher), 0.242 (Törnqvist)
  - Geometric Young: -0.959 (Fisher), -0.981 (Törnqvist)
- Geometric means of formulas:
  - GYoung-Young: -0.249 (Fisher), -0.272 (Törnqvist)
  - GYoung-Lowe: 0.368 (Fisher), 0.345 (Törnqvist)
  - GLowe:Young: 0.366 (Fisher), 0.343 (Törnqvist)
  - GLowe:Lowe: 0.986 (Fisher), 0.963 (Törnqvist)
- Lent-Dorfman (η using lag):
  - GYoung:Young: -0.339 (Fisher), -0.375 (Törnqvist)
  - GYoung:Lowe: -0.196 (Fisher), -0.219 (Törnqvist)
  - GLowe:Young: 0.339 (Fisher), 0.316 (Törnqvist)
  - GLowe:Lowe: 0.581 (Fisher), 0.558 (Törnqvist)
- Note: Table 2 covers January 2004 to December 2010.

### Conceptual and practical implications
- The widely-used arithmetic Laspeyres-type aggregation formulas at higher (weighted) levels for CPIs—Young and Lowe—have little justification in theory and practice and show upward drift against Laspeyres; Lowe drifts more.
- Lowe (and Laspeyres) has the advantage of ease of interpretation as a fixed quantity basket index but provides a “well-defined, but biased, result.”
- L-D index has a conceptual basis as a Taylor approximation to the Törnqvist (or Fisher) index and can be calculated in real time; Lent and Dorfman (2009) note the method continuously picks up changes in consumer buying patterns without iterative numerical procedures and is easily automated.
- The Geometric Lowe vs. Geometric Young difference is formally decomposed and found to be based on long-run unidirectional price changes (equations (10) and (11)), making Geometric Lowe unreliable for a predictable relationship with Geometric Young/Törnqvist.

### Policy recommendations and operational suggestions
- Debate moving on from Laspeyres-type indexes is warranted; consider calculating retrospective indexes to identify the extent of substitution bias.
- CPI users would be better served by a real-time measure that more closely tracks a superlative index; public acceptance of a more complex formula may follow demonstration of superior performance.
- Statistical agencies can compile L-D approximations and simple geometric-means adjustments in real time using existing prices databases to substantially reduce Lowe’s bias.
- The Geometric Young:Lowe and related L-D adjustments are practical candidate formulas for immediate improvement while maintaining real-time computability and operational simplicity.

*Italic source attribution: Excerpted content from _wp12105 - 2010.  Lowe has the largest bias of about 1.7 percent from a superlative index and Young*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12105.pdf_