Editing Price index (section)

=== Bilateral indices ===
These indices compare two periods or locations using prices and quantities from both, aiming to reduce bias from the single-period weighting of unilateral indices. They incorporate substitution effects by blending data symmetrically or averaging across periods, unlike unilateral indices that fix quantities and miss consumer adjustments.

==== Marshall-Edgeworth index ====
Credited to [[Alfred Marshall]] (1887) and [[Francis Ysidro Edgeworth]] (1925),<ref>PPI manual, Chapter 15, p. 378.</ref> it averages quantities:

: <math>P_{ME} = \frac{\sum [p_{c,t_n} \cdot (q_{c,t_0} + q_{c,t_n})]}{\sum [p_{c,t_0} \cdot (q_{c,t_0} + q_{c,t_n})]}</math>

It uses a simple arithmetic mean of base and current quantities, making it symmetric and intuitive. However, its use can be problematic when comparing entities of vastly different scales (e.g., a large country’s quantities overshadowing a small one’s in international comparisons).<ref>PPI manual, 620.</ref><ref>PPI manual, Chapter 15, p. 378</ref>

==== Superlative indices ====
Introduced by [[Walter Erwin Diewert|W. Erwin Diewert]] in 1976,<ref name="Diewert197642">{{cite journal |last=Diewert |first=W. Erwin |year=1976 |title=Exact and Superlative Index Numbers |journal=Journal of Econometrics |volume=4 |issue=2 |pages=115–145 |doi=10.1016/0304-4076(76)90009-9}}</ref> superlative indices are a subset of bilateral indices defined by their ability to exactly match flexible economic functions (e.g., cost-of-living or production indices) with second-order accuracy, unlike the Marshall-Edgeworth index, which uses a basic arithmetic average lacking such precision. They adjust for substitution symmetrically, making them preferred for inflation and GDP measurement over simpler bilateral or unilateral indices.<ref name="Hill200442">{{cite journal |last=Hill |first=Robert J. |year=2004 |title=Superlative Index Numbers: Not All of Them Are Super |url=https://doi.org/10.1016/j.jeconom.2004.08.018 |journal=Journal of Econometrics |volume=130 |issue=1 |pages=25–43 |doi=10.1016/j.jeconom.2004.08.018|url-access=subscription }}</ref><ref>Export and Import manual, Chapter 18, p. 23.</ref>

===== Fisher index =====
Named for [[Irving Fisher]], it’s the geometric mean of Laspeyres and Paasche:<ref>{{cite book |last1=Lapedes |first1=Daniel N. |url=https://archive.org/details/mcgrawhilldictio00iona/page/367 |title=Dictionary of Physics and Mathematics |publisher=McGrow–Hill |year=1978 |isbn=0-07-045480-9 |page=[https://archive.org/details/mcgrawhilldictio00iona/page/367 367]}}</ref>

: <math>P_F = \sqrt{P_L \cdot P_P}</math>

It balances Laspeyres’ base-period bias (overstating inflation) and Paasche’s current-period bias (understating it), offering greater accuracy than Marshall-Edgeworth’s arithmetic approach. It requires data from both periods, unlike unilateral indices, and in chaining, it multiplies geometric means of consecutive period-to-period indices.<ref>PPI manual, p. 610</ref>

===== Törnqvist index =====
{{Main|Törnqvist index}}
A [[geometric mean]] weighted by average value shares:<ref>PPI manual, p. 610</ref><ref>[http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/tornqvist+index+and+other+log-change+index+numbers "Tornqvist Index"] {{Webarchive|url=https://web.archive.org/web/20131224111339/http://www2.stats.govt.nz/domino/external/omni/omni.nsf/wwwglsry/tornqvist+index+and+other+log-change+index+numbers|date=24 December 2013}}</ref>

: <math>P_{T} = \prod_{i=1}^{n} \left(\frac{p_{i,t}}{p_{i,0}}\right)^{\frac{1}{2} \left[\frac{p_{i,0} \cdot q_{i,0}}{\sum (p_{0} \cdot q_{0})} + \frac{p_{i,t} \cdot q_{i,t}}{\sum (p_{t} \cdot q_{t})}\right]}</math>

It weights price relatives by economic importance (average expenditure shares), providing precision over Marshall-Edgeworth’s simpler averaging, but it’s data-intensive, needing detailed expenditure data.<ref>PPI manual, p. 610</ref>

===== Walsh index =====
Uses geometric quantity averages:<ref>PPI manual, p. 610</ref>

: <math>P_{W} = \frac{\sum (p_{t} \cdot \sqrt{q_{0} \cdot q_{t}})}{\sum (p_{0} \cdot \sqrt{q_{0} \cdot q_{t}})}</math>

It reduces bias from period-specific weighting with geometric averaging, outperforming Marshall-Edgeworth’s arithmetic mean in theoretical alignment, though it shares superlative data demands.<ref>PPI manual, p. 610</ref>