Inflation measurement in the presence of stockpiling and consumption smoothing

on Aue , Ludwig
A icle — Published Ve sion
In la ion measu emen in he p esence o s ockpiling and
consump ion smoo hing
Re iew o Income and Weal h
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John Wiley & Sons
Sugges ed Ci a ion: on Aue , Ludwig (2024) : In la ion measu emen in he p esence o s ockpiling
and consump ion smoo hing, Re iew o Income and Weal h, ISSN 1475-4991, Wiley Pe iodicals,
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Re iew o Income and Weal h
Se ies 71, Numbe 1, Feb ua y 2025; e12682
DOI: 10.1111/ oiw.12682
INFLATION MEASUREMENT IN THE PRESENCE OF STOCKPILING
AND CONSUMPTION SMOOTHING
BY LUDWIG VON AUER*
Uni e si ä T ie
A chained p ice index is said o su e om chain d i bias i i indica es an o e all p ice change, e en
hough he p ices and quan i ies in he cu en pe iod ha e e e ed back o hei le els o he base
pe iod. The empi ical ele ance o his bias is well documen ed in s udies ha apply sub-annual chaining
o scanne da a. The e i is shown ha s ockpiling can lead o downwa d chain d i bias. The p esen
pape d aws a en ion o he ac ha smoo hing consump ion causes subs an ial upwa d chain d i . In
addi ion, his s udy in oduces a s ochas ic simula ion app oach ha is consis en wi h bo h, s ockpiling
and consump ion smoo hing. A “s ess es ” is conduc ed ha examines whe he olling window a ian s
o mul ila e al indices (GEKS, TPD, and GK) e ec i ely cu ail he chain d i p oblem.
JEL Codes: C43, E31, E52
Keywo ds: bias, chain d i , mul ila e al, p ice index, simula ion
1. INTRODUCTION
The con en ional app oach o compu e he p ice change be ween a base pe iod
0 and a compa ison pe iod Tuses only he p ices and quan i ies o hese wo pe i-
ods. Such index o mulasa eknown asbila e al p ice indices (o di ec p iceindices).
In he ollowing, hey a e deno ed by PT∕0. As an al e na i e o he bila e al index,
PT∕0, one may compu e he p ice change be ween pe iods 0 and T om a sequence
o o e lapping adjacen bila e al p ice indices. The elemen s o his sequence can
be linked by mul iplica ion: P1∕0P2∕1…PT∕T−1. Such p oduc s a e known as chain
indices and i s ac o s as chain links.
When he i em uni e se emains cons an o e ime and all p ices and quan-
i ies in he cu en pe iod T e e back o hei le els in he base pe iod 0, a
p ice index compa ing he cu en pe iod wi h he base pe iod should indica e
ha no p ice change occu ed. All easonable bila e al p ice indices, PT∕0,sa is y
his equi emen . Howe e , chained bila e al p ice indices usually iola e i . In he
p esen pape , his iola ion is deno ed as chain d i bias. I is usually a ibu ed
No e: A o me e sion o his s udy was p esen ed a he “O awa G oup Mee ing, 2019” in Rio de
Janei o and a he con e ence “Messung de P eise, 2019” in E u . Help ul commen s om pa icipan s
a e g a e ully acknowledged. The pape also bene i ed om aluable sugges ions by Sebas ian Weinand
as well as by h ee anonymous e e ees and he edi o .
*Co espondence o: Ludwig on Aue , Uni e si ä T ie , Fachbe eich IV–VWL, Uni e si ä s ing
15, 54296 T ie , Ge many ( onaue @uni- ie .de).
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In e na ional Associa ion o Resea ch in Income and Weal h.
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion-NonComme cial-
NoDe i s License, which pe mi s use and dis ibu ion in any medium, p o ided he o iginal wo k is
p ope ly ci ed, he use is non-comme cial and no modi ica ions o adap a ions a e made.
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Re iew o Income and Weal h, Se ies 71, Numbe 1, Feb ua y 2025
o sales igge ing nons anda d subs i u ion beha io o consume s. Feens a &
Shapi o (2003, p. 135) examine scanne da a on canned una and compile a weekly
chained Tö nq is index ha exhibi s upwa d chain d i caused by sales, while de
Haan (2008, p. 19), s udying scanne da a on de e gen s, inds ha sales lead o
downwa d chain d i , e en hough bo h s udies apply he same index o mula and
equency o chaining.
These con adic o y esul s sugges analyzing he causes o chain d i in a
mo e sys ema ic way. The p esen pape pu s i s ocus on chained bila e al indices
ha include he quan i ies o bo h, he base and he cu en pe iod (e.g. he indices
o Tö nq is , Fishe , Ma shall–Edgewo h, and Walsh). I is shown ha he chain
d i o such chained bila e al p ice indices is caused by quan i y and p ice changes
ha a e no pe ec ly synch onized in ime.1Such asynch onous p ice and quan i y
changes can a ise om di e en o ms o in e empo al op imiza ion beha io o
consume s.
A well-known example o such consume beha io is s ockpiling. Du ing a
sale, he consume s inc ease bo h hei consump ion and hei s ocks. The e o e,
he pu chases “o e shoo ” consump ion. As soon as he p ice e u ns o no mal,
he pu chases o e shoo consump ion in he downwa d di ec ion because he con-
sume s i s use up hei ex a s ocks. When in he subsequen pe iods he p ice
emains on i s no mal le el, pu chased quan i ies g adually align wi h consumed
quan i ies. Fo simplici y, hese consequences o s ockpiling a e deno ed he e as
o e shoo ing quan i y eac ions o p ice changes.2Also empo a y p ice spikes ig-
ge o e shoo ing quan i y eac ions.
The e a e o he o ms o in e empo al op imiza ion beha io leading o asyn-
ch onous p ice and quan i y changes. P obably he mos impo an ones a e delayed
quan i y adjus men s o p ice changes. O en, his smoo hing is caused by sea ch
and adjus men cos s. E en so he p ice o a consume ’s a o i e p oduc may ha e
pe manen ly inc eased o he p ice o a compe ing p oduc may ha e pe manen ly
allen, he consume may show no o only a mode a e immedia e change in he usual
pu chasing beha io . Only a some la e poin o ime, a e acqui ing in o ma ion
abou he quali a i e ea u es o he handling o al e na i e p oduc s, she may pa ly
o comple ely swi ch o e o such al e na i es. O he easons o delayed quan i y
eac ions a e consump ion habi s caused by ha m ul addic ions (e.g. nico ine) o
by pas in es men s in inc eased enjoymen om consumed goods (e.g. abili y o
cook as y and heal hy ood).3Fo simplici y, all delayed quan i y adjus men s a e
deno ed he e as s icky quan i y eac ions o p ice changes.
Se e al s udies o scanne da a (e.g. de Haan, 2008, p. 18; de Haan & an de
G ien , 2011, p. 43) show ha o e shoo ing quan i ies cause downwa d chain d i .
1Hill (2006, pp. 314-315) eaches a simila conclusion o he chained Laspey es index and he
chained Paasche index.
2Hayashi (1985) iden i ies a simila e ec o du ables: “A highe le el o expendi u e means a la ge
s ock o consump ion, which will dep ess expendi u e in he nex pe iod i households beha e in a way
o smoo h ou consump ion ( a he han expendi u e) o e ime (p. 1092).”
3Muellbaue (1988) emphasizes he ele ance o habi o ma ion o mac oeconomic models. His
empi ical e idence appea s o a o myopic habi s, ha is, consume s who a e no awa e o he impac
o cu en consump ion decisions on u u e consump ion decisions.
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Re iew o Income and Weal h, Se ies 71, Numbe 1, Feb ua y 2025
The chain d i a ising om s icky quan i ies, howe e , wen la gely unno iced.4
The e o e, he p esen pape ’s i s con ibu ion is o show ha s icky quan i ies
lead o upwa d chain d i .
As a solu ion o he chain d i p oblem, I ancic e al. (2011) ad oca e a olling
window a ian o he Gini–Él e ˝
o–Kö es–Szulc (GEKS) app oach. The GEKS
index is ee o chain d i . Howe e , i s olling window a ian (R-GEKS) in ol es
a mechanism ha links he cu en window o pas windows. This special o m o
chaining is usually deno ed as splicing. I canno be uled ou ha splicing gene a es
chain d i .
By now, se e al a ian s o such R-GEKS indices ha e been de eloped.5
Besides R-GEKS indices, many o he mul ila e al app oaches ha e been p oposed
o o e come he chain d i p oblem. These include olling window a ian s o
he ime-p oduc dummy me hod (R-TPD) and o he Gea y–Khamis app oach
(R-GK).6Again, hese me hods equi e some o m o splicing. Consequen ly, hey
migh also su e om chain d i bias.
How can one in es iga e his suspicion? A meaning ul examina ion equi es
an unassailable benchma k such ha he de ia ion o an index numbe om ha
benchma k ep esen s chain d i bias. This benchma k di ec ly ollows om he
de ini ion o chain d i : when all p ices and quan i ies e e back o hei o me
le els, a p ice index should indica e ha no p ice change occu ed. Un o una ely,
he p ices and quan i ies o eal wo ld da a (e.g. scanne da a) ne e e u n o hei
o iginal le els. The e o e, an analysis o chain d i ha uses eal wo ld da a mus
do wi hou his na u al benchma k.
By con as , in a simula ion app oach he esea che can gene a e p ice–
quan i y scena ios whe e all p ices and quan i ies e u n o hei o iginal le els.
Howe e , he simula ion mus ensu e ha he p ice–quan i y scena ios e lec
ea u es o eal wo ld pu chasing beha io . This equi es p ice scena ios wi h sales
and o dina y p ice changes as well as quan i y scena ios ha cap u e o e shoo ing
and s icky quan i ies a ising om consume beha io such as s ockpiling and
smoo hing o consump ion.
Using such a simula ion app oach, his s udy’s second con ibu ion is a
quan i a i e analysis o he chain d i bias o he R-GEKS, R-TPD, and R-GK
app oaches. I is shown ha hese index me hods educe chain d i bias, bu hey
canno elimina e i . Fu he mo e, some a ian s a e mo e e ec i e han o he s.
As a i s s ep, Sec ion 2in oduces he no ion o chain d i and explains why
o e shoo ing quan i ies and s icky quan i ies cause di e en di ec ions o chain
d i . Sec ion 3b ie ly discusses he implica ions o chain d i o p ice measu e-
men elying on scanne da a and explains why a simula ion app oach is app op ia e
o a sys ema ic examina ion o he a ious index me hods’ esilience o chain d i
bias. Sec ion 4p esen s a simula ion-based “s ess es ” ha examines whe he he
4A no able excep ion is T iple (2003, p. 152) who poin s ou ha s o age, sea ch, and in o ma ion
cos may gene a e measu emen p oblems o scanne da a p ice indices.
5Fo compac su eys, see, o example, Diewe & Fox (2022, pp. 357-359), Fox e al. (2022,
pp. 9-12), Van Loon & Roels (2018, pp. 7-8), o Online Appendix Bo he p esen pape .
6Fo ecen su eys o he a ious me hods see, o example, Chessa e al. (2017), Chessa (2019), de
Haan & K sinich (2014), o Diewe & Fox (2022).
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Re iew o Income and Weal h, Se ies 71, Numbe 1, Feb ua y 2025
R-GEKS, R-TPD, and R-GK me hods cu ail chain d i and whe he some a i-
an s a e be e sui ed han o he s. Sec ion 5concludes.
2. CHAIN DRIFT AND ITS SOURCES
Le he in ege s i=1,…,N ep esen he Ni ems o an economy. All i ems
a e a ailable du ing he base pe iod ( =0) and he compa ison pe iod ( =T)and
also du ing all in e media e pe iods (0 < <T). The pe iod ec o o p ices is
(p
1,…,p
N)and he co esponding ec o o quan i ies is (x
1,…,x
N). I is cus-
oma y o in e p e a bila e al p ice index, P, as a mapping o he N-dimensional
ec o s (p0
1,…,p0
N),(x0
1,…,x0
N),(pT
1,…,pT
N),and(xT
1,…,xT
N)in o a single
posi i e numbe , PT∕0, ha measu es he “o e all p ice change” be ween pe iods 0
and T. All bila e al p ice indices conside ed in his s udy a e lis ed in he Online
Appendix A.
Conside some sequence o ime pe iods, =0,…,T.
De ini ion 1. A chain index ha compa es pe iod T o he base pe iod 0 is de ined by
(1) 
PT∕0=P1∕0P2∕1…PT−1∕T−2PT∕T−1.
When he p ices and quan i ies in pe iod T e u n o hei le els o pe iod 0, he
chain index, 
PT∕0, should gi e he same index numbe as he bila e al index, PT∕0,
namely 1 (e.g. I ancic e al., 2011, p. 26). Acco dingly, in his speci ic p ice–quan i y
scena io, he de ia ion om uni y is an app op ia e measu e o he ex en o chain
d i bias (e.g. Ribe, 2012, p. 3; Diewe , 2022; Diewe & Fox, 2022, p. 557). This
in e p e a ion o chain d i bias can be o malized in he ollowing way:
De ini ion 2. The chain d i es o he bila e al p ice index, P ∕ −1, pos ula es ha
(2) P1∕0P2∕1…PT−1∕T−2P0∕T−1=1.
Condi ion (2) can be ound in Walsh (1901, p. 401).7No e ha o T=1 he chain
d i es simpli ies o he ime e e sal es : P1∕0=1∕P0∕1.8
7Walsh la e calls condi ion (2) he ci cula i y es (see Diewe , 1993, p. 40). In he cu en p ice
index li e a u e, his label is ese ed o he condi ion P2∕0=P2∕1P1∕0. This condi ion is s ic e han
he al e na i e o maliza ion o chain d i because i does no p esc ibe ha he p ices and quan i ies
o pe iods 0 and 2 coincide. In he con ex o spa ial p ice compa isons, ci cula i y is usually deno ed as
ansi i i y. Diewe (1993, p. 40) coins he chain d i es as he “mul ipe iod iden i y es ”. No e, how-
e e , ha he chain d i (o mul ipe iod iden i y) es conside s he case whe e he quan i ies e e se o
hei base pe iod alues, while in he iden i y es he e olu ion o he quan i ies is comple ely i ele an .
In ac , a chain d i es wi hou he quan i y e e sal pos ula e is conside ed in de Haan (2008, p. 10).
He calls i he “in a iance o p ice bouncing es ”.
8An anonymous e e ee poin ed ou ha o bila e al p ice indices ha sa is y he ime e e sal es ,
one ge s PT∕T−1=P0∕T−1=1∕PT−1∕0and condi ion (2) becomes (P1∕0P2∕1…PT−1∕T−2)∕(PT−1∕0)=
1. Acco ding o his al e na i e o maliza ion, a bila e al p ice index su e s om chain d i when he
a io o he chained index and he co esponding bila e al index, 
PT−1∕0∕PT−1∕0, de ia es om uni y.
Se e al w i e s use his al e na i e o maliza ion o chain d i (e.g. Fo sy h & Fowle , 1981, p. 234;
F isch, 1936, p. 8; Hill, 2006, pp. 14-15; Len , 2000, p. 314; Pe sons, 1921, p. 109; Pe sons, 1928, p. 101).
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Re iew o Income and Weal h, Se ies 71, Numbe 1, Feb ua y 2025
Figu e 1. Synch onous Quan i y Reac ions o P ice Changes (Scena io 1)
In s anda d mic oeconomic consume heo y, consume s ins an aneously
adjus hei cu en pu chases o he cu en p ices. The usual assump ion is
ha consume s subs i u e away om p oduc s ha ha e become ela i ely mo e
expensi e. The la ge he p ice elas ici y o demand, he mo e p onounced a e hese
quan i y eac ions. Figu e 1 ansla es his s anda d heo y in o a highly s ylized
example. The igu e depic s he p ices and quan i ies o some i em du ing nine
consecu i e pe iods, =0,…,8. In he depic ed Scena io 1, he p ices o he i em
can ake only he alues “low”, “no mal”, and “high”, and he quan i ies sold can
be “small”, “no mal”, o “la ge”.
P ice changes a e ma ked by he weigh icons. The p ice s a s a a no mal le el,
d ops in pe iod 1 o he lowe le el, s ays he e o ano he pe iod, e u ns o no mal
in pe iod 3, and s ays he e also du ing pe iod 4. In pe iod 5, he p ice inc eases o
high, s ays he e o ano he pe iod, be o e i d ops back o no mal du ing pe iod 7,
and emains he e du ing pe iod 8. The quan i ies pu chased mo e exac ly in e sely
o he p ices; ha is, he consume s’ quan i y eac ions a e pe ec ly synch onous o
he p ice changes. As soon as he p ice e u ns o no mal, he quan i y also e u ns
o no mal. In imes o cons an p ices also he quan i ies emain cons an . When
he p ice elas ici y o demand is less han uni y, expendi u e sha es and p ices a e
posi i ely co ela ed.
Many p ice indices a e expendi u e-weigh ed a e ages o in e empo al p ice
a ios (pT
i∕p0
i) whe e he impac o he wo pe iods 0 and Ton he expendi-
u e weigh s is symme ic (e.g. Tö nq is , Walsh, Ma shall–Edgewo h, Theil,
Sa o–Va ia).9Due o his symme ic impac , Scena io 1 gene a es weigh s such
ha he p ice decline be ween pe iods 0 and 1 (i s weigh is ela ed o he expendi-
u es du ing pe iods 0 and 1) and he p ice inc ease be ween pe iods 2 and 3 (i s
9The Fishe index and he gene alized uni alue (GUV) indices o Bane jee, Da ies, and Leh
( on Aue , 2014, pp. 848–852) exhibi a simila ype o “in e empo al symme y”. A mo e com-
ple e classi ica ion o “symme ic bila e al p ice indices” is p o ided by on Aue & Shumskikh
(2024, sec . 2).
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Figu e 2. Downwa d Chain D i Bias om O e shoo ing Quan i y Reac ions Caused by Sales o
P ice Spikes (Scena io 2)
weigh is ela ed o he expendi u es du ing pe iods 2 and 3) exac ly o se each
o he . G aphically, his balanced weigh ing is indica ed by he equal-sized weigh
icons loca ed a he p ice decline be ween pe iods 0 and 1 and a he p ice inc ease
be ween pe iods 2 and 3. The same is ue o he p ice inc ease be ween pe iods
4 and 5 and he p ice decline be ween pe iods 6 and 7. As a consequence o his
balanced weigh ing o p ice inc eases and p ice declines, symme ically weigh ed
indices a e immune o chain d i om scena ios like Scena io 1 and one ge s
P1∕0⋅P2∕1⋅…⋅P8∕7=1.10
Scena io 1 co esponds o s anda d consume heo y as p esen ed in in oduc-
o y mic oeconomics ex books. Real- wo ld consume beha io , howe e , is mo e
complex. One impo an aspec igno ed by s anda d consume heo y is s ockpil-
ing. Fo example, sales usually lead o inc eased pu chases, pa o which a e s o ed.
I in he nex pe iod he p ice e u ns o i s no mal le el, he pu chased quan-
i y alls below i s no mal le el, because consume s i s use up hei ex a s ock.
Only a e he ex a s ock is deple ed, he pu chased quan i y e u ns o i s no -
mal le el. In he ollowing, such a scena io is deno ed as an o e shoo ing quan i y
esponse o sales.11 Du ing he sales pe iod, acquisi ions exceed consump ion (i.e.
s ocksinc ease),while igh a e he sales pe iod, consump ion exceeds acquisi ions
(s ocks dec ease).
Pe iods 0–3 o Scena io 2 (see Figu e 2) depic his case in a highly s ylized
o m. In such a scena io, symme ically weigh ed indices exhibi chain d i . As a
esul o he o e shoo ing quan i y esponse, he weigh a ached o he p ice educ-
ion be ween pe iods 0 and 1 is la ge han he weigh a ached o he p ice inc ease
10Chain d i a ises, howe e , o he Laspey es and Paasche index. This issue is add essed in a i-
ous s udies including Fo sy h & Fowle (1981, pp. 234-235), Szulc (1983, pp. 540-541), and Hill (2006,
pp. 314-315).
11This ype o scena io is desc ibed, o example, in I ancic e al. (2009, p. 4), de Haan & an de
G ien (2011, p. 39), and Ribe (2012,p.3).
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Re iew o Income and Weal h, Se ies 71, Numbe 1, Feb ua y 2025
be ween pe iods 1 and 2. The e o e, he p ice index le el o pe iod 2 is below ha
o pe iod 0. A he end o pe iod 2, he cus ome s’ in en o y is back o i s no -
mal le el. The e o e, he pu chases in pe iod 3 inc ease o hei no mal le el, e en
hough he p ice is cons an . In he li e a u e, he e is a a he b oad consensus ha
e e y alid bila e al index o mula mus sa is y he iden i y es . This es says ha ,
in he absence o any p ice changes, he p ice index is uni y, ega dless o any quan-
i y changes. All p ice indices lis ed in he Online Appendix Asa is y he iden i y
es . The e o e, he p ice index le el o pe iod 3 emains on he le el o pe iod 2 and,
hus, below ha o pe iod 0. In o he wo ds, downwa d chain d i a ises.
Pe iods 0–3 o Scena io 2 desc ibe a s o able i em ha consume s keep in
s ock. In imes o unusually low p ices (sales) he consume s add o hei o dina y
s ock some ex a s ock and hey deple e ha ex a s ock when he p ice e e s o
i s no mal le el. This is he s anda d na a i e o s ockpiling.
Howe e , s ockpiling is ele an no only in imes o sales bu also in imes
o p ice spikes. Du ing such spikes, consume s can plunde hei o dina y s ock
and es ock i as soon as he p ice e u ns o no mal. Again, o e shoo ing quan i y
eac ions a ise. This is depic ed by pe iods 4–8 o Scena io 2. The p ice inc eases in
pe iod 5. Many consume s swi ch o using up hei s ocks. This leads o a nega i e
quan i y eac ion ha is la ge han o i ems ha canno be s o ed. In pe iod 6,
he p ice e u ns o i s no mal le el. This gi es he consume s he oppo uni y o
e esh hei in en o ies. A he same ime, hey e u n o hei no mal consump-
ion. The e o e, he o al quan i y pu chased exceeds he no mal quan i y. As a
esul o hese o e shoo ing quan i y esponses, he weigh a ached o he p ice
inc ease be ween pe iods 4 and 5 is smalle han ha a ached o he p ice educ-
ion be ween pe iods 5 and 6. This leads again o downwa d chain d i . A he
end o pe iod 6, he s ock is back o i s s anda d le el, such ha he pu chases in
pe iod 7 e u n o hei no mal le el e en hough no p ice change occu s be ween
pe iods 6 and 7.
In sum, o e shoo ing quan i ies igge ed by sales o by p ice spikes wo k in
he same di ec ion. Bo h gene a e downwa d chain d i . O e shoo ing quan i ies,
howe e , a e only one d i e o chain d i . Ano he d i e ope a es in he opposi e
di ec ion; ha is, i causes upwa d chain d i .
In he ield o indus ial o ganiza ion he e is ex ensi e li e a u e on sea ch
and adjus men cos s and hei implica ions o ma ke s.12 Also in he ield o
p ice measu emen i is well known ha sea ch and adjus men cos s a e el-
e an in eal- wo ld consump ion decisions and ha hey c ea e p oblems o
p ice measu emen pu poses (e.g. Reinsdo , 1994, p. 137; T iple , 2003, p. 152).
Such cos s can delay he consume s’ subs i u ion beha io , such ha pa o he
quan i y esponse o he comple e quan i y esponse happens in a la e pe iod
han he unde lying p ice change. This is pa icula ly ue when he leng h o a
pe iod is ela i ely sho (e.g. 1 week o 1 mon h). Ano he cause o delayed quan-
i y esponses a e ha m ul o bene icial consump ion habi s ha he consume s
ha e de eloped o e ime. Delayed quan i y esponses a e deno ed he e as s icky
quan i ies.
12A su ey o his li e a u e is Fishe Ellison (2016).
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Re iew o Income and Weal h, Se ies 71, Numbe 1, Feb ua y 2025
Figu e 3. Upwa d Chain D i Bias Caused by S icky Quan i y Reac ions o P ice Changes (Scena io
3)
Figu e 3illus a es he consequences o s icky quan i ies. In he depic ed Sce-
na io 3, demand is comple ely p ice inelas ic in he sho un bu p ice elas ic in he
long un. Mo e speci ically, he comple e quan i y eac ion o each p ice change is
delayed by one pe iod.
In pe iod 1 he p ice d ops, while he quan i y emains unchanged. The e o e,
he obse ed expendi u e du ing pe iod 1 is smalle han i would be wi h he usual
quan i y eac ion o an elas ic demand. As a consequence, also he weigh a ached
o he p ice decline is smalle han i would be wi h an elas ic demand. G aphically,
his diminished weigh is indica ed by he small weigh icon a he p ice decline
be ween pe iods 0 and 1.
The comple e quan i y eac ion o he educed p ice in pe iod 1 occu s one
pe iod delayed, ha is, in pe iod 2. The p ice in pe iod 2 emains on he le el o
pe iod 1. All popula p ice indices sa is y he iden i y es . The e o e, hei p ice
index le el does no change be ween pe iods 1 and 2.
Because demand is inelas ic in he sho un, he p ice inc ease be ween pe iods
2 and 3 occu s wi hou a quan i y educ ion. The e o e, he weigh a ached o his
p ice inc ease is la ge han i would be wi h he usual quan i y educ ion. The la ge
weigh icon a he ansi ion om pe iods 2 o 3 highligh s his in la ed weigh . The
p ice inc ease be ween pe iods 4 and 5 is analogous o ha be ween pe iods 2 and
3. Again, he p ice inc ease ecei es an in la ed weigh . The p ice decline be ween
pe iods 6 and 7 shows he same pa e n as ha be ween pe iods 0 and 1. This p ice
decline ecei es a diminished weigh . O e all, his unbalanced weigh ing o p ice
declines and p ice inc eases leads o upwa d chain d i .
Theo e ically, one can hink o quan i y eac ions ha a e an eda ed by one
pe iod. This could be ega ded as a “nega i e delay”. The weigh ing e ec s a e
exac ly opposi e o hose o Scena io 3. P ice inc eases ecei e a diminished weigh ,
whe eas p ice declines ecei e an in la ed weigh . As a esul , downwa d chain d i
would a ise. In eali y, such an icipa ing consump ion beha io is unlikely unless
s ockpiling is in ol ed. Then s ocks a e deple ed in an icipa ion o he sale. This
ends o agg a a e he o e shoo ing quan i ies e ec and he esul ing downwa d
chain d i . An icipa ed p ice spikes ha e he same e ec .
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TABLE 1
CHAIN DRIFT OF SEVERAL BILATERAL PRICE INDICES WITH SYMMETRIC EXPENDITURE WEIGHTS (IN %)
O e shoo ing S icky Hyb id
Tö nq is −8.51 9.63 4.90
Ma shall–Edgewo h −7.96 9.13 4.69
Walsh −8.68 9.46 4.86
Theil −8.60 9.48 4.86
Sa o–Va ia −8.60 9.48 4.86
TABLE 2
CHAIN DRIFT OF SOME ADDITIONAL BILATERAL PRICE INDICES (IN %)
O e shoo ing S icky Hyb id
Fishe −7.95 9.12 4.68
Bane jee −8.60 9.48 4.86
Da ies −7.96 9.13 4.69
Leh −9.19 9.60 4.96
la ge downwa d chain d i ha a ies be ween he indices. Also he upwa d chain
d i alues a ising om s icky quan i ies a e simila o hose lis ed in Table 1.
The ables show ha chained bila e al indices gene a e he expec ed esul s.
O e shoo ing quan i ies lead o conside able downwa d chain d i , while s icky
quan i ies gene a e conside able upwa d chain d i . O e all, he bila e al p ice
indices o Ma shall–Edgewo h, Fishe , and Da ies sligh ly ou pe o m he o he
ones. This is ue o o e shoo ing quan i ies as well as o s icky quan i ies.
GEKS indices a e ansi i e and, hus, immune o chain d i bias a ising
om o e shoo ing quan i ies. The e o e, R-GEKS indices ha e been p oposed
as a emedy o chain d i bias. Howe e , hese indices in ol e some o m o
splicing which, in u n, can cause chain d i bias. The pe o mance o he a ious
R-GEKS indices may depend on he choice o he splicing me hod and he window
leng h (e.g. Melse , 2018, p. 517). The e o e, Table 3p esen s he esul s o a ious
splicing me hods and o window leng hs o 4, 8, 12, and 24 pe iods. These esul s
a e de i ed om R-GEKS indices ha use he Tö nq is index as hei bila e al
base index.
The index numbe s in he i s wo ows o Table 3show ha he di ec mean
splice and he mean splice p oduce i ually he same R-GEKS index numbe s.
O e all, he numbe s ein o ce he a gumen s o Diewe & Fox (2022, pp. 560-561)
and de Haan (2015, pp. 25-26) in a o o a mo e “balanced” splicing app oach
han he mo emen splice o window splice. Wi h a su icien ly la ge window, he
R-GEKS app oach in conjunc ion wi h he di ec mean splice, mean splice, o
hal splice e ec i ely cu ails chain d i bias a ising om o e shoo ing quan i ies
(uppe le pa o Table 3). The hal splice wi h a window leng h o 12 mon hs
pe o ms bes . The uppe middle pa o Table 3 e eals ha he p e ious indings
ca y o e o he scena io o s icky quan i ies.
In he hyb id scena io, h ee qua e s o he households a e smoo hing house-
holds. The esul ing R-GEKS index numbe s a e lis ed in he uppe igh pa o
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TABLE 3
CHAIN DRIFT BIAS OF R-GEKS, R-TPD, AND R-GK INDICES (IN %) FOR DIFFERENT SCENARIOS,SPLICING METHODS,AND WINDOW LENGTHS (4,8,12,AND 24 MONTHS)
O e shoo ing S icky Hyb id
Window
leng h 4 8 12 24 4 8 12 24 4 8 12 24
R-GEKS
Di ec mean −2.41 −0.60 −0.35 −0.14 3.49 0.99 0.61 0.25 2.03 0.60 0.38 0.16
Mean −2.41 −0.60 −0.34 −0.13 3.49 0.99 0.59 0.24 2.03 0.60 0.37 0.15
Mo emen −2.56 −1.28 −0.80 −0.33 3.38 1.47 0.97 0.41 1.90 0.79 0.54 0.23
Hal −2.11 −0.23 −0.01 −0.15 3.72 0.59 0.20 0.28 2.27 0.39 0.15 0.18
Window −2.56 −1.28 −0.78 −0.32 3.38 1.47 0.96 0.39 1.90 0.79 0.53 0.22
R-TPD
Di ec mean −1.65 −0.21 −0.01 0.22 3.14 0.65 0.33 −0.03 2.00 0.44 0.25 0.03
Mean −2.44 −0.60 −0.33 −0.12 3.33 0.98 0.58 0.23 1.96 0.60 0.36 0.14
Mo emen −0.16 0.09 0.46 0.68 2.70 0.60 0.17 −0.29 1.97 0.47 0.24 −0.06
Hal −2.21 −0.23 −0.01 −0.14 3.45 0.61 0.22 0.27 2.16 0.40 0.16 0.17
Window −4.91 −2.62 −2.01 −1.29 3.86 2.30 1.71 1.04 1.76 1.10 0.81 0.47
R-GK
Di ec mean −4.03 −1.34 −1.24 −1.22 1.23 −0.45 −0.91 −1.46 −0.04 −0.67 −0.99 −1.41
Mean −2.67 −0.64 −0.35 −0.13 3.63 1.03 0.60 0.23 2.13 0.62 0.37 0.14
Mo emen −6.88 −4.18 −4.00 −3.35 −3.48 −3.58 −4.17 −4.23 −4.34 −3.74 −4.13 −4.02
Hal −2.30 −0.22 0.03 −0.16 3.79 0.60 0.19 0.27 2.38 0.40 0.14 0.16
Window 1.36 1.46 2.35 2.75 11.12 6.96 6.46 5.26 8.77 5.62 5.46 4.65
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Table 3. The chain d i bias caused by he smoo hing households seems o domi-
na e he chain d i bias caused by he s ockpiling households. Howe e , his e ec
is d i en by he pa ame e s ha de e mine he households’ desi e o s ockpiling
and smoo hing as well as by he sha e o s ockpiling households. When ha sha e
o he desi e o s ockpiling is su icien ly inc eased o he desi e o smoo hing is
su icien ly educed, he dominance would be e e sed (no shown in he able). The
o he quali a i e esul s a e no a ec ed by such changes. The di ec mean splice,
mean splice, and hal splice emain he leas biased op ions when applied wi h a
su icien ly la ge window leng h.
R-GEKS indices a e no he only app oach o cu b he chain d i p oblem.
Among he al e na i es a e he R-TPD and he R-GK app oaches. The e o e,
he same s ess es wi h he same splicing op ions and window leng hs has been
conduc ed o he R-TPD and he R-GK app oaches. Table 3also p esen s hese
esul s. Fo he R-GK app oach, he choice o window leng h and splicing me hod
causes mo e a ia ion han o he R-GEKS and he R-TPD app oaches.27 When
he window splice is a oided, he R-TPD app oach sligh ly ou pe o ms he
R-GEKS app oach which, in u n, ou pe o ms he R-GK app oach. When he
mean splice o he hal splice is applied, he choice be ween R-GEKS, R-TPD, and
R-GK is o mino ele ance.28
I should be kep in mind ha in he applied simula ion no i em a i ion occu s.
Wi h i em a i ion, la ge windows may gene a e assignmen o asso men bias (e.g.
on Aue , 2017, pp. 83–84). In a con ex o i em a i ion, Melse & Webs e (2021,
pp. 777-783) iden i y in hei own simula ions li e-cycle p icing and, in pa icula ,
un-ou sales as an impo an d i e o chain d i . The simula ion amewo k de el-
oped in he p esen s udy could be modi ied and elabo a ed o gain u he insigh s
in o he issue o i em a i ion and i s e ec on p ice measu emen . To his end, i
would be use ul o iden i y he basic s uc u e o p oduc u no e in eal- wo ld
scanne da a and o ans e his s uc u e o he simula ions.
Such a simula ion may also add ess ano he impo an issue. To mi iga e chain
d i bias, one may y o agg ega e he weekly p ices in o mon hly o e en qua e ly
uni alues (say, e.g. Diewe , 2007, p. 3). In he simula ion one could sys ema ically
s udy he e ec s o such a s a egy.
5. CONCLUDING REMARKS
Sales and he associa ed s ockpiling gi e ise o “o e shoo ing quan i-
y” mo emen s. I is well known ha o e shoo ing quan i ies c ea e p oblems o
sub-annual chaining o bila e al p ice indices because such quan i ies gene a e
downwa d chain d i bias. The p esen s udy a gues ha o e shoo ing quan i ies
27This is in line wi h indings in Fox e al. (2022, pp. 17-22), Lambo ay (2021, pp. 13-17) and Van
Loon & Roels (2018, pp. 9-13). No e ha he o me s udy uses household-le el scanne da a, while he
la e wos udies use poin -o -salesscanne da a.None o hese s udies di e en ia es be weens ockpiling
and consump ion smoo hing.
28The compa able pe o mance o he R-GEKS and R-GK app oaches is also epo ed in Fox
e al. (2022).
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Re iew o Income and Weal h, Se ies 71, Numbe 1, Feb ua y 2025
a e only pa o he chain d i p oblem. O he impo an causes o chain d i bias
a e sea ch and adjus men cos s as well as habi s. They imply ha p ice changes
lead o delayed quan i y changes. The esul ing “s icky quan i ies” gene a e upwa d
chain d i .
In he li e a u e, R-GEKS, R-TPD, and R-GK app oaches ha e been p oposed
as a emedy o chain d i bias. Howe e , i is unclea whe he hese app oaches
me ely educe chain d i bias o e en elimina e i . Fu he mo e, some app oaches
may be mo e e ec i e han o he s. The p esen pape answe s all hese ques ions.
To his end, i de elops a no el simula ion amewo k ha is consis en wi h house-
holds ha ha e a desi e o s ockpiling (o e shoo ing quan i ies) and/o o smoo h-
ing quan i y adjus men s o e ime (s icky quan i ies).
Building on his amewo k, a s ess es ha examines he esilience o di -
e en p ice indices agains chain d i is de eloped. This s ess es is applied o
a ious bila e al p ice indices and o se e al splicing a ian s and window leng hs o
R-GEKS, R-TPD, and R-GK indices. The bila e al p ice indices show he expec ed
esul s. O e shoo ing quan i ies gene a e downwa d chain d i bias, while s icky
quan i ies gene a e upwa d chain d i bias. R-GEKS, R-TPD, and R-GK indices
educe his bias, bu do no elimina e i . Sho e window leng hs end o gene a e
mo e chain d i bias han longe ones. The window splice is clea ly ou pe o med
by he hal splice and mean splice. When he la e wo splicing a ian s a e used,
he choice be ween R-GEKS, R-TPD, and R-GK indices is o mino ele ance.
ACKNOWLEDGEMENT
Open Access unding enabled and o ganized by P ojek DEAL.
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SUPPORTING INFORMATION
Addi ional suppo ing in o ma ion may be ound in he online e sion o his
a icle a he publishe ’s web si e:
Appendix S1: Suppo ing In o ma ion.
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