Rational Seasonality

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Rational Seasonality Travis D. Nesmith 2007-04 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Rational Seasonality TravisD.Nesmith (cid:3) BoardofGovernorsoftheFederalReserveSystem November28,2006 Abstract Seasonal adjustment usually relies on statistical models of seasonality that treat seasonal (cid:3)uctuations as noise corrupting the ‘true’ data. But seasonality in economic series often stems from economic behavior such as Christmas-time spending. Such economic seasonality invalidates the separability assumptions that justify the construction of aggregate economic indexes. To solve this problem, Diewert (1980, 1983, 1998, 1999) incorporates seasonal behavior into aggregation theory. Using duality theory, I extend these results to a larger class of decision problems. I also relax Diewert’sassumptionofhomotheticity. IprovidesupportforDiewert’spreferredseasonally-adjustedeconomicindexusingweakseparabilityassumptionsthatareshowntobesuf(cid:2)cient. JELClassi(cid:2)cation: C43;D11;E31 (cid:3) 20th&CSts.,NW,MailStop188,Washington,DC20551;travis.d.nesmith@frb.gov IwishtothankWilliamBarnett,ErwinDiewert,BarryJones,andHeinzScha¤ttlerforhelpfuldiscussionsandcomments.Theviewspresentedaresolelymyownanddonotnecessarilyrepresentthose oftheFederalReserveBoardoritsstaff. 1

1 Introduction Economic indexes are often treated as given; the complicated aggregation theory underlying the construction of the index is ignored in empirical research. But aggregation and statistical index number theory has returned the favor and largely ignored the consensus that seasonal (cid:3)uctuations, due to by phenomena such as seasonal patterns in the growing cycle, Christmas shopping, etc., are endemic to economic time series including economic indexes. Relatively little work has attempted to incorporate seasonal (cid:3)uctuations into the theory, even though seasonality can invalidate the separability assumptions that justify the construction of aggregateeconomicindexes. Seasonalityhasusuallybeenaddressedeconometrically. Standardeconometric approaches view seasonality as an undesirable characteristic of the data. Consequently,thebulkoftheresearchonseasonalityhastreatedseasonal(cid:3)uctuationsas noisethatiscorruptingtheunderlyingsignal. Econometricresearchhasfocusedon howtosmoothorremoveseasonal(cid:3)uctuations. Econometricseasonal-adjustment techniques(cid:151)ranging from the inclusion of seasonal dummies in regression analysis to the complicated procedures, such as the X-12 procedure, implemented by statistical agencies to produce seasonally adjusted data(cid:151)rely on statistical models of seasonality. No matter how statistically sophisticated, these models share a fundamentalweaknessinthattheyhavelittleornoconnectiontoeconomictheory. Diewert(1996b,page39)describessuchmodelsas(cid:147)moreorlessarbitrary.(cid:148) Withfewexceptions(Ghysels,1988;Miron&Zeldes,1988;Miron,1996;Osborn, 1988), researchonseasonal-adjustmentexplicitlyorimplicitlyassumesthat seasonality is not the result of economic behavior. Grether and Nerlove (1970) acknowledge that seasonal phenomena in economic data is generated by customs and institutions, and should be expected to be more complex than meteorological phenomena. Nevertheless, the main approaches to econometric seasonal adjustmentarebasedonunobservedcomponentmodelshistoricallydevelopedtomodel astronomicalphenomena. InaseriesofpapersDiewert(1980,1983,1996b,1998, 1999) argues that much of the seasonality in economic time series is produced by the behavior of economic agents, and that such behavior should consequently be modeledwitheconomicsratherthaneconometrics. 2

Diewertfocusesonthefactthatmanyeconomictimeseriesareconstructedas statisticalindexnumbers. Theconstructionofstatisticalindexnumbersisjusti(cid:2)ed, in the economic approach to index number theory, by their connection to speci(cid:2)c economicmodels. Diewertstressesthatthesemodelsdonotaccountforbehavior thatvariesacrossseasons,and,consequently,theeconomicindexesarenotvalidin thepresenceofseasonality. Heexaminestwodifferentwaysthatseasonalbehavior ofeconomicagentscanberationalizedinaneoclassicalframework,andconcludes that only one of these possibilities is consistent with the economic approach to constructingindexnumbers. Seasonalbehaviorcanberationaliftheagentisoptimizingatimevaryingobjective function. However, a time varying objective function generally cannot be trackedbyaneconomicindex,1 Alternatively,theagent’sobjectivefunctionisnot separableattheobservedseasonalfrequency. Theimplicationsofthislackofseparability on the functional structure of the agent’s decision is more amenable to analysisthangeneraltimevariability. Diewert(1980)concludesthatresearchinto seasonalbehaviorshouldfocusondecisionproblemsthatarenottimeseparableat seasonal frequencies; in his subsequent papers he adapts a standard utility maximizationproblemtoaccountforseasonality. Hisseasonaldecisionproblemcanbe used to construct economic indexes from data that contains seasonality; Diewert (1998,page457)describeshisresearchas(cid:2)llingagap: (cid:147)The problem of index number construction when there are seasonal commodities has a long history. However, what has been missingisanexpositionoftheassumptionsontheconsumer’sutilityfunctionthatarerequiredtojustifyaparticularformula. Wesystematically list separability assumptions on intertemporal preferences that can be usedtojustifyvariousseasonalindexnumberformulasfromtheviewpointoftheeconomicapproachtoindexnumbertheory.(cid:148) Diewert’sapproachdeseasonalizesstatisticalindexnumbersbytheirconstruction. Diewert’s models and the resulting indexes have an obvious advantage over 1Time-varyingpatternsthatareonlyafunctionoftheseasonasinOsborn(1988)areaspecial caseofseaonalinseparabilitywhichisthesecondapproachDiewertexamines. 3

econometric models; their connection to economic theory obviates the development of econometric criteria for evaluating different adjustment methods. Economictheorydirectlyjusti(cid:2)estheindexnumberapproachtoseasonaladjustment. In this paper, I further extend the aggregation theory approach to seasonal adjustment. WhilethispaperextendsDiewert’slineofresearch,thefocusisslightly different. Ifocusmoreonde(cid:2)ningseasonalaggregates,ratherontheresultingindexformulae, asde(cid:2)ning aggregatesislogicallypriortode(cid:2)ningtheindexesthat trackthem. Inaddition,althoughIamweakeningtheconditionsnecessarytorationalizeseasonalaggregatestheresultingindexesarethesameasinDiewert(1999) sofocusingontheindexformulaewouldberedundant. Usingdualitytheory,IextendDiewert’sresultstoalargerclassofdecisionproblems. IalsorelaxDiewert’s assumption of homotheticity. The most novel result is a justi(cid:2)cation of Diewert’s moving year index, which is his preferred seasonal index, using only separability assumptions. Thederivation, whichfollowsfromatheoremofGorman(1968), is notonlysuf(cid:2)cient,itisalsoshowntobenecessary. Theremainderofthispaperisorganizedasfollows. Section2brie(cid:3)ydiscusses econometricseasonaladjustmentmethods. Section3reviewstheindexnumberapproach to seasonal adjustment developed by Diewert. Section 4 presents different typesofseparabilityfortheexpenditureanddistancefunctions. Section5provides the necessary conditions supporting the construction of seasonal indexes. In particular,Diewert’smovingyearindexisderivedfromaseparabilityassumption. An argument for why these particular separability assumptions are reasonable is also advanced. Although no empirical analysis of the index number approach to seasonality is provided, Section 6 comments on some empirical implications of the theory. Thelastsectionconcludes. 2 Econometric Adjustment This section brie(cid:3)y discusses econometric adjustment techniques; see Nerlove, Grether, and Carvalho (1979); Bell and Hillmer (1984); Hylleberg (1992) and Miron (1996) for more extensive reviews. The discussion focuses on how dif(cid:2)cult it is to establish criteria for determining how to econometrically adjust series forseasonality. Thelackofcriteriamakesthechoiceofwhichmethodtousesub- 4

jective. The situation is similar to the dif(cid:2)culties in choosing a statistical index numberformulasolelyonthebasisoftheiraxiomaticproperties. The majority of seasonal adjustment techniques are based on decomposing a series,ormultipleseries,intounobservedcomponents. GretherandNerlove(1970, page686),indiscussingthe(cid:147)desiderata(cid:148)ofseasonaladjustment,notethattheunobservedcomponentsmethodsoriginatedinastronomy,andstate, (cid:147)It is of course, quite debatable whether the idea of unobserved components, appropriate as it may be in the analysis of astronomical observations, is usefully applied to economic data or even to meteorologicaldata. Nonetheless,webelievethatthisidealiesbehindboth present methods of seasonal adjustment and the desire for seasonally adjustedtimeseries.(cid:148) Grether and Nerlove (1970) and Nerlove et al. (1979) show that ‘optimal’ econometricseasonaladjustmentdependsonboththemodelofseasonalityandthe modelinwhichthedataaretobeused. Nerloveetal. (1979,page171)conclude, (cid:147)...itisclearthat(a)nosinglemethodofadjustmentwillbebestforallpotential usersofthedataand(b)itisessentialtoprovideeconomictimeseriesdatainunadjustedform.(cid:148)Theyadditionallyconcludethatdespitetheincreaseinsophistication of the econometric techniques, (cid:147)...in terms of modeling explicitly what is going on, thereseemstohavebeenremarkablylittleprogress.(cid:148)Thisconclusionremains valid. The lack of an empirical standard leads Bell and Hillmer (1984) to conclude that seasonal adjustment methods should be judged on whether the model of seasonality implicit in the method is consistent with the observed seasonality in the data. This would suggest that different adjustment methods should be applied to differentdataseries,sothereisnounique‘optimal’method. Seasonaladjustment hasalsobeencharacterizedasasignalextractionprobleminthefrequencydomain. Grether and Nerlove (1970) argue against evaluating adjustment methods using empiricalcriteriabasedonthespectralpropertiesoftheadjustment,althoughthey donotdiscountitsusefulnessforcharacterizingtheeffectsofdifferentmethodologies. The lack of a consensus on how to seasonally adjust has led some authors to focus on the effects on the statistical properties of the data when the data is 5

seasonallyadjustedusingthewrongstatisticalmodel(Wallis,1974,discussesthis issue). Several authors, notably Lovell (1963, 1966) and Jorgenson (1964) try to deriveasetofaxiomsthataseasonaladjustmentmethodshouldsatisfy. Jorgenson’s approach is to specify that the adjustment method should satisfy the properties of theuniqueminimuminvariance, linear, estimator. Whilethisseemsreasonable, it stillprovidesanindeterminatesolution,becauseotherstatisticalmodels,forexampleaminimumdistanceestimatororweightedleastsquares,arejustassensible. In addition, Lovell (1966) showed that Jorgenson’s method does not satisfy Lovell’s orthogonality axiom, so the adjusted series is correlated with the seasonal adjustmentcomponent. TheapproachinLovell(1963)isperhapsthemostintriguingrelativetoDiewert’sapproach,becauseitisreminiscentoftheaxiomaticapproachtoindexnumber theorydevelopedinFisher’s(1922)seminalwork. Justliketheaxiomaticapproach toindexnumbers,2thisaxiomaticapproachtoseasonaladjustmentis(cid:3)awedbythe factthatsensiblesetsofaxiomsareinconsistentwitheachother. Lovellisup-front about the dif(cid:2)culty. Lovell (1963, page 994) shows in Theorem 2.1 that the only operatorsthatpreservesums,inthesensethat xa ya .x y /a,andpreserve t C t D t C t products, in the sense that xaya .x y /a are trivial in that either xa x or t t D t t t D t xa 0. Thesetwoaxiomsareintuitivebecausethe(cid:2)rstoneimpliesthataccountt D ing identities are unchanged by the adjustment, and the second one implies that the relationship between prices, quantities, and expenditure are not altered by the adjustment. Consequently, this result shows that two of the most intuitive axioms forseasonaladjustmentareinconsistent;Lovell(1963,page994)characterizesthis resultas‘disturbing’andconcludesthat(cid:147)itsuggeststhattwoquitesimplecriteria rule our the possibility of a generally acceptable ‘ideal’ technique for adjusting economictimeseries.(cid:148) Thesolutiontotheinconsistencyoftheaxiomaticapproachtoindexnumbers istheeconomicapproach. Theeconomicapproachallowsevaluationofindexformulaebyappealingtotheory. Indexesthathaveastrongerconnectiontoeconomic theoryunderweakerassumptionsarejudgedtobesuperior. Theusefulnessofsuch 2SeeSwamy(1965) 6

criteria can be seen in how superlative indexes (Diewert, 1976) are now accepted as the de(cid:2)nitive approach to constructing index numbers, not only by theorists, but also by statistical agencies. Diewert’s approach to constructing seasonal indexnumbersbyde(cid:2)ningseasonaleconomicaggregatescansimilarlyanswerhow indexesshouldbeseasonallyadjustedbyappealingtoeconomictheory. 3 Review of Diewert’s Approach Diewert (1996b, 1998, 1999) treats the problem of seasonality as part of the economicapproachtoconstructingbilateralindexnumbersandjusti(cid:2)esdifferentseasonal index number formulae on the basis of different separability assumptions.3 The theoretical basis of this work allows it to be used as a standard for seasonal adjustment. This section reviews Diewert’s approach. It focuses on three of his de(cid:2)nitions of seasonal indexes: Annual, Year-over-year, and Moving Year. The notation largely follows Diewert’s, but a different separability de(cid:2)nition will be used. Forexposition,anumberofsimplifyingassumptionsaremade. First,theconsumptionspacewillbeassumedtobeofconstantdimensionineachseason. Diewert(1998)dividesseasonalcommoditiesintotype-1andtype-2. Type-1commodities are goods that are not available in every season. These are type of goods are particularlyproblematicforindexnumbertheory. Theassumptionthatthedimensionofthecommodityspacedoesnotchangeinaseasonmeansthattype-1goods are not allowed to be randomly missing. Note that if a good is not consumed, it doesnotnecessarilymeanitwasunavailable. Itcouldbethatthepriceofthegood wasaboveitsreservationprice. Thiscaseisobservationallyequivalenttothe(cid:2)rst, however,andinaggregatedataitseemsreasonabletoassumethatifagoodisnot consumeditisunavailable,sothefocusisontype-2goods. Diewert (1998) also assumes that type-2 seasonal commodities can be further divided into type-2a and type-2b commodities. Type-2a commodities are commodities whose seasonal (cid:3)uctuations correspond to rational optimizing behavior overasetofseasonswhereprices(cid:3)uctuatebutpreferencesforthecommodityre- 3Themodelcanbeeasilyadaptedtorepresentarepresentative(cid:2)rmthatproducesasingleoutput frommultipleinputs.Multipleoutput(cid:2)rmsintroducefurthercomplications(Fare&Primont,1995). 7

mainunchanged. Type-2bcommoditiesarethosewherethisdoesnotapply. Type- 2a commodities can be aggregated under normal assumptions. In the following, I donotdifferentiatebetweenthetype-2sub-commodities; anygroupofcommodities is assumed to contain at least one type-2b commodity so that aggregation of thegrouprequiresfurtherassumptionstorationalizetheseasonalbehavior. Theeffectofin(cid:3)ationisalsoignored. Consequently, currentperiodpricesare usedratherthanspotprices. Thus,thecostindexesarefuturespriceindexesrather thanspotpriceindexes(Pollak,1975). AsnotedbyInalowin(cid:3)ationenvironment using the current period prices is not a major concern and it removes a level of complexityfromtheexposition. Thesimplifyingassumptionscanberelaxedwithoutmuchdif(cid:2)cultyfollowing Diewert(1998,1999). Somenotationisneededtode(cid:2)netheseasonaldecisionproblem: Notation 1. Let m 1,...,M denote the season, where M is the number of sea- D sons, typically be 4 or 12. Each season m has N commodities for each year m t 0;1;:::;T . Let ptm [ptm;:::; ptm] be the vector of positive prices and 2 f g D 1 Nm qtm [qtm;:::;qtm]bethevectorofcommoditiesconsumedinseasonm ofyeart. D 1 Nm Annual vectors of prices and consumption are de(cid:2)ned by pt [pt1;:::; ptM] and D qt [qt1;:::;qtM], respectively. Let (cid:127) denote the complete consumption space D which is equal to R T.N1 C(cid:1)(cid:1)(cid:1)C NM/: Let x y denote the standard inner product for (cid:1) vectors. The(representative)agentisassumedtohaveatransitive,re(cid:3)exive,complete, andcontinuouspreferenceorderingon(cid:127):Preferencesarealsoassumedtobenondecreasing and convex. Under these assumptions, preferences can be represented byareal-valuedutilityfunctionU : (cid:127) Rthatsatis(cid:2)es: ! Condition2. Continuity,positivemonotonicity,andquasi-concavity. The following decision problem then represents a basic utility maximization problemadaptedtotheseasonalnotation: Problem3(UtilityMaximization). The(representative)agentsolvesthefollowing intertemporal utility maximization problem where the utility function U. / satis- 8

(cid:2)es: T max U.x0;x1;:::;xT/ (cid:27) pt xt W (1) t x0;x1;:::;xTf j (cid:1) (cid:20) g t 0 XD where xi has the same dimension asqi, pt xt M ptm xtm, (cid:27) is a strictly (cid:1) D m 1 (cid:1) t D positive discount factor, and W is the discounted present value of intertemporal P wealth at t 0. Assume the vector [q0;:::;qT] solves the intertemporal utility D maximizationproblem. ThenW T (cid:27) pt qt. D t 0 t (cid:1) D P Remark 4. Theassumptionsonpreferencesimplythatthesuperiorset,de(cid:2)nedas S.u/ q q (cid:127) U .q/ u , is closed and convex. These properties of the (cid:17) f j 2 ^ (cid:21) g superiorsetareimportantforduality,astheyimplythatpreferencescanbeequivalently represented by a expenditure function. The dual representation is valid, becauseaclosedconvexsetcanbeequivalentlyrepresentedbytheintersectionof theclosedhalf-spacesthatcontainit(Luenberger,1969,Theorem5). Diewert makes a series of structural assumptions on this general utility maximization problem to de(cid:2)ne annual, year-over-year, and Moving Year seasonal aggregatesandeconomicindexes. Inordertode(cid:2)neannualeconomicindexes,Diewert(1998,1999)assumesthat theutilityfunctionin(1)takestheform U.x0;x1;:::;xT/ F.f.x0/; f.x1/;:::; f.xT// (2) D where f. / is positively linearly homogeneous (PLH) and satis(cid:2)es Condition 2. The annual aggregator function f. / treats each good in a different season as a differentgood. FromTheorem5.8inBlackorby,Primont,andRussell(1978,pages206(cid:150)207), the annual aggregator functions satisfy additive price aggregation, and de(cid:2)ne annualeconomicquantityaggregatesbecauseoftheirhomogeneity. Thedualunitexpenditure function is the annual economic price aggregate. Annual Konu¤s (1939) true cost-of-living indexes and Malmquist (1953) economic quantity indexes can bede(cid:2)ned: De(cid:2)nition5(Annualeconomicindexes). Annualeconomicpriceandquantityin- 9

dexesarede(cid:2)nedby e.pt/ f.qt/ KA.t;s/ and MA.t;s/ : D e.ps/ D f.qs/ for0 s < t T,wheree. /istheunitexpenditurefunction.4 (cid:20) (cid:20) Remark 6. The assumptions necessary to de(cid:2)ne the annual economic index are the weakest that address seasonality. These annual indexes can be tracked using standard index number theory. The resulting index number is deseasonalized by construction. The problem is that the index only provides a single measure per year,whichisnotfrequentenoughformanyapplications. The deseasonalization of the annual index is a by-product of the time aggregation that takes place. The annual indexes also represents Diewert’s (1980) preferredmethodfortimeaggregatingeconomicdata,astheseassumptionsplacethe fewest restrictions on intertemporal preferences. This method was implemented in constructing annual indexes from monthly data in Anderson, Jones, and Nesmith (1997a); the annual indexes calculated from seasonally adjusted and nonseasonallyadjusteddataareindistinguishable.5 Year-over-year indexes, which were suggested by Mudgett (1955) and Stone (1956), give a measure for each season, but require further assumptions. Diewert (1999, page 50) assumes that the annual aggregator function, for each t 2 0;:::;T ;takestheform f g f.xt1;xt2;:::;xtM/ h[f1.xt1/; f2.xt2/;:::; fM.xtM/] (3) D where fm. /form 1;:::;M isaseasonalaggregatorfunction,withdimension D N , of the annual aggregator function f. /. Under this assumption f. / is an m annual aggregator function over seasonal aggregator functions, fm. /. Note that since f. / is PLH, so are the seasonal aggregators. The fm. / are, clearly valid seasonalaggregates,andareusedtode(cid:2)neyearoveryearseasonalindexes: 4Homogeneityoftheannualaggregatorfunctionsimpliestheexistenceofannualunitexpenditure functions. 5ThedataareavailablefromtheMSIdatabaseonFREDatwww.stls.frb.org. 10

De(cid:2)nition 7 (Year-over-year seasonal economic indexes). For every season, denoted by m 1;:::;M , year-over-year seasonal economic price and quantity 2 f g indexesarede(cid:2)nedby em.ptm/ fm.qtm/ Km.t;s/ and Mm.t;s/ D em.psm/ D fm.qsm/ for 0 s < t T, where em. / is the dual unit expenditure function for the (cid:20) (cid:20) season. Remark 8. The seasonal indexes are still comparing one season to a season in a previousyear. The separability assumptions imply that the solution achieved by solving the generalproblemin(1)willalsobethesolutiontothefollowingmultistagedecision problem: inthe(cid:2)rststage,theconsumerchoosestheoptimalamountofwealthto allocate to each year to maximize the overall utility function U. /; in the second stage, for each year, the consumer chooses the optimal amount of the allocated wealth from the (cid:2)rst stage to allocate to expenditure in each season to maximize h. /; and in the third stage, the consumer chooses the optimal quantities of the differentseasonalgoodssubjecttotheallocatedwealthtomaximize fm. /. The multi-stage decision justi(cid:2)es de(cid:2)ning annualized year-over-year indexes, by (cid:2)rst constructing year-over-year indexes, suitably normalized in the base period, and then constructing an annual index from the seasonal indexes.6 Clearly, theannualindexcalculatedinstagesgenerallyrequiresstrongerassumptionsthan the actual annual indexes. Superlative indexes constructed in such a two-stage algorithm will not generally equal a superlative annual index, because superlative indexesonlyapproximatelysatisfyconsistencyinaggregation(Diewert,1978). Diewert’s(1999)lasttypeofindexisthemovingyearindex.7 Diewertmakes 6Thedualpriceindexcanbecalculatedbyfactorreversal.Theeffectofdiscountingisignoredin thisdiscussion. Inpracticetheeffectofintertemporaldiscountingcouldbeminimizedbychaining theindices. 7InSection3ofhispaper,Diewert(1999)discussesshort-termseason-seasonindexes,whichare de(cid:2)nedoversubsetsofnon-seasonalcommodities. Sinceseasonalbehaviorisexcludedfromthese indexes,theyarenotcoveredhere. 11

theadditionalassumptionthatU. /satis(cid:2)es U.x01;:::;x0M ::: xT1;:::;xTM/ 1 T M (cid:12) [fm.xtm/] (4) I I D (cid:0) f t 0 m 1 m g D D X X where (cid:12) are positive parameters that allow cardinal comparison of the transm formed seasonal utilities [fm.xtm/] and [ ] is a monotonic function of one positivevariablede(cid:2)nedby z(cid:11); if(cid:11) 0 .z/ f .z/ 6D (5) (cid:11) (cid:17) (cid:17) 8 lnz; if(cid:11) 0. < D This assumption implies that the intertem:poral utility functionU. / is a constant elasticity of substitution (CES) aggregator of the seasonal aggregator functions fm. /. ItalsoimpliesthattheannualaggregatorfunctionsareCESintheseasonal aggregatorfunctions: h[f1.xt1/; f2.xt2/;:::; fM.xtM/] 1 M (cid:12) [fm.xtm/] (6) D (cid:0) f m 1 m g D X fort 0;:::;T. Awell-knownresultinindexnumbertheory,duetoSato(1976), D isthattheSato-VartiaquantityindexisexactfortheCESfunctionalform.8 UndertheCESassumptions,thechangeintheannualaggregatescanbetracked through the same two-stage method discussed previously. The difference is that at the second stage the aggregator functions are assumed to have the restricted CES form. As Diewert (1996a) notes, the strong assumption that U. / is CES mightbepuzzling. Itsusefulnessisthataggregationtobeextendedtonon-calendar years; under the CES assumption there exists an annual aggregator function for any sequential run of the M seasons. Thus, for each season, an annual index can be calculated from the previous M 1 seasons (e.g. in July, an index could be (cid:0) calculated over the monthly data from the previous August through July). These movingyearannualindicesarealreadyseasonallyadjustedbyconstruction. Thenotationwillbesimpli(cid:2)edbythefollowinglagfunction: 8TheSato-Vartiaindexwas(cid:2)rstde(cid:2)nedbyVartia(1976a,1976b)astheVartiaIIindex. 12

De(cid:2)nition9(LagFunction). Thefunctionfortimet isde(cid:2)nedby t if x 0, Lt.x/ (cid:21) D 8 t 1 if x < 0. < (cid:0) WiththisfunctiontheMovingye:arindexescanbewrittenasfollows: De(cid:2)nition 10 (Moving year annual seasonal economic indexes). For each season m inyeart,movingyearannualseasonaleconomicpriceandquantityindexesare de(cid:2)nedby M (cid:12) 1[ei.pLt.m i/i/] K.m;t;s/ f i 1 i (cid:0) (cid:0) g (7) D D M (cid:12) 1[ei.pLs.m i/i/] f Pi 1 i (cid:0) (cid:0) g D and P 1 M (cid:12) [fi.qLt.m i/i/] M.m;t;s/ (cid:0) f i 1 i (cid:0) g (8) D D 1 M (cid:12) [fi.qLs.m i/i/] (cid:0) f Pi 1 i (cid:0) g D for 0 s < t T, where em. / isPthe dual unit expenditure function for the (cid:20) (cid:20) season. Remark 11. The moving year indexes provide an annual measurement for each season. Similar indexes could also be constructed for shorter or longer sequential runs. Asafurthersophistication,Diewert(1999)suggestscenteringthenon-calendar years. Thelagfunctionisnolongersuf(cid:2)cient;thefollowingcenteringfunctionwill beused: De(cid:2)nition12(Centeringfunction). Thefunctionfortimet andnumberofseasons M isde(cid:2)nedby t 1 if x > M, C Ct .x/ 8 t if0 x M, M D > > (cid:20) (cid:20) > < t 1 if x < 0. (cid:0) > The following de(cid:2)nition cente> >rs the moving year indexes assuming there are : an even number of seasons as is the norm. Effectively, the index is calculated by taking the M 1 terms centered around the season m and adding half of the (cid:0) value of two extra terms: m M=2 seasons ahead and m M=2 seasons prior. C (cid:0) 13

Notationally,thisaccomplishedbyaveragingtwosequencesof M termswherethe secondsequenceislaggedoneseasonrelativetothe(cid:2)rst. De(cid:2)nition13(Centeredmovingyearannualseasonaleconomicindexes). Foreach season m in year t, centered moving year annual seasonal economic price and quantityindexesarede(cid:2)nedby f 1 2 i M 1 (cid:12) i (cid:0) 1[ei.pC M t .m C M=2 (cid:0) i/i/] KC.m;t;s/ C 1 2 Pi M D 1 D (cid:12) i (cid:0) 1[ei.pC M t .m (cid:0) 1 C M=2 (cid:0) i/i/] g (9) D f P 1 2 i M 1 (cid:12) i (cid:0) 1[ei.pC M s .m C M=2 (cid:0) i/i/] C 1 2 Pi M 1 D (cid:12) i (cid:0) 1[ei.pC M s .m (cid:0) 1 C M=2 (cid:0) i/i/] g D P and (cid:0) 1 f2 1 i M 1 (cid:12) i [fi.qC M t .m C M=2 (cid:0) i/i/] MC.m;t;s/ C 1 2 i M P D 1 (cid:12) D i [fi.qC M t .m (cid:0) 1 C M=2 (cid:0) i/i/] g (10) D (cid:0) 1 P f 1 2 i M 1 (cid:12) i [fi.qC M s .m C M=2 (cid:0) i/i/] C 1 2 i M P1 (cid:12) D i [fi.qC M s .m (cid:0) 1 C M=2 (cid:0) i/i/] g D for 0 s < t T, where em.P/ is the dual unit expenditure function for the (cid:20) (cid:20) season. Remark14. Ifthereareanoddnumberofseasons,thenotationforacenteredindex ismuchsimpler. Remark 15. Inpractice,Diewert(1999)suggestscalculatingtheannualindicesas superlative indices also, as they can provide a second-order approximation to any aggregatorfunctionincludingtheCESspeci(cid:2)cation. Also,superlativeindexesare usuallychained,sothatthereferenceperiodadvancesandisalwaysonelagofthe currentperiod.9 Theseasonalindexesreviewedinthissectionareconnectedtoeconomictheory by their derivation from the utility maximization problem. The various indexes werederivedbyassumingmoreandmoreaboutthestructureoftheutilityfunction. ThesectionfollowedDiewert’sinthatseparabilitywasonlyimplicitlymentioned astherationaleforthefunctionalstructures. Thisseemingoversightisjusti(cid:2)edby 9SeeAnderson,Jones,andNesmith(1997b)foradiscussionofchaining. 14

theassumptionthatthenestedutilityfunctionsarehomothetic. Implicitly,Diewert isassumingthatpreferencesarehomotheticallystrictlyseparableattheannualand seasonal frequencies. Homotheticity, which Swamy (1965) called a ‘Santa Claus assumption,’allowsthemosteleganttreatmentofaggregationandstatisticalindex numbertheory. But,homotheticityisastrongassumptionandnotnecessary. 4 Duality and Separability Diewertimplicitlyconnectedtheseasonalstructuresandindexesde(cid:2)nedinSection 3 to an agent’s preferences through assuming homothetic strict separability. The seasonal indexes that Diewert derived can be supported under weaker conditions thanheused;weakeningDiewert’sconditionsprovidesbroadertheoreticalsupport fortheseasonalindexesandhelpsinoculatethetheoreticalapproachtoseasonality fromcriticismthatclaimstheassumptionsareunrealistic. As telegraphed at the end of the previous section, the (cid:2)rst step to weakening Diewert’s conditions is to weaken the homotheticity assumption. Relaxing homotheticity leads naturally to focusing on the expenditure and distance function representation of preferences. The bene(cid:2)t of beginning with the expenditure and distance function is twofold. First, these two dual representations are always homogeneous in prices and quantities respectively. This property led Konu¤s (1939) tode(cid:2)nethetruecostoflivingindexthroughtheexpenditurefunction. Similarly, Malmquist (1953) originally used the distance function to de(cid:2)ne economic quantity indexes. The weakest conditions that support the various seasonal structures arenaturallyspeci(cid:2)edonthefunctionsthatareusedinthede(cid:2)nitionoftheindexes. Clearly,thisarguesforusingtheexpenditurefunction;thesimilarargumentforthe distance function is obscured by Diewert’s assumption of homotheticity. Second, thedualitybetweentheexpenditureanddistancefunctionsisstrongerthanbetween otherrepresentationofpreferences. Imposingfunctionalstructureontheexpenditure function implies that the distance function will have the same property and vice versa. This is not generally true for other representations of preferences. In particular,assumingtheutilityfunctionhasaseparablestructuredoesnotgenerally imply that the expenditure function will have the same structure, unless homotheticityisalsoimposed. Thesestatementsareclari(cid:2)edinthe(cid:2)rstsubsection,which 15

presentstheexpenditureanddistancefunctionanddiscussestheirduality. The second step to weakening Diewert’s conditions is to make the separabilityassumptionsexplicit. ThiswillmakeclearwhatDiewertisimplicitlyassuming whenderivingthedifferentfunctionalstructuresthataccountforseasonaldecisionmaking. The second subsection presents a variety of de(cid:2)nitions of separability. Usingthesede(cid:2)nitions,theweakestconditionsthatrationalizetheannualandseasonal indexes can be established. These de(cid:2)nitions also set up the subsequent sectionwhichdiscussesthemovingyearindexes. 4.1 TheDualExpenditureandDistanceFunctions The strong connection between the expenditure and distance function stems from the fact that they are both conic representation of preferences. The expenditure function, which is the negative of the support function, and the distance function areequivalentmathematicalrepresentationsofaconvexset. Forautilitylevel,both functions are positive linearly homogeneous convex functions.10 Gorman (1970, page105)referstothepairas‘perfect’dualsastheyalwayssharethesameproperties. Section2.3.3inBlackorbyetal. (1978,pages26(cid:150)33)providessomefurther intuition for the strong connection between the expenditure and distance function by showing that the functions switch roles with regard to the indirect utility function; the distance function can be viewed as an indirect expenditure function and the expenditure function can be viewed as an indirect distance function. The two functionscanbede(cid:2)nedintermsoftheutilityproblemin(1)asfollows: Problem 16 (Expenditure Minimization Problem). Let R.U/ denote the range of U. /withthein(cid:2)mumexcludedand(cid:127) thepositiveorthantof(cid:127). Theexpenditure C function,e : (cid:127) R.U/ R ,thatisdualtotheutilityfunctionin(1)isde(cid:2)ned C(cid:2) ! C as T M e p0;:::; pT;u min (cid:27) p xtm U.x0;:::;xt/ u (11) t tm D x0;:::;xt( (cid:1) j (cid:21) ) t 0 m 1 (cid:0) (cid:1) XD XD Problem 17 (Distance Minimization Problem). The distance function, d : (cid:127) C (cid:2) 10Inthetheoryofconvexfunctions,suchfunctionsarecalledgaugefunctions(Eggleston,1958). 16

R.U/ Risde(cid:2)nedby ! d.q0;:::;qT;u/ min (cid:21) R U q0=(cid:21);:::;qT=(cid:21) u (12) D (cid:21) 2 C (cid:21) (cid:8) (cid:12) (cid:0) (cid:1) (cid:9) Although the distance function has been(cid:12)used in economics since at least Debreu (1954), it is less familiar. For a given u, the distance function measures the amountthatq (cid:127)mustbescaledupordownsuchthatq isintheboundaryofthe 2 superior set: i.e. q=(cid:21) @S.u/. For more discussion, see Deaton and Muellbauer 2 (1980). Given the prior assumptions made on preferences in de(cid:2)ning the utility function, the expenditure function will have the following properties: continuity in .p;u/; non-decreasing, and concave in p; and increasing in u; where p (cid:127) 2 C and u R.U/. The expenditure function has an additional property that is ex- 2 tremely useful in aggregation and statistical index number theory, positive linear homogeneity(PLH)in p;whichmeansthat (cid:18) > 0; .p;u/ (cid:127) R.U/ e.(cid:18)p;u/ (cid:18)e.5;u/: 8 8 2 C(cid:2) D ThePLHoftheexpenditurefunctionholdswithoutanysuchsimilarpropertyholdingforU. /. Thepropertiesoftheexpenditurefunctionarereferredtoas: Condition 18. Joint continuity in .p;u/, strict positive monotonicity in u, and positivemonotonicity,positivelinearhomogeneity,andconcavityin p. Asperourdiscussion,thedistancefunctionhasthesamepropertiesexceptthat itisstrictlynegativelymonotonicinu andq takestheroleof p: Condition 19. Joint continuity in .q;u/, strict negative monotonicity in u, and positivemonotonicity,positivelinearhomogeneity,andconcavityinq. The duality of the expenditure and distance functions can be made clearer using the fact that away from points of global satiation, U .q/ u if and only if (cid:21) d.q;u/ 1. Consequently,theexpenditurefunctioncanbede(cid:2)nedas (cid:21) e.p;u/ min p q q (cid:127) d.q;u/ 1 D q f (cid:1) j 2 ^ (cid:21) g 17

Similarly,thedistancefunctioncanbede(cid:2)nedas d.q;u/ min p q p (cid:127) e.p;u/ 1 : D p f (cid:1) j 2 ^ (cid:21) g Thetwofunctionshaveidenticalfunctionalformexceptthattherolesofpricesand quantitiesarereversed. Therelationshiptotheutilityfunctionisclari(cid:2)edbynoting thatifpreferencesarehomotheticthen e.p;u/ e.p;1/u and d.q;u/ d.q;1/u: D D The homotheticity of U. / implies that the unit distance function d.q;1/ is the PLHcardinalizationofU. /andisitselfautilityfunction. As noted, the key properties of the expenditure and distance functions is their PLH and their functional equivalence. This (cid:2)rst implies that separability assumptionscanbeappliedtosupporttheconstructionofseasonalindexeswithoutassuming homotheticity. Without homotheticity, separability assumptions do not necessarily commute from one representation of preferences to another. The second propertyenablesthistobeavoided. 4.2 Separability Thebasicde(cid:2)nitionofseparabilityusedhereisoriginallyduetoBliss(1975). This de(cid:2)nitionismoregeneralthanthefamiliarde(cid:2)nitiondevelopedindependentlyby Sono(1961)andLeontief(1947a,1947b)asitdoesnotrequiredifferentiability. In addition,strictseparability(Stigum,1967)isused;strictseparabilityisequivalent toGorman’s(1968)de(cid:2)nitionofseparability. Finally,complete(strict)separability is de(cid:2)ned. Homothetic versions of the various forms of separability are also discussed. The de(cid:2)nitions will be presented for the expenditure function. Equivalent de(cid:2)nitions exist for the distance function with quantities replacing prices. Similar de(cid:2)nitions also exist for the utility function, but let me reiterate that imposingseparabilityontheutilityfunctiondoesnotgenerallyimplyanythingabout theexpenditurefunctionandviceversa. Thede(cid:2)nitionofseparabilitydependsontheexistenceofacollectionofsubsets being nested. Let B B ;B ;::: be a collection of subsets of some set. The 1 2 D f g 18

collection is nested if B ;B B either B B or B B . Some further i j i j i j 8 2 (cid:18) (cid:19) notationisrequired: Notation20. Let I 1;2;:::;T.N N / denotethesetofintegersthat 1 M D f C(cid:1)(cid:1)(cid:1)C g identify variables over which preferences are de(cid:2)ned. De(cid:2)ne a n-partition of the set I tobeadivisionof I inton subsetssuchthat: n I[n] I.1/;I.2/;:::;I.r/;:::;I.n/ I.j/; (13) D D j 1 (cid:8) (cid:9) [D where j;k I.j/ I.k/ ; and j I.j/ . Corresponding to I[n], (cid:127) can be 8 \ D ; 8 6D ; expressedastheCartesianproductofn subspaces: n (cid:127) (cid:127).j/ D j(cid:2)1 D where for every j, the cardinality of (cid:127).j/ is given by I.j/.11 The goods vector can then be written as q q.1/;q.2/;:::;q.n/ and the price vector p D D p.1/; p.2/;:::; p.n/ where the n categories denote general sectors, which are (cid:0) (cid:1) years or seasons in this paper; if q is in the kth sector then q is a component (cid:0) (cid:1) i i ofq.k/ (cid:127).k/ and p isacomponentof p.k/ (cid:127).k/. i 2 2 C Forsimplicity,onlythecasewhereasectorisseparablefromitscomplementin (cid:127)ispresented. Thegeneralitylostbymakingthisassumptionisnotaproblemfor theseasonalstructures. Theassumptionmeansthatthepartitionusedinde(cid:2)nitions ofseparabilityassumesn 2ratherthanthefullygeneralcasewheren 3:The D D generalcasecanbefoundinBlackorbyetal. (1978). Thefollowingfunctionwill beused: De(cid:2)nition 21. De(cid:2)ne (cid:13)r : (cid:127) R.U/ }.(cid:127).r//; where } denotes the power C (cid:2) ! C set,tobeamappingwhoseimageis (cid:13)r.p.j/; p.r/;u/ p.r/ (cid:127).r/ e.p.j/; p .r/ ;u/ e.p.j/; p.r/;u/ : (14) N D O 2 C j O N (cid:21) N n o 11Thegoodsaretriviallyassumedtobeconvenientlyorderedsothat(cid:127)isequaltotheCartesian product. 19

Thisfunctionde(cid:2)nesacollectionofsubsets 0r.u/ (cid:13) p.j/; p.r/;u p.j/ (cid:127) .j/ p.r/ (cid:127).r/ (15) N D N j 2 C ^ 2 C n o (cid:0) (cid:1) fora(cid:2)xedscalaru R.U/. N 2 Propertiesofthesetsde(cid:2)nedin(15)areusedtode(cid:2)neseparability, strictseparability,andcomplete(strict)separability: De(cid:2)nition 22 (Separability). The set of variables indexed by I.r/ is separable in e. /fromitscomplementin I[n] if0r.u/isnestedforeveryu R.U/; N N 2 De(cid:2)nition 23 (Strict separability). The set of variables indexed by I.r/ is strictly separableine. /fromitscomplementin I[n] if (cid:13)r p.j/; p.r/;u (cid:13)r p.j/; p.r/;u D Q (cid:0) (cid:1) (cid:0) (cid:1) forall.p.j/; p.r/;u/ (cid:127) .j/ (cid:127).r/ R.U/;and 2 C (cid:2) C (cid:2) De(cid:2)nition 24 (Complete (strict) separability). The expenditure function is completely(strictly)separableinthepartition I[m] I[n] ifeverypropersubsetof I[m] (cid:18) is(strictly)separablefromitscomplementin I[m]. Remark25. Separabilityisimpliedbyeitherstrictseparabilityorcompleteseparability,buttheconverseisnotgenerallytrue. Similarly,strictseparabilityisimplied bycompletestrictseparabilitybutnottheconverse. Remark 26. Multiple separable sectors are not precluded; de(cid:2)ning multiple separablesectorssimplyrequiresrepeatedapplicationoftheappropriatede(cid:2)nition. Remark27. Thede(cid:2)nitionofcomplete(strict)separabilityissensibleonlyifthere are at least three separable sectors in I[m]. Consequently, the de(cid:2)nition of (strict) separability is implicitly being applied at least three times to de(cid:2)ne at least three (strictly)separablesectorsin I[m] priortoconsideringtheirpropersubsets. De(cid:2)ning homothetic (strict) separability for the expenditure function is more complicated as the function is already PLH in p. Note that both separability and strict separability de(cid:2)ne a preference ordering on (cid:127).r/ for every u R.u/; p.r/ is C N 2 O 20

preferredto p.r/ on(cid:127).r/ conditionallyonu ife.p.j/; p.r/;u/ e.p.j/; p.r/;u/for every p.j/ (cid:127) .j/. Ifa C consumerisindiffer N entbetween p.r/ N an (cid:21) d p.r/ fort O hisc N ondi- 2 C O tionalpreferenceordering,theyarealwaysindifferentbetween(cid:21)pr and(cid:21)pr forev- O ery(cid:21) > 0. Toseethis,supposeitisnottrue. Thenthereexists p.r/, p.r/,and(cid:21) > 0 O suchthattheconsumerconditionallystrictlypreferseither(cid:21)p.r/ or(cid:21)p.r/ although O they are indifferent between p.r/ and p.r/. Without loss of generality, suppose O (cid:21)p.r/ is strictly preferred to (cid:21)p.r/. This implies that that there exists a p.j/, such O O Q thate.p.j/;(cid:21)p.r/;u/ > e.p.j/;(cid:21)p.r/;u/. Multiplybothsidesby1=(cid:21). Homogene- Q N Q O N ity of the expenditure function implies that e.1p.j/; p.r/;u/ > e.1p.j/; p.r/;u/. (cid:21) Q N (cid:21) Q O N Thisviolatestheassumptionthattheconsumerisindifferentbetween p.r/and p.r/. O Consequently, indifference between p.r/ and p.r/ implies that the consumer is in- O differentbetween(cid:21)p.r/ and(cid:21)p.r/ forevery(cid:21) > 0forall.p.r/; p.r// (cid:127).r/ (cid:127).r/ O O 2 C (cid:2) C andtheconditionalpreferenceorderingon(cid:127).r/ isalwayshomotheticforapartic- C ularu R.U/. N 2 There is however a sensible de(cid:2)nition of homothetic (strict) separability for the expenditure function. Generally, the conditional preference ordering on the rth sector depends on u. If it does not depend on u, the sector is de(cid:2)ned to be homothetically(strictly)separable. Thede(cid:2)nitionwillneedthefollowing: Notation28. De(cid:2)ne I I.0/ I 0;1;2;:::;T.N N / . Let 0 1 M D [ D f C(cid:1)(cid:1)(cid:1)C g n I[n] I.j/ 0 D j 0 [D representanextendedpartition. Thefollowingde(cid:2)nitionusesthisextendedpartition: De(cid:2)nition 29 (Homothetic (strict) separability). The rth sector is homothetically (strictly)separableifitis(strictly)separablefromitscomplementin I[n]. 0 Thisconditionimpliesthattheconditionalpreferenceorderingisnotdependent onu:Therationaleforcallingthisconditionhomothetic(strict)separabilityisthat itimplies,andisimpliedby,homothetic(strict)separabilityoftheutilityfunction. Asnotedpreviously,homotheticseparabilityisanexceptiontothestatementthat, in general, (strict) separability of one of the representations of preferences has no 21

implicationsforseparabilityoftheotherdualrepresentations. Itshouldbeapparent thatDiewertisimplicitlyassuminghomotheticseparability,sothathecanusethe dualunitexpenditurefunction. 5 Seasonal Decision-making With the separability apparatus developed in the previous section, the seasonal indexesdevelopedinSection3canberevisited. First,notethatacurrentyearannualaggregatecanbede(cid:2)nediftheexpenditure functionisseparable. Theorem30. Lete. /satisfyCondition18. Thene. /isseparablein I[m] I[n] (cid:18) ifandonlyifthereexistm 1: C er : (cid:127).r/ .U/ R r 1;:::;m; C (cid:2)R ! C D and m e : . .er/ (cid:127) .j/ .U/ R O r(cid:2)1R (cid:2) C (cid:2)R ! C D each satisfying the following regularity conditions in prices only, i) continuity, ii) positive monotonicity, iii) positive linear homogeneity, and iv) concavity,12 such that e.p;u/ e.e1.p.1/;u/;:::;er.p.r/;u/;:::;em.p.m/;u/; p.c/;u/: (16) D O Furthermore if e. / is strictly separable in I[m] I[n], e. / is continuous and (cid:18) O thereexistsanappropriatenormalizationoftheexpenditurefunctionsuchthat er.p.r/;u/ e.p1;:::; p.r 1/; p.r/; p.r 1/;:::; p.m/; p.c/;u/ (cid:0) C D N N N N N where p.j/ (cid:127) .j/ ; j 1;:::;r 1;r 1;:::;m;c are arbitrary reference 2 C D (cid:0) C vectors. Moreover,thefollowingapply: 1. e. ;u/isincreasing; O (cid:1) 12e. /satis(cid:2)esthesepropertiesifithasthesepropertiesin.e1.p.1/;u/;:::;em.p.m/;u//. O 22

2. eacher. /satisfyCondition18; 3. e. ;u/andeacher. ;u/inherit(partial)differentiabilityinprices;13 and O (cid:1) (cid:1) 4. eacher.p.r/; /inherits(strict)convexityandpositivelinearhomogeneityin (cid:1) u. Proof. FollowsfromTheorems3.4andCorollaries3.5.2and4.1.4inBPR(pages 70,80,112). Bythestrongdualityoftheexpenditureanddistancefunctions,theequivalent theoremholdsforthedistancefunction. Thefollowingcorollaryisimmediate: Corollary 31. The expenditure and distance functions are separable at annual frequenciesifandonlyif e p0;:::; pT;u e e0.p0;u/;e1.p1;u/;:::;eT.pT;u/;u D O (cid:0) (cid:1) (cid:0) (cid:1) and d q0;:::;qT;u d d0.q0;u/;d1.q1;u/;:::;dT.qT;u/;u : D O (cid:0) (cid:1) (cid:0) (cid:1) An annual separability assumption on either the expenditure or distance function is suf(cid:2)cient to de(cid:2)ne annual economic indexes, albeit ones that depend on u. This is signi(cid:2)cantly weaker than assuming homothetic strict separability. Unfortunately, separability is not quite enough, because the annual economic indexes cannot be guaranteed to satisfy weak factor reversal under only separability even though strong factor reversal holds. Consequently, separability of the expenditure function is suf(cid:2)cient to de(cid:2)ne economic aggregates, but strict separability is necessary (and suf(cid:2)cient) to de(cid:2)ne annual economic indexes. Not surprisingly, since thedifferencebetweenseparabilityandstrictseparabilitydisappearsunderhomotheticity,ifhomotheticseparabilityisassumedweakfactorreversalholds(cid:151)infact homotheticitymakesweakandstrongfactorreversalequivalent. Strictseparability isstillamuchweakerassumptionthanhomotheticseparability. 13Iftheparentfunctionisdirectionallydifferentiablein p,thenthesectoralfunctionsarepartially differentiablein p.r/. 23

Similar assumptions rationalizes year-over-year aggregates. De(cid:2)ne a partition I[M;t] I.1;t/;I.2;t/;:::;I.M;t/ where for every t, I.m;t/ indexes ptm for D f g m 1;:::;M . Furthermore,de(cid:2)nethepartition I[M;0];C whereC indexesthe 2 f g f g complementof I[M;0]. Corollary 32. The expenditure and distance functions are separable at seasonal frequencies if and only if the expenditure function is separable in I[M;0] from its complementsothat e p0;:::; pT;u D (cid:0) e e01.p01;u(cid:1)/;:::;e0M.p0M;u/;:::;eT1.pT1;u/;:::;e TM .pTM;u/;u O (cid:16) (cid:17) andthedistancefunctionisseparablein I[M;0] fromitscomplementsothat d q0;:::;qT;u D (cid:0)d d01.q01;u/(cid:1);:::;d0M.q0M;u/;:::;dT1.qT1;u/;:::;d TM .qTM;u/;u : O (cid:16) (cid:17) Thediscussionaboutannualaggregatesandindexesisappropriatehereaswell. Consequently,yearoveryearseasonalindexescanberationalizedbyonlytheimposition of strict separability. Note that there is another generalization here. The seasonalindexesarenotnestedinsideanannualindex,sotheassumptionsonpreferences are relaxed somewhat. Of course, the same result could be applied to De(cid:2)nition7. Inthiscase,theseasonalpatternofthedecisionproblemdoesnotimplythatitisnotseparableatfrequencieshigherthanayear. Itsimplyimpliesthat preferencesarenotstationary.14 Thistypeoftimevariationcanbehandledbythe index approach, at least to some extent. At least some of the time-varying utility functions that have been used to model seasonal behavior (cid:2)t into this framework: forexample,Osborn(1988). Theresultssofardemonstratethathomotheticityisnotnecessarytorationalize constructing seasonal index numbers. As discussed previously, separability in the utility and expenditure functions are not generally related to each other. Conse- 14Itmightbesensibletorefertothesekindofpreferencesascyclostationary. SeeGardnerand Franks(1975)forade(cid:2)nitionofcyclostationarityforrandomvariables. 24

quently, the developments in this section extend the class of preferences that can beusedtojustifyseasonalaggregatesandindexes. Themostinterestingextension addressestheMovingYearSeasonalIndexes,however. InordertoderivetheMovingYearIndex,Diewertwasassumedthattheutility functionhadaconstantelasticityform. Diewert’smodusoperandiwasto(cid:147)systematically list separability assumptions on intertemporal preferences(cid:148) to rationalize theindexnumbers. TheCESassumptiondoesnotseemtofollowfromanyseparabilitycondition,sotheassumptionseemsoutofplace. Thisapparentproblemcan berecti(cid:2)ed. Eitheroneofthefollowingconditionsissuf(cid:2)cient: Condition 33. e. / is differentiable such that i; @e.p;u/=@p > 0 for all p i 8 2 (cid:127) ,andthateachsectoralfunctioner. /canbechosentobedifferentiable;or C Condition34. Forallprices, p.r/ (cid:13)r p.j/; p.r/;u implies O 2 N (cid:0) (cid:1) e p.j/; p.r/;u < e p.j/; p.r/;u ; (cid:3) (cid:3) O N N (cid:0) (cid:1) (cid:0) (cid:1) forall p.j/ (cid:127).j/ andforeachu .U/wheree . /denotestheextensionofthe (cid:3) 2 N 2 R expenditurefunctiontotheboundarybycontinuityfromabove.15 Thefollowingtheoremgivesarepresentationforcompletestrictseparability: Theorem 35 (Complete Strict Separability Representation). Let the expenditure function, e. / be completely strictly separable in I[n].16 If e. / satis(cid:2)es Condition 18 and either Condition 33 or Condition 34 then there exists a function 0 : .U/ R andn functions, R ! C er : (cid:127).r/ .U/ R C (cid:2)R ! C allsatisfyingregularityconditions1-4fromTheorem30,suchthateither 1=(cid:26).u/ n e.p;u/ 0.u/ er.p.r/;u/(cid:26).u/ 0 (cid:26).u/ 6 1 D 6D ! r 1 XD 15Thisconditionrulesoutthickindifferencecurvesfortheconditionalpreorderingon(cid:127).r/. 16Noticethatthecomplementoftheunionoftheseparablesectorsisofzerodimension. 25

or n e.p;u/ 0.u/ er.p.r/;u/(cid:26)r.u/ (cid:26)r.u/ > 0 r D 8 ! r 1 YD where m (cid:26)r.u/ 1. r 1 D D ToPprovethistheorem,thefollowinglemmaisneeded: Lemma 36. Assume that e. / is continuous and that the commodities indexed by I.r/ are separable from their complement. Then letting e . /; e . /; and er . / (cid:3) (cid:3) (cid:3) O denotetheextensionsofe. /, e. /, ander. /in(16), respectively, to(cid:127) .U/, O (cid:2)R (cid:127).j/ .er / .U/,(cid:127).r/ .U/bycontinuityfromabove, p (cid:127), (cid:3) (cid:2)R (cid:2)R (cid:2)R 8 2 e .p;u/ e .p.j/er .p.r/;u/;u/: (17) (cid:3) (cid:3) (cid:3) D O Moreover,er . /satis(cid:2)esconditions1-4fromTheorem30. (cid:3) Proof. Supposethat(17)isfalseundertheassumptionsfor p @.(cid:127)/. Foragiven 0 2 arbitraryu,let p beasequencein p (cid:127) e.p;u/ e.5;u/convergingto s 0 f g f 2 C j (cid:21) p. Then 0 lim e.5 ;u/ lim e..5c/ ;er..5r/ ;u/;u/ s s s s D s O !1 e ! .l 1 im.5c/ ;er.lim.5r/ ;u/;u/ s s D O s s !1 !1 e ..5c/;er ..5r/;u/;u/ (cid:3) 0 (cid:3) 0 D O e .5;u/ (cid:3) 0 6D which contradicts the continuity of e . / from above. Since u was arbitrary, this (cid:3) establishes (17). The properties of er . / follow from the properties of er. / by a (cid:3) similarargument. CompleteStrictSeparabilityRepresentation. Under Condition 33 the result followsfromTheorem4.9ofBPR(pages143-147). ToprovethetheoremunderCondition34,notethatcompletestrictseparabilityofe. /in I[n]implies,byCorollary 4.8.4inBPR(page142)thate. /canbewrittenas: n e.p;u/ e . er .p.r/;u/;u/; (18) (cid:3) (cid:3) D r 1 XD 26

wheree . /isincreasingandeacher . /ishomothetic. ByTheorem30, (cid:3) (cid:3) e.5;u/ e.e1.p.1/;u/;:::;en.p.n/;u/;u/ D O By repeated application of Lemma 36, this representation can be extended to the boundary of (cid:127) . The condition implies that the representation extended to the C boundary can be taken to be strictly separable rather than just separable. Consequently, r thesectoralutilityfunctioncanbechosenas 8 er.p.r/;u/ e.p.r/;0c;u/ D where 0c is the zero element of the complement of (cid:127).r/.17 From the properties of theexpenditurefunction,thisequationimpliesthater. /isPLHin p.r/. Substitutingfrom(18)intothisequationproduces n er.p.r/;u/ e . er .0s;u/ er .p.r/;u/;u/ r 1;:::;m: (19) (cid:3) (cid:3) (cid:3) D C D s 1 XsDr 6D n Let er .0s;u/ a .u/forr 1;:::;n. Then(19)canbewrittenas (cid:3) r D D s 1 sDr P6D er.p.r/;u/ e .er .p.r/;u/ a .u/;u/ (20) (cid:3) (cid:3) r D C Sinceer. ;u/isPLH,thisimpliesthat, (cid:21) > 0and r 1;:::;n, (cid:1) 8 8 D e .er .(cid:21)p.r/;u/ a .u/;u/ (cid:21)e .er .p.r/;u/ a .u/;u/: (21) (cid:3) (cid:3) r (cid:3) (cid:3) r C D C Homotheticityofeacher . ;u/impliesthat (cid:3) (cid:1) (cid:30)r.er.p.r/;u// er .p.r/;u/ r 1;:::;n; (22) (cid:3) P D D where each (cid:30)r. / is increasing and each er. / is PLH. Substituting this equation P 17This is a slight abuse of notation as e. / and er. / are now refering to the extension to the boundary;thisabusewillbecontinuedthroughoutthisproof,asitsimpli(cid:2)esnotation. 27

into(21)yields,foreachr, e .(cid:30)r.(cid:21)er.p.r/;u/ a .u//;u/ (cid:21)e .(cid:30)r.er.p.r/;u// a .u/;u/: (23) (cid:3) r (cid:3) r P C D P C Letting(cid:21) 1 er.p.r/;u/,foreachr, (cid:0) D P 1 e .(cid:30)r.1 a .u//;u/ e .(cid:30)r.er.p.r/;u// a .u/;u/: (24) (cid:3) C r D er.p.r/;u/ (cid:3) P C r P Rearrangingterms,thisimplies,foreachr, e .(cid:30)r.er.p.r/;u// a .u/;u/ er.p.r/;u/e .(cid:30)r.1 a .u//;u/: (25) (cid:3) r (cid:3) r P C D P C Calltheright-handsideof(25)e.5r;u/. Inverting(25)for(cid:2)xedu,yields Q (cid:30)r.er.p.r/;u// e (cid:3)(cid:0) 1 .er.p.r/;u/;u/ a r .u/ (26) P D Q (cid:0) Using(22),substitute(26)into(18)toget n e.p;u/ e (cid:3) . e (cid:3)(cid:0) 1 .er.p.r/;u/;u/ A.u/;u/; (27) D Q C r 1 XD n where A.u/ a .u/. Thefactthate. /andeache. /arePLHinusercosts r D (cid:0) Q r 1 impliesthat P D n (cid:21)e (cid:3) . e (cid:3)(cid:0) 1 .er.p.r/;u/;u/ A.u/;u/ Q C D r 1 XD n e (cid:3) . e (cid:3)(cid:0) 1 .er.(cid:21)p.r/;u/;u/ A.u/;u/ Q C r 1 XD foreachu. Sinceu wasarbitrary,thisargumentholdsforeveryu .U/,which 2 R impliesthate. /isaquasi-linearPLHfunctionofthearguments P .er.51;u/;:::;er.51;u//: (28) P P ThiscompletestheproofbyatheoremofEichhorn(1974,page24). 28

Theimmediatecorollaryprovidesabasisforthemovingyearseasonalindexes usingcompletestrictseparability: Corollary37. De(cid:2)nethepartition I[M;T] I[1;0];:::;I[M;0];:::;I[m;t];:::;I[1;T];:::;I[M;T] : D f g Assume, in addition to satisfying Condition 18, the expenditure function satis(cid:2)es either: Then it has a CES functional form in the seasonal aggregates if it is completelystrictlyseparablein I[M;T]. These suf(cid:2)cient conditions may not be necessary. Nevertheless, they would seemtobetheweakestseparabilityconditionssuf(cid:2)cienttorationalizethemoving yearindexesthatcanbeexpectedtohold. ThisresultgivessomeinsightintothediscussioninSection3. Itisnotsurprisingthatanindexcanbede(cid:2)nedfornon-calendaryearsifitsiscompletelyseparable inseasons. Rememberthatcompleteseparabilitymeansthatanysubsetofthepartitionisalsocompletelystrictlyseparable. Thus,completestrictseparabilityallows ustode(cid:2)neaggregatesoverarbitrarypartitionsoftheseasons. Non-calendaryears are just one of the possibilities. For example, econometric seasonal adjustment is oftendoneusing(cid:2)ltersthatcontainmorethanjust12leadsorlags. TheCESoralternativelythecompletestrictseparabilityassumptionmayseem overlystrong,butthereisasensibleargumentforthiscondition. Thecalendaryear is not necessarily intrinsically special. For example, the (cid:2)scal year may be more important economically. In the discussion at the start of Section 2, the consumer was normally assumed to re-optimize or re-plan at the beginning of the period. I adaptedthistore-optimizingatthebeginningofeachyearinorderto(cid:2)nessehow strongtheseparabilityconditionsneededtobe. However,thereisnothingintrinsic toseasonalitytosuggestthattheconsumercannotstillbeallowedtore-planevery periodratherthanstickingwithhisplanforanentireyear. The presumption that seasonality in the data implies that decision problem is not separable at periods shorter than a year still seems reasonable. So the model is that the consumer solves a problem in say the (cid:2)rst month of the year, where his or her preferences are separable over the year but not for any shorter timeperiod. Then in the next period, the consumer resolves a problem, where his or 29

her preferences are separable over the year but not for any shorter time-period. This is called ‘rolling plan optimization.’ But, these are not the same years. The (cid:2)rstyearrunsfromJanuarythroughDecemberandthesecondrunfromFebruary through January. If this is viewed as being embedded in a larger, possibly in(cid:2)nite horizon problem, then this implies the existence of overlapping separable sectors. IfeachsectorisstrictlyseparableandJanuaryineitherthe(cid:2)rstorsecondyearsis strictly essential, then Gorman’s (1968) overlapping theorem implies that January commodities in year 0, February through December commodities, and January year commodities in year 1 are all strictly separable. In fact, the theorem states that they are completely strictly separable. If this thought experiment is iterated, it implies that each month’s commodities are completely strictly separable. To the extent that it seems sensible that consumer’s plan over an annual horizon and re-planthroughouttheyeartheCESassumptionseemsplausible. 6 Empirical Implications This paper contains no empirical analysis, but there are some interesting implications of the index number approach to seasonality. First, Diewert (1999) brings up some practical reasons to favor the index number method. First, the method is perhapslessarcanethancurrenteconometricpractices,andcouldbeappliedmore easily. Second,thedataindexescouldbeproducedinatimelymanner. Third,the data would be subject to fewer historical revisions, perhaps only those associated withswitchingtothecenteredversionfromapreliminarynon-centeredindexafter six months. These are cogent arguments for using index number method. However, most statistical agencies will require substantial empirical analysis before they would consider switching methods, so a few suggestions for future research seemwarranted. Anaturalwaytoanalyzeseasonaladjustmentisinthefrequencydomain. Examiningtheindexformulaadvancedheshouldmakeitapparentthattheseformula remove all power at frequencies higher than annual. Consequently, it might be interesting to view the index numbers as acting like a low pass (cid:2)lter. This is in contrast to some seasonal adjustment methods, which are more like a notch (cid:2)lter: see Nerlove (1964). One well-known problem in (cid:2)nite (cid:2)ltering theory is that the 30

optimallowpass(cid:2)lterisnotrealizable. Aninterestingquestioniswhetherseasonal indexes approximate the ideal (cid:2)lter by effectively pooling data. This is almost a stochastic index number viewpoint. Given the perspective that the index number formula effectively clips all higher order power, the indexes should be relatively smooth. Consequently,themovingyearindexes,whichaveragetheseseasonalindexes should be expected to be exceptionally smooth. The moving year indexes shouldbeexpectedtobeisolatinglargelythelong-runtrend. Inaddition,theeconometricadjustmentliteratureoftentakeslinearityasadesirable property for seasonal adjustment, despite the fact that the X-12 procedure, its predecessors and related methods are not, generally, linear. A fair amount of workhasbeenundertakentryingtodemonstratethattheseproceduresareapproximately linear. The index number method suggests that the linearity criterion is misguided. Clearly,theindexnumberadjustmentisnon-linear. Infact,lookingat Lovell’s axioms, it preserves products by de(cid:2)nition, so it cannot preserve sums in general. Aninterestingquestioniswhetherornotalinearmethodcanapproximate the index number methods. If not, an open question would be whether there are non-lineareconometricmethodsthatcanapproximatetheindexnumberapproach. Finally, the fact that the index number approach satis(cid:2)es the product preservingaxiomsuggeststhateconomicindexes,ifnotadjustedusingtheindexnumber methods, should be adjusted by techniques that are also product preserving rather thansumpreserving. Furthermore,manyeconomictimeseriesarenotindexes,so the index number approach is not applicable. Consequently, the development of econometric techniques that approximate the output of index number methods as closelyaspossiblewouldbeusefultomaintainconsistency. 7 Conclusion This paper has further developed the rational behind the index number approach. The class of preferences that can rationalize the seasonal indexes advocated by Diewert (1998, 1999) were extended. In particular, suf(cid:2)cient conditions for the movingyearindexbasedonaseparabilityassumptionwerepdeveloped. Additionally, a heuristic argument was proposed based on Gorman’s (1968) overlapping theorem that supports this separability assumption if an agent reoptimizes over a 31

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