A Parametric Approach for Estimating Core Inflation and Interpreting the Inflation Process *

Transcription

1 A arameric Approach for Esimaing Core Inflaion and Inerpreing he Inflaion rocess * Mikael Apel, er Jansson Sveriges Riksbank, S Sockholm, Sweden April 1999 Absrac In his paper we propose a new parameric approach for measuring core inflaion and analysing he inflaion process. In he model, measured inflaion may change because of changes in hree basic facors: long-run condiions, ransiory oupu, and special facors. The special facors include supply shocks and oher facors ha affec inflaion over and above changes in long-run condiions and ransiory oupu. None of he hree basic facors can be direcly observed, bu each facor is economerically idenified. We show ha our approach can be used o derive esimaes of core inflaion ha parallel hree differen views found in he lieraure -- as long-run inflaion, demanddriven inflaion, and inflaion excluding cerain undesired special facors. The approach is illusraed using Swedish quarerly daa covering he ime period 1970:1-1998:1. Key Words: Core inflaion, Kalman filer, srucural ime-series models, underlying inflaion, unobserved-componens models. JEL Classificaion Sysem Numbers: C3, E31. * We are graeful for useful commens from colleagues a he cenral bank of Norway and from seminar paricipans a Sveriges Riksbank and he 1999 BIS meeing of Cenral Bank Model Builders and Economericians, in paricular Øyvind Eirheim, Andreas Fischer, Hans Lindblad, and Anders Vredin. The views in his paper are hose of he auhors and do no necessarily reflec hose of Sveriges Riksbank. 1

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3 1. Inroducion In he las decade, inflaion argeing has become a widely used framework for boh heoreical analysis and pracical design of moneary policy. 1 In his framework, he primary objecive of he cenral bank is o keep inflaion in line wih he arge, mainly by affecing real economic aciviy hrough appropriae adjusmens of is insrumen rae. This ask may seem sraighforward bu is in pracice associaed wih considerable difficulies. One problem perains o he idenificaion and selecion of an appropriae arge variable. The common view of inflaion-argeing cenral banks seems o be ha no all movemens in he general price level are equally imporan from a moneary policy poin of view. For example, if an increase or decrease in inflaion is perceived o be sufficienly emporary, a policy response may no be regarded as necessary. This suggess ha cenral banks wish o avoid basing heir moneary-policy decisions on inflaion changes ha are no par of he pure inflaionary process, and raher focus on he underlying, or core, rae of inflaion. Unforunaely, underlying inflaion is a variable ha canno be direcly observed. 3 Anoher problem is relaed o he deerminaion of he componen of real economic aciviy ha he cenral bank can affec hrough is policy. According o he widely acceped noion of long-run neuraliy of money, he cenral bank can affec real economic aciviy only emporarily. However, like underlying inflaion, he ransiory componen of oupu ha can be affeced by moneary policy is no observable. An inerpreaion of he ask of he cenral bank is herefore ha i has o conrol an unobservable variable underlying inflaion mainly hrough he effecs of ineres raes on anoher unobservable variable a ransiory componen of oupu. This is obviously a raher inricae ask. Somewha surprisingly, and adding o he complexiy of he problem, he concep of core inflaion appears o have no clear heoreical definiion. As indicaed above, i is usually inerpreed as some more persisen componen of measured inflaion, bu differen approaches seem o refer o differen pars of persisen inflaion. In he lieraure, i is possible o idenify a leas hree differen views on core inflaion. The firs, proposed by Ecksein (1981), inerpres core inflaion as he rae [of inflaion] ha would occur on he economy s long-erm growh pah, provided he pah were free of shocks, and he sae of demand were neural in he sense ha markes were in long-run equilibrium. (Ecksein, 1981, p. 8.) In wha follows we label his view on core inflaion long-run inflaion, reflecing he fac ha i in his view is seen as a seady-sae concep. 4 A second view, inroduced by Quah & Vahey (1995), looks a core inflaion as ha componen of measured inflaion ha has no (medium- o) long-run impac on [real] oupu. (Quah & Vahey, 1995, p ) An alernaive way of puing his is as ha componen of inflaion ha is generaed by shocks wih no (medium- o) long-run effecs on real oupu. Because shocks wih no long-run effecs on real oupu are ofen referred o as demand shocks (Blanchard & Quah, 1989), he Quah-Vahey view on core inflaion may alernaively be inerpreed as approximaely corresponding o he demanddriven componen of inflaion. A hird view, which in wha follows is referred o as he cenral-bank view, seeks o capure core inflaion by eliminaing or reducing he influence of cerain facors, ypically paricularly volaile and erraic componens (see for example Blinder, 198a). Since demand shocks are no in general considered o be among hese undesirable componens, his view on core inflaion differs from ha of Ecksein, in which he sae of demand, as noed above, is required o be neural a he core rae of inflaion. 5 I also seems o differ from he Quah-Vahey inerpreaion since no only demand shocks are assumed o maer for core inflaion. 1 Surveys are given in, for example, Leiderman & Svensson (1995), Haldane (1995, 1997), Debelle (1997), and Mishkin & osen (1997). In his paper, he erms core inflaion and underlying inflaion are used synonymously. 3 In his conex i is however imporan o noe ha he policy implicaions from argeing he core or headline rae of inflaion no necessarily need o be differen. An inflaion-argeing cenral bank usually bases is acions on a forecas of inflaion. Only o he exen ha he forecas of headline inflaion differs from he forecas of core inflaion will he policy acions hen differ. This will happen if here are foreseeable effecs, for example emporary effecs, which affec he forecas of headline inflaion bu no he forecas of core inflaion. The difference beween he policy acions is hence likely o depend on he cenral bank s arge horizon. 4 Scadding (1979) suggesed a similar inerpreaion. 5 See Blinder (198b). 3

4 In his paper we propose a new parameric approach for measuring core inflaion and inerpreing he inflaion process. The approach akes he unobservabiliy of boh core inflaion and is deerminans explicily ino consideraion and esimaes hese unobservable componens simulaneously. In he model, measured inflaion may change because of changes in hree basic facors: long-run condiions, ransiory oupu, and special facors. The special facors include supply shocks and oher facors ha affec inflaion over and above changes in long-run condiions and ransiory oupu. None of he hree basic facors can be direcly observed, bu each facor is economerically idenified and hus possible o esimae. Our approach has several ineresing feaures. Firsly, because i explicily idenifies he deerminans of inflaion wihin a heoreical model, i allows a decomposiion of inflaion ino economically inerpreable componens. This faciliaes he undersanding and analysis of he inflaion process. Secondly, and as a corollary of he above-menioned aspec, we are able o derive esimaes of core inflaion ha parallel he above-discussed differen views on his variable. A measure closely relaed o Ecksein s (1981) approach is obained by leing core inflaion correspond o he par of inflaion generaed by long-run condiions, ha is when he influences of ransiory oupu and special facors are eliminaed. Given ha ransiory oupu is assumed o reflec he sae of aggregae demand, which is he common inerpreaion in his ype of model, a measure corresponding o Quah & Vahey s (1995) approach is obained by leing he par of inflaion generaed by ransiory oupu represen core inflaion. The cenral-bank measure, finally, is obained by merely excluding he effecs of cerain of he special facors from he measured inflaion rae. Thirdly, because our approach is parameric, differen specificaions of he processes of he deerminans of inflaion (and hence also of core inflaion) may be considered. This may help us o improve our undersanding of he inflaion process in Sweden (which is he counry ha we sudy), bu i also may make he approach usable for applicaions o oher counries. The remainder of he paper is srucured as follows. Secion gives a brief review of he approaches o esimaing core inflaion ha can be found in he lieraure. Secion 3 presens our parameric model of he inflaion process and discusses how i relaes o he above-menioned views on core inflaion. The empirical illusraions are presened and discussed in Secion 4. Secion 5, finally, provides concluding remarks.. Differen Views on Core Inflaion.1. Long-Run Inflaion According o he framework in Ecksein (1981), inflaion can be divided ino hree componens: core inflaion, a componen relaed o aggregae demand, and a shock componen. Core inflaion is inerpreed as he inflaion rae ha would occur on he economy s long-erm growh pah in he absence of shocks and a a neural sae of demand ha is, as he inflaion rae ha would occur in long-run equilibrium; long-run inflaion for shor. 6 Ecksein develops an economeric model of he US economy, which he uses o decompose acual inflaion ino hese hree componens. arkin (1984) shows ha his concep of core inflaion essenially coincides wih he expeced rae of inflaion in a radiional expecaions-augmened hillips (or aggregae supply) curve. 7 Despie he fac ha an immense number of hillips curves have been esimaed in differen conexs, hillips-curve specificaions have rarely been used o explicily esimae core inflaion inerpreed in his way. A possible explanaion is ha his seady-sae inerpreaion of core inflaion seems o be rarely used ouside (and possibly also inside) he academic sphere and is probably no wha people in general have in mind when referring o he erm. 6 This long-run inerpreaion of he core-inflaion concep can also be found in macroeconomic exbooks. See, for example, Burda & Wyplosz (1993) and Romer (1996). 7 See also Scadding (1979, p. 8) who argues ha core (underlying) inflaion presumably comes close o he heoreical noion of he perceived rae of inflaion. 4

5 .. The Quah-Vahey Approach An alernaive approach for esimaing core inflaion was inroduced by Quah & Vahey (1995). Like he Ecksein framework, his approach esablishes a link beween core inflaion and oher economic variables. Core inflaion is seen as he componen of measured inflaion ha has no (medium- o) longrun impac on real oupu. This resricion is in Quah & Vahey (1995) implemened in a bivariae oupu-inflaion vecor-auoregressive (VAR) sysem by assuming ha here exis (permanen) shocks ha do no affec real oupu in he long run. These shocks are hen assumed o be he shocks ha generae core inflaion. In he lieraure, shocks wih no long-run impac on oupu have ofen been inerpreed as demand shocks. Thus, in he Quah-Vahey framework, core inflaion may be inerpreed as demand-driven inflaion (alhough Quah & Vahey hemselves do no explicily make his inerpreaion). This view of core inflaion seems o differ from oher views on he concep. Looking a core inflaion as he componen of measured inflaion ha has no (medium- o) long-run impac on real oupu implies ha core inflaion is associaed wih ransiory movemens of real oupu ou of longrun equilibrium. This conrass wih he inerpreaion of Ecksein, who assumes ha core inflaion is he rae of inflaion ha would occur when he real economy is a is long-run equilibrium. The view also differs from he cenral-bank view, which does no assume ha core inflaion only depends on demand shocks. Alhough no cenral bank o our knowledge is currenly using an esimae derived from he Quah-Vahey approach as is official esimae of underlying inflaion, he approach has cerainly gained widespread use among analyss of moneary policy (see for example Blix, 1995, Fase & Folkersma, 1997, Bjørnland, 1997, Claus, 1997, Dewacher & Lusig, 1997, and Garner & Wehinger, 1998)..3. The Cenral-Bank View The approaches for esimaing core inflaion emanaing from he cenral-bank view may be loosely described as various ways of eliminaing or reducing differen undesirable effecs on he measured inflaion rae. Typically, measured inflaion is adjused for highly volaile componens and price developmens considered o be represening one-off shifs in he price level, such as changes in indirec axes. Someimes, measured inflaion is also adjused wih respec o he direc, more or less definiional, adverse effecs of he cenral bank s own acions. In many counries, componens direcly relaed o ineres-rae changes are lef ou of he inflaion measure since, for example, a ighening of moneary policy will hrough hese componens increase measured inflaion auonomously. Clearly, such an adverse shor-erm effec is hardly an adequae reason for furher moneary ighening. 8 A common feaure of he pracical implemenaions of he cenral-bank approaches is ha hey, unlike he approaches of Ecksein (1981) and Quah & Vahey (1995), do no esablish an explici link beween core inflaion and oher economic variables. Hence, hey end o have a weaker heoreical under-pinning and may herefore be viewed as more mechanical. On he oher hand, hey are less complicaed and hus easier for he general public o undersand, a leas in he sense ha he operaions made o arrive a he core-inflaion esimae are quie sraighforward. Daa on he differen aggregae price index componens (in pracice, CI componens) are ofen used as he saring poin for he analysis of core inflaion according o he cenral-bank view. One commonly used approach aemps o make measured inflaion reflec he underlying rae more accuraely by removing he esimaed effecs of specific disurbances and evens on a case-by-case basis. 9 The mos common example of his ype of correcion is adjusmen for he effecs of changes in indirec axes. Oher evens ha someimes are believed o moivae an adjusmen are significan changes in he erms of rade or differen ypes of naural disasers causing large price increases on cerain iems. 10 This procedure requires adequae informaion regarding he source, magniude, and iming of he disurbance on he price series concerned, which may ofen be difficul o obain. 8 See, for example, Roger (1994). 9 See, for example, Roger (1994) and Ravnkilde Erichsen & van Rie (1995). 10 The escape clauses of he insiuional moneary-policy framework of New Zealand (see for example Mishkin & osen, 1997, p. 38) may be viewed as a ype of (implici) case-by-case adjusmen in ha hey allow he cenral bank o emporarily disregard cerain inflaion impulses and o accommodae firs-round effecs on prices, bu no o allow he passing on of hese effecs o a second round. 5

6 Adjusmen is primarily made wih respec o he firs-round effecs, which may be less uncerain han he successive feed-hrough effecs of he shocks. In he case of indirec-ax adjusmens, firs-round effecs are ofen calculaed by simply using he change in he ax rae and he weigh in he CI of he iems in quesion. However, even firs-round effecs may be difficul o deermine since hey may vary over ime, for example due o varying opporuniies for firms o absorb price shocks in he profi margins, and i may be unclear exacly which iems ha are affeced. Furhermore, a seemingly emporary price shock may affec inflaion expecaions and hereby feed hrough ino he more persisen pars of inflaion. Hence, case-by-case adjusmen necessarily conains a judgmenal ad-hoc elemen and may, as a resul, someimes be viewed as a less ransparen mehod. Anoher frequenly used approach inended o make measured inflaion correspond more closely o underlying inflaion is he so-called excluding-food-and-energy approach which implies ha cerain price series are compleely removed from he aggregae price index. For example, he inflaion rae relevan for moneary-policy decisions in he US excludes changes in food and energy prices while he inflaion-arge variable in he UK is adjused for morgage ineres paymens. Conrary o he case-bycase approach, adjusmens are made sysemaically according o a pre-specified rule and hey may herefore be regarded as more ransparen. 11 A disadvanage of he excluding-food-and-energy approach is ha i requires an ex ane idenificaion of he price series o be excluded, which may no always be an easy ask. This is illusraed by he finding in Cecchei (1997) ha he CI excluding food and energy is, in fac, no less volaile han he CI iself. Furhermore, one can hardly be cerain ha he excluded price series never conain informaion on core inflaion. Changes in excluded price series may for example a some poin in ime and under cerain circumsances affec inflaion expecaions and hence feed ino he more persisen pars of inflaion in he same way as some of he disurbances eliminaed in he case-by-case approach. I is also possible ha he composiion of he group of iems whose price behaviour differs from he behaviour of prices in general changes over ime. The once-and-for-all choice of he iems o be excluded herefore runs he risk of generaing an esimae of underlying inflaion ha over ime becomes misleading. The basic idea in he case-by-case and excluding-food-and-energy approaches is ha because he overall price index is calculaed as a weighed mean of he prices of individual iems, he imporance of emporary disurbances will be oversaed. This is also he poin of deparure for an approach using so-called limied-influence esimaors (LIEs) o analyse core inflaion. One ype of LIE, suggesed by Bryan & ike (1991), is he weighed median across he number of individual prices. 1 The median will differ from he mean when he disribuion of individual price changes is skewed. This may be he case when, for example, a period of poor weaher raises he price of cerain iems emporarily. The skewed disribuion generaes a ransiory increase in he mean whereas he median may no be affeced (or, a leas, less affeced). Bryan & Cecchei (1994) provide a heoreical jusificaion for he use of LIEs, based on he framework in Ball & Mankiw (1995). In he absence of shocks, Bryan & Cecchei assume ha firms raise heir prices in line wih underlying inflaion. When a relaive price shock (or cos shock) occurs, he firms affeced have o decide wheher or no o change heir prices a a rae differing from he underlying rae. Changing he price is assumed o be associaed wih an adjusmen cos (menu cos), which implies ha he shocks have o be sufficienly large o rigger such a price change. If he cos is large enough, hen he firms will choose no o reac o he shock and we would as a resul find a spike in he cross-secional price-change disribuion a he rae of inflaion represening core inflaion. Furhermore, above and below cerain cu-off poins deermined by he adjusmen coss we would find a lower and an upper ail represening firms hi by shocks large enough o induce deviaing price changes despie he adjusmen coss. If he disribuion of he underlying shocks is, for example, skewed o he righ, we would in he disribuion of realised price changes expec o find an upper ail ha is larger han he lower ail. The mos common inflaion measure he mean of realised price changes -- would be influenced by boh he spike and he ails and would hence over-esimae core inflaion. The median, on he oher hand, would only regard he spike, which, according o he assumpions, represens core inflaion. 11 I should be noed ha he excluding-food-and-energy approach is ofen used as a complemen o he case-by-case approach. I is for example common o adjus for changes in indirec axes and a he same ime exclude cerain volaile price series. 1 The weighed median is obained by ordering he individual iems in he aggregae index wih respec o he magniude of he price change, accumulaing he weighs and picking he price increase of he iem corresponding o an accumulaed weigh of half of he oal weigh. 6

7 An advanage of he weighed median compared o he case-by-case and excluding-food-andenergy approaches is ha i is compleely sysemaic in he sense ha no arbirary judgemen concerning wha shocks o adjus for or wha price series o disregard from is needed. Furhermore, Bryan & Cecchei (1994) conclude ha among a number of differen esimaes of core inflaion, he weighed median performs bes in many respecs, for example regarding he abiliy of he esimae o forecas fuure price changes. Anoher LIE is he rimmed mean, suggesed by, for example, Bryan & Cecchei (1994). This esimaor is compued by rimming a percenage from he ails of he disribuion of individual price changes, and averaging wha is lef. Thus, he weighed median may be seen as a special case of he rimmed mean where 50 percen has been removed from each ail of he disribuion of price changes. Bryan, Cecchei & Wiggins (1997) and Cecchei (1997) invesigae he efficiency properies of differen esimaors on US daa. They find ha he mean wih around 10 percen rimmed from each ail is he mos efficien esimaor of core inflaion. 13 The choice of how much o rim from he ails is however no obvious. Shorcomings ha he LIEs share wih he above-discussed oher cenral-bank esimaes are ha i is difficul o give he esimaes an explici economic conen (for example, how hey relae o changes in demand and supply) and ha here is a risk of excluding poenially imporan informaion. 3. A arameric Model of he Inflaion rocess As menioned above, he approach ha we propose is based on he idea ha he link beween inflaion and oher economic variables can be summarised in a model where inflaion is a funcion of hree basic facors: long-run condiions, ransiory real oupu, and special facors. 14 The approach may be regarded as an applicaion of he so-called srucural ime-series or unobserved-componens (STM/UC) mehodology. 15 In his secion we presen he model and discuss some of is properies and implicaions. Le π be he measured (CI) inflaion rae, π long-run inflaion, y he relevan ransiory componen of real oupu, Z a vecor of special facors (o be defined below) normalised so ha E ( Z ) = 0, and an IID error wih zero mean and consan variance σ. The key equaion of our model may hen be wrien as: π π = α ( π π ) α ( π π + β y 1 p p p ) β y βq y q + δ 0Z + δ 1Z δ m Z m, + or, equivalenly, α ( ( π π ) = β ( y + δ ( Z +, (3.1) i where L is he lag operaor, L x = x i for any variable x. Thus, he shor-run componen of measured inflaion ( π π ) depends on a ransiory componen of real oupu ( y ) and a vecor of special facors ( Z ), or, equivalenly, measured inflaion ( π ) depends on long-run inflaion ( π ), y, and Z. A formal jusificaion of our inerpreaion of π as he inflaion rae ha occurs in he long run will be given below. Equaion (3.1) bears a raher close resemblance o radiional hillips (or aggregae supply) curves (see, for example, Gordon, 1997, and Hall & Mankiw, 1994). In some respecs, however, he equaion differs from radiional hillips-curve specificaions. One difference is ha long-run inflaion 13 A hiry-six monh cenred moving average of acual inflaion is used as a represenaion of core inflaion. 14 Our framework is hence concepually similar o ha of Ecksein (1981). 15 See, for example, Harvey (1989) for a general reference. Examples of oher applicaions of he STM/UC mehodology are Apel & Jansson (1997, 1998) and Gerlach & Smes (1997). 7

8 eners (3.1) as an explicily idenified componen which is allowed o vary over ime. Usually, he erm ha we label long-run inflaion (which in radiional expecaion-augmened hillips-curve specificaions corresponds o expeced inflaion) is assumed o be consan (capured by he mean rae of inflaion) or equal o he inflaion rae in he previous period (whereby he change of inflaion eners he lef-hand side of equaion (3.1)). The equaion hus allows for a separae idenificaion of he dynamics associaed wih changes in shor-run and long-run inflaion (he dynamics of long-run inflaion will be discussed below). In a specificaion wih no long-run inflaion dynamics (a consan π ), he acual persisence of measured inflaion has o originae from one or several of he following sources: (1) he auoregressive lags of measured inflaion; () he ransiory oupu erms; (3) he special facors included in he Z vecor. Equaion (3.1) adds o hese sources of persisence by allowing for differen dynamics of ineria wih respec o shor- and long-run inflaion. In his ype of specificaion, he Z vecor is generally regarded as a vecor of supply-shock proxies, inended o capure shifs in he hillips curve (see, for example, Gordon, 1997). 16 Ignoring he influence of supply changes is likely o give rise o mis-specificaion problems (see Apel & Jansson, 1997, for furher discussions of his poin). In he presen applicaion i will be useful o divide he Z vecor ino wo sub-caegories. The firs caegory ( Z, 1) conains he undesirable componens ha he cenral bank wishes o exclude when making is analysis of core inflaion (see he discussion in Secion.3). The second caegory ( Z, ) includes (oher) supply-shock proxies ha improve he fi of he equaion bu ha here have no direc implicaions for he esimaes of core inflaion. From he discussion in he previous secion i is clear ha he cenral-bank approach o esimaing core inflaion in pracice involves a subsanial judgmenal elemen when i comes o deciding on wha disurbances and/or price series o adjus for. Hence, in our implemenaion of he procedure, here are several possible candidaes for variables o include in Z, 1. In general, of course, he choice of variables in Z, 1 (as well as Z, ) is a non-rivial issue ha involves boh heoreical and empirical consideraions. In our empirical applicaion, we le he procedures and esimaes of he Swedish cenral bank (which are similar o hose of oher cenral banks) serve as guidelines when deciding on wha variables o include in Z, 1, and how hey are allowed o ener equaion (3.1). In his vecor we herefore include daa on changes in shor-erm nominal ineres raes, changes in (he log of) nominal oil prices and nominal impor prices, and dummies represening changes of indirec axes. Furhermore, only he conemporaneous effecs of hese variables are considered. The Z, vecor conains changes in (he log of) labour produciviy and relaive oil prices. 17 Equaion (3.1) can hen be re-wrien as: α ( ( π +, (3.) π ) = β( y + δ 0,1Z,1 + δ ( Z, i where δ 0, 1 capures he conemporaneous impac of he variables in Z, 1 and δ ( = δ i,l. Alhough he problem of selecing appropriae Z variables is presen also in his parameric approach, some ineresing differences compared o many of he previously discussed mehods can be idenified. To elaborae somewha on his poin, i is useful o rewrie equaion (3.) as: m i = 0 16 If (3.1) is viewed as an aggregae supply relaionship, hen is also usually inerpreed as a supply shock. 17 For purposes of idenificaion, i is assumed ha changes in real oil prices do no have an immediae impac on measured inflaion. Alhough no problem-free, his assumpion does no appear unreasonable in ligh of he fac ha behavioural changes presumably show up wih some lags. The daa are quarerly and run from 1970:1 o 1998:1. For furher deails, see Appendix 1. 8

9 π = π α( + α( β ( y + α( δ 0,1Z δ ( Z + α(. 18 (3.3), Firsly, because we measure he variables effecs on inflaion economerically raher han using he weighs of he iems in he CI, we may, a leas o some exen, be able o capure he inerdependence beween differen iems. For example, an increase in he price of oil may give rise o conemporaneous price increases in a large number of iems in he CI baske. By esimaing he average impac of changes in he price of oil (capured in δ 0, 1 ) raher han jus using he weigh of oil in he baske, i may be possible o obain a more accurae measure of hese effecs on overall CI inflaion. Secondly, he fac ha he specificaion conains dynamics of shor-run inflaion implies ha i is possible o ake ino accoun poenial dynamic feed-hrough effecs of changes in he Z, 1 variables (as well as, of course, of changes in Z, and y ). These effecs are in his model refleced in he α ( erm. For example, a change in an indirec ax may, due o inflaion ineria, have effecs in several consecuive periods. Jus considering he firs-round effec, possibly by simply using he change in he ax rae and he weigh in he CI of he iems hi by he ax, may herefore give misleading resuls. 19 In hillips-ype equaions i is common o inerpre he ransiory componen of oupu, y, as an esimae of he oupu gap. Since changes in aggregae demand are frequenly regarded as he main source of business-cycle flucuaions, he oupu gap (or he unemploymen gap) is ofen regarded as a measure of excess demand. The sae of demand may hence be considered o be neural when y = 0. In mos empirical sudies, he ransiory par of oupu (or unemploymen) is calculaed separaely and insered as an exogenous variable in he hillips-curve specificaion. In he presen STM/UC approach, however, i is possible o rea he ransiory par of oupu as an endogenous variable and esimae i simulaneously wih long-run inflaion and he parameers of he model. In he preceding secion, hree differen views on core inflaion were described. Equaion (3.3) can be used o illusrae hese views. In long-run equilibrium a a neural sae of demand and in he absence of shocks, equaion (3.3) implies ha π = π. 0 This provides a jusificaion for our inerpreaion of π as he rae of inflaion ha occurs in he long run. A core inflaion esimae closely relaed o he one proposed by Ecksein (1981) hen obains as: π = π. 1 (3.4) CORE,1 + π π, does no conain 18 In going from (3.) o (3.3) i is assumed ha he polynomial α ( is inverible so ha shor-run inflaion, any uni roos. 19 Such dynamic feed-hrough effecs could also have been allowed for by including lags of he variables in Z,1. However, a a concepual level, we find i in our case more naural o model hese as arising because of inflaion ineria. More generally hen, one may wish o consider differen ses of AR parameers associaed wih shor-run inflaion changes depending on he source of he change of inflaion. Because our empirical applicaions are foremos mean as illusraions we have chosen no o address his issue furher, bu we noe ha i is an ineresing generalisaion (alhough presumably no problem-free from a echnical poin of view) o be considered in fuure applicaions. 0 The precise saemen is: y = Z = Z = = 0 for all π = π provided α ( 1) 0., 1, 1 I deserves here o be noed ha he measuremen of core inflaion according o a sric inerpreaion of Ecksein s approach canno be deal wih wihou knowing he precise naure of he sources of ime variaion ha are prevalen in he process of long-run inflaion. For example, if long-run inflaion is driven by some sochasic shocks, hen core inflaion needs o be measured condiionally on he effecs of hese shocks (in order o fulfil he in-absence-of-shocks crierion). In our applicaions, we shall generally allow he esimaes of core inflaion inerpreed as corresponding o he Ecksein view o depend on sochasic shocks, bu i is emphasised ha he esimaes corresponding o he sricer inerpreaion obain as simple special cases in which resricions on cerain parameers are imposed (ha is, zero resricions on he variances of long-run inflaion). If all he sources of ime variaion are regarded as ulimaely originaing from shocks, hen, inuiively, he Ecksein approach o core inflaion has o predic ha core inflaion always is consan. In (3.3) hen, core inflaion is equal o he consan uncondiional expecaion of acual inflaion. Bu, in his case, of course, here is no esimaion problem. 9

10 In he framework of Quah & Vahey (1995), core inflaion is basically inerpreed as he demand-driven componen of inflaion. In our model, his would correspond o he second erm of he righ-hand side of equaion (3.3): π CORE = α( L ) 1 β( y. (3.5) The cenral-bank view, in which he influence of differen undesirable effecs on measured inflaion is reduced or eliminaed, is in his framework mos naurally approximaed by subracing he conemporaneous erm δ 0,1Z, 1 from measured inflaion. Hence, core inflaion according o he cenral-bank view is aken o be: π CORE ( α( = π δ Z = π + α( ( y 0,1,1 β,1, 1) δ Z + α( δ Z + α(. (3.6a) 0,1 An advanage of our parameric approach, which is clear from he second equaliy in equaion (3.6a), is ha he resuling cenral-bank esimae of core inflaion, in conras o more mechanically derived esimaes in his radiion, can be decomposed ino economically inerpreable componens. As discussed previously, i is wihin his approach also possible o derive a dynamically adjused cenral-bank esimae of core inflaion: π π CORE = π α( δ 0,1Z,1 = + α( β( y + α( δ Z + α(. (3.6b) Thus, he quaniy ( α( ) δ 01, Z, 1 may be regarded as a measure of he imporance of he dynamic feed-hrough effecs associaed wih changes in Z, 1. To be able o esimae equaion (3.), i is necessary o specify a parameric process for long-run inflaion, π. In STM/UC applicaions, he mos common assumpion is ha permanen unobservable long-run variables follow random walks. In our case his would mean ha π is I(1) and saisfies: π = π +, (specificaion A) (3.7a), + where is an IID error erm wih ( E ) = 0 and a consan variance σ. However, in he case of Sweden, one may quesion wheher a non-saionary I(1) process like (3.7a) provides a reasonable approximaion of he behaviour of long-run inflaion during he enire sample period. From he early 1970s unil he beginning of he 1990s, recurren cos crises in Sweden were accommodaed by several devaluaions (and a depreciaion when he fixed exchange rae was abandoned in November 199). Boh he mean and he variance of inflaion were high and one may well argue ha he Swedish economy during his period lacked a reliable nominal anchor. Shorly afer he swich o a floaing exchange-rae regime in 199, an explici inflaion arge of percen was inroduced. Since hen inflaion has been low and reasonably sable. I is of course difficul o ranslae he srucural VAR framework of Quah & Vahey o he hillips-curve framework used in his paper in a fully saisfacory way. One imporan difference is ha he demand-generaed par of inflaion is assumed o be an I(1) process in Quah & Vahey whereas i in our specificaion as equaion (3.5) makes clear is an I(0) variable. We emphasise ha he I(1) assumpion used by Quah & Vahey is no a necessary condiion for applying he Blanchard-Quah idenificaion scheme of demand shocks. If acual inflaion insead is assumed o be I(0), hen he bivariae VAR sysem would be driven by a permanen and a purely ransiory shock. No furher idenifying assumpions would be needed o achieve exac idenificaion. A saionary demand-driven componen of inflaion, consisen wih he idenificaion scheme of Blanchard & Quah, could hen be compued by seing all permanen shocks equal o zero. This procedure, presumably, would generae a Quah-Vahey core inflaion esimae which, a leas from an empirical poin of view, would be easier o compare wih he esimae derived from equaion (3.5). 10

11 Agains his background, we consider he following wo alernaive specificaions of he process of π : π = π = π + u µ + u µ 1 + η µ + η,1,,1, 199 : 4, > 199 : 4 (specificaion B) (3.7b) 199 : 4, > 199 : 4 (specificaion C) (3.7c) where µ, µ 1, and µ are consans and u, 1, u,, η, 1, and η, innovaions ha are assumed o be IID wih E( u, 1) = E( u, ) = E( η,1) = E( η, ) = 0 and consan variances σ u 1, σ u, σ η 1, and σ η. Thus, he swich o a regime wih an explici inflaion arge is assumed o have affeced he process for long-run inflaion. In specificaion B, π is assumed o follow a random walk in he period before he swich and o flucuae randomly around a consan hereafer. In specificaion C, long-run inflaion is assumed o flucuae randomly around a consan in boh periods, bu he consans may be differen for he wo periods. Noe ha in (3.7c), long-run inflaion will be equal o he consan uncondiional expecaion of acual inflaion as µ 1 = µ and ση = σ 1 η = 0 (see foonoe 1). I is by no means obvious ha he regime shif occurred exacly a he poin in ime assumed above. Differen ypes of daa give a mixed guidance. For example, survey daa on inflaion expecaions of households and agens on he money marke show ha households sared o revise heir expecaions downwards already before he swich o he floaing exchange-rae regime while agens on he money marke did no sar o revise heir expecaions downwards unil afer he swich. Thus, his evidence suggess he possibiliy of a smooh, raher han discree, ransiion o he new regime. However, given he considerable echnical difficulies associaed wih modelling a smooh ransiion o he new regime, we have in his illusraive applicaion chosen o resric ourselves o processes ha imply a discree shif. Given his, i seems quie reasonable o le he shif coincide wih he swich o he floaing exchange-rae sysem and he inroducion of he explici inflaion arge. The relaionships ha are used o complee he sysem are γ ( = (3.8) y and y = λ + y +, (3.9) where y is he permanen par of oupu (ha is, y y y ), λ a consan drif parameer, and and innovaions ha are assumed o be IID wih E( ) = E( ) = 0 and consan variances σ and σ. All he roos associaed wih he lag polynomial ( 1 γ L γ L γ =... γ L n 1 n are assumed o lie ouside he uni circle so ha y is I(0). The permanen componen of oupu y, on he oher hand, is I(1) wih a linear rend. To summarise, he model consiss of he four equaions given by (3.), (3.7a) (or (3.7b) or (3.7c)), (3.8), and (3.9). All shocks of he sysem are assumed o be muually uncorrelaed. For purposes of esimaion, i is convenien o re-wrie he model in sae-space form. Once he model has 11

12 been pu in sae-space form, one can apply he Kalman filer and maximum likelihood o obain esimaes of he unknown parameers and he unobserved variables π, y, and y Empirical Illusraions When esimaing he specificaions A, B and C in heir basic form, he predicion errors associaed wih real oupu urn ou no o be serially uncorrelaed. The correlogram of he predicion errors reveals ha he auocorrelaion problem is due o a significan correlaion a lag one. To handle his problem, he error-process in equaion (3.9) is replaced by e = ρ +, ρ < 1, (4.1) e where ~ IID (0, σ ). We noe ha while his generalisaion of he process for he errors in (3.9) leads o sequences of predicion errors ha appear o be free of serial correlaion, he main resuls of our empirical analysis are no affeced. 4 The parameer esimaes for specificaion A, in which long-run inflaion is assumed o follow a random walk during he enire sample period, have in general he expeced signs (he resuls are shown in he firs column of Table A1 in Appendix ). The variance of long-run inflaion σ is significanly differen from zero a he 11 percen level. As argued above, however, one may quesion wheher he random-walk assumpion accuraely describes he behaviour of long-run inflaion during he whole sample period. In specificaion B, longrun inflaion follows a random walk during he period before he moneary-policy regime shif bu flucuaes randomly around a consan hereafer. This reflecs our belief ha an explici inflaion arge is a more reliable nominal anchor han a fixed, bu frequenly devalued, exchange rae. As is shown in Table 1, he maximised value of he log likelihood improves by approximaely 10 unis when using specificaion B insead of specificaion A. 5 Furhermore, his resul obains for boh unresriced and resriced versions of he wo specificaions. I should however be noed ha because he models are no nesed, a formal es canno be underaken in he usual way. Table 1. Informaion crieria and maximised log-likelihood values for he hree specificaions Spec. AIC UR SC UR HQ UR l UR AIC R SC R HQ R l R A B C Noes: The informaion crieria are defined as follows; Akaike s crierion: AIC = T ( + l) ; Schwarz crierion: SC = T ( log( ) + l); Hannan & Quinn s crierion: HQ = T ( log(log( )) + l). Here, T is he number of usable observaions, is he number of parameers included in he sysem, and l is he maximised value of he log likelihood. In he able, he sub-index UR indicaes ha he sysem is unresriced while he subindex R indicaes ha some parameers of he sysem have been assumed o be equal o 0. More specifically, in hese models, all parameers ha are no significanly differen from 0 a he 10 percen level of significance have in general been resriced o be equal o 0. Numbers in bold indicae a minimum. 3 For full echnical deails see, for example, Hamilon (1994) or Harvey (1989). The log likelihood is maximised in predicion-error decomposiion form using a derivaive-free SIMLEX algorihm available in he program-package RATS. The program used for esimaion is available from he auhors upon reques. 4 The resuls for he basic specificaions are available upon reques. 5 In he case of specificaion B, here are some (weak) signs of auocorrelaion in he predicion errors of inflaion (see he boom rows of Table A1 in Appendix ). Adjusing for his problem using an equaion similar o (4.1) does no change he main resuls for his specificaion. 1

13 An informal way o discriminae beween he differen non-nesed specificaions is o rank hem on he basis of differen informaion crieria. The resuls in Table 1 show ha he informaion crieria hroughou favour specificaion B over specificaion A. Hence, he conclusion drawn upon direcly comparing he specificaions log-likelihood values does no change. I also appears from he evidence in Table 1 ha he fi can be furher improved upon by using specificaion C. In his specificaion, long-run inflaion flucuaes randomly around a consan in boh sub-periods, bu he consan in he second period may differ from ha in he firs period. This suggess ha he apparen lack of a reliable nominal anchor during he 1970s and 1980s empirically does no require he use of a non-saionary process for long-run inflaion during hese years. Raher, a saionarily flucuaing long-run inflaion is preferred by all informaion crieria and hus seems sufficien. In he remainder of he paper we herefore concenrae our discussion on specificaion C. Before proceeding, however, i may be informaive o show he esimaes of long-run inflaion for he hree specificaions. This is done in Figure 1, where he inflaion measures as in he res of he figures in he paper are ploed as annual raes derived from he unresriced versions of he specificaions. Figure 1. Acual inflaion and esimaed long-run inflaion for differen specificaions Specificaion A year Specificaion B year Specificaion C year long-run inflaion acual inflaion The difference beween he developmen in recen years and ha in he 1970s and 1980s regarding he level of long-run inflaion is apparen for all specificaions, even hough specificaion A depics he ransiion o a low-inflaion regime as a raher drawn-ou process. The fac ha he fis are beer for he specificaions wih a discree deerminisic shif suggess ha he ransiion process was faser han indicaed by specificaion A (he p values for esing he null hypohesis of no shif in long-run inflaion are well below 1 percen for boh specificaions B and C). A second resul worh noing is ha when inroducing a discree deerminisic shif bu allowing for differen variances of long-run inflaion before and afer he shif, only he variance of long-run inflaion in he firs sub-period becomes significanly differen from 0. Boh hese resuls suppor he view ha here has been a shif in he Swedish economy from a regime wih high inflaion and a less reliable nominal anchor o a regime wih low inflaion and a more reliable explici inflaion arge. 13

14 4.1. Esimaes of Core Inflaion In Secion 3, i was shown ha he approach may be used o derive counerpars o hree differen esimaes of core inflaion used in he lieraure long-run inflaion, demand-driven inflaion, and inflaion excluding cerain undesired special facors. Figure displays hese esimaes as obained from equaions (3.4), (3.5), and (3.6a), respecively, using specificaion C. As expeced, he esimae ha follows acual inflaion mos closely is he one based on he cenral-bank view. In his paricular case, inflaion has been adjused wih respec o all variables in he Z,1 vecor; ha is, wih respec o conemporaneous changes in nominal ineres raes, nominal oil prices, nominal impor prices, and dummies represening changes of indirec axes (below we discuss alernaive cenral-bank esimaes where adjusmens are made wih respec o only some of hese variables). Deviaions beween acual inflaion and he cenral-bank esimae of core inflaion occur for example during he oil crises and in connecion wih he abandonmen of he fix exchange rae in lae 199 when impor prices increased considerably as a resul of he depreciaion of he krona. Figure. Acual inflaion and differen esimaes of core inflaion acual inflaion long-run inflaion inflaion excluding "special facors" year demand-driven inflaion year 14

15 Demand-driven inflaion is in his model esimaed as a series ha flucuaes saionarily around zero raher erraically. Since his esimae of core inflaion is a linear funcion of y (see equaion (3.5)), his implies ha he (endogenously derived) ransiory componen of oupu has a similar shape. Even hough his resul is no in line wih he common view on he evoluion of cyclical economic aciviy (or he oupu gap), i remains a fac ha his is he way a seemingly reasonable model prefers o describe he relaionship beween real oupu and inflaion when allowing for a simulaneous esimaion of he ransiory componen of oupu and long-run inflaion. I should be noed ha his feaure is robus across all specificaions considered (see Table A1 in Appendix ). Furhermore, he esimaes of he β i parameers in equaion (3.) do no appear numerically unreasonable, and are in mos cases significan a he convenional es levels. 6 Like many cenral banks, he Swedish cenral bank calculaes differen esimaes of underlying inflaion. The esimaes ha are published in he quarerly inflaion repor are obained by using a combinaion of he previously described case-by-case and excluding-food-and-energy approaches. A measure called UND1 is obained by excluding house morgage ineres coss and axes and subsidies. UND is equal o UND1 excluding peroleum and perol prices. UNDINH is calculaed by also excluding prices of goods ha are mainly impored. I may be ineresing o compare hese esimaes wih he parameric cenral-bank esimaes ha can be derived using our model. Figures 3 o 5 show acual inflaion along wih he Swedish cenral bank s esimae and he closes corresponding parameric esimae ha can be derived from he esimaed equaions (called parameric UND1, UND, and UNDINH). Figure 3. Acual inflaion, he Swedish cenral bank s esimae of underlying inflaion (UND1), and he closes corresponding parameric esimae acual inflaion parameric UND1 UND year 6 Some furher insigh ino his issue may be gained by sudying how he explanaory power (as measured by he R saisic from a regression analysis) of (a version of) he inflaion equaion (3.1) relaes o he degree of persisence in he ransiory componen of oupu. Transiory componens of oupu wih differen degrees of persisence may be generaed by filering acual oupu wih he H filer, using a wide range of values of he smoohing parameer in he filering procedure. The resuls confirm ha here seems o exis a saionary highfrequency componen of oupu ha produces a good fi for equaion (3.). 15

16 Figure 4. Acual inflaion, he Swedish cenral bank s esimae of underlying inflaion (UND), and he closes corresponding parameric esimae acual inflaion parameric UND UND year Figure 5. Acual inflaion, he Swedish cenral bank s esimae of underlying inflaion (UNDINH), and he closes corresponding parameric esimae acual inflaion parameric UNDINH UNDINH year Boh ses of esimaes smooh he acual inflaion series and are in mos cases on he same side of acual inflaion. However, occasionally hey differ subsanially. One obvious explanaion is ha he variables included in Z, 1 -- dummies for changes in indirec axes, changes in shor-erm nominal ineres raes, oil prices, and impor prices -- do no exacly mach he iems excluded from he CI baske in he cenral bank s calculaions of underlying inflaion. 7 For example, he effecs of he Swedish ax reform in he beginning of he 1990s are reaed quie differenly in he wo ses of esimaes. As concerns direc effecs, he parameric esimaes are only affeced by his reform hrough is effecs on indirec axes, while addiional adjusmens have been underaken for he cenral bank s esimaes. 7 Hisorical daa on he price developmens of he differen componens in he CI baske are no readily available. 16

17 Anoher explanaion may as emphasised above -- be ha simply excluding an iem from he CI does no guaranee ha he iem s full impac on he CI is eliminaed. If a price change of an iem affecs he prices of oher iems, hen is oal effec will be broader han refleced by is relaive weigh in he CI. A parameric approach can, a leas poenially, ake his ino consideraion. Anoher propery of he parameric esimaes ha is worh emphasising is ha hey explicily are ensured o flucuae saionarily around long-run inflaion. This implies ha our parameric esimaes have an explicily defined, and economically inerpreable, low-frequency behaviour, which he esimaes of underlying inflaion from he case-by-case and excluding-food-and-energy approaches do no have. So far we have only repored he conemporaneously adjused parameric cenral-bank esimaes of core inflaion (according o equaion (3.6a)). Above we argued ha i is possible o ake ino accoun poenial feed-rough effecs of he Z, 1 variables (using equaion (3.6b)). The wo alernaive esimaes of UNDINH are shown in Figure 6. Figure 6. Acual inflaion and conemporaneously and dynamically adjused esimaes of core inflaion acual inflaion parameric UNDINH, conemporaneous adjusmen parameric UNDINH, dynamic adjusmen year As can be seen, he difference beween he conemporaneously and dynamically adjused series is raher subsanial. This suggess ha he feed-hrough effecs of he variables in Z, 1 may be quie imporan. I needs however o be recalled ha our procedure probably only provides a very crude approximaion of he imporance of such effecs, and he resuls have hus o be inerpreed wih care (see he discussion in foonoe 19). 5. Concluding Remarks In his paper we have suggesed an approach ha generaes parameric esimaes of core inflaion using an empirical macroeconomic model in which long-run inflaion and he sae of aggregae demand (as measured by a ransiory componen of real oupu) are deermined endogenously. The key equaion of he model is a hillips-ype inflaion equaion in which acual inflaion depends on a riparie se of basic facors: he wo above-menioned facors ha is, long-run inflaion and demand and a se of special facors including proxies for supply shocks. The probably mos imporan advanage of he approach is ha, because i is based on an empirical macroeconomic model, i can be used o analyse he inflaion process and o generae esimaes of core inflaion ha are economically inerpreable and saisically well-defined. Alhough he approach does of course no solve all problems associaed wih he concep of core inflaion, i appears as an ineresing alernaive or complemen o oher procedures. 17