External UNIT. time. Discussion Paper No.13. UK Consumers' Habits. by Ryan Banerjee and Nicoletta Batini

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1 p Exernal UNIT ime Discussion Paper No.13 UK Consumers' Habis by Ryan Banerjee and Nicolea Baini

2 Exernal MPC Uni Discussion Paper No. 13 * UK Consumers Habis By Ryan Banerjee and Nicolea Baini May 2003 ISSN: Copyrigh Bank of England 2003 (*) Disclaimer: These Discussion Papers repor on research carried ou by, or under supervision of he Exernal Members of he Moneary Policy Commiee and heir dedicaed economic saff. Papers are made available as soon as pracicable in order o share research and simulae furher discussion of key policy issues. However, he views expressed are hose of he auhors and do no represen he views of he Bank of England or necessarily he views of Exernal Members of he Moneary Policy Commiee.

3 UK Consumers Habis 1 Ryan Banerjee* and Nicolea Baini** May 2003 Absrac We follow Fuhrer (2000) in esimaing via Maximum Likelihood a log-linear consumpion funcion on UK daa. In doing so we consider various habi formaion assumpions. We show ha a model of purely exernal habis as in Fuhrer (2000) fis he UK daa remarkably well, and possibly in a superior way han US daa where, according o our esimaes, consumers habis look more inernal in ha hey appear indexed o pas average consumpion of only a subse of (peer) consumers in he economy, raher han oal pas per capia consumpion. We also find ha for abou one sevenh of UK consumers, curren consumpion equals curren incomea srong violaion of he permanen income hypohesis. Embedded in a sicky price-sicky inflaion open-economy moneary model, he model ha we esimae helps mimic he hump-shaped response of he oupu gap o income and ineres rae shocks observed in he UK. Esimaes of oupu Euler equaions for he UK using a similar mehod agree wih our general resuls. The consumpion and oupu models ha we esimae forecas significanly beer han unresriced open-economy VARs. Keywords: consumpion, habi formaion, oupu gap channel JEL Classificaion Codes: D12, E52, E43 *Economis, Exernal MPC Uni, Bank of England, Threadneedle Sree, London EC2R 8AH, Unied Kingdom. Tel: Fax: ryan.banerjee@bankofengland.co.uk ** Research Adviser, Exernal MPC Uni, Bank of England, Threadneedle Sree, London EC2R 8AH, Unied Kingdom. Tel: Fax: nicolea.baini@bankofengland.co.uk (1) We are profundly graeful o Jeff Fuhrer for providing us wih his Malab code and paienly inroducing us o is use. We also hank Emilio Fernandez-Corugedo, Francesco Giavazzi, Seve Nickell, Simon Price, Gregory Thwaies and paricipans o a Bank of England seminar for useful commens on a previous draf. All errors and omissions are our own. The views expressed herein do no necessarily reflec hose of he Bank of England or of he Bank of England s Moneary Policy Commiee.

4 1. Inroducion Hall s (1978) well-known Euler equaion for consumpion assumes ha consumers are raional in ineremporally allocaing heir wealh on consumpion. In his model, household opimisaion leads o a firs order condiion such ha consumpion oday can be expressed as a funcion of omorrow s expeced consumpion, he shor-erm ex ane real ineres rae and an innovaion error. On one hand, Hall s model is compaible wih he older consumpion funcion lieraure, namely he Modigliani-Brumberg (1954, 1980) life-cycle hypohesis (LCH) and Friedman s (1957) permanen income hypohesis (PIH), which asser a sable relaionship beween consumpion and wealh (lifeime resources for LCH; permanen income for PIH). Hall s model is also immune o he Lucas criique because i incorporaes he represenaive agen hypohesis and so solves he expecaion problem faced by previous heories. 1 On he oher hand, however, simple descripions of consumer behaviour as in Hall s REPIH (raional expecaions permanen income hypohesis) model have proved o have a difficul ime maching key dynamic feaures of aggregae daa (see for example Deaon, 1987, 1992; Campbell and Deaon, 1989; Muellbauer, 1983, 1994; and more recenly, Fuhrer, 2000, on US daa; and Fernandez- Corugedo and Price, 2002, on UK daa). This is because hese descripions assume ha consumpion jumps immediaely and fully in response o curren news abou lifeime resources, a consequence of he pure forward-looking behaviour of consumpion in Hall s simples Euler equaion specificaion. To resolve his issue, since Hall s seminal work, researchers have suggesed various modificaions o he basic REPIH model. These include allowing for precauionary savings moives 2 (see e.g. Carroll, 1992, 2001; and Deaon, 1992), for liquidiy consrains (see e.g. Hall and Mishkin, 1982; and Campbell and Mankiw, 1989, 1991) and for habi formaion in consumpion (see e.g. Carroll, Overland and Weil, 2000; and Fuhrer, 2000). 3 In his paper we focus on he las kind of hybrid models ha generalise he REPIH by inroducing habis. These models allow for inerial consumpion behaviour by assuming ha households uiliy depends no only on he level of consumpion in each period, bu also on he level in previous periods. In oher words, hey allow for habi formaion implying ha consumers preferences are no imeseparable in consumpion. This helps models replicae he excess smoohness ypically observed in consumpion daa relaive o income daa, hereby improving he empirical fi of Hall s original formulaion. 1 In addiion, he aggregae consumpion model is derived from he represenaive agen s ineremporal opimising behaviour. 2 Allowing for precauionary savings does no require modifying he REPIH model, bu simply assuming more general (non-quadraic) uiliy preferences. So, in fac, i implies a generalisaion raher han a modificaion of Hall s original model. 3 Duesenberry (1949) pioneered he idea of habi formaion in consumpion. 1

5 In paricular we follow Fuhrer (2000) in esimaing via Maximum Likelihood a log-linear consumpion funcion wih habi formaion on UK daa. 4 This model encompasses various ineresing alernaives, including: Hall s basic REPIH model, 5 he basic REPIH model wih a fracion of consumers whose curren consumpion equals curren income ( rule-of-humbers ), and he REPIH modified o allow for habi formaion in spending paerns. When modelling habis, we consider wo differen habi formaion assumpions. Firs, as in Fuher (2000), we assume ha consumers index heir consumpion habis o pas per capia aggregae consumpion. This behaviour is described in he lieraure as caching up wih he Joneses (see, among ohers, Abel, 1990) or exernal habi formaion (Campbell and Cochrane, 1999), in ha i is aggregaeraher han individual consumpionha maers for he habi. A hear, his assumpion implies ha he consumers reference level for he habi is a funcion of everybody s pas consumpion, viz, of he spending of boh forward-looking and rule-of-humb consumers in he pas. Second, we assume ha consumers form heir habi using as a benchmark pas average consumpion of he forward-looking consumers, i.e. consumpion of heir peer group, bu no average consumpion of he rule-of-humbers. Alhough i does no imply ha he habi is purely inernal (i.e. uniquely indexed o individual raher han aggregae consumpion), his second way of specifying he habi allows for he possibiliy ha consumers are paricularly concerned abou caching up wih he spending paerns of heir peer group, raher han caching up wih anybody s average consumpion. Looking a wo insead of one habi model enables us o es he robusness on UK daa of he habi specificaion originally used for he US. I also allows us o hrow some ligh on wheher ha specificaion is indeed he bes on US daa. In line wih Fuhrer s resuls for he US, we find ha, in general, habi formaion in consumpion is an economically imporan deerminan in he uiliy funcion of Briish households. This is consisen wih previous findings in Baini, Harrison and Millard (2001), which indicae ha augmening a sochasic general equilibrium model calibraed on UK daa o allow for habis in consumpion helps capure he hump-shaped response of he oupu gap o a moneary policy innovaion observed in empirical models esimaed on ha daa. 6 Wih regard o differen habis assumpions, we show ha 4 There is a debae in he lieraure on wheher i is possible o esimae Euler equaions for consumpion. Carroll (2001), for insance, showed ha he sandard cross-secion procedures for microeconomic esimaion of consumpion Euler equaion do no work. This follows from he fac ha hese procedures imply esimaing he firs-order approximaed or log-linearised versions of he Euler equaion, ignoring he fac ha higher-order erms are endogenous wih respec o he firs-order erms and omied variables. On he oher hand, Fuhrer (2000) examined several measures of he accuracy of he approximaion used in his paperwhich is he one used hereand finds ha he linear model offers a very reliable approximaion of he behaviour implied by he nonlinear model. Also, parly in response o Carroll s claims, Aanasio and Low (2000) showed ha i is neverheless viable o obain consisen esimaes of he preference parameers in an Euler equaion for consumpion using ime-series esimaion, as long as uiliy is isoelasic and he sample of daa used in esimaion covers a long enough span of ime. These condiions hold in our case, and our model is like Fuhrer s (2000), so we assume ha Carroll s poenial approximaion errors do no preven proceeding wih he esimaion. 5 Throughou we refer o Hall s model as one where he real ineres rae is allowed o vary. Hall s (1978) original formulaion assumed for simpliciy ha he real ineres rae is consan, bu his assumpion is unrealisic in pracice and so, as Fuhrer (2000) does, we relax i. 6 As shown using US daa by Abel (1990), Jermann (1998) and Campbell and Cochrane (1999), among ohers, models wih generalised habi formaion can go a long way in explaining he equiy premium puzzle ha undermines he empirical consisency of consumpion capial asse pricing model (C-CAPM). Clerc, Harrison 2

6 for he UK a model of habis indexed o per capia aggregae consumpion as in Fuhrer (2000) fis he UK daa remarkably well, and possibly in a superior way han US daa where, according o our esimaes, consumers habis look more inernal in ha hey appear indexed o pas average consumpion of only a subse of (peer) consumers in he economy, raher han oal pas per capia consumpion. More specifically, our esimaes sugges a level of habis and a curvaure of he uiliy funcion ha are similar o hose found by Fuhrer (2000) for he US for a comparable habi assumpion. However, we find a more mued response of he consumpion o income raio o real ineres raes in he UK relaive o he US, and one more in line wih he consrains implied by he srucural model. This may sugges ha moneary policy in he UK works more by influencing inflaion expecaions and hrough he exchange rae channel han i is he case in he US. We also find ha for abou one sevenh of UK consumers, curren consumpion equals curren incomea srong violaion of he permanen income hypohesis. This is consisen wih he evidence on credi resricions in he UK (cf. Jappelli and Pagano,1989; Campbell and Mankiw, 1989, 1991; Muellbauer, 1994), since i is usually assumed ha consumpion is equal o curren income for he credi-consrained (see Hall and Mishkin, 1982). I also squares wih findings in Muellbauer and Laimore (1995) and Bakhshi (2000), which sugges ha aggregae consumpion in he UK is more sensiive o changes in human wealh (and labour income) han prediced by he PIH. The excess sensiiviy of aggregae consumpion o curren income possibly indicaes, in fac, ha some UK consumers are unable o borrow off heir fuure labour income, ha is, hey are credi-consrained. 7 We show how he Fuhrer (2000) model wih habis indexed on pas per capia aggregae consumpion and is various nesed alernaives are good a capuring auo and cross-correlaion funcions observed in UK daa. In general, as in Fuhrer (2000), we find ha he habi formaion is successful a capuring he persisence in he own correlaion of consumpion, as well as he persisen dynamic correlaions beween consumpion and income, ineres raes and inflaion. This is in conras wih he pure REPIH model ha canno replicae he dynamics of he UK spending and inflaion daa. We progressively add resricions o he unconsrained VAR equaions used in he habi formaion model o predic leads of regressors o Fuhrer s per capia consumpion equaion. Our VAR differs from Fuhrer s original one in ha we include changes in he real exchange rae in he se of endogenous variables in consideraion of he openness of he UK economy. Ye, as in Fuhrer s case, our consrained models largely replicae he dynamic behaviour of he unconsrained open-economy VAR. Finally, we analyse he implicaions ha habis in consumpion have on oupu in he UK. We do so by esimaing a parallel oupu Euler equaion which allows for habi formaion, as done in Fuhrer and Rudebusch (2002). Habis give a raionale for including boh forward and backward-looking componens in he Euler equaion for oupu. We find mixed resuls regarding he imporance of forward-looking variables in modelling oupu dynamics in he UK. However, ou-of-sample forecas and Neiss (2001) show ha his puzzle is presen in UK daa as well, so our model may lend suppor o he presence of habi formaion in he UK oo. 7 For a discussion of his finding on US daa see Campbell and Mankiw (1989, 1991) and Darby and Ireland (1994). 3

7 exercises sugges ha models wih forward-looking componens improve significanly upon forecass from purely backward-looking models. The paper is organised as follows. In Secion 2 we briefly lay ou he habi formaion model ha we employ, and compare is inuiion wih ha behind he radiional Hall s Euler equaion for consumpion. A modified version of his model incorporaes he possibiliy ha some consumers are rule-of-humbers. In his secion we also look a an alernaive, more inernal habi formaion assumpion. In Secion 3 we show and discuss empirical resuls obained when esimaing he model on UK daa when habis are as in Fuhrer (2000). In his secion we also discuss how good are his habi formaion model and is nesed alernaives a porraying observed auo and cross-correlaion funcions of UK daa. We hen look a he fi of UK and US daa on he alernaive, more inernal habi model. In Secion 4 we add resricions o he unconsrained VAR used in Secion 2 and examine he abiliy of he consrained VAR o mach he dynamic behaviour of is unconsrained version. In Secion 5 we exend our analysis o esimae an Euler equaion for oupu as in Fuhrer and Rudebusch (2002). Finally, in Secion 6, we compare he forecasing accuracy of he various consumpion and oupu models analysed in his paper. Secion 7 concludes. We append o he paper a descripion of he daa used in esimaion and deails abou he resricions ha we impose in Secion Hall s model and he reamen of habis 2.1 Hall s REPIH consumpion model Hall s sandard model of opimal consumer behaviour assumes ha sociey is populaed by a coninuum of households. Each household is infiniely lived and has idenical preferences defined over consumpion. The aim of he represenaive household is o maximise is life-long uiliy funcion: T E (1 ) 0 u( C ) (1) subjec o he ineremporal budge consrain: T (1 r) ( C W ) 0 A (2) where E is he mahemaical expecaion operaor condiional on all informaion available in period, is he subjecive ime preference rae, r is he real rae assumed consan over ime, where r, T is he lengh of he represenaive agen s life, here equal o infiniy, u is he insananeous uiliy funcion which is assumed o be sricly concave, C is consumpion in period, W are earnings in period, and finally, A are asses apar from human capial (i.e. non-human wealh). C 4

8 Subsiuing he ensuing firs order condiion for equaion for consumpion: C ino ha for C 1, gives Hall s familiar Euler E u C uc r (3) Equaion (3) says ha for he consumpion plan o be opimal, oday s marginal uiliy mus equal nex period s marginal uiliy, adjused for he raio of subjecive and marke discoun raes. Relaxing Hall s arbirary assumpion of a consan real rae, and approximaing he represenaive consumer s firs order condiion wih is log-linear approximaion abou he seady sae value for C, we ge: c E c 1 r (4) where c is he naural logarihm of consumpion, C ; he real rae, r, is now ime-varying; and he parameer > 0 represens he ineremporal elasiciy of subsiuion in privae spending, alias, in his simple case, he real ineres elasiciy of consumpion. Equaion (4) implies ha consumpion oday can be expressed as a funcion of omorrow s expeced consumpion, he shor-erm ex ane real ineres rae and an innovaion error,. Relaxing he assumpion of consan real rae implicaes ha Hall s model can no longer be mapped ino a random-walk process for consumpion. Neverheless, also wih a ime-varying real rae, consumpion ends o jump immediaely in response o unanicipaed shocks, changes in consumpion being fully elasic o curren news abou lifeime resources Inroducing habis In pracice, as shown by Deaon (1987, 1992) and Campbell and Deaon (1989), consumpion does no respond immediaely o news, bu raher, is behaviour hrough ime is remarkably smooh; his is enirely in conflic wih he exra responsiveness of consumpion o shocks implied by Hall s model. One direc implicaion of his is ha he REPIH basic model of consumpioneven augmened wih Campbell-Mankiw rule-of-humb behaviouris unable o capure adequaely he dynamic ineracion of consumpion, income and ineres raes observed in he daa. Looking a US daa, for insance, Fuhrer (2000) poins ou ha he REPIH model is paricularly unsuccessful a capuring he humpshaped response of consumpion o income shocks ha appear o characerise aggregae (US) daa. To address his puzzle, Carrol, Overland and Weil (2000) firs, and Fuhrer (2000) laer, follow Abel (1990) and modify he expression for period uiliy u () o include habi formaion. In essence, habi formaion implies ha preferences are no ime-separable in consumpion, so ha households uiliy depends no only on he level of consumpion in each period, bu also on heir 5

9 level in he previous period(s). Inuiively, i is plausible ha consumers follow persisen paerns in heir spending decisions, and ha hey, hence, aemp o smooh no jus heir level of spending bu also he rae a which hey change heir spending from one period o anoher. And he lieraure on he equiy premium puzzle boh in he US (Abel, 1990; Consaninides, 1990; Jermann, 1998; Campbell and Cochrane, 1999; and Wacher, 2002, among ohers) and in he UK (Allais, Cadiou and Dees, ), lends suppor o he view ha habis may play an imporan role in consumpion/saving decisions by consumers in hese wo counries. Imporanly, because habi formaion conveys a uiliy-based smoohing moive for boh changes and levels of consumpion, assuming habi behaviour enhances he abiliy of he model o mach he dynamic response of consumpion o shocks. Fuhrer (2000) illusraes how including habi formaion beers he empirical fi of he consumpion equaion on US daa. Similarly, using UK daa, Baini, Harrison and Millard (2001), show ha augmening a calibraed sochasic general equilibrium model o allow for habis in consumpion helps capuring he hump-shaped response of he oupu gap o a moneary policy innovaion of an empirical model esimaed on UK daa. Carroll e al (2000) and Fuhrer s (2000) model wih habis implies a re-specificaion of he uiliy funcion such ha consumers -period uiliy like: 1 1 C u 1 (5) Z where Z is he habi formaion reference consumpion level, defined as: Z C 1 Z (6) 1 1 So in conras wih Hall s model in equaion (1), where -period uiliy depended merely on he level of ime- consumpion, here uiliy depends on curren consumpion relaive o pas consumpion (given by he habi-formaion reference level). According o his modified specificaion of he uiliy funcion, uiliy is no longer ime-separable because he decision on how much o consume oday affecs he fuure habi reference level in all fuure periods uiliy from omorrow onwards. Noe ha equaion (6) implies ha each represenaive agen indexes consumpion o he average of pas consumpion of all agens in he economy. We discuss his assumpion furher in Sub-secion 2.3 when we inroduce he possibiliy of rule-of-humb behaviour. 8 News abou lifeime resources are news abou permanen income, defined as ha income level which, susained over he life cycle, has he same presen value as he acually expeced income sream. 9 Allais e al (2000) invesigae empirically he abiliy of he consumpion capial asse pricing model (C-CAPM) o solve his puzzle once assumed ha consumpion behavior presens habi formaion. From he esimaion of he model's parameers for he G-7 counries, and show ha he consumpion model wih habi formaion is able o accoun for financial asse reurns wih more reasonable preferences. See also Clerc, Harrison and Neiss (2001) and May, Larsen and Talbo (2002) for a discussion of he role of habis in explaining his puzzle in UK calibraed C-CAPM models. 6

10 As discussed in Fuhrer (2000), his simple habi formaion specificaion is convenien because i nealy parameerises wo key aspecs of habi formaion. Firs, he parameer measures he imporance of habi formaion in he uiliy funcion. If 0, Hall s original REPIH model obains. This implies a version of he opimising consumpion equaion as ha used iniially in opimising moneary models by Kerr and King (1996), Woodford (1997) and McCallum and Nelson (1999b). If 1, uiliy hen only depends on consumpion relaive o pas consumpion. is capped a 1 o ensure non-saiaion in consumpion. Second, as discussed in Fuhrer (2000), he parameer measures he persisence or memory in he habi formaion reference level. If 0, hen uiliy only depends on las period s consumpion. For beween 0 and 1, he closer ges o uniy, he wider he hisorical ime span over which he reference level is deermined. measures he curvaure of he uiliy funcion, bu in he presence of habis ( 0 ) is no longer inerpreable simply as he inverse of he ineremporal elasiciy of subsiuion. For values of greaer han one, he higher he expeced growh in he reference habi level, he lower consumpion is relaive o income (i.e. he lower he consumpion-o-income raio). Combining (5) and (6) wih a sandard ime-varying ineres rae budge consrain, and following he manipulaion in Campbell and Mankiw (1989), 10 Fuhrer (2000) shows ha i is possible o obain he approximae log-linear consumpion funcion: c p p a z z j y E y j a1 j1 j 2 j1 j r j1 j1 (7) wih p defined as: p E p 1 b1c b2 z (8) As shown in equaion (8), p summarises he relaionship of he marginal uiliy of curren consumpion wih fuure consumpion. This resuls from he fac ha oday s consumpion choices impac fuure consumpion reference levels, and is he origin of non-ime separabiliy in his model. In equaion (7), r is he ex ane real ineres rae. I is expressed as he discouned weighed average of model-consisen forecass of shor erm real ineres raes (d below being equal o D/(1+D) where D is, as in Fuhrer, 2000, he duraion of he implici long-erm real bond, se equal o 40 quarersen yearsin his and our paper) as follows: 10 Campbell and Mankiw (1989) combine he Euler equaion for consumpion of he REPIH wih he household s infinie horizon budge consrain o ge a consumpion funcion relaing consumpion, wealh, and expeced fuure reurns on wealh. They hen move from he wealh-based consumpion funcion o an incomebased consumpion funcion by expressing he marke value of wealh in erms of fuure expeced reurns and he fuure expeced income flows from wealh. Noe ha a model wih an income-based consumpion funcion would no capure wealh revaluaions (due o shocks o echnology shocks, for example) if dividends are no paid oua poin made o us by Francesco Giavazzi. However, his should presumably be less of an issue in modelling UK consumpion as non-human wealh here is predominanly invesed in housing sock, conrary o he US where equiies consiue a larger porion of households asses. 7

11 i1 i i1 i r (1 d) d E f (9) Above, lower-case leers denoe logs, y denoes disposable income, and he parameers a 1, a 2, and are non-linear funcions of he deep parameers,, and of he consumers opimisaion problem. 11 is he degree of forward-looking behaviour, and capures how much weigh consumers aach o expecaions of fuure variables driving heir decision o consume. This is disinc from he discoun facor, which insead governs he subjecive preference of wheher o consume oday or in fuure periods, bu is also bounded, such ha 0 1. If is uniy, hen consumers weigh forecas informaion from ime +1 equally o informaion from ime + n, where n > 1. Finally, measures he impac of ineres raes on consumpion. A priori assumpions abou how ineres raes changes affec consumpion, sugges ha has a negaive sign for all plausible values of he underlying uiliy parameers. Because z is a funcion of lags of consumpion (see equaion (6)) consumpion oday depends on expeced fuure and lagged consumpion (see McCallum and Nelson, 1999a; and Baini, Harrison and Millard, 2001, for a derivaion of he log-linear approximae consumpion funcion under habis in he simpler case where u C, C 1 ). This has he effec of making consumpion sicky, hereby eliminaing he fronloaded response of consumpion o income and ineres rae shocks ha feaures in Hall s original REPIH model. 2.3 Rule-of-Thumb Consumers Equaion (7) assumes ha all consumers are idenical, ha hey are all raional in ineremporally allocaing heir wealh on consumpion, and ha no credi resricions apply. In pracice, some consumers may be unable o borrow off heir fuure labour income because of liquidiy consrains. In his case, for some of hem, consumpion will simply equal curren income, insead of he ne presen value of fuure income sreams, in violaion of he permanen income hypohesis. Is here evidence of liquidiy consrains for UK consumers? Jappelli and Pagano (1989), Campbell and Mankiw (1989, 1991), Muellbauer (1994), and Muellbauer and Murphy (1990, 1997) all find evidence ha a proporion of consumers in he UK are credi consrained. In paricular, Jappelli and Pagano (1989) find ha hese consrains have ighened since he lae 1950s. Corroboraive evidence on credi resricions in he UK comes from Muellbauer (1983, 1994), Muellbauer and Laimore (1995) and Bakhshi (2000), 12 who show ha aggregae consumpion in he UK is more sensiive o changes in human wealh (and labour income) han is prediced by he REPIH. The excess sensiiviy of aggregae consumpion o curren income possibly indicaes ha some UK consumers are indeed credi-consrained. Likewise, Iacoviello and Minei (2000) find evidence for several European counries, including he UK, ha households' aggregae borrowing 11 More specifically: a1 = (((1 )(1))/), a 2 = ((1))/, = [((1)(1)1)/]. However, laer in he esimaion sep, is no fully consrained. In our analysis,, he subjecive discoun facor, is se equal o 0.99 hroughou. 8

12 coss vary wih aggregae balance shee srengha sign of liquidiy consrains. Finally, by looking a a number of sylised facs, Aoki e al (2002) also infer ha credi fricions may be an imporan elemen in Briish households consumpion plans, and hence examine he direc impac of house prices on consumpion via credi marke effecs. 13 Allowing for he possibiliy of rule-of-humb consumers, ha is consumers whose curren consumpion equals curren income because of liquidiy consrains, yields he following log-linear consumpion funcion: c y j 1 E E y j a p j p j a z j z j r j c 1 j1 (10) Where, which lies beween 0 and 1, is he proporion of rule-of-humb consumers in he populaion who consume curren income in each period. c is he srucural innovaion in he consumpion equaion, usually inerpreed as he innovaion o lifeime resources. When income is modelled explicily as in Secion 2 below, however, his innovaion capures emporary changes in preference parameers. Once more, as in Fuhrer (2000), equaion (10) encompasses various aracive alernaives, including: Hall s basic REPIH model (for 0, 0 ); he basic REPIH model wih a fracion of consumers whose curren consumpion equals curren income ( rule-of-humbers ) ( 0 ); and he REPIH modified o allow for habi formaion in spending paerns ( 0 ). Two addiional poins are worh menioning abou equaion (10). Firs, allowing for he possibiliy of rule-of-humb behaviour as in equaion (10) implies ha here are now wo ypes of represenaive consumers ( forward-looking and rule-of-humb ). Thus, aggregae consumpion can now be divided ino he consumpion of he forward-looking agens and consumpion of he rule-of humbers. This poses he quesion of whose consumpion defines he habi reference level z for he forward-looking consumers, who plan heir spending paerns according o equaion (10). As specified in Fuhrer (2000) and in equaion (6), z is a funcion of pas per capia aggregae consumpion. So, according o his specificaion, he reference level for he habi of forward-looking consumers is a funcion of everybody s pas consumpion, viz, of he spending of boh forward-looking and rule-of-humb consumers in he pas. An alernaive specificaion, no ye considered in he lieraure, is ha forward-looking consumers form heir habi using as a benchmark pas average consumpion of he forward-looking consumers alone, and ignore average consumpion of he rule-of-humbers. Alhough i does no imply ha habis become purely inernal, his second 12 See also surveys by Deaon (1992). 13 Aoki e al (2002) poin in paricular o he fac ha, in he UK, he spread of morgage raes over he risk-free ineres rae varies wih he collaeral posiion of each household, and unsecured borrowing raes, which are he marginal source of finance once collaeral has been exhaused, are much higher han morgage raes. 9

13 way of specifying he habi allows for he possibiliy ha consumers are paricularly concerned abou caching up wih he spending paerns of heir peer groupnamely forward-looking consumersraher han caching up wih anybody s consumpion. The simple way in which Fuhrer (2000) specifies rule-of-humb behaviour allows us o recover he proporion of consumpion consumed by he forward-looking agens, and herefore o re-specify he habi o allow for his consideraion. The alernaive, more inernal habi formaion model urns ou o be: Z C Y Z (11) Equaion (11) now suggess ha forward-looking consumers wish o cach up wih he pas average consumpion of oher forward-looking consumers, bu are indifferen owards pas spending paerns of he rule-of-humbers. Second, i can be argued ha equaion (10)irrespecive of how habis are specifiedis suied o capure consumpion behaviour in an open economy like he UK. In fac, given our assumpion ha consumers are infiniely lived and ha he discoun facor of he represenaive agen is exogenously deermined, he exernal asse posiion of he economy is irrelevan for consumpion behaviour. The evidence indeed suggess ha, a he business cycle frequency, he dynamics of consumpion are generally no much changed by he inroducion of hese feaures (see Kollmann, 1991, and Kim and Kose, 2000). On he oher hand, since openness makes impors a componen of he consumer price index, he inclusion of a separae exchange rae erm in he Phillips curve canno be overlooked. We reurn o his issue when we esimae he unconsrained VAR ha we use in conjuncion o equaion (10) o proxy for leads of he regressors of equaion (10). Below we esimae he modified model for consumpion as in equaion (10) and es wheher habis and rule-of-humb behaviour are imporan o model Briish consumers behaviour. In doing so we presen esimaion resuls obainined when we use boh Fuhrer s (2000) and our more inernal habi assumpion. In laer secions we examine wheher he bes performing habi model among hese wo helps capure observed dynamics in UK daa beer han he basic REPIH model. Throughou we compare findings on UK daa wih Fuhrer s and/our own findings on US daa. 3. Esimaion resuls We break he discussion of our resuls in hree subsecions. In Sub-secion 3.1 we describe he mehodology ha we employ o esimae equaion (10). We hen review our empirical findings when we apply ha mehodology o UK daa over a sample period ha goes from 1970 Q1 o 2002 Q2. We compare hese resuls wih Fuhrer s (2000) resuls on US daa. In Sub-secion 2.2 we show how resuls change for boh he UK and he US when we assume ha habis are formed according o our more inernal habi model in equaion (11). In Sub-secion 3.3 we illusrae how good is our besfiing consumpion model a capuring UK consumpion dynamics. 10

14 3.1 Mehodology and Maximum Likelihood Esimaes on UK and US daa Throughou he paper we use he mehodology in Fuhrer (2000). So we sar by esimaing an unresriced VAR of he key elemens driving consumpion. Since he UK economy is open, we use an open-economy VAR hroughou he analysis. More specifically, our VAR is esimaed from 1970 Q1 o 2002 Q2 on quarerly daa for inflaion in he consumpion deflaor, he nominal wo-week repo rae, he log of real consumpion of non-durables and services, he log of non-consumpion GDP, he log of real disposable income, and finally, he firs difference in he log of he real effecive exchange rae index (ERI). We derend he consumpion, income and non-consumpion GDP daa wih a segmened linear rend wih breaks in 1980, 1990 and To orhogonalise he innovaions in he VAR we apply a Choleski decomposiion, assuming ha he variables follow he causal ordering: consumpion of non-durables and services RPIX inflaion real ERI repo rae derended non-consumpion GDP real disposable income. This is in line wih he ordering of he open-economy VAR in Baini and Nelson (2001), which allows he ineres raebu no he exchange raeo respond conemporaneously o moneary policy innovaions. And i follows Fuhrer (2000), by allowing he ineres rae o reac conemporaneously o inflaion bu no o non-consumpion GDP or income. As a second sep, we hen combine he consumpion equaion (10) wih he definiion equaions for z (eq. (6)), p (eq. (8)) and he ex ane real ineres rae r (eq. (9)) wih he VAR equaions for income, inflaion and he shor-erm nominal ineres rae ino a linear raional-expecaions model. We cas he model as a se of sochasic linear difference equaions, solve for expecaions in his model, and hen re-express hem in erms of curren and lagged variables following he soluion procedure of Anderson and Moore (AIM, 1985). This ensures ha all raional-expecaions resricions implied by he consumpion model are imposed in esimaion; i also allows us o specify he sysem as a se of consrained decision rules in observable variables, where he original srucural shocks are idenified. Finally, we se up he likelihood for his sysem, which we esimae via a numerical, derivaive-based maximisaion mehod. As Fuhrer (2000) poins ou, his mehod has he advanage ha i sars off wih an unconsrained linear VAR, which ness all oher successive models, and progresses owards increasingly consrained linear models. Once we have esimaed he deep parameers of equaion (10), we can go back o he VAR and impose furher resricions o he iniially unconsrained equaion. In he end his delivers a moneary model, nesed in he iniial unresriced VAR, on which we can assess he abiliy of he suggesed consumpion framework o successfully porray consumpion dynamics observed in he UK. 14 These daes seem sensible when inspecing he paern of consumpion since Indeed, he iming of hese splis is suppored by resuls from Chow breakpoin ess. Also, regressing he log of consumpion on five lags of iself, a consan, plus inercep and slope dummies for hese hree daes, we can rejec a he 5% confidence level he resricion of no srucural change [ F-saisic = 1.94, pvalue = 0.051]. The resricion ha he parameer non-consancy is confined o he equaion s inercep is can also be rejeced a ha confidence level [F-saisic = 1.54, pvalue = 0.16]. For a discussion of breaks in he UK GDP series see Nelson and Nikolov (2002). 11

15 Table 1 below shows Maximum Likelihood parameer esimaes of he consumpion model laid ou above for boh he UK and Fuhrer s (2000) esimaes for he US, for comparison. Esimaion samples are 1970 Q12002 Q2 and 1966 Q11995 Q4, respecively. Table 1: US and UK ML parameer esimaes Coefficien UK esimae Sandard error US esimae Sandard error Various imporan poins emerge from Table 1. Firs, he memory in he habi reference level is he only parameer no significanly differen from zero a he 5% level, for boh he UK and he US. This indicaes ha only las period s consumpion (i.e. consumpion a 1) defines he habi level. Fuhrer (2000) gives a deailed discussion of why his migh be he case. He noes ha presumably habis are formed over a horizon longer han one quarer. However, since he derended level of consumpion displays significan auocorrelaion, las period s consumpion level conains informaion abou consumpion of many previous periods. Therefore, he argues, he reference level of he single lag in consumpion may be sufficien o impar smoohness o he changes in consumpion expendiure absen in he sandard permanen income model where consumpion depends on leads bu no lags of iself. In he UK case, he finding ha only he firs lag of consumpion is informaive in predicing curren consumpion accords, for insance, wih empirical esimaes of he consumpion funcion of he Bank of England s MTMM model. 15 Second, habi persisence seems imporan in explaining boh UK and US aggregae consumpion daa. More specifically, we find ha also on UK daa > 0, ha is, no only oday s consumpion maers for Briish households curren consumpion plans, bu also he level of oday s consumpion relaive o pas consumpion. Resuls in Table 1 indicae ha he imporance of oday s consumpion relaive o pas consumpion in he uiliy funcion looks similar across he wo counries. Third, he proporion of rule-of-humb consumers appears o be sizeable no jus in he US bu also in he UK. Campbell and Mankiw (1991) esimae in heir basic model, = for he US and = for he UK wih seasonally adjused daa. Our esimaes of he rule-of-humb/habi formaion model on UK daa also sugges ha he fracion of rule-of-humb consumers is lower in he UK han in he US. However, we find ha only one sevenh (as opposed o Campbell and Mankiw s finding of 15 See Bank of England (1999), Economic Models a he Bank of England, and (2000), Economic Models a he Bank of England: Sepember 2000 Updae. 12

16 one fifh) of consumers may consume all heir curren income in each period insead of smoohing heir consumpion ineremporally. Fourh, ou of he consumers who are no rule-of-humbers, we find ha he horizon over which consumers compue heir forecass is long, and i is longer in he UK ( = 0.999) han in he US ( = 0.996). We also find ha he esimaes of and are quie differen in magniude beween he UK and he US. Typically, is he coefficien of relaive risk aversion and he inverse of he coefficien of ineremporal subsiuion in he sandard REPIH model. However, as already poined ou, he inroducion of economically significan habis breaks his explici link. So here is jus an index of he curvaure of he uiliy funcion. Our esimaes indicae ha he uiliy funcion of he UK represenaive agen exhibis less curvaure han ha of he represenaive agen in he US. Ye for boh counries, is greaer han one, implying ha he higher is expeced growh in he habis reference level, z, he lower is he log consumpion-o income-raio in boh counries. Wih regard o, he expeced level of fuure ineres raes has a negaive effec on ime- consumpion as prediced (remember ha in he consumpion model in equaion (10) is preceded by a negaive sign). As Fuhrer (2000), we do no fully consrain he real ineres rae in esimaion. However, for he UK, he magniude of he freely esimaed is quie in line wih he fully consrained coefficien given esimaes of he consrain ( = 0.77 relaive o a conrain-implied value of 0.47). The opposie is rue for he US, where he magniude of he freely esimaed is considerably larger han he fully consrained coefficien given esimaes of he consrain ( = 28 relaive o a consrain-implied value of 0.5). This is an ineresing resul highlighing ha consumpion in he UK may be (much) less affeced by expeced discouned real ineres raes han consumpion in he US. A corollary of his finding is ha moneary policy in he UK may work mainly by influencing direcly inflaion expecaions and/or hrough he exchange rae channelraher han srongly hrough he oupu gap channel as i seems possible in he US. Nesed ess of habi formaion and rule-of-humb behaviour in our UK-esimaed model confirm he imporance of hese wo assumpions as modificaions of he REPIH o capure consumer behaviour in he UK. For insance, he hypohesis ha habis do no maer in decisions on privae spending in he UK (i.e. he hypohesis ha = 0) is overwhelmingly rejeced by he daa. The 2 likelihood raio es for his single resricion akes he value of , wih p-value Analogously, he 2 likelihood raio es for he resricion ha for no consumer consumpion equals curren income ( = 0) akes he value 21.87, wih p-value Crucially, he likelihood raio es for he consrained baseline model, which embeds 24 zero and cross-equaion resricions as implied by he srucure in equaion (10) and by he assumpion ha a (1-) porion of consumers behaves raionally, akes he value of 46.1, no significan a even he 10 % level. This accords wih resuls for he US in Fuhrer (2000), where he ess ake he corresponding values of: = 0 LR= 21.4; p-value ; = 0 LR= 12.6; p-value ; all model resricions LR= 32.8). In his sense, boh he UK and US resuls are a infrequen example in which join resricions from an opimising model wih 13

17 raional agens canno be rejeced relaive o he unconsrained model in which he resriced model is nesed. 3.2 Esimaes using he alernaive, more inernal habi model In his secion we re-esimae he consumpion model of he previous sub-secion, bu assume ha habis are formed according o he model in equaion (11). In essence, equaion (11) suggess ha forward-looking consumers wish o cach up wih pas average consumpion of oher forwardlooking consumers, bu are indifferen owards pas spending paerns of he rule-of-humbers when indexing heir habis. Table 2 shows he parameer esimaes ha we obain on boh UK and US daa once we inroduce his modificaion over he same sample period as above: Table 2: Parameer esimaes from he pure inernal habi model Coefficien UK esimae Sandard error US esimae Sandard error Modelling he habi formaion process wih habis defined as in equaion (11) gives differen esimaes for some key parameers for boh counries. In paricular, assuming ha forward-looking consumers care of heir own average pas consumpion bu no of hose of he rule-of-humbers, provides ha: i. The esimaed proporion of rule-of humb consumers,, drops relaive o he case when we assume purely exernal habis. In he UK he esimae is around 6% as opposed o 14% (one sevenh) when using he original Fuhrer (2000) habi formaion model. In he US he proporion of rule-of-humbers also drops subsanially (falling o around 11% from 25%). ii. Parameer esimaes obained on UK daa are more unsable across model specificaions han i is he case for esimaes obained on US daa. For example, he curvaure of he uiliy funcion,, increases significanly in he UK case when we use he alernaive, more inernal habi model. This, in urn, reduces he implied equilibrium consumpion-o-income raio for he UK. Conversely, our esimaes of on US daa are saisically indisinguishable across models. iii. In he UK case he coefficien on he ex ane ineres rae,, is now much greaer in he UK case han originally esimaed in Table 1. However, he esimae of his coefficien is lower in he US case han ha arrived a in Table 1. 14

18 On he oher hand, for boh specificaions, esimaes of he parameer indexing he imporance of habis,, are similar o he original ones for boh counries. In addiion, boh in he UK and in he US case, habis sill seem o depend only on las period s consumpion ( 0) when we use he alernaive habi model. Overall, he consumpion model obained when we assume he alernaive, more inernal habi specificaion seems less good han Fuhrer s (2000) original model a replicaing he dynamics exhibied by consumpion in boh impulse response funcions and auocovariances for he UK. 16 This accords wih he resul ha he log-likelihood (LL) of he sysem is lower under he alernaive habi assumpion presened here han under he original habi formaion assumpion in Fuhrer (2000) (LL = compared wih LL = for he model wih Fuhrer s, 2000, habis). This suggess ha he alernaive, more inernal habi specificaion where consumers ry o cach up wih consumpion of heir own peer group alone is a poorer represenaion of UK consumpion han ha where consumers ry o cach up wih everybody s consumpion. Conversely, he model wih more inernal habis seems o slighly ouperform he model where habis are formed relaive o oal pas consumpion when i comes o US consumpion dynamics, as i emerges by looking a he impulse response funcions of he alernaive habis model esimaed on US daa. 17 In addiion, alhough resuls from he auocovariances indicae here is lile difference beween he wo models, he log-likelihood of he sysem is higher when we fi he model wih he alernaive, more inernal habi assumpion, han when we fi he model wih he original habi assumpion discussed in Fuhrer (2000) (esimaes of he alernaive, more inernal habi model gives LL = compared wih a log-likelihood of from he original habi specificaion). Therefore, on balance, i seems as if he alernaive, more inernal habis model may offer a beer represenaion of he behaviour of US consumers, he opposie of wha we find for he UK. Undersanding why he wo habi formaion models fi differenly in he UK and he US requires a sysemaic analysis of he preferences of consumers in hose wo counries, as well an exploraion of he psychology of spending across Briish and American consumer groups ha plan looking forward in ime. Such analyses are beyond he scope of his paper. However, possible causes ha come o mind may be, iner alia: (i) he abiliy in boh counries o observe consumpion paerns of differen peer groups. Facors like he lower densiy of populaion in he US relaive o he UK, and he fac ha here income groups end o live in homogenous-income neighbourhoods more han i is perhaps he case in he UK may imply ha consumpion of non-peers is less easily and/or less frequenly observed, and so less used as a benchmark for habi formaion; (ii) he markeing of consumpion goods and services may be more focussed o paricular income and/or social groups in he US han in he UK, which could induce a sronger fragmenaion of demandpossibly a consequence of he fac ha US consumpion akes place via caalogue orders, he Inerne and via elesales more han i is he case in he UK, due o he geographical remoeness of suppliers. 16 These are available from he auhors on reques. 17 Ibidem. 15

19 Given ha UK consumpion seems beer porrayed by a model wih habis indexed on oalas opposed o forward-looking agens onlyconsumpion, in he res of he paper we focus on his model alone. So, in wha follows, when we refer o habis or habi formaion model we mean he original exernal habis model discussed in Fuhrer (2000). 3.3 Do habis and rule-of-humb behaviour help capure UK daa? A comparison of he unresriced VAR and habi formaion model auo and cross-correlaion funcions esimaed on UK daa indicaes ha our model implies sensible daa-consisen dynamic correlaions. This is shown in Char 2 below. The char shows ha he habi formaion model capures well he auocorrelaion in consumpion suggesed by he unresriced VAR. The char also shows ha he model is capable of replicaing he persisen dynamic correlaions beween consumpion and ineres raes, income and inflaion observed in he VAR, wihou worsening he abiliy of he model o capure ineracions of consumpion wih oher variables. Char 3 provides furher evidence ha he parameer esimaes ha we ge help obaining a reliable model of consumpion behaviour, one which can mach reasonably well he hump-shaped response of consumpion o a uni income shock. This conras wih ypical resuls from he basic REPIH model, which is unable o replicae ha responseas suggesed by he excess smoohness lieraure on UK consumpionand by he hin solid line in he char, ploing he response of he basic REPIH model wih no habis and no rule-of-humb behaviour. Finally, Char 4 illusraes how a model wih habis and rule-of -humb behaviour helps maching he delayed and hump-shaped response of consumpion spending o a nominal ineres rae shockalhough i suggess ha consumpion roughs following he shock sooner han wha implied by he VARconrary o he basic REPIH model. This is consisen wih previous findings in Baini, Harrison and Millard (2001), which indicae ha augmening a sochasic general equilibrium model calibraed on UK daa o allow for habis in consumpion helps capuring he hump-shaped response of he oupu gap o a moneary policy innovaion observed in empirical models esimaed on ha daa. 4. Consrained versus unconsrained VAR resuls In Secion 3 we have esimaed he parameers of he consumpion funcion wih habis (eq. (10)), where habis are indexed o pas per capia aggregae consumpion. This model seemed o offer a superior fi han he alernaive, more inernal habis one. Focussing on he former model, we now add successive resricions o he shor-erm nominal ineres rae, inflaion and he real exchange rae equaions in he unresriced vecor auoregression of Secion 3. 16

20 Char 2: Auo and Cross-correlaion funcions Habi Formaion Model vs. VAR Noes: X-axis: Lags in quarers, Y-axis: Correlaion Funcion Solid Blue Lines: Unresriced VAR, Doed Green Lines: Esimaed Habi formaion Model Char 3: Impulse response of consumpion o a shock in real disposable income: model vs. VAR Esimaed Habi model Unresriced VAR REPIH Quarers

21 Char 4: Impulse response of consumpion o a shock of he repo rae: model vs. VAR Esimaed Habi model Unresriced VAR REPIH Quarers We sar by imposing a shor-erm nominal ineres rae, which follows a simple feedback rule as in Taylor (1993). This akes he form: f r f f, i i, j j y (12) f, i i1 j0 k0 2 y, k k f We esimae equaion (12) from 1970 Q1 and 2002 Q2 using FIML. Esimaes of he parameers are shown in he Resuls Appendix. Dynamic correlaions of he habis model embedded in a VAR wih his addiional resricion vis a vis dynamic correlaions of he unresriced VAR are ploed in Char 5 below. As expeced, he auo and cross-correlaions of he repo rae wih he oher variables change somewha. However, he char indicaes ha differences beween dynamic correlaions of consumpion for his model and he one wihou he Taylor rule resricion on he moneary reacion funcion are quaniaively small. As a second resricion, we consrain he price process and impose an open-economy adapaion of he Fuhrer and Moore (1995) conracing process, similar o ha derived in Baini and Haldane (1999). The difference wih ha equaion is ha here, as Fuhrer (2000), we use a longer lag-lead specificaion equal o he lag lengh of he unresriced VAR. This implies ha he equaion for inflaion under his specificaion akes he form: k i i y y 1 (1 ) q Eq 1 ( 1/ 2k) (13) i1 p 18