UK Consumption in the Long Run: the Determinants of Consumer Spending

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1 UK Consumpion in he Long Run: he Deerminans of Consumer Spending Andrew P. Blake, Gonzalo Camba-Mendez and Marin Weale Naional Insiue of Economic and Social Research 2 Dean Trench Sree London SWP 3HE Unied Kingdom February 998 Absrac Sudy of long runs of economic daa makes i possible o disinguish beween uni roo processes and deerminisic, bu broken rends. We find ha mos of he variables o be used in a consumpion funcion have deerminisic rends. We esimae a modified life-cycle model over he period , finding saisfacory and sable model parameers ogeher wih long-run residuals from which we can exclude broken rends, indicaing ha we have found a corending and co-breaking relaionship. By conras, he long-run residuals from a model which explains consumpion in erms of income and inflaion exhibi a broken rend. KEYWORDS Consumpion, life-cycle model, uni roo, co-rending, co-breaking JEL Reference Nos: C4, E2 This research has been financed by he ESRC under gran no R

2 . Inroducion. Much of he recen work on he deerminans of consumer spending in he Unied Kingdom and elsewhere has focussed on aemps o idenify impacs of financial liberalisaion which began in he lae 970s in he UK and he US, and slighly laer in oher counries (Bacchea and Gerlach, 997; Sefon, 998). Here we ake a differen perspecive, looking a consumer spending over he period , in order o esablish wheher here is any degree of sabiliy in eiher shor-erm or long-erm relaionships. Sabiliy over a long period is perhaps he bes indicaor ha a funcional relaionship represens underlying behaviour. Mos analysis is conduced on he assumpion ha boh consumpion and he explanaory variables used in a consumpion funcion are I(). This means ha any long-run relaionship has o be a co-inegraing vecor, linking a number of I() variables in a linear relaionship which has I(0) residuals. However, here are now a number ess which make i possible o discriminae beween series which are I() and hose which are rend saionary bu whose rend rae of growh may change from ime o ime, and which may also experience occasional breaks in heir rend. I is necessary o esablish wheher he variables ypically used in a consumpion funcion are rend-saionary; if his is he case a search for a co-inegraing vecor is plainly inappropriae. Insead, o use he erminology of Hendry (995), he long-erm consumpion funcion should be a co-rending and co-breaking relaionship (see also Chapman and Ogaki, 993). While consumpion and he series used o consruc he consumpion funcion may be rended wih rends which change from ime o ime and wih breaks in heir rends, he residuals of any long-run relaionship should be saionary and should no exhibi any breaks. In any sudy including he 940s, he effecs of he Second World War are a dominan feaure of he daa. There was a large rise in GDP and a large fall in privae consumpion as he supply side was expanded bu privae spending was resrained hrough a mixure of axaion and raioning. There is a risk ha his migh disor any findings abou he deerminans of consumpion and we have o address his issue. We begin he paper by describing our daa and summarising he way in which we used missing observaion echniques o deal wih he ouliers generaed by he Second World War and he ime-series properies of our daa. We hen presen a sandard life-cycle consumpion funcion modified o ake accoun of rule of humb consumers and es i o verify ha i represen a co-breaking and co-rending

3 relaionship. The model is conrased wih a radiional parial adjusmen model and conclusions are drawn. An appendix describes our daa in more deail, gives he deails of he missing observaion mehods used and summarizes he resuls of our ess for deerminisic versus sochasic rends. In appendix wo he derivaion of our life cycle consumpion funcion wih rule of humb consumers is explained. A hird appendix explains our esimaion echnique and he ess we use for parameer and equaion sabiliy. Finally appendix four explains he mehods by which we es wheher we have found co-rending and co-breaking relaionship. 2. Consumpion, Income and Wealh, : he Daa and heir Properies The Daa The flow daa needed for our model are hose presened by Sefon and Weale (995). Tha daase covers he period wih he various residuals in he naional accouns allocaed using he mehod of leas squares (Sone, Champernowne and Meade, 942). Solomou and Weale (997) provide esimaes of personal secor wealh for he period o link in wih oher daa and o provide a coninuous series from An imporan aspec of our work is ha we redefine consumpion and income reflecing he fac ha a sock of durable goods provides a flow of services over ime, raher han being used up on or immediaely afer purchase. We consruced our own esimae of he sock of durable goods using he perpeual invenory mehod (see appendix ). We hen redefined consumpion by deducing purchases of durables and adding on he esimaes of depreciaion derived from he perpeual invenory model. Our esimaes of personal wealh were increased by he esimaed value of he sock of durable goods because he exising measure of disposable income is gross of he depreciaion of he capial of unincorporaed businesses. To mainain consisency wih ha view, we added on our esimaes of he depreciaion of he sock of durable goods. This means ha our measure of saving is gross of he depreciaion of he sock of durable goods as well as of business capial. Our fundamenal consumpion equaion is expressed in erms of he level of consumpion. We scale by lagged income in order o reduce he heeroscedasiciy generaed from economic growh. The daa are per capia and in 985 consan prices. Variables will be defined as follows: 2

4 GDP C G X YD YL RW r π Gross Domesic Produc Consumpion Expendiure Public Consumpion Expors Disposable Income Labour Income Ne Wealh Real Ineres Rae Inflaion A full descripion of he daa is provided in appendix. The second world war was a major disrupive influence o he economy and we have o consider how o deal wih his. One migh inser dummies ino regression equaions, bu we prefer o esimae suiable ime-series models represening he series of ineres, and use hese o calculae missing values for he war years (Harvey, 989). This has he advanage ha i makes full use of he daase while preserving he ime-series properies of he peace-ime years. I does no, herefore, prejudge he issue wheher each series incorporaes a uni roo or a deerminisic rend. Deails are again provided in appendix. The applicaion of hese echniques gave us a filered series from which he direc effecs of he war had been removed, and gave us he saring poin o research wheher he ime series were driven by sochasic or deerminisic rends. As an illusraion figure below shows he series of consumpion expendiure before and afer removing he impac of he war upon i. Uni Roos or Broken Trends I was noed by Wes (988), ha when a uni roo process conains a linear rend or a drif, is variabiliy is dominaed by a quadraic or a linear rend. This implies ha an appropriaely normalised variabiliy will converge o a consan. There will no be convergence o a consan if he uni roo process does no have a linear rend or a uni roo. In ha case he variabiliy of Blake, Camba-Mendez and Weale (998) compare hese ess applied o real GDP per capia wih he war filered ou as agains he raw daa. They provide a es which shows ha i is beer o work wih he filered daa and rea he war years as ouliers han o regard hem as par of he peace-ime saisical process. 3

5 Figure a. Log Consumpion (985 prices) Raw Daa Figure b. Log Consumpion (985 prices) Filered Daa. 4

6 he uni process will converge o a funcion of Brownian moions. Based on his fac, Dolado e al (990) proposed he following sraegy o idenify uni roos: i) Tes he hypohesis γ = 0 in equaion () below. If he Dickey-Fuller ables allow rejecion of ha hypohesis, we conclude ha he series has a uni roo; if he hypohesis is no rejeced we proceed o nex sage; y = a + a + γy + β y + u () 0 2 i i i = k ii) Tes he hypohesis a 2 = 0 given γ = 0 using he -suden disribuion; if a 2 is significan use he normal disribuion o es he hypohesis γ = 0 in (); if i is no significan go o nex sage; iii) Esimae equaion () and es he hypohesis γ = 0 given a 2 = 0 using he Dickey-Fuller ables; if he hypohesis canno be rejeced go o nex sage; iv) Tes for he significance of he consan erm given a 2 = γ = 0; if he consan erm is significan hen use he normal disribuion in he uni roo es of sep iii) in order o accep or rejec he null hypohesis. The crierion followed o deermine he number of lags for he ADF es (i.e. k in equaion ()) is ha suggesed by Perron (989); namely choose a lag lengh k such ha he saisic associaed wih ha lag is significan, bu he saisic of lag k+ in a regression using k+ lags is no significan. Following Perron, we rea a parameer as significan when he saisic associaed wih i is larger han.6. Perron (989) argued ha uni roo ess are biased owards non rejecion in he presence of a break in he series. Zivo and Andrews (992) proposed a sequenial ADF es o es he uni roo hypohesis agains he alernaive of a saionary process around a broken rend. The broken rend can be of hree ypes: i) a change in he level of he rend (he Crash Model); ii) a change in he slope of he rend (The Changing Growh Model); and finally iii) boh a shif in he level and a change in he slope. Table displays a summary of he es for he presence of a uni roo for some UK macroeconomic series. Full resuls of hese ess are presened in appendix. Three crieria have been used: i) he ADF es; ii) he sraegy suggesed by Dolado e al.; iii) The Zivo and Andrews es in he presence of an unknown break. Afer using hese ess, i appears ha he uni roo hypohesis can be rejeced for mos series. In fac i is he Zivo and Andrews es 5

7 which rejecs he uni roo hypohesis for mos series. However, his is no he case eiher for GDP or for labour income (ne of ax bu including ransfers). The Dolado e al crieria allows rejecion of he presence of a uni roo for GDP. Some economiss neverheless argue ha he Dolado crieria may be very sof in small samples, and hey favour he use of he ADF levels of significance. Blake, Camba-Mendez and Weale (998) showed ha he series of GDP for he period under sudy in his paper has no one bu wo breaks in is deerminisic rend; his is preferred o he hypohesis of a uni roo. We have conduced a similar exercise o ha in Blake e al (998) for he labour income series, and, as for GDP, once we allow for wo breaks he uni roo hypohesis can be rejeced (hese resuls are presened in appendix ). Therefore in view of he resuls in able we favour he view ha he real macroeconomic series for he period under sudy are rend saionary, where he rends may display break poins. The inflaion rae, however, is no direcly affeced by he real economy, and could be I() even when real variables have deerminisic rends 2. Table. Uni Roo Tes. Period Series ADF Crieria Dolado e al. Crieria Unknown Break Crieria Two Breaks Resul Log GDP I(0) Log C I(0) Log G I(0) Log X I(0) Log YD I(0) Log YL I(0) Log RW I(0) r I(0) π Uni Roo When esing for an unknown break, use is made of he Zivo and Andrews (992) es for all series bu for real ineres rae and inflaion for which use of he Perron and Vogelsang (992) was he chosen mehod. 2 We also find some evidence of a level shif in he real ineres rae in 979 (able A4). However, examinaion of he daa suggess ha his is caused by ouliers arising from he high inflaion of he mid-970s. We herefore assume ha here is a sable mean o his series. If he assumpion were incorrec we migh expec i o lead o parameer insabiliy in our model which assumes a consan expeced real rae of ineres. 6

8 This has imporan implicaions for our subsequen research sraegy. If he series of ineres were believed o be I() Johansen s mehod would be used o idenify co-inegraing vecors. Bu wih series which are driven by deerminisic rends, we need o idenify insead a coinegraing and co-breaking vecor. We firs presen and esimae our basic model. We hen apply a es o he long-run residuals o idenify wheher he variables represened in our consumpion funcion are co-rended and co-broken. 3. Dynamic Adjusmen and a Life-Cycle Model. Our basic model is a form of Blanchard s (985) perpeual youh life-cycle model; he only modificaion is ha here is assumed o be a consan level of subsisence consumpion enering ino he uiliy funcion. For each consumer he probabiliy of surviving from one period o he nex is assumed o be p. Because his is independen of age, he behaviour of each consumer is independen of is dae of birh, and depends only on is holding of wealh and expeced labour income. The marke rae of ineres is r ; each consumer invess wealh in an annuiy which disribues he asses of hose who die in any cohor o he survivors. This means ha invesed by a paricular cohor in one period rises o + r in he nex period. This is disribued among he fracion p surviving, so ha he reurn o he survivors is (+r )/p. Uiliy is assumed o be discouned a a rae θ. When calculaing he presen value of a fuure consumpion pah, due accoun has o be aken of he probabiliy of survival. If uiliy is logarihmic, u = log(c - c ) wih c individual consumpion and c individual subsisence consumpion, hen he expeced uiliy of a consumpion sream is τ ( θ p) log( cτ cτ) τ = (2) Here c τ and c τ describe he per capia consumpion of a paricular cohor. The age of his cohor is no indicaed in order o make he noaion idier. I does no affec he subsequen calculaions. Boh he subsisence consumpion and he rend labour income of each individual are assumed o decay a a rae ψ. The derivaion of he consumpion funcion is se ou in Appendix 2. A dynamic consumpion funcion, consisen wih he assumpion of raional expecaions, can be derived from he model using he approach suggesed by Hayashi (982). We aemped his 7

9 bu found, even when, again following Hayashi and Campbell and Mankiw (989,99), we assumed ha some expendiure akes place ou of curren income, ha he parameers were highly unsable. The model gave saisfacory parameer values for he period bu no when he sample included a much longer period. We herefore looked a an alernaive, ha consumers made assumpion abou g he growh of rend average labour income per capia, YL, and ρ he rae a which any deviaion from rend, DYL =YL -YL is correced 3. In common wih oher models of his ype, we assumed ha he expeced real rae of ineres is consan a r 4. ( ) YL+ = + g YL { } ( ) E DYL + = + g ρ DYL (3) The model is weakly raional if hese assumpions are consisen wih he daa. We in fac explored a number of differen assumpions o calculae he resuling value of human capial as a funcion of boh acual labour income and is rend. In Appendix 2 we derive he following expression for he average per capia consumpion, C, of he opimizing consumers. C C RW g YL ( g) DYL C = µ µ µ α (4) µ + µ ρ + µ Here µ = (+r )/ψp- is he ineres rae used o discoun fuure labour income and α=-θp Rule of Thumb Consumers This gives us he basis for a family of final models. Like Hayashi (982) and Campbell and Mankiw (989, 99) we assume ha a proporion κ of disposable income accrues o rule of humb consumers; we posi heir per capia long-run consumpion funcion o be 3 Appendix 2 describes he relaionship beween he evoluion of he labour income of each individual and ha of he per capia average 4 We did in fac address he quesion of variaions in he expeced ineres rae, assuming ha i was expeced o rever o is mean following an AR() process; we used a linearizaion o idenify he effecs of changes in he expeced real rae of ineres on he value of human capial and he discouned value of subsisence consumpion. The exercise did no yield meaningful resuls. We aribue his o he difficuly of idenifying a suiable real rae of reurn, boh because here is a large number of possibiliies and because here may well have been confusion abou he effecs of inflaion for much of he period. 8

10 C R = C + φ YD (5) The subsisence consumpion of each rule of humb consumer is assumed o be he same as he average of he opimizing consumers. Here he propensiy o consume has o be consisen wih he assumpion ha a consan proporion of income accrues o rule of humb consumers and herefore, implicily ha hey own a consan proporion of oal wealh. If ζ is he average raio of wealh o income and g inc is he rend rae of growh of income 5, hen he propensiy o consume ou of income has o be -g inc ζ. Even his is an approximaion, since he subsisence consumpion erm means ha he share of wealh owned by rule of humb consumers will only asympoically approach is long-run value. Given hese wo classes of consumers we arrive a a final long-run funcion for per capia consumpion C C ( ) YD RW g YL ( g) DYL C = µ µ µ κφ κα +υ (6) µ + µ ρ + µ where υ is a random error erm. A Dynamic Consumpion Funcion The funcion se ou above is a long-erm funcion which makes no aemp o model dynamics. Having rejeced he resricion imposed by he assumpion of raional expecaions, we insead looked a a parial adjusmen model. Such a model can be jusified for opimizing consumers in he manner suggesed by Anderson and Blundell (983). They showed ha his specificaion was obained if uiliy-maximizaion defined a long-run level of consumpion and ha his level was approached in a manner defined by quadraic adjusmen coss. We denoe by C long, he value of consumpion defined by (6) before he random error is aken ino accoun. We have o address he quesion of scaling. The esimaion mehod (GMM) should be robus o heeroscedasiciy as well as serial correlaion bu i sill seemed sensible o reduce he degree of heeroscedasiciy by scaling he equaion by our definiion of personal disposable income (including depreciaion of durables). We herefore looked a he model 9

11 ( C Clong ) C YD C long,, = β + β2 + β3 + η YD YD YD YD (7) wih C long defined by equaion (6). This is closely relaed o he funcion used by Campbell and Mankiw (99) who also assumed a consan expeced real ineres rae. They allowed for a ime rend in β and, of course, did no include he erms in β 2 or β 3. We prefer his scaled equaion o he logarihmic specificaions suggesed by Church e al (994) and Muellbauer and Laimore (995). We could no find sable specificaions of logarihmic models. 4. Consumpion in he UK, The model requires us o esimae he rae a which labour income is expeced o rever o is rend. Equaion (7) is herefore esimaed simulaneously wih he equaion for he deviaion of labour income from is rend (equaion 3). However, in esimaing he model we have o consider wha values o use for he rend pah of labour income and for is expeced growh rae. The mos coheren sance would be o use he broken rend for labour income per capia o define boh is rend and is expeced growh rae. We found an increase in he rend rae of growh in 952 and a reducion in 964. This would carry he implicaion ha he acceleraion and slow-down were no anicipaed bu were recognized as such immediaely hey happened; in ha sense expecaions would be weakly raional. However, we waned o look a he possibiliy ha consumers expecaions of growh in labour income were deermined no by he rend in labour income iself bu raher by he rend in GDP, he argumen being ha his is a beer indicaor of rising prosperiy 6. The rae of growh of GDP also increased in 952, bu i did no slow again unil 972 (see able 2). Table 2. Raes of Growh of GDP and YL. GDP % 2.35%.65% YL % 3.28% 2.254% 5 We invesigaed he effecs of changes in he rend rae of growh of income during he sample period bu found hem unimporan. 6 GDP growh and labour income growh can diverge no only because of variaions in he share of labour income and he ax ake bu also because of movemens in he erms of rade. 0

12 Finally here is he possibiliy ha he breaks we have idenified do no help in he esimaion of he consumpion funcion; if his is rue he assumpion ha an unbroken rend is used as he reference poin for labour income and ha he expeced rae of growh is consan should be expeced o give resuls beer han hose found in specificaions which ake accoun of he broken rend. Accordingly we looked a six possibiliies. Firs we looked a a case where he expeced rae of growh of labour income was consan. In he firs case (model ) we used he acual rend rae of growh of labour income per capia while in he second case we used he rend rae of growh of GDP per capia over he period. In hese cases we also assume ha he rae of regression of shocks in labour income o he rend was calculaed wih reference o an unbroken rend and ha he unbroken rend line used o define YL was also used in he regression equaion for evaluaing ρ (equaion 3). Then in models 3 and 4 we assumed ha he rend value of labour income was calculaed wih he breaks in 952 and 964 for evaluaion of boh YL and ρ. In he firs case expeced growh of labour income was calculaed using he acual rae, while in he second case i was calculaed using he rend rae of growh of per capia GDP. Finally we looked a he case where he rend in labour income was calculaed assuming he break ook place in 972. In he firs case we used he growh rae of GDP o give he expeced rae of growh of labour income while in he second case we used he growh rae of labour income, calculaed assuming breaks in 952 and 972. These six models are summarized in able 3. Table 3 Assumpions abou he expeced Growh and Trend Pah of Labour Income. Model Model 2 Model 3 Model 4 Model 5 Model 6 Expeced growh of labour income calculaed from Lab. Income GDP Lab. Income GDP GDP Lab. Income Breaks in expeced growh of labour income None None Breaks in rend labour income (YL ) None None

13 A preliminary sudy suggesed ha while he funcion was sable, here were wo large residuals in he long-run equaion. The firs was a he end of he second world war, while he second was in he lae 980s. We addressed he firs by inroducing a variable, War, represening he sock of consumpion foregone. This was calculaed by cumulaing he difference beween observed consumpion and he value in our series from which he direc effecs of he war were filered ou. The sock of missing consumpion was augmened by he shorfall in each of he war years bu was assumed o decay a 5% p.a. boh during and afer he war. I was assumed o have no impac in 946, half is impac in 947 and full impac from 948 onwards. We were unable o find a similar raionale for a dummy in he lae 980s and have herefore lef he laer residuals as hey sand. The final definiion of long-erm consumpion (6) once we incorporae he War variable is: ( ) C = C + κφyd + β 0 War + RW g YL ( g) DYL C + + µ µ µ κα µ + µ ρ + µ + ν wih he dynamic funcion again defined by (7). (6 ) Parameer Esimaes Table 4 shows he model parameers esimaed for hese six models and also he values of he J-saisic, he minimand when esimaing by he generalized mehod of momens. I can be seen ha he parameers are no grealy affeced by he choice. However he models wih broken rends clearly do beer han models and 2 wih consan rends. This provides some evidence ha some accoun should be aken of influence of acual and expeced changes in he rend rae of growh of labour income when rying o explain consumpion over he period Model 5 has he lowes value of he J saisic and we herefore chose i as our reference poin. This assumes ha consumers expecaions of growh in labour income were deermined by rend GDP growh raher han by acual rend labour income growh. 2

14 Table 4. Esimaion Resuls for Models o 6. Parameer Model Model 2 Model 3 Model 4 Model 5 Model 6 α κ C β µ ρ β β β J Sa In each case he equaion are (6 ), (7) and ( g ) YL YL = ρ + YL YL YL YL Here g is he acual rae of growh of rend labour income, calculaed for each of he subperiods idenified in he las row of able 3. The models differ because differen values of g, g and YL are used as explained on able 3. + ε We are able o impose he resricions β =0 and β 2 = (χ 2 2=3.03). This implies ha changes in income have no influence beyond ha represened by our long-erm consumpion funcion, and ha in he shor erm changes in consumpion occur for wih changes in he deerminans of long-run consumpion. Table 5 shows he model parameers esimaed wih hese resricions in place. I also shows he resuls of he sabiliy ess. The las wo columns of he firs par of he able show he value of he Lagrange muliplier saisic comparing he model ha he parameer in quesion changes a he year indicaed wih he alernaive ha i is sable hroughou. The values shown are for he years which deliver he highes saisics and herefore come closes o rejecing he hypohesis of parameer sabiliy. The significance levels are shown in he noes; none of he saisics are significan; nor is he same es when applied o es he sabiliy of he equaion as a whole, raher han he values of individual parameers. I does, however, have is peak value in 986; we have already noed he large posiive residuals which develop in he long-run equaion in he lae 980s (figure 3). 3

15 Table 5 GMM Esimaion Resuls of Model LM Sabiliy Tes Parameer value Sandard deviaion level of signif. Saisic Year α [0.000] κ [0.000] C [0.000] β [0.000] µ [0.000] ρ [0.000] β β β [0.007] J-Saisic LM Sabiliy Tes χ 2 () signif. Saisic Year [0.939] Noes: For he compuaion of he weighing marix used has been made of he Newey Wes (987) covarince marix wih a bandwih of 0. The Insrumen used for GMM purposes are: consan, ime rend, g, g -, g -2, g -3, g -4, x, x -, x -2, x -3, x -4, r -, r -2, r -3, (/YD - ), (/YD -2 ), (YL - - YL ) Where g and x are public consumpion and expor; divided by he scaling facor (/YD ), and YL is he rend componen of YL. The 0%, 5% and % level of significance of he Superior LM es for he parameers are 6.05, 7.5 and 0.9 respecively. The equivalen levels of significance for overall sabiliy are 7.74, 20.0 and We regard he model parameers hemselves as very saisfacory. The propensiy o spend ou of wealh is 2.43% p.a. The discoun rae applied o fuure labour income is 7.6% p.a. This is high enough o sugges ha here is more involved han simply he risk of dying idenified by Blanchard (985), bu i is considerably below he figure of 3.2% or 7.3% suggesed by Hayashi (982) depending how he resrics his model. Disposable income is found o be spli almos equally beween consrained and opimizing consumers. This is a lower proporion going o opimizers han some auhors have suggesed bu is wihin he range of 0.44 o 0.66 for he US quoed by Campbell and Mankiw (989) when he works wih insrumenal variables and also he raher wide range Campbell and Mankiw (99) find for he UK (0.20 o 0.66 depending on he esimaion mehod used). Sefon (forhcoming) finds a raher lower proporion of income accruing o consrained consumers falling from 30% unil 985 asympoically o 5%. The decline is no, however, saisically significan. The level of 4

16 subsisence consumpion idenified is 429 per capia in 985 prices. This compares wih he lowes acual consumpion in our daase of 400 per capia in 920. Co-rending and Co-breaking Figures 2 o 4 show firs, he acual and long-run fied daa 7, secondly he long-run residuals and hirdly he residuals of he shor-erm equaion. We are ineresed in wo aspecs of he long-run residuals. Firs of all, if he underlying assumpion ha he daa follow deerminisic raher han sochasic rends is wrong, hen we would wan he long-run residuals no o show a uni roo. Table 6 shows he DF and ADF saisics. Engle and Granger (987) quoe 5% significance levels of for he DF saisic and -3.7 for he ADF saisic. As is usual, we rejec he uni roo hypohesis more firmly wih he DF es han he ADF es; we regard he finding as saisfacory given our iniial view ha uni roos are absen. Table 6. Uni Roo Tes on Long-run Residuals. No inercep wih inercep DF ADF(4) DF ADF(4) Table 7 shows he relevan es saisics wih he underlying assumpion presumed correc; deails of he es mehodology are given in Appendix 4. In his case we can rejec he hypohesis ha here are rends or level shifs in he residuals and hus confirm ha we have found a co-breaking co-rending vecor linking consumpion and he explanaory variables. The use of GMM means ha he shor-erm equaion residuals do no need o be free of serial correlaion for he resuls o be saisfacory. On he oher hand we may wan o know wheher he parial adjusmen process accommodaes he dynamics saisfacorily. In GMM we do no have an equivalen of he Lagrange Muliplier es. Bu as an informal check we looked a he Durbin-Wason saisic. A figure of.76 does no indicae any cause for concern. 7 Scaling means ha he variable shown in figure 2 is C/YD- he average propensiy o consume. This is lower han in he convenional daa because we have included o depreciaion of he sock of durable goods as a componen of income in order o rea his on he same fooing as he depreciaion of oher capial goods owned by he personal secor. More deails are given in Appendix. 5

17 Table 7. Tes for he presence of a rend in he residuals. (Unbroken Trend) e = δ + δ 2 + u Parameer value Sandard deviaion level of signif. Bandwih δ 2 (NW) 0.229x x0-3 [0.89] 2 (Unknown Broken Trend) e = δ + δ 2 d + δ 3 + δ 4 d + u Exogenous Level of Significance Endogenous Hypohesis Saisic Break 5% 0% 5% 0% δ 2 =δ 3 =δ 4 = Figure 2. Daa and he Fied Long Run. 6

18 Figure 3. Long-run Residuals. Figure 4. Shor Run Residuals Model 5 (resriced). 7

19 While we find a sable long-run consumpion funcion, he we have noed he presence of high residuals in he 980s. The fac ha hese proved emporary and ha he consumpion revered o he prediced value in he 990s suggess ha he surge was caused by a emporary facor raher han a permanen fac, such as financial liberalisaion. The findings herefore are consisen wih he suggesion of Aasanio and Weber (994), ha he surge in consumpion was caused by a emporary increase in he expeced rae of growh of labour income; i is perfecly possible for such a emporary change o have occured wihou being idenified by our parameer sabiliy ess, where i would have been ransmied ino insabiliy in µ. 5. Comparison wih a Tradiional Dynamic Adjusmen Model. The performance of our model can helpfully be compared wih a radiional parial adjusmen model derived from he sudy by Davidson e al. (978). Their model is expressed in erms of logarihmic variables and percenage changes. The logarihmic growh of consumpion is relaed o he logarihmic growh of income, a erm in he log of he raio of lagged consumpion o income and erms in he rae of inflaion, π and is derivaive. One worry abou his equaion is ha he inflaion rae appears o be I(), whereas we find boh consumpion and income o be I(0). Bu we follow he pas procedure of esimaing a model by GMM and hen looking a he long-run residuals. In order o mainain comparabiliy wih our earlier work we prefer o esimae a sysem scaled by lagged income, raher han ake logarihms. Thus our alernaive equaion is C YD YD C War = β + β2 + β3 + β4 + β5 π + β6π + β7 + η (8) YD YD YD YD As before War is a dummy variable represening he effec of consumpion missed during he war on spending in he pos-war years. I is described in secion 4. Only β 2 is saisically significan. There is insabiliy in parameers β 4, β 5 and β 6 a a 5% level, and we also rejec he hypohesis ha he model as a whole is sable. One migh re-esimae seing β 6 o zero on he grounds ha inflaion appears o be I() while he oher variables have deerminisic rends. Bu if his is done he long-erm relaionship beween consumpion and 8

20 income, represened by β 3 remains insignifican casing doub on he presence of a co-breaking relaionship. As a shor-erm equaion i has a higher sandard error han our previous model, as compared o However, when we look a he long run, even of he equaion as i sands, i is clear ha i is no a enable model. Table 9. GMM Esimaion Resuls of Equaion LM Sabiliy Tes Parameer value Sandard deviaion level of signif. Saisic Year β [0.2] β [0.000] β [0.9] β [0.255] β [0.658] β [0.56] β [0.438] J-Saisic LM Sabiliy Tes χ 2 (0) signif Saisic Year [0.93] Noes: For he compuaion of he weighing marix used has been made of he Newey Wes (987) covarince marix wih a bandwih of 0. The Insrumen used for GMM purposes are: consan, ime rend, g, g -, g -2, g -3, g -4, x, x -, x -2, x -3, x -4, r -, r -2, r -3, (/YD - ), (/YD -2 ) Where g and x are governmen expendiure and expor; divided by he scaling facor (/YD ). The 0%, 5% and % level of significance of he Superior LM es for he parameers are 6.05, 7.5 and 0.9 respecively. The equivalen levels of significance for overall sabiliy are 7.74, 20.0 and Figure 5 shows he fied values calculaed from our esimaed parameers of equaion 8. Visual comparison wih figure 2, showing he fied values of our previous equaion is enough o indicae ha his model is subsanially worse han ha wih sronger heoreical roos. If one akes he view ha, despie our ess, consumpion and income are I() processes, hen he residuals of he long-run equaion should be saionary. The ess for uni roos (DF = -2.83, ADF(4) wih inercep) is less saisfacory. On he oher hand, if consumpion and income are assumed o follow deerminisic rends, hen he residuals should be free of rends and breaks. 9

21 Figure 5. Fied Consumpion o Disposable Income Raio from Equaion 8. Applying he ess discussed above, we find ha here is a significan rend presen, leading us o rejec he idea ha consumpion, income and inflaion are co-rended and co-broken. We can accep he hypohesis ha here is no a uniform rend in he long-run residuals. Bu once we es for a broken rend, wih he break-poin seleced arbirarily, we have o rejec he view ha such a break poin is absen. I follows ha he hree variables in he long-run equaion are no co-rended. However Figure 6 suggess ha he paern of he residuals becomes much more sable afer 955. This may explain why researchers looking only a he recen pas have ended o find he model saisfacory. 20

22 Figure 6. Long-run Residuals from Equaion 8. Table. Tes for he presence of a rend in he long run residuals. (Unbroken Trend) e = δ + δ 2 + u Parameer value Sandard deviaion level of signif. Bandwih δ 2 (NW) [0.82] 3 (Unknown Broken Trend) e = δ + δ 2 d + δ 3 + δ 4 d + u Level of Significance Exogenous Endogenous Hypohesis Saisic Break 5% 0% 5% 0% δ 2 =δ 3 =δ 4 =

23 6. A Consumpion Funcion or an Accouning Ideniy? Economic heory indicaes ha consumpion is linked by a behavioural relaionship o income and wealh. Bu accouning indicaes ha here is a separae accouning ideniy linking hese hree variables. For a consumpion funcion o be saisfacory one should be confiden ha i is indeed a behavioural relaionship raher han say a linear combinaion of behavioural and accouning relaionships 8. A sufficien condiion for his o be he case is ha he esimaed funcion is orhogonal o he accouning ideniy (alhough of course he underlying behavioural relaionship need no mee his condiion). The accouning relaionship corresponding o our daa is W + YD W C = ( + g ) + YD YD where g is he capial gain measured as a proporion of wealh in period. If we look a he funcion W + YD W C + = ε YD YD where ε is reaed as a random variable (no necessarily of zero mean) we can hen es wheher ε is orhogonal o he residuals of our co-breaking vecor. We find we can accep his hypohesis a a convenional 5% confidence level ( 68 =.88, p=0.065). However, he near-significan value is obained because in 988 here were large posiive residuals in boh he accouning equaion and he behavioural equaion. If we include a dummy for 988 when regressing he long-run residuals of he consumpion funcion on ε we find ha he -saisic falls o We can herefore comforably accep he hypohesis ha our equaion is a behavioural equaion and is no conaminaed by he accouning ideniy. 7. Conclusions An economeric relaionship such as he consumpion funcion should be sable over long periods and have sensible parameer values if i is o be saisfacory. If he funcion embodies boh shor and long-erm relaionships, hen he long-erm relaionships should be wellbehaved. If mos economic variables represening flows of goods and services are aken o be 22

24 I(), hen he requiremen ha he long-erm residuals should be well-behaved implies ha hey should be I(0). Bu if he variables have deerminisic, possibly broken, rends and perhaps shifs in heir levels from ime o ime, hen such rends and shifs should be absen from he residuals. In oher words he variables represening he long-erm relaionship should be corended and co-broken. The model should, also, of course pass ess for he sabiliy of boh individual parameers and he overall equaion. In a preliminary analysis of he variables o be used in a long-erm consumpion funcion, we concluded ha hey exhibi deerminisic raher han sochasic rends and one should herefore look for evidence of co-rending and co-breaking raher han co-inegraion. We were however, able o find a consumpion funcion for he UK for whose long-run residuals appeared o be co-broken and co-rended. However, because he original view ha he daa were driven by deerminisic raher han sochasic rends may have been incorrec, we also esed for a uni roo and rejeced he hypohesis one was presen in he residuals. The model we adoped was a closed-form of Blanchard s (985) life-cycle model; he dynamic adjusmen owards his was, however, represened by a parial-adjusmen process and was no consisen wih he assumpion of raional expecaions. The model conrass wih a radiional parial adjusmen model in which, in he long run consumpion is driven only by income. This model has a saisfacory long-erm performance only afer 960, perhaps explaining why i has been adoped as a resul of he sudy of shorer ime periods. 8 We are graeful o Mike Wickens for drawing our aenion o his poin. 23

25 7. References. Anderson, G. and R. Blundell. (983). Tesing Resricions in a Flexible Dynamic Demand Sysem: an Applicaion o Consumers Expendiure in Canada. Review of Economic Sudies. Vol L. pp Andrews, D.W.K. (99): Heeroskedasiciy and auocorrelaion consisen covariance marix esimaion. Economerica, Vol. 59, pp Andrews, D.W.K. (993): Tess for parameer insabiliy and srucural change wih unknown change poin. Economerica, Vol. 6, pp Aasanio, O.P. and G. Weber. (994). The UK Consumpion Boom of he Lae 980s:Aggregae Implicaions of Microeconomic Evidence. Economic Journal. Vol. 04. Pp Bacchea, P. and Gerlach, S. (997): 'Consumpion and Credi Consrains: Inernaional Evidence'. Journal of Moneary Economics, Vol. 40, pp Blake, A.P.; Camba-Mendez, G. and Weale M. (998): Growh and slowdown: rends in UK GDP Naional Insiue of Economic and Social Research, Working Paper. Blanchard, O.J. (985). Deb, Deficis and Finie Horizons. Journal of Poliical Economy. Vol. 93. Pp Cambell, J.Y. and N.G. Mankiw (989). Consumpion, Income and Ineres Raes: Reinerpreing he ime-series evidence. In Olivier J. Blanchard and Sanley Fischer (eds), NBER Macroeconomic Annual 989, Cambridge, Mass.: MIT Press, pp Campbell J.Y and N.G. Mankiw (99) The Response of Consumpion o Income: a crosscounry invesigaion. European Economic Review. Vol. 35, pp Chapman, A.L. and Knigh, R. (953): Wages and Salaries in he Unied Kingdom Sudies in he Naional Income and Expendiure of he Unied Kingdom, Vol. 5. Cambridge Universiy Press, Cambridge. Chapman, D. A. and Ogaki, M. (993): 'Corending and he saionariy of he real ineres rae'. Economics Leers, Vol. 42, pp Church, K.B., Michell, P.R., Smih, P.N. and Wallis, K.F. (994): Wealh Financial Deregulaion, expecaions and consumer behaviour. Governmen Economic Service Working Paper No. 22, London. Davidson, J.E.H., Hendry, D.F., Srba, F. and Yeo, S. (978): Economeric Modelling of he aggregae ime series relaionship beween consumer s expendiure and income in he Unied Kingdom. Economic Journal, Vol. 88, pp Dolado, J.J., Jenkinson, T. and Sosvilla-Rivero, S. (990): Coinegraion and Uni Roos. Journal of Economic Surveys, Vol. 4, pp Feinsein, C.H. (972): Naional Income, Expendiure and Oupu of he Unied Kingdom, Sudies in he Naional Income and Expendiure of he Unied Kingdom, Vol. 6. Cambridge Universiy Press, Cambridge. 24

26 Hansen, L.P. (982): Large sample properies of generalized mehod of momens esimaors. Economerica, Vol. 50, pp Harvey, A.C. (989): Forecasing, Srucural Time Series Models and he Kalman Filer. Cambridge Universiy Press, Cambridge. Harvey, A.C. and Jaeger, A. (993): Derending, sylized facs and he business cycle. Journal of Applied Economerics, Vol. 8, pp Hayashi (982): 'The Permanen Income Hypohesis: Esimaion and Tesing by Insrumenal Variables'. Journal of Poliical Economy, Vol. 90, pp Hendry, D.F. (995). A Theory of Co-breaking. Mimeo. Nuffield College. Oxford. HMSO (97): Briish Labour Saisics: Hisorical Absrac Deparmen of Employmen and Produciviy. London, HMSO. Michell, B.R. (988): Briish Hisorical Saisics. Cambridge Universiy Press, Cambridge. Muellbauer, J. and Laimore, R. (995): 'The Consumpion Funcion: A Theoreical and Empirical Overview'. In M.H. Pesaran and Wickens eds. Handbook of Applied Economerics, Blackwell, Oxford. Newey, W.K. and Wes, K.D. (987): A simple posiive semi-definie heeroskedasiciy and auocorrelaion consisen covariance marix. Economerica, Vol. 55, pp Perron, P. (989): The Grea Crash, he Oil Price Shock and he Uni Roo Hypohesis. Economerica, Vol. 57(6), pp Perron, P. and Vogelsang, T.J. (992): Nonsaionariy and Level Shifs wih an Applicaion o Purchasing Power Pariy. Journal of Business and Economic Saisics, Vol. 0, pp Poliis, D.N. and Romano, J.P. (994): The Saionary Boosrap. Journal of he American Saisical Associaion, Vol. 89, pp Sefon, J. A. (998): 'Consumpion and Wealh: an Inernaional Comparison'. Mancheser School, forhcoming. Sefon, J.A. and Weale, M.R. (995): Reconciliaion of Naional Income and Expendiure: Balanced Esimaes of Naional Income for he UK, , Cambridge, Cambridge Universiy Press. Solomou, S. and Weale, M. (997): Personal Secor Wealh in he Unied Kingdom, Review of Income and Wealh. Series 43, pp Sone, J.R.N, D.G. Champernowne and J.E. Meade (942). The Precision of Naional Income Esimaes. Review of Economic Sudies. Vol. 9, pp Wes, K. (988): Asympoic normaliy when regressors have a uni roo. Economerica, Vol. 56, pp Williams, G. (997): Non-Financial, Non-Housing Wealh in he Unied Kingdom Personal Secor: New Esimaes London Business School. Mimeo. Zivo, E. and Andrews, D.W.K. (992): Furher Evidence on he Grea Crash, he Oil-Price Shock, and he Uni Roo Hypohesis. Journal of Business and Economic Saisics, Vol. 0, pp

27 Appendix. The Daa. Daa Descripion The series: - Consumpion Expendiure. (CO) - Consumpion Expendiure in 985 prices. (CO85) - Gross Domesic Produc in 985 prices. (GDP85) - Disposable Income. (DI) - Income from Employmen. (IE) - Income from Self Employmen. (ISE) - Rens Dividends and Ineress. (REDI) - Toal Income. (TI) - Transfers from he Governmen o he Personal Secor. (TGOV) - Oher Transfers o he Personal Secor. (OT) - Public Consumpion in 985 prices. (G85) - Expors in 985 prices. (X85) have been aken from Sefon and Weale for he period , and from CSO sources for he period Populaion. (PO). Daa for he period are from Michell (988); noe ha he figure for N. Ireland for is compued as a percenage of he figure of Ireland populaion, i is assumed ha he share of N Ireland in all-ireland populaion is equal o ha of 922. Daa for are from he CSO Blue Book. Toal Employmen. (TE). Daa for he period are from Feinsein (972), able 57; Daa for are from he CSO Blue Book. Self Employmen (SE). Daa for he periods and were compued as he difference beween oal employmen TE and employees in employmen. The figure of employees in employmen was aken from Chapman and Knigh (953) for he period , able. For he period i was consruced from he HMSO (97) using ables 5 'number of insured employees' and able 55 'Srenghs of he arm forces and women's services'. Figures have been adjused backwards o make hem coheren, as he concep of 'insurable worker' differs for cerain periods. Daa for he period are 26

28 from HMSO (97) able 8. Daa for he period are from he Blue Book, or daasream code ukempsemf. Housing Wealh. (RBW). Daa for from Solomou and Weale (997) Daa for he period from he Blue Book, or Daasream, code ukalln... Personal Secor Impued ren of owner occupied dwellings. (OOI). Daa for he period from he CSO Blue Book. For he period he daa was esimaed. The raio of OOI over housing wealh remains fairly consan over he period The average raio for his period is muliplied by he housing wealh over he period o complee he series. Labour Income. The series is compued as follows: TE LDI = IE TE SE ( τ ) + TGOV + OT where τ = TI DI IE + ISE + REDI OOI all variables defined as above, and τ is he average ax rae on non-gran personal income. This is calculaed assuming ha no ax is levied on impued ren (sricly rue only afer 960) or gran income. The calculaion is made before disposable income is adjused for he depreciaion of consumer durables. Nominal Ineres Rae. (IB). Daa for he period are he average of he maximum and minimum monhly values given in Michell (988, p ). Daa for are he monhly average as given by Michell. Daa for are he monhly average as given by Daasream uk3mhine. Daa for he period are he hree monh London Inerbank raes (end of period), aken from Daasream ukinerb. Ne Personal Wealh. (NW). I refers o Deposis, Equiy Holdings, Housing Wealh, Deb hold by residens, less personal liabiliies (i.e. morgages, credis and loans). Use is made of he series compued by Solomou and Weale (997). Ne Sock of Consumer Durables. (DURW). Esimaes are produced applying a perpeual invenory o he daa on purchases of consumpion goods given by Solomou and Weale (995). The depreciaion raes were raised as compared o hose used by Solomou and Weale (997) so as o give esimaes very similar o Williams (997). 27

29 Depreciaion of Ne Sock of Consumer Durables. (DDURW). The perpeual invenory also allowed us o calculae he depreciaion of he sock of consumpion goods. The calculaions were done in consan prices and revalued using he deflaors appropriae o he specific caegories of goods. Using hose series above, he variables in he paper are consruced as follows: GDP GDP85/PO C G X YD YL RW (CD85 - DUR85 + DDURW)/PO G85/PO X85/PO {DI(CD85/CO) + DDURW}/PO {(LDI/PO)(CD85/CO)}/PO (NW(CD85/CO) + DURW)/PO r IB-{(CO CD85 - )/(CO - CD85 )}00 π ln(co /CD85 )-ln(co - /CD85 - ) The Esimaion of Missing Observaions for he War Years We consider he following srucural ime series model: y = Z α + ε where: d Iw M0 M0 M0 M0 d ' µ M0 0 0 µ α β β = = ' M ' ϕ M0 0 0 ρcosλc ρsin λc ϕ ϕ M 0 0 ρsin λ ρcosλ ϕ M0 η + ξ. κ κ ' 0 c c Z is an vecor, Z ( z ) = 0 0 where z is a 7 vecor wih elemens z k for k =,2,,7 where he elemens z k are zero excep for z,939, z 2,940, z 3,94, z 4,942, z 5,943, z 6,944 and z 7,945 which ake he value. The model is an inervenion model wih a 7 vecor of dummies d o remove he ransiory effec of he war years, wih a separae dummy for each war year. I w is a 7 7 ideniy marix and M 0 a 7 vecor of zeros. This model is in effec ha suggesed by Harvey and Jaeger (993), augmened o accoun for he war years. 28

30 Each series, y o which he model is applied, is assumed o have a rend componen µ, a cyclical componen ϕ, an irregular componen ε and he seven dummies d. The noise componens ε, η, ξ κ and κ 2 are NID( 0 ) 2 (, ) 2 2 σ z NID( 0, σ η ), NID( 0 ),, 2 σ ξ NID( 0 ),,,σ κ and NID 0 σ κ respecively. I is furher assumed ha he saisical properies of κ are he same as hose of κ. The parameer ρ akes a value ha is inside he uni circle, and λ c is a facor ha gives he frequency of he cycle in radians. The rend componen in he model above follows an ARIMA(0, 2, ) process. Noe however ha if σ ξ = 0 he rend reduces o a random walk wih drif; and if boh σ ξ and σ η are zero hen he rend is deerminisic. Hence i encompasses boh uni roo processes and deerminisic rends. The model is a ime varying coefficiens models in sae space form and can be esimaed using he Kalman Filer (see Harvey, 989). We use his model o filer boh hose series which are direcly required for our sudy of consumer spending, and also hose which are poenial insrumenal variables. Table A presens he esimaion resuls of filering he war from all series. Noe ha here is a minor exension compared o he model presened above. Namely, an exra dummy variable has been added o he series of governmen expendiure for 946. This maches beer wih he daa, as governmen expendiure coninued o be very high in

31 Table A. Esimaion Resuls of he Srucural Time Series Models. All series in 985 prices Series GDP G X YL r C YD RW π a ( ) ρ (0.0206) λ (0.0380) σ ε ( ) σ ξ ( ) σ -2 η ( ) σ κ ( ) dum ( ) dum ( ) dum ( ) dum ( ) dum ( ) dum ( ) dum ( ) ( ) ( ) ( ) ( ) ( ) 0.06 ( ) ( ) ( ) ( ).426 ( ).380 ( ).457 ( ).083 ( ) ( ) dum ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.00) (0.006) (0.007) ( ) ( ) ( ) 0.86 (0.085) (0.0265) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.3002) (0.078).4063 (0.0406) (0.09).7498 (0.0466) ( ) (0.0498) (0.3789) (0.4203) (0.477) (0.44) (0.423).7938 (0.487).3900 (0.3737) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ( ) (0.000) (0.068) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).0836 (0.068) ( ) ( ) ( ) ( ) ( ) (0.002) (0.08) (0.07) (0.007) ( ) ( ) ( ) (0.027).2449 (0.09) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) π/λ Wald Tes p.v. [0.000] [0.000] [0.000] [0.579] [0.002] [0.000] [0.64] [0.386] [0.000] 30

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