Volume 31, Issue 1. Pitfall of simple permanent income hypothesis model

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1 Volume 31, Issue 1 ifall of simple permanen income hypohesis model Kazuo Masuda Bank of Japan Absrac ermanen Income Hypohesis (hereafer, IH) is one of he cenral conceps in macroeconomics. Single equaion version of IH is ofen appeared in exbooks and academic papers. Bu, even in single equaion version of IH Romer(2006) suggesed, o ge economic insighs from esimaion, we need o consider he addiional income deerminaion equaion and hen we can' ignore Simulaneous equaions bias. In his noe, we examine his Simulaneous equaions bias effec heoreically and empirically. Our resuls sugges ha ignoring his bias will lead o he wrong esimaes and conclusion. More aenion should be given on simulaneous equaions approach. Conac: Hongokucho, Nihonbashi, Chuo-ku, okyo , Japan. hone: Fax: kazuo.masuda@boj.or.jp We appreciae commens from anonymous referee. he views expressed in his paper are hose of he auhor and do no necessarily reflec he official views of he Bank of Japan. Also, all he remaining errors belong o he auhor. Ciaion: Kazuo Masuda, (2011) ''ifall of simple permanen income hypohesis model'', Economics Bullein, Vol. 31 no.1 pp Submied: Sep ublished: January 04, 2011.

2 1. Inroducion As Friedman(1957) suggesed, ermanen Income Hypohesis (hereafer, IH) is one of he cenral conceps in macroeconomics. And, some exbooks inroduce and explain he simple, single equaion version of IH, and many papers wih his model are published in academic journals (see Romer[2006], DeJuan and Seaer[2006], for example). Bu, are he resuls brough by single equaion IH consisen wih formal economeric mehodology? o be more precise, does his single equaion version of IH successfully bring he consisen esimaor when we consider he simulaneous relaion beween necessary variables such as consumpion and income in IH? In his noe, we examine he effec of Simulaneous equaions bias on single equaion version of IH heoreically and empirically. Simulaneous equaions bias is he ypically classical concep in economerics, and is known o violae he classical assumpion of OLS esimaors (he independence of regressors from he disurbance erm) as many economerics exbook such as Johnson and DiNardo(1997) shows. In such a case, he applicaion of OLS will give biased and inconsisen esimaes. his noe is organized as follows. In secion 2, afer we inroduce he single equaion version of IH, we examine is relaion o Simulaneous equaions bias and show he classical soluion, Insrumenal Variable mehod. In secion 3, we show he empirical evidence wih OLS and IV, and discuss how simulaneous equaions bias is imporan pracically. And in secion 4, we summarize our conclusion. 2. IH, Simulaneous Equaions Bias and Is Classical Soluion (1) exbook Model Single Equaion Version of IH his secion is based on Romer(2006), and DeJuan and Seaer(2006). Suppose he nex simple IH model. Consumpion is equal o permanen income: C and income equaion has 2 pars: permanen income par and ransiory income par, permanen income.. ransiory income reflecs differences of curren income from In consumpion equaion, consumpion is affeced by curren income: C u, and, are parameers and u follows Gaussian whie noise wih mean 0 and variance 2 u. Noe ha he assumed saisics of ransiory income is a mean 0, uncorrelaed wih permanen income and E ( ) 0 u holds for any since is assumed o be exogenous. So, income is deermined by C (Clearly, his model is simulaneous equaions model, as we discuss

3 soon afer his paragraph). hen, from he well-known resul of OLS regression, in he special case of univariae regression such as our consumpion funcion, as Romer(2006) shows, we have ˆ (1 ), ˆ ˆ Var( ). ˆ is 1 if here is no Var( ) Var( ) ransiory income and Var( ) is zero. his is he single equaion version of ermanen Income Hypohesis. Bu, as long as we use he above simulaneous equaion model o derive he resul of single equaion version of IH, his simple resul is wrong because of he exisence of Simulaneous equaions bias. Nex, we show he evidence of Simulaneous equaions bias heoreically. (2) Effec of Simulaneous Equaions Bias and Is Classical Soluion We follow Ban e al.(2006) o derive he nex resul. 1 2 A firs, suppose ha plimi ( ) Var( ). hen, he seup of model in his secion (1) leads o ˆ Var( u) plimi (1 ). Var( ) Var( u) 1 his resul shows he exisence of Simulaneous equaions bias in he above model. In such a case, i is widely known ha he insrumenal variable is effecive. In our model, insrumenal variable resolves his bias and we ge consisen ˆ. In shor, he inerpreaion of coefficien based on his secion (1) does no hold. In he following secion, we invesigae how large his simulaneous equaion bias is and how i will lead he wrong conclusion based on real daa. 3. Empirical Evidence (1) Daa Descripion Daa are Household consumpion expendiure (including Non-profi insiuions serving households) and Gross Domesic roduc (GD) of Japan, UK and US a consan 1990 price in naional currency from C1970 o C2008, downloaded from Naional Accouns Main Aggregaes Daabase in he websie of Unied Naions Saisics Division. (2) Esimaion Resuls Since all of hese daa show uni roo by augmened Dickey-Fuller es, for OLS and IV esimaion, we ake he firs order difference of independen, dependen and insrumenal variables. Before ha, we need o separae he permanen income and

4 ransiory income from income daa. o do his, we use Hodrick-resco filer wih he smoohing parameer lambda 100. Hodrick-resco filer disinguishes smoohed and cycle series, and we use he firs order difference of cycle series as insrumenal variable. Noe ha cycle series of Hodrick-resco filer has mean zero and he problem is wheher i is exogenous from error erm in consumpion equaion or no. he esimaion resul is as follows. Esimaed Bea wih OLS and IV OLS IV Wald es(null Hyp.:IV esimaes is no differen from OLS esimaes) Japan UK US Rejec Accep Rejec Noe: 1. Esimaes of OLS and IV are in 1s row, and heir sandard errors are in 2nd row of each counry. 2. 2nd row of Wald es column in each counry shows -value. Accep/Rejec decision is judged a 5% significance level. he Wald es resul in he above able, in which null hypohesis is ha INS is equal o, shows ha we ge wrong resuls in Japan and US cases if we use OLS. See he nex able. his able shows he weak exogeneiy es (Wu-Hausman es) resul of our cycle series. Weak Exogeneiy es Wu-Hausman es Saisics 5% Cri. Value of χ(1) Japan UK US he null hypohesis of weak exogeneiy es is plimi u 0. hen, his n Weak Exogeneiy es able shows, a leas in Japan and US, he null hypohesis is rejeced. In oher words, he assumpion E ( u ) 0 of our IV esimaes of Japan and US is successfully saisfied. And, our IV esimaes of Japan and US cases is valid. On he oher hand, regarding for UK case, we can accep OLS resul. hen, how

5 should we inerpre hese resuls? Firs, subsiuing C for in consumpion equaion, we have C ) INS ( C u. And solving for C, hen we have he following equaliy: esimaed INS 1 INS C INS 1 wih OLS and IV. INS 1 INS INS 1 1 INS u. herefore, in our model, he is used o calculae he sensiiviy of consumpion o ransiory income,. his sensiiviy is calculaed in he following able, Esimaed Sensiiviy Esimaed Sensiiviy wih OLS and IV OLS IV Japan UK US he resul in Esimaed Sensiiviy wih OLS and IV means ha he sensiiviy of consumpion o ransiory income by IV esimae is higher han ha by OLS, in Japan and US. In shor, he rue sensiiviy is underesimaed by OLS. And, Japan is relaively less sensiive o ransiory income, while US is relaively more sensiive o ransiory income. If we only use he OLS resuls, we have he wrong conclusion ha UK is mos sensiive o ransiory income, bu is sensiiviy is slighly higher han US. However, if we use he IV esimaor, we reach he rue resul ha US is mos sensiive, and is sensiiviy is apparenly higher han UK. Acually, our IV esimaes resul is consisen wih Campbell and Mankiw(1991). Also wih IV, hey esimae he share of Rule of humb consumers in some counries who do no follow ermanen Income Hypohesis and decides heir consumpion based on heir curren income. hey repor ha his ype s share is 0.351, and in US, UK and Japan. Implicaion of heir resul is ha consumpion depends on ransiory income in some degree and is magniude is ranked wih he following order: US, UK and Japan. his conclusion is consisen wih our resul considering Simulaneous equaions bias. Clearly, hese resuls show he exisence of Simulaneous equaions bias and i should no be ignored. One imporan problem of Simulaneous equaions bias is ha i will affec esimaed in boh posiive and negaive direcions and we canno predic his direcion in advance (see secion 2[2]). For example, DeJuan and Seaer(2006) ries o measure differen s by IV assuming measuremen errors in

6 variables which is a differen seing from us 1. Wihou such careful consideraions, we may ge wrong esimaes because of simulaneous equaions bias as we show in his noe. herefore, we canno ignore his bias heoreically and empirically. Noe ha, in our resul, however, IV mehod is no imporan in UK case since our esimaed ransiory income, cycle series of Hodrick-resco filer, does no show weak exogeneiy in UK case. his fac may sugges ha we should consruc ransiory income more rigorously by superior saisical mehod o Hodrick-resco filer or larger scale simulaneous equaions model o include he relaionship beween errors in consumpion funcion and ransiory income in UK. However, in his noe, we sop exploring beer mehods since our resul is sufficienly robus and exploring beer mehods surely leads o oher differen opics and obscure our aim. 4. Conclusion Our resuls sugges ha Simulaneous equaions bias is imporan boh heoreically and empirically. We should ake much care of reaing single equaion version of ermanen Income Hypohesis. Ignoring his bias will lead o he wrong conclusion, paricularly when measuring is he main purpose as we see in his noe. Furhermore, we should no limi our aenion only o ermanen Income Hypohesis. Our conclusion abou single equaion version of IH is only one suggesion, and here may be many oher single equaion examples. More aenion should be given on simulaneous equaions approach. References [1] Ban, Kanemi, Jiro Nakamura, and Naosumi Aoda, Economerics, uhikaku, 2006 (in Japanese). [2] Campbell, John., and Gregory Mankiw, he Response of Consumpion o Income A Cross Counry Invesigaion, European Economic Review 35, pp , [3] DeJuan, Joseph., and John J. Seaer, A Simple es of Friedman s ermanen Income Hypohesis, Economica, 73, pp.27-46, [4] Friedman, Milon, A heory of he Consumpion Funcion, rinceon Universiy ress, [5] Johnson, Jack, and John DiNardo, Economeric Mehods Fourh Ediion, McGraw Hill, [6] Romer, David, Advanced Macroeconomics hird ediion, McGraw Hill, In our case, o be more precise, we consider SLS and simulaneous equaions model. Bu, since heoreically IV includes SLS, we mainly use he word IV in his noe.