Universiy of Preoria Deparmen of Economics Working Paper Series Is he Permanen Income Hypohesis Really Well-Suied for Forecasing? Rangan Gupa Universiy of Preoria Emmanuel Ziramba Universiy of Souh Africa Working Paper: 2009-09 March 2009 Deparmen of Economics Universiy of Preoria 0002, Preoria Souh Africa Tel: +27 12 420 2413 Fax: +27 12 362 5207
Is he Permanen Income Hypohesis Really Well-Suied for Forecasing? Rangan Gupa * and Emmanuel Ziramba # Absrac This paper firs ess he resricions implied by Hall s (1978) version of he permanen income hypohesis (PIH) obained from a bivariae sysem of labor income and savings, using quarerly daa over he period of 1947:01 o 2008:03 for he US economy, and hen uses he model o forecas changes in labor income over he period of 1991:01 o 2008:03. Firs, our resuls indicae he overwhelming rejecion of he resricions on he daa implied by he PIH. Second, we found ha, when compared o univariae and bivariae versions of classical and Bayesian Vecor Auoregressive (VAR) models, he PIH model, in general, is ouperformed by all oher models in erms of he average RMSEs for one- o eigh-quarers-ahead forecass for he changes in labor income. Finally, as far as forecasing is concerned, we found he mos igh Gibbs sampled univarie Bayesian VAR o perform he bes. In sum, we do no find evidence for he US daa o be consisen wih he PIH, neiher does he PIH model perform beer relaive o alernaive aheoreical models in forecasing changes in labor income over an ou of sample horizon ha was characerized by high degree of volailiy for he variable of ineres. JEL Classificaions: E17, E21, E27, E37, E47. Keywords: Permanen Income Hypohesis; Forecas accuracy; Gibbs Sampling 1. Inroducion This paper firs ess he resricions implied by Hall s (1978) version of he permanen income hypohesis (PIH) obained from a bivariae sysem of labor income and savings, using quarerly daa over he period of 1947:01 o 2008:03 for he US economy, and hen uses he model o forecas changes in labor income over he period of 1991:01 o 2008:03. The forecas performance of he PIH model is compared using he Roo Mean Squared Error for one o eigh-quarers-ahead forecass generaed from he univariae and bivariae versions of boh classical and Bayesian varians of he Vecor Auoregressive (VAR) model. In addiion o his, given ha he period of 1991:01 o 2008:03 was characerized by high levels of variaion in he changes in labor income, as indicaed in Figure 1, we also use a Gibbs-sampled version of he Bayesian VAR (BVAR) model characerized by heeroscedasic disurbances o capure he nonconsan variance in he daa over he ou-of-sample horizon. [INSERT FIGURE 1] * To whom correspondence should be addressed. Associae Professor, Deparmen of Economics, Universiy of Preoria, Preoria, 0002, Souh Africa, Email: Rangan.Gupa@up.ac.za. Phone: +27 12 420 3460, Fax: +27 12 362 5207. # Senior Lecurer, Deparmen of Economics, Universiy of Souh Africa, P.O. Box 392, Preoria, 0003, Souh Africa, Email: zirame@unisa.ac.za, Phone: +27 12 429 4486. 1
The moivaion of our sudy emanaes from he work of Ireland (1995). In his paper, he auhor firs esed for he resricions implied by Hall s (1978) version of PIH, as derived by Campbell (1987), over he period of 1959:01 o 1994:03 and hen used he PIH model o forecas changes in labor income over he period of 1971:01 o 1994:03. Ireland (1995) failed o accep he resricions, bu indicaed ha he PIH mus be judged on is abiliy o forecas beer han alernaive economeric models. In his backdrop, he objecive of his paper is simply o revisi he forecasing abiliy of he PIH model, relaive o sandard alernaive economeric models, over an ou-of-sample horizon characerized by high degree of volailiy in he variable of ineres, namely, he changes in he aggregae labor income, based on longer daa se. To he bes of our knowledge, his is he firs aemp o compare he abiliy of a PIH model in forecasing change in labor income, relaive o alernaive Bayesian models, and in paricular, Gibbs sampled BVARs ha accouns for non-consan variance in he variable of ineres. 1 The permanen income hypohesis has is origins in Irving Fisher s (1907) heory of ineres rae. In he nex secion, we sar of by reviewing Fischer s analysis, followed by Hall s (1978) version of he permanen income hypohesis. Secion 3 presens he empirical esimaes and he es for he resricions implied by he PIH model, following he works of Campbell (1987) and Ireland (1995). Secion 4 oulines he basics of he alernaive models used for he forecasing exercise, while Secion 5 presens he forecasing resuls. Finally, Secion 6 concludes. 2. Tracing he Hisory of PIH 2.1 Fisher s Theory of ineres and he PIH Fisher (1907) considers an iner-emporal household uiliy maximizaion over wo periods. The represenaive household receives income in boh periods makes is consumpion and savings decisions. The household s preferences a he wo ime periods are represened by he indifference curves. The household seeks o maximize uiliy subjec o a budge consrain. The iner-emporal price is represened by he fixed ineres rae (r) which he household faces. The slope of each indifference curve is deermined by he household s marginal rae of ineremporal subsiuion or he rae a which i is willing o exchange goods a ime 1 for goods a ime 0. In order o maximize uiliy he household equaes he is marginal rae of iner-emporal subsiuion o he slope of he budge line (1+r). The poin of angency gives he consumpion pair (c 0, c 1 ) on indifference curve U 0 as he opimal combinaion. The household s savings will be given by s 0 =y 0 -c 0. All income poins along a given budge consrain have he same presen value, equal o PV = y 0 y1 + (1 + r) (1) One implicaion of Fisher s heory is ha he household s consumpion choice depends only on 1 See Rao (2005) and Rao and Sharma (2007) for wo recen sudies on he applicaion of Hall s (1978) version of PIH using daa from Fiji, and Fiji and Ausralia respecively. Also refer o Dejuan e al. (2004) for a direc es of he PIH applied o he US economy. 2
he presen value of is income pair (y 0,y 1 ), no on y 0 and y 1 separaely. If second period income increases he presen value of he household s income pair will increase. I shifs he consumer s budge line o he righ. The household will also reduce is ime 0 savings o s 0 = y 0 - c 0. This confirms he second implicaion of Fisher s heory, ha he household saves more when i expecs fuure income o be high. This suggess ha daa on household savings may be used o forecas fuure income. Friedman s (1957) permanen income hypohesis generalizes Fisher s analysis o a model in which here are more han wo periods and he consumer faces uncerainy abou is fuure income prospecs. In his seup he presen value becomes PV Ey = = (1 + r 0 ), (2) where y represens he household s expeced income a ime. Friedman defines he household s permanen income y p as he consan income level ha, if received wih cerainy in each period, has he same presen value as he household s acual income pah. Tha is, y p saisfies y Ey p = PV = = 0 ( 1+ r) = 0 (1 + r). (3) Equaion (3) simplifies o r y p = PV 1 + r (4) Friedman expresses he household s dae 0 consumpion as a funcion of permanen and also says ha he household borrows o increase consumpion oday when i expecs higher income in he fuure. Thus, i saves less when i expecs higher fuure income. 2.2 Hall s version of he PIH Hall (1978) develops a mahemaical version of he permanen income hypohesis ha indicaes how daa on savings can be used o forecas changes in income. He assumes an infiniely- lived represenaive household whose fuure income is uncerain and has expeced uiliy given by E = 0 β u( c ), (5) where E once again represens he household s expecaion, u(c ) measures is uiliy from consuming amoun c a ime, and β is he discoun facor whose value lies beween zero and one. The represenaive household sars period wih asses of value A. I receives ineres on hese 3
asses a rae r equal o y k. I also receives labor income y l during period. A he end of period, he household divides is oal income y = y k +y l beween consumpion c and saving s =y -c. The household is able o borrow agains fuure income labor income a he ineres rae r. Is borrowing is, however, consrained in he long run by he requiremen ha A lim (1 + r) = 0. (6) Equaion (6) can be used o derive equaion (7). c + j y l + j A = j + 1 j = 0 (1 + r) (7)Equaion (7) shows ha he household mus repay any deb owed oday (-A ) by seing fuure consumpion c +j below E c + j E y l + j A =, j + 1 j + 1 j = 0 fuure labor income y l+j. This equaion implies ha ( 1 + r) j = 0 (1 + r) (8) Equaion (8) saes ha he household s curren level of asses A mus be adequae o cover any shorfall beween he presen value of expeced fuure consumpion and he presen value of expeced fuure labor income. The uiliy maximizaion opimaliy condiion is given by equaion (9). This simply represens he angency beween he budge consrain and he household s indifference curve. I says ha he household ses is expeced marginal rae of iner-emporal subsiuion o is gross rae of ineres (1+r). ' ' u ( c ) = β (1 + r) Eu ( c + 1). (9) The analysis assumes ha he uiliy funcion is quadraic and separable in ime and in equilibrium r = β. Given equaion (1) and he assumpions above he Hall equaion is given by: C = +1 C + ε or C = ε (10) where ε is whie noise. Hall (1978) argued ha if expecaions of life-ime income are raional, he permanen income hypohesis implies ha he change in consumpion should be a random walk and his conribuion has become known as he Random Walk Hypohesis. Equaion (10) implies ha all he available informaion is used in period o predic fuure consumpion C +1. Equaion (10) can also be represened as 4
= Ec+1 c (11) c Subsiuing equaion (11) ino equaion (8) yields (12) = ra + r E y l + j j 1 + r j = 0 (1 + r) The righ-hand side of equaion (12) represens he household s permanen income and his equaion implies ha consumpion is deermined by permanen income. Using y k = ra and s = y k +y l -c and denoing he change in labor income by y l =y l -y l-1 we can rewrie equaion (12) as s = E y l + j j j = 1 (1 + r) (13) Equaion (13) shows ha he household s curren savings s equals he presen value of he expeced fuure decline in is labor income. Thus, he household saves more when i expecs fuure decline in income, ha is, negaive values for y l+j. This resul is he second implicaion of he permanen income hypohesis and suggess ha daa on savings help o forecas fuure changes in labor income. 3. Tesing he PIH: Campbell (1987) showed exacly how Hall s version of he PIH can be used o formulae an economeric model for forecasing. Since he PIH implies ha daa on savings will help o forecas fuure changes in labor income, Campbell (1987) sared wih a bivariae vecor auoregression (VAR) for change in labor income ( y l ) and savings (s ) of he following form:, (14) where a(l)= a 1 +a 2 L+a 3 L 2 + +a p L p-1, L is he lag operaor, and u 1 and u 2 are serially uncorrelaed errors. Campbell (1987) hen showed ha he relaionship beween savings and fuure labor income ha was idenified by Hall ranslaes ino a se of parameer resricions which mus hold on he VAR, and hey ake he following form 2 : a 1 = c 1,..,a p = c p, d 1 -b 1 = (1+r), b 2 = d 2,.,b p = d p. (15) 2 Please refer o Appendix A of Ireland (1995) for furher deails. 5
Using quarerly daa, obained from he Bureau of Economic Analysis (BEA), for he period 1947:1 o 2008:3, and following Ireland (1995) o generae he labor income and savings variable 3, we esimae he VAR as specified in equaion (14), boh wih and wihou he PIH resricions. The esimaed models include hree lags of each variable on he righ-hand side. 4 [INSERT TABLE 1] In Table 1, panel (b) shows he equaion of he labor income growh when permanen income consrains (15) are imposed on he VAR. Following Ireland (1995) he esimaes assume r = 0.01. Panel (a) shows he unconsrained equaion for labor income growh. In boh cases, he marginally negaive sum of he coefficiens on lagged savings indicaes ha a decrease in savings ranslaes ino a forecas of faser income growh, as implied by he permanen income hypohesis. An F-es rejecs (in boh cases) he null hypohesis ha savings daa do no help o forecas fuure income growh; he coefficiens on he lags of savings are joinly significan a he 99 percen confidence level. Though, saisically significan, given ha he sum of coefficiens adds upo values of for he unconsrained and consrained models respecively, he role played by lagged savings in defining he process of labor income growh is economically insignifican. Noe ha he coefficiens of he consrained equaion is quie similar o hose of he unconsrained equaion, providing mild evidence of he daa being consisen wih he permanen income hypohesis. However, a saisical es rejeced he resricions in equaion (15) a he 95 percen confidence inerval. 5 Ireland (1995) argued ha he permanen income hypohesis mus be judged on is abiliy o forecas he daa beer han alernaive models. Thus, he nex secion discusses he alernaive models used in forecasing he changes in labor income, while he following secion is devoed o he comparison of he PIH model wih he oher models in forecasing he change in he labor income. 4. Alernaive Models: The alernaive models which were used for forecasing he changes in he labor income includes he univariae and bivariae version of he VAR, BVAR and he Gibbs sampled BVAR. In wha follows, we presen a brief ouline of hese models. 6 VAR models, hough aheoreical, are paricularly useful for forecasing purposes. An unresriced VAR model, as suggesed by Sims (1980), can be wrien as follows: y = C + A( L) y + ε (16) 3 Please refer o Appendix B of Ireland (1995) for furher deails on he daa and he mehodology followed o generae he wo variables of our ineres. 4 The choice of 3 lags is based on he unanimiy of he Akaike informaion crierion (AIC), he final predicion error (FPE) crierion and he Hannan-Quinn (HQ) informaion crierion applied o a sable VAR esimaed wih he wo variables of concern. Noe, sabiliy, as usual, implies ha no roos were found o lie ouside he uni circle. 5 King (1995) poined ou ha formal hypohesis ess seldom fail o rejec he implicaions of deailed mahemaical models, and, hence, he rejecion of he resricions of Hall s (1978) model, should no be surprising. 6 The discussion in his Secion relies heavily on he discussion available in LeSage (1999) and Gupa and Sichei (2006), Gupa (2006, 2007). 6
where y is a ( n 1 ) vecor of variables being forecased; A(L)is a ( n n) polynomial marix in he 2 p backshif operaor L wih lag lengh p, i.e., A(L) = AL 1 + AL 2 +... + ApL ; C is a ( n 1 ) vecor of consan erms, and ε is a ( n 1 ) vecor of whie-noise error erms. The VAR model uses equal lag lengh for all he variables of he model. One drawback of VAR models is ha many parameers are needed o be esimaed, some of which may be insignifican. This problem of overparameerizaion, resuling in mulicollineariy and loss of degrees of freedom leads o inefficien esimaes and large ou-of-sample forecasing errors. One soluion, ofen adaped, is simply o exclude he insignifican lags based on saisical ess. Anoher approach is o use near VAR, which specifies unequal number of lags for he differen equaions. However, an alernaive approach o overcome his overparameerizaion, as described in Liermann (1981), Doan e al (1984), Todd (1984), Liermann (1986), and Spencer (1993), is o use a BVAR model. The Bayesian mehod imposes resricions on hese coefficiens by assuming ha hey are more likely o be near zero han he coefficiens on shorer lags. However, if here are srong effecs from less imporan variables, he daa can override his assumpion. The resricions are imposed by specifying normal prior disribuions wih zero means and small sandard deviaions for all coefficiens wih he sandard deviaion decreasing as he lags increase. The excepion o his is, however, he coefficien on he firs own lag of a variable, which has a mean of uniy. Lierman (1981) used a diffuse prior for he consan. This is popularly referred o as he Minnesoa prior due o is developmen a he Universiy of Minnesoa and he Federal Reserve Bank a Minneapolis. Formally, as discussed above, he Minnesoa prior means ake he following form: 2 2 β ~ N(0, σ )and β ~ N(0, σ ) (17) i β i j β j where β i denoes he coefficiens associaed wih he lagged dependen variables in each equaion of he VAR, while β j represens any oher coefficien. In he belief ha lagged dependen variables are imporan explanaory variables, he prior means corresponding o hem are se o uniy. However, for all he oher coefficiens, β j s, in a paricular equaion of he VAR, a prior mean of zero is assigned, o sugges ha hese variables are less imporan o he model. 2 2 The prior variances σ β i and σ β j, specify uncerainy abou he prior means β i = 1, and β j = 0, respecively. Because of he overparameerizaion of he VAR, Doan e al. (1984) suggesed a formula o generae sandard deviaions as a funcion of small numbers of hyperparameers: w, d, and a weiging marix f(i, j). This approach allows he forecaser o specify individual prior variances for a large number of coefficiens based on only a few hyperparameers. The specificaion of he sandard deviaion of he disribuion of he prior imposed on variable j in equaion i a lag m, for all i, j and m, defined as S(i, j, m), can be specified as follows: ˆ σ i Si (, jm, ) = [ w gm ( ) f(, i j (18) )]ˆ σ j d wih f(i, j) = 1, if i = j and k ij oherwise, wih ( 0 k ij 1), g(m) = m, d > 0. Noe ha ˆ σ i is he esimaed sandard error of he univariae auoregression for variable i. The raio ˆ σ / ˆ i σ j scales he variables so as o accoun for differences in he unis of measuremen and, hence, causes 7
specificaion of he prior wihou consideraion of he magniudes of he variables. The erm w indicaes he overall ighness and is also he sandard deviaion on he firs own lag, wih he prior geing igher as we reduce he value. The parameer g(m) measures he ighness on lag m wih respec o lag 1, and is assumed o have a harmonic shape wih a decay facor of d, which ighens he prior on increasing lags. The parameer f(i, j) represens he ighness of variable j in equaion i relaive o variable i, and by increasing he ineracion, i.e., he value of k ij, we can loosen he prior. 7 Following Gupa (2007), we choose 0.1, 0.2 for he overall ighness (w) and 1 and 2 for he harmonic lag decay parameer (d). Moreover, as in Dua and Ray (1995), we also repor our resuls for a combinaion of w = 0.3 and d = 0.5. Finally, a symmeric ineracion funcion, f(i, j), is assumed wih k ij = 0.5, as in LeSage (1999). Tradiionally, The Bayesian varians of he classical VARs are assumed o have an error srucure ha follows he Gauss-Markov assumpions and is esimaed using Theil's (1971) mixed esimaion echnique, which involves supplemening he daa wih prior informaion on he disribuion of he coefficiens. In an arificial way, he number of observaions and degrees of freedom are increased by one, for each resricion imposed on he parameer esimaes. The loss of degrees of freedom due o over- parameerizaion associaed wih a VAR model is, herefore, no a concern in he BVAR model. However, he esimaion of he BVARs using Gibbs sampling allows us o incorporae ouliers or non-consan variances. Noe, wih heeroscedasic errors, i.e., 2 ε ~ N(0, σ V), V = diag( v1, v2,..., vn ) wihv being a n nmarix, wih he relaive variance erms v1, v2,..., v n assumed o be fixed bu unknown, we could have used generalised leas squares o implemen Theil s (1971) mixed esimaion. Bu i is ofen he case ha wih he Gibbs sampling mehodology, complex esimaion problems, such as hese, are simplified considerably by condiioning on unknown parameers, by assuming ha hese values are known. 8 Moreover, as Kadiyala and Karlsson (1997) poins ou, amongs differen ways of carrying ou he Mone Carlo inegraion, he Gibbs sampling algorihms are less adversely affeced by model size, and hence, performs no worse han oher imporan sampling mehods. 9 5. Evaluaion of Forecas Accuracy: Table 2 repors on he forecasing performance of he univariae and bivariae versions of he VAR and BVAR, wih he laer also including he Gibbs sampled models 10, relaive o he PIH model. We firs esimaed all he models including he consrained VAR wih daa from 1947:1 up o 1990:4 and hen used he models o generae ou-of sample forecass for he change in labor income for one- o eigh-quarers-ahead. Recall ha, he differen models are esimaed wih hree lags for each variable(s). Since we use hree lags, he iniial hree quarers of he sample, 1947:01 o 1947:03, are used o feed he lags. We generae dynamic forecass, as would naurally be achieved in acual forecasing pracice. The models are re-esimaed each quarer 7 For an illusraion, see Dua and Ray (1995). 8 See secion 3 for furher deails. 9 See Kadiyala and Karlsson (1997) and he references cied herein for furher deails. 10 The parameer esimaes from he Gibbs sampled BVARs are based on 1000,000 ieraions, wih iniial 100,000 discarded o preven issues of iniializaion. Based on he 1000,000 random draws for he 7 parameers in each equaion, hoss of alernaive convergence ess, oulined in LeSage (1999), were carried ou o ensure ha he sampler converges in he limi. The convergence ess have no been repored o save space. However, all he resuls are available upon reques from he auhors. 8
over he ou-of-sample forecas horizon in order o updae he esimae of he coefficiens, before producing he 8-quarers-ahead forecass. This ieraive esimaion and 8-seps-ahead forecas procedure was carried ou for 71 quarers, wih he firs forecas beginning in 1991:01. This experimen produced a oal of 71 one-quarer-ahead forecass, 71-wo-quarers-ahead forecass, and so on, up o 71 8-sep-ahead forecass. The RMSEs 11 for he 71, quarer 1 hrough quarer 8 forecass are hen calculaed for he change in aggregae labor income. The model ha produces he lowes average value for he RMSE is seleced as he opimal model for forecasing he change in labor income. In Table 2, we compare he raios of he RMSEs of one- o eigh-quarers-ahead ou-of-sampleforecass for he period 1991:01 o 2008:03, generaed by he abovemenioned alernaive models, relaive o he PIH model. A his sage, a few words need o be said regarding he choice of he evaluaion crierion for he ou-of-sample forecass generaed from Bayesian models. As Zellner (1986: 494) poins ou he opimal Bayesian forecass will differ depending upon he loss funcion employed and he form of predicive probabiliy densiy funcion". In oher words, Bayesian forecass are sensiive o he choice of he measure used o evaluae he ou-of-sample forecas errors. However, Zellner (1986) poins ou ha he use of he mean of he predicive probabiliy densiy funcion for a series, is opimal relaive o a squared error loss funcion and he Mean Squared Error (MSE), and hence, he RMSE is an appropriae measure o evaluae performance of forecass, when he mean of he predicive probabiliy densiy funcion is used. This is exacly wha we do below in Table 2, when we use he average RMSEs over he one- o eigh-quarer-ahead forecasing horizon. The conclusions on he forecas performance of alernaive models for he change in aggregae labor income, based on he relaive average oneo eigh-quarers-ahead RMSEs, from Table 2 can be summarized as follows: (i) (ii) (iii) In general, unlike Ireland (1995) 12, wih he excepions of he univariae BVAR under w = 0.3, d = 0.5, he bivariae BVAR wih w = 0.1, d = 0.1 and w = 0.1, d = 2.0, and he Gibbs sample bivariae BVAR under w = 0.2, d = 1.0, he PIH model is ouperformed by all oher models in erms of average RMSEs for forecasing he changes in labor income; 13 Amongs he univariae models, he Gibbs sampled univariae BVAR wih he mos igh priors i.e., w = 0.1, d = 2.0 performs he bes, while, amongs he bivariae models, he Gibbs sampled BVAR, wih he second mos igh prior srucure (w = 0.2, d = 2.0) sands ou. However, he opimal Gibbs sampled univariae BVAR ouperforms he opimal Gibbs sampled bivariae BVAR; Overall, he resuls ends o indicae he imporance of inroducing a heeroscedasic error srucure for forecasing changes in labor income over an ou-of-sample horizon ha was characerized by high degree of volailiy in changes in labor income. Moreover, he fac ha he mos igh Gibbs sampled univarie BVAR is he sand ou performer, ends o corroborae our findings in Secion 3, where we 11 Noe ha if A + ndenoes he acual value of a specific variable in period + n and F + n equals he forecas made N ( ) 2 1 n n in period for + n, he RMSE saisic equals he following: F + A + N where N equals he number of forecass. 12 Ireland (1995) found he PIH model o consisenly ouperform he classical versions of he univariae and bivariae VARs. 13 Ineresingly, for he one-quarer-ahead forecass he PIH model ouperforms all he univariae and bivariae versions of he VAR and BVAR based on he homoscedasic error srucure, bu is, in urn, ou done by boh he univariae and bivariae versions of he Gibbs sampled BVAR, he excepion being he Gibbs sampled bivariae BVAR under w = 0.2, d = 1.0. 9
indicaed ha he impac of lagged savings on labor income is virually negligible (in magniude, even hough he impac is significan when aken ogeher). 6. Conclusions [INSERT TABLE 2] This paper presens Hall s (1978) version of he PIH obained from a bivariae sysem of labor income and savings, using quarerly daa over he period of 1947:01 o 2008:03 for he US economy, and hen uses he model o forecas changes in labor income over he period of 1991:01 o 2008:03. Our resuls indicae he overwhelming rejecion of he resricions on he daa implied by he PIH. Given ha Ireland (1995) argued ha he permanen income hypohesis mus be judged on is abiliy o forecas he daa beer han alernaive models, we compared he forecas performance of he PIH model relaive o he forecass generaed from he univariae and bivariae versions of boh classical and Bayesian varians VAR model. Realizing ha he period of 1991:01 o 2008:03 was characerized by high levels of variaion in he changes in labor income we also used a Gibbs-sampled version of he BVAR model characerized by heeroscedasic disurbances o capure he non-consan variance in he daa over he ou-ofsample horizon. We found ha, in general, he PIH model is ouperformed by all oher models in erms of he average RMSEs for one o eigh quarers ahead forecass for he changes in labor income, and overall i is he mos igh Gibbs sampled univarie BVAR ha performs he bes. So o conclude, we do no find evidence for he US daa o be consisen wih he PIH, neiher does he PIH model perform beer relaive o alernaive aheoreical models in forecasing changes in labor income over an ou-of-sample horizon ha winessed high volailiy in he changes in labor income. However, i mus be realized, ha he resuls are perhaps an indicaion of he simplisic naure of he PIH model. As Ireland (1995) poins ou Hall s (1978) model makes very resricive assumpions abou he ineres rae and he household s uiliy funcion. The saisical rejecion of he consrains and he poor forecasing performance may hus be reflecing he failure of hese addiional assumpions o hold in he daa, raher han a more general inapplicabiliy of he permanen income hypohesis iself. Fuure research would hus aim a developing more profound models of PIH along he lines of Hansen and Singleon (1982), Campbell and Mankiw (1989), and more recenly, Corradini (2005). References Campbell, J.Y., (1987) Does saving anicipae declining labor income? An alernaive es of he permanen income hypohesis, Economerica, vol. 55 (November), pp. 1249-73. and Mankiw, N.G., (1989) Consumpion, income and ineres raes: reinerpreing he ime series evidence, Working Paper No. 2924, Naional Bureau of Economic Research, Cambridge, MA. Corradini,R. 2005. "An Empirical Analysis of Permanen Income Hypohesis Applied o Ialy using Sae Space Models wih non zero correlaion beween rend and cycle," Economerics 0509009, EconWPA. 10
Dejuan, J.P., Seaer, J.J., and Wirjano, T.S., (2004) A direc es of permanen Income Hypohesis wih an Applicaion of he U.S. Saes, Journal of Money Credi and Banking, vol. 36, No. 6, pp. 1091-1103. Doan, T. A., Lierman, R. B. and Sims, C. A. (1984). Forecasing and Condiional Projecions Using Realisic Prior Disribuions. Economeric Reviews, vol. 3, 1-100. Dua, P. and Ray, S. C. (1995). A BVAR Model for he Connecicu Economy. Journal of Forecasing, vol. 14, 167-180. Fisher, I. (1907) The Rae of Ineres. New York: MacMillan Company. Friedman, M., (1957) A Theory of Consumpion Funcion, Princeon, NJ: Princeon Universiy Press. Gupa, R., (2007) Forecasing he Souh African economy wih GIBBS sampled BVECMs, Souh African Journal of Economics, Volume 75(4), 631-643. Gupa, R., (2006) Forecasing he Souh African economy wih VARs and VECMs, Souh African Journal of Economics, Volume 74(4), 611-628. Gupa, R. and Sichei, M.M., (2006) A BVAR Model of he Souh African Economy, Souh African Journal of Economics, vol. 74(3), pp. 391-409. Hall, R.E., (1978) Sochasic Implicaions of he Life Cycle-Permanen Income Hypohesis: Theory and Evidence, Journal of Poliical Economy, pp.971-987. Hansen, L. P. and Singleon, K. J. (1982) Generalized Insrumenal Variables Esimaion of Nonlilear Raional Expecaions Models, Economerica, vol. 50, pp. 1269-86. Ireland, P.N., (1995) Using he Permanen Income Hypohesis for Forecasing, Federal Reserve Bank of Richmond Economic Quarerly, volume 81/1, pp. 49-63. Kadiyala, K. R. and Karlsson, S. (1997). Numerical Mehods for Esimaion and Inference in Bayesian VAR- Models, Journal of Applied Economerics, vol. 12, 99-132. King, R.G., (1995) Quaniaive Theory and Economerics. Manuscrip, Universiy of Virginia. LeSage,J.P.,(1999). Applied Economerics Using MATLAB, www.spaial-economerics.com. Lierman, R. B. (1981). A Bayesian Procedure for Forecasing wih Vecor Auoregressions. Working Paper, Federal Reserve Bank of Minneapolis. Lierman, R. B. (1986). Forecasing wih Bayesian Vecor Auoregressions Five Years of Experience. Journal of Business and Economic Saisics, vol. 4 (1), 25-38. Rao, B.B., (2005) Tesing Hall s permanen income hypohesis for a developing counry: he case of Fiji, Applied Economics Leers, 12, 245-248. and Sharma, K. L., (2007) Tesing he permanen hypohesis in he developing and developed counries: A comparison beween Fiji and Ausralia. MPRA paper number 2725, 11
hp://mpra.ub.uni-muechen.de/2725/. Sims, C. A. (1980). Macroeconomics and Realiy. Economerica, vol. 48, 1-48. Spencer, D. E. (1993). Developing a Bayesian Vecor Auoregression Model. Inernaional Journal of Forecasing, vol. 9, 407-421. Theil, H. (1971). Principles of Economerics. New York: John Wiley. Todd, R. M. (1984). Improving Economic Forecasing wih Bayesian Vecor Auoregression. Quarerly Review, Federal Reserve Bank of Minneapolis, Fall, 18-29. Zellner, A.. (1986). A Tale of Forecasing 1001 Series: The Bayesian Knigh Srikes Again. Inernaional Journal of Forecasing, vol. 2, 494-494. 12
Figure 1: Changes in Labor Income Table 1. Esimaed Changes in Labor Income Equaion from he PIH Model: a) Unconsrained Model Y l = 0.000021 +0.216 Y l-1 (0.0802) +0.242 Y l-2 (0.0803) -0.331S -1 +0.204S -2 (0.0619) (0.0829) b) Consrained Model Y l = +0.166 Y l-1 +0.187 Y l-2 0.000012 (0.0803) (0.0804) -0.349S -1 +0.260S -2 (0.0620) (0.0830) Noe: Sandard errors are in parenheses. -0.0119 Y l-3 (0.0619) +0.126S -3 (0.0659) -0.0179 Y l-3 (0.0620) +0.088S -3 (0.0659) 13
Table 2. RMSEs Relaive o PIH Model: QA 1 2 3 4 5 6 7 8 Average UVAR 1.046 0.965 0.977 0.969 0.975 0.971 0.975 0.974 0.981 MVAR 1.014 0.979 0.989 0.984 0.987 0.983 0.997 0.992 0.990 UBVAR 1.013 0.998 1.000 0.998 0.999 0.999 0.998 0.999 1.001 w=0.3, d=0.5 UBVARG 0.005 0.014 0.003 0.006 0.016 0.193 0.420 0.053 0.089 MBVAR 1.025 0.974 0.986 0.980 0.986 0.982 0.997 0.992 0.990 MBVARG 0.815 0.568 0.764 0.554 0.548 1.100 1.088 2.163 0.765 UBVAR 1.08 0.96 0.98 0.97 0.97 0.97 0.97 0.97 0.98 w=0.2, d=1.0 UBVARG 0.03 0.11 0.08 0.09 0.18 0.18 0.41 0.05 0.02 MBVAR 1.05 0.97 0.99 0.98 0.99 0.98 1.00 0.99 0.99 MBVARG 129.50 0.57 2.22 5.54 18.63 61.59 200.66 659.30 121.59 UBVAR 1.165 0.956 0.987 0.963 0.978 0.970 0.975 0.974 0.995 w=0.1, d=1.0 UBVARG 0.136 0.222 0.139 0.249 0.088 0.401 0.321 0.671 0.028 MBVAR 1.146 0.963 1.000 0.974 0.993 0.983 1.001 0.993 1.006 MBVARG 0.592 0.568 0.556 0.554 0.548 1.100 2.719 5.949 1.599 UBVAR 1.078 0.962 0.978 0.967 0.975 0.970 0.974 0.973 0.984 w=0.2, d=2.0 UBVARG 0.124 0.256 0.439 0.897 1.687 3.767 7.553 15.609 3.876 MBVAR 1.046 0.971 0.987 0.978 0.987 0.983 0.998 0.992 0.992 MBVARG 0.592 0.568 0.556 0.554 1.096 0.550 1.088 0.608 0.626 UBVAR 1.165 0.956 0.987 0.963 0.978 0.970 0.975 0.974 0.995 w=0.1, d=2.0 UBVARG 0.002 0.074 0.209 0.180 0.065 0.103 0.110 0.203 0.012 MBVAR 1.146 0.963 1.000 0.974 0.993 0.983 1.001 0.993 1.006 MBVARG 0.814 0.568 0.556 0.762 1.096 0.550 1.088 2.163 0.765 Noes: QA: QUARTERS AHEAD; UVAR: UNIVARIATE VAR; MVAR: BIVARIATE VAR; UBVAR: UNIVARIATE BVAR; UBVARG: GIBBS SAMPLED UBVAR; MBVAR: BIVARIATE BVAR; MVARG: GIBBS SAMPLED MBVAR. 14