preliminary and incomplee Inerpreing Permanen Shocs o Inflaion When Money Growh is Endogenous John W. Keaing * Universiy of Kansas Deparmen of Economics 213 Summerfield Hall Lawrence, KS 6645 web page: hp://www.people.u.edu/~jeaing/ e-mail: jeaing@u.edu phone: (785)864-2837 fax: (785)864-527 Sepember 27, 24 Key Words: superneuraliy, vecor auoregression, idenificaion resricions, moving average represenaion, permanen-ransiory shoc decomposiion JEL Codes: C32, C52, E31, E5 Absrac: Permanen inflaion shocs in VAR models have been used o ry o es for superneuraliy, he hypohesis ha a permanen change in he growh rae of money has no permanen effec on cerain real variables. A permanen increase in inflaion was found o be associaed wih a significan permanen increase in he level of oupu for a number of low inflaion counries. These resuls have been inerpreed as evidence of Mundell-Tobin effecs. However, if cenral ban behavior has made money growh endogenous, ha inerpreaion of he findings is quesionable. This paper shows ha endogenous policy implies hese are downward biased esimaes of he inended effec. An imporan implicaion is ha his evidence can no be explained by reverse causaion, bu i can sill be inerpreed as suppor for Mundell- Tobin effecs. If policy is endogenous, he acual Mundell-Tobin effecs are larger han he esimaed effecs. Anoher ineresing resul is ha endogenous moneary policy does no provide a good explanaion for he finding ha counries wih higher inflaion raes end o have smaller long-run oupu responses o permanen inflaion shocs. Variaion in he effec on oupu of exogenous changes in money growh does, however, serve as a reasonable explanaion for ha empirical finding. In addiion, he cross counry variaion in impulse response of oupu o permanen inflaion shocs sugges ha exogenous changes in money growh have smaller shor-run effecs on oupu when he inflaion rae is higher and/or moneary policy is more endogenous in counries wih higher inflaion. The paper also shows how o exend his analysis of a bivariae sysem o he general muli-variae case, and hen applies his resul o a paricular rivariae srucural VAR model ha was used o assess superneuraliy and Fisher s heory of ineres rae deerminaion. * Thans o seminar paricipans a he Missouri Economics Conference for helpful suggesions on a preliminary version. The auhor assumes all responsibiliy for any errors and omissions. Suppor from he General Research Fund a he Universiy of Kansas is graefully acnowledged.
Inroducion Macroeconomic heory ofen generaes long-run neuraliy proposiions. Long-run neuraliy of exogenous changes in he money soc is a feaure of almos all modern moneary heories. Consequenly, much of he early wor wih srucural VARs used neuraliy assumpions o idenify a model. Shapiro and Wason (1988), Blanchard and Quah (1989) and Gali (1992), for example, assumed aggregae demand shocs have no long-run effecs on oupu. Also, heory is someimes able o show ha a permanen changes in he growh rae of he money supply will have no long-run effec on cerain real variables, When ha is he case, money is said o be superneural. King, Plosser, Soc and Wason (1991) idenified a sysem wih permanen shocs o balanced growh, inflaion, and he real rae of ineres. They allowed inflaion shocs o affec he real rae of ineres, and as a resul hese shocs could influence he percenage of oupu devoed o consumpion and o invesmen. (These effecs were found o be small and saisically insignifican.) An ineresing feaure of heir model is ha i allows inflaion o be affeced in he long-run by balanced growh shocs which hey aribue o permanen changes in produciviy. This ype of response can be moivaed by he assumpion ha he cenral ban allows money growh o permanenly change in response o a supply shoc. However, King e.al. resric inflaion shocs o have no long-run oupu effec. Thus because he assumpion of superneuraliy wih respec o oupu is used o idenify he model, hey were unable o es ha hypohesis abou oupu. Subsequenly, a number of papers (REF) have aemped o esimae he effec on oupu from a permanen change in he growh raes of nominal variables. Superneuraliy is an ineresing hypohesis o es because i does no hold under all srucural assumpions. The resuls are quie varied wih a permanenly higher money growh rae yielding a posiive effec, a negaive effec or no effec on oupu in he long run depending on paricular feaures of he heory. 1
A number of papers use inflaion raher han money growh in empirical models designed o address quesions abou superneuraliy. One reason for his subsiuion is ha here have been serious disagreemens abou which measure is bes o use. The well-received principle ha inflaion in he long run is a moneary phenomenon allows economiss o subsiue inflaion for money growh when formulaing ess of superneuraliy proposiions. This noion receives a fair amoun of empirical suppor from cross-counry sudies ha compare long-run averages of inflaion and money growh as well as ime series sudies ha have found money growh and inflaion are coinegraed. This evidence suppors he hypohesis ha permanen shocs o money growh and inflaion are really one and he same, or if I may borrow from perhaps Milon Friedman s mos famous quoe: Permanen shocs o inflaion are everywhere and always caused by permanen shocs o he growh rae of money. An imporan corollary of his idea is ha permanen shocs o inflaion are exogenous if and only if permanen shocs o money growh are exogenous. As is well documened by Fisher and Seaer (1993) and King and Wason (1997) (following imporan wor by Sargen (1971) and Lucas (19??) ) ha ime series ess of long-run proposiions require permanen shocs of some ind in he sample period. Much of he recen empirical lieraure uses linear models and specifies permanen changes by assuming he daa have a uni roo. Bullard and Keaing (1995), for example, sudied wheher inflaion and real oupu each had a uni roo for a large number of counries. When boh series have a uni roo, hey show how o esimae he long-run effec on oupu from a permanen change in he inflaion rae. They use hese esimaes o infer wheher changes in he money growh rae are non-neural wih respec o oupu in he long run. Their ey idenificaion assumpion is ha permanen changes in inflaion are exogenous. Fisher and Seaer (1993) idenify permanen changes in US money growh as exogenous, and economic heory ells us ha hese wo exogeneiy assumpions are really he same assumpion. Bullard and Keaing found ha he long-run effec of inflaion on oupu is inversely relaed o he 2
in-sample average level of inflaion across counries. For low inflaion counries a permanenly higher rae of inflaion ends o have a significan posiive effec on he level of oupu. However, as he average inflaion rae increases, his esimae becomes zero or negaive, wih he only significanly negaive long-run esimae obained for he counry wih he highes average rae of inflaion. Bullard and Keaing inerpreed heir resuls as being mosly consisen wih superneuraliy, excep primarily for some low inflaion counries where he inflaion shoc permanenly and significanly raised he level of oupu. Ohers have used differen variables o invesigae superneuraliy. Crosby and Oo (2) find evidence of Mundell-Tobin effecs for cerain low inflaion counries by using he capial soc insead of oupu. Rapach (23) uses a model wih hree variables and hree shocs and obains a similar finding for oupu. His model also permis him o es he hypohesis ha inflaion and nominal ineres raes move one for one in he long run. The assumpion ha permanen shocs o inflaion are exogenous is no, however, impervious o criicism. There are various ways ha cenral bans may have reaced o real disurbances. For example, he oil price shocs in he 197s are ofen cied as he source of persisenly high inflaion during ha period. Economic heory shows ha if money growh is held consan, a permanen supply shoc can have a permanen effec on he price level, bu only a ransiory effec on he inflaion rae. (assuming supply shocs affec he level of oupu and no he long-run growh rae of oupu). Hence, money growh mus permanenly change in response o oil price shocs for hese shocs o have permanen effecs on inflaion. Ohers have claimed he Fed pursued a policy of opporunisic disinflaion in he 199s, waiing for beneficial supply shocs ha would allow he cenral ban o disinflae wih less chance of insigaing a recession. There is also a long lieraure on how and why governmens or cenral bans may choose o purchase deb raher han issue i o he public. An adverse supply shoc will cause ax revenues o decline. If a fracion of he consequen budge defici is purchased by he moneary auhoriy money may grow a a faser rae growh and ha could lead o more inflaion evenually. Any of hese policies could 3
permi supply shocs o cause permanen changes in he rae of inflaion. If moneary policy is endogenous and policy reacs o he economy in such a way ha money growh is permanenly affeced by real shocs, hen empirical models associaing permanen changes in inflaion wih exogenous changes in money growh will be of quesionable relevance. In oher words, i is possible ha evidence agains superneuraliy from hese models migh be explained by reverse causaion whereby permanen changes in real oupu cause permanen changes in inflaion, and no he oher way. The fac ha many naions had heir highes poswar inflaion raes following he 197s oil price shocs seems consisen wih a reverse causaion inerpreaion. This paper invesigaes wha can be learned abou economic srucure from permanen shocs o inflaion when money growh is endogenous. I develop a srucural ime series model ha is driven by wo shocs: Exogenous echnology disurbances and exogenous money growh shocs. For nearly all of he analysis his paper assumes ha non-superneuraliy is possible. Permanen exogenous shocs o aggregae supply arise from shocs o he producion funcion, wih changes in he relaive price of energy a paricular source of such disurbances. I assume he cenral ban may allow money growh o rise in response o he decline in real oupu caused by adverse supply shocs. I consruc he saisical model of permanen and ransiory shocs o inflaion under his se of srucural assumpions. I show ha if superneuraliy holds and long-run money growh is endogenous o supply shocs, a permanen posiive shoc o inflaion is associaed wih a permanen decline in he level of oupu. This resul is inconsisen wih he posiive effecs, also nown as Mundell-Tobin effecs, ha have been found in he empirical lieraure. Therefore, superneuraliy combined wih endogenous moneary policy can no explain he finding ha permanen shocs o inflaion are associaed wih a permanen increase in oupu for a number of low inflaion counries. Then I exend he previous analysis by augmening he model of endogenous moneary policy wih non-superneural effecs from changes in he money growh rae. Under hese assumpions, I show 4
ha he esimaed Mundell-Tobin effecs are downward-biased esimaes of he long-run oupu effec of a permanen exogenous increase in he money growh rae. In oher words, if moneary policy has an imporan endogenous componen, hen he long-run effec on oupu of an exogenous permanen increase in he growh rae of money exceeds he Mundell-Tobin effec esimaes in Bullard and Keaing. Exogenous money growh shocs may acually have a sronger long-run effec on oupu han is implied by heir esimaes. Then... A Model of Permanen and Transiory Shocs o Inflaion The bivariae decomposiion of inflaion ino permanen and ransiory shocs can be consruced essenially by replacing oupu wih inflaion in Blanchard and Quah s (1989) decomposiion. The choice of second variable depends on he quesion under invesigaion. Bullard and Keaing (1995) use oupu growh as he second variable because hey wan o deermine if a permanen change in he inflaion rae affecs he level of oupu, paricularly in he long run. While his paper emphasizes heir model, he mehods developed here are suiable for inerpreing empirical resuls from models ha use anoher variable in place of oupu. Crosby and Oo (2), for example, use capial soc as he second variable in a permanen-ransiory shoc decomposiion for inflaion. Furhermore, he appendix shows how he basic mehod can be applied o a VAR model wih an arbirary number of variables, and Secion? applies he mehod o he rivariae srucural VAR used by Rapach (23) o invesigae he effec of permanen shocs o inflaion on oupu and nominal ineres raes. Bullard and Keaing s (1995) bivariae decomposiion of inflaion ino permanen and ransiory shocs can be wrien as: 5
πp πt P π R (L) R (L) u = yp yt T y R (L) R (L) u (1) where = 1-L is he firs difference operaor, i i j = R j L j= R (L) for =π,y and i=p,t, permanen shocs are defined as u P and emporary shocs by u T. Consans and deerminisic funcions of ime have been omied wihou loss of generaliy. This saisical model is idenified by wo assumpions: (i) permanen and ransiory shocs are uncorrelaed: Eu u = P T ; (ii) a emporary shoc has no long-run effec on inflaion: j= R = πt j The saisical model can be wrien as: X = r(l)u P T where X = ( π,y ), u = (u,u ), πp πt R (L) R (L) r(l) = yp yt R (L) R (L) (2) and given ha he shocs are uncorrelaed, i is convenien o normalize he shoc variances o one and hus mae he shoc covariance marix an ideniy: Eu u = Σ = I. Recursive subsiuion allows u us o rewrie he saisical model as: 1 X X Q u = + = (3) 6
This las equaion implies ha: where Q = rj. (4) j= X = u j= r j for =, 1, 2..., -1. (5) By leing go o infiniy, he equaion yields: X lim = rj r(1) u j=, (6) where r(1) is he sum of parameers in r(l). This marix provides he long run effec of each shoc on each variable. From his las equaion, we also see ha he second idenifying assumpion implies: π πt lim R T = j = u j=, (7) which is of course wha i means for a shoc o no have a permanen effec on inflaion. Hence, his second idenifying assumpion maes r(1) a lower riangular marix: RπP r(1) = RyP R yt. (8) Idenifying he Parameers in he Saisical Model To obain he saisical model s coefficiens, he idenifying assumpions are applied o a reduced-form vecor auoregression (VAR): β(l) X = e (9) where e is he vecor of residuals o he inflaion and oupu equaions and β(l) = I - β 1 L - β 2 L 2 -... - β R L R represens VAR coefficiens in which he ideniy marix, I, and each β j for j=1,2,...,r are 2 2 marices 7
and R is he number of lags in he VAR. In general, a VAR represenaion exiss and is unique. The saisical model and he VAR are relaed by wo equaions: β (L) = r r(l) 1, (1) and e = r u (11) which are obained by mapping he saisical model ino he VAR. Defining β(1) as he sum of VAR coefficiens marix, he firs of hese wo equaions implies: β (1) = r r(1) 1. (12) Then defining Σ e as he covariance marix for residuals, he second of hese wo equaions along wih he assumpion of an ideniy covariance marix for shocs o he saisical model implies: Σ =rr. (13) e Combining he las wo equaions we obain he following relaionship for long-run parameers in he saisical model: r(1)r(1) =β(1) Σβ(1) 1 1 e. (14) Given ha r(1) is a riangular marix, is parameers are uniquely deermined as is shown by Hamilon *1994,p.?). The appropriae Cholesy decomposiion of he righ hand side of his equaion would ypically be used o esimae r(1). Given hese long-run parameers, he VAR is used o calculae he dynamic response of each variable o emporary and permanen shocs (from 3 equaions on he previous page): 1 r(l) =β(l) β(1)r(1). (15) Bu raher han use his relaionship o generae he dynamic response of each variable o each ype of 8
shoc, economiss generally ae an easier approach and jus simulae he ime series model. Empirical Evidence on Permanen Shocs o Inflaion The primary focus of Bullard and Keaing (1995) is he long run relaionship beween oupu and inflaion. They esimae values of R yp R π P from VAR models for counries ha saisfy he necessary condiion ha a uni roo in boh inflaion and oupu can no be rejeced. This raio of parameers can be inerpreed as an esimae of he long run effec on oupu from a 1 percenage poin increase in he rae of inflaion. Mos of he empirical resuls are common across counries. Transiory shocs o inflaion cause he price level o permanenly rise, and he level of oupu o permanenly fall. By consrucion his shocs has no long-run effec on inflaion. These effecs are consisen wih a permanen aggregae supply shoc inerpreaion. And permanen shocs o inflaion cause inflaion o be higher, in general. The responses of oupu o shocs ha permanenly increase inflaion yield four ineresing findings. Firs, for mos of he counries here is no saisically significan long-run relaionship beween permanen inflaion shocs and he level of oupu. This finding comes from he counries for which heir VAR mehod is appropriae and also for a number of counries where inflaion has a uni roo bu oupu is rend saionary. Fisher and Seaer s (1993) long-run derivaive is used o inerpre hese es resuls as evidence of superneuraliy. Inuiively, if an economy has permanen shocs o inflaion bu no permanen shocs o oupu, hen he permanen inflaion shocs mus be having no long-run effec on oupu. Since permanen inflaion movemens are he resul of permanen changes in money growh, he absence of a uni roo in oupu would imply ha superneuraliy is a feaure of an economy. Second, he primary excepions o he firs finding are 4 counries wih low average raes of inflaion for which a permanen increase in inflaion has a permanen posiive effec on he level of oupu. There is also one counry ha had a significanly negaive long-run oupu response o a permanen increase in inflaion, 9
and ha counry had he highes average inflaion rae in he sample period. Third, he oupu effec of a permanen inflaion shoc is relaed o he average inflaion rae. For 9 of he 1 counries wih inflaion averaging less han 11 percen, a permanen increase in inflaion has a long-run posiive effec on oupu whereas ha effec is zero or negaive for all 6 counries wih inflaion averaging 15 percen or more. The US is he only low inflaion counry (averaging roughly 5 percen inflaion over he sample period) for which he poin esimae is negaive, alhough i is no saisically significan. And fourh, he dynamic response of oupu o a permanen inflaion shoc is inversely relaed o he average rae of inflaion. As he inflaion rae rises he enire impulse response ends o be lower. In fac, for he highes inflaion counries (wih average inflaion raes of? and?), he oupu response o a permanen increase in inflaion is non-posiive a all poins. Bullard and Keaing inerpre he firs resul as some suppor for superneuraliy. The second resul is inerpreed o mean ha superneuraliy fails o hold in some cases, and in paricular ha a Mundell-Tobin effec is operaional in a number of low inflaion counries. The hird resul can be inerpreed as evidence consisen wih heoreical wor of Azariadis and Smih (1996) in which a Mundell-Tobin effec exiss when inflaion is low bu weaens, possibly disappearing, when he inflaion rae rises beyond a cerain level. Their resul is derived from a heory of credi mare imperfecions. Lagos and Rocheeau (23) obain a similar resul using search heory. Bullard and Keaing do no discuss he fourh resul. Oher evidence...???? A Srucure wih Endogenous Moneary Policy 1
The ey idenifying assumpion for he saisical model is ha permanen inflaion shocs are associaed wih exogenous changes in money growh. However, i is easy o mae he case ha changes in money growh have someimes been endogenous. The narraive hisory of moneary policy (Romer and Romer?) shows how cenral bans have reaced o supply shocs and possibly made inflaion exhibi permanen, or a leas highly persisen changes as a resul. The high inflaion of he 197s and early 198s, for example, is ofen aribued o adverse supply shocs. Economic heory predics ha if a cenral ban holds he growh rae of money consan, a permanen change in oil prices will have a mos a emporary effec on inflaion, alhough he price level would be permanenly higher. Cenral bans apparenly increased he growh rae of money in an aemp o couner balance he negaive oupu effecs resuling from hese adverse supply shocs (REF). Some have argued (REF) ha in he 199s a policy of opporunisic disinflaion was followed whereby a cenral ban would lower inflaion only when doing so would no be oo painful for he real economy. For example, when he economy experiences beneficial supply shocs, a sillful cenral baner migh be able o gradually bring he rae of inflaion down wihou causing a period of negaive or even wea oupu growh. Money growh may also be endogenous when a cenral ban moneizes governmen deb. Aggregae supply shocs cause oupu o fall, reducing ax revenues and forcing he defici o rise. If some porion of he deb is financed by an increase in he money supply and if he deficis persis hen a policy of moneizing deb can lead o a persisenly higher money growh rae. This mechanism for endogenous money growh may be more relevan for less developed economies which have ended o experience he mos severe fiscal problems and also have ended o have relaively underdeveloped financial mares. In such cases, a large and persisen defici may yield more deb han he mare will bare a a reasonable cos of financing, possibly forcing a governmen or is cenral ban o purchase all of he new deb. According o his analysis, he cenral ban s reacion o supply shocs may cause he growh rae of money o move in opposie direcion o he supply shoc s effec on real oupu. Should hese 11
responses lead o permanen changes in he money growh rae, hen every permanen movemen in inflaion would clearly no be he resul of exogenous changes in money growh. This plausible criicism of empirical models ha assume permanen shocs o inflaion are exogenous leads o fundamenal quesions. Is here any value in decomposing inflaion ino permanen and ransiory componens? Do hese decomposiions provide any informaion abou he srucure of he economy? I urns ou ha even when we ae accoun of endogenous money growh he answer o each quesion is yes. Assume ha he economy is described by a srucure wih exogenous shocs o money growh, µ, and exogenous shocs o echnology, λ: π (L) (L) µ πµ πλ = yµ yλ y (L) (L) ε (16) where i i j (L) j L j= = for =π,y and i=µ,λ specifies he dynamic responses of variables o he srucural shocs. The srucure can be wrien as: X = a(l) ε (17) = ( π ) ( ) X,y where, ε = µ, λ and. πµ πλ (L) (L) a(l) = yµ yλ (L) (L) And using he common assumpion in he srucural VAR lieraure ha hese shocs are uncorrelaed, we can normalize he srucural shoc variances o be uniy and wrie he covariance marix of hese shocs as: Eεε =Σ = I. Using recursive subsiuion on he srucure we obain: ε Laer i will be convenien o wrie his marix: X 12 1 = X + ϕ ε = ϕ = j j= (18) where a. (19)
where vs Φ = i= vs i Φ ϕ = for v=π,y and s=µ,λ. Φ πµ πλ yµ yλ Φ Φ (2) Hence, he dynamic effec of each shoc on oupu and inflaion is given by: X ε = a =ϕ j= j. (21) And by leing go o infiniy, he long-run effec of each shoc on each variable is obained: X lim = a j = a(1) ε j=. (22) where he las equaliy comes from seing L=1 in a(l). The srucure can also be mapped ino he VAR and so obain a relaionship for he VAR coefficiens: β (L) = a a(l) 1 (23) he sum of VAR coefficiens: β (1) = a a(1) 1 (24) and he VAR residuals: e = a ε (25) From he las equaion and he assumpion abou variances and covariances for he srucural shocs, he covariance marix for residuals is: Σ e =aa. (26) Applying he equaion for he sum of VAR coefficiens o he las equaion, one obains: 13
a(1)a(1) =β(1) Σβ(1) 1 1 e. (27) How is he Saisical Model Relaed o he Srucure? Now we will derive wo imporan relaionships beween he saisical model and he srucure. The firs relaionship comes from combining equaions (14) and (27): r(1)r(1) = a(1)a(1). (28) This equaion describes how he long-run properies of he saisical model are relaed o parameers ha characerize srucure in he long run. The second imporan equaion comes from insering (23) and (24) ino (15) o obain: ( )( ) 1 1 1 1 r(l) =β(l) β (1)r(1) = a(l)a a(1)a r(1) = 1 a(l)a(1) r(1) (29) or equivalenly: 1 rj = a ja(1) r(1) for all j. (3) Inser (3) ino (4), he equaion describing responses from he saisical model, and hen use he definiion of srucural responses from (19) o obain: 1 1 = j = j =ϕ j= j= Q r a a(1) r(1) a(1) r(1). (31) Equaion (31) characerizes he relaionship beween he saisical model s impulse response funcion and he srucure s dynamics. The long-run effec marix for he srucure can be wrien as: πµ πλ a(1) = yµ y λ 14. (32)
Economic heory ofen ells is ha some of hese srucural parameers saisfy cerain inequaliy consrains. These consrains will help us deermine he srucural implicaions of he effecs of permanen inflaion shocs. For example, heory says ha an exogenous increase in he growh rae of money will raise inflaion in he long run: πµ >. In mos modern macroeconomic heories his long-run parameer is equal o one because a one percenage poin increase in money growh nearly always increases long-run inflaion by one percenage poin, holding all else consan. Theory also suppors he assumpion ha a permanen beneficial echnology shoc raises he level of oupu in he long run: yλ >. If he cenral ban allows long-run money growh o respond o supply shocs, i will raise he growh rae of money when he economy experiences an adverse supply shoc and lower money growh when here is a beneficial supply shoc: πλ <. Oherwise πλ =. A posiive value for his srucural parameer is ruled ou because ha would imply a beneficial supply shoc leads o a permanenly higher inflaion and a negaive supply shoc causes permanen lower inflaion. Such implicaions are no observed in he daa and also a posiive value for his parameer is inconsisen wih he way moneary policy should behave. (Laer on I will prove ha when πλ =, he permanen shocs in he saisical model will idenify he dynamic effecs of he exogenous money growh shocs and he emporary shocs will idenify he effecs of exogenous echnology shocs. As we saw previously, he emporary inflaion shocs from he empirical models behave lie supply shocs wih oupu and price moving in opposie direcions. Hence, his robus empirical finding is consisen wih he simple heory s predicion.) The long run effec of money growh on oupu, yµ, is wha economiss would lie o esimae 15
using he permanens inflaion shocs. A primary moivaion for empirical wor on his issue is ha economic heory gives a variey of predicions abou his srucural parameer. Someimes heory yields superneuraliy ( yµ = ), bu ofen a heory implies non-superneuraliy wih eiher a Mundell-Tobin effec ( yµ > ) or a reverse Mundell-Tobin effec ( yµ < ). Thus we mae no assumpion abou he value of his srucural parameer. Implicaions of Endogenous Policy for Empirical Findings from Bivariae Models Combining equaions (8) and (32) in equaion (28) yields: RπP RπP RyP πµ πλ πµ y µ R R = R yp yt yt yµ yλ πλ yλ (33) from which he saisical model s long-run coefficiens are relaed o srucural parameers: 2 2 R π P = πµ + πλ R yµ πµ yλ πλ,, (34) yp + = + 2 2 πµ πλ R yt = + yλ πµ yµ πλ 2 2 πµ πλ Noe here are wo soluions for each parameer in r(1), one is posiive, he oher negaive value and each has he same absolue value. The posiive square roo for R πp is used because he empirical lieraure focuses on measuring he effecs of a permanen 1 percenage poin increase in inflaion. If > yλ πµ yµ πλ hen he soluion above for R yt is also posiive. This inequaliy seems liely o hold. For example, i is guaraneed o hold if eiher πλ is equal o zero, or if yµ is non-negaive. For his inequaliy o be reversed, i is necessary for yµ and πλ o boh have relaively large negaive values. However, if boh are negaive, a sufficien condiion for he previous inequaliy o hold would be > πµ > πλ if and. These inequaliies hold if echnology shocs explain a larger share yλ yµ 16
of he long-run oupu variance and exogenous money growh shocs explain mos of he long-run variance of inflaion, respecively. While i is reasonable o believe ha he value of R yt given above is posiive, wheher or no his is rue has no effec on resuls ha we obain for permanen inflaion shocs. Of course, he impulse responses for emporary shocs are affeced when he previous inequaliy fails o hold. Bu since his paper pus no emphasis on resuls for emporary shocs his is irrelevan. If he inequaliy sign is flipped and we normalized he impulse responses so ha he emporary inflaion shoc raises oupu in he long run, hen ha would flip he response of inflaion o emporary shocs abou is zero response line. The saisical model idenifies emporary shocs ha behave lie adverse supply shocs, which is wha he srucural model implies will occur when πλ =. Hence, his evidence provides some suppor for he inequaliy, bu wheher or no he inequaliy holds is of no consequence for our resuls. Inerpreing he Long-run Effecs on Oupu from Permanen Inflaion Shocs From equaion (34) we can see how he raio of parameers from he saisical model is relaed o he underlying srucure: R R + yp yµ πµ yλ πλ = 2 2 πp πµ +πλ. (35) In general, he saisical model will no idenify srucural parameers, bu firs we invesigae condiions under which i will. Proposiion 1: If he money growh rae does no permanenly change in response o real shocs, hen he permanen shoc o inflaion will idenify he long-run effec on oupu of an exogenous change in money growh. 17
This resul is easily seen by seing πλ = in equaion (35) and obaining =. This resul R yp R π P is expeced because he saisical model s idenificaion assumpions are equivalen o he resricions implied by ha srucure. No only are he long-run effecs srucural bu also he dynamic responses of each variable o exogenous money growh and exogenous echnology shocs are idenified by his permanen-ransiory decomposiion of inflaion. (This resul illusraes a necessary condiion for he empirical model o idenify a srucure.) (Somewhere, maybe here, menion he ohers). Hence, if he idenificaion assumpions are valid srucural assumpions, Bullard and Keaing (1995) are correc in inerpreing heir empirical model as a means of esing for superneuraliy. The saisical model will idenify wha we wan when policy is endogenous as long as his endogeneiy does no induce a permanen change in money growh following a supply shoc. The primary concern of he paper is ha permanen changes in money growh may no be exogenous for a hos of possible reasons. Such a policy may mae inflaion endogenous o aggregae supply shocs in he long run, and herefore he economic srucure would be inconsisen wih he saisical model s principal idenifying assumpion. One possible concern is ha exogenous changes in money growh are superneural and endogenous moneary policy causes permanen inflaion shocs o be associaed wih permanen oupu movemens. The period of highes inflaion for many counries coincides wih he 197's oil price shocs and ha evidence seems o suppor his hypohesis. However, he following proposiion rejecs his explanaion for significan posiive esimaes obained for a number of low inflaion counries. yµ πµ Proposiion 2: If exogenous money growh shocs are long-run superneural wih respec o oupu and he cenral ban permanenly raises he money growh rae in response o adverse supply shocs, hen a permanen 18
shoc o inflaion will cause he level of oupu o fall in he long run. By seing = in equaion (35), we ge. The denominaor is posiive and he yµ R R yp yλ πλ = 2 2 πp πµ + πλ srucural assumpions mae he numeraor negaive. Thus < when = and <. R yp R π P yµ Someone who believes in superneuraliy and ha permanen inflaion shocs are primarily he resul of moneary policy being endogenous o real disurbances would expec o find mosly negaive esimaes of R yp R π P, insead of he posiive esimaes found for nearly all he low inflaion counries in Bullard and Keaing s sample of counries for which hey can apply heir mehod. Proposiion 2 is imporan because i shows ha endogenous moneary policy, by iself, is unable o explain he posiive esimaes. However, his hypohesis migh explain negaive esimaes obained for mos of he high inflaion counries and for one low inflaion counry. This leads direcly o a second quesion abou he saisical model: Do posiive esimaes for his raio of parameers have srucural implicaions? πλ yp Proposiion 3: Posiive values for imply yµ >. R R π P yp If < hen is negaive because he numeraor in (35) is negaive (and he denominaor is clearly yµ R R π P posiive). The only way for he raio of parameers o be posiive is if here is a long-run posiive effec of money growh on oupu ha ouweighs he negaive effec resuling from long-run money growh being endogenous. Hence a posiive long-run oupu response o a permanen increase in inflaion implies a Mundell-Tobin effec. 19
Proposiion 2 says ha he esimae of R yp R π P is downward biased when here is superneuraliy and he cenral ban maes money growh respond in he long run o supply. In fac, downward bias is no limied o srucures ha have superneuraliy. Proposiion 4: For all non-negaive values and a poenially wide range of negaive values for yµ, if long-run money growh responds counercyclically o supply shocs, hen he long-run oupu effec of a permanen inflaion shoc will be smaller han he long-run effec on oupu of an exogenous permanen shoc o money growh. To prove his proposiion we need o deermine he condiion for yµ such ha: Insering equaion (35) ino his inequaliy yields: R yp yµ < R πp πµ. + < yµ yλ πλ 2 πµ πµ y µ 2 πµ πλ 1+ πµ. Afer some algebraic manipulaion his inequaliy can be wrien as: < 2 yλ πµ πλ yµ πλ, and since he firs erm is posiive, i implies 2
yλ πµ yµ > πλ. This inequaliy ses a non-posiive lower bound for he long-run effec of money growh on oupu such ha in he long run he oupu response o a permanen inflaion shoc is smaller han he effec on oupu from an exogenous shoc o money growh. For economies wih Mundell-Tobin effecs or where superneuraliy holds, he esimae is clearly downward biased. Thus Bullard and Keaing s conclusion ha some low inflaion counries experienced Mundell-Tobin effecs is reinforced by endogenous policy. Proposiion 3 shows ha he posiive esimaes imply Mundell-Tobin effecs are presen and Proposiion 4 shows ha when Mundell-Tobin effecs are presen, endogenous money growh forces he long-run oupu effec from permanen inflaion shocs o be smaller han he effec one would lie o measure using permanen inflaion shocs. Bu he esimae may be downward biased even when reverse Mundell-Tobin effecs are presen. To ge a rough measure of wha his lower bound is, assume he long-run variances of oupu and inflaion explained by echnology shocs are equal o one anoher: = yλ πλ. Economic heory nearly always yields πµ = 1, and combining hese wo condiions wih he previous inequaliy, we ge ha he esimae of R yp R π P is downward biased for any value of yµ > -1. This range encompasses mos, if no all, of he values ha macroeconomic heory provides for his srucural parameer. And if echnology shocs explain more of he long-run variance of oupu han of inflaion, ha would serve o increase he range of parameers over which his bias occurs. The inequaliy implies ha he esimaes of long-run oupu response o permanen inflaion shocs will ypically, if no always, be downward biased esimaes of he inended effec. 21
The nex issue is o deermine wha can explain he finding ha almos all he low inflaion counries in Bullard and Keaing had posiive esimaes and none of he higher inflaion counries had posiive esimaes. Can his cross-counry relaionship be explained by variaion in he long-run effec of exogenous money growh on oupu? Proposiion 5: Holding all oher srucural parameers consan, he long-run response of oupu o a permanen inflaion shoc varies wih he long-run effec of exogenous money growh on oupu. R yp The parial derivaive of wih respec o yµ is: R π P πµ 2 2 πµ πλ +. Since ha derivaive is posiive, he raio of parameers will vary wih he long-run effec of money growh on oupu, and so as yµ falls so will R yp R π P. Bullard and Keaing s finding of differen effecs of permanen inflaion shocs on oupu for low and high inflaion counries is consisen ha yµ depending on average inflaion. For nearly all of he low inflaion counries we now yµ > given Proposiions 3 and 4. The evidence in combinaion wih Proposiion 5 suggess ha his parameer is ypically smaller for higher inflaion counries. Therefore, Bullard and Keaing s inerpreaion of his cross-counry relaionship as evidence ha he Mundell-Tobin effecs decline as inflaion ges higher is no rejeced by endogenous money growh. The previous resul is obained by holding fixed he long-run response of money growh o supply shocs. However, πλ could be negaively relaed o he rae of inflaion. Cenral bans in higher inflaion 22
counries end o be less independen, and so hese policymaers may experience more pressure o simulae he economy when a negaive supply shoc sries. Also, counries ha have experienced higher inflaion are ypically small open economies ha usually have relaively less-developed financial mares. Such counries ofen have difficuly issuing a large quaniy of public deb o he privae secor, and herefore have been forced o use he inflaion ax o finance a defici. Can endogenous money growh explain he difference beween low and high inflaion counries in erms of he long-run oupu effec from a permanen inflaion shoc? Proposiion 6: The long-run response of oupu o a permanen inflaion shoc is no clearly relaed o πλ.. Taing he derivaive of R yp R π P wih respec o πλ and simplifying, we obain: R = + yp 2 2 R P πµ y λ πλy λ 2 π yµ πµ πλ 2 πλ 2 2 ( πµ πλ ). The sign of his derivaive is ambiguous; The denominaor is posiive, bu he numeraor consiss of a posiive erm, a negaive erm and one wih he sign deermined by he sign of yµ, respecively. Therefore we can say ha he smaller esimaes from high inflaion counries obain because higher inflaion counries have more endogenous long-run money growh. In fac, he opposie effec is quie possible. If he values for πλ and yµ become increasingly negaive as inflaion rises, hen his derivaive is more liely o be negaive for high inflaion counries. And such were he case, a more endogenous moneary policy would mae he esimae of R yp R π P rise wih inflaion which, of course, would be inconsisen wih 23
he evidence. Inerpreing he Impulse Responses of Oupu o Permanen Inflaion Shocs Ineresing findings were also observed for he impulse responses o permanen inflaion shocs. The impulse responses are obained by firs calculaing: a(1) 1 r(1) = πµ πλ πλ πµ 2 2 πµ πλ + (36) and hen combining his equaion and equaion (2) in (31) o obain: πµ πλ Φ Φ πµ πλ yµ yλ X Φ Φ πλ πµ = Q = u 2 2 πµ +πλ. (37) From his equaion he response of oupu o a permanen inflaion shoc is: y yµ yλ Φ πµ +Φ πλ = P 2 2 πµ πλ u +. (38) To inerpre his impulse response we mus mae assumpions abou he dynamic responses of oupu o he wo srucural shocs: A1: Oupu rises for some ime following an exogenous increase in money growh: y µ yµ = Φ > for <<K wih K>; 24
A2: Oupu responds posiively o a beneficial supply shoc: y λ yλ = Φ > œ. Assumpion A1 does no resric he sign or magniude of. Assumpions abou he dynamic responses yµ of inflaion o hese shocs are unnecessary. (However, o answer quesions abou dynamic responses of inflaion one would need o mae assumpions abou he dynamic effecs of srucural shocs on inflaion. Mos heories imply ha an exogenous permanen increase in money growh will cause inflaion o rise and ha an exogenous permanen increase in produciviy will cause inflaion o fall alhough his decline would only be emporary if πλ =. ) Bullard and Keaing observe ha a permanen increase in inflaion never has a posiive effec on oupu in counries wih he highes inflaion raes. Under our assumpions abou he dynamic responses of oupu o exogenous money growh and echnology shocs, his finding implies ha he long-run effec of a echnology shoc on money growh is negaive for hese counries. Proposiion 7: If he impulse response of oupu o a permanen inflaion shoc is never posiive, hen πλ <. (38) implies: When he response of oupu o a permanen inflaion shoc is non-posiive for all <K, equaion πλ Φ yµ πµ yλ Φ. The srucural assumpions guaranee ha he righ side is negaive, and herefore non-posiive impulse 25
responses imply ha money growh was endogenous o real shocs in he counries from Bullard and Keaing s sample wih very high raes of inflaion. The finding ha high inflaion counries have πλ < is consisen wih he fac ha mos run away inflaions have been caused by ou of conrol fiscal policy, resuling in deficis ha had o be financed by he prining press. The wo high inflaion counries, Chile and Argenina, ran huge budge deficis in he 7s and 8s, in par a consequence of adverse oil price shocs, and hey moneized much of heir exploding deb load. One would expec πλ o be significanly less han zero for hese wo counries. Anoher finding is ha he impulse responses of oupu o permanen inflaion shocs are relaed o inflaion. This can be seen from Figure? in Bullard and Keaing (1995), where he dynamic response of oupu o a permanen shoc ends o fall as average inflaion rises. Can he hypohesis ha money growh becomes more endogenous as inflaion rises explain his finding? Proposiion 8: The dynamic response of oupu o a permanen inflaion shoc shifs down as πλ becomes more negaive. yp The parial derivaive of Q wih respec o πλ is: Φ Φ 2 yλ yµ πµ πµ πλ 2 2 3/2 ( πµ +πλ ). Since his is posiive, he dynamic response of oupu o a permanen shoc o inflaion shifs lower as πλ becomes more negaive. Therefore more endogenous moneary policy can explain he finding ha impulse responses become lower as inflaion ges higher. Alesina and Summers (1992) and ohers have shown an inverse empirical relaionship beween cenral ban independence and inflaion. Cenral ban independence should be posiively correlaed wih πλ because less independen cenral bans will be under greaer pressure o respond o adverse supply 26
shocs. This cross-counry relaionship beween independence and inflaion provides furher suppor for he hypohesis ha πλ ends o be more negaive for high inflaion counries. Bu cross counry variaion in he endogeneiy of moneary policy may no be he only reason why impulse responses vary wih he rae of inflaion. Anoher possibiliy is ha as he average rae of inflaion rises, inflaion responds faser o a permanen change in he growh rae of money while oupu becomes less responsive. Ball, Maniw and Romer (1989) derive such a relaionship from a heory of sicy price adjusmen and hen subjec his heory o empirical ess. They invesigae he relaionship beween inflaion and he responsiveness of inflaion and oupu o an aggregae demand shoc, and find ha heir heoreical predicion is suppored: The dynamic effec of an aggregae demand shoc on oupu becomes smaller as inflaion rises. Can his idea explain he relaionship beween inflaion and dynamic impulse responses? Proposiion 9: When he dynamic response of oupu o an exogenous permanen money growh shoc is smaller, he dynamic response of oupu o a permanen shoc shifs downward. Φ yµ The parial derivaive wih respec o of oupu s response o a permanen shoc,, is Φ yµ πµ 2 2 πµ πλ + posiive. Thus if falls wih inflaion, consisen wih he predicion of Ball, Maniw and Romer, hen he dynamic response funcion of oupu o a permanen shoc will shif downward wih inflaion, holding all oher srucural parameers fixed. Hence, he cross counry variaion of impulse responses wih inflaion can be explained by dynamic responses of oupu o permanen inflaion shocs falling wih inflaion and/or by cenral bans in higher inflaion counies being less independen. 27
Exending he Resuls o Sysems wih more han Two Variables The relaionship beween he saisical model and o a srucure can be exended o an arbirary number of variables. The Appendix derives he general resuls. Here I presen hese resuls in a somewha simpler form, bu hey are equivalen. Assume we have n variables, 1 2 n v, v,..., v, each of which is difference-saionary and we have esimaed a VAR wih all variables in firs differences. (I is concepually sraigh-forward o modify he analysis for saionary variables and coinegraion). Suppose we are ineresed in he effecs on all variables of a permanen shoc o one of hese variables, and call ha variable p v wih p(1,2,...,n), and assume ha his shoc explains all of he long-run variance of p v ε, ε,..., ε 1 2 n. If here are as many srucural shocs,, as variables, he impulse response of variable i o a permanen shoc o variable p is given by: n j= 1 Φ n j= 1 viε j 2 v ε P v ε P j j for i =1,2,... n. (39) Once can use his equaion o inerpre permanen changes in any variable in a sysem, no maer how many variables here are in he empirical model. Leing in he las equaion yields he long run effec of his permanen shoc o on p v each variable: n j= 1 n v ε j= 1 v ε i j P j 2 v ε P j for i =1,2,... n. (4) 28
And when i=p, his equaion becomes: n j= 1 2 v ε P j (41) Clearly he number of inequaliies describing economic srucure needed o inerpre he effecs of his permanen shoc on a paricular variable will increase wih he number of variables in he model. Rapach (22) uses a rivariae VAR model and daa from a number of counries o invesigae he long-run effecs on oupu and ineres raes from permanen inflaion shocs. This framewor allows him o joinly address wo hypoheses in a single empirical model for each counry: (1) Long-run superneuraliy wih respec o oupu; and (2) he Fisher hypohesis ha in he long run a permanen change in inflaion of a given percenage poin change will yield precisely he same change in nominal ineres raes. He finds some evidence agains boh hypoheses. The evidence agains superneuraliy poins o Mundell-Tobin effecs. The evidence agains Fisher s hypohesis finds ha he long-run nominal raes response o a permanen change in inflaion is significanly less han one, boh saisically and in quaniaive erms. Using our noaion, he variables in Rapach are given by nominal ineres rae, he shocs are given by P 2 3 ( ) u = u,u,u ( ) X = π,i,y where i is he where I use 2 and 3 o index he second and hird shocs, respecively, in his model. He resrics he second and hird shocs o have only emporary effecs on inflaion, and wries he long-run effecs of shocs on variables as: r πp = ip i2 r(1) r r ryp ry2 r y3 Rapach argues he second shoc is o preferences and he hird one is a shoc o echnology. If he acual srucure is lower riangular, he idenificaion resricions are valid and his model can be used o obain. 29
consisen esimaes of all srucural parameers. A ey idenificaion assumpion is ha permanen shocs o inflaion explain all he long-run variance of inflaion, similar o Bullard and Keaing (1995) and ohers. I modify Rapach s srucural assumpions by allowing money growh o be endogenous o supply shocs, similar o he previous analysis in his paper. In his case we can wrie he vecor of srucural shocs as ε = ( µ ρ λ ) where ρ is he shoc o preferences and he srucural parameer marix is wrien as:,, a(1) = iµ iρ πµ πλ yµ yρ yλ. The general resuls from above mae i simple o solve for he relaionship beween parameers in r(1) and parameers in a(1). Ineresingly, he soluions for r πp and r yp in erms of srucural parameers are precisely he same as we obained earlier for he bivariae case, and so he raio of hese wo parameers repored in Rapach is subjec o he same analysis ha was already done in his paper. In paricular, his evidence of Mundell-Tobin effecs is no refued by endogenous moneary policy.... More on wha he finds... The long-run effec of permanen inflaion shocs on he nominal ineres rae is: r ip πµ iµ = 2 2 πµ + πλ Dividing by r πp allows us o calculae he effec of a 1 percenage poin increase in he rae of inflaion on he nominal ineres rae: iµ ip πµ i µ πµ = = 2 2 2 πp πµ πλ πλ 1 2 πµ r r + +. If an exogenous permanen increase in money growh of 1 percenage poin ulimaely raises inflaion by 3
1 percenage poin and he Fisher hypohesis is rue, hen is equal o one. Consequenly, will iµ πµ r ip r π P be less han one for all non-zero values of πλ. In general, when money growh is endogenous, he longrun effec on he nominal ineres raes of a permanen shoc o inflaion is biased downward from he effec of an exogenous permanen increase in money growh when money growh is endogenous. Endogenous moneary policy does no inerfere wih he inerpreaion of permanen posiive oupu effecs from permanen changes in inflaion as evidence of Mundell-Tobin effecs. This conclusion is a consequence of he proposiions and discussions abou Mundell-Tobin effecs made previously in he paper. However, Rapach s (22) finding ha in he long run nominal ineres raes move less han one for one following a permanen inflaion shoc (and wha abou ohers wih a similar finding?) does no necessarily mean ha Fisher s hypohesis is rejeced. We can rule ou he hypohesis ha endogenous money growh is forcing he long-run response of ineres raes o a permanen inflaion shoc o be smaller han 1. Fuure empirical sudies of Fisher s heory should conrol for endogenous money growh so ha empirical ess won be biased oward incorrecly rejecing he hypohesis. Conclusion The resuls of his paper are briefly summarized. When money growh is no permanenly affeced by supply shocs, permanen inflaion shocs may be used o es hypoheses abou superneuraliy. If money growh is superneural wih respec o oupu and moneary policy allows longrun money growh o be endogenous o supply shocs, he long-run oupu effec of a permanen inflaion shoc will end o be negaive. Hence, he posiive esimaes of long-run oupu effecs from permanen increase in inflaion for low inflaion counries can no be explained by his simple reverse causaion 31
sory. (An ineresing quesion is wheher high inflaion counries would sill obain zero or negaive poin esimaes if we explicily allowed for endogenous moneary policy.) This evidence can be inerpreed as Mundell-Tobin effecs. If long run money growh is affeced by supply shocs, hen he long-run effec on oupu of a permanen exogenous increase in he growh rae of money exceeds he long-run oupu effec of an permanen shocs o inflaion for all non-negaive values and a poenially large range of negaive values for he parameer describing he long-run oupu effec from an exogenous money growh shoc. This resul raises he possibiliy ha Mundell-Tobin effecs may exis in virually all counries, and i is he bias from using permanen inflaion shocs combined wih he possibiliy ha he Mundell-Tobin effec becomes smaller wih inflaion explain why he esimaes are zero or negaive for high inflaion counries. The fac ha higher inflaion counries have smaller long-run oupu responses o permanen movemens in inflaion can be explained by long-run oupu responses o exogenous money growh shocs ha fall wih inflaion. I is no rue, in general, ha his finding can be explained by long-run money growh ha ends o be more endogenous as inflaion rises. Bu since endogenous policy can no be ruled ou, his suggess he need o esimae models ha allow for endogenous moneary policy in order o ge consisen esimaes of he his effec in he low inflaion counries and also o deermine wheher or no high inflaion counries may acually be experiencing Mundell-Tobin effecs. Anoher moivaion for more elaborae models of policy behavior is ha he dynamic responses sugges endogenous moneary policy has been a facor. In paricular, he finding ha he dynamic response of oupu o a permanen shoc is always negaive implies ha long-run money growh was endogenous o aggregae supply in he very high inflaion counries ha Bullard and Keaing mehod could be applied o. Anoher finding is ha he dynamic responses of oupu end o be lower as inflaion rises. This resul can be explained by long-run money growh ending o become more endogenous as inflaion increases or by dynamic responses of oupu o money growh shocs ha end o become smaller as he rae of inflaion increases. 32