Interpreting Permanent and Transitory Shocks to Output When Aggregate Demand May Not Be Neutral in the Long-run

Transcription

1 Inerpreing Permanen and Transiory Shocs o Oupu When Aggregae Demand May No Be Neural in he Long-run John W. Keaing * Universiy of Kansas Deparmen of Economics 213 Summerfield Hall Lawrence, KS eaing@u.edu phone:(785) fax:: (785) March 27, 24 Key Words: vecor auoregression, idenificaion resricions, moving average represenaion, aggregae supply and demand heory, permanen-ransiory shoc decomposiion JEL Codes: C32, C52, E3 Absrac: I examine he saisical model of permanen and ransiory shocs o oupu under he following srucural assumpions: An aggregae supply shoc ha raises oupu will cause he price level o fall and an aggregae demand shoc ha iniially raises oupu will cause he price level o rise. No assumpion is made abou he long-run effec of aggregae demand on oupu. Based on hese assumpions I obain hree primary resuls. Firs, if a permanen increase in oupu is associaed wih an increase in he price level, hen aggregae demand shocs mus have a posiive long-run effec on oupu. Second, oupu variance explained by permanen shocs will exceed he variance aribuable o aggregae supply when aggregae demand shocs have a posiive effec on oupu in he long run. Third, permanen and ransiory shocs will affec price and oupu in qualiaively he same way as aggregae supply and aggregae demand shocs, respecively, from exboo macro heory over a range of posiive and negaive values for he srucural parameer describing he long-run effec of aggregae demand on oupu. The resuls in his paper are used o inerpre findings from empirical research and o moivae direcions for furher invesigaion. * Thans o Shigeru Iwaa, Rober Rasche, Ellis Tallman, Tao Zha, and seminar paricipans a he Alana Fed, he Universiy of Kansas, he Midwes Economerics Group Meeing held a he Universiy of Chicago and he Missouri Economics Conference for helpful commens on preliminary versions of his paper. The auhor assumes all responsibiliy for any errors and omissions. Suppor from he General Research Fund a he Universiy of Kansas is graefully acnowledged.

2 1. Inroducion For many years economiss have been using saisical models ha decompose ime series ino permanen and ransiory componens o invesigae macroeconomic relaionships. Much of his research has deal wih quesions abou aggregae real oupu. While iniial sudies employed univariae models, 1 more recen wor has almos universally used ime series models wih muliple variables. Mulivariae models are hough o be preferable because more informaion is used in he decomposiion, muliple srucural relaionships can be esimaed, and he permanen and ransiory shocs can be orhogonal o one anoher. 2 Blanchard and Quah (1989) developed one of he firs mulivariae models in his lieraure, and here have been numerous applicaions and exensions of heir approach. An imporan feaure of Blanchard and Quah s decomposiion of oupu is ha he permanen and ransiory shocs may be able o idenify he effecs of aggregae supply and demand. Necessary condiions for his srucure o be idenified wih heir decomposiion include: (i) he aggregae supply curve is verical; (ii) aggregae demand shocs do no affec supply in he long run; (iii) he dynamic srucure is inverible; 3 and (iv) he shocs o supply and demand are independen. Some researchers have quesioned Blanchard and Quah s bivariae approach. Wih only a single aggregae supply shoc and a single aggregae demand shoc, one concern is ha heir model may ignore oher imporan srucural shocs, 4 in which case heir specificaion could be misspecified. In fac, Blanchard and Quah (1989) show in an appendix condiions under which heir approach will successfully idenify he effecs of he aggregae supply and aggregae demand shocs even when an economy experiences muliple inds of supply and demand shocs. Faus and Leeper (1997) elaboraed on ha poin and exended he discussion in a number of imporan direcions. Anoher concern involves he use of unemploymen rae daa in he decomposiion. Some 1

3 economiss have replaced unemploymen wih price daa. 5 This subsiuion is based on he fac ha exboo aggregae supply and demand heory is used o usify he saisical model and his heory is mos ofen formulaed in erms of oupu and he price level. To explain he behavior of unemploymen requires ha a labor mare srucure be appended o aggregae supply and demand. Anoher advanage of using price daa in he saisical model is ha heory usually predics all inds of supply shocs will affec oupu and he price level in qualiaively he same way and also ha each variable will respond o he various aggregae demand shocs in qualiaively similar ways. Supply shocs ypically cause oupu and price o move in opposie direcions while demand shocs ypically cause oupu and price o move in he same direcion, a leas for some porion of ime afer he shoc occurs. On he oher hand, he unemploymen rae responds in a fundamenally differen way o differen ypes of supply shocs. For example, an increase in labor supply raises oupu and he unemploymen rae whereas an increase in labor demand raises oupu bu lowers he unemploymen rae. Resuls from Blanchard and Quah (1989) and from Faus and Leeper (1997) show ha if here is more han one ype of supply shoc and each ype has fundamenally differen effecs on oupu and he unemploymen rae, he bivariae decomposiion will no idenify he effecs of aggregae supply and aggregae demand shocs. Consequenly, I will focus on models ha use price daa in place of he unemploymen rae. In conras o possible concerns abou muliple inds of srucural shocs, his paper considers he possibiliy ha aggregae demand shocs are no neural in he long run. A subsanial number of macroeconomic heories imply long-run non-neuraliy, and if hey are relevan, ha would invalidae he primary srucural assumpion ha usifies Blanchard and Quah s decomposiion. This paper examines he effecs of his alernaive assumpion on impulse responses and variance decomposiions obained wih heir saisical model. One moivaion of his paper is o deermine if srucural inerpreaions can be given o four findings from he empirical lieraure. Firs, impulse responses for poswar economies are ypically 2

4 consisen wih exboo heory; Permanen shocs behave lie aggregae supply shocs and emporary shocs behave lie aggregae demand shocs. However, his resul is no robus o differen ime periods. The second empirical finding is ha a permanen increase in oupu is associaed wih an increase in he price level for mos of a sample of prewar economies in Keaing and Nye (1998). The erm prewar is used o describe he period before World War I. These prewar responses o permanen shocs are inconsisen wih he aggregae supply shoc inerpreaion. Third, ha same sudy finds ha he amoun of finie horizon oupu variance explained by permanen shocs ends o be larger in he pre-1914 period han in he poswar. Keaing and Nye (1999) use he unemploymen rae, in accord wih Blanchard and Quah (1989), and obain a similar resul. Fourh, he immediae effec on oupu of a permanen shoc exceeds he long-run effec in mos esimaes wih pre-world War I daa. This characerisic is described as shorrun overshooing. While common in he prewar, shor-run overshooing of oupu responses o permanen shocs is no observed in he poswar esimaes. I use a se of plausible srucural assumpions o inerpre hese saisical findings. Bu insead of he long-run neuraliy assumpion, I use inequaliy consrains on he dynamic responses of variables o srucural shocs. Specifically I assume an aggregae supply shoc ha raises oupu causes he price level o fall and an aggregae demand shoc ha iniially raises oupu causes he price level o rise. The longrun effec of aggregae demand on oupu is no consrained. Based on hese srucural assumpions, he aforemenioned empirical findings can be given srucural inerpreaions. The poswar findings sugges ha exboo aggregae supply and demand heory provides a good descripion of economies in ha period. However, he findings wih prewar daa ha permanen oupu shocs cause prices and oupu o move in he same direcion, exhibi shor-run overshooing, and explain more oupu variance han hese shocs do in he poswar, suppor he hypohesis ha aggregae demand shocs had long-run posiive oupu effecs in he earlier sample period. Finally I show ha even if impulse responses are consisen wih exboo heory, as is ypically he case wih poswar daa, 3

5 aggregae demand shocs migh sill be having permanen effecs on oupu. I show ha permanen shocs behave lie aggregae supply and ransiory shocs lie aggregae demand over a range of posiive and negaive values for he parameer describing he long-run effec of aggregae demand on oupu. The paper concludes by briefly summarizing some economic heories in which aggregae demand may be nonneural in he long run, discussing wheher or no each of hese heories is a plausible explanaion for he differences beween prewar and poswar esimaes and recommending poenially useful areas for fuure research. 2. The Srucure and he Saisical Model A saisical model provides a means of poenially discovering a srucural relaionship. This secion will describe a saisical model and a srucure in erms of he dynamic responses of variables o shocs, moving average represenaions (MARs) as hey are called in ime series analysis. The srucural MAR describes he dynamic response of each variable o each srucural shoc. For he saisical MAR, I will use Blanchard and Quah s decomposiion of oupu ino emporary and permanen shocs. This decomposiion is obained by imposing a paricular se of idenificaion resricions on he reduced-form parameers of a VAR. This secion inroduces he VAR model and characerizes each of hese MARs. 2.1 The VAR In general, he VAR represenaion exiss and is unique, and can be wrien as: β(l) X = e (1) where X = (Y,P ) is he vecor of variables, e is he vecor of residuals, = 1-L is he firs 4

6 difference operaor and 2 (L)=I- 1L- 2L...- L κ β β β β κ represens he coefficiens in he VAR wih he ideniy marix and each β for =1,2,...,κ a 2 2 marix and κ he number of lags in he VAR. Deerminisic feaures such as consans, deerminisic rends or dummy variables ha migh be essenial for conducing a valid empirical analysis have been omied wihou loss of generaliy. The firs difference of Y will ensure here are permanen shocs o oupu in a linear model wih consan coefficiens. This specificaion is differen han Blanchard and Quah because he second variable is P, wih P he logarihm of he price level, insead of he unemploymen rae. Hence, P is (approximaely) equal o he rae of inflaion. If he choice of second variable does no aler he idenificaion resricions for he saisical model of oupu or he heoreical assumpions, hen he resuls derived in his paper for oupu are independen of ha choice. 2.2 The Srucural Moving Average Represenaion Assume he economic srucure has he following MAR: X =θ(l) ε (2) where ε = ( ε, ε ) S D is a vecor of shocs o aggregae supply and aggregae demand, respecively, and = θ (L) =θ +θ L +θ L +... = θ L srucural shocs wih θ = θ θ PS PD θ θ specifies he dynamic responses of Y and P o hese for all. If we assume supply and demand shocs are independen, a sandard assumpion in he srucural VAR lieraure, hen i is convenien o normalize hese shocs o have variances equal o one: Eε ε = I. Appendix A shows how recursive subsiuions are used o ransform equaion (2), he sysem for 5

7 X, ino he sysem in erms of X: X = X +θ ε + ( θ +θ ) ε + ( θ +θ +θ ) ε (3) from which responses of X o srucural shocs are obained: X = θ Φ ε =. (4) The las equaliy defines Φ as he sum of he firs parameer marices in θ(l), a definiion ha will be convenien in laer analysis. Noe ha Φ is a 2 2 marix: where vi Φ = θ = vi for v=y,p and i=s,d. While economiss wan o esimae hese srucural responses, he paper is concerned wih condiions under which he saisical model may be unable o (5) obain consisen esimaes of he dynamic srucure. The long-run responses of variables o shocs are obained by leing go o infiniy in (4): Φ Φ Φ = PS PD Φ Φ (6) where he las equaliy comes from seing L=1 in θ(l). The θ(1) marix represens he long-run mulipliers for srucural shocs, 6 and i can be wrien as: Θ Θ θ (1) =. (7) ΘPS ΘPD 2.3 The Saisical Model s Moving Average Represenaion Le he MAR for he saisical model be wrien as: X = C(L) µ (8) 6

8 where µ = ( µ, µ ) P T is he vecor of permanen and ransiory shocs, respecively, and = C(L) = C + C L + C L +... = C L are he impulse responses of X o hese shocs wih C a 2 2 marix for all non-negaive ineger values of. In all applicaions of he bivariae framewor, he permanen and ransiory shocs are assumed independen, and herefore he variance of each shoc in he saisical model can be normalized o one for convenience: Eµ µ = I. Equaion (8) is in firs differences, bu i can be ransformed ino a relaionship in erms of X using recursive subsiuion, following he same procedure ha was used wih (2) o generae (3): X = X + C µ + (C + C ) µ + (C + C + C ) µ (9) Hence, he impulse responses of Y and P o permanen and ransiory shocs are obained: X = µ = C. (1) Leing go o infiniy in (1) yields he long-run effecs of hese shocs on he variables: X lim C C(1) = = µ =, (11) where he las equaliy comes from evaluaing C(L) a L=1. C(1) represens he sum of coefficiens in C(L), and i can be wrien as: CYP C(1) = CPP C PT (12) where C vi is he long-run response of price or oupu, v(y,p), o a permanen or ransiory shoc, i(p,t). C(1) is made lower riangular by seing C YT =, he resricion ha forces emporary shocs o no have a permanen effec on oupu, as seen from equaion (11): Y lim CYT. T = = µ 7

9 3 Relaionships beween he Saisical and Srucural MARs The saisical MAR will no be he same as he srucural MAR unless he idenificaion resricions in he saisical model are valid srucural resricions. An easy way o see how he wo MARs are relaed is o map each of hem ino he VAR. A VAR is a sysem of equaions in which each variable is a funcion of lagged endogenous variables and a serially uncorrelaed error. The saisical decomposiion is ransformed ino he VAR represenaion by muliplying equaion (8) by C C(L) -1. The srucure is ransformed ino he VAR by muliplying equaion (2) by θ θ(l) -1. These mappings deermine how VAR residuals: e = C µ =θ ε (13) and VAR coefficiens: β (L) = C C(L) =θ θ(l) 1 1 (14) are funcions of parameers from boh he saisical model and he srucure. I use hese equaions o describe a well-nown mehod of calculaing coefficiens in he saisical decomposiion and, more imporanly, o characerize he way ha coefficiens from he saisical model are relaed o srucural parameers when Θ. Given equaion (13) and he ideniy covariance marices assumpion for he shocs in each MAR, he covariance marix for residuals: Σ = CC =θ θ e, (15) is a funcion of shor-run parameers from he srucure and also a funcion of shor-run coefficiens from he saisical decomposiion. A relaionship beween he saisical decomposiion, he srucure and β(1), he marix describing he sum of VAR coefficiens, is obained by seing L=1 in equaion (14): 8

10 β (1) = C C(1) =θ θ(1) 1 1. (16) The firs ideniy in equaion (16) yields: C = β(1)c(1). Inser his expression ino he firs ideniy in equaion (15) and simplify: C(1)C(1) =β(1) Σ β(1) 1 1 e. (17) This equaion illusrae a popular mehod for esimaing parameers in he saisical decomposiion. Given ha C(1) is a riangular marix, he C(1) parameers can be obained by Cholesy decomposiion of he righ-hand side of equaion (17). Inser he esimae of C(1) ino he firs equaliy from equaion (16), solve for C, inser he value for C ino he firs equaliy in equaion (14) and solve for C(L), he dynamic responses of variables o permanen and ransiory shocs. Now we invesigae how he saisical decomposiion is relaed o srucure. Noice ha he las idenify in (14) can be manipulaed o yield: C = θ θ(1) -1 C(1). Inser his equaion ino he second equaliy of (15) and simply, o obain: C(1)C(1) = θ(1) θ(1). (18) The sandard assumpion from exboo heory is ha aggregae demand shocs are long-run neural which is given by Θ = in he srucural model. This condiion, along wih he assumpions ha he srucure is inverible and ha he srucural shocs are orhogonal o one anoher, implies ha he permanen and ransiory shocs idenify he dynamic effecs of supply and demand, respecively. The θ(1) marix is lower riangular when Θ =, and because he lower riangular facor of a symmeric marix is unique, 7 equaion (18) yields C(1) = θ(1). Hence, C(1) idenifies he long-run effecs of aggregae demand and aggregae supply on Y and P. This las ideniy is combined wih equaion (14) which maps hese wo represenaions ino he VAR coefficiens o show ha C(L)=θ(L). As one migh expec, he saisical model idenifies he complee dynamic responses of each variable o each srucural 9

11 shoc when he idenificaion resricions are valid. This paper is concerned wih he more general case in which Θ is a possibiliy. Using he second ideniies from equaions (14) and (16), i is easy o derive he relaionship beween he saisical model s impulse responses and he srucural responses: or equivalenly: 1 CL ( ) = θ( L) θ() 1 C() 1 (19) C = θθ() 1 1 C() 1 for all. (2) Using he definiion of θ(1) from (7) and he definiion of C(1) from (12) in equaion (18), i is sraighforward o calculae he relaionship beween he long-run coefficiens from he saisical model and long-run srucural parameers: 2 2 ( ) 1 2 C = Θ +Θ YP C PS PD, and. PP Θ Θ +Θ Θ = C YP C PT Θ Θ Θ Θ = C PD PS YP Using hese hree ideniies, he following resul is easily obained: 1 θ (1) C(1) = Θ Θ C Θ YP Θ. (21) Inser (2) ino (1), he equaion describing responses from he saisical model, and hen use he definiion of srucural responses from (4) o obain: X µ 1 1 = C = θ θ() 1 C() 1 = Φ θ() 1 C() 1 = =. (22) Equaion (22) characerizes he relaionship beween he saisical model s impulse response funcion and he srucure s dynamics. Subsiuing (21) and (5) ino (22) yields: 1

12 Φ Φ Θ Θ PS PD X Φ Φ Θ Θ = µ C YP (23) Equaion (23) shows how he impulse responses for he saisical model are a funcion of he srucural parameers. If Θ, hen he saisical model s coefficiens are nonlinear funcions of srucural parameers, no consisen esimaes of he srucure. This resul suggess ha when he saisical model uses inappropriae idenificaion resricions, i will be uninformaive abou he srucure of he economy. However, if oher assumpions abou he economic srucure are available, his misspecified saisical model migh sill be able o infer imporan facs abou he underlying economic srucure. Equaion (23) will be used o answer quesions ha moivae his paper. 4. Srucural Assumpions If we are unable or unwilling o ae a sand on some feaures of he srucure, i is impossible, in general, o give srucural inerpreaions o empirical models. 8 More o he poin, economiss are unable o infer how he impulse responses o permanen or ransiory shocs are relaed o he srucure wihou assumpions of some ind concerning how he economy operaes. Blanchard and Quah, along wih many ohers, have aen he posiion ha aggregae demand is long-run neural o inerpre heir saisical model, bu wheher or no his srucural hypohesis is correc, Blanchard and Quah s decomposiion is able o idenify a saisical model wih permanen and ransiory shocs. If we are unwilling o assume ha aggregae demand is long-run neural wih respec o real oupu, hen we will need alernaive srucural assumpions o give an economic inerpreaion o he saisical model. Forunaely oher assumpions are available. Economic heory ofen places bounds on he qualiaive responses of variables o srucural shocs. 9 For example, mos heories predic ha a 11

13 beneficial aggregae supply shoc will raise oupu 1 and have a negaive effec on he price level: Y S = Φ > A1: for all ; ε P S PS = Φ < A2: for all. ε These assumpions are wea enough o allow for he possibiliy ha supply shocs also shif he aggregae demand curve. If supply shocs cause boh curves o shif in he same direcion, hen assumpion A2 requires ha he demand curve no shif by as much as supply. An example of an aggregae supply facor ha could shif boh curves in he same direcion is a permanen increase in produciviy. Aggregae demand would also shif o he righ because a permanen increase in produciviy increases invesmen demand by raising he expeced fuure marginal produc of capial. There is some debae in he lieraure abou he long-run effecs of aggregae demand on oupu. Since he long-run aggregae supply curve is verical in virually all modern macroeconomic heories, ha means a shif in he aggregae demand curve will no have a long-run effec on oupu unless i has a permanen effec on some facor in he aggregae supply curve. There are a number of differen heories ha predic aggregae demand may be non-neural in he long run. Some heories predic an increase in aggregae demand will cause oupu o rise in he long run while oher heories find he opposie effec. A lis of prominen examples from he lieraure includes: 1. Non-Superneuraliy: A permanen increase in he growh rae of money may raise or lower oupu in he long-run depending on paricular feaures of he srucure; 2. Long-Run Fiscal Policy Effecs: An increase in governmen spending may crowd-ou or crowd-in invesmen in he long-run, affecing he soc of capial and consequenly long-run aggregae supply, and changes in marginal ax raes may have supply-side effecs; 12

14 3. Hyseresis: The naural rae of unemploymen may depend on pas levels of he unemploymen rae, and so aggregae demand may affec he naural rae and as a resul he long-run level of oupu; 4. Coordinaion Failures: Coordinaion problems may yield muliple equilibria, allowing aggregae demando poenially affec he long-run equilibrium posiion aained by he economy; 5. Desabilizing Price Flexibiliy: Raher han bring abou general equilibrium, price flexibiliy may desabilize he economy causing aggregae demand o have persisen effecs on oupu. Irrespecive of he ulimae consequence for oupu, I assume ha a posiive aggregae demand shoc raises oupu for a leas K periods afer he shoc occurs: Y D = Φ > A3: for =,1...,K wih < K < 4. ε This assumpion allows for he possibiliy ha afer K periods oupu may fall below is pre-shoc level in response o a beneficial aggregae demand shoc. There are a leas wo ways ha his migh occur. The firs is if aggregae demand has a negaive long-run effec on oupu. Obviously, wih Θ <, he response of oupu o a posiive aggregae demand shoc mus evenually become negaive. A second way would be if oupu exhibis damped cycles around is seady sae. When an economy experiences his sor of behavior, oupu may fall below is iniial level as he economy dynamically aduss. Such dynamics are more liely o cause a decline in oupu if aggregae demand is neural in he long-run, bu negaive oupu responses may occur even when demand has a posiive long-run oupu effec. The lielihood of his effec is a funcion of he cyclical ampliude relaive o he long-run effec on oupu. I also assume ha aggregae demand shocs which iniially cause oupu o rise will have a posiive effec on he price level: 11 P D PD = Φ > A4: for all. ε 13

15 If a posiive aggregae demand shoc also shifs he long-run aggregae supply curve o he lef, Θ <, hen A4 will hold because he movemen of each curve raises he price level. On he oher hand, if a posiive aggregae demand shoc shifs long-run aggregae supply o he righ, Θ >, hen he supply curve mus no shif by as much as demand does for A4 o hold. Aggregae demand neuraliy may be hough of as a reasonable woring hypohesis, bu i is inconsisen wih many differen macroeconomic heories. And while assumpions A1 hrough A4 may no hold for every conceivable srucure, hese assumpions are consisen wih mos economic heories. Furhermore, hese assumpions do no rule ou he possibiliy ha aggregae demand shocs are neural in he long run because no assumpion is made abou Θ. 5. Resuls Using permanen-ransiory shoc decomposiions wih pre-world War I daa for 1 counries ha have relaively long ime series, Keaing and Nye (1998) find ha a permanen increase in oupu is associaed wih an increase in he price level for 8 counries. In 5 of hese cases, he effec is saisically significan. 12 This evidence srongly reecs he exboo srucure which underlies Blanchard and Quah s (1989) decomposiion, a leas for he prewar sample, because if permanen shocs are supply shocs hey should cause price and oupu o move in opposie direcions. Is a reecion of he srucural hypohesis all ha can be inferred or does his empirical evidence ell us somehing more abou he srucure of prewar economies? Proposiion 1 provides an answer o his quesion. Proposiion 1: Given assumpions A1, A2 and A4, if a permanen increase in oupu is associaed wih an increase in he price level, hen aggregae demand mus have a posiive effec on oupu in he long run. 14

16 Proof of Proposiion 1: From equaion (23), he response of price o a permanen increase in oupu is PS PD = P CYP P Φ Θ +Φ Θ µ. The condiion on Θ ha maes his response posiive is: Θ > Φ Θ Φ PS PD and he srucural assumpions guaranee ha he righ side is posiive. Q.E.D. Keaing and Nye (1998) speculaed ha heir evidence wih pre-1914 daa migh suppor heories in which aggregae demand shocs have posiive long-run effecs on oupu, and Proposiion 1 provides formal heoreical suppor for his inerpreaion. Since he price level rises a all poins on he impulse responses for 8 counries in he pre-1914 sample, Θ mus be large enough o saisfy he inequaliy for all in hese economies. These price responses reec he srucural model used o usify he idenificaion resricions, bu hey also imply, under arguably more plausible srucural assumpions, ha aggregae demand had a posiive long-run effec on oupu in a number of prewar economies. Anoher finding in Keaing and Nye (1998) is ha impulse responses are fundamenally differen across he wo ime periods. The immediae effec on oupu of a permanen shoc in prewar daa is larger han he long-run effec for 7 of he 1 counries used in he sudy. 13 Shor-run overshooing responses are no observed in poswar daa from any of hese counries. Based on economic heory and properies of he saisical model, I argue ha he only plausible srucural explanaion for his overshooing is ha aggregae demand shocs had permanen posiive effecs on oupu. The response of oupu o a permanen shoc is aen direcly from equaion (23): 15

17 = P CYP Y Φ Θ +Φ Θ µ. (24) If Θ = hen he response of oupu o a permanen shoc is idenical o he response of oupu o supply. I am no familiar wih any economic heory in which shor-run overshooing characerizes he response of oupu o a permanen supply shoc. Theory ypically shows a gradual oupu adusmen o a permanen supply shoc, wih he possibiliy of cyclical dynamics as he economy approaches a seady sae. Thus Θ = is unable o explain shor-run overshooing. Now consider he Θ < case. This assumpion along wih A1, A3 and equaion (24) implies: Y µ P < Φ for small, because when Θ is negaive, he coefficien on Φ in (24) is posiive and less han one and he second erm is negaive. Furhermore, for any non-zero Θ : Y lim C ( ) P µ 2 2 YP 1 = = Θ +Θ 2 >Θ. The implicaion is ha when Θ is negaive, permanen shocs have a smaller shor-run effec and a larger long-run effec on oupu han aggregae supply shocs. Therefore, a permanen shoc o oupu will no exhibi shor-run overshooing if aggregae demand has a negaive long-run effec on oupu, given ha he response of oupu o aggregae supply can no experience shor-run overshooing. This leaves Θ > as he only possible srucural explanaion for shor-run overshooing. In general, he response of oupu o a permanen shoc is a linear combinaion of oupu responses o Φ aggregae supply and demand, and when Θ is posiive, he coefficien on in equaion (24) is posiive. Macroeconomic heories ofen predic an aggregae demand shoc will have is pea effec on oupu wihin a year, an effec ha is qualiaively similar o he shor-run overshooing observed in 16

18 models wih annual prewar daa. If aggregae demand has a long-run posiive oupu effec and his effec is sufficienly large, he response of oupu o a permanen shoc could inheri shor-run over-shooing behavior from he dynamic response of oupu o aggregae demand. Mos of he research wih permanen and ransiory decomposiions has used poswar daa. Empirical resuls from his sample period are ypically consisen wih he aggregae supply inerpreaion of permanen shocs and he aggregae demand inerpreaion of ransiory shocs. Therefore, economiss ofen reach he conclusion ha he exboo aggregae demand and supply srucure provides a good descripion of poswar economies. Anoher ineresing finding is ha he amoun of variance explained by permanen shocs o oupu ends o be larger in he pre-1914 period han in he pos-world War II period. This finding is obained by Keaing and Nye (1999) who followed Blanchard and Quah and used he unemploymen rae and also by Keaing and Nye (1998) who used inflaion in place of he unemploymen rae. This relaionship can only occur a finie horizons because as he forecas horizon goes o infiniy permanen shocs explain 1% of he variance of oupu by consrucion. While here may be a variey of significan differences beween prewar and poswar economies, i would be ineresing o deermine wheher a significan difference in he long-run oupu effec of aggregae demand, by iself, could explain observed differences in variance decomposiion across hese wo periods. The following proposiion addresses his quesion: Proposiion 2. If he aggregae supply and demand srucure applies o wo economies, demand shocs o Economy A are long-run neural, demand shocs o Economy B may have a long-run effec on oupu, and his is he only difference beween hese wo economies, hen he fracion of finie horizon oupu variance associaed wih permanen shocs is larger for Economy B if and only if aggregae demand has a long-run posiive effec on oupu in Economy B. 17

19 Proof of Proposiion 2: See Appendix B. Le Economy A be a poswar economy for which empirical resuls are usually consisen wih exboo heory. Le Economy B be a pre-1914 economy. Oupu variance explained by permanen shocs ends o be larger in he earlier period for he sample of counries sudied by Keaing and Nye (1998,1999). If neuraliy holds in he poswar economies and he only significan difference beween hese wo sample periods is ha aggregae demand may be non-neural in he prewar, hen Proposiion 2 ells us ha a posiive long-run effec of aggregae demand on oupu in he prewar period by iself can explain why permanen shocs accoun for more oupu variance in his earlier sample period. Hence, Proposiion 2 provides furher suppor for he hypohesis ha some economies in he lae 19 h and early 2 h cenuries experienced a permanen increase in oupu from aggregae demand shocs. As is clear from he appendix, proving Proposiion 2 is equivalen o showing ha permanen shocs will explain more oupu variance han he amoun aribuable o supply, if and only if he aggregae demand shocs have a long-run posiive effec on oupu. I urns ou ha his proof requires he assumpion ha a permanen shoc always has a posiive effec on oupu. This assumpion is srongly suppored by empirical resuls in Keaing and Nye (1998,1999) and is used o rule ou exremely negaive values of Θ. A criicism of he permanen-ransiory decomposiion lieraure is ha few sudies have provided formal ess of he idenificaion resricions. While researchers using poswar daa have ofen found ha heir saisical models are consisen wih exboo heory, failure o reec a heory, of course, does no mean ha a heory is necessarily correc. In mos cases, consisency wih a heory is based on qualiaive properies of impulse response funcions. An ineresing quesion is: How are he qualiaive feaures of a saisical model s impulse responses affeced by differen values for he srucural parameers? 18

20 Proposiion 3: Impulse responses o permanen and ransiory shocs are consisen wih he qualiaive effecs of exboo aggregae supply and aggregae demand shocs, respecively, over a range of values for Θ. Proof: Calculae bounds on Θ such ha he saisical model finds ha he shocs ha permanenly increase oupu cause price o fall and emporary shocs ha iniially increase oupu cause price o rise. The following discussion presens and inerpres hese bounds. Based on equaion (23), i is easy o show ha he emporary shoc causes oupu and he price level o rise, respecively, when PD >Θ > PS Φ Φ Φ Θ Φ Θ. Assumpions A1 hrough A4 imply ha if Θ is greaer han he negaive value on he righ for all and less han he posiive number on he lef for small values of (small meaning non-negaive ha are less han K), hen he saisical model will yield impulse responses o emporary shocs ha appear lie he aggregae demand shocs from exboo macro heory. If we ever observe oupu and price responses o emporary shocs moving in opposie direcions for small, ha would imply Θ falls ouside he range of values given above. Unforunaely, observing his sor of response would no ell us wheher he longrun effec of demand is posiive or negaive. 14 Now examine he responses o a permanen shoc. Based on equaion (23), a permanen oupu shoc has a negaive effec on price when: PS PD Φ Θ >Θ Φ Given A1, A2 and A4, his expression ses a posiive upper bound on Θ such ha he permanen shoc. 19

21 o oupu will cause a drop in he price level. Proposiion 1 addressed he eigh pre-world War I economies for which his inequaliy was violaed. Nex deermine condiions under which he permanen shoc will always have a posiive effec on oupu. This response is posiive a long horizons, by consrucion, for any value of Θ. Consider firs he case when Φ is posiive. Under his assumpion he permanen shoc o oupu is posiive afer periods when Θ > Φ Θ Φ, implying ha Θ is greaer han his negaive number, given A1, A3 and he currenly mainained assumpion ha Φ is posiive. Wha if Φ is negaive for cerain values of? Then he permanen shoc will have a posiive effec on Y afer periods when: Θ < Φ Θ Φ. There are wo imporan differences beween his inequaliy and he previous one. Firs, he inequaliy is reversed because of division by Φ, which is now assumed negaive, and he second difference is ha he righ-hand side of he inequaliy is now a posiive number. This new condiion ses a posiive upper bound on Θ. Thus we have esablished a range of posiive and negaive values for Θ ha permi he permanen shoc o have a posiive effec on oupu for all. Based on he large number of empirical sudies ha assume neuraliy of aggregae demand, his assumpion would seem o have a significan amoun of credibiliy in he economics profession. I is no surprise ha empirical findings consisen wih he exboo model are inerpreed as suppor for his 2

22 heory. Bu in proving Proposiion 3, we have deermined a range of values for Θ ha permi he responses o permanen and ransiory shocs o appear consisen wih he effecs of aggregae supply and demand shocs, respecively. While Θ = is included in his range of values, he qualiaive properies of impulse responses do no serve as a reliable basis for udging ha Θ is essenially equal o zero. And if Θ is no very close o zero, he impulse responses from he saisical decomposiion of oupu may differ subsanially from he dynamic effecs of srucural shocs. 6. Discussion and Conclusions This paper aemps o provide srucural inerpreaions for a number of empirical resuls. One imporan conclusion is ha aggregae demand had a permanen posiive effec on oupu in prewar economies. This conclusion derives from he endency for he prewar period s permanen oupu shocs o move prices and oupu in he same direcion, obain oupu responses ha in he shor run overshoo he long-run response, and explain more oupu variance han he permanen shocs in he poswar. An imporan area for fuure research is o invesigae possible srucures ha may have caused aggregae demand o be non-neural in pre-world War I economies. While quie a few poenial explanaions exis, a number of hem seem unliely based on observable differences beween economies of hese wo periods. For example, non-neural long-run effecs from permanen changes in he growh rae of money are no eviden in he prewar period. For non-superneuraliy o have been a significan facor, here would need o have been persisen or permanen changes in he growh rae of he money supply. Such changes would be expeced o show up as uni roo behavior in inflaion, money growh and oher nominal variable growh raes, even if anoher ind of non-saionariy provides a beer descripion of he daa generaing process. In fac, a uni roo for inflaion can be reeced in daa from he pre-1914 sample period. I is more difficul o reec his hypohesis for inflaion in he poswar period, and so long- 21

23 run non-superneuraliy sands a beer chance of being a facor in poswar economies. 15 Crowding-ou or crowding-in effecs from fiscal spending and permanen oupu effecs from ax rae policy are also poenial explanaions of permanen oupu effecs from aggregae demand. Bu once again hese possibiliies are more liely a facor in poswar economies. Tax raes and governmen s share of oupu were boh very low in he prewar period. Excep for imes of war, large-scale governmen involvemen in he macroeconomy did no occur unil afer he Grea Depression. Hyseresis heories 16 of he labor mare provide anoher mechanism hrough which aggregae demand may have long-run effecs on he level of oupu. For example, if recessions cause a permanen loss in he soc of human capial, hen he marginal produc of labor will decline. This would cause a permanen decline in he demand for labor which could increase he naural unemploymen rae. Hence, an adverse shoc o aggregae demand would no only induce a recession, bu also would cause a reducion in full-employmen oupu if hyseresis is a facor. And posiive aggregae demand shocs would cause he opposie effec. Consequenly, he unemploymen rae would liely exhibi permanen changes resembling uni roo behavior. While ess wih poswar unemploymen rae daa have some difficuly reecing a uni roo, a uni roo is easily reeced for unemploymen raes from he pre-1914 sample period. This weaens he case for he hyseresis explanaion of permanen oupu effecs from aggregae demand in prewar economies. To summarize, hyseresis, long-run non-superneuraliy of money and permanen oupu effecs from fiscal policy do no appear o be imporan facors in prewar economies. Each of hese effecs has a beer chance of being relevan during he poswar, bu i is his period for which impulse responses are mos ofen consisen wih sandard exboo macro heory. Coordinaion failure heories provide a more plausible explanaion of permanen oupu effecs from aggregae demand in pre-world War I economies. A coordinaion failure may occur when economic decisions have sraegic complemenariies or spillover effecs. 17 Simple examples of his effec are when 22

24 he liquidiy of a financial mare depends on he number of oher mare paricipans or when he uiliy of a communicaions device (a elegraph, a elephone, he Inerne, ec.) depends on he number of oher users of ha same echnology. Coordinaion failure economies ofen have muliple equilibria. Consequenly, a posiive aggregae demand shoc may push he economy o a new equilibrium ha has a higher level of economic aciviy, and a negaive shoc could do he opposie. The observaion ha ransacions coss were falling hroughou he nineeenh and wenieh cenuries is consisen wih his heoreical explanaion. 18 Transacions coss for businesses and individuals were lowered by expansion of commercial baning and financial inermediaion services, advances in ransporaion and improvemens in communicaions echnology. I is plausible ha a reducion in ransacions coss over ime may have ransformed economies from coordinaion failure srucures in he 19 h cenury o modern srucures for which exboo macro heory is a good descripion. Desabilizing price flexibiliy is anoher plausible explanaion for permanen oupu effecs from aggregae demand. While macroeconomic heory ypically ells us ha more rapid price adusmen causes aggregae demand o have smaller oupu effecs, here are some heories in which falling prices may acually push he economy away from full employmen. For example, Fisher (1933) describes how deflaion and deb may produce an adverse aggregae oucome. Keynes (1936), Tobin (1975) and DeLong and Summers (1986) have emphasized how falling prices migh raise real ineres raes and reduce spending, 19 an effec ha may be paricularly significan when nominal ineres raes are close o he zero bound. Consisen wih hese ideas is he evidence (discussed briefly in foonoe 11) ha prices were more flexible in he pre-1914 period han in he poswar and he fac ha many economies experienced periods of subsanial deflaion during he prewar. 2 While here is some evidence consisen wih desabilizing price flexibiliy and some ha is consisen wih coordinaion failures, his in no way proves ha eiher heory is correc. Formal ess wih prewar daa are required for hese hypoheses or any ohers ha migh be offered o explain he empirical 23

25 resuls. Deermining why aggregae demand is no neural in many pre-world War I economies is imporan in is own righ. And i is also possible ha he same srucural mechanisms may have caused aggregae demand shocs o have permanen oupu effecs in modern economies. While he poswar resuls are largely consisen wih exboo heory, Proposiion 3 shows ha his evidence does no necessarily mean he mechanisms causing prewar non-neuraliy of aggregae demand have become irrelevan. Bu if hese prewar effecs coninue o be imporan, heir influence is now relaively smaller given ha price levels almos never rise following a permanen increase in oupu in he poswar. Finding such mechanisms are sill relevan would enable us o develop a beer undersanding of he srucure of modern economies. Of course, here is no guaranee he mechanisms ha affeced pre-1914 economies are sill imporan. Fundamenally differen srucural mechanisms ha invalidae long-run neuraliy of aggregae demand may be a wor in he poswar. An implicaion of Proposiion 3 is ha economiss should es he hypohesis ha aggregae demand is neural, raher han asser neuraliy as is commonly done wih srucural VARs idenified by long-run resricions. Some ess of long-run neuraliy proposiions have been performed, bu more ess are required along wih new esing mehods or improvemens o currenly available procedures. 21 If he ess generally fails o reec neuraliy, hen he vas empirical research employing neuraliy assumpions can be given srucural inerpreaions. However, wide-spread reecion of neuraliy would call for new empirical wor on a wide array of macroeconomic quesions. The basic ideas in his paper can be applied o a bivariae permanen and ransiory shoc decomposiion for any variable. 22 The srucural assumpions will, of course, depend on he variable ha is being decomposed ino hese wo ypes of shocs as well as he oher variable in he model. New insighs migh be obained by exending his bivariae analysis o a seing wih more han wo shocs. Wih his exension we could invesigae he srucural implicaions of models ha idenify muliple permanen shocs o oupu and muliple emporary shocs, for example. 23 Taing he analysis 24

26 from his paper o such models is no, however, a simple exension. One complicaion will be marix algebra ha is cerainly more edious. A more difficul problem would be if he number of inequaliies increases o he poin where i becomes difficul or impossible o obain unambiguous resuls. Neverheless, because of he many papers ha have idenified muliple permanen and/or muliple ransiory shocs, his exension may well be worh pursuing This paper provides a deeper undersanding of he relaionship beween permanen-ransiory shoc decomposiions of oupu and economic srucure. In general, when he ey idenificaion assumpion is no a valid srucural resricion, his decomposiion will obain inconsisen esimaes of he srucural responses. Neverheless, if plausible srucural assumpions are available, he permanenransiory shoc decomposiion may sill be used o infer imporan facs abou he economy. 25

27 Appendix A: Recursive Subsiuion Equaion (2) can be wrien as: This relaionship holds for any ime period, and so: X = X +θ(l) ε 1 X = X +θ(l) ε Subsiuing his equaion for X -1 ino he firs equaion yields: X = X +θ(l) ε +θ(l) ε 2 1 Then a similar subsiuion is made for X -2, followed by X -3, ec., o obain: This equaion can be wrien as: 1 X = X + θ(l) ε = X = X + ( θε +θε +θε +...) + ( θε +θε +θε +...) ( θ ε +θ ε +θ ε +...) Maching up he coefficiens on each g yields equaions (3) and (4). And precisely he same mehod is used wih equaion (8) o obain equaions (9) and (1)

28 Appendix B: Proof of Proposiion 2. The hree srucural assumpions in his proposiion are: I. For Economy A, he exboo aggregae supply and demand model describes he srucure, hence he permanen-ransiory shoc decomposiion idenifies srucural effecs. II. For Economy B, aggregae demand shocs have permanen effecs on oupu, hence he permanenransiory shoc decomposiion is unable o idenify effecs of aggregae supply and demand. III. Boh economies have idenical shor-run and inermediae-run srucures. Calculae he -sep forecas error for each economy: For Economy A, equaion (3) gives he permanen-ransiory shoc decomposiion, and herefore he - sep forecas error is: 1 1 S Φ Φ ε X E X = Φε = PS PD D = = Φ Φ ε. (i) For economy B, equaion (9) yields he decomposiion because aggregae demand has a permanen effec on oupu. Hence, he -sep forecas error for his economy is: 1 P Φ Φ Θ Θ µ 1 PS PD T 1 = Φ Φ Θ Θ µ = Φθ µ = 1/2 = X E X (1)C(1) 2 2 ( Θ +Θ ) Alhough Φ can be he same for all because he wo economies are differen in he long run, hese srucural parameers are he same for each economy for some range of finie because of Assumpion III. Recall ha oupu is he firs elemen in X. Then, he -sep forecas variance for oupu associaed wih permanen shocs for Economy A is obained from equaion (i): 1 1 = ( Φ ) ( Φ ) + ( Φ ) = and for Economy B is obained from equaion (ii): 1 ( ) ( ) = Φ Θ + Φ Θ + Φ Θ Φ Θ 1 = 2 2 ( Φ ) ( ) ( Φ Θ +Θ) (ii), (iii). (iv) The quesion is wha condiions on Θ will guaranee ha permanen shocs explain a larger fracion of oupu variance in Economy B? Muliplying by posiive denominaors and collecing erms, we can show ha expression (iv) is greaer han expression (iii) when: 27

29 ( ) ( ) 2 Θ Θ Φ Φ +Θ Φ Φ > = = I is convenien o divide by Θ squared, and facor he expression as follows: 2 ( ) ( ) > 1 1 Θ 2 2 Θ Φ Φ + Φ Φ Θ = Θ = From assumpions A1 and A3 we now ha ( Φ ) ( Φ ) = 1 = Φ Φ is posiive. However,. (v) may be zero, negaive or posiive depending on he relaive imporance of supply and demand shocs for oupu during he firs periods. Consider each of hese hree cases. When ( Φ ) ( Φ ) = is: Θ Case 1: zero, > saisfies (v); Θ Θ = Case 2: negaive, < < > ; saisfies (v); 1 Θ 2 2 posiive, (v) is saisfied by eiher: Θ = 1 2 Φ Φ ( Φ ) ( Φ ) Θ Case 3a: > ; or Θ Θ = Case 3b: < <. Θ = 1 2 Φ Φ ( Φ ) ( Φ ) For Cases 1, 2 and 3a, is posiive, and herefore Θ is posiive. However, Case 3b yields a Θ negaive value for his parameer. If we can show ha hese negaive Θ values are irrelevan, ha will complee a proof ha Θ is posiive. 28

30 The ey o ruling ou he negaive values is o noe ha permanen oupu shocs always have a posiive effec on oupu in he esimaes. This posiive oupu response places a lower bound on Θ, a negaive number ha is no as negaive as he values of Θ in Case 3b. Hence, he negaive values of Θ ha saisfy (v) are so negaive ha a permanen shoc would cause oupu o fall a some poin in he impulse response, a condiion ha is ruled ou by he evidence. The response of oupu o a permanen shoc is given by equaion (24), and if he firs responses are posiive his equaion implies: Θ Φ > for =,1,...,-1. Θ Le Φ ρ = Φ Φ. Each ρ is posiive because of assumpions A1 and A3. Define ρ * as he minimum over all ρ for =, 1, 2,..., -1. Based on he previous inequaliy we can see ha if were smaller han Θ -ρ *, some porion of oupu s response o a permanen shoc would be negaive. Since oupu does no fall in response o a permanen shoc, -ρ * ses a lower bound for are ruled ou if: 1 = 1 = 2 Φ Φ ( Φ ) ( Φ ) To show ha he inequaliy in (vi) holds, eliminae ( Φ ) ( Φ ) = Since ρ is posiive for all and 2 2 Φ Θ Θ Θ. The negaive values in Case 3b < -ρ *. (vi) using he definiion of ρ, and remembering is posiive for his analysis, manipulae (vi) ino he following inequaliy: 1 = ( Φ ) 2 * * (2 ρ ρ +ρ ρ ) ρ ρ* holds, ruling ou Case 3b and compleing he proof of Proposiion 2. 2 >. for all, his inequaliy is unquesionably rue. Therefore (vi) 29

31 Noes 1. See Campbell and Maniw (1987), Clar (1987), Cochrane (1988) and Wason (1986), for example. 2. Quah (1992) presens heoreical resuls showing poenial advanages of a mulivariae approach. 3. Lippi and Reichlin (1993) develop mehods for handling non-inverible srucures. Blanchard and Quah (1993) discuss relevancy and implicaions of non-inveribiliy in heir reply. 4. Shapiro and Wason (1988), King Plosser, Soc and Wason (1991), Gamber and Jouz (1993) and Amed, Ices, Wang and Yoo (1993) are early examples of models wih more han wo shocs. 5. Bordo (1993), Bayoumi and Eichengreen (1994), Karras (1994) and Keaing and Nye (1998) use inflaion insead of he unemploymen rae. 6. While i is rue ha Φ 4 =θ(1), i is useful o disinguish beween finie horizon effecs, Φ for finie values of, and long-run effecs, θ(1), in he analysis o follow. 7. See Hamilon (1994, p.91). 8. This poin is relaed o he fac ha correlaion does no always imply causaion. 9. Faus (1998) and Uhlig (1999) employ sign resricions o esimae srucures. While Waggoner and Zha (23) discuss hese wo papers in ligh of problems ha may arise from inappropriaely normalizing equaions in a mulivariae sysem, hey also poin ou ha recursive models are no subec o such problems. Bu I only use sign resricions for inerpreaion of he saisical model. The permanenransiory shoc decomposiion is recursive, and is herefore no subec o hese normalizaion concerns. 1. Basu, Fernald and Kimball (1999) is a rare excepion ha heoreically shows oupu iniially falling afer a echnological improvemen. I am, however, unaware of any empirical evidence for such effecs. 11. I could have made he even weaer assumpion ha his price response is non-negaive. Permiing PD Φ = allows us o inerpre resuls under he assumpion ha prices are sicy for some ime following an aggregae demand shoc. When his response is zero, here is only one new insigh: A permanen increase in oupu will unambiguously lower he price level. This is ineresing because here is some evidence ha prices are sicy in he poswar and in nearly all poswar esimaes price falls wih a permanen increase in oupu. Furhermore, in all bu wo of he prewar esimaes price rises wih a PD permanen increase in oupu. This finding reecs Φ =, which is consisen wih he view of some economiss ha price adusmen was comparaively fas in he prewar. See he discussion in Calomiris and Hubbard (1989) and heir references o differences in he speed of price adusmen. 12. Secion 4 in heir paper discusses how problems wih he qualiy and consisency of pre-1914 daa are unable o explain his unusual finding. 3