THE UNIVERSITY OF KANSAS WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS

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1 THE UNIVERSITY OF KANSAS WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS INTERPRETING PERMANENT AND TRANSITORY SHOCKS TO OUTPUT WHEN AGGREGATE DEMAND MAY NOT BE NEUTRAL IN THE LONG RUN John W. Keaing Deparmen of Economics Universiy of Kansas July 24 THE UNIVERSITY OF KANSAS W P S T A E ORKING APERS ERIES IN HEORETICAL AND PPLIED CONOMICS WORKING PAPER NUMBER 248

2 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) July 6, 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 Economeric Sociey Summer Meeings a Brown Universiy, he Midwes Economerics Group Meeing a he Universiy of Chicago, he Midwes Macroeconomic Conference a Iowa Sae Universiy, and he Missouri Economics Conference a he Universiy of Missouri for helpful commens on various 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.

3 1. Inroducion Economiss have for many years 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 mehods, 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 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) shocs o supply and demand are uncorrelaed. 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 and as a resul heir specificaion could be misspecified. In fac, Blanchard and Quah (1989) show in an appendix he condiions under which heir approach will successfully idenify he effecs of he aggregae supply and aggregae demand shocs even when an economy experiences muliple ypes of each shoc. Faus and Leeper (1997) elaboraed on ha poin and 1

4 exended he discussion in a number of imporan direcions. Anoher concern involves he use of unemploymen rae daa in he decomposiion. Some 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 ofen predics ha all inds of supply shocs will affec oupu and he price level in qualiaively he same way and ha each of hese variables 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. The 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 qualiaively differen effecs on oupu and he unemploymen rae, hen a bivariae decomposiion based on unemploymen will no be able o idenify he effecs of shocs o supply and demand. Consequenly, I will focus on models ha use price daa in place of he unemploymen rae. The primary concern of his paper is 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 any of hem are relevan, he primary srucural assumpion ha usifies Blanchard and Quah s decomposiion would be invalid. This paper analyically examines he effecs of his alernaive assumpion on he 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 2

5 findings from he empirical lieraure. Firs, impulse responses for poswar economies are ypically consisen wih exboo heory; Permanen shocs behave lie aggregae supply shocs and emporary shocs behave lie aggregae demand shocs when economiss use daa ses ha begin afer World War II. However, his resul is no robus o all 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 he prewar economies in Keaing and Nye (1998), where prewar describes a sample period ha ends us before World War I. These prewar responses o permanen shocs are inconsisen wih he aggregae supply shoc inerpreaion of permanen shocs o oupu. Third, ha same sudy finds ha he 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 of he impulse responses is ermed shor-run overshooing. While common in he prewar, shor-run overshooing of oupu responses o permanen shocs is no observed in poswar esimaes. I use a se of plausible srucural assumpions o inerpre he permanen-ransiory shoc decomposiion. Bu insead of he sandard assumpion ha aggregae demand shocs are neural in he long run, 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 long-run 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 resuls sugges ha exboo aggregae supply and demand heory provides a good descripion of economies in ha period. However, he endency for permanen oupu shocs in prewar daa o cause prices and oupu o move in he same direcion, yield shor-run overshooing for oupu 3

6 responses, and explain more oupu variance han hese shocs do in he poswar, suppors he hypohesis ha aggregae demand shocs had long-run posiive oupu effecs for some counries 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, aggregae demand shocs migh sill be having permanen effecs on oupu. I is found 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 economic heories in which aggregae demand may be non-neural 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 opics for fuure research. 2. The Srucure and he Saisical Model Saisical models can provide a means of discovering srucural relaionships. This secion will describe a saisical model and a srucure in erms of he dynamic responses of variables o shocs, or moving average represenaions (MARs) as hey are nown 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 4

7 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 2 difference operaor and β(l)=i-β1l-β2l...-β L κ 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 use of firs differences is a common way of modeling ime series subec o permanen shocs. 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 or he heoreical assumpions used o inerpre he model, hen he resuls 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 and each θ is a 2 2 marix of srucural parameers, specifies he dynamic responses of Y and P o hese θ = θ θ PS PD θ θ assume supply and demand shocs are uncorrelaed, a sandard assumpion in he srucural VAR for all. If we 5

8 lieraure, hen i is convenien o normalize hese shocs o have variances equal o one: Eεε = I. Appendix A shows how recursive subsiuions are used o ransform equaion (2), he sysem for 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 in (4) defines Φ as he -h parial sum of 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 Φ Φ Φ = PS PD Φ Φ for v=y,p and i=s,d. While economiss would lie o esimae he srucural responses in (4), his paper is concerned wih condiions under which he saisical model may be unable o obain consisen esimaes of he dynamic srucure. The long-run responses of variables o shocs are obained by leing go o infiniy in (4): (5) X lim = θ = θ( 1) ε = (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 6

9 Le he MAR for he saisical model be wrien as: X = C(L) µ (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 have been assumed uncorrelaed, and herefore he variance of each shoc in he saisical model can be normalized o one for convenience: Eµ µ = I. Following he same recursive subsiuion procedure ha was used wih (2) o generae (3), equaion (8) can be ransformed ino a relaionship in erms of X: X = X + C µ + (C + C ) µ + (C + C + C ) µ , (9) yielding he impulse responses of Y and P o permanen and ransiory shocs: 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 7

10 no have a permanen effec on oupu, as seen from equaion (11): Y lim CYT. T = = µ 3 Relaionships beween he Saisical and Srucural MARs The saisical MAR can no be idenical o he srucural MAR when he idenificaion resricions are no valid srucural resricions. The easies 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 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 saisical model and srucural parameers. I will use hese equaions o describe a wellnown mehod of calculaing coefficiens in he saisical decomposiion and o characerize he way ha coefficiens from he saisical model are relaed o srucural parameers when Θ. Given equaion (13) and he ideniy covariance marix 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), 8

11 he marix describing he sum of VAR coefficiens, is obained by seing L=1 in equaion (14): 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 illusraes a popular mehod for esimaing parameers in he saisical model. Given ha C(1) is riangular, he C(1) parameers can be obained by he appropriae Cholesy decomposiion of he righ-hand side of equaion (17). Then he dynamic responses of variables o permanen and ransiory shocs can be obained by insering he esimae of C(1) ino he firs equaliy from equaion (16), solving for C, insering C ino he firs equaliy in equaion (14), and solving for C(L). 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. This is seen by noing θ(1) is lower riangular when Θ =, and herefore equaion (18) yields C(1) = θ(1) because he lower riangular facor of a symmeric marix is unique. 7 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). 9

12 As one would expec, he saisical model idenifies he complee dynamic responses of each variable o each srucural 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: ( ) 1 2 C = Θ +Θ YP C PS PD, and. PP Θ Θ +Θ Θ = C Using hese hree ideniies, he following resul is easily obained: YP C PT Θ Θ Θ Θ = C PD PS YP 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

13 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 seems o sugges ha when he saisical model uses inappropriae idenificaion resricions, i will no be able o ell us anyhing useful abou he srucure of an economy. However, I will show ha if oher assumpions abou he economic srucure are available, his misspecified saisical model migh sill be used o infer imporan facs abou he underlying economic srucure. 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. 9 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 do no wish o assume aggregae demand is long-run neural wih respec o real oupu, hen alernaive srucural assumpions are required o provide an economic inerpreaion of he saisical model. 1 Forunaely oher assumpions are available. Economic heory ofen places bounds on he qualiaive responses of variables o srucural shocs. 11 For example, mos heories predic ha a 11

14 beneficial aggregae supply shoc will raise oupu 12 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 affecing aggregae supply. 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

15 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 he firs 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 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 demand is neural in he long-run. Bu even if demand has a posiive long-run effec on oupu, i is possible for damped cycling around he seady sae o generae a negaive oupu response as he economy aduss o is long run posiion. The lielihood of ha is a funcion of how large he cyclical ampliude is relaive o he long-run effec of demand on oupu. I also assume ha aggregae demand shocs which iniially cause oupu o rise will have a posiive effec on he price level: 13 13

16 P D PD = Φ > A4: for all. ε If, in addiion o shifing demand, 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 also shifs long-run aggregae supply o he righ, Θ >, hen for A4 o hold he supply curve mus no shif by as much as demand does. 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 of hese counries. 14 In 5 cases, his effec is saisically significan. 15 This evidence srongly reecs he exboo srucure which underlies Blanchard and Quah s (1989) decomposiion because if permanen shocs are supply shocs hey should move price and oupu 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 14

17 wih an increase in he price level, hen aggregae demand mus have a posiive effec on oupu in he long run. 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 findings wih pre-1914 daa migh suppor heories in which aggregae demand shocs have posiive long-run effecs on oupu, and Proposiion 1 provides formal usificaion for ha 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 previous inequaliy for all in each of hese economies. Hence, while hese price responses reec he srucural model, hey also imply, under arguably more plausible srucural assumpions, ha aggregae demand had a posiive longrun 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. 16 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 will argue ha he only plausible srucural explanaion for his overshooing is ha 15

18 aggregae demand shocs had permanen posiive effecs on oupu. The response of oupu o a permanen shoc is aen direcly from equaion (23): = 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 supply shoc. Theory ypically shows ha oupu gradually rises in adusmen o a permanen beneficial supply shoc, wih he possibiliy of cyclical dynamics as he economy approaches he seady sae. Thus Θ = is unable o explain shor-run overshooing. Now consider he case of Θ <. Along wih A1, A3 and (24), his assumpion 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 in (24) is negaive. Furhermore, for any non-zero Θ : Y lim C ( ) P µ 2 2 YP 1 = = Θ +Θ 2 >Θ. The implicaions are 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 does 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 aggregae supply and demand. When Θ is posiive, he coefficien on Φ in equaion (24) is 16

19 posiive. Macroeconomic heories ofen predic an aggregae demand shoc will have is pea effec on oupu afer abou a year or so, somehing ha is qualiaively similar o he shor-run overshooing observed in models wih annual prewar daa. Thus 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 shorrun 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 follow Blanchard and Quah and use he unemploymen rae and also by Keaing and Nye (1998) who use inflaion in place of he unemploymen rae. Of course, ha difference can only occur a finie horizons because as he forecas horizon goes o infiniy he permanen shocs explain 1% of he variance of oupu by consrucion. While here may be a variey of imporan 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 he differences in variance decomposiion. 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 17

20 demand has a long-run posiive effec on oupu in Economy B. 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 his earlier period for he 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 long-run non-neural in he prewar period, hen Proposiion 2 ells us ha a posiive long-run effec of aggregae demand on oupu in he prewar period by iself could explain why permanen shocs accoun for more oupu variance in his earlier sample period. Hence, he variance decomposiion resuls provide 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. This proof employs he assumpion ha a permanen shoc always has a posiive effec on oupu, which is used o rule ou exremely negaive values of Θ. This assumpion is srongly suppored by he empirical resuls of Keaing and Nye (1998,1999). A criicism of he permanen-ransiory decomposiion lieraure is ha few sudies have provided formal ess of he idenificaion resricions. Using poswar daa researchers have ofen found ha heir empirical models are consisen wih exboo heory, bu of course he failure o reec a heory does no mean he heory is necessarily correc. In mos cases, consisency wih heory is based on qualiaive properies of impulse response funcions. An ineresing quesion is: How are he qualiaive feaures of 18

21 he saisical model s impulse responses affeced by differen values for long-run srucural parameers? 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 he shocs ha permanenly increase oupu cause price o fall and he 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. (25) 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 (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 in (25). 17 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. 19

22 Given A1, A2 and A4, his expression ses a posiive upper bound on Θ such ha he permanen shoc 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 when Θ > Φ Θ Φ, implying ha Θ is greaer han a negaive number, given A1, A3 and he currenly mainained assumpion ha Φ is posiive. Wha if Φ is negaive for cerain values of >K? 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 sign is reversed because of division by Φ, which is now assumed negaive. The second difference is ha he righ-hand side of he inequaliy has become 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 long-run neuraliy of aggregae demand, his srucural assumpion would seem o have a significan amoun of credibiliy. Hence, i is no surprise ha empirical findings consisen wih he exboo model are inerpreed as suppor for his 2

23 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, his proposiion shows ha he qualiaive properies of impulse responses do no serve as a reliable basis for udging wheher or no Θ is essenially equal o zero. And if Θ is no very close o zero, he impulse responses from he saisical decomposiion of oupu will almos cerainly differ by a subsanial margin from he dynamic effecs of srucural shocs. 6. Discussion and Conclusions This paper aemps o provide srucural inerpreaions for a number of empirical findings using permanen-ransiory shoc decomposiions of oupu. The resuls developed here imply ha aggregae demand had a permanen posiive effec on oupu in a number of 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 heir long-run responses, 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 growh raes for oher nominal quaniies, even if anoher ype of non-saionariy would provide a more 21

24 accurae descripion of he daa generaing process. In fac, a uni roo for inflaion can be reeced in daa from he pre-1914 sample period for he 1 counries sudied in Keaing and Nye (1998). I is more difficul o reec his hypohesis for inflaion in he poswar period, and so long-run non-superneuraliy may sand a beer chance of being a facor in poswar economies. 18 Crowding-ou or crowding-in effecs from governmen spending and supply-side effecs from changes in ax raes 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 shares of oupu were boh very low in he prewar period, and large-scale governmen involvemen in he macroeconomy did no occur unil afer he Grea Depression. Hyseresis heories 19 of he labor mare provide anoher mechanism hrough which aggregae demand may have long-run effecs on he level of oupu. For example, if a recession causes 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, if hyseresis is a facor, an adverse shoc o aggregae demand would no only induce a recession, bu also could cause a reducion in full-employmen oupu. And a posiive aggregae demand shoc would have he opposie effec. Consequenly, he unemploymen rae would liely exhibi permanen changes resembling uni roo behavior. 2 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 for counries sudied in Keaing and Nye (1999). This evidence 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 seems o have a beer chance of being relevan during he poswar, bu i is his period for which impulse responses are ypically consisen wih sandard exboo macro heory. 22

25 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. 21 Simple examples of his effec are when he liquidiy of a financial mare depends on he number of oher mare paricipans or when he uiliy of a communicaions device (e.g. elegraph, elephone, ec.) depends on he number of oher users of ha same echnology. Coordinaion failure economies ofen exhibi 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 fell hroughou he nineeenh and wenieh cenuries is consisen wih his heoreical explanaion. 22 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. Fisher (1933) describes how deflaion may combine wih deb o produce adverse aggregae oucomes. Keynes (1936), Tobin (1975) and DeLong and Summers (1986), for example, have emphasized how falling prices migh raise real ineres raes and reduce spending, 23 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 endnoe 13) ha prices were more flexible in he pre-1914 period han in he poswar and he fac ha a number of economies experienced periods of subsanial deflaion during he prewar period

26 While here is some evidence consisen wih desabilizing price flexibiliy and some consisen wih coordinaion failures, his in no way proves ha eiher heory is correc. Tess are required for hese hypoheses or any ohers ha migh be offered o explain he prewar resuls. Deermining why aggregae demand is no neural in a number of pre-world War I economies is imporan in is own righ. I is also possible ha he same srucural mechanisms may sill be relevan in modern economies. While poswar resuls are largely consisen wih exboo heory, we now from Proposiion 3 ha his evidence does no necessarily mean he mechanisms ha caused prewar non-neuraliy of aggregae demand have since become irrelevan. However, if hese prewar effecs coninue o be imporan, heir influence mus now be relaively smaller, given ha price levels almos never rise following a permanen increase in oupu in he esimaes wih poswar daa. Finding paricular non-neural mechanisms o sill be 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. The mos imporan implicaion of Proposiion 3 is ha economiss should formally es he hypohesis ha aggregae demand is neural. This sands in conras o he fac ha he vas maoriy of long-run srucural VAR models have assumed neuraliy is a valid srucural resricion. Some ess of long-run neuraliy proposiions have been developed, bu i would be beneficial o have new esing mehods or improvemens o currenly available procedures. 25 If he ess generally fail o reec neuraliy, hen he vas empirical research employing neuraliy assumpions can be given srucural inerpreaion. However, wide-spread reecion of neuraliy would call for new empirical wor on a wide array of macroeconomic quesions. The basic mehods of his paper can be applied o a bivariae permanen and ransiory shoc decomposiion for any variable. 26 The srucural assumpions will, of course, depend on he variable ha is decomposed ino hese wo ypes of shocs and also on he oher variable used in he saisical model. 24

27 And new insighs migh be obained by exending his bivariae analysis o a seing wih more han wo shocs. Wih his exension we could invesigae, for example, he srucural implicaions of empirical models ha idenify muliple permanen shocs o oupu and muliple emporary shocs. 27 Taing he analysis from his paper o such models is no, however, a simple exension. One complicaion almos cerain o arise is more edious algebra. An even more difficul problem would be if he number of srucural response inequaliies increases o he poin where i becomes difficul or impossible o obain unambiguous resuls. Bu given he large number of 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 economic srucure and a permanen-ransiory shoc decomposiion of oupu. 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 oher plausible idenifying assumpions are available, resuls from he decomposiion may sill be used o infer imporan facs abou he srucure of an economy. 25

28 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 expression 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 also be wrien as: 1 X = X + θ(l) ε = X = X + ( θε +θε +θε +...) + ( θε +θε +θε +...) ( θ ε +θε +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).. 26

29 Appendix B: Proof of Proposiion 2. Srucural assumpions: 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 27. (i) For economy B, equaion (9) yields he permanen-ransiory shoc 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 ) Given ha he economies are differen in he long run, heir Φ parameers can all be he same. However, from Assumpion III we now ha for some finie range of hese srucural parameers are he same for hese wo economies. The -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

30 fracion of oupu variance in Economy B? Muliplying by posiive denominaors and collecing erms, equaion (iv) is greaer han equaion (iii) when: ( ) ( ) 2 Θ Θ Φ Φ + Θ Φ Φ > = = I is convenien o divide by Θ squared, and facor he expression as follows: 1 1 Θ 2 2 Θ 2 Φ Φ + ( Φ ) ( Φ ) > Θ = Θ = From assumpions A1 and A3 we now ha 1 ( ) ( ) 1 = Φ Φ is posiive. However,. (v) 2 2 Φ Φ 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 ) = Θ Θ Case 1: Zero, hen > saisfies (v); Θ Θ is: Case 2: Negaive, hen < < > saisfies (v); 1 = Case 3: Posiive, hen (v) is saisfied by eiher: Θ 3a: > ; or Θ Θ 1 1 = 2 Φ Φ ( Φ ) ( Φ ) = 3b: < <. Θ Θ Θ = 2 Φ Φ ( Φ ) ( Φ ) 2 2 For Cases 1, 2 and 3a, is posiive, and herefore Θ is posiive. However, Case 3b yields a negaive value for his parameer. If we can show ha hese negaive Θ values are irrelevan, ha will 28

31 complee a proof ha Θ is posiive. The ey o ruling ou he negaive values is o noe ha permanen oupu shocs always have a posiive effec on oupu in he model esimaes. This posiive oupu response places a lower bound on Θ, a negaive number ha we will show 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 responses for - 1 periods 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 ) 2 2 To show ha (vi) holds, use he definiion of ρ o eliminae ( Φ ) ( Φ ) = Since ρ is posiive for all and Θ Θ Φ Θ. The negaive values in Case 3b < -ρ *. (vi) 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

32 1. Wason (1986), Campbell and Maniw (1987), Clar (1987) and Cochrane (1988), 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. In heir reply Blanchard and Quah (1993) discuss he relevancy and implicaions of non-inveribiliy. 4. Shapiro and Wason (1988), King Plosser, Soc and Wason (1991), Gamber and Jouz (1993) and Amed, Ices, Wang and Yoo (1993) are examples of models wih more han wo shocs. 5. Bordo (1993), Bayoumi and Eichengreen (1994), Karras (1994) and Keaing and Nye (1998) use he inflaion rae 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, and long-run effecs, θ(1), in he analysis o follow. 7. See Hamilon (1994, p.91). Noes 8. The following soluions ae he posiive roos for C YP and C PT because economiss ypically use he model o sudy he effecs of posiive shocs o supply and demand, respecively. Noe ha o obain his calculaion of C PT I have assumed ΘΘPD ΘΘPS. This inequaliy holds for any posiive value of Θ, given srucural assumpions A1, A2 and A4 found in he nex secion of he paper. However, i is possible ha Θ could ae a value so negaive ha he inequaliy does no hold. In his case, o mae C PT a posiive number, we would have o muliply he soluion for C PT in he ex by -1. This would have no effec whasoever on responses o permanen shocs, bu would affec he resuls for emporary shocs, which means primarily equaion (25). The mos ineresing implicaion of Θ being his negaive is ha a emporary oupu shoc will cause oupu o fall iniially and he price level o rise in he long run. These effecs are inconsisen wih virually all heories abou how he economy responds o aggregae demand shocs. 9. This poin is a corollary of he fac ha correlaion does no always imply causaion. 1. Cover, Enders and Hueng (forhcoming) permi aggregae demand o possibly have a permanen oupu by allowing supply and demand shocs o be correlaed. 11. Faus (1998) and Uhlig (1999) employ sign resricions, bu in conras o his paper, use hem o esimae a srucure. Waggoner and Zha (23) discuss hese wo papers in ligh of problems ha may arise from inappropriaely normalizing he equaions in a simulaneous sysem. They poin ou ha recursive models are no subec o such problems. I use sign resricions o inerpre a saisical model ha is recursive, and herefore my analysis is no subec o he normalizaion problems Waggoner and Zha have observed. 12. Basu, Fernald and Kimball (24) is a rare excepion. They describe a srucural mechanism by which oupu may iniially decline following a echnological improvemen. I am unaware, however, of evidence 3