On Phase Shifts in a New Keynesian Model Economy. Joseph H. Haslag. Department of Economics. University of Missouri-Columbia. and.

Transcription

1 On Phase Shifs in a New Keynesian Model Economy Joseph H. Haslag Deparmen of Economics Universiy of Missouri-Columbia and Xue Li Insiue of Chinese Financial Sudies & Collaboraive Innovaion Cener of Financial Securiy Souhwesern Universiy of Finance and Economics Absrac: The purpose of his paper is focus direcly on he phase shif. For one hing, we ask wheher a sicky-price model economy can accoun for boh counercyclical prices and procyclical inflaion. We presen findings in which he price level is counercyclical and he inflaion rae is procyclical. We proceed o use he model economy as an idenificaion mechanism. Wha se of individual shocks are sufficien o accoun for he phase shif? Tha se is empy. Nex, we ask wha se of shocks are necessary o accoun for he phase shif. This se conains echnology shocks and moneary policy shocks. The resuls are imporan as a building block. We infer ha price sickiness is an imporan model feaure; wihou price sickiness, we are in he real business cycle economies ha Cooley and Hansen sudied. Bu, i raises furher quesions. For insance, is price sickiness of he Roemberg form he one used here necessary o explain he phase shif? Keywords: Phase shif, counercyclical price, procyclical inflaion, necessary and sufficien shocks, Bayesian esimaion JEL Codes: E31, E32 The auhors wish o hank William Brock, Jesus Fernandez Villaverde, and Parick Minford for helpful commens. Any remaining errors are solely he auhors. 1

2 1. Inroducion Business-cycle facs are represened by comovemens, ypically wih real GDP, and by volailiies. When represening comovemens, researchers ofen characerize he lead-lag relaionship, or phase shif, beween ime series. Since Friedman and Schwarz (1963), researchers have sudied he relaionship beween he price level and oupu. As researchers applied saisical echniques o exrac he cyclical componen from he observed ime series, here were wo poswar correlaions ha are presened: he price level is counercyclical and he inflaion rae is procyclical. 1 There is a simple way o reconcile hese wo facs; here is a phase shif in he relaionship beween price level and oupu. To illusrae he phase shif, consider a case in which he cyclical componens of price level and oupu had he same periodiciy. Noe ha if he price level and oupu were in phase in he sense of he peak of he cycle in oupu occurred simulaneously wih he rough in he cycle of he price level, hen he rae of change in he price level ha is, he inflaion rae would be negaively correlaed wih oupu as he price level is. However, if he price-level waveform were shifed o he lef horizonally, hen i is possible for he price level o be counercyclical and he inflaion rae o be procyclical. I is jus such a shif in he price level relaive o oupu ha we are looking for in a DSGE model economy. In broad erms, business-cycle explanaions are divided ino wo camps. The New Keynesian camp and he Real-Business Cycle camp are similar in many ways. The consumpion-saving choice is an essenial dynamic force hrough which shocks are propagaed. In addiion, he aggregae echnology is indisinguishable beween he wo camps. And, expecaions are raionally formed. Of course, some forms of marke power and price sickiness are imporan differences beween he wo camps. Anoher is he way in which he New Keynesian model economies specified moneary policy; Woodford (2003) and ohers focused on moneary policy as means of seing he nominal ineres rae. In doing so, he so-called cashless model economies were being analyzed. Cenral banks specify nominal ineres rae arges and he Taylor rule is jusified on ha pracice. By specifying a Taylor rule, however, he price level can be ignored, or even discarded. Hence, counercyclical prices are essenially excluded from he se of facs considered by researchers specifying New Keynesian model economies. 1 The evidence is presened in Cooley and Ohanian (1991) for he poswar period and in Cooley and Hansen (1995). We see evidence of he dividing lines in wo quoes. In view of he counercyclical price level, Kydland and Presco (1990) sae: We cauion ha any heory in which procyclical prices figure crucially in accouning for poswar business cycle flucuaions is doomed o failure. (p.17, 1990). Laer den Haan (2000) counered, saying a heory in which prices do no have some procyclical feaure is, a bes, missing a par of he explanaion of U.S. business cycle flucuaions. (p. 5, 2000). 2

3 Modern macroeconomics has used he price level and he inflaion rae as a dividing line. Real Business cycle researchers can accoun for counercyclical price level by relying on he echnology as he driving force behind business cycle flucuaions. Alernaively, researchers in he New Keynesian camp can accoun for procyclical inflaion by relying on aggregae demand shocks. In doing so, he evidence ha a phase shif exiss coninues o be ignored. One previous aemp o accoun for he phase shif in raional-expecaions economies failed. Cooley and Hansen (1995) use sandard RBC models, bu find boh he price level and he inflaion rae are counercyclical. 2 So, hey calibrae a nominal conracing model. The sicky price resuls indicae ha he price level is acyclical in heir model economy. Thus he sicky price economy can generae a phase shif when he economy is subjeced o a echnology shock, bu he evidence suggess he phase shif is oo small. 3 The purpose of his paper is focus direcly on he phase shif. For one hing, we ask wheher a sicky-price model economy can accoun for boh counercyclical prices and procyclical inflaion. We presen findings in which he price level is counercyclical and he inflaion rae is procyclical. We proceed o use he model economy as an idenificaion mechanism. Wha se of individual shocks are sufficien o accoun for he phase shif? Tha se is empy. Nex, we ask wha se of shocks are necessary o accoun for he phase shif. This se conains echnology shocks and moneary policy shocks. The resuls are imporan as a building block. We infer ha price sickiness is an imporan model feaure; wihou price sickiness, we are in he real business cycle economies ha Cooley and Hansen sudied. Bu, i raises furher quesions. For insance, is price sickiness of he Roemberg form he one used here necessary o explain he phase shif? Brock and Haslag (2016) presen alernaive o he nominal conracing approach. They consruc a model economy in which prices are sicky because here exiss a se of echnologies used o forecas fuure price levels. The mos expensive echnology is perfec foresigh while cheaper opions use he hisory of price levels o forecas fuure values. In addiion, hey specify a model economy ha is difference saionary. In firs differences, he cyclical componens of oupu and he price level are represened by smaller phase shif in he sense ha he price level is counercyclical and he inflaion rae is acyclical. For economies wih some posiive measure of agens using low-cos forecas echnology ha 2 Brock and Haslag (2016) argue ha i is difficul o generae counercyclical price level and procyclical inflaion in a model economy wih echnology shocks and raional expecaions. Wih raional expecaions, he price level responds rapidly o he informaion ha a persisen echnology shock has occurred. The upsho is ha he price movemen is in phase wih he oupu movemen. 3 Haslag and Hsu (2012) consruc addiively sochasic sine funcions o demonsrae he parameric values of he phase shif ha will yield a negaive correlaion in levels and a posiive correlaion in he rae of change of one variable and he level of he oher. 3

4 relies on pas observaions of he price level o forecas he fuure price level, Brock and Haslag are able o accoun for hese wo difference-saionary facs. In his paper, our conribuion is wofold. Firs, we use a sicky-price model economy o deermine if i can accoun for boh counercyclical prices and procyclical inflaion. By examining he phase shif, we use he naural feaure ha he wo facs are par of one fac; he inflaion rae is he rae of change in he price level. The working hypohesis is ha sicky prices are an imporan ingredien so ha he equilibrium laws of moion will generae he phase shif. Furher, noe ha Brock and Haslag sugges ha some kind of sickiness is necessary. Second, he model economy allows one o ask wheher all, or some subse of, shocks are able o accoun for he phase shif. The model economy is aken from Ireland (2003) and prices are sicky because price changes are cosly a la Roemberg (1982). In addiion, we consider a rich se of shocks, including shocks o producion echnology, invesmen efficiency, consumpion demand, money demand, and a moneary policy shock. We ask wheher any of he shocks in he model economy are sufficien o accoun for he observed phase shif. The sufficiency condiion is checked by looking a he model economy wih only one shock operaing. If he phase shif is found, hen ha shock is sufficien o accoun for he phase shif. In addiion, we check o see if a necessary condiion is saisfied. Here, we consider model economies in which only one shock is no operaing; essenially, if, for example, he no echnology shock economy canno accoun for he phase shif, hen he echnology shock is necessary for he phase shif. Our resuls are easily summarized. We begin by reviewing he daa. For he poswar sample hrough 2016, we verify ha he price level is counercyclical and he inflaion rae is procyclical. Hence, he evidence of a phase shif beween he price level and oupu is suppored wih he longer sample. Nex, we use Bayesian mehods o esimae he model economy. 4 The numerical resuls indicae ha he model economy can capure boh he counercyclical price level and procyclical inflaion rae. Moreover, he resuls indicae ha he cross-correlaion funcion maches he acual paern insofar as he peak crosscorrelaion occurs beween he price level and wo-quarer-ahead oupu. Thus, he primary numerical resul is ha he model wih sicky prices can accoun for he phase shif and correcly accoun for he naure, or direcion, ha he phase shif akes. To check for sufficiency, we consider five versions of he model economy. Each one is disinguished by seing he sandard deviaion of four shocks equal o zero. If, for example, he phase shif is observed in he model economy in which only he echnology shock is operaional, hen he 4 See DeJong, Ingram and Whieman (1996) for a descripion. 4

5 echnology shock is sufficien for he phase shif, condiional on he model economy. We find ha none of he shocks saisfy he sufficiency condiion. The necessary condiion is a check of he saemen ha no A implies no B. In oher words, suppose ha we consider five versions of he model economy. Here, each one is disinguished by he sandard deviaion of only one shock se equal o zero. For example, if he preference shock is no operaional implies ha he phase shif is no observed in he model economy, and hen he preference shock is necessary for he phase shif. We find ha he echnology shock and he moneary policy shock are candidaes ha saisfy his definiion of a necessary condiion. The paper is organized as follows. We review he empirical evidence supporing he exisence of he phase shif beween he price level and oupu in Secion 2. The economic model is specified in Secion 3. In Secion 4, we define and characerize he equilibrium. We presen he numerical analysis of he model economy in Secion 5. In Secion 6, we deermine wheher any of he shocks saisfy he sufficien or he necessary condiions. We repor he impulse responses funcions for each shock in Secion 7. In Secion 8, we offer a brief summary and conclusion. 2. Evidence Throughou his analysis, he daa are quarerly and run from 1959:Q1 hrough 2016:Q2. The variables are per-capia real GDP (oupu) and he chain-weigh deflaor (price level). 5 Our firs aim is o repor evidence regarding he naure of he phase shif beween he price level and oupu. Throughou our analysis, we use logs of oupu and he price level and he inflaion rae is he firs-difference values of he log price level. The H-P filer is applied o each series wih Table 1 repors he sandard deviaion for derended levels of oupu (y), he price level (p), and he inflaion rae (π), as well as he conemporaneous price-oupu correlaion and inflaion-oupu correlaion. The es saisic is compued following Ashley, Granger and Schmalensee (1980) and indicaes ha one can rejec he null hypohesis ha each conemporaneous correlaion coefficien is equal o zero. I follows ha he conemporaneous covariance beween he inflaion rae and oupu depends on wo covariances; ha is, he conemporaneous covariance beween he oupu and he price level and he covariance beween oupu and one lagged value of he price level. Formally, 5 Per-capia real GDP is obained by real GDP (chained 2009 dollars) divided by he civilian noninsiuional populaion, age 16 and over. 5

6 y y y cov, cov,p cov,p. 6 The evidence indicaes ha he covariance beween oupu 1 and he conemporaneous price level is negaive. Thus, if he covariance beween oupu and he lagged price level is negaive and less han he conemporaneous covariance beween oupu and he price level, hen he conemporaneous correlaion beween oupu and inflaion will be posiive. In work by Haslag and Hsu (2012), he auhors explain how a phase shif beween he price level and oupu can accoun for he wo conemporaneous correlaion coefficiens. More specifically, le he cyclical componen of oupu be he reference line as ploed wih respec o ime. We refer o he relaionship beween he cyclical componens of he price level and oupu as a lef phase shif if he cyclical componen of he price level ends o peak and rough before oupu. Formally, if sin and sin y x y 2 2 p x where ~ N(0, ) and ~ N(0, ), hen a lef phase p shif is a value of 0. Haslag and Hsu derive values, for which he price level is counercyclical and he inflaion rae is procyclical. Wih, and 0, a lef phase shif is consisen wih movemens in he cyclical componen of he price level leading movemens in he cyclical componen of oupu. For 0, he phase shif is oo small for he inflaion rae o be counercyclical. By coninuiy, here exiss ', where ', such ha prices are counercyclical and he inflaion rae is acyclical. A simple way o verify he lef-phase shif in he daa would be o conduc a Granger causaliy es. Consider a bivariae VAR wih oupu and he price level. Formally, we esimae y y p p y a 5 b y 5 d p 0 i i i i y (1) i1 i1 p a 5 f y 5 g p 1 i i i i p (2) i1 i1 wih 5 lags chosen by likelihood raio es. In his case, we are ineresed in wo hypohesis ess. Firs, we es wheher movemens in oupu Granger cause movemens in he price level; hence, H0, yp : fi 0i 1, 2,3, 4,5. The p-value for his es is One canno rejec he null hypohesis ha movemens in oupu Granger cause movemens in he price level a normal confidence level. Nex, we ask wheher movemens in he price level Granger cause movemens in oupu; ha is, he null 6 See Ross (2014) for Lemma and he proof on p.122 for he complee derivaion of his saemen. 6

7 hypohesis is H0, py : di 0i 1, 2,3, 4,5. The p-value for H0, p y is Hence, we can rejec he null hypohesis and he daa are consisen wih he noion ha here is a lef-phase shif characerizing he relaionship beween oupu and he price level a business cycle frequencies. For he full poswar sample, he evidence suggess ha here is a phase shif beween movemens in he cyclical componens of oupu and he cyclical componen of he price level. Ohers have presened evidence ha is consisen wih he phase shif in he poswar daa, including Kydland and Presco (1990) and Cooley and Ohanian (1991). In relaion o he findings presened in hese papers, our resuls are no surprising. The evidence presened here confirms, over a longer poswar sample, wha previous researchers have found. Oher researchers have scruinized he conemporaneous relaionship beween he price level and oupu. Even wih all he scruiny, perhaps he bigges concern is ha he phase shif is no sable. For example, Cooley and Ohanian have daa going back he middle of he 19 h Cenury. They divide he sample ino subperiods. According o Cooley and Ohanian, he phenomenon ha he price level is counercyclical and inflaion rae is procyclical does occur in he poswar sample period. In he inerwar period, for example, boh he price level and he inflaion rae are procyclical. In his paper, our analysis will rea he period afer World War II as having a sable relaionship beween he price level and oupu. 3. Economic Environmen In his secion, we use a version of he economy developed in Ireland (2003). There is a echnology ha implemens price changes in his economy. The Roemberg model inroduces a nominal rigidiy ha is consisen wih he ideas of ineria pu forward by Sims (2003). 8 There are an infinie number of discree ime periods. Le ime be indexed by 0,1,2,... he commodiy space consiss of labor, he capial good, a uni measure coninuum of inermediae goods and a single finished consumpion good. The economy is populaed by four ypes of agens: a large number of idenical households, a coninuum of firms producing he inermediae good, a large number of idenical firms producing he final consumpion good, and a cenral bank. 7 In his seup, here is no es for heeroscedasiciy in he error erms. Noe ha he evidence presened by applying he sandard F-es for Granger causaliy is consisen wih a Wald es and a heeroskedasic-robus covariance esimaor. See Haslag and Hsu (2012) for hose resuls. 8 Woodford (2003) assers ha price sickiness is an imporan deerminan ha affecs equilibrium resource allocaion. Sims approaches he problem as a cogniive echnology ha limis informaion processing. See also Maćkowiak and Wiederhol (2009), and Moscarini (2004) for oher examples of informaion heoreic approaches o sicky prices. 7

8 The represenaive household offers labor and capial o firms producing he inermediae good, purchasing final consumpion goods from he final-goods producer. The represenaive final goods producer purchases he array of inermediae goods from he inermediae-goods producer, ransforming hem ino he final consumpion good. The cenral bank deals wih he represenaive household hrough lump-sum ransfers (axes), prining (desroying) fia money. A dae 0, he represenaive household is endowed wih money balances equal o M, 0 an iniial sock of pure discoun bonds, B, 0 and an iniial sock of capial goods, K. 0 A each dae 0, he represenaive household is endowed wih one uni of ime ha be divided beween labor and leisure. A represenaive household begins each period wih money balances, bonds, and capial, capured by he riple M1, B 1, K 1. Capial is rened o inermediae-goods producing firms, receiving renal paymens equal o Q unis of money per uni of capial. Discoun bonds carried over from he previous period maure, paying B 1 unis of money. In addiion, each household receives T unis of money as a lump-sum paymen from he cenral bank. Labor income is equal o W unis of money per uni of labor supplied o inermediae-good producing firms. Any profis by inermediae-goods producers are paid o he represenaive household. The represenaive household chooses he consumpion good, real balances, leisure, bonding holdings and invesmen o maximize expeced lifeime uiliy subjec o he period budge consrain. Inermediae-good producing firms have access o a echnology ha combines labor and capial o produce an inermediae good. Le i 0,1 be he index for inermediae-good producing firm. Capial rened by firm i is denoed by firms saisfies 1 i K and labor used by firm i is denoed l. Thus, aggregaing across di and l l K K 0 i 1 0 i, di, represening he oal capial sock rened o all firms and oal labor supplied o all firms, respecively. Finally, nominal profis are paid o he represenaive household. Le D 1 0 i inermediae-good producing firms. D be he oal nominal profis paid o represenaive households by The represenaive household faces disribuion of differen shocks in his economy. The household seeks o maximize expeced lifeime uiliy represened by i 8

9 M E 0 u C.,1l 0 P (1) where he momenary uiliy funcion is represened by u a C e l M 1 P. ln ln 1 (2) Equaion (2) inroduces wo preference shocks. Here, a sands for a preference shock ha affecs he household s consumpion growh rae responsiveness o he changes in real ineres rae. There is a separae shock o he household s money demand capured by e. Throughou his paper, we assume ha boh preference shocks follow a saionary AR(1) process. Le where [0,1) j a a and e e e ln ln ln 1 ln ln a 1 a e e 1 e 2 and j N j demand parameer. 0, for j a,e. Here, e denoes he seady sae value of he money The household uses consumpion good purchases o consume and o ransform he consumpion good ino he capial good. We assume here is an adjusmen cos associaed wih changes in he capial sock represened by K k 2 gk 1 2 K. 1 (3) Le k sand for he capial adjusmen cos parameer and g is he seady sae value of he capial growh rae. The household s period budge consrain is given as 2 B M M 1 B 1 Q K 1 Wl D T k K r C I 1 K 1 P 2 gk 1 P (4) 9

10 Noe ha discoun bonds are purchased a price 1 r where r is he gross nominal ineres rae ha applies for daes o 1. In equaion (4), he lef-hand-side is he real value of resources available for he household o spend while he righ-hand-side is he real value of purchases. Formally, The law of moion for he capial sock includes a shock o he marginal efficiency of capial. 1 K 1 K x I. (5) We assume he efficiency shock erm follows a saionary AR(1) as follows 2 x x x 1 x wih x N x ln ln 0,. (6) The household s problem, herefore is o maximize (1) subjec o (4) and (5), aking prices as given. Inermediae-goods producing firms produce a firm specific, perishable goods. For convenience, le firm i produce inermediae good i. The echnology used by he inermediae good firm is a consan reurns o scale Cobb-Douglas producion funcion given by 1. Y K g z l (7) i i i Here, is he share of income paid o he capial facor, z is a produciviy shock ha follows a saionary AR(1) process depiced by he following: 2 z z z z z1 z wih z N z ln 1 ln ln 0,. (8) Here, z sands for he seady-sae level of produciviy. The inermediae-good producing firm i sells is good o he final-good producer a nominal price P. Each inermediae-good producer possesses some marke power in a monopolisically compeiive i marke for inpus. Each firm ses he price, facing a price-adjusmen cos given by i P P 1 Y. 2 Pi 1 2 (9) 10

11 Here, P is he price-adjusmen cos parameer and is he seady sae inflaion rae. Each inermediae-good producer seeks o maximize he discouned sum of profis. 9 Thus, each firm chooses capial, labor, and is price o maximize D i 0 0 P (10) E where is he Lagrange muliplier ha applies o he represenaive household s budge consrain. The dae- dividends paid o he household is given by 1 P 2 1 Di Pi Yi Q Ki Wli T Pi 1 Y. P P Pi 2 (11) The represenaive final-good producer uses a consan-reurns o scale echnology. The quaniy of final-goods produced is given by i. (12) Y Y di 0 The final goods marke is perfecly compeiive. Wih no barriers o enry, he zero-profi condiions will be saisfied in equilibrium. From previous work, he price of he final good will be P P. 0 i di (13) A each, he cenral bank creaes (desroys) fia money a rae. Hence, he law of moion for he money supply is M. M 1 Changes in he money are passed o he represenaive household. The nominal budge consrain for he cenral bank is given by 1 1 T M M 1 M. (14) 9 This approach is consisen wih maximizing he household s uiliy derived from real dividend paymens. 11

12 Throughou our analysis, we assume ha he pah for he money supply he value of --is deermined by a ype of Taylor rule. The moneary policy rule is given by r y Y r y ln ln ln ln ln v (15) where r,,, and y are seady sae values of he nominal ineres rae, he money growh rae, he inflaion rae, and derended oupu; ha is, y Y, respecively. In equaion (15), g is he weigh on money growh relaive o ha on he nominal ineres rae. Furher, and are he policy responses o log-deviaions of inflaion and derended oupu from heir seady sae values. Finally, v is a moneary policy shock which follows a saionary AR(1) process given by 2 v v v 1 v wih v N v ln ln 0,. (16) y 4. Equilibrium In his model economy, a symmeric compeiive equilibrium is defined as a sequence of M D C, I, K, l,, allocaions P P 0 and a sequence of prices, P, Q, W, r such ha: i. he represenaive household maximizes expeced lifeime uiliy, aking prices as given; ii. each inermediae-good producer chooses he price so as o maximize expeced discouned sum of profis, aking all inpu prices as given; iii. he represenaive final-good producer maximizes profis, aking prices as given; iv. he cenral bank s budge consrain is saisfied; v. he marke for all he inermediae goods, he marke for he final good, he capial marke, he labor marke, he bond marke, and he money marke clear. In he symmeric equilibrium, each inermediae-good producer will choose he same price. By equaion (13), we obain i P. Similarly, i follows ha K K, l l, and D P Equaion (12) implies ha Y Y. i i i i D. 12

13 5. Numerical analysis Wih his version of a sicky-price, DSGE model economy, here are 24 parameers. We follow Ireland (2003) in choosing values for,, and. Le 1.5, which is consisen wih he represenaive household spending one-hird of heir ime in marke producive aciviy. The quarerly depreciaion rae is se such ha annual depreciaion is approximaely en percen; hence, The price elasiciy of he demand for inermediae goods is se wih 6 corresponding o a seady sae markup equal o 20 percen. We use Bayesian mehods o obain values for he remaining 21 parameers. We use a Mone- Carlo base opimizaion rouine o compue he mode value for he oher 21 parameers. Wih he hree calibraed parameers and 21 esimaed parameers, we solve he firs-order Taylor expansion around he seady sae. Because he model economy s dynamics are subjec o five exogenous shocks, we use five observed series o avoid sochasic singulariy. The five observable variables are: real personal consumpion expendiures, real gross privae domesic invesmen, real M2 balances, he chain-weigh GDP deflaor, and he 3-monh U.S. Treasury bill rae. 10 Table 2 repors he resuls of he Bayesian esimaion using quarerly daa spanning he period 1959:Q2 hrough 2016:Q2. The parameer values are broadly divided ino four groups: ases and producion, adjusmen coss, moneary policy, and he five exogenous shocks. For parameers ha lie in he zero-one inerval, we specify a Bea disribuion as he prior disribuion. The firs group of parameers conains he ases and producion parameers. The mean of he poserior disribuion of he discoun facor ells us ha implying ha he represenaive household exhibis a low subjecive rae of ime preference. The mean of he poserior disribuion for implies he ineres elasiciy of money demand is For he capial share, he mean of he poserior disribuion of capial share is which The second group of parameers focuses on he cos adjusmen parameers. Noe ha he mean values of he poserior disribuions indicae greaer price sickiness in he inermediae goods secor han in he capial goods secor. The value of he cos adjusmen parameer for he inermediae goods is more han four imes he value of he cos-adjusmen parameer in he capial goods secor. 10 Following Ireland (2003), we use per-capia measures of consumpion, invesmen, and real money balances o esimae he model. Per-capia measures are obained by dividing he aggregae value by he civilian noninsiuional populaion, age 16 and over. 13

14 The moneary policy rule uses he seady sae money growh rae. We esimae he growh rae from he daa. The mean of he poserior disribuion is which corresponds o an annual average growh rae of 4.4 percen. The average annual growh rae of he M2 money sock is 6.8 percen for he U.S. during he sample period. The esimaed value is reasonably close o he acual average growh rae. For he Taylor rule parameers, he Bayesian esimaes indicae ha moneary policy is slighly acive wih respec o he movemens in he oupu gap. Wih , he calibraed response of he nominal ineres rae is fairly small for a given change in he oupu gap. We also see ha , implying ha he Federal Reserve sems inflaion. The above Bayesian esimaes of he policy coefficiens are in line wih findings in he lieraure and indicae he sabilizaion propery of he esimaed Taylor rule. The fourh group of parameers describes he persisence and volailiy of he five shocks ha are in he model economy. We find ha he esimaed coefficiens on lagged values of mos shocks preference, money demand, invesmen efficiency, and produciviy are beween and 0.991, implying a high level of persisence is presen in hese shocks. The Bayesian esimae for , implying moderae persisence is presen in he moneary policy shock. Among he five shocks, Table 2 repors ha he Bayesian esimaes of he sandard deviaion in invesmen efficiency shock is abou en imes he value of he oher four shocks. 5.1 Business cycle properies Wih he model economy and is parameer values se, we repor he key properies. Table 3 repors he sandard deviaions for he acual and simulaed economies for five differen values; specifically, oupu, he price level, he inflaion rae, M2, and he nominal ineres rae. 11 y v Excep for he nominal ineres rae and oupu, he model economy repors slighly greaer volailiy for he price level and he inflaion rae compared wih he acual volailiy. M2, however, is abou 1 1/2 imes more volaile in he model economy han i is in he acual economy. Table 4 repors he conemporaneous correlaion coefficien wih oupu for he price level, he inflaion rae, M2, and he nominal ineres rae. Focus on he firs wo columns. The model economy generaes a se of ime series in which he price level is counercyclical. However, he evidence indicaes ha he relaionship beween oupu defined as he sum of consumpion and invesmen and he 11 Since we do no use he daa on real GDP for esimaion, here acual oupu is measured as he sum of consumpion and invesmen. So he volailiy of acual oupu in Table 3 and he conemporaneous price-oupu and inflaion-oupu correlaions in Table 4 are differen from hose in Table 1. 14

15 inflaion rae is significan a he jus below he en percen level. Hence inflaion is acyclical in he measure of oupu consising of resources consumed by consumers and firms. 12 Noe ha he phase shif embodied in he correlaion coefficien beween inflaion and oupu is quaniaively very close when comparing he model economy and he acual economy; 0.1 and 0.14 are saisically insignifican from one anoher. Therefore, we inerpre he model economy wih sicky prices as capable of accouning for a phase shif beween he price level and oupu 6. Necessary and Sufficien Shocks Wih five differen shocks affecing he model economy, we consider wo alernaive approaches ha may help shed some ligh on wheher here is a se of eiher necessary or sufficien shocks ha can accoun for he phase shif beween he price level and oupu. We begin by examining model economies in which only one shock is operaing. The approach yields a kind of sufficiency condiion; if, for example, here is only a moneary policy shock, do we observe counercyclical price level and procyclical inflaion in he model economy? If yes, hen he moneary policy shock is sufficien for he lef phase shif observed in he daa. Table 5 presens conemporaneous cross correlaions for five simulaions of he model economy. In each simulaion, he shock lised in he firs column is he only one wih a nonzero sandard deviaion. We repor he median value of he conemporaneous correlaion coefficien from 5000 simulaions of he model economy. In four of he five cases, he median value of he conemporaneous correlaion coefficien has he same sign for boh he price level and he inflaion rae. In he case of he consumpion preference shock, he price level is acyclical while he inflaion rae is srongly procyclical. Thus, here is evidence of a phase shif presen when here is a shock o preferences for curren consumpion. However, based on he resuls presened in Table 5, none of he shocks are sufficien o accoun for he lef phase shif observed in he daa. Even wih cosly price changes imparing some price sickiness ino he model economy, he equilibrium we consider are raional expecaions. Brock and Haslag (2014) spend some ime deriving 12 The implicaion is ha he correlaion beween prices and privae spending is consisen wih a phase shif ha saisfies ',. The phase shif generaed by he more narrow measure of oupu economy is a fracional shif in he sense ha i is oo small ha is, less han --o accoun for he counercyclical price level and procyclical inflaion. 15

16 condiions in which here could be a phase shif in a Woodford-syle (2003) model economy. 13 Basically, in a model economy wih raional expecaions, he price level responds quickly o he new informaion such he relaionship beween he price level and oupu is in phase. Price sickiness helps, bu for he simulaed model economy presened here, he numerical resuls indicae ha any phase shif is insufficien when a shock is considered in isolaion. The moneary policy was considered a good candidae ha could accoun for he lef phase shif. In paricular, he equilibrium inflaion rae depends on he iniial impac on deviaions in consumpion. In a model economy wih capial, Ruper and Susek (2016) argue ha consumpion can be perfecly smoohed, a leas iniially, o a moneary policy shock. Wih an unexpeced increase in he nominal ineres rae, he iniial response is for he equilibrium inflaion rae (and hence he price level) o decline. As he persisence of he moneary policy shock dissipaes, he inflaion rae approaches is saionary value from below. Wih a decline in inflaion rae and fuure expeced inflaion raes, i follows ha oupu mus also iniially decline; he decline in oupu is mached by a decline in invesmen spending since consumpion spending is iniially unchanged. Like he inflaion rae, oupu will approach is saionary value from below. Table 5 suppors Ruper and Susek s inuiion; indeed; Table 5 repors a very high conemporaneous correlaion beween oupu and he inflaion rae when here is a moneary policy shock only. Sill we need o explain why here is, on average a negaive correlaion beween he price level and oupu. Well, as oupu increases following he iniial negaive shock, he inflaion rae is also increasing. Wih he inflaion rae sill below is saionary value, however, he price level is declining. In oher words, afer he iniial response in which he price level and oupu move in he same direcion, here is period during he convergence o he saionary seings in which he price level is coninuing o decline while oupu is increasing. Based on he simulaion findings, he resuls indicae ha he srengh of he iniial price level-oupu response swamps he negaive correlaion of he responses ha follow. We plo he cross-correlaion funcion for he model economy wih all shocks operaing and wih each shock operaing separaely in Figure 1. In each panel (A hrough F), he reader can see p, y i in he lef-hand column and, y for 5, 4,...,4,5 i i for i 5, 4,...,4,5 in he righ-hand column. The cross-correlaion funcion for he simulaed economy and he acual daa are ploed ogeher. The reader can ge a sense of he goodness-of-fi of he model economy in Panel A. The lef-phase shif is 13 Brock and Haslag (2016) provide a deailed exposiion on he role ha raional expecaions plays in keeping he price level and oupu in phase. In shor, hey find ha he price-level response is essenially oo quick for he phase shif o be eviden in a version of Woodford s (2004) money-in-he-uiliy funcion model wih compleely flexible prices. 16

17 represened by he maching of he cross-correlaion funcion beween he price level and oupu; in he baseline economy, he median correlaion coefficien indicaes ha movemens in he price level emporally precede movemens in oupu as observed in he acual daa. Panels B hrough F depic crosscorrelaion funcions ha are markedly differen han he acual funcion for he price level and oupu. In paricular, Figure 1 shows ha for each of one-shock-only model economies, he cross-correlaion funcions exhibi more pronounced swings compared wih he acual cross-correlaion funcions. Noe ha for he echnology shock, he money demand shock, he invesmen efficiency shock, and he moneary policy shock, he model economy s cross-correlaion funcion indicaes ha he price level and oupu are eiher in phase (conemporaneous correlaion is he larges) or ha movemens in oupu emporally precede movemens in he price level (fuure price level correlaion is he larges). Even in he case of he preference shock (Panel C), we observe a symmery in he cross-correlaion funcion beween he price level and oupu; he funcion is upward sloping over he range from -5 (price leads oupu) o 5 (oupu leads price). Overall, we find some evidence ha consumpion preference shock can accoun for a phase shif. However, he phase shif is no sufficien o mach he observed paern of counercyclical price level and procyclical inflaion. We conclude, herefore, ha here is no a single shock o he model economy ha is sufficien o accoun for he phase shif observed in he daa. Nex, we examine he model economy wih four shocks operaing and only one shock se such ha is sandard deviaion is equal o zero. In his way, we can ask wheher one shock is necessary for he lef phase shif we observe. To illusrae he quesion, suppose we se he sandard deviaion of one shock equal o zero and he price level is counercyclical and he inflaion rae is procyclical in he model economy. We know ha he shock is no necessary for he model economy o be able o accoun for he lef phase shif observed in he daa. If he sandard deviaion is se o zero and he model economy is no able o mach he empirical observaion, hen we know he shock is necessary. Table 6 repors he median conemporaneous correlaion coefficiens for he price level and oupu and he inflaion rae and oupu for five model economies. Each model economy is idenified by he firs column in Table 6; specifically, he column idenifies he shock for which sandard deviaion is se o zero. The resuls indicae ha in hree cases he consumpion preference, he money demand, and he invesmen efficiency shocks he price level is counercyclical and he inflaion rae is (marginally) procyclical. Consequenly, hese hree shocks are no necessary o accoun for he lef phase shif. In he case of he produciviy shock and he moneary policy shock, Table 6 repors ha he signs of he wo correlaion coefficiens are he same; ha is y p y sgn, sgn, 0 for he produciviy 17

18 shock and y p y sgn, sgn, 0for he moneary policy shock. Thus, he resuls sugges ha he echnology shock and he moneary policy shock are necessary for he model economy o accoun for he lef phase shif. Figure 2 plos he cross-correlaion funcions for wo model economies. In boh models, we plo he funcions for he price level and he inflaion rae. For one model economy, he sandard deviaion of he echnology shock is se equal o zero (Panel A) and he oher is cross-correlaion funcion for he model economy in which he sandard deviaion for he moneary policy shock is se equal o zero (Panel B). In boh panels, he median value of he cross-correlaion coefficien is repored for a se of 5000 simulaions. Comparing Panel A o Panel B, Figure 2 shows ha he cross-correlaion funcion is a worse fi wih he echnology shock off han when he moneary policy shock is off. The doed lines are 90 percen confidence bands consruced from he 5000 simulaions. Wih no echnology shock, we observe ha he price-oupu cross correlaion funcion for he model economy is significanly differen from he acual cross correlaion funcion for i 2, 1,0,1,2,3,4. Indeed, he correlaion values have a differen sign han he acual coefficiens for i 1,0,1,2,3,4. Noe ha in he inflaion rae-oupu crosscorrelaion funcion, he median value for he model economy is significanly differen from he acual correlaion coefficien for i 5, 4, 3, 2, 1,0,3,4,5. For he moneary policy shock, he price-oupu cross-correlaion is significanly differen han he acual correlaion for i 5, 4, 3. In he moneary policy shock case, however, he correlaion coefficiens have generally he same sign as he acual correlaions. Noe ha inflaion-oupu correlaion coefficiens are no significanly differen from he acual correlaions when he moneary policy shock is no operaional. The numerical evidence suppors he noion ha boh moneary policy shocks and echnology shocks are necessary for he model economy o be able o accoun for he observaion ha here is a phase shif in he price level. In his DSGE model, we provided he inuiion for why he moneary policy would be able o accoun for he phase shif in he discussion of he sufficiency condiions. In he case of he echnology shock, Freeman and Huffman (1991) provide an elegan inuiion ha helps us undersand why a echnology shock will resul in he phase shif. In he case of a posiive echnology shock, for example, he immediae effec is for incomes o rise and money demand o rise. For a given money supply pah, he price level iniially declines. Here is where price sickiness plays an imporan role. The price decline is parial relaive o wha i would have been in a seing wih no price sickiness. I simply akes ime for he price level o respond fully o he echnology shock. Meanwhile, as he price level converges back o is saionary pah, he inflaion rae will increase while oupu is also increasing. Thus, he iniial price level response accouns for why he price level and oupu are negaively correlaed. The price 18

19 sickiness impars a propagaion propery o he rae of change in he price level ha accouns for he inflaion rae is procyclical in he model economy. Overall, he resuls provide a deeper undersanding of how he DSGE model economy can accoun for he phase shif in he price level relaive o oupu. While no one shock is sufficien o accoun for he phase shif, we presen evidence ha boh echnology and moneary policy shocks are necessary o accoun for he phase shif. 7. Impulse Response Funcions as a Means of Idenificaion How he economy responds o each shock can shed ligh on conribuion o he phase shif beween he price level and oupu. Figures 3 hrough 7 repor he impulse response funcions for nine endogenous variables over 40 quarers for each exogenous shock. Here, we will use he fac ha he inflaion rae is he rae of change in he price level. So, we will concenrae on he impulse response funcions for oupu and he inflaion for each of he five shocks in he model economy. The general picure obained from he impulse response funcions are consisen wih conemporaneous correlaion coefficiens repored in Panels B hrough F in Figure 1. Noe ha he immediae responses o a paricular shock repored for oupu and he inflaion rae (and he price level) are consisen wih he conemporaneous cross correlaion repored in Figure 1 for each individual shock case. For example, in Panel B of Figure 1, he conemporaneous cross correlaion beween oupu and he inflaion rae is negaive. The immediae impulse response o a echnology shock is an increase in oupu and a decline in he inflaion rae as shown in Figure 6. In he following discussion, we consider a more deailed descripion of he impulse responses and cross-correlaion funcions. Figure 3 plos he impulse responses o a one sandard deviaion posiive shock o preferences. The immediae response is for oupu and inflaion o increase. The conemporaneous cross correlaion indicaes he price level is acyclical and he inflaion rae is procyclical in he model economy wih only a preference shock operaing. Noe furher ha boh he oupu response and he inflaion rae response are nonmonoonic. The peak response in oupu occurs only one quarer afer he shock while he peak response in he inflaion rae occurs four quarers afer he shock. Hence, he impulse responses are nonsynchronous. The implicaion is ha oupu is posiively correlaed wih fuure inflaion as observed in he cross correlaion funcions (see Panel C in Figure 1). When oupu increases, boh inflaion and he price level coninue o increase for several more quarers. Figure 4 plos he impulse responses when here is a one sandard deviaion posiive change in he money demand shock. The immediae response indicaes ha boh he oupu response and he inflaion 19

20 rae response are negaive and are hus consisen wih he procyclical price level and procyclical inflaion rae repored in Panel D of Figure 1. Moreover, boh he oupu response and he inflaion rae responses are monoonically declining in magniude over he range of he impulse response funcion. The movemens in he fuure inflaion rae are small enough so ha he correlaion beween oupu and fuure inflaion vanishes. Noe ha afer an iniial decline, inflaion rebounds o is seady-sae level wihin wo years afer he shock and hen reaches and says a a level slighly higher han is seady-sae value. Such a response paern indicaes ha he price level is coninuing o increase afer he shock. And he crosscorrelaion funcion in Panel D of Figure 1 shows ha here is a posiive correlaion beween curren oupu and fuure price level when he money demand shock is operaing. Conversely, movemens in he price level are negaively correlaed wih fuure oupu. In Figure 5, we consider he impulse responses o a one sandard deviaion posiive shock o he invesmen efficiency. The immediae response indicaes ha boh oupu and he inflaion rae increase, which is consisen wih he evidence in Panel E of Figure 1; ha is, he price level and inflaion rae are procyclical. In erms of he cross-correlaion funcion, he relaionship beween curren oupu and fuure inflaion is indicaed by he impulse responses as follows: as oupu coninues o increase we see ha inflaion sars o decline, approaching he seady sae from above. Thus, he correlaion beween curren oupu and fuure inflaion becomes negaive. Because he inflaion rae is posiive wihin five quarers afer he shock, however, he price level is increasing. Hence, he correlaion beween curren oupu and fuure price level is posiive. We can observe he relaionship in he impulse response funcions, given an invesmen efficiency shock, as being refleced in he cross-correlaion funcions for oupu and he price level and oupu and he inflaion rae when only he invesmen efficiency shock is operaional. Figure 6 represens he impulse response funcions for a one sandard deviaion posiive produciviy shock. The immediae reacion is a posiive response by oupu and a negaive response by he inflaion rae. The immediae responses can accoun for counercyclical price level and counercyclical inflaion observed in Panel B of Figure 1. The relaionship beween curren oupu and fuure inflaion owes chiefly o he fac ha he impulse response indicaes ha inflaion is increasing over ime as i approaches seady sae from below. The posiive correlaion shows up in he cross correlaion funcion as he correlaion beween curren oupu and fuure inflaion becomes posiive. Because he inflaion rae response is increasing, bu negaive, he correlaion beween curren oupu and fuure price level remains negaive. We furher observe ha he curren price level is declining, bu he hump-shaped paern in he oupu response indicaes ha evenually oupu declines. Thus, he impulse response can accoun for a posiive correlaion beween curren price level and fuure oupu. 20

21 Lasly, Figure 7 plos he impulse response funcions given a one sandard deviaion posiive shock o moneary policy. The immediae effec is ha oupu declines and he inflaion rae declines. Panel E of Figure 1 indicaes ha price level is procyclical and he inflaion rae is procyclical. As Ruper and Šusek explain, he moneary policy shock iniially resuls in a higher real ineres rae, reducing he demand for curren consumpion and oupu iniially declines in he face of an unexpeced increase in he nominal ineres rae. The impulse response shows ha oupu approaches seady sae from below, so y is increasing. The impulse response also indicaes ha he inflaion approaches seady sae rapidly, so ha inflaion is also increasing. This is why we see a rapid decline in he cross-correlaion funcion. We also can accoun for why he curren price level is negaively relaed o fuure oupu; as inflaion approaches seady sae from below, he price level is declining while he fuure oupu response is posiive afer he iniial decline. 8. Summary and conclusion In his paper, we examine a New Keynesian model economy wih Roemberg-ype sicky prices, aemping o quaniaively assess he model economy s abiliy o accoun for a phase shif eviden in he relaionship beween he price level and oupu. Here, he phase shif is consisen wih he observaion ha he price level is counercyclical and he inflaion rae is procyclical. We are able o demonsrae ha he sicky price model can accoun for he phase shif. This resul may no be surprising o many. We invesigae furher, and our main conribuion lies in looking for he source of he model economy s abiliy o accoun for he lef phase shif in he daa. Is one ype of shock sufficien o accoun for he observed paern in he daa? The answer is no. We consider five versions of he model economy in which each is disinguished by seing he sandard of deviaions of four shocks equal o zero. In each one-shock-only model economy, he price-oupu and inflaion-oupu conemporaneous correlaion coefficiens do no mach he acual correlaion coefficiens. Hence, we conclude ha no one shock is sufficien o accoun for he observed paern. In addiion, we examine wheher one shock is necessary o be able o accoun for counercyclical price level and procyclical inflaion. We find ha he moneary policy shock and echnology shock boh mee a simple crierion o be considered necessary for he lef phase shif. Specifically, we consider separae model economies in which he echnology shock or he moneary policy shock is no operaional. In boh cases, he model economies canno accoun for boh counercyclical price level and procyclical inflaion. In shor, no echnology shocks imply no a lef phase shif and no moneary policy shocks imply no a lef phase shif. Therefore, we conclude ha here is a se of necessary shocks o accoun for he observaion. 21

22 Our resuls are firs, imporan sep ha could possibly explain one of he main modelling differences ha is presen in analyses of business cycle flucuaions. The wo principal camps sudying business cycle flucuaions have divided beween emphasizing he relaionship beween inflaion and oupu versus emphasizing he relaionship beween he price level and oupu. By focusing on he phase shif in he relaionship beween he price level and oupu, some noion of sicky prices is imporan. As researchers move forward, he phase shif provides a focal poin. 22