Inside and Outside Bounds: Threshold Estimates of the Phillips Curve. Michelle L. Barnes and Giovanni P. Olivei*

Transcription

1 Inside and Ouside Bounds: Threshold Esimaes of he Phillips Curve Michelle L. Barnes and Giovanni P. Olivei* Over he pas 30 years, debaes abou he usefulness of he Phillips curve for explaining inflaion behavior have been ongoing. A he end of he 1970s, he Phillips relaionship was heavily criicized for is apparen inabiliy o characerize inflaion dynamics in he face of oil shocks. The relaionship was also deemed inappropriae as a policy ool because of is poenial insabiliy in he face of changes in he economic environmen. Subsequen empirical work in he early 1990s showed ha such criicism was largely unjusified. By conrolling for supply shocks as shifers in he Phillips relaionship, hese sudies provided evidence of a sable and significan radeoff beween inflaion and unemploymen. 1 Indeed, he Phillips relaionship coninues o feaure prominenly in several macroeconomeric models used for policy analysis. 2 More recenly, hough, as a resul of he sriking combinaion of declining unemploymen wih no aendan inflaionary pressures during he second half of he 1990s, criicism has mouned once again. This developmen has led many researchers o quesion again he viabiliy of he Phillips curve and is usefulness as a macroeconomic policy ool. While he debae coninues, he fall in inflaion coupled wih significan slack in he economy since lae 2002 seems consisen wih a sandard Phillips curve radeoff. 1 See, for example, Fuhrer, 1995 and Tooell,

2 One of he reasons for he recurring debaes abou he exisence of an inflaion and unemploymen radeoff is ha here have been several insances when large movemens in he unemploymen rae have elicied lile response in he inflaion rae. Figure 1 shows he behavior of inflaion, measured by he change in he core personal consumpion expendiures (PCE) deflaor, and he unemploymen rae for each of he four decades, beginning in he 1960s. In he 1960s (Char A), he unemploymen rae flucuaed beween 4 percen and 7 percen for many years, wih lile change in he inflaion rae. The radeoff was eviden only a he end of he decade, when he unemploymen rae dropped below 4 percen. Afer he oil price shock of he early 1970s and he associaed sagflaion (Char B), he years 1976 o 1979 saw he unemploymen rae decrease noiceably, wih only a modes change in inflaion. Then, he relaionship seemed o re-esablish iself; he recession of he early 1980s (Char C) saw he expeced radeoff, wih increasing raes of unemploymen associaed wih a sharp decline in inflaion. owever, his radeoff was no apparen from 1985 o 1989, even hough he unemploymen rae dropped appreciably. Similarly, he recession of he early 1990s (Char D) winessed a large fall in inflaion, bu since 1994 and hrough lae 2002, large movemens in he unemploymen rae, a firs decreasing and hen increasing, have been associaed wih relaively small changes in inflaion. 2 Blinder (1998), for example, noes ha a linear Phillips curve fis he daa exremely well. No everyone would agree wih such a saemen. See,for example, Akeson and Ohanian (2001). 2

3 In principle, hese episodes of horizonal movemen -- ha is, episodes of inflaion sabiliy coupled wih large changes in he unemploymen rae -- are consisen wih a Phillips curve relaionship. They jus require he curve o shif in he same direcion as he movemen in he unemploymen rae. For example, he experience of he second half of he 1990s can be reconciled, a leas parly, wih a Phillips curve radeoff, by arguing ha over his period he naural rae of unemploymen declined. Economeric represenaions of he Phillips relaionship usually incorporae facors ha can cause he Phillips curve o shif over ime. Typical conrol variables are food and energy shocks, inflaion expecaions, and, in some insances, a changing naural rae of unemploymen. 3 So far, however, he lieraure has no provided a es of wheher such conrols are sufficien o explain he episodes of horizonal movemen. In his paper, we es he explanaory power of a piecewise linear specificaion of he Phillips relaionship agains a simple linear specificaion. The piecewise linear specificaion allows he inflaion and unemploymen radeoff o vary wih he level of he unemploymen rae. Such a specificaion mainains ha he radeoff may vary, depending on wheher he unemploymen rae lies wihin or ouside some range. If he usual shifers are sufficien o characerize periods of horizonal movemen, hen a piecewise linear specificaion should no improve upon he sandard linear Phillips curve. 4 3 Some would argue ha he acceleraion of produciviy in he second half of he 1990s was an imporan conribuing facor o low inflaion. There is lile compelling evidence, hough, supporing he inclusion of produciviy as a conrol variable in he Phillips relaionship. 4 This is no he firs aemp o characerize a changing radeoff over differen levels of he unemploymen rae, bu i has he advanage of parsimony. In 1958, A. W. Phillips originally drew a nonlinear curve 3

4 Insead, our findings srongly suppor a piecewise linear Phillips relaionship. The evidence indicaes ha he radiional shifers in he relaionship are insufficien o characerize he episodes of horizonal movemen. Apparenly, he gap beween he unemploymen rae and he naural rae of unemploymen mus be larger or smaller han some hreshold values before riggering a response in inflaion. When he unemploymen gap lies wihin he range defined by he hresholds, here is no evidence of a significanly and economically relevan radeoff beween inflaion and unemploymen. Graned ha some feaures of he Phillips curve remain heoreically puzzling, 5 wha facors migh explain a piecewise linear specificaion? Bargaining beween firms and workers implies ha he negoiaed wage lies wihin a range defined by he reservaion wage for he firm and he reservaion wage for he worker. This range will shif when economic condiions change. Typically, he bargaining process has he feaure ha a he ime of renegoiaing he wage, he new wage changes by he smalles possible amoun necessary o bring i wihin he range defined by he reservaion wages of he worker and he firm (Thomas and Worrall 1988). As a resul, changes in economic condiions usually lead o small changes in wages and prices. I is only when changes become very large ha a discree jump in he wage is necessary o bring i wihin he new range. This, in urn, will lead o a discree jump in prices (all 2003). Oher explanaions for a piecewise linear Phillips curve relae o he shape of he demand curve faced by firms. Suppose such ha low unemploymen raises wage inflaion more han high unemploymen lowers i. Oher ypes of nonlineariies in he Phillips relaionship have been explored by Eisner, See Blanchard and Fischer,

5 firms face kinked demand curves, wih he demand for heir produc decreasing sharply if firms increase heir price and increasing lile if firms decrease heir price. Then i is possible o show ha shifs in demand can be accompanied by lile or no movemen in prices, unless such shifs become very large (Woglom 1982). The res of he paper coninues wih a descripion of he linear and piecewise linear Phillips curve represenaions, along wih an explanaion of he esing and esimaion mehodology. The empirical resuls hen follow, along wih a conclusion ha relaes he findings o he mos recen inflaion experience. 5

6 I. Linear vs. Piecewise Linear Phillips Curves The sandard approach o esimaing he Phillips curve posis a linear shor-run radeoff beween inflaion and he level of an indicaor of he srengh of demand in he economy, such as he unemploymen rae. In is linear form, a general Phillips curve is given by he following specificaion: s k l ξ jz j + j= 1 j= 0 j= 0 α jπ j + βu + δ j u j + π = µ + ε. (1) The dependen variable π is he rae of inflaion. Lagged inflaion capures he ineria in he way inflaion expecaions are formed. Inflaion expecaions ener he Phillips relaionship because workers, concerned wih real wages, ake ino accoun expeced changes in prices when conracing changes in nominal wages. Equaion (1) hen posis ha expeced inflaion, π e, is formed as a weighed average of pas inflaion, wih he weighs α j esimaed on acual daa. 6 The variable u is he indicaor of he inensiy of demand in he economy, which in he presen analysis akes he form of he unemploymen rae. The unemploymen rae eners equaion (1) boh in levels and in firs differences (wih he firs difference u u 1 denoed by u). Thus, he coefficien ß measures he effec of he level of he unemploymen rae on inflaion, while he sum of coefficiens δ j measures he effec of curren and lagged changes in unemploymen on inflaion (ofen called he speed limi effec). The crucial parameer of ineres in he Phillips relaionship is, of course, ß. In he empirical secion ha follows, however, we will also 6 The sum of he coefficiens on lagged inflaion is usually consrained o uniy, so ha in he long run acual inflaion equals expeced inflaion. 6

7 menion some resuls concerning he speed limi effec. Finally, he Phillips curve relaionship incorporaes curren and lagged supply shock variables, here summarized by he vecor Z (normalized so ha Z = 0 indicaes absence of supply shocks), while ε is a serially uncorrelaed error erm. The prima facie evidence on he relaionship beween inflaion and unemploymen discussed in he previous secion suggess much horizonal movemen in he Phillips curve. In erms of equaion (1), his horizonal movemen could be accouned for by he supply shocks Z, or by changes in inflaion expecaions. Figure 2 shows a scaerplo of he parial correlaion beween inflaion and he level of he unemploymen rae ha arises from esimaing equaion (1) over he period from he hird quarer of 1961 hrough he fourh quarer of The esimaion uses observaions for which he unemploymen rae u ranges only beween 4.0 percen and 7.5 percen over he period considered. The figure shows ha wihin he specified range of he unemploymen rae, lile inverse relaionship beween inflaion and unemploymen is apparen even afer conrolling for supply shocks and inflaion expecaions. If we insead esimae equaion (1) using all he observaions -- and no jus he observaions for which he unemploymen rae lies beween 4.0 percen and 7.5 percen -- he esimaed radeoff beween inflaion and unemploymen becomes saisically significan and economically relevan. One poenial explanaion for his phenomenon is ha he radeoff 7 Deails abou he esimaion are given in Secion III. The measure of inflaion used in deriving Figure 2 is he change in he core PCE deflaor. 7

8 beween inflaion and unemploymen differs a differen levels of he unemploymen rae. Broadly speaking, he radeoff could be more pronounced for high and low values of he unemploymen rae han for normal values. To assess wheher his is indeed he case, we consider a more general specificaion of equaion (1) ha akes he following form: s k l ξ jz j + j= 1 j= 0 j= 0 α jπ j + ßu + d j u j + π = µ + ε, (1 ) where µ I O = µ I( u ) + µ (1 I( u )), ß = β I( u ) + β (1 I( u )), and d = δ I ( u ) + δ (1 I( u )). I O I The indicaor variable I u ) akes he value of 1 when he ( unemploymen rae u lies wihin a specified inerval γ ], and O [ L γ he value of 0 when u lies ouside ha inerval. Simply pu, equaion (1 ) allows he inercep µ and he coefficiens on he level and firs difference of he unemploymen rae, β and δ j, o ake differen values according o wheher u lies inside or ouside a specified range. The piecewise linear form of equaion (1 ) is illusraed in Figure 3, which depics he parial correlaion beween inflaion and he level of unemploymen. The radeoff beween inflaion and unemploymen akes he value of β I when he unemploymen rae is inside he inerval beween γ L and γ, and i akes he value of β O when he unemploymen rae is ouside his inerval. In principle, here is no reason o consrain he radeoff beween inflaion and unemploymen o be he same for low and high values of he unemploymen rae. In equaion (1 ), we do 8

9 consrain he coefficiens µ, β, and δ j o be he same when he unemploymen rae is below hreshold γ L or above hreshold γ only o preserve degrees of freedom a he esimaion sage. Since we are mainly ineresed in assessing he srengh of he radeoff beween inflaion and unemploymen when he unemploymen rae is no paricularly high or low, his simplificaion is no crucial. Moreover, o he exen ha he piecewise linear relaionship in (1 ) is saisically beer han he linear relaionship (1), hen, a foriori, a piecewise linear relaionship ha is even more flexible han (1 ) will perform beer han he linear relaionship (1). Anoher poenial explanaion for he weak radeoff beween inflaion and unemploymen depiced in Figure 2 is ha he linear relaionship in equaion (1) omis some imporan explanaory variable ha acs as a shifer o he inflaionunemploymen radeoff. An obvious candidae is a ime-varying naural rae of unemploymen. 8 As shown in he previous secion, during he years 1994 o 1999, he unemploymen rae fell from 6.5 percen o 4.1 percen, while he rae of price inflaion acually declined slighly. Insead of reflecing a lack of a significan radeoff over his range of he unemploymen rae, he failure of inflaion o increase over his period could be due o a decline in he naural rae of unemploymen. Such a decline would lead o a shifing in of he Phillips curve, wih poenially lile effec on inflaion. If his is indeed he case, hen he suggesed piecewise linear represenaion (1 ) would grossly misrepresen he naure of he inflaion- 8 Equaion (1) assumes ha, absen supply shocks, here is a consan naural level of he unemploymen rae ha is consisen wih a 9

10 unemploymen radeoff, even if i were o provide a beer approximaion o he daa han he linear relaionship (1). For his reason, we consider an alernaive formulaion of he linear Phillips curve relaionship ha explicily allows for a changing naural rae of unemploymen: = s k l N α jπ j + β u u ) + δ j u j + ξ jz j + j= 1 j= 0 j= 0 π ( ε (2) where u N denoes he ime-varying naural rae of unemploymen. Facors relaed o labor demand and supply, such as demographics and produciviy, could in principle lead o changes in he naural rae of unemploymen. Unforunaely, he naural rae of unemploymen is no known, and he series u N needs o be esimaed. None of he mehods commonly used for esimaing a ime-varying naural rae of unemploymen is foolproof, and he uncerainy surrounding he esimaed u N is usually large. Thus, while equaion (2) is poenially more informaive han (1), esimaes of a Phillips relaionship of he form of (2) should sill be regarded wih cauion. The piecewise linear counerpar o he linear specificaion (2) is given by: s k l ξ jz j + j= 1 j= 0 j= 0 N α jπ j + ß( u u ) + d j u j + π = µ + ε, (2 ) where N N N N µ = µ I( u u ) + µ (1 I( u u )), ß = β I( u u ) + β (1 I( u u )), and I N N d = δ I( u u ) + δ (1 I ( u u )). I O O The inerpreaion of he coefficiens in equaion (2 ) is he I O consan level of inflaion. This consan naural level of he unemploymen rae is esimaed as -µ/ß. 10

11 N same as in equaion (1 ), bu now I( u u ) akes he value of 1 when he unemploymen gap -- i.e., he deviaion of he unemploymen rae from he ime-varying naural rae of unemploymen -- lies wihin a specified inerval γ ], and i [ L γ akes he value of 0 when he unemploymen gap lies ouside ha inerval. If he presence of a changing radeoff beween inflaion and he level of unemploymen over differen ranges of he unemploymen rae is largely he resul of a ime-varying naural rae of unemploymen, hen equaion (2 ) should no provide a beer represenaion of he daa han he linear specificaion (2). In he nex secion, we proceed o esimae he piecewise linear, or hreshold, relaionships (1 ) and (2 ) and heir nesed linear versions, equaions (1) and (2), respecively. In so doing, we assess wheher he difference beween a linear and a hreshold represenaion of he Phillips relaionship is meaningful, from boh a saisical and an economic sandpoin. 11

12 II. Esimaion Mehod and Daa Descripion In order o evaluae empirically a hreshold relaionship such as (1 ) or (2 ), i is necessary o esimae, in addiion o all oher parameers in he relaionship, he hreshold parameers γ L and γ. Esimaion is grealy simplified by noing ha for given values of γ L and γ, i is possible o esimae he hreshold relaionship by ordinary leas squares (OLS). As a resul, he mehod for esimaing a hreshold relaionship consiss of a sequenial process ha performs OLS on (1 ) or (2 ) over differen values of he pair γ, ). The hreshold ( L γ esimaes are hen given by he pair γ ˆ, ˆ ) for which he OLS ( L γ regression of (1 ) or (2 ) yields he minimum sum of squared residuals. The oher parameers esimaes in relaionship (1 ) or (2 ) resul from he OLS regression ha uses γ ˆ, ˆ ) as he pair of hreshold esimaes. ( L γ The linear relaionships (1) and (2) of he Phillips curve can be hough of as resriced versions of (1 ) and (2 ), respecively. 9 In paricular, in he linear case he parameers are resriced o be equal inside and ouside he inerval [ γ L γ ] for any value of γ L and γ. 10 Since he hreshold specificaion ness is linear version, he hreshold model will always provide a leas as good a fi as he linear model. The quesion is wheher he fi is significanly beer from a saisical 9 Of course, relaionships (1) and (2) can be esimaed direcly by OLS. 10 If he parameers are resriced o be equal inside and ouside a given inerval γ ], hen he resricion will apply o any inerval [ γ L [ γ ], as one can easily infer from Figure 3. As a resul, when he L γ linear resricion holds, he hreshold values γ L and γ are no idenified. 12

13 sandpoin. To address his issue, we perform an F-es ha uses he sum of squared errors from he (resriced) linear and he (unresriced) hreshold model, respecively. Because under he null hypohesis of lineariy he values of he upper and lower hresholds γ L and γ from he piecewise linear alernaive are no idenified, he F-es has a non-sandard disribuion. The disribuion heory for such a es, however, is now well developed (ansen 1996 and 2000), and we rely on he exan lieraure o consruc p-values for he es. Box 1 provides addiional informaion abou esimaion and inference of hreshold models -- including informaion on how o obain confidence inervals for he esimaes γˆ L and γˆ. The daa used in he esimaion are quarerly and cover he period from he hird quarer of 1961 hrough he fourh quarer of We consider hree basic measures of inflaion: he change in he core componen of he chain-weighed deflaor for personal consumpion expendiures, he change in he core componen of he Consumer Price Index (CPI), and he change in he chain-weighed GDP deflaor. 11 Our measure for u is given by he civilian unemploymen rae. When evaluaing relaionships (2) and (2 ), we also need o form an esimae of he ime-varying naural rae of unemploymen, u N Budge Office (CBO) measure of u N. We use he Congressional, and we discuss alernaive measures laer in he secion. The supply shock variables included in Z are he change in he relaive price of food and energy and Gordon s (1982) series for wage and price conrols. 11 Denoing he price index by p, inflaion is defined as π = 100*(( p / p 1) 1). 4 13

14 When he measure of inflaion is given by he change in he GDP deflaor, Z also includes he change in he rade-weighed dollar. Daa sources for all series are given in he daa appendix. 14

15 III. Empirical Findings We firs discuss esimaion resuls for he linear and he piecewise linear forms of he Phillips curve wih a consan naural rae of unemploymen, equaions (1) and (1 ), respecively. In he piecewise linear case, he hreshold parameers γ L and γ are esimaed over a wide range of values aken by he unemploymen rae during he sample period. Table 1 summarizes he main findings for he hree measures of inflaion considered. The able repors esimaes for he main parameer of ineres in he Phillips relaionship, he coefficien β on he level of he unemploymen rae, for boh he linear and he piecewise linear specificaions. In he piecewise linear specificaion, he coefficien akes he value β I when he unemploymen rae lies inside he inerval γ ], and he value [ L γ β O when he unemploymen rae is eiher below γ L or above γ -- ouside he inerval. The able also repors esimaes of he hresholds γ L and γ, and, in square brackes, he 95 percen confidence inerval associaed wih hese esimaes. The las column of he able shows he p-value of an F-es of he null hypohesis of a linear model agains he esimaed hreshold alernaive. As usual, he esimaed coefficien on he level of he unemploymen rae in he linear model is highly significan and economically relevan for all hree measures of inflaion. The piecewise linear represenaion of he Phillips curve shows ha he radeoff beween inflaion and unemploymen is only significan for values of he unemploymen rae below γˆ L or above γˆ. For all hree measures of inflaion, γ L and γ are esimaed 15

16 a abou 4 percen and 7.5 percen, respecively. There is no evidence of a radeoff beween inflaion and he level of he unemploymen rae when he unemploymen rae lies beween 4 percen and 7.5 percen. In addiion, i is possible o show ha speed limi effecs are esimaed o be insignificanly differen from zero when he unemploymen rae lies inside he inerval γ ˆ ˆ ]. owever, for values of he unemploymen rae [ L γ below γˆ L or above γˆ, hese effecs end o be significan and larger han he ones esimaed by means of a linear specificaion. 12 The las column in he able shows ha, when inflaion is measured by eiher he core PCE deflaor or he core CPI index, he null hypohesis of a linear model is rejeced in favor of a hreshold specificaion. The small p-values indicae ha he piecewise linear represenaion of he Phillips curve is, from a saisical sandpoin, highly significan. When inflaion is insead measured by he GDP deflaor, he null hypohesis is rejeced in favor of a hreshold specificaion only marginally a he asympoic 5 percen level. This, however, has o do wih he presence of wo influenial observaions. 13 As shown in he las row of he able, when hese wo observaions are excluded from he sample, he piecewise linear represenaion of he Phillips curve again becomes highly significan. 12 This is rue when inflaion is measured eiher by he core CPI index or by he GDP deflaor. owever, when inflaion is measured by he core PCE deflaor, speed limi effecs are esimaed o be insignificanly differen from zero in boh he linear and he piecewise linear specificaions. 13 These observaions are for he second quarer of 1981 and he firs quarer of

17 Confidence inervals for he esimaed γˆ L and γˆ in he able are ofen large and run ino he lower and upper bounds for he unemploymen rae, 3.9 percen and 7.6 percen, respecively, over which we are searching for hreshold effecs. Sill, i is ineresing o noe ha, when inflaion is measured by eiher he core PCE deflaor or he core CPI index, he -saisic associaed wih he esimaed coefficien β I is never greaer han 2 for any pair γ, ) wihin he 3.9 percen o 7.6 percen ( L γ range. When inflaion is measured by he GDP deflaor and we drop he wo ouliers from he sample, he maximum -saisic for β I is 2.1, wih he -saisic above 2 in only 4 of he approximaely 800 differen pairs γ, ) we consider over he ( L γ menioned range. Overall, he findings in he able lend suppor o a piecewise linear version of he Phillips curve, wih a saisically significan and economically relevan radeoff beween inflaion and unemploymen only for paricularly high or low values of he unemploymen rae. We nex ask wheher hese findings coninue o hold over a shorer sample period. Table 2 repors esimaion resuls for he same specificaions of he Phillips relaionship as in Table 1, bu over he period from he hird quarer of 1961 hrough he fourh quarer of There are wo reasons o be ineresed in his paricular sample period. Firs, some have argued ha he naural rae of unemploymen was quie sable over he 1960s, 1970s, and 1980s. 14 If so, hen esimaing relaionships (1) and (1 ) over he period 1961 o 1986 should alleviae he criicism ha he relaionships are misspecified by no allowing for a 14 See, for example, Fuhrer (1995), Gordon (1998), and Tooell (1994). 17

18 ime-varying naural rae of unemploymen. Second, over he period 1987 o 2002, he unemploymen rae has been ouside he range of 4 percen o 7.5 percen only in hree quarers. I seems naural, hen, o ask o wha exen he findings in Table 1 are driven by he experience of he pas 15 years. Overall, he resuls in Table 2 confirm our previous findings. For he piecewise linear specificaion, we find evidence of a saisically significan radeoff beween inflaion and he level of he unemploymen rae only ouside he esimaed inerval γ ˆ ˆ ] [ L γ. The poin esimaes for γ L and γ are essenially he same as before for all hree measures of inflaion. The null hypohesis of a linear model is rejeced in favor of he hreshold alernaive when inflaion is measured by he core CPI index or he GDP deflaor. One canno rejec he hypohesis of a linear specificaion when inflaion is measured by he core PCE deflaor, bu he -saisic associaed wih he esimaed coefficien β I is never greaer han 2 for any pair ( L γ γ, ) wihin he 3.9 percen o 7.6 percen range. I is possible o compare he performance of he hreshold specificaion (1 ) vis-à-vis he linear specificaion (1) by means of a dynamic simulaion, in which prediced values of inflaion for he curren period are fed ino subsequen periods as lagged values. In essence, he simulaion is a muli-period in-sample forecas of inflaion ha does no refer o an acual inflaion rae over he simulaion horizon. Figure 4 depics he resul of his exercise for core CPI inflaion over he period 1961 o 1986, using he esimaes repored in Table 2. Considering ha he simulaion runs over 25 years, boh he linear and he piecewise linear model perform well. I is 18

19 eviden, hough, ha he hreshold model racks acual inflaion much beer han he linear model. Unforunaely, he repored success of he piecewise linear specificaion is mirrored by is failure o rack inflaion in a dynamic simulaion over he period 1987 o 2002, using he esimaes in eiher Table 1 or Table 2. The reason is ha he hreshold γ is usually esimaed a abou 7.5 percen. The hreshold model largely fails o capure he decline in inflaion associaed wih he recession and he slow iniial recovery of he early 1990s, when he unemploymen rae was above he 7.5 percen mark in only wo quarers. 15 This observaion, per se, does no mean ha a piecewise linear represenaion of he Phillips curve canno capure recen inflaion dynamics. Indeed, i is possible o show ha he piecewise linear relaionship (1 ) esimaed over he period 1987 o 2002 is highly significan, wih he null hypohesis of a linear relaionship (1) always rejeced in favor of he hreshold alernaive. Bu he esimaed inerval over which here is lile or no radeoff beween inflaion and he unemploymen rae now ranges from abou 4 percen o 6.5 percen. Wih such an esimae of γ, i is no surprising ha he hreshold relaionship performs very well in a dynamic simulaion of inflaion over he pas 15 years. Ye, we find i somewha unappealing o resor o a change in γˆ in order o explain recen inflaion behavior. Parameer insabiliy could in fac be he resul of having specified equaion (1 ) incorrecly. For his reason, in he res of his secion we explore wheher a 15 As a resul, in a dynamic simulaion over he period 1987 o 2002, he piecewise linear model would predic core CPI inflaion in he range of 5 percen o 6 percen over he pas decade. 19

20 Phillips curve relaionship ha allows for a ime-varying naural rae of unemploymen provides a more sable guidance for inflaion dynamics. There are several approaches o esimaing he pah of a ime-varying naural rae of unemploymen. One approach esimaes a changing naural rae of unemploymen from a linear Phillips relaionship in which he inercep is allowed o vary over ime. Anoher approach akes a consan naural rae of unemploymen esimae from a linear Phillips curve such as equaion (1), bu hen adjuss he esimae o accoun for demographic facors. Changes in hese facors cause he demographically adjused esimae of he naural rae of unemploymen o change over ime. An example of he laer approach is he Congressional Budge Office (CBO) measure of he naural rae of unemploymen, which we ake as our esimae of he ime-varying naural rae of unemploymen in wha follows. In essence, such a measure varies over ime because he shares of differen demographic groups (broken down by age, sex, and race) in he labor force vary. 16 Figure 5 depics he CBO esimae of he naural rae of unemploymen, he acual value of he civilian unemploymen rae, and he unemploymen gap, which is he difference beween he unemploymen rae and he naural rae of unemploymen. The figure shows an increase in he esimae of he naural rae of unemploymen from abou 5.5 percen a he beginning of he 1960s, o abou 6.25 percen in he lae 1970s. Since hen, he 16 The CBO esimaes, by means of a linear Phillips curve such as equaion (1), a naural rae of unemploymen for married males. This naural rae of unemploymen in urn is used o esimae a naural rae of unemploymen for differen demographic groups. The overall naural rae of unemploymen is hen compued as a weighed average of he naural rae of unemploymen for he differen demographic groups, wih he weighs se equal o each group's share of he labor force. 20

21 naural rae of unemploymen has declined, mos noably in he pas decade, o close o 5 percen. The increase in young workers -- who have higher unemploymen raes han older workers -- accouns for much of he rise in he naural rae of unemploymen before 1980, while he recen decrease in he number of young workers explains much of he recen fall in he naural rae of unemploymen. Because changes in he esimaed naural rae of unemploymen occur very gradually over ime, he unemploymen gap depiced in he figure racks he acual unemploymen rae closely. Table 3 summarizes esimaion resuls for he unemploymen gap version of he linear and piecewise linear Phillips curves, equaions (2) and (2 ), respecively. In he piecewise linear specificaion, he hreshold parameers γ L and γ are now esimaed over a wide range of values aken by he unemploymen gap during he sample period. Thus, he radeoff beween inflaion and he unemploymen gap is given by β I when he unemploymen gap lies inside he inerval γ ] [ L γ he unemploymen gap is eiher below γ L or above γ., and by β O when In he linear specificaion, esimaes of he radeoff, βˆ, are significan and economically relevan for all hree measures of inflaion. In he piecewise linear specificaion, we find evidence of a significan radeoff only for values of he unemploymen gap below γˆ L or above γˆ. In order o observe his radeoff, he unemploymen gap mus be below 1.4 percen or above 1.4 percen, approximaely. I is ineresing o noe ha over he period 1961 o 2002, values of he gap below 1.4 percen usually map ino values of he unemploymen rae a or 21

22 below 4 percen, and values of he gap above 1.4 percen usually map ino values of he unemploymen rae a or above 7.5 percen. Such a mapping is precise, bu no perfec. In paricular, in he early 1990s, he gap was above 1.5 percen for six consecuive quarers, whereas he unemploymen rae was above 7.5 percen in jus wo quarers. Sill, here is a close similariy beween he findings in Table 1 for he consan naural rae of unemploymen specificaion (1 ) and he findings in Table 3 for he gap specificaion (2 ). The las column in Table 3 shows ha he null hypohesis of a linear model is srongly rejeced in favor of a hreshold specificaion when inflaion is measured by he core CPI index or by he GDP deflaor. The linear model is rejeced marginally when inflaion is measured by he core PCE deflaor. In his case, however, he -saisic associaed wih he esimaed coefficien β I is always considerably smaller han 2 for any of he differen pairs γ, ) we consider wihin he 1.4 percen o ( L γ 1.4 percen range. Overall, we conclude ha he piecewise linear model coninues o be significan from a saisical sandpoin even when we allow for a ime-varying naural rae of unemploymen. One could argue ha such a finding is an arifac of our chosen measure of he ime-varying naural rae of unemploymen. For his reason, we experimened wih oher esimaes of he naural rae of unemploymen -- specifically wih esimaes ha exhibi more variabiliy han he CBO measure -- bu we reached similar conclusions. 17 In general, he more variable he esimaed 17 For example, we have esimaed specificaions (2) and (2 ) using a long-run, wo-sided moving average of he acual unemploymen rae as he esimae for he ime-varying naural rae of unemploymen (see 22

23 naural rae of unemploymen, he smaller he esimaed inerval [ γ L γ ] over which here is an insignifican radeoff beween inflaion and unemploymen. A more variable naural rae of unemploymen racks he acual unemploymen rae more closely, and his leads o a decline in he average size of he unemploymen gap. Bu hen he esimaed inerval γ ] over [ L γ which here is no significan radeoff, while smaller, sill conains almos wo-hirds of he observaions, as is he case wih he findings in Table 3. Before concluding, le us briefly revisi he issue of wheher he piecewise linear model can rack inflaion over he years 1987 o In order o do so, we now esimae equaions (2) and (2 ) for he period 1961 o 1986, and we use hese esimaes o forecas inflaion over he nex 16 years. The resuls of his dynamic simulaion for core CPI inflaion are displayed in Figure 6. In he firs half of he simulaion, he linear and hreshold specificaions perform equally well. Since he unemploymen gap is above he esimaed hreshold value γ of 1.4 percen for six quarers during he early 1990s, he hreshold model is able o capure he downurn in inflaion ha occurred a ha ime. No surprisingly, he hreshold model does significanly beer han he linear model in he laer half of he simulaion. The naural rae of unemploymen, while declining over his period, is no declining fas enough o close he negaive unemploymen gap. As a resul, he linear model predics a surge in inflaion ha never maerialized. owever, since he Saiger, Sock and Wason 2001). We have also derived an esimae of he ime-varying naural rae of unemploymen from a linear Phillips curve specificaion such as (1) ha allows for a ime-varying inercep (see Gordon 1998). 23

24 negaive gap is never below he esimaed γ L for exended periods of ime, he hreshold model performs reasonably well. 24

25 IV. Concluding Remarks Overall, we find ha a piecewise linear specificaion of he Phillips curve provides a good characerizaion of inflaion dynamics over he pas 40 years. The esimaed relaionship implies a range of values for he unemploymen rae, or he unemploymen gap, over which we find no significan radeoff beween inflaion and unemploymen. Ouside of his range, a significan radeoff obains. ow should we inerpre our findings in ligh of he mos recen inflaion and unemploymen experience? As is well known, core inflaion measures have dropped significanly over he period from he fourh quarer of 2002 hrough he second quarer of 2003, wih he unemploymen rae rising o 6.4 percen by he end of he firs half of Unless he decline in inflaion is emporary, i is difficul o reconcile his recen experience wih he esimaes obained from a piecewise linear Phillips relaionship using a consan naural rae of unemploymen. In such a specificaion, he esimaed radeoff beween inflaion and unemploymen is insignifican when he unemploymen rae ranges from abou 4.0 percen o 7.5 percen. I is likely, hough, ha while his range is well suied o characerize he pah of inflaion over he years 1960 o 1989, i is no appropriae for he mos recen period. The experience of he second half of he 1990s appears, in fac, o be consisen wih a value of he naural rae of unemploymen ha is lower han he average naural rae of unemploymen of he previous hree decades. For his reason, he esimaes ha arise 25

26 from an unemploymen gap specificaion of he piecewise linear Phillips relaionship may have more bearing on he curren siuaion. If he naural rae of unemploymen is now near 5 percen, hen, according o our upper hreshold esimae of 1.4 percen, we should expec inflaion o sar dropping when he unemploymen rae is close o 6.5 percen. This is broadly consisen wih recen experience, in ha inflaion has been relaively sable over mos of he period from 2000 o 2002, when he unemploymen gap was small. By conras, i sared o decline noiceably only when he unemploymen gap became relaively large. From a moneary policy sandpoin, he finding of a range of values for he unemploymen rae or he unemploymen gap over which here is no significan radeoff beween inflaion and unemploymen means ha here is scope for argeing a level of he unemploymen rae a which inflaion is sable. Clearly, he lower he argeed level of he unemploymen rae associaed wih no inflaionary pressures, he higher sociey s welfare. 18 This is an argumen for moneary policy o ry cauiously o drive he unemploymen rae lower unil he lower limi of he sable inflaion range is reached. Such a sraegy, however, is complicaed by he fac ha he lower level of he unemploymen rae or unemploymen gap associaed wih no inflaionary pressures is esimaed wih uncerainy. As a resul, 18 The Federal Reserve Ac specifies ha he Federal Reserve Sysem and he Federal Open Marke Commiee should seek o promoe effecively he goals of maximum employmen, sable prices, and moderae long-erm ineres raes. 26

27 policymakers would have o deermine when o sop by being aler o incipien signs of acceleraing inflaion. Thus, i remains an open quesion wheher a rigger sraegy in which moneary policy aims o drive he unemploymen rae or unemploymen gap o he lower hreshold and hen responds vigorously would deliver a saisfacory sabilizaion performance. The answer depends on one s judgmen concerning policymakers abiliy o noe he emergence of incipien inflaion-acceleraion indicaors, heir abiliy o respond rapidly wih appropriae measures, and he lead-ime for such measures o ake effec. 27

28 * Economis and Senior Economis, respecively, Federal Reserve Bank of Boson. We hank several colleagues a he Boson Fed for helpful commens and suggesions. References Akeson, Andrew, and Ohanian, Lee E Are Phillips Curves Useful for Forecasing Inflaion? Federal Reserve Bank of Minneapolis Quarerly Review 25 (1) Winer: Blanchard, Olivier Jean, and Fischer, Sanley Lecures on Macroeconomics. Cambridge, MA: MIT Press. Blinder, Alan S Cenral Banking in Theory and Pracice. Cambridge, MA: MIT Press. Eisner, Rober A New View of he NAIRU. In Improving he Global Economy; Keynesianism and he Growh in Oupu and Employmen, edied by Paul Davidson and Jan A. Kregal. Lynne, N: Edward Elgar. Fuhrer, Jeffrey C The Phillips Curve Is Alive and Well. New England Economic Review March/April: Gordon, Rober J Price Ineria and Ineffeciveness in he Unied Saes, Journal of Poliical Economy 90(6) December: Gordon, Rober J Foundaions of he Goldilocks Economy: Supply Shocks and he Time-Varying NAIRU. Brookings Papers on Economic Aciviy 0 (2): all, Rober E Wage Deerminaion and Employmen Flucuaions. NBER Working Paper No. w9967, Sepember. ansen, Bruce E Inference When A Nuisance Parameer Is No Idenified Under The Null ypohesis. Economerica 64(2): ansen, Bruce E. 2000, Sample Spliing and Threshold Esimaion. Economerica 68(3): Phillips, A. W The Relaion beween Unemploymen and he Rae of Change of he Money Wage Raes in he Unied Kingdom. Economica 25 November: Saiger, Douglas, Sock, James. and Wason, Mark W Prices, Wages and he U.S. NAIRU in he 1990s. In The Roaring Nineies: Can Full Employmen Be Susained?, edied by Alan B. Krueger and Rober M. Solow, pp. 3-60, New York : Russell Sage Foundaion, Cenury Foundaion Press. 28

29 Thomas, Jonahan and Worrall, Tim Self-Enforcing Wage Conracs. Review of Economic Sudies 55(4): Tooell, Geoffrey M. B Resrucuring, he NAIRU, and he Phillips Curve. New England Economic Review Sepember/Ocober: Woglom, Geoffrey Underemploymen Equilibrium wih Raional Expecaions. Quarerly Journal of Economics 97(1):

30 Daa Appendix Price Series (π): Core CPI: CPI-U, All Iems Less Food and Energy (SA, =100), Bureau of Labor Saisics, Quarerly Percenage Change, annualized. GDP Deflaor: Gross Domesic Produc: Chain-ype Price Index (SA, 1996=100), Bureau of Economic Analysis, Quarerly Percenage Change, annualized. Core PCE Deflaor: PCE less Food and Energy: Chain Price Index (SA, 1996=100), Bureau of Economic Analysis, Quarerly Percenage Change, annualized. Real Aciviy Variables (u): Civilian Unemploymen Rae: 16 yr + (SA, %), Bureau of Labor Saisics. Non-acceleraing Inflaion Rae of Unemploymen {CBO} (%), Congressional Budge Office. Oher Variables (Z): Relaive Price of Food and Energy: (CPI /Core CPI ), where CPI is CPI-U: All Iems (SA, =100), Bureau of Labor Saisics, Quarerly Percenage Change. U.S.: Nominal Effecive Exchange Rae (1995=100), Inernaional Financial Saisics (IMF), Quarerly Percenage Change. Gordon (1982) Wage and Price Conrol Variable: equal o 0.8 from 1971:Q3 hrough 1972:Q3, -0.4 for 1974:Q2, -1.6 from 1974:Q3 hrough 1974:Q4, -0.4 for 1975:Q1, and 0 for all oher daes. 30

31 Box 1 A general and compac way of represening he piecewise linear or hreshold model in he ex is: y = β ( γ ) + ε x wih β = β I, β ), γ = γ L, γ ), and ( O ( x I ( γ L q γ ) x ( ) = γ, x (1 I( γ L q γ )) where x is he vecor of righ-hand side variables, q is he hreshold variable, γ L is he lower hreshold value of he hreshold variable, γ is he upper hreshold value of he hreshold variable, and I is he indicaor funcion which akes on a value of one when he expression in parenhesis is rue. Le S 0 represen he sum of squared residuals under he null hypohesis of a linear model, and S ( ) he sum of squared residuals of he piecewise linear model as a funcion of γ. In his case an F-es of he null hypohesis of a linear Phillips curve is based on: 1 γ F 1 S ˆ 0 S1( γ ) = n, S ( γˆ ) 1 where S 1 ( γ ˆ) is he minimum sum of squared residuals for he piecewise linear model obained by searching over a grid of possible upper and lower hreshold values. ansen (1996) shows ha he asympoic saisical disribuion for his es saisic is nonsandard and sricly dominaes he χ 2 disribuion. This nonsandard disribuion can be approximaed o he firs order by a boosrap procedure, and p-values consruced from he boosrap for he es of he linear null 31

32 agains he hreshold alernaive hypohesis are asympoically valid. In order o es he hypohesis 0 : γ = γ 0, he likelihood raio es is o rejec for large values of LR1( γ 0), where: LR S ( γ) S ( γˆ ) 1 1 1( γ ) = n. S ˆ 1( γ ) In addiion, he asympoic disribuion of LR1( γ 0) can be used o form valid asympoic confidence inervals abou he esimaed hreshold values. These confidence inervals are he se of values of γ such ha he likelihood raio lies below he criical values abulaed by ansen (2000) for he desired level of confidence. Finally, he esimaor β = β (γ ) depends on he hreshold esimae. Since he dependence on he hreshold esimae is no of firs-order asympoic imporance, inference on β can proceed as if he hreshold esimae were he rue value. 32

33 Table 1 Phillips Curve Esimaes wih Consan Naural Rae of Unemploymen: 1961 o 2002 s k l ξ jz j + j= 1 j= 0 j= 0 α jπ j + βu + δ j u j + π = µ + ε Linear Model Threshold Model Inflaion Measure n βˆ βˆ I βˆ O γˆ L γˆ p-value of F-es Core PCE Deflaor ** (.0484).1008 (.0792) ** (.0607) 3.95 [3.90, 5.30] 7.40 [5.80, 7.60].016 Core CPI ** (.0745).0926 (.1053) ** (.0974) 3.95 [3.90, 4.80] 7.45 [7.30, 7.60].001 GDP Deflaor ** (.0587) (.0953) ** (.0739) 3.95 [3.90, 5.50] 7.35 [6.00, 7.60].048 GDP Deflaor excluding ouliers ** (.0573) (.0823) ** (.0783) 3.95 [3.90, 5.50] 7.45 [7.30, 7.60].001 Noe: Esimaion period is 1961:Q3 o 2002:Q4. For he core PCE deflaor and he Core CPI index, he esimaed equaions include seven lags of inflaion, he conemporaneous unemploymen rae, he conemporaneous and wo lags of he firs difference of he unemploymen rae, wo lags of he change in he relaive price of food and energy, and he Gordon variable. For he GDP deflaor, he esimaed equaions include eigh lags of inflaion, one lag of he unemploymen rae, four lags of he firs difference of he unemploymen rae, one lag of he change in he relaive price of food and energy, wo lags of he change in he exchange rae, and he Gordon variable. For he core PCE deflaor and he Core CPI index, he hreshold variable is he conemporaneous unemploymen rae, while for he GDP deflaor i is he firs lag of he unemploymen rae. ** indicaes significance a he 5 percen level.

34 Table 2 Phillips Curve Esimaes wih Consan Naural Rae of Unemploymen: 1961 o 1986 s k l ξ jz j + j= 1 j= 0 j= 0 α jπ j + βu + δ j u j + π = µ + ε Linear Model Threshold Model Inflaion Measure n βˆ βˆ I βˆ O γˆ L γˆ p-value of F-es Core PCE Deflaor ** (.0589).2687 (.1434) ** (.0664) 3.95 [3.90, 5.50] 7.25 [5.80, 7.60].205 Core CPI ** (.1045).1513 (.1888) ** (.1223) 3.95 [3.90, 4.80] 7.45 [7.00, 7.60].004 GDP Deflaor excluding ouliers ** (.0782) (.1354) ** (.0957) 3.95 [3.90, 5.50] 7.45 [7.40, 7.60].048 Noe: Esimaion period is 1961:Q3 o 1986:Q4. For he core PCE deflaor and he Core CPI index, he esimaed equaions include seven lags of inflaion, he conemporaneous unemploymen rae, he conemporaneous and wo lags of he firs difference of he unemploymen rae, wo lags of he change in he relaive price of food and energy, and he Gordon variable. For he GDP deflaor, he esimaed equaion includes eigh lags of inflaion, one lag of he unemploymen rae, four lags of he firs difference of he unemploymen rae, one lag of he change in he relaive price of food and energy, wo lags of he change in he exchange rae, and he Gordon variable. For he core PCE deflaor and he Core CPI index, he hreshold variable is he conemporaneous unemploymen rae, while for he GDP deflaor i is he firs lag of he unemploymen rae. ** indicaes significance a he 5 percen level.

35 Table 3 Phillips Curve Esimaes wih a Time-Varying Naural Rae of Unemploymen: 1961 o 2002 s k l ξ jz j + j= 1 j= 0 j= 0 N α jπ j + β( u u ) + δ j u j + π = µ + ε Linear Model Threshold Model Inflaion Measure n βˆ βˆ I βˆ O γˆ L γˆ p-value of F-es Core PCE Deflaor ** (.0551) (.0933) ** (.0612) [-1.60, ] 1.30 [1.20, 1.60] 0.06 Core CPI ** (.0849) (.1366) ** (.0978) [-1.40, -1.30] 1.40 [1.30, 1.45] GDP Deflaor excluding ouliers ** (.0626) (.1034) ** (.0719) [-1.60, -0.80] 1.56 [1.20, 1.60] Noe: Esimaion period is 1961:Q3 o 2002:Q4. For he core PCE deflaor and he Core CPI index, he esimaed equaions include seven lags of inflaion, he conemporaneous unemploymen gap, he conemporaneous and wo lags of he firs difference of he unemploymen rae, wo lags of he change in he relaive price of food and energy, and he Gordon variable. For he GDP deflaor, he esimaed equaion includes eigh lags of inflaion, one lag of he unemploymen gap, four lags of he firs difference of he unemploymen rae, one lag of he change in he relaive price of food and energy, wo lags of he change in he exchange rae, and he Gordon variable. For he core PCE deflaor and he Core CPI index, he hreshold variable is he conemporaneous unemploymen gap, while for he GDP deflaor i is he firs lag of he unemploymen gap. ** indicaes significance a he 5 percen level.

36 Figure 1 Char A Inflaion and Unemploymen: 1960 o inflaion, Q4/Q Q unemploymen rae Char B Inflaion and Unemploymen: 1970 o inflaion, Q4/Q Q unemploymen rae

37 Figure 1 (coninued) Char C Inflaion and Unemploymen: 1980 o Q1 inflaion, Q4/Q unemploymen rae Char D Inflaion and Unemploymen: 1990 o Q1 inflaion, Q4/Q unemploymen rae

38 Figure 2 Parial Correlaion Beween Inflaion and he Unemploymen Rae when he Unemploymen Rae is beween 4 and 7.5 percen: 1960 o Inflaion X Unemploymen Rae X

39 Figure 3 Piecewise Linear Phillips Curve π x slope = β O slope = β I slope = β O 0 γ L γ u x

40 Figure 4 Dynamic Simulaion of Core CPI Inflaion: 1961 o Acual Core CPI Inflaion Simulaed Inflaion wih Piecewise Linear Model 14 Simulaed Inflaion wih Linear Model Jan-61 Jan-63 Jan-65 Jan-67 Jan-69 Jan-71 Jan-73 Jan-75 Jan-77 Jan-79 Jan-81 Jan-83 Jan-85

41 Figure 5 Acual Unemploymen Rae, CBO Naural Rae of Unemploymen, and Unemploymen Gap: 1959 o Jan-59 Jun-64 Dec-69 Jun-75 Nov-80 May-86 Nov-91 May-97 Oc CBO Esimae of Naural Rae of Unemploymen Civilian Unemploymen Rae Unemploymen Gap -8.00

42 Figure 6 Dynamic Simulaion of Core CPI Inflaion: 1987 o Acual Core CPI Inflaion 6 Simulaed Inflaion wih Piecewise Linear Model 5 Simulaed Inflaion wih Linear Model Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03