A Working Solution to the Question of Nominal GDP Targeting

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1 A Working Soluion o he Quesion of Nominal GDP Targeing Michael T. Belongia Oho Smih Professor of Economics Universiy of Mississippi Box 1848 Universiy, MS and Peer N. Ireland Deparmen of Economics Boson College 140 Commonwealh Avenue Chesnu Hill, MA January 2013 Absrac: Alhough a number of economiss have ried o revive he idea of nominal GDP argeing since he financial crisis of 2008, very lile has been said abou how his objecive migh be achieved in pracice. This paper adops and exends a sraegy firs oulined by Holbrook Working (1923) and laer employed by Hallman, e al. (1991) in he P-Sar model. I presens a series of heoreical and empirical resuls o argue ha Divisia moneary aggregaes can be conrolled by he Federal Reserve and ha he rend velociies of hese aggregaes exhibi he sabiliy required o make long-run argeing feasible. JEL codes: E58, E52, E51 The auhors wish o hank William Barne, David Beckworh, and Sco Sumner for exremely helpful commens and Yan Li for able research assisance.

2 A Working Soluion o he Quesion of Nominal GDP Targeing Alhough sabilizing nominal GDP has been suggesed before as an objecive for moneary policy acions, an increasing number of economiss have ried o revive he idea since he financial crisis of 2008 and he apparen ineffeciveness of manipulaing he federal funds rae when he zero bound consrain has been me. Bu while he meris of nominal GDP sabilizaion as a final objecive for moneary policy have been emphasized in recen discussions, very lile has been said abou how his goal migh be achieved in pracice. Indeed, whereas earlier discussions offered explici sraegies and esablished linkages beween, for example, nominal GDP and he moneary base (see, e.g., McCallum (1988) and Melzer (1987)) or a broader moneary aggregae (Feldsein and Sock (1994)), he recen discussions have been relaively srong on he goal and relaively silen on how a pah o he goal migh be implemened. 1 Indeed, wih he recen innovaions of paymen of ineres on reserves and unusual behavior of he moneary base in he afermah of he financial crisis, even houghs of reviving some of he older, well-ariculaed sraegies have been pu on hold. Thus, for all of he aenion ha nominal GDP argeing has received as a poenial goal for moneary policy, a pracical means of achieving ha end has ye o be offered. 2 In his paper we propose a sraegy for nominal GDP argeing based on a framework firs oulined by Holbrook Working (1923) and used, wih only minor modificaions, by Hallman, e al. (1991) in he P-Sar model. In hese earlier applicaions, a policymaker is able o evaluae wheher a value for he money sock is consisen wih long-run price sabiliy. Using essenially he same derivaion found in Working s original paper and making appropriae changes o he pracical adapaions employed by Hallman e al., we find a pah for money ha is consisen wih any desired long-run rajecory for nominal GDP. Unlike 1 An excepion is Sumner (1989, 1995) and he suggesion of implemening moneary policy hrough he use of a nominal GDP fuures marke. For a survey of issues regarding nominal GDP argeing, see Bean (1983). Clark (1994) offers some evidence on lagged adjusmen v. forecas adjusmen rules when NGDP argeing is implemened. 1

3 previous applicaions of his framework, we employ Divisia moneary aggregaes in esablishing a pah for money ha he cenral bank should ry o mainain and use a one-sided filering algorihm ha can be implemened in real ime o conrol for slow-moving rends in velociy. 3 In wha follows, we firs explain he basic analyics of Working s framework and how we have adaped i o nominal GDP. Then, afer reproducing Hallman e al. s regression resuls o show ha movemens in he Divisia aggregaes consisenly anicipae movemens in nominal income over a sample period ha exends from 1967 hrough he presen, we compare acual pahs for he Divisia moneary aggregaes o alernaive rajecories ha, according o our framework, would have been consisen wih more sable nominal GDP growh since Afer using his comparison o discuss, in paricular, he sance of curren moneary policy, we examine how he Fed migh conrol he behavior of hese Divisia aggregaes wihin an inermediae argeing sraegy. Overall, we conclude ha if nominal GDP is chosen as he cenral bank s objecive (a quesion on which his paper akes no posiion), he sraegy oulined in his paper has several virues: I is ransparen o ouside observers, i is forwardlooking, and ye i can be implemened in a fairly sraigh-forward manner. 4 In fac, one migh speculae ha one reason for he demise of he P-sar model was he possample insabiliy of he velociy of M2, somehing which can be raced o he financial innovaions era bu can be aribued more specifically o he problems inheren in simple sum aggregaion mehods ha fail o inernalize pure subsiuion effecs and would have been a consequence of such hings as he paymen of ineres on deposis, he availabiliy of a broader array of deposi accouns, and he greaer subsiuion among hese accouns by consumers in response o changes in user coss. In his conex, i is ineresing o noe ha Working, nearly niney years ago, devoed an appendix of his paper o an aemp o creae an Index for a Medium of Exchange. Even hough he was wriing long before he era of financial innovaions and he paymen of ineres on checkable deposis, he inuied ha differen componens of a moneary aggregae should be weighed differenly and in his appendix he made an early aemp o do jus ha. One reason we ake no posiion on he desirabiliy of NGDP argeing is he resuls in Wes (1986). Using a model presened in Bean (1983), Wes demonsraed ha he preference of NGDP argeing over, say, money supply argeing depends on values of cerain parameers and, a priori, here is no clear reason o believe why hose should ake a value ha would lead a policymaker o prefer one opion over he oher. We also ake no posiion on wheher argeing he level of nominal GDP is o be preferred o argeing he growh rae of NGDP. Throughou, our purpose is derive a pracical approach o argeing he level of NGDP if ha is o become he cenral bank s adoped goal. 2

4 Working s Framework Working s (1923) objecive was o find a value for he money supply ha would be consisen wih long-run price sabiliy. A he ime of his wriing, many ohers had invesigaed Quaniy Theory relaionships empirically. 5 From his research, sraegies o sabilize he price level emerged bu even Fisher s (1920) plan did no incorporae a mehod for dealing wih lags in he process. Thus, Working s innovaion was o recognize he role of lags and o esablish a policy framework ha embedded a long-run desired pah for price sabiliy. A cenral bank hen could compare he curren price level agains he desired long-run pah and evaluae wheher he sance of policy was oo accommodaive or oo resricive. Using Quaniy Theory relaionships, Working re-wroe he basic expression as (V/T) = (P/M). Because (P/M) did no have a definie concepion, Working deal wih is reciprocal. To find a long-run pah for i, he esimaed a rend value for he price level using a regression of he log value of he price level on ime, ime squared, and ime cubed; fuure values for he price level were exrapolaions from his rend regression. Wih his informaion, Working hen could plo, on a log scale, values for (M/P) o illusrae he value of circulaing medium ha would be consisen wih his long-run rend pah for he aggregae price level. In adaping Working s framework for he P-Sar model, Hallman, e al. (1991) expressed heir basic relaionship as: (1) P * = (M2 V * )/Q *. In his expression, P * is he long-run arge value for he price level a ime, V *, is he longrun equilibrium value for velociy, aken by Hallman, e al, o be he sample mean for M2 velociy, and Q * is he value for poenial real GDP a ime. 6 Rearranging erms so as o apply For more background, see he surveys in Humphrey (1973) and Laidler (2011). Alhough Taylor (1993) published his famous paper on a rule for he implemenaion of moneary policy afer he P-sar paper was published, he did no cie i. Noneheless, he had his o say abou an alernaive rule in ha paper (pp ): Since he mid-1970s moneary arges have been used in many counries o sae arges for inflaion. If money velociy were sable, hen, given an esimae of poenial oupu growh, money arges would imply a arge for he price level; given velociy and a real oupu arge, he arge price level would obviously fall ou algebraically from he money supply arge. Even hough he 1980s 3

5 he framework more direcly o nominal income argeing and making some desirable changes in empirical choices, he framework o be employed in his paper is: (2) PQ * = M V *, where PQ * is he long-run arge value for nominal GDP, M is he value of a Divisia moneary aggregae, and V * is rend velociy for ha chosen moneary aggregae. Equaion (2) highlighs one key advanage of any nominal income argeing scheme, relaive o he price-level or inflaion argeing framework implied by he P-sar model in (1): Nominal income argeing allows one o sidesep he challenge of esimaing accuraely poenial oupu in real ime. Meanwhile, he use of a Divisia moneary aggregae in (2) in place of simple-sum M2 in (1) is moivaed by Barne s (1980) classic work, which inroduced moneary economiss o he logic behind, and he pracical benefis of, Divisia moneary aggregaion; his empirical choice also disinguishes our approach from ha of Feldsein and Sock (1994), which like he P-sar model, uses simple sum M2 as an inermediae arge wihin a nominal GDP argeing sraegy. 7 In (2), we also depar from he P-sar framework in ye anoher way, by calculaing rend velociy V * using he one-sided version of he Hodrick-Presco (1997) filer described by Sock and Wason (1999). Figure 1 uses quarerly daa o compare he acual velociies of Divisia M1 and MZM o he rend values obained wih his one-sided H-P filer. 8 The choice, boh here and below, o focus on M1 and MZM ogeher allows us o assess he robusness of our findings o he choice of narrow versus broad moneary aggregaes. Our series on Divisia have shown ha money velociy is no sable in he shor run, he long-run sabiliy of he velociy of some moneary measures allows one o sae arges for he price level. For example, wih an esimaed secular growh of real oupu of 2.5 percen and a seady velociy, a money growh range of 2.5 percen o 6.5 percen he Fed s arges for 1992 would imply ha he price level arge grows a 0 o 4 percen per year. Given biases such as index number problems in measuring prices, he 2-percen per year implici arge inflaion rae is probably very close o price sabiliy or zero inflaion. 7 For a more recen discussion and survey of he exensive lieraure on he Divisia moneary aggregaes, see Barne (2012). The MZM aggregae money, zero mauriy includes hose asses in M2, less small ime deposis, plus insiuion-only money marke muual funds. I firs was discussed in deail by Moley (1988), who referred o i as non-erm M3. I laer picked up he label of MZM. 4

6 M1 and MZM are drawn from he Federal Reserve Bank of S. Louis FRED daabase; Anderson and Jones (2011) describe heir consrucion in deail. Wih quie similar resuls, no shown, we also replicaed he analysis using Anderson and Jones Divisia M2 series, as well as he much broader, Divisia M4 aggregae provided by he Cener for Financial Sabiliy and described in Barne, e al. (2012). The graphs in Figure 1 reveal quie clearly he shifing, bu slow-moving, rends in velociies ha, Reynard (2007) finds, mus be accouned for in idenifying he long-run linkages beween money and prices, no jus in he U.S. bu in Swizerland and he Euro Area as well. 9 Our one-sided version of he H-P filer imposes he same seing λ = 1600 for he smoohing parameer as is commonly used in he wo-sided H-P filer for quarerly daa. I produces a similar, bu somewha more volaile, measure of he rend, reflecing he fac ha, unlike he sandard H-P filer, he one-sided varian only uses daa up hrough period in consrucing he value for he rend a period. This feaure, however, is precisely wha allows our algorihm o be implemened in real ime and also makes our measure suiable for use in he forecasing equaions described below. An added advanage of his one-side filer is ha once he parameer λ is fixed, no addiional parameers need o be esimaed or calibraed in consrucing he series for rend velociy: As explained by Sock and Wason (1999, p. 301), values for he rend can be generaed quickly and easily using he equaions of he sandard Kalman filer. Oherwise, equaion (2) parallels (1) for he P-sar model by depicing he nominal GDP arge PQ * for ime as one ha is implied by he level of he Divisia moneary aggregae M for ha period, given he value of V *, and by suggesing ha he acual value for nominal income PQ should end o graviae, over ime, owards he arge PQ *. To es his hypohesis, we esimae a se of regression equaions ha mirror Hallman e al. s (1991, p. 847) in heir 9 Along he same lines, i is ineresing o noe once again ha Working (1923) himself found i necessary o conrol for slow-moving shifs in rends by including ime squared and cubed in addiional o ime iself in his regression equaions. Since Working s regression-based approach migh well be considered an early version of he modern, hough only slighly more elaborae, filering procedures used here, we find i especially useful o race he origins of our own approach back o his as well as o he more familiar P-sar model. 5

7 specificaion. Specifically, Hallman e al. find ha in quarerly daa running from hrough , inflaion ends o rise when he long-run price arge P * implied by (1) is above he acual price level P ; likewise, inflaion falls when P * is below P. They confirm he saisical significance of his resul by regressing he change in inflaion on four of is own lags and he lagged value of he price gap, defined as he difference beween p * and p, he naural logarihms of P * and P, and rejecing he null hypohesis ha he coefficien on he lagged price gap equals zero. Here, similarly, we regress Δ 2 pq, he change in nominal income growh (and hence he analog o Hallman e al. s Δ = Δ 2 p, he change in he inflaion rae) on four of is own quarerly lags and on he lagged value of he nominal income gap, defined as he difference beween pq *, he naural log of he nominal income arge in (2), and pq, he log of he acual value of nominal GDP during period. Alhough he availabiliy of daa on he Divisia moneary aggregaes pushes he saring dae for our own quarerly sample ahead o , we can now exend ha sample well beyond Hallman e al. s, all he way hrough Our esimaes, wih he absolue value of he associaed saisic below each coefficien, are Δ 2 pq = 0.605Δ 2 pq Δ 2 pq Δ 2 pq Δ 2 pq (pq * -1 pq -1) (8.3) (4.4) (3.5) (1.0) (3.7) for Divisia M1 and Δ 2 pq = 0.612Δ 2 pq Δ 2 pq Δ 2 pq Δ 2 pq (pq * -1 pq -1) (8.3) (4.4) (3.5) (1.0) (3.4) for Divisia MZM. 10 In boh cases, he large and saisically significan coefficien on he lagged nominal GDP gap indicaes ha nominal income growh acceleraes when he gap is posiive and deceleraes when he gap is negaive, so ha acual nominal GDP converges over ime o he long-run arge defined in (2). Table 1 shows, addiionally, ha he lagged nominal GDP 10 Again following Hallman e al. (1991), quarerly changes in nominal GDP growh are muliplied by 400, so ha hey are expressed in annualized percenage poins, and he nominal GDP gap is muliplied by 100, so ha i is measured in percenage poins, in hese regressions. A consan erm, shown in able 1 bu no in he equaions as displayed here, is also included in each regression. 6

8 gap reains is significance across subsamples running from hrough and from hrough Thus, a nominal income arge se wih reference o eiher a narrow or a broad Divisia moneary aggregae proves useful in forecasing fuure nominal GDP growh, even in he mos recen daa. Mos imporanly from a pracical perspecive, no breakdowns of he forecasing equaion are observed, and no special shif-adjusmens beyond accouning for he slow-moving rends in V * using he one-sided filer are needed o mainain he sabiliy of hese empirical relaionships. Equaion (2) follows he approach in Hallman e al. (1991) by defining he long-run arge for nominal GDP in erms of he observed value of he moneary aggregae and he rend value of velociy. I is equally useful, however, o urn he equaion around, and use i o idenify he pah for a moneary aggregae ha is consisen wih a desired rajecory for nominal GDP. Towards his end, le (3) M * = PQ * /V * define he arge M * for money ha is consisen wih a chosen arge PQ * for nominal income, given he long-run value for velociy V *. In he Unied Saes beween 1985 and 2007, in fac, nominal GDP grew a an average annual rae of almos exacly 5.5 percen. The op panel of Figure 2 plos he acual series for he logarihm of nominal GDP agains a rend line wih his slope, fied via a leas-squares regression over he 23-year period. The boom panel, meanwhile, shows deviaions of nominal GDP from his rend, highlighing he modes swings experienced during he Grea Moderaion as well as he much more pronounced gap ha opened during he mos recen recession and coninues o widen oday. As noed by Woodford (2012), nominal GDP now lies more han 15 percen below a rend line esimaed wih daa from he period before he financial crisis. Inerpreing he rend line in Figure 2 as a arge pah for nominal GDP ha exends hrough 2012:3, Figure 3 plos he gaps beween he logs of acual Divisia M1 and MZM and he corresponding arge values for money implied by equaion (3). Wih he regression resuls from above in mind, one can view posiive values for hese money gaps as puing upward pressure on nominal GDP growh and negaive values as puing downward pressure on 7

9 nominal GDP; he gaps hereby indicae wheher moneary policy was oo accommodaive, oo resricive, or appropriaely neural during any given period. In fac, negaive values for boh he M1 and MZM gaps are observed jus before he recession of , and boh series decline, while remaining slighly posiive, before he recession of Larger posiive gaps, meanwhile, appear during he economic recoveries of he middle 1980s and early 1990s. Mos significanly, however, boh panels of Figure 3 sugges ha he sance of moneary policy shifed gradually from ease o ighness owards he middle of he las decade and, in fac, began o exer a considerable drag on nominal income growh in 2005 and 2006, hereby supporing Hezel s (2012) claim ha Federal Reserve policy was iself a key facor in riggering he iniial slowdown and severe recession ha followed. Wha s more, boh figures sugges ha despie he Federal Reserve s effors o lower ineres raes and increase dramaically he supply of bank reserves, insufficien growh in he moneary aggregaes, paricularly agains he backdrop of heighened demand for safe and highly liquid asses refleced by he downward movemens in rend velociy shown in Figure 1, coninues o severely depress nominal GDP in he U.S. economy oday. 11 Overall, he picure ha emerges from Figure 3 is one of persisen volailiy in he sance of moneary policy, swiching from periods of ease o conracion and back again. 12 This volailiy is no enirely unexpeced, however, because, under a regime of ineres-raeargeing, a cenral bank will have o change he quaniy of reserves (and money) o mainain is ineres rae peg. Thus, in addiion o offering a perspecive on wheher moneary policy has been relaively easy or resricive a various poins in ime, Figure 3 also can be inerpreed as offering evidence on one consequence implemening moneary policy hrough an ineres rae 11 Once again, he resuls shown in Figure 3 appear similar when he analysis is applied o Divisia M2 and M4, excep ha weakness in large ime deposis, repurchase agreemens, and commercial paper highly liquid money marke insrumens included in he M4 aggregae bu no in M1 or MZM make moneary policy look even more resricive hroughou he period since Hezel (2008, Chaper 23, and 2012, Chaper 8) characerizes hese variaions as sop-go moneary policy and offers a deailed explanaion for why i may have evolved in his manner over he pas five decades. 8

10 arge: Judged in reference o a smooh pah for nominal GDP, argeing he federal funds rae apparenly has creaed an inheren insabiliy in moneary policy. In summary, he foregoing discussion has ried o esablish ha moneary policy has he poenial o hi a long-run pah for nominal GDP if i can conrol he behavior of a Divisia moneary aggregae ha would keep nominal GDP on such a arge pah. I is o his quesion of moneary conrol we now urn. Money Mulipliers for Simple Sum and Divisia Aggregaes Spind (1983) exends Barne s (1980) work by deriving general expressions for he mulipliers of Divisia moneary aggregaes. Here, hese expressions are reproduced for he special case of aggregaes formed from currency and a single ype of ineres-bearing deposi. The resuls make clear how he appearance of user-cos erms in he budge-share weighs of he Divisia index can and seemingly do help dampen volailiy in he behavior of is companion muliplier. A series of numerical examples, based on hese expressions ogeher wih a model of he demand for currency and deposis drawn from Belongia and Ireland (2012), reveals ha for a wide range of plausible parameerizaions, he muliplier for he Divisia moneary aggregae is likely o more sable han he muliplier for he corresponding simple sum measure. We find ha his same paern appears in he U.S. daa. Le D, C, and The simple sum moneary aggregae R denoe he dollar values of deposis, currency, and bank reserves. s M and he moneary base H are hen defined by (4) M = D + C s and (5) H = R + C. Following he usual roue owards obaining an expression for he money muliplier of he simple sum aggregae, le (6) = / k C D 9

11 denoe he currency-deposi raio and (7) = / r R D denoe he reserve raio. Using (4)-(7), he simple sum muliplier can be calculaed as (8) s M D + C 1+ k m = = =. H R + C r + k s Equaion (8) depics he exbook resul ha he money muliplier depends inversely on boh he currency-deposi raio and he reserve raio. Because Divisia indexes are growh rae indexes, however, i is useful for he sake of comparison o express he muliplier for he simple sum aggregae in is less familiar growh rae form as well. Spind (1983) accomplishes his ask using he approximaions (9) w + w w + w Δ = Δ + Δ 2 2 D D C C s 1 1 ln( M ) ln( D) ln( C) and (10) v + v v + v Δ = Δ + Δ 2 2 R R C C 1 1 ln( H) ln( R) ln( C) for he growh raes of he simple sum aggregae and he money base, where (11) w D D 1 = = = s M D + C 1+ k D and (12) w C C k = = = s M D + C 1+ k C represen he quaniy shares of deposis and currency in he simple sum aggregae and, analogously, (13) v R R r = = = R H R + C r + k and 10

12 (14) v C C k = = = C H R + C r + k represen he quaniy shares of reserves and currency in he moneary base. Equaions (9)- (14) combine o yield (15) Δ 1 = k + k Δ s 1 ln( m ) ln( k) 2 1+ k 1+ k 1 1 r r 1 1 k k 1 + Δln( r) + Δ ln( k), 2 r + k r 1+ k 1 2 r + k r 1+ k 1 resaing (8) in growh rae form. Meanwhile, he growh rae of he Divisia quaniy aggregae currency is defined in discree ime by d M of deposis and (16) s + s s + s Δ = Δ + Δ 2 2 D D C C d 1 1 ln( M ) ln( D) ln( C), where (16) replaces he quaniy shares ha appear in (9) wih expendiure shares on he moneary services provided by deposis and currency. These shares are compued using Barne s (1978) formulas for he user coss D u and C u of deposis and currency: (17) u B D ρ ρ = B 1+ ρ D and B ρ = 1 + ρ C (18) u, B B where ρ denoes he rae of reurn on a benchmark asse ha provides no moneary services, D ρ denoes he own-rae of reurn of deposis, and (18) reflecs he fac ha currency does no pay ineres. Le (19) u u ρ C B = = D B D u ρ ρ 11

13 denoe he raio of he user cos of currency o he user cos of deposis. Using his expression ogeher wih he formula (6) defining he currency-deposi raio, he expendiure shares appearing in (16) may be compued as (20) s D u D 1 = = u D + u C 1+ uk D D C and s C uc uk = = u D + u C 1+ uk C (21). D C Equaions (10), (13), (14), (16), (20), and (21) combine o yield an expression for he growh rae of he money muliplier d m for he Divisia aggregae: (22) Δ 1 = uk + u k Δ d 1 1 ln( m ) ln( k) 2 1+ uk 1+ u 1k 1 1 r r 1 1 k k 1 + Δln( r) + Δ ln( k). 2 r + k r 1+ k 1 2 r + k r 1+ k 1 Spind (1983) shows how (22) exends o he more general case, wih muliple ypes of deposis and reserve asses. Comparing (15) and (22) reveals ha he relaive user cos erm u defined in (19) eners ino he money muliplier formula for he Divisia aggregae bu no for he corresponding simple sum measure. Inuiively, he wo mulipliers coincide when u = 1; in his case, deposis pay no ineres, implying ha an opimizing agen will be indifferen beween he moneary services provided by an addiional dollar in deposis and he moneary services provided by an addiional dollar in currency and will, in ha sense, view deposis and currency as perfec subsiues a he margin. Equaions (15) and (22) indicae ha movemens in he reserve raio r affec he mulipliers for he Divisia and simple sum aggregaes symmerically. Since u > 1 whenever deposis do pay ineres, however, he firs erm inside brackes on he righ-hand side of (22) will ypically be a larger posiive number han he corresponding erm in 12

14 (15), suggesing ha, in paricular, a decrease in he currency-deposi raio k ha increases he money muliplier for he simple sum aggregae will end o produce a smaller-sized increase in he money muliplier for he Divisia aggregae. Two observaions, however, force us o sop shor of using his comparison beween (15) and (22) alone o claim ha he money muliplier for he Divisia aggregae will surely be more sable han he money muliplier for he simple sum measure. Firs, while he growh rae formula (15), like he more familiar level formula (8), implies ha a fall in he currency-deposi raio will always cause he money muliplier for he simple sum aggregae o rise, sufficienly large values of he relaive user cos variable response o he same change in u may cause he money muliplier o fall in k : Under such circumsances, ha are larger, in absolue value, han he corresponding changes in d m could exhibi movemens s m. Second, if mos changes in he currency-deposi raio reflec underlying changes in user coss brough abou by exogenous shocks or moneary policy acions ha change eiher he benchmark ineres B rae ρ or he spread beween he benchmark rae and he own rae on deposis hen u will vary ogeher wih ρ B ρ, k, producing movemens in he money muliplier for he Divisia aggregae ha are difficul o pin down from an inspecion of (22) alone. To resolve hese ambiguiies, we combine (15) and (22), which are, by hemselves, simply accouning formulas ha idenify he more fundamenal deerminans of he money mulipliers, wih elemens drawn from he more deailed, general equilibrium model of he demand for moneary asses presened in Belongia and Ireland (2012). In his model, a represenaive household economizes on shopping ime using an aggregae D a M of moneary services obained from currency C and deposis D, where he moneary aggregaor akes he consan elasiciy form (23) M = [ v C + (1 ν ) D ] a 1/ ω ( ω 1)/ ω 1/ ω (1 ω)/ ω ω/( ω 1) 13

15 and he parameers saisfy 0< ν < 1 and ω > 0 opimally chooses he currency-deposi raio variable u defined above, in (19). In paricular,. Wih his specificaion, he household k as a funcion of he same opporuniy cos (24) k ν 1 = 1 ν u ω, a relaion ha associaes an increase in he opporuniy cos of currency relaive o deposis wih a decline in currency-deposi raio. Equaion (24) can be combined wih eiher (15) or (22) o obain a model of how he money muliplier for eiher he simple sum or he Divisia aggregae changes in response o movemens in he currency-deposi raio ha are ulimaely driven by changes in he user cos variable u. Monhly daa covering he period from hrough guide us in calibraing his model: The sample s saring dae marks he beginning of he era in which consumers have had access o a wide range of ineres-earning deposis, while he erminal dae ensures ha he figures are no influenced unduly by he exreme flucuaions in moneary variables winessed (and shown, for insance, in our own Figure 3 from above) during and since he financial crisis. Over his period, he average raio of Federal Reserve Bank of S. Louis adjused reserves o deposis was for M1 and for MZM; hence, in evaluaing (15) and (22), he reserve raio is fixed a eiher r = 0.12 or r = The average raio of currency o deposis was for M1 and for MZM; hence, in (24), he parameer ν is chosen o mach a value of k = 0.60 or k = As noed in Belongia and Ireland (2012), he price aggregaor (25) ρ ρ ν ρ ν ρ ρ B a B 1 ω B D 1 ω 1/(1 ω) = [ ( ) + (1 )( ) ], 13 Since (24) implies ha he average currency-deposi raio also depends on he elasiciy of subsiuion parameer, he seing for ν is adjused as ω varies across he range of examples considered below o mainain hese consan values of k. 14

16 a is dual o he quaniy aggregaor (23), where ρ denoes he own rae of reurn on he rue moneary aggregae M and ρ and a B D ρ are, as in (17)-(19), he benchmark reurn and he own rae on deposis. Daa provided hrough he Cener for Financial Sabiliy and also B described by Barne, e al. (2012) include readings on benchmark raes of reurn ρ as well as on ineres rae aggregaes for Divisia M1 and Divisia MZM ha can serve as measures of B ρ. Average values over he period in hese daa are ρ = for he a B a benchmark rae, ρ ρ B a = for he M1 aggregae, and ρ ρ = for MZM. We use D hese figures, ogeher wih (25), o back ou implied values for ρ, he average own rae on deposis in each moneary aggregae, hen subsiue he average benchmark and deposi raes ino (19) o obain an iniial seing for u when evaluaing (15), (22), and (24) numerically. Wih he model hereby calibraed for boh M1 and MZM cases, Table 2 shows values of s d he derivaives ln( m ) / u and ln( m ) / u compued numerically using (15), (22), and (24), for various values of he parameer ω measuring he elasiciy of subsiuion beween currency and deposis. Thus, each enry in he able quanifies he response of he money muliplier for eiher he simple sum or he Divisia aggregae o a shock or moneary policy acion ha increases he relaive user cos of currency and hereby leads, hrough (24), o a decrease in he currency-deposi raio. In every case, he resuls confirm he inuiion suggesed, earlier, by a direc comparison of (15) and (22). The posiive values repored for s ln( m ) / u indicae ha a shock ha causes he currency-deposi raio o fall causes he d simple sum muliplier o rise; bu he values repored for ln( m ) / u sill posiive, ye disincly smaller in magniude show ha he Divisia muliplier rises as well, bu by a smaller amoun. Thus, while i is possible o concoc examples in which he opposie is rue, his realisically calibraed model consisenly suggess ha he money muliplier for a Divisia aggregae is likely o be more sable han he money muliplier for he corresponding simple sum measure. 15

17 Table 3 shows ha he relaionship prediced by he model also holds rue in he U.S. daa. For he same period used in he calibraion exercise, and for hree addiional sample periods considered in he forecasing exercises below, he money muliplier for he Divisia M1 or MZM aggregae has a sandard deviaion ha is smaller han ha of he muliplier for he corresponding simple sum measure. Wheher hese smaller monh-o-monh movemens in he Divisia money muliplier are also forecasable is he subjec of he nex secion. Forecasing Experimens The muliplier relaionships explored above sugges several hypoheses and relaed experimens ha would updae he resuls repored by Spind (1984). Because one of he poenial errors ha could move GDP off he arge pah would be conrol errors ha resul from an inabiliy o forecas movemens in he Divisia money muliplier ou-of-sample, our specific goal here is o evaluae, wihin he conex of a nominal GDP argeing framework, which pair of insrumen and moneary aggregae would be mos likely o keep nominal GDP on a arge pah. Wih wo Divisia aggregaes M1 and MZM as he basis for calculaing a fuure pah for money, he cenral bank mus decide which insrumen is mos closely linked o he behavior of hese measures. In he forecasing exercise below, we will consider mulipliers derived from four poenial insrumens of conrol: Adjused reserves, oal reserves, non-borrowed reserves, and he adjused moneary base. Across he inerval , we firs esimae univariae ARMA models for each muliplier series over hree sub-samples. The resuls of hose esimaions hen are used o calculae errors from saic, ou-of-sample forecass over horizons of hree years following he erminal daa poin of he esimaion inerval. The sub-samples were chosen o evaluae he effecs of noable insiuional changes, hereby confroning he models wih heir greaes challenge. The firs forecas period covers he period of he Fed's experimen wih moneary argeing ( ). The second esimaion period ends a he ime of he Y2K injecion of reserves such ha he forecas period covers a sample period when he Fed was draining reserves from he sysem and hen dealing wih a recession ha may have been caused by is 16

18 excessively resricive acions pos-y2k. 14 The hird esimaion period spans he Grea Moderaion and ends jus prior o he onse of he mos recen downurn; mos noably, however, his is a period in which any emphasis on money and moneary conrol had disappeared from discussions of moneary policy. Before proceeding wih he forecasing experimen, i is insrucive o presen he daa in broad overview. Also, because of he wholesale changes in financial markes ha occurred in he early 1980s, hese summary saisics are repored for hree sample periods: , , and he enire period under sudy. Alhough he daa in Table 4 reveal very broad similariies across alernaive money mulipliers and over ime, he muliplier derived from non-borrowed reserves exhibis a sandard deviaion ha is subsanially larger han ha of he base or adjused reserves; somewha surprisingly, his resul prevails even in he sample period prior o he adven of financial innovaions. On is face his does no mean ha non-borrowed reserves canno be used as he cenral bank s insrumen of conrol or ha movemens in his muliplier canno be forecased ou-of-sample, bu is consisenly larger sandard deviaion is somehing o noe as he forecasing exercises are underaken. The resuls of he saic forecass are repored in Table 5. Because he foregoing examples for nominal GDP examined only Divisia M1 and MZM we limi our analysis o hose variables bu hese analyics could be applied o oher Divisia aggregaes as well. Again, we conduc he forecasing experimen over hree differen periods of ime o minimize he chances ha any paricular resul is due o happensance. 15 Variables chosen o represen he cenral bank's policy insrumen (H) in each able include adjused reserves (ADJ RES), nonborrowed In he middle of his esimaion period, he Fed reduced reserve requiremens on demand deposis from welve o en percen in April 1992 and eliminaed reserve requiremens on nonpersonal ime deposis in December For example, he relaively low and sable raes of base/reserves/money growh over he las decade may inroduce an "illusion" of more precise moneary conrol. Sabiliy in inflaion and ineres raes coupled wih generally sable real growh also could conribue o his illusion. Or, hese resuls may sugges ha he sandard money muliplier model be re-examined in he conex of modern insiuional arrangemens wih special aenion o changes ha would end o enhance moneary conrol. 17

19 reserves (NBR), oal reserves (TOT RES), and he adjused moneary base (BASE) as repored by he Federal Reserve Bank of S. Louis. The cell enries include wo error saisics: Roo mean squared error (RMSE) and mean absolue error (MAE). We firs discuss resuls for each moneary aggregae in urn, hen aemp o draw more general conclusions by reviewing he resuls as a group, and conclude wih a final se of experimens ha speak direcly o he possibiliy of using a Divisia moneary aggregae as an inermediae arge agains he backdrop of he financial crisis of 2008 and he insiuional disrupions and changes ha followed. The Divisia M1 Aggregae The resuls for he Divisia measure of M1 and he four variables used o represen he Fed's policy insrumen indicae ha, in all cases and across all sample periods, he moneary base muliplier is associaed wih he smalles MAE and RMSE. Moreover, in many cases, he error saisics for he base muliplier are an order of magniude smaller han hose of he nex closes compeior. Thus, if he Fed were o implemen his paricular approach o NGDP argeing wih Divisia M1 as is guide, he moneary base would appear o be he policy insrumen ha would generae he smalles conrol error. Wih respec o he general resuls over sample periods, i is ineresing o noe ha, for he mos par, he forecas errors are no markedly differen across ime. This resul is surprising because he inroducion of new bank liabiliies no subjec o reserve requiremens, he increasing use of sweep aciviies by banks, and he reducion in reserve requiremens more generally should have made moneary conrol subjec o larger errors. The Divisia MZM Aggregae Resuls for he MZM mulipliers indicae ha, as for Divisia M1, he muliplier derived from he moneary base produces he lowes forecas errors for each of he hree sample periods and hose errors are lower by a subsanial margin compared o he hree oher alernaives. Also, as in he case of Divisia M1, he nonborrowed reserves insrumen produces he highes MAE and RMSE values. Finally, i is ineresing o noe ha he conrol errors for he much broader MZM liabiliies grouping are similar o hose for he narrow M1 aggregae. 18

20 Thus, while one reason o choose beween a narrow and broad inermediae arge ofen is how closely i is associaed wih he cenral bank s insrumen of conrol, here is nohing in Table 5 ha would lead one o prefer srongly one Divisia measure o he oher; i seems as if he cenral bank could use he moneary base o influence he pah of eiher wih comparable success. The Recen Financial Crisis and Moneary Conrol The foregoing experimens all were conduced over sample periods prior o he recen financial crisis and responses o i by he Federal Reserve ha have made, in he minds of many observers, reserves and he moneary base uninformaive indicaors of cenral bank acions. Moreover, he inroducion of paymen of ineres on reserves would have weakened, if no severed, any link beween radiional measures of cenral bank liabiliies and money o a degree ha discussions of moneary conrol would be all bu a moo poin, pos Taken a face value, hese poins migh seem correc. For pracical purposes, however, he quesion facing a cenral bank always becomes one of wha i wishes o accomplish. For example, here is lile doub ha sweep accouns represen an effor by banks o evade reserve requiremens and his evasion complicaes measuremen of he money supply. As a bank regulaor, however, he Federal Reserve has a number of opions o conrol or eliminae his behavior if, in fac, greaer conrol and more accurae measuremen of he money supply were a policy objecive. 16 Wih regard o he pos-2008 environmen, a similar logic applies. While i is rue ha he Federal Reserve has added a large volume of asses o is porfolio and begun o pay ineres on reserves, hese acions have no necessarily disored all linkages beween he Fed s balance shee and he aggregae quaniy of money. Taom (2011), for example, has derived boh balance shee and muliplier relaionships in he afermah of he financial crisis and found ha a relaively sraighforward adjusmen subracing excess reserves from he 16 Feldsein and Sock (1994, pp ) make a similar poin wih respec o heir NGDP argeing framework based on simple sum M2. They argue ha he Federal Reserve could exercise igher conrol over sum M2 by re-exending reserve requiremens o he non-m1 componens of he broader moneary aggregae and paying ineres on reserves as well. 19

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