Maccini, Louis J.; Schaller, Huntley; Moore, Bartholomew. Working Papers, The Johns Hopkins University, Department of Economics, No.

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1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Maccini, Louis J.; Schaller, Hunley; Moore, Barholomew Working Paper The ineres rae, learning, and invenory invesmen Working Papers, The Johns Hopkins Universiy, Deparmen of Economics, No. 5 Provided in Cooperaion wih: Deparmen of Economics, The Johns Hopkins Universiy Suggesed Ciaion: Maccini, Louis J.; Schaller, Hunley; Moore, Barholomew (004) : The ineres rae, learning, and invenory invesmen, Working Papers, The Johns Hopkins Universiy, Deparmen of Economics, No. 5 This Version is available a: hp://hdl.handle.ne/049/7039 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 April 004. The Ineres Rae, Learning, and Invenory Invesmen Louis J. Maccini Barholomew Moore (Corresponding auhor) Deparmen of Economics Deparmen of Economics Fordham Universiy Johns Hopkins Universiy 44 Eas Fordham Road 3400 N. Charles Sree The Bronx, NY 0458 Balimore, MD 8 (78) (40) Hunley Schaller Deparmen of Economics Carleon Universiy 5 Colonel By Drive Oawa, ON KS 5B6 (63) [JEL Classificaion: E. Keywords: Invenories, Ineres Raes, Learning] We hank Heidi Poruondo for research assisance. For helpful commens we hank Larry Ball, Bob Chirinko, Ricardo Fiorio, Jim Hamilon, Brad Humphries, Serena Ng, Soo-Bin Park, Bob Rossana, Ken Wes, and seminar paricipans a Brandeis Universiy, he Johns Hopkins Universiy, Ene Einaudi, he AEA Meeings, and he XII symposium of he Inernaional Sociey for Invenory Research. Schaller hanks he SSHRC for is financial suppor of his research program and he Economics Deparmen a MIT for providing a simulaing environmen in which o conduc a significan porion of his research.

3 The Ineres Rae, Learning, and Invenory Invesmen Absrac Economic heory predics a negaive relaionship beween invenories and he real ineres rae, bu previous empirical sudies (mosly based on he older sock adjusmen model) have found lile evidence of such a relaionship. We derive parameric ess for he role of he ineres rae in specificaions based on he firm s opimizaion problem. These Euler equaion and decision rule ess mirror earlier evidence, finding lile role for he ineres rae. We presen a simple and inuiively appealing explanaion, based on regime swiching in he real ineres rae and learning, of why ess based on he sock adjusmen model, he Euler equaion, and he decision rule all of which emphasize shor-run flucuaions in invenories and he ineres rae are unlikely o uncover a relaionship. Our analysis suggess ha invenories will no respond much o shor-run flucuaions in he ineres rae, bu hey should respond o long-run movemens (regime shifs; e.g., beween low real raes in he 970s and high raes in he early 980s). Boh simple and sophisicaed ess confirm our predicions and show a highly significan long-run relaionship beween invenories and he ineres rae, wih an elasiciy of abou -.5. Furhermore, a formal model of our explanaion yields a disincive, esable implicaion. This implicaion is suppored by he daa.

4 I. Inroducion In heir survey of invenory research, Blinder and Maccini (99) observe ha an imporan puzzle in he empirical research on invenories is ha he real ineres rae seems o have lile impac on invenory invesmen. Bu, very lile research on his issue has been conduced over he las decade. As an indicaor of he lack of work, he recen survey by Ramey and Wes (999) barely menions he effec of ineres raes on invenories. The lack of an effec of he real ineres rae on invenory invesmen is a puzzle for wo reasons. Firs, i severs one of he convenional channels hrough which moneary policy influences spending. Moneary policy is implemened by changing shor-erm ineres raes, which sandard heory predics should influence invenory invesmen spending. Second, he financial press is replee wih saemens by business people assering ha higher ineres raes induce firms o cu invenory holdings. Alhough i is someimes no clear wheher he ineres rae under discussion is real or nominal, here is noneheless he percepion of an inverse relaionship beween invenory invesmen and ineres raes. Ye, almos no evidence exiss for such an effec. Earlier empirical work wih invenories uilized flexible acceleraor or sock adjusmen models. Among he key issues invesigaed in his lieraure was he relaionship beween invenory invesmen and ineres raes. See Akhar (983), Blinder (986a), Irvine (98), Maccini and Rossana (984), and Rossana (990) for relaively recen sudies using his approach. These sudies ypically assumed ha desired socks depend on he curren expeced real ineres rae as well as curren expeced sales and Sock adjusmen models were firs used in empirical work wih invenories by Lovell (96). See Akhar (983) for a survey of he relevan lieraure prior o he eighies.

5 expeced inpu prices. Curren expeced real raes were relaed o acual raes hrough disribued lag relaionships and empirical work proceeded. These sudies, however, generally failed o esablish subsanial and sysemaic evidence of a relaionship beween he real ineres rae and invenory invesmen, especially wih finished goods invenories in manufacuring. Furhermore, he lieraure was subjec o he criicism ha i lacked a basis in explici opimizaion. In he eighies, he linear-quadraic model of invenory behavior combined wih raional expecaions began o be applied o empirical work on invenories. The linearquadraic model of invenory behavior was developed by Hol, Modigliani, Muh and Simon (960), and was revived in he economics lieraure in he eighies by Blinder (98, 986b) who used he model in analyical work. When combined wih raional expecaions, he model is a very fruiful framework for empirical work. However, he relaionship beween he real ineres rae and invenory movemens was no a key issue under invesigaion in empirical research wih he linear-quadraic model. Raher, research focused on aemps o explain why producion seemed o flucuae more han sales, which conradics he producion-smoohing moive for holding invenories, and why invenory socks seemed o exhibi such persisence. Two approaches were employed in empirical research. One approach esimaed Euler equaions using generalized mehods of momens echniques. Conribuions using his approach include Durlauf and Maccini (995), Eichenbaum (989), Kashyap and Wilcox (993), Kollinzas (995), Krane and Braun (99), Ramey (99), and Wes (986). In his work, i proved difficul o esimae he srucural parameers of ineres when he discoun facor, which is defined by he real ineres rae, is allowed o vary.

6 3 Hence, researchers invariably assumed ha he discoun facor is a given, known value, which of course eliminaes by assumpion any effec of he real ineres rae on invenory invesmen. Given he finie sample problems wih generalized mehods of momens applied o Euler equaions, anoher approach used in empirical work wih he linear-quadraic model was o solve for he opimal choice of invenories, ha is, for he decision rule, and o esimae i using maximum likelihood echniques. See Blanchard (983), Fuhrer, Moore and Schuh (995), and Humphreys, Maccini and Schuh (00). However, solving for he decision rule requires an Euler equaion ha is linear in is variables. Since Euler equaions are nonlinear in he discoun rae, and herefore in he real ineres rae, researchers again resored o a consan and known discoun rae for reasons of racabiliy. A few sudies depared from he linear-quadraic framework and allowed real ineres raes o vary. Miron and Zeldes (988) uilize an approach wih a Cobb-Douglas producion funcion ha emphasizes cos shocks and seasonal flucuaions. Kahn (99) and Bils and Kahn (000) focus on a more rigorous reamen of he sockou avoidance moive. In hese sudies, when a check was made, no effec on he empirical implicaions of he model was found when he real ineres rae was allowed o vary or was held consan. 3 An excepion is Ramey (989) who developed a model ha reas invenory socks as facors of producion, and found evidence of ineres rae effecs hrough relevan impued renal raes. As is well known, he decision rule for opimal invenories can be convered ino a sock adjusmen model. A key difference beween his approach and he earlier sock adjusmen principles is ha he desired sock now depends on expeced fuure sales, inpu prices, and real ineres raes as well as curren expeced values. 3 See Miron and Zeldes (988) and Kahn (99).

7 4 Given he lack of srong evidence of a relaionship beween he real ineres rae and invenory movemens, a number of auhors conjecured ha he problem wih he sandard model is ha i assumes perfec capial markes, so ha firms may borrow or lend as much as hey wan a given ineres raes. Raher, hey argued ha capial marke imperfecions arising from asymmeric informaion will impose finance consrains on he firm s invenory decision. These consrains sugges ha he cos of exernal finance o he firm is inversely relaed o he firm s inernal financial posiion, as measured by liquid asses or cash flow. See Kashyap, Lamon and Sein (994), Gerler and Gilchris (994), and Carpener, Fazzari and Peersen (994) for conribuions o his approach. They find ha such financial variables do have an influence on invenory movemens of small firms bu no of large firms. This leaves open he relaionship beween he real ineres rae and invenory movemens for large firms and in he aggregae. The purpose of his paper is o ake a fresh look a he relaionship beween he real ineres rae and invenory invesmen. This is obviously an imporan issue for empirical work on he ransmission and effeciveness of moneary policy, and complemens he empirical work underway wih ineres rae rules as descripions of moneary policy. We begin by exending he ypical approach aken wih he linearquadraic model, specifically o obain a specificaion in which he ineres rae appears in a separae erm wih is own coefficien. Essenially, his involves an appropriae linear approximaion of he Euler equaion in he real ineres rae. This enables us o solve for he opimal level of invenories as a linear funcion of he real ineres rae and oher variables.

8 5 Using he linearized Euler equaion as a saring poin, we are able o derive specificaions ha allow us o paramerically esimae he effec of he ineres rae on invenories. We use wo approaches he linearized Euler equaion and he firm s decision rule for invenories (which can be derived from he linearized Euler equaion). We underake empirical work wih monhly daa on invenories for he nondurable aggregae of U.S. manufacuring for he period The resuls reinforce he exising puzzle: hese specificaions reveal no significan effec of he ineres rae. Why don he Euler equaion and decision rule show an effec of he real ineres rae on invenories? We sugges ha he answer lies in he behavior of he ineres rae, which displays ransiory variaion around highly persisen mean values (e.g., persisenly negaive real ineres raes in he 970s). In oher words, real ineres raes appear o ener regimes ha exhibi sabiliy for exended periods wih emporary variaion around a persisen level wihin each regime. Regime changes are infrequen. In fac, careful economeric sudy has provided evidence ha he real ineres rae is well described by Markov regime swiching (Garcia and Perron 996). If he mean real ineres rae is highly persisen, firms may largely ignore shorrun ineres rae flucuaions, alering heir ypical invenory level only when here seems o be a persisen change in he real ineres rae. Under hese condiions, economeric procedures ha focus on shor-run flucuaions in invenories and he ineres rae such as he older sock adjusmen or he newer Euler equaion and decision rule specificaions may find lile evidence of a relaionship. On he oher hand, firms will adjus heir invenory posiions if hey believe here has been a change in he underlying ineres-rae regime. This suggess ha esimaion of

9 6 he long-run relaionship beween he invenories and he real ineres rae may be fruiful. We use wo approaches. The firs is simple and inuiive: we divide our sample ino ineres rae regimes high, medium, and low and calculae he mean level of (derended) invenories in each regime. We find ha invenories are significanly higher when he ineres rae is low. The second approach is more sophisicaed. Using he linearized Euler equaion as a saring poin, we derive he coinegraing relaionship beween invenories and he ineres rae. Coinegraion ess show ha invenories and he ineres rae are coinegraed. Esimaes of he coinegraing vecor uncover a srong long-run effec of he ineres rae on invenories in aggregae daa. This finding is especially sriking in view of he failure o find such a relaionship using specificaions ha focus on shor-run flucuaions. We proceed o formally model he implicaions of regime swiching in he ineres rae. Of course, i is someimes difficul o disinguish beween a ransiory shock and a shif o a new persisen regime. To capure his difficuly, we assume ha firms mus learn he unobservable regime from observable movemens in he real ineres rae. Under he assumpion of regime swiching and learning, he model of opimal invenory choice yields a disincive implicaion: invenories should be based on he firm s assessmen of he probabiliy ha he economy is currenly in a given ineres rae regime. In paricular, under he assumpion of regime swiching and learning, he probabiliies of being in eiher he high or low ineres rae regime should replace he ineres rae in he coinegraing vecor for invenories. We es his implicaion and find saisically significan evidence ha he long-run behavior of invenories is linked o hese probabiliies in aggregae daa.

10 7 In addiion o he aggregae daa, we es he disincive implicaion of he model of regime swiching and learning in wo-digi indusry daa. In wo-hirds of he indusries, here is significan evidence ha he regime probabiliies influence invenories. In he indusry daa, evidence from coinegraing regressions ha include he regime probabiliies is sronger han he evidence from coinegraing regressions ha include he ineres rae. Overall, he evidence from he wo-digi indusry daa provides addiional suppor for he hypohesis ha regime swiching in he real ineres rae and learning play an imporan role in invenory behavior. The nex secion presens he firm s opimizaion problem. Secion III examines he shor-run relaionship beween invenories and he real ineres rae, inroducing he new ess for he role of he ineres rae based on he Euler equaion and he decision rule. Secion IV analyzes and ess he long-run relaionship beween invenories and he real ineres rae. Secion V inroduces he formal model of regime swiching and learning, derives he disincive implicaion of he model, calculaes he probabiliies, and ess he implicaion. Secion V also presens simulaions of he model. The simulaions illusrae why i is difficul o find a shor-run relaionship beween invenories and he ineres rae: he cos of no adjusing o a ransiory ineres rae shock is abou 00 imes smaller han he cos of no adjusing o a regime shif in he ineres rae. Secion VI presens robusness checks, and Secion VII concludes.

11 8 II. The Firm s Opimizaion Problem We begin by assuming a represenaive firm ha minimizes he presen value of is expeced coss over an infinie horizon. Real coss per period are assumed o be quadraic and are defined as C θ γ δ = ξw ( ) ( ) Y + Y + Y + N αx () where θ,γ,δ,ξ,α > 0. C denoes real coss, Y, real oupu, N, end-of-period real finished goods invenories, X, real sales, and W, a real cos shock, which we will associae wih real inpu prices. (We do no include unobservable cos shocks, since hey are no direcly relevan o he relaionship beween invenories and he ineres rae. In Secion IV.B, we discuss how he modeling of unobservable cos shocks would affec he coinegraing vecor.) The level of real sales, X, and he real cos shock, W, are given exogenously. The firs wo erms capure producion coss. The hird erm is adjusmen coss on oupu. The las erm is invenory holding coss, which balance sorage coss and sockou coss, where αx is he arge sock of invenories. Le β be a variable real discoun facor, which is given by β = + r, where r denoes he real rae of ineres. The firm s opimizaion problem is o minimize he presen discouned value of expeced coss, E 0 = 0 j= 0 β j C, () subjec o he invenory accumulaion equaion, which gives he change in invenories as he excess of producion over sales,

12 9 N N = Y X. (3) The Euler equaion ha resuls from his opimizaion problem is { θ β+ + γ β+ + β+ β+ + ( ) ( ) E Y Y + Y Y + Y (4) ( W + W+ ) + ( N X+ )} + ξ β + δβ α = 0 where from (3) Y = N N + X. Observe ha (4) involves producs of he discoun facor and he choice variables and producs of he discoun facor and he forcing variables. Linearizing hese producs around consan values, which may be inerpreed as saionary sae values or sample means, yields a linearized Euler equaion: { θ( β + ) γ ( β + β + ) ξ β + E Y Y + Y Y + Y + ( W W ) (5) ( N X+ ) r+ c} + δβ α + η + = 0 where η = β( θy + ξw) > 0, c = rβ( θy + ξw) < 0, β = + r, and a bar above a variable denoes he saionary sae value. This linearized Euler equaion will serve as a basic relaionship ha we will use in he empirical work.

13 0 III. The Shor-Run Relaionship beween Invenories and he Real Ineres Rae A. Euler Equaion Esimaion A common approach in empirical work on invenories is o apply raional expecaions o eliminae unobservable variables and hen use Generalized Mehods of Momens echniques o esimae he Euler equaion. We firs invesigae wheher a shorrun relaionship beween invenories and he real ineres rae can be found using his approach. Assume ha sales, X, he cos shock, W, and he real ineres rae, r, obey general sochasic processes. Then, use raional expecaions o eliminae expecaions from (5) o ge ( ) ( ) θ Y βy + γ Y β Y + β Y + ξ( W βw ) (6) δβ( N αx ) + ηr + c= κ I + + I where κ is a forecas error. Since no all he srucural parameers of he Euler equaion are idenified, we adop he widely used normalizaion and se δ equal o. We esimae he Euler equaion by GMM, 4 using a consan, Y -, W -, N -, X - and r - as insrumens. 5 All of he variables are linearly derended. 6 Invenory Euler equaions have been esimaed by GMM by many auhors, including Durlauf and Maccini (995), Eichenbaum (989), Kashyap and Wilcox (993), Kollinzas (995), Krane and Braun (99), Ramey (99), 4 As discussed by Wes (995), esimaion by GMM is valid boh in he case where sales are I(0) and in he case where hey are I(), as long as (in he laer case) hey are coinegraed. See paricularly he discussion on pages The ineres rae is included because i appears in he Euler equaion specificaion ha allows for variaion in he ineres rae, and i is desirable o use a consisen se of insrumens across specificaions. 6 The resuls are qualiaively similar if he Euler equaion is esimaed wihou derending.

14 and Wes (986). A few papers have allowed for ineres rae variaion, for example, Bils and Kahn (000), Miron and Zeldes (988), Kahn (99), and Ramey (989). To he bes of our knowledge, however, his paper is he firs o esimae a coefficien on he ineres rae in a invenory Euler equaion. Esimaes of he Euler equaion under he assumpion of a consan ineres rae are presened in he firs column of Panel A of Table. (The version of he Euler equaion relevan o hese esimaes is equaion (4) wih β se equal o a consan.) The arge invenory-sales raio, α, is very precisely esimaed and is approximaely four weeks of sales, which is plausible. Furher, he esimae of he slope of marginal cos, θ, is posiive and significan, indicaing rising marginal cos. These esimaes are consisen wih hose found by esimaing analogous Euler equaions in he recen lieraure. 7 Ineresingly, ξ, he parameer associaed wih observable cos shocks, is posiive (as heory predics) bu insignificanly differen from zero. The esimaed adjusmen cos parameer, γ, is posiive, a resul ha is consisen wih he exisence of adjusmen coss, bu γ is imprecisely esimaed and no significanly differen from zero. Esimaes of he Euler equaion under he assumpion of a variable ineres rae are presened in he second column of Table, Panel A. The saisic on η provides a simple es for he effec of he ineres rae on invenories. The poin esimae of η has he wrong sign and is insignificanly differen from zero. Even if η is no significanly differen from zero, i is possible ha allowing for a variable ineres rae could improve esimaes of he oher parameers and, more generally, improve he fi of he Euler equaion. Informally, a comparison of he firs and 7 See, for example, Durlauf and Maccini (995).

15 second columns of Panel A suggess ha allowing for a variable ineres rae makes some quaniaive difference in he esimaes of he oher parameers bu lile qualiaive difference. A formal es procedure, which is based on a comparison of he overidenifying resricions beween he wo models, is described by Newey and Wes (987). The inuiion for he es is sraighforward. If a model is incorrecly specified, he J saisic for he model will end o be large; he difference in J saisics beween wo models provides a es of wheher he improvemen in specificaion is saisically significan. The difference in J saisics is disribued as a χ, wih degrees of freedom equal o he number of omied parameers, here equal o one. 8 The Newey-Wes es saisic is.75, so i is no possible o rejec he consan ineres rae resricion. In he remaining panels of Table, we check he robusness of he resuls o changes in he specificaion of he model. In he empirical invenory lieraure, here is mixed evidence on he imporance of adjusmen coss and observable cos shocks. 9 Panel B presens Euler equaion esimaes from a specificaion ha includes observable cos shocks bu ses γ o zero. Esimaes of he oher parameers (θ, ξ, and α ) are no dramaically affeced. As in he Panel A resuls, he ineres rae eners wih he wrong sign and is insignificanly differen from zero. Also as in Panel A, he Newey-Wes es fails o rejec he consan ineres rae specificaion. Panel C presens esimaes of a specificaion ha allows for adjusmen coss bu ses ξ equal o zero (so observable cos shocks do no ener). Panel D presens esimaes 8 For he es, he same weighing marix should be used; we use he weighing marix from he variable ineres rae specificaion, since i is he "unresriced" model. 9 See, e.g., he surveys by Blinder and Maccini (99), Ramey and Wes (999), and Wes (995).

16 3 of a specificaion ha excludes boh adjusmen coss and observable cos shocks. In neiher case is here saisically significan evidence of a role for he ineres rae. Overall, he Euler equaion resuls presened in his secion show no evidence of a saisically significan relaionship beween invenories and he ineres rae. This is consisen wih much earlier research, which has ypically found ha invenories are no significanly relaed o he ineres rae. B. Decision Rule Esimaion An alernaive approach in empirical work wih invenories is o esimae he decision rule. We nex explore wheher his approach can deec a shor-run relaionship beween invenories and he real ineres rae. In an appendix, we show ha he linearized Euler equaion, (5), may be wrien as a fourh-order expecaional difference equaion in N. Le λ and λ denoe he sable roos of he relevan characerisic equaion. The firm s decision rule can be expressed as βλ λ j EN N N ( ) ( ) E (7) + j+ = ( λ + λ) λλ + βλ βλ + j ( λ λ ) Ψ j= 0 where Ψ = X + θ γ ( β) X θ γ ( β) αδβ X γβ γβ + j + + j + + j + j ξ η c + X ( W + j βw++ j) r ++ j. (8) β γβ γβ γβ

17 4 Assume ha he firm carries ou is producion plans for ime, so ha EY = Y. Then equaion (3) implies ha ( N E N ) ( X E X ) =, which means in effec ha invenories buffer sales shocks. Define u x ( X E X ) as he sales forecas error. Assuming ha sales, real inpu prices, and he real ineres rae follow independen AR() processes, and ha he firm s curren informaion se includes lagged values of sales, and curren and lagged values of inpu prices and he ineres rae, equaions (7) and (8) give ( ) N =Γ+ λ + λ N λλ N +Γ X +Γ W +Γ r+ u. (9) x 0 X W r > wih Γ 0, Γ < 0, and Γ < 0. < X W r The coefficien on sales is, in general, ambiguous, as i balances producion smoohing and sockou avoidance. Based on prior empirical work, we expec a posiive coefficien, which implies ha sockou avoidance dominaes. I follows from Γ W < 0 ha an increase in real inpu prices should cause a decline in invenories. Γ r < 0 implies ha an increase in he real ineres rae should induce he firm o reduce invenories. Under he assumpion made above, ha sales is AR(), he decision rule is jus idenified and can be esimaed by OLS. (In laer secions, we specifically assume a uni roo process, bu he resul ha he decision rule is jus idenified and can be esimaed by OLS holds generally for any AR() process.) Inference can be carried ou wih sandard disribuions, regardless of wheher sales are I(0) or I(). 0 As noed above, a number of 0 See Wes (995) for a deailed discussion of esimaion and inference issues. Our esimaion procedure for he decision rule is also valid if here are unobservable cos shocks (including serially correlaed shocks)

18 5 auhors have esimaed he decision rule, including Blanchard (983), Fuhrer, Moore, and Schuh (995), and Humphreys, Maccini, and Schuh (00). Based on previous sudies ha allow for ineres rae variaion, i would be mildly surprising if we found ha he coefficien on he ineres rae was of he heoreically prediced sign and saisically significan. On he oher hand, o he bes of our knowledge, no one has previously repored esimaes of he decision rule for invenories ha allow for a variable ineres rae. Prior sudies ha have allowed ineres rae variaion have esimaed Euler equaions (alhough hese sudies, as noed above, did no include a coefficien on he ineres rae and herefore did no paramerically esimae he effec of he ineres rae on invenories). We begin by considering he mos general specificaion of he decision rule (allowing for boh adjusmen coss and observable cos shocks) in Panel A of Table. Under he assumpion of a consan ineres rae, he coefficiens on all of he variables in he decision rule are significanly differen from zero. Sales has a posiive coefficien, indicaing ha he sockou avoidance moive dominaes, and real inpu prices have a negaive coefficien, consisen wih he predicions of he heory. Under he assumpion of a variable ineres rae, he second column of Panel A shows ha he esimaed coefficien on he ineres rae is posiive, a resul ha is conrary o he implicaions of he linear-quadraic model, alhough he coefficien on he ineres unless he shocks are I(). We consider he case of I(0) observable cos shocks in his subsecion and I() observable cos shocks in Secion VI. These sudies have ypically esimaed srucural parameers via nonlinear maximum likelihood procedures, whereas in his paper we are in effec esimaing reduced form parameers. However, since he relevan srucural parameer conneced o he real ineres rae is η, and since η appears only in he reduced form parameer Γ r, he resuls for he role of he ineres rae are unlikely o be improved by esimaing srucural parameers using nonlinear procedures.

19 6 rae is insignifican. Furher, allowing for a variable ineres rae has no effec on he signs or significance of he coefficiens on sales, real inpu prices, or lagged invenories. Panel B presens decision rule esimaes from a specificaion ha includes observable cos shocks bu excludes adjusmen coss. This changes he specificaion of he decision rule, leading o he omission of he second lag of invenories. Panel C presens esimaes of a specificaion ha allows for adjusmen coss bu excludes observable cos shocks. The specificaion in Panel D excludes boh adjusmen coss and observable coss shocks. The esimaed coefficien on he ineres rae always has a posiive sign, alhough i is never significan. The signs and significance of he oher variables are lile affeced by wheher or no he ineres rae is variable or consan. Summarizing he resuls from our esimaion of he decision rule, here is no evidence ha he real ineres rae has a saisically significan effec on invenories. Again, his is consisen wih earlier research. IV. The Long-Run Relaionship beween Invenories and he Real Ineres Rae. Why is i ha esimaes of he Euler equaion and decision rule show no saisically significan effec of he real ineres rae on invenories? A possible clue lies in he behavior of he ineres rae. Figure plos he ex-pos real ineres rae over he period As he figure illusraes, here are long periods when he ineres rae is cenered on a given mean. For example, he real ineres rae is cenered on a value jus below % for much of he 960s. In he early 970s, here is a shif in he mean real ineres rae o a value of abou -%. The real ineres rae rises sharply around 980 and A minor excepion is ha he coefficien of sales is a bi more significan when he ineres rae is variable. Also, noe ha excluding observable cos shocks seems o reduce he saisical significance of sales, suggesing ha excluding cos shocks creaes an omied variable bias problem.

20 7 remains high (around 5% on average) for much of he 980s. In he lae 980s and for much of he 990s, he real ineres rae reurns o a mean value ha is close o is 960s level. Wha implicaion does his behavior of he ineres rae have for empirical esimaes of he effec of he ineres rae on invenories? If he mean ineres rae is highly persisen, firms may largely ignore shor-run ineres rae flucuaions, adaping average invenory levels only when here appears o be a persisen change in he opporuniy cos of holding invenories. If his is he case, hen economeric procedures ha focus on shor-run flucuaions in invenories and he ineres rae may find lile evidence of a relaionship. In his secion, we examine he long-run relaionship beween invenories and he ineres rae. We consider wo ess. The firs is a simple, inuiive es: are invenories lower on average during he high ineres rae regime? The second is a more sophisicaed es: we derive and esimae he coinegraing vecor for invenories under he assumpion of variable ineres raes. A. Tess of Means Our firs es of he long-run relaionship beween invenories and he ineres rae divides he period ino regimes characerized by differen mean ineres raes. Specifically, we classify he period 97:0-980:0 in he low-ineres-rae regime, 980:-986:04 in he high-ineres-rae regime, and he remaining observaions in he medium-ineres-rae regime. 3 3 This is an inuiive definiion of he regimes; we inroduce a formal procedure for idenifying regimes in Secion V.

21 8 The resuls are presened in Table 3. Invenories are highes in he low-ineresrae regime and lowes in he high-ineres-rae regime. The level of derended invenories is abou 8% higher in he low-ineres-rae regime han in he high-ineresrae regime. The difference in means beween he high and low-ineres-rae regimes is highly significan. To he bes of our knowledge, his is he firs ime ha his simple es for he long-run effec of he ineres rae on invenories has been repored. B. Derivaion of he Coinegraing Vecor A more sophisicaed es of he long-run relaionship beween invenories and he ineres rae is o derive he coinegraing relaionship beween invenories, he ineres rae, and any oher relevan variables. The advanage of he coinegraion approach over he simple es of means presened earlier is ha i accouns for he effec of variables such as sales and observable cos shocks on he long-run level of invenories. To derive he coinegraing vecor, i is helpful o rewrie he basic Euler equaion, (5), in such a way as o pu mos of he variables in he form of firs differences 4, 5 : 4 See Kashyap and Wilcox (993) and Ramey and Wes (999) for derivaions of he coinegraing relaionship for invenories under he assumpion ha he ineres rae is consan. 5 We are agnosic on he issue of unobserved cos shocks, which are no of primary ineres in his paper. As a resul, we do no include a erm comparable o U c in Hamilon (00) or Ramey and Wes (999) in our model and hus do no address he issue of wheher unobservable cos shocks are bes modeled as I(0) or I(). Hamilon (00) argues ha, if one were o include unobservable cos shocks under his preferred assumpions, he same variables would appear in he coinegraing vecor, bu he coefficiens would be alered. I is possible o show ha his argumen can be generalized o our model, which includes a imevarying ineres rae and observable cos shocks. Under assumpions similar o Hamilon s, he same variables appear in our coinegraing vecor. The coefficiens are alered, bu signs of he coefficiens remain he same.

22 { γ ( β + + β + ) βθ( ) + θ βδα + βξ + + η + E Y Y Y N X N X W r 9 θ( β) + βδ N α X + ( β ) ξw + ηr + c = 0. βδ (0) where Y is again given by (3). [For he derivaion of (0) from (5), see he appendix.] Noe ha we can express equaion (0) as { } E χ + = for he appropriae definiion of 0 χ +. Raional expecaions implies ha he expecaion error { } φ χ E χ will be serially uncorrelaed and herefore canno have a uni roo; in oher words φ + is I(0). Since { } 0 E χ, χ + = φ +. Since φ + is I(0), + = χ + will also be I(0). Suppose for he momen ha N, X, W and r are I(). Then he saionariy of χ + implies ha invenories, sales, he cos shock, and he real ineres rae will be coinegraed, wih coinegraing vecor θ( β) ξ( β) η, α,,. βδ βδ βδ I is useful o noe ha ADF ess show N, X, W and r o be I() variables (in he usual sense ha he ess fail o rejec he null hypohesis of a uni roo). 6 From he above derivaion, i follows ha he coinegraing vecor is he same regardless of wheher adjusmen coss are included in or excluded from he model. To see why, consider he firs erm in parenheses in equaion (0), which reflecs adjusmen 6 An alernaive procedure for deriving he coinegraing relaionship beween invenories and he forcing variables is o express he decision rule for opimal invenories in he form of a sock adjusmen principle in which case he coinegraing relaionship is defined by he desired or he equilibrium sock of invenories. Such a procedure yields an idenical coinegraing vecor.

23 0 coss. Noe ha all he elemens in his erm ener in he form he elemens in his erm will be I(0). Y. Thus, if Y is I(), all The parameers α, θ, δ, ξ are assumed o be posiive, and we have shown, below equaion (5), ha η is posiive. When he coinegraing vecor is expressed in he form of a regression, X, W and r will be on he righ hand side of he equaion, so heir coefficiens will have signs opposie o hose shown in he coinegraing vecor above. In oher words, in he long run, we expec invenories o be inversely relaed o he cos shock and he real ineres rae, and, if he acceleraor moive for holding invenories dominaes he producion smoohing moive, we expec invenories o be posiively relaed o sales. C. Coinegraion Tess and Esimaes of he Coinegraing Vecor Johansen-Juselius ess of coinegraion beween invenories, sales, observable cos shocks, and he ineres rae are presened in Table 4 for levels, logs, and linear derending of he variables. The evidence is consisen wih he heory: he ess rejec he null hypohesis of no coinegraion. We urn nex o esimaion of he coinegraing vecor. Our esimaion procedure is DOLS as described by Sock and Wason (993). In conras o SOLS esimaion of coinegraing vecors, DOLS correcs for biases ha can arise (excep under raher srong assumpions) in finie samples. In addiion, Sock and Wason (993) find ha DOLS has 7, 8 he minimum RMSE among a se of poenial esimaors of coinegraing vecors. 7 DOLS essenially adds leads and lags of he firs differences of he righ hand side variables o he coinegraing regression o ensure ha he error erm is orhogonal o he righ hand side variables. (For a brief descripion, see, e.g., Hamilon (994), p ) In heory, he number of leads and lags could be

24 Table 5 presens esimaes of he coinegraing vecor. The ineres rae eners he coinegraing relaionship wih a negaive sign, and he -saisic is greaer han five, which is very srong evidence of a long-run relaionship beween invenories and he ineres rae. This is a very sriking resul, especially when compared wih he resuls repored above indicaing no evidence of a shor-run relaionship beween invenories and he ineres rae. The poin esimae of he coefficien on he ineres rae (based on linearly derended variables) is and he difference in he ineres rae beween he highineres-rae regime and he low-ineres-rae regime is abou 6.8%. The esimaed coefficien in Table 5 herefore implies a decrease in invenories of abou % as he economy moves from he high o he low ineres rae. 9 I is ineresing o compare hese resuls wih he findings in Table 3. The simple comparison of means in Table 3 shows ha invenories are abou 8% lower in he highineres-rae regime han in he low-ineres-rae regime. Conrolling for oher variables, specifically sales and observable cos shocks -- as he coinegraing regression does -- leads o a qualiaively similar bu slighly larger effec. Invenories are abou % lower in he high-ineres-rae regime han he low-ineres-rae regime. infinie, bu his is impracical. There is Mone Carlo evidence (for he case of fixed invesmen) ha relaively high numbers of leads and lags are he mos effecive in reducing bias. Caballero (994, p. 56) finds ha he bias is smalles when he number of leads and lags is 5 for a sample size of 0. We se he number of leads and lags o 4. (Recall ha we are using monhly daa.) 8 In esimaing a coinegraing regression, he appropriae economeric procedure is o allow for he possibiliy of a deerminisic rend. We do his boh by derending he daa (in he specificaions ha use linearly derended daa) and by always allowing for a deerminisic rend in he coinegraing regression. 9 The only previously repored coinegraing vecors for invenories which allow for a variable ineres rae of which we are aware are in Rossana (993), which uses a raher differen approach. Insead of including he ex pos real ineres rae in he coinegraing vecor, he eners he nominal ineres rae and he inflaion rae as separae variables and (using wo-digi indusry-level daa) ess he resricion ha he coefficiens are equal in magniude and of opposie signs. I is herefore no sraighforward o deermine from he resuls he repors wheher invenories have an economically or saisically significan relaionship wih he real ineres rae.

25 Ineresingly, cos variables ener he coinegraing regression wih he heoreically prediced sign and a coefficien ha is significanly differen from zero, a relaionship ha many empirical sudies ha focus on shor-run flucuaions fail o uncover. The esimaed elasiciy of invenories wih respec o observable cos shocks is beween -0.8 and -.0. V. Formally Modeling Regime Swiching and Learning In he previous secion, we sugges an inuiively appealing explanaion for he lack of a relaionship beween invenories and he real ineres rae. The behavior of he real ineres rae is srongly suggesive of regime shifs, wih ransiory variaion around persisen mean ineres raes. Firms may largely ignore he shor-run ransiory variaion in he ineres rae and only respond o changes in he ineres rae ha appear o signal a change in he persisen regime. Consisen wih his explanaion, wo ess confirm a highly significan long-run relaionship beween invenories and he ineres rae. In his secion, we go a sep furher. We formally model regime shifs in he real ineres rae and show how his affecs invenory behavior. This leads o a disincive implicaion of regime swiching and learning for he long-run behavior of invenories. In subsecion D, we es his implicaion. A. Regime Swiching and Learning Regime swiches can be modeled by assuming ha he real ineres rae follows r = r +σ ε () S S where ε ~ i.i.d. N(0,) and S is he ineres rae regime (wih he mnemonic S for sae ). Regime-swiching in he real ineres rae has been sudied by economericians.

26 3 In paricular, Garcia and Perron (996) show ha he real ineres rae in he U.S. is well described by a hree-sae Markov swiching model. We herefore assume ha S {,, 3} follows a Markov swiching process 0. Le r < r < r3, so ha when S = he real ineres rae is in he low-ineres-rae regime, when S = he real ineres rae is in he medium-ineres-rae regime, and when S = 3 he real ineres rae is in he high-ineresrae regime. S and ε are assumed o be independen. Denoe he ransiion probabiliies governing he evoluion of S by pij = Prob( S = j S = i). Collecing hese probabiliies ino a marix we have p p p P p p p 3 = 3 p p p Ineres rae regimes are no direcly observable. No one announces o firms ha he economy has jus enered he low-ineres-rae regime. Insead, firms mus make inferences abou he underlying regime from heir observaions of he ineres rae. In oher words, firms learn abou he ineres rae regime. To be precise, we assume ha he firm knows he srucure and parameers of he Markov swiching process bu does no know he rue ineres-rae regime. The firm mus herefore infer S from observed ineres raes. We denoe he firm s curren probabiliy assessmen of he rue sae by π. Tha is, π Prob( S = Ω) π = π Prob( S ) = = Ω π Prob( S = 3 Ω ) 3, 0 For a comprehensive discussion of Markov swiching processes, see Hamilon (994, Chaper ).

27 4 where he firm s informaion se, Ω, includes he curren and pas values of r. Here, π i is he firm s esimae a dae of he probabiliy ha he real ineres rae is in regime i. To undersand he learning process, consider how he firm uses is observaion of he curren real ineres rae o develop is probabiliy assessmen, π. Beginning a he end of period - he firm uses π ogeher wih he ransiion probabiliies in P o form beliefs abou he period ineres rae sae prior o observing r. Tha is he firm evaluaes Prob( S i ) = Ω for i =,, 3 using π i π π = Pπ π 3 () Once he firm eners period and observes r, i uses he prior probabiliies from () ogeher wih he condiional probabiliy densiies, f S = i = i (r ) exp (r r ) for =,, 3, i σ σ i π i (3) o updae π according o Bayes rule. Specifically, π i π i = i = 3 j= π f(r S = i) f(r S = j) j for,,3. (4) Thus, he firm uses Bayes rule and is observaions of he real ineres rae o learn abou he underlying ineres rae regime. Given π, he expeced real ineres rae is given by Er = r Pπ = γπ + γπ + γπ (5) + v 3 3 where r v = [r,r,r 3 ], γ pr+ pr + p3r3, γ pr+ pr + p3r3, and

28 5 γ p r + p r + p r. Since π + π + π3 = by definiion, we can eliminae π from he righ hand side of (5) o obain ( ) ( ) E r = γ γ π + + γ γ π + γ (6) 3 3 Now, o isolae he expeced real ineres rae in he linearized Euler equaion, pariion (5) so ha { θ ( β + ) + γ ( β + + β + ) + ξ β + E Y Y Y Y Y ( W W ) (7) } + δβ( N αx ) + ηer + c= Then, subsiue (6) ino (7) o ge { θ ( β + ) + γ ( β + + β + ) + ξ β + E Y Y Y Y Y ( W W ) (8) } + δβ ( N α X ) + η( γ γ ) π + η( γ γ ) π + ηγ + c= To summarize his sub-secion, we have inroduced a formal model of regime swiching and learning. The key variable in he model is π i, he firm's assessmen of he probabiliy of being in ineres rae regime i. In he model of opimal invenory choice under he assumpion of regime swiching and learning, hese probabiliies replace he ineres rae in he Euler equaion. B. Derivaion of he Disincive Implicaion of Regime Swiching and Learning We follow he same approach as in Secion IV.B o derive he long-run implicaion of regime swiching and learning. Re-wrie equaion (8) so ha mos of he variables are in firs differences:

29 6 ( ) E { γ Y β Y + β Y βθ( N + X ) + θ N βδα X βξ W θ( β) + βδ N α X + ( β ) ξw } βδ (9) + η( γ γ ) π + η( γ γ ) π + ηγ + = c Suppose now ha N, X, W, π and π 3 are I(). Then invenories, sales, he cos shock, and he probabiliies will be coinegraed wih coinegraing vecor θ( β) ξ( β) ηγ ( γ ) ηγ ( 3 γ), α,,,. βδ βδ βδ βδ Again, noe ha, when he coinegraing vecor is expressed in he form of a regression, X, W, π and π 3 will be on he righ-hand-side of he equaion, so heir coefficiens will have signs opposie o hose shown in he coinegraing vecor above. We have shown earlier ha 0 η >. ( γ γ ) and ( γ γ ) 3 are complicaed funcions of he elemens of P and r v, so i is no possible o sign hem unambiguously for all mahemaically feasible values of P and r v. They can, however, be signed for he empirically relevan values. Using our esimaes of he elemens of P and r v, we obain ( γ γ ) < and ( γ γ ) 0 >, so he model predics ha he coefficien on π will be 3 0 posiive and he coefficien on π 3 will be negaive. Since π and π 3 have a resriced range, one migh wonder wheher i is beer o model hem as I(0) or I(). We noe wo poins. Firs, in careful applied economeric research, variables wih resriced ranges, such as he nominal ineres rae, are modeled as I() variables when hey are highly persisen. (See, e.g., Sock and Wason (993) and Caballero (994).) Second, uni roo ess indicae ha π and π 3 are I(). See he nex sub-secion.

30 7 This accords wih our inuiion of how he probabiliies should affec invenories. If, for example, here is an increase in π, he firm believes ha he economy is enering a persisen low-ineres-rae regime. This will lower he expeced opporuniy cos of holding invenories and should herefore lead o an increase in N. Looking a he coinegraing vecor, we can see his effec. Wih η( γ γ) < 0, an increase in π will lead o an increase in N (since π will be on he righ hand side of he coinegraing regression). Similarly, since η( γ3 γ) > 0, an increase in π 3, which indicaes ha he firm believes he economy is enering a persisen high-ineres-rae regime, will lead o a decrease in N. Thus, he disincive implicaion of regime swiching and learning is ha invenories will be coinegraed wih he probabiliies π and π 3 and ha he coefficien on π will be posiive and he coefficien on π 3 will be negaive. C. Calculaing he Probabiliies π and π 3 In order o es he disincive implicaion of regime swiching and learning, we mus consruc he probabiliies π and π 3. This can be done using he echniques described in Hamilon (989 and 994, Chaper ). We esimae he parameers of a hree-sae Markov swiching process for he real ineres rae over our sample period. Our esimaes of he elemens of he ransiion probabiliy marix are p = 0.98 p = 0.0 p3 = 0.00 P= p 0.0 p 0.98 p = = =. p3 = 0.00 p3 = 0.0 p33 = 0.96

31 8 Our esimaes of r, r, and r 3 (annualized) are -.7,.6, and 5.5, and our esimaes of σ, σ, and σ 3 are.90, 0.80 and.96, respecively. Two feaures of he behavior of he real ineres rae sand ou from hese esimaes. Firs, since p, p, and p 33 are all close o one, he ineres rae regimes are highly persisen. For example, hese esimaes indicae ha, if he economy is in he low-ineres-rae regime his period, here is a 98% probabiliy ha i will be in he lowineres-rae regime nex period. Similarly, if he economy is in he high-ineres-rae regime his period, here is a 96% probabiliy ha i will be in he high-ineres-rae regime nex period. This suggess ha changes in he ineres-rae regime will occur infrequenly. Furhermore, once he firm comes o believe ha he economy has enered a paricular ineres rae regime, i will anicipae ha he curren regime will persis for some ime. Second, noe ha he difference beween he mean ineres raes of any wo regimes is large relaive o he sandard deviaions. For example, r r = 3.3, which is.7 imes as large as he sandard deviaion of he whie noise shock in regime one. This suggess ha, wihin a given regime, whie noise shocks ha are sufficienly large o be misaken for a regime change will no be common. Figure plos he behavior of π, π, and π 3 as obained by applying he filer in equaions (), (3), and (4) o he real ineres rae daa in our sample. ( π is no required for subsequen ess, since he sum of he hree probabiliies is, bu we illusrae π in Figure for compleeness.) This figure confirms ha, when viewed from he perspecive of he Markov swiching model, mos of he shor-run variaion in he real ineres rae consiss of emporary flucuaions around he mean ineres rae for he

32 9 curren regime. For he mos par, he probabiliy of being in a given ineres rae regime is close o 0 or. Only occasionally does he Markov swiching model idenify shifs in he mean real ineres rae. D. Tess of he Disincive Implicaion of Regime Swiching and Learning In Secion V.B, we show ha regime swiching and learning have a disincive and esable implicaion: invenories will be coinegraed wih π and π 3, he probabiliies ha he economy is in he low and high-ineres rae regime, respecively. Johansen-Juselius ess of coinegraion beween invenories, sales, observable cos shocks, and he probabiliies π and π 3 are presened in Table 6 for levels, logs, and linear derending of he variables. The evidence is consisen wih regime swiching and learning: he ess rejec he null hypohesis of no coinegraion. Table 7 presens esimaes of he coefficiens in he coinegraing vecor beween invenories, sales, cos shocks, and he probabiliies π and π 3. Consisen wih regime swiching and learning, he coefficien on π 3 is negaive and highly significan, implying ha an increase in he probabiliy of he high-ineres-rae regime reduces invenories in he long run. The poin esimae of he coefficien on π 3 implies ha an increase in he ineres rae from.6% (he mean ineres rae in he medium-ineres-rae regime, which is he poin of reference) o 5.% (he mean ineres rae in he high-ineres-rae regime) reduces invenories by abou 7%. The coefficien on π is posiive as prediced. Alhough less precisely esimaed, he poin esimae implies a change in invenories, moving from he low-ineres-rae o he medium-ineres-rae regime, ha is similar o

33 30 he change implied by he esimaes in Table 5, a decrease in invenories of abou 5%. The esimaed cumulaive effec of a move from he low-ineres-rae regime o he highineres-rae regime is a decrease in invenories of abou %. E. Simulaions of he Cos Funcion. We have argued ha firms will largely ignore ransiory shocks and respond only o ineres-rae movemens ha signal a change in he persisen mean. To undersand why his is rue noe from equaion () ha he curren realizaion of he ransiory shock, ε, affecs he curren ineres rae bu has no effec on fuure ineres raes. If he firm knew wih cerainy ha a movemen in r was caused by a purely ransiory shock, ha movemen would have no effec on firm s choice of invenories. 3 Er + + j, and would, herefore, no affec he However, because he firm canno direcly observe he ineres-rae sae, i will aemp o infer, from he size and direcion of an observed ineres rae movemen, wheher or no ha movemen was caused by a change in regime. If he movemen is large enough o signal a change in he persisen sae, i alers expeced fuure ineres raes and, herefore, he firm s opimal choice of invenories. Transiory shocks are ypically small relaive o he differences beween he ineres rae means. From our esimaes in Secion V.C, he difference beween he mean ineres rae in he mediumineres-rae regime and in he low-ineres-rae regime is 3.3. The difference beween he mean ineres rae in he high-ineres rae regime and in he medium-ineres-rae 3 Noe from he decision rule, equaion (7), and from he ineres rae erms in Ψ + j ha N depends on r++j expeced fuure ineres raes, E, bu does no direcly depend on he curren ineres rae, r.

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