Inventory Behavior with Permanent Sales Shocks

Transcription

1 Invenory Behavior wih Permanen Sales Shocks Louis J. Maccini Barholomew Moore (Corresponding auhor) Deparmen of Economics Deparmen of Economics Fordham Universiy Johns Hopkins Universiy 44 Eas Fordham Road 400 N. Charles Sree Bronx, NY 0458 Balimore, MD 8 (78) (40) bmoore@fordham.edu maccini@jhu.edu Hunley Schaller Deparmen of Economics Carleon Universiy 5 Colonel By Drive Oawa, ON KS 5B6 (6) hunley_schaller@carleon.ca April, 0 We hank Marin Eichenbaum, James Kahn and Adrian Pagan for commens on an earlier draf of he paper and seminar paricipans a he American Economic Associaion, Bank of Canada, Economeric Sociey, Fordham Universiy, Insiue for Advanced Sudies (Vienna), Inernaional Sociey for Invenory Research, The Sociey for Compuaional Economics, he Universiy of Kenucky, Mark Blanchee for excellen research assisance, and he SSHRC for financial assisance.

2 Invenory Behavior wih Permanen Sales Shocks Absrac Empirically, sales are I(). We derive a new model of invenories based on his fac. Our heory implies hree sarling resuls. Firs, he variance of producion is equal o he variance of sales in he long run. Second, his resul holds regardless of he srengh of producion smoohing, sockou avoidance, or cos shocks. Third, a business cycle horizons, he condiional variance of producion is greaer han sales. Our heory leads o a differen way of esimaing, esing, and calibraing invenory models. The calibraed model simulaneously accouns for four radiional invenory puzzles and hree puzzles abou invenories and moneary policy. [JEL Classificaion Codes: E, E] Keywords: Invenories, Producion Smoohing, Sockou Avoidance, Coinegraion, Moneary Policy Effecs April, 0

3 I. INTRODUCTION Invenory movemens are imporan. In he recession, invenories accouned for one-hird of he fall in US GDP, a huge amoun for such a small componen of oupu. This is ypical: Invenory movemens accoun for a wildly disproporionae share of macroeconomic flucuaions in mos poswar US recessions and in oher counries, oo. Despie he imporance of invenory flucuaions, here are large gaps in our undersanding of he basic economics of invenories. Equally seriously, here are sharp conradicions beween he predicions of sandard heory and he response of invenories o he main macroeconomic policy ool, moneary policy. I has long been hough ha invenories ac as a shock absorber for flucuaions in aggregae demand. Sandard economic heories imply ha invenories are used o smooh producion. A long-sanding puzzle is why producion varies more han sales in he daa. A variey of heoreical explanaions have been proposed. Our heory implies ha hese explanaions all urn ou o be irrelevan, a leas in he long run. In he daa, sales are I(). We develop a new heory of invenories in which his fac plays a cenral role. Our model implies hree sarling new analyical resuls. Firs, in he empirically relevan case, he variance of producion is equal o he variance of sales in he long run. Second, his resul holds regardless of he srengh of producion smoohing, sockou avoidance, or cos shocks. Third, a business cycle horizons, he condiional variance of producion is greaer han sales. This implies ha invenories amplify sales shocks during business cycles, raher han dampening shocks as producion smoohing would imply. Our heory leads o a differen way of esimaing, esing, and calibraing invenory models. When sales are I(), our heory implies ha invenories will also be I(), an implicaion ha is consisen wih he daa. This means ha we need o derive a coinegraing relaionship. 4 This is more difficul han i migh iniially appear, because we need o linearize he firm s Euler equaion around saionary variables, a ask ha has no been explicily addressed in he exising lieraure. The workhorse linear-quadraic invenory model does no lend iself o he ask. We According o NIPA daa, over he six quarers 008:-009:, he cumulaive change in invenory invesmen was 4.8% of he cumulaive change in GDP. See Blinder and Maccini (99) and Ramey and Wes (999). On he heory side, a pioneering paper is Wes (990), who, in he conex of oher issues, obains a weak inequaliy on he relaive variance of producion and sales, allowing for boh saionary and I sales. We build on Wes's work and obain more specific resuls for I sales. 4 Hamilon (00), Kashyap and Wilcox (99), Ramey and Wes (999), and Rossana (99, 998) are early papers ha consider he coinegraing relaionship.

4 presen a model ha capures much of he flavor of he linear-quadraic model bu ha can be linearized around saionary variables. We es wheher oher variables ha migh affec invenories (e.g., inpu coss) are I(), as well as esing wheher he variables ha are assumed o be saionary in our heory are, in fac, I(0). Our heory leads o a specific coinegraing relaionship ha we esimae on aggregae US daa. Pas effors o esimae he effecs of he deerminans of invenories, based on I(0) economerics, have ofen suggesed ha key variables, such as inpu coss and he ineres rae, have coefficiens wih he wrong sign or saisically insignifican coefficiens. In conras, he coefficiens in he coinegraing relaionship implied by our heory all have he heoreically prediced signs and are srongly significan. Our heory shows how he underlying srucural parameers can be calculaed from he coinegraing relaionship implied by he model. Using his novel approach o calibraion, which flows direcly from our heory, we simulae our model. Simulaions of he calibraed model show ha i provides a unified explanaion for four radiional invenory puzzles. In addiion, he model accouns for hree puzzles abou moneary policy and invenories ha have been documened in he previous lieraure. An imporan radiional puzzle ha has plagued he lieraure for decades is he Variance raio puzzle. If producion coss are convex, hen firms wan o smooh producion in response o demand shocks. This has long been hough o imply ha producion should vary less han sales. Bu, empirically, producion ends o vary more han sales. As noed above, our heoreical model implies ha he variance of producion should be equal o he variance of sales in he long run. Why hen do empirical sudies ypically find ha producion varies more han sales? Simulaions of our model reveal ha small sample bias is he culpri. The simulaions indicae ha, in samples of he size used in empirical sudies in he invenory lieraure, convenional procedures will sugges ha producion moves more han sales. Asympoically, however, he variance of producion is equal o he variance of sales. Oher radiional puzzles include he following. Slow adjusmen puzzle: As an influenial survey of he invenory lieraure pus i, "One major difficuly wih sock-adjusmen models is ha adjusmen speeds generally urn ou o be exremely low; he esimaed adjusmen speed is ofen less han 0 percen per monh. This is implausible when even he wides swings in invenory socks amoun o no more han a few days of producion. [Blinder and Maccini (99,

5 page 8)]". Wen puzzle: Wen (005) disinguishes beween he movemens of oupu and sales a shor horizons (less han hree quarers) and medium horizons (abou 8-40 quarers). A medium horizons, he finds ha producion is more volaile han sales. More surprisingly, he finds ha producion is less volaile han sales a shor horizons. Wen argues ha hese sylized facs consiue a "limus es" for invenory heory and concludes ha none of he exising explanaions for he variance raio puzzle -- sockou avoidance, cos smoohing, or increasing reurns o scale -- can accoun for he behavior of oupu and sales a boh shor and medium horizons. Inpu cos puzzle: When coss are low, firms have an incenive o produce more and build up heir invenories. I has been surprisingly difficul, however, o find evidence of an empirical relaionship beween observable coss and invenories. Simulaions of our calibraed model enable us o explain hese radiional invenory puzzles as well. Our heory yields a closed-form soluion for he condiional variance raio he variance of oupu relaive o ha of sales over he shor and medium run. The insighs from he formula for he condiional variance raio play a key role in he simulaneous soluion of hree of he radiional puzzles -- he variance raio, slow adjusmen, and Wen puzzles. One of he mos imporan insighs is ha convexiy of producion coss, which provides he moive for producion smoohing, is consisen wih a higher medium-run variance for producion han for sales. The formula for he condiional variance raio shows ha he relaive imporance of wo key moives -- producion smoohing and sockou avoidance -- depends on he seady-sae real ineres rae. Previous heoreical work has no elucidaed his role of he ineres rae. Anoher insigh is ha, despie he fac ha he variance of producion exceeds he variance of sales a business cycle horizons, he model is consisen wih slow adjusmen speeds in invenory invesmen equaions due o he high convexiy of producion cos and hus he srong incenive for he firm o smooh producion. Furher, we are able o explain he Wen puzzle by calculaing, using he calibraed model, ha he producion smoohing moive dominaes high frequency invenory movemens, bu he sockou avoidance moive dominaes business cycle movemens. Finally, our coinegraing regression provides srong empirical evidence of he effec of inpu coss on invenories in he long-run, which explains he inpu cos puzzle. Despie he srong evidence of he effec of inpu coss on invenories, simulaions of our calibraed model indicae ha inpu coss have lile effec on he condiional variance raio.

6 4 Here are he hree moneary policy puzzles. Mechanism puzzle: Moneary policy changes he ineres rae and should affec invenories, since he ineres rae represens he opporuniy cos of holding invenories. In fac, VAR sudies find ha moneary policy affecs invenories. Bu 40 years of empirical research on invenories based on I 0 economerics has generally failed o find any significan effec of he ineres rae on invenories. Sign puzzle: Simulaive moneary policy lowers he ineres rae. A decrease in he ineres rae should increase invenories. VAR sudies find ha he shor-erm effec of simulaive moneary policy is jus he opposie -- invenories decline. Timing puzzle: Expansionary moneary policy induces a ransiory decline in he ineres rae. The effec of moneary policy on he ineres rae largely disappears wihin one year. Bu invenories begin o rise only afer he ransiory shock o he ineres rae has largely dissipaed. In our model, he firm's response o an ineres rae movemen depends on he exen o which he firm believes he movemen is persisen. This makes he ransiional dynamics of he invenory response o a change in he ineres rae complex and nonlinear and herefore difficul o deec using 0 economerics. In conras, when we use economerics specifically, he coinegraing regression implied by our model he daa provide srong evidence of he role of he ineres rae. The combinaion of model and empirical evidence provides he soluion o he mechanism puzzle. The key o our model's success in explaining he sign puzzle is he role of invenories in buffering demand shocks. A simulaive moneary policy shock lowers he ineres rae and increases sales, bu he firm canno immediaely raise producion, so invenories fall a he same ime ha moneary policy is pushing he ineres rae down. Two elemens of our model explain he iming puzzle. Firs, he firm akes ime o learn wheher a movemen in he ineres rae is persisen (i.e., represens a regime swich). This delays he firm's response o he ineres rae movemen. Second, producion smoohing plays a role. A simulaive moneary policy shock lowers he ineres rae and increases he desired level of invenories. Bu, because of he convexiy of he producion cos funcion, he firm is relucan o adjus producion oo sharply, so he change in invenories is gradual. The paper is organized as follows. Secion II inroduces he model wih I sales. Secion III saes he decision rule for invenories implied by he model. Secion IV presens

7 5 hree sarling resuls regarding he relaive variance of oupu ha emerge from he model. Secion V describes he model s implicaions for esimaion and esing. Secion VI oulines our innovaive approach o calibraion, which flows from he model. Secion VII explains how he model resolves four radiional invenory puzzles -- slow adjusmen, variance raio, Wen, and inpu cos. Secion VIII examines how he model accouns for he hree moneary policy puzzles. Secion IX provides a summary and conclusion. II. THE MODEL The lieraure on invenory models has been dominaed by he use of linear-quadraic approximaions of an underlying cos funcion originally advanced in Hol, e al (960) 5. In his paper, we depar from he linear-quadraic lieraure by assuming a consan elasiciy approximaion o an underlying cos funcion. We uilize a consan elasiciy approximaion o ensure ha he equilibrium condiions can be expressed in erms of saionary raios. The represenaive firm is assumed o minimize he presen value of is expeced coss over an infinie horizon. 6 Real coss per period consis of producion coss and invenory holding coss. Producion coss, PC PC, are defined as AY W () wih,, 0, where Y is real oupu and W is real inpu coss, which we will measure wih real inpu prices of variable facors of producion, and A is a shif variable ha capures he sae of echnology and fixed facors of producion. 7 Observe ha average producion coss, J, are 5 Sudies in he lieraure ha have used he linear-quadraic model in work on invenories include, for example, Blanchard (98), Blinder (986-b), Wes (986), Miron and Zeldes (988), Eichenbaum (989), Durlauf and Maccini (996), Hamilon (00), Humphreys, e al (00), Kashyap and Wilcox (99), and Wen (005). 6 We assume ha he firm minimizes discouned expeced coss and hereby absrac from marke srucure issues, because our key innovaion is o recognize ha sales are I and o analyze he implicaions of his empirical fac for he long-run behavior of invenories. See, e.g., Bils and Kahn (000), Chang, Hornsein and Sare (009) and Jung and Yun (0, 0) for models ha deal wih marke srucure issues. Even hough we absrac from marke srucure issues, our model is quie successful in capuring many aspecs of he behavior of invenories. An ineresing quesion for fuure research is wheher our characerizaion of invenory behavior a business cycle horizons can be refined by incorporaing marke srucure issues ino he model. 7 In he empirical work, we allow o be freely esimaed wihou imposing he assumpion ha, hough is required for posiive and rising marginal producion coss. A producion cos funcion wih rising marginal producion coss, due o eiher he presence of fixed facors of producion or diminishing reurns o scale, has been

8 6 PC J AY W () Y and marginal producion coss are J. Invenory holding coss, N X HC, are HC X N () wih 0, 0, and 0, where N is he sock of finished goods invenories a he end of period, and X is he level of real sales, which is given exogenously. 8 Invenory holding coss N consis of wo basic componens. One, X, which we refer o as sockou avoidance X coss, capures he idea ha, given sales, higher invenories reduce coss in he form of los sales because hey reduce sockous. 9 The oher, N, which we refer o as sorage coss, capures he idea ha higher invenories raise holding coss in he form of sorage coss, insurance coss, ec. 0 widely used in he invenory lieraure o capure he producion smoohing moive. See, for example, he papers lised in foonoe 4 as well as Kashyap and Wilcox (99) and Hamilon (00) who, as we do, use coinegraion mehods in heir empirical work. 8 The assumpion ha sales are exogenous is empirically consisen wih he pioneering work on invenories and coinegraion by Granger and Lee (989), who conclude (page S5) ha, "The sales series may be hough of as being largely exogenously deermined." Theoreically, sales can be endogenized by specifying an inverse demand funcion. Indusry equilibrium can be analyzed wih such a demand curve, as in Eichenbaum (989). Alernaively, Chrisiano and Eichenbaum (989) and Wes (990) derive such a linear inverse demand curve in general equilibrium. In linear-quadraic invenory models, his leads o a decision rule ha is similar o he case wih exogenous sales. See, e.g., Ramey and Wes (999, Secion 4). An alernaive approach o endogenizing sales is o incorporae invenories ino a general equilibrium model. See Jung and Yun (005), Khan and Thomas (007), Wen (008), Wang and Wen (009), Iacoviello, Schianarelli and Schuh (007), among ohers. A poenially ineresing opic for fuure research is o ake he model of firm behavior developed here and incorporae i ino a general equilibrium model. 9 See Bils and Kahn (000) for a model ha uilizes a consan elasiciy specificaion of he benefis of holding finished goods invenories, wih he benefis embedded on he revenue side of he firm. As discussed in foonoe 6, here are benefis o absracing from marke srucure issues if he objecive is o ake accoun of he fac ha sales are I () and analyze he long-run behavior of invenories. See Maccini and Pagan (009) for a recen paper ha uses a specificaion of he benefis of holding finished goods invenories ha is similar o equaion (). 0 These wo componens underlie he raionale for he quadraic invenory holding coss in he sandard linearquadraic model. The formulaion above separaes he componens and assumes consan elasiciy funcional forms which faciliaes log-linearizaion around seady-sae raios. Observe ha () implies a arge sock of finished goods invenories ha minimizes finished goods holding coss. The arge sock is N TS / X so ha he implied sock is proporional o sales. This is analogous o he arge sock assumed in he sandard linear-quadraic

9 7 The firm's informaion se a ime,, includes he curren and pas values of all relevan variables, bu when he firm chooses N is informaion se is. This assumpion, which is sandard in he invenory lieraure, capures he idea ha invenories buffer demand shocks; for example, he firm may mee an unanicipaed increase in sales ou of is sock of invenories. Le be a variable real discoun facor, which is given by / r, where r denoes he real rae of ineres. The firm s opimizaion problem is o minimize he presen discouned value of expeced oal coss, where E E. 0 0 E, and 0 j j C, (4) C PC HC A Y W N X X N, (5) subjec o he invenory accumulaion equaion, which gives he change in invenories as he excess of producion over sales, and N N Y X. (6) The opimaliy condiions ha resul from his opimizaion problem are E A Y W 0 (7) E 0 N X where is he Lagrange muliplier associaed wih he consrain (6). (8) To inerpre he opimaliy condiions, eliminae he Lagrange muliplier o reduce he opimaliy condiions o E A Y W N E X E A Y W model. Noe ha he arge sock is no he seady-sae sock of finished goods invenories. The seady-sae sock minimizes oal coss in seady sae whereas he arge sock merely minimizes invenory holding coss.

10 8 Now, E A Y W is he marginal cos of producing a uni of oupu oday, E A Y W is he discouned marginal cos of producing a uni of oupu omorrow, and E N / X is he discouned marginal holding cos. The Euler equaion hus saes ha he firm should equae he marginal cos of producing a uni of oupu oday and carrying i in invenories o he discouned marginal cos of producing he uni of oupu omorrow. In Appendix B, we show ha linearizing he opimaliy condiions around seady-sae values yields a linearized Euler equaion: E J lny lny J lnw lnw ln ln 0 A N X Jr Ju c where J is seady-sae average producion cos, J is seady-sae marginal producion cos, is seady-sae value of average sockou avoidance coss, which are defined by RN x, is seady-sae marginal sockou avoidance coss, (9) R N is he seady-sae invenory/sales raio, / r, r is he uncondiional mean real ineres rae, x is he seady-sae growh rae of sales, u A is a saionary shock, c is a consan, and a bar above a variable denoes a seady-sae value. In he daa, sales are I(), as shown in Table, which shows ha N is also I(). As we will show, he fac ha sales are I() has sarling implicaions. See Hamilon (00) for a careful discussion of he saionariy properies of marginal producions coss ha are implied by invenory models. In paricular, Hamilon (00) shows how saionariy of marginal producion coss arises naurally when sales, coss, oupu, ec. are nonsaionary. Noe ha is average seady sae sockou avoidance coss, no average oal invenory holding coss. The laer is, which includes boh sockou avoidance coss and sorage coss.

11 9 Table Uni Roo Tess and Esimaed Coinegraing Regression Panel A: Uni Roo Tess Variables in Coinegraing Regression N X W [0.94] -.0 [0.084] -.6 [0.469] -.7 [0.] -.0 [0.06] Panel B: Uni Roo Tess Raios Assumed o be Saionary N / X Y / X J [0.0] [0.000] [0.000] [0.04] N is invenories, X is sales, Y is oupu, W is inpu coss, J is average producion coss, is relaed o marginal sockou avoidance coss, is he probabiliy of being in he low-ineres-rae sae, and is he probabiliy of being in he highineres-rae sae. All variables are log-linearly derended. The cell enries are ADF ess for uni roos, p-values in brackes. (The number of lags in he ADF ess was chosen using a sandard crierion; i.e., he lag lengh ha minimizes he AIC plus. All of he uni roo ess include a consan and a deerminisic rend.) Since sales and invenories are I(), we need o linearize he opimaliy condiions around saionary variables. We assume ha he raios, RN N / X, RY Y X, J, and, are saionary. Table presens uni roo ess. Sandard ADF ess rejec he null hypohesis of a uni roo for each of he four raios. We assume furher ha he real ineres rae follows a hree-sae Markov swiching process. Specifically, we assume ha he real ineres rae follows r r (0) S S where ~ i.i.d. N(0,) and where S {,, } follows a Markov swiching process. Le r r r, so ha, when S,,, he real ineres rae is in he low-ineres-rae, moderaeineres-rae, and high-ineres-rae regime, respecively. S and are assumed o be This is consisen wih empirical paerns in real ineres raes; see Garcia and Perron (996) and Maccini, Moore and Schaller (004). The laer paper describes how he firm uses is observaions of he real ineres rae o develop is probabiliy assessmens. For a comprehensive discussion of Markov swiching processes, see Hamilon (994, Chaper ).

12 0 independen. Denoe he ransiion probabiliies governing he evoluion of S by p Prob( S j S i). Collecing hese probabiliies ino a marix we have ij p p p P p p p p p p. () The firm is assumed o know he srucure and parameers of he Markov swiching process bu does no know he rue real 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, Prob( S ) Prob( S ). () Prob( S ) Given, he erm E r v in equaion (9) can be compued as E r r P () and where r v [ r, r, r ] r'p. Since by definiion, we v can eliminae from he righ hand side of () o obain E r. (4) Then, subsiuing (4) ino (9) yields E J lny lny J lnw lnw (5) A lnn ln X Ju J c 0 which is he log-linearized Euler equaion incorporaing he firm s learning process. III. DECISION RULE The log-linearized Euler equaion implied by he model, equaion (5), may be wrien as a second-order expecaional difference equaion. Solving his difference equaion yields a decision rule, which is saed in he following proposiion.

13 Proposiion. Decision Rule: The model implies ha he firm's decision rule is where ln N ln N ln X lnw u (6) 0 X W r 0 r 4, (7-a) R J R Y N >0, (7-b) 0 is a consan, u is a saionary shock, and RY and RN are he seady sae values of RY and RN respecively. Proof: See Appendix C. The coefficien on lagged invenories,, is he sable roo of he relevan characerisic equaion ha emerges from solving he second-order expecaional difference equaion ha is implied by he Euler equaion. As we shall see below, defines he speed of adjusmen ha governs he fracion of he gap beween desired and acual invenories ha is closed each period by invenory invesmen. The decision rule shock, u, arises from unanicipaed flucuaions in sales and oupu. In he shor run, invenories ac as a buffer, absorbing hese unanicipaed flucuaions. The resuling invenory movemens involve only ransiory deviaions from he level of invenories dicaed by he variables in he decision rule, so u is saionary. The coefficiens on sales, inpu coss and ineres-rae-regime probabiliies are defined in he following proposiions. Proposiion. Decision Rule Coefficien on Sales: The coefficien on sales in he decision rule, X, is X Furher, 0 as R Y r J RN r X r J. (8)

14 Proof: See Appendix C. The erm is he convexiy of sockou avoidance coss and J is he convexiy of producion coss. Hence, an increase in sales will induce he firm o produce enough addiional oupu o raise invenory holdings so long as he presen value of he change in marginal sockou avoidance coss exceeds he change in marginal producion coss, and viceversa. To undersand he inuiion for he sign of X, suppose he firm is making a marginal decision abou oupu in response o an increase in sales. The cos of producing one more uni of oupu is a one-ime cos. The benefi of an addiional uni of invenory is he presen value of he reducion in sockou avoidance coss. If he laer dominaes, he firm will produce enough addiional oupu o increase is invenory holdings. Bu, if he former dominaes, he firm will increase oupu by less han he increase in sales, so is invenory holdings will fall. Proposiion. Decision Rule Coefficien on Inpu Coss: The coefficien on inpu coss in he decision rule, W, is r R Y W <0 (9) RN r Proof: See Appendix C. The model implies ha he coefficien on inpu coss in he decision rule W will be negaive. Inuiively, an increase in inpu coss raises producion coss, which induces he firm o cu producion and hereby reduce invenory holdings. Proposiion 4. Decision Rule Coefficiens on he Ineres-Rae-Regime Probabiliies: The model implies ha he decision rule coefficiens on he ineres-rae-regime probabiliies are R Y I P R r (0-a) N 0 0 R Y I P R r (0-b) N where. Furhermore, if p p 0.5 (-a)

15 p p 0.5 (-b) p, (-c) p 0 hen, 0 and 0. Proof: See Appendix C. Observe ha he assumpions ha p p 0.5 and p p 0.5 mean ha he ineres rae regimes are persisen, and he assumpion ha p p 0 means ha he economy moves hrough he medium-ineres-rae regime on is way from he high- o lowineres-rae regime, and vice versa. Inuiively, if he ineres rae regimes are persisen (and hey are in he daa, as repored in (0) below), lower ineres raes oday imply ha fuure ineres raes will be lower, reducing he opporuniy cos of holding invenories and hereby inducing he firm o hold more invenories. IV. THE RELATIVE VARIANCE OF OUTPUT: THREE IMPLICATIONS A key quesion abou invenories is wheher hey amplify or dampen demand shocks. In a model wih sales shocks, Wes (990) obains an inequaliy Var Y Var X ha applies boh when sales follow a saionary sochasic process and when sales are I. 4 We are able o esablish a more specific resul for he I case. To sar, we derive he decision rule for oupu, which akes he form of a firs-order linear difference equaion. 5 Using assumpions abou he sochasic processes for sales and inpu coss ha are consisen wih our earlier assumpion ha hey are I variables, we solve he difference equaion backwards o obain an equaion for oupu as a funcion of oupu and sales a a fixed dae in he pas and subsequen sales and cos shocks. By aking he variances of lny and ln X, condiional on oupu and sales a a fixed dae in he pas, we obain Proposiion 5. Proposiion 5. Condiional Variance Raio: 4 More precisely, Wes assers ha uncondiional mean, EX Y 0. Under his assumpion ha all variables have zero E X Y E X E Y Var X Var Y, so his resul implies ha Var Y Var X. 5 To focus on producion smoohing, sockou avoidance, and cos shocks, we here assume a consan ineres rae.

16 VarlnY lny n X Var ln X ln X n n n n W n W X X n n where Var lny lny n is he variance of ln Y condiional on lny n he variance of ln X condiional on ln X n, (), Varln X ln X n is X is he variance of he sales shock, W is he 4 variance of he cos shock, and X R N R X Y J r ( r ) 0 W R N R W r Y r 0, (-a) (-b) where X and W are he elasiciies of oupu wih respec o sales and inpu coss, respecively. Proof: See Appendix C. In he empirical lieraure, auhors frequenly compue wha is referred o as he "variance raio". The calculaion akes he daa for oupu and sales over a sample, compues he variance of each, and hen akes he raio of he variance of oupu o he variance of sales. The variance raio is hus an uncondiional saisic. Mahemaically, we can obain he uncondiional variance raio from he model by aking he limi of he condiional variance raio, (), as n. Proposiion 6 saes he resul. Proposiion 6. Uncondiional Variance Raio: lim n VarlnY lny n (4) Var ln X ln X n Proof: Proposiion 6 follows from aking he limi of equaion () as n, noing ha 0. QED

17 5 Proposiion 6 is our key resul: When sales are I, he variance raio is. This means ha he variance of producion is equal o he variance of sales in he long run. The inuiion for our key resul flows from he persisence of he shocks o sales. Suppose a firm has a convex cos funcion and faces uncerain demand. If he shocks o demand are ransiory, i is opimal for he firm o produce a an inermediae level of oupu raher han o someimes produce a a low level and someimes a a high level. Bu suppose here is a permanen shock o sales. Then he firm increases oupu by he same amoun as he permanen shock, because a permanen shock relocaes he opimal level of oupu. Proposiion 7. Effec of Producion Smoohing, Sockou Avoidance, and Cos Shocks: The srucural parameers,, and he variance parameer in he sochasic process for inpu coss W have no effec on he uncondiional variance raio. Proof: Proposiion 7 follows direcly from Proposiions 5 and 6. The srucural parameers,, and ener he second and hird erms of equaion () hrough, X, W -- and W direcly eners he hird erm, bu none of hese parameers eners he firs erm. As n, he second and hird erms approach 0. QED Proposiion 7 means ha he srengh of producion smoohing (refleced in ), sockou avoidance (refleced in ), and cos shocks (measured by W and refleced in ) play no role in he long-run response of producion o sales. V. ESTIMATION AND TESTING Since Panel A of Table shows ha he key variables are I, we begin by deriving he coinegraing regression. 6 Proposiion 8. Coinegraing Regression: The model in Secion II implies ha invenories, sales, inpu coss, and he ineres-rae-regime probabiliies are coinegraed, wih coinegraing regression where ln N b b ln X b lnw b b, (5) 0 X W,, 6 Kashyap and Wilcox (99) and Ramey and Wes (999) provide earlier derivaions of a coinegraing vecor for invenories (boh under he assumpion of a consan real ineres rae).

18 6 b r J r J b X W b r r (6-a) J J b, (6-b) b 0 is a consan, and v is a saionary error erm. Proof: See Appendix C. The equaions in (6) are of crucial imporance in undersanding invenory behavior because hey provide a mapping beween he coinegraing regression coefficiens and he underlying srucural parameers. Proposiion 8 suggess an immediae es of he model, since i saes ha he variables in equaion (5) will be coinegraed. The daa are consisen wih Proposiion 8: The Johansen- Juselius es rejecs he null hypohesis of no coinegraing vecor, wih a es saisic of 97.9 (pvalue=0.00). 7 We esimae equaion (5) using he Sock and Wason (99) Dynamic OLS (DOLS) esimaor. Sock and Wason (99) show how DOLS reduces he small sample bias in OLS esimaes of coinegraing regressions. 8 Applied o esimaing he coinegraing regression in (5), he DOLS empirical specificaion (including consan and ime rend) is: ln N b b b ln X b lnw b b 0 T X W,, p p p p B ln X B ln W B B. X, s s W, s s, s, s, s, s s p s p s p s p The resuls are presened in Table. (7) 7 For reasons of daa availabiliy, he sample is 959:0 o 004:08. The number of lags used in he es is se o minimize he AIC. 8 See also Caballero (994), Caballero (999), and Schaller (006) on he small sample bias in OLS esimaes of coinegraing regressions. Our own Mone Carlo simulaions show ha he OLS bias is severe enough in our conex o yield esimaes of b X wih he wrong sign. Based on our Mone Carlo simulaions, we se p 48 in equaion (7).

19 7 Table Esimaed Coinegraing Regression Consan Time b X.589 (.89).5E-0 (0.9) 0.50 (.098) b W b (-5.44) DOLS esimaes of he coinegraing vecor wih -saisics in parenheses (0.974) b (-4.6) The signs of he coefficiens of he coinegraing regression ha are implied by he model are presened in he nex proposiion. Proposiion 9. Signs of he Coefficiens in he Coinegraing Regression: A. b X 0 B. b W 0 as r J C. If (-a), (-b), and (-c) hold, hen b 0, and b 0. Proof: See Appendix C. Proposiion 9-A saes ha, in he long run, an increase in sales will increase (decrease) invenories if he presen value of he convexiy of sockou avoidance coss exceeds (is less han) he convexiy of producion coss. The inuiion is he same as ha for Proposiion. As Table shows, our resuls from esimaing he coinegraing regression yield an esimae of b X ha is posiive and saisically significan (The -saisic is..), so our resuls indicae ha empirically he sockou avoidance moive dominaes he producion smoohing moive. Proposiion 9-B saes ha an increase in inpu coss will reduce long-run invenories. Table shows ha he daa are consisen wih Proposiion 9-B. The esimae of b W is negaive and srongly saisically significan. (The -saisic is -5..) Proposiion 9-C saes ha a higher probabiliy of he low-ineres-rae regime increases invenories, and a higher probabiliy of he high-ineres-rae regime reduces invenories. Table shows ha he daa are consisen wih he predicions of Proposiion 9-C. The esimae of posiive and srongly saisically significan. (The -saisic is abou.0.) The esimae of negaive and also saisically significan. (The -saisic is -4..) b is b is

20 8 VI. CALIBRATION Our approach o calibraion is innovaive and flows direcly from he model. Since sales (and he oher key variables) are I (), we use he coinegraing regression o calibrae he srucural parameers of he model. Noe from he definiions of b and b ha b X r b r. (8) Since r,, and are given from our esimaes of he Markov swiching model, we can use our X esimaes of definiions of b and b o obain a unique value for from (8). Similarly, noe from he X b and b ha W bw b r r. (9) We can hus use our esimaes of b and b o obain a unique value for from (9). W We obain r,, and from our esimaes of he parameers of he sochasic process for he real ineres rae, ha is, from our esimaes of he elemens of P and r v. From our esimaion of he hree-sae Markov-swiching model, hose esimaes are p 0.98 p 0.0 p 0.00 P p 0.0 p 0.96 p 0.05 p 0.00 p 0.0 p 0.95 v r r r. Togeher hese esimaes imply ha he and r uncondiional mean of he monhly real ineres rae is r =0.00, which gives (0) Finally, noe from he definiions of b r J R x ( ) N b in (6-b) and recalling ha R x N ()

21 9 Using he esimae of b, he normalizaion 9, and given values 0 for RN, x, and J, equaion () gives a single resricion on he value of. We have assumed ha 0. We herefore search numerically over (,0] o find he value of ha saisfies (). Thus, we obain a unique value for each of he model s srucural parameers here are no free parameers. The values ha we obain are repored in Table, Panel A. The calibraed parameers are consisen wih our heoreical predicions. In paricular,, 0, and 0. Table Calibraed Srucural Parameers and Decision Rule Coefficiens Panel A: Cos Funcion Parameers Panel B: Decision Rule Coefficiens X W As shown in equaion (), and are he elasiciies of producion cos wih respec o oupu and wih respec o inpu coss, respecively. As shown in equaion (), is he elasiciy of sockou avoidance coss wih respec o he invenory/sales raio. As is common in he invenory lieraure, is normalized o. Consequenly,, and, are measured relaive o. (The sorage cos parameer is no included in he able because i does no affec he decision rule coefficiens.) The coefficiens,, X, W, and are he coefficiens in he firm's decision rule on lagged invenories, sales, inpu coss, and he Bayesian probabiliies of he low-ineres-rae and highineres-rae regimes, respecively. Similarly, equaions (8), (9) and (0) provide he mapping from he srucural parameers o he decision rule coefficiens. These can be used o derive calibraed values of he 9 This normalizaion implies ha we can only evaluae he relaive magniude of oher srucural parameers such as and. A comparable siuaion exiss wih linear-quadraic invenory models, where he relaive magniude of key srucural parameers deermines he behavior of invenories. See, e.g., Ramey and Wes (999), p R,, and N x J are seady-sae raios. For R and x, we herefore use he sample mean values of N N X and X X, respecively, which gives RN and x Noe from J A Y W Y ha J denoes he seady-sae value of average producion coss. Based on daa from he 99 Census of Manufacuring, we esimae producion coss o be 7.4% of oal oupu and se J Our numerical search shows ha only one value of saisfies ().

22 0 decision rule coefficiens. The calibraed values of he decision rule coefficiens are repored in Table, Panel B. From Proposiion, X will be posiive if he presen value of he change in marginal sockou avoidance coss exceeds he change in marginal producion coss. The calibraed srucural parameers imply ha X is indeed posiive. Furher, from Proposiions and 4 and he sufficien condiions saed in (), he model implies ha W 0, 0 and 0. Tha is, our model implies ha an increase in coss or an increase in he probabiliy ha he economy is in he high-ineres-rae regime lowers invenories. The calibraed srucural parameers imply calibraed values of he decision rule coefficiens ha are consisen wih hese predicions. VII. TRADITIONAL PUZZLES A. Slow Adjusmen Puzzle In early empirical work on invenories, a common specificaion was he sock-adjusmen equaion. Lovell (96), for example, developed a model which yielded an invenory invesmen relaionship of he form: * N N N N u N () where u N is a shock. In he Lovell framework, invenory invesmen is proporional o he gap beween he acual and desired sock of invenories. The proporionaliy facor,, measures he speed of adjusmen, as i capures he fracion of he deviaion beween desired and acual invenories ha is closed each period. The slow adjusmen puzzle is ha esimaed values of appear o be implausibly low. Blinder and Maccini (99, page 8) summarize he puzzle as follows, "Theory srains o explain low adjusmen speeds unless he incenive o smooh producion is exremely srong, which is hard o reconcile wih he fac ha producion is more variable han sales. So he puzzle remains." A number of possible explanaions have been pu forward o explain he slow adjusmen puzzle. One explanaion emphasized economeric problems -- eiher omied variables or problems wih he economeric procedure see Maccini and Rossana (984) and Blinder (986a). Anoher explored he effec of aggregaion bias see Chrisiano and Eichenbaum (989), Seiz (988), Blinder (986a), and Coen-Pirani (004). See Blinder and Maccini (99) and Ramey and Wes (999) for surveys.

23 To derive an invenory invesmen relaionship in our model, subrac ln N from boh sides of (6) o ge where * ln N ln N ln N ln N u () N X W * ln X ln W ln o (4) is he desired sock of (log) invenories. Comparing () o () we see ha measures he speed of adjusmen. Using he definiion of in (7-a), his speed of adjusmen erm can be wrien as 4 r r where is defined in (7-b). (5) Sraighforward calculaions reveal he relaionship beween he speed of adjusmen, he convexiy of producion coss, and he convexiy of sockou avoidance coss. To see his, differeniae (5) and use (7-b) o ge / 0, which saes ha greaer convexiy of producion coss reduces he speed of adjusmen. Similarly, / 0, which saes ha greaer convexiy of sockou avoidance coss increases he speed of adjusmen. Inuiively, greaer convexiy of producion coss increases he incenive o smooh producion, which makes he firm slow o change he level of producion. Wih respec o sockou avoidance, he inuiion is as follows. When he firm pays a cos as a resul of no having enough invenory, he firm wans o increase invenories when sales go up, so long as sales are posiively serially correlaed. The desired invenory level rises immediaely when sales increase. The sronger he sockou avoidance moive, he more rapidly he firm wans o adjus oupu. Table 4 illusraes he effec of and on he speed of adjusmen. In general, i is no possible o recover he ransiion dynamics of a variable from a coinegraing regression. Inuiively, his is because he coinegraing regression capures he long-run behavior of he variable, absracing from ransiion dynamics. Our model is an excepion, because he sickiness of invenories arises from he srucure of he model, raher han from an ad hoc adjusmen cos funcion. By deriving he coinegraing regression from he

24 model, we are able o recover he srucural parameers from he coinegraing regression coefficiens. The srucural parameers (which are repored in Table ) imply ha he speed of adjusmen This is consisen wih esimaes from he large empirical lieraure on he speed of adjusmen of invenories. (See Blinder and Maccini (99) and Ramey and Wes (999).) The slow speed of adjusmen ha flows from he model implies ha he incenive o smooh producion is quie srong. Table 4 Producion Smoohing, Sockou Avoidance, and he Speed of Adjusmen The enry in each cell is he speed of adjusmen,. The calibraed values are and Equaions () and (4) and Table 4 help o explain why I( 0) economerics has largely been a failure in he sudy of invenories. When he speed of adjusmen of invenories is slow, i akes so long for movemens in he variables ha deermine long-run desired invenories, * N, o have an effec on curren invenories, N, ha radiional I( 0) approaches, such as adding 6 or lags, ypically fail o pick up he effecs of variables like he ineres rae and inpu coss. B. Variance Raio Puzzle Proposiion 6 shows ha, when sales are I() (as hey are in he daa), he variance raio is. Why hen do empirical researchers ypically obain esimaes of he variance raio ha are greaer han? Our simulaions of he calibraed model reveal ha his occurs because of small sample bias in he variance raio, as shown in Table 5. Our empirical resuls are based on a sample of 548 monhly observaions, a relaively large number of observaions (and long ime A similar poin has been made in he lieraure on fixed capial by Caballero (994, 999) and Schaller (006).

25 span) for empirical work in macroeconomics. The simulaions show ha he variance raio, calculaed over his number of monhly observaions, is abou.0. For a sample of 500 monhly observaions, he variance raio is sill slighly greaer han, abou.0. I is only wih a sample size of 5000 observaions, roughly an order of magniude larger han he usual sample size, ha he variance raio converges o.00. In our discussion of he Wen puzzle below, we explain why his small sample bias arises. Table 5 Simulaion Evidence on Small Sample Bias in he Variance Raio Sample Size (Number of Monhly Observaions) Median Variance Raio This able is based on simulaions of he calibraed model for differen sample sizes. The second row repors he median variance raio over 000 repeiions of he simulaion. Srucural parameers are calibraed o he daa as shown in Table. C. Wen Puzzle and he Condiional Variance Raio i. Accouning for he Wen Puzzle Wen (005) disinguishes beween he movemens of producion and sales a medium horizons (abou 8-40 quarers) and shor horizons (less han hree quarers). A medium horizons, he finds ha producion is more volaile han sales. More surprisingly, he finds ha producion is less volaile han sales a shor horizons. His empirical work shows ha hese sylized facs hold for he US, a number of oher indusrialized counries (Ausralia, Ausria, Canada, Denmark, France, Finland, Grea Briain, Japan, he Neherlands, and Swizerland), Europe as a whole, and he OECD as a whole. Wen (005, p. 5) argues ha, "The sylized fac ha producion and invenories exhibi drasically differen behaviors a he high- and low-cyclical frequencies offers a limus es for [invenory] heories." 4 Define he empirical condiional variance raio as 4 Wen s argumen runs as follows. The shor-horizon behavior of oupu and sales is consisen wih producion smoohing bu no wih sockou avoidance. The medium-horizon behavior of oupu and sales is consisen wih sockou avoidance bu no wih producion smoohing. The medium-horizon behavior of oupu and sales is consisen wih increasing reurns o scale (i.e., concaviy of he producion cos funcion), bu he shor-horizon behavior is no. Finally, if cos shocks are incorporaed ino a model wih a producion-smoohing moive, cos shocks can make oupu more variable han sales, bu: ) cos shocks make oupu more variable han sales a boh shor and medium horizons; or ) when non-negaiviy consrains on invenories dominae, cos shocks have no effec on he correlaion beween invenory invesmen and sales. Wen (005) hus concludes ha none of he exising explanaions for he variance raio puzzle -- sockou avoidance, cos shocks, or increasing reurns o scale -- can simulaneously accoun for he behavior of oupu and sales a boh shor and medium horizons.

26 4 Var lny lny CVR Var ln X ln X n n. (6) To moivae he empirical CVR, noe ha ln X, condiional on ln X n, is he sum of subsequen sales shocks: n X ln n j j ln X X u. (7) Thus, he variance of log sales, condiional on ln X n, is Varln X ln X n Var ln X ln X n n X. (8) In our daa, he empirical CVR shows he paern documened by Wen (005). As shown in he firs row of Table 6, a shor horizons, CVR <; for example, for n monh, CVR = 0.6; for n monhs, CVR = 0.7. A business cycle horizons, CVR >; for example, for n 50 monhs, CVR =.0; for n 70 monhs, CVR =.0. Using Proposiion 5, we can calculae he CVR in he model as a funcion of he horizon n. The solid line in Figure shows he CVR from our calibraed model which is less han a shor horizons and greaer han a business cycle horizons. The second row of Table 6 repors he numerical magniudes, based on he calibraed model, for seleced n. The model is successful in accouning for he Wen puzzle. Table 6 Condiional Variance Raio for Seleced n Shor Horizons Business Cycle Horizons n n n 50 n 70 Daa Model The row labeled Daa repors he empirical condiional variance raio, which is defined in equaion (6). The row labeled Model repors he condiional variance raio from he model, which is calculaed from equaion () wih he srucural parameers calibraed o he daa as shown in Table. The inuiion for he model's explanaion of he Wen puzzle involves he balance beween he producion smoohing and sockou avoidance moives. The solid line in Figure shows he CVR from he model when he srucural parameers are calibraed o he daa. If he producionsmoohing moive were sronger (relaive o he sockou avoidance moive), he CVR would be lower a all horizons. The dashed line in Figure illusraes he case where is subsanially

27 5 higher han he value in he daa. 5 This makes he producion smoohing moive sronger; in fac, so much sronger ha he producion smoohing moive dominaes a all horizons and oupu varies less han sales a all finie horizons. The explanaion for he Wen puzzle herefore runs as follows. The firm engages in boh producion smoohing and sockou avoidance. A shor horizons, producion smoohing dominaes, so he variance raio is less han. A business cycle horizons, sockou avoidance dominaes, so he variance raio is greaer han. Figure Condiional Variance Raio Monhs The solid line shows he condiional variance raio (CVR) calculaed from equaion (), where he srucural parameers are calibraed using he coinegraing regression. The dashed line shows he condiional variance raio for equal o.5 imes he calibraed value. A larger value of implies greaer convexiy of he producion cos funcion and hus a sronger producion-smoohing moive. The horizonal axis shows he horizon (n) in monhs. Our analysis of he Wen puzzle provides insigh ino why empirical researchers ypically find ha he (uncondiional) variance raio is greaer han. In he model, saring a abou he 9- monh horizon, he condiional variance raio is greaer han. As n (i.e., as he horizon ges very long), he variance raio approaches. Bu, in a finie sample, empirical researchers are effecively aking he average of he condiional variance raio, which is above for mos values of n. The resul is he finie sample bias ha we documen above. 5 The relevan srucural parameers, and, ener and X in equaion () as a raio. See equaion (7) for and how i eners, and equaion (-a) for X. Thus, seing higher is equivalen o seing proporionaely lower.

28 6 ii. Reconciling he Condiional Variance Raio wih Slow Adjusmen The solid line in Figure shows he pah of sales in response o a one-ime permanen sales shock. The line wih open circles shows he response of producion, based on he calibraed model. In he long run, producion moves by he same amoun as sales. In he shor run, producion moves more han sales. This shor-run movemen of producion relaive o sales is he issue highlighed by Blinder and Maccini (99): How can we reconcile he fac ha producion varies more han sales wih he slow adjusmen of invenories, which, according o he sandard analysis, is he resul of a srong incenive o smooh producion? If he producion smoohing moive were he only force, producion would change by less, in he shor run, han he amoun of a sales shock. Why, hen, does producion vary more han sales? The answer is he balance beween producion smoohing and sockou avoidance. The line wih riangles in Figure illusraes wha would happen if he producion smoohing moive were subsanially sronger. In he long run, producion would sill rise by he same amoun as sales, bu, in he shor run, producion would rise by less han sales. Figure Response of Producion o a Sales Shock Monhs The solid line shows he pah of sales in he wake of a one-ime, one-sandard-deviaion permanen sales shock. The line wih open circles shows he response of oupu in he model when he srucural parameers are calibraed using he coinegraing regression for invenories. The line wih solid riangles shows wha he response of oupu would be if were.5 imes is calibraed value. iii. Role of he Ineres Rae In a model wih a sockou avoidance moive, he firm has a long run desired sock of invenories, * N, which is shown in equaion (4). Proposiion gives he mahemaical