Evidence on the Link between Firm-Level and Aggregate Inventory Behavior

Evidence on the Link Between Firm-Level and Aggregate Inventory Behavior Scott Schuh1 Federal Reserve Board November, 1996 120th & Constitution Ave., N.W., Mail Stop 61, Washington, D.C. 20551 (Internet: sschuh@frb.gov). This paper is a revised version of parts of my Ph.D. dissertation and was previously circulated under the title “Inventory Models and Aggregation.” I thank my advisors, Louis Maccini, John Haltiwanger, and David Bizer, as well as Charlie Bates, Steve Blough, Ricardo Caballero, Carl Christ, Steve Davis, Mark Doms, Timothy Dunne, Steve Durlauf, Robert McGuckin, Steve Olley, Ruth Runyan, and Thomas Stoker for helpful comments. I thank the U.S.Census Bureau's M3Branch for the data and Center for Economic Studies for financial, computing, and administrative support. Census employees Steve Andrews, Robert Bechtold, LisaFeldman, Louise Fry, Sam Jones, CyrLinonis, Kathy Menth, Jim Monahan, Eddie Salyers, andRobertTaylorwereespecially helpful. Ithank SteveBaldwin, Leonard Loebach, GregWon,andMary YoungattheU.S.BureauofEconomicAnalysisfordeflatorsandaggregatedata. MikeFratantoniandBrian Doyleprovidedexcellentresearchassistance. Theresearchinthispaperwasconductedwhiletheauthorwas a research associate at the Center for Economic Studies, U.S. Bureau of the Census. Research results and conclusions expressed arethose oftheauthor anddonotnecessarily indicate concurrence bytheBureauof theCensus,theCenterforEconomicStudies,ortheFederalReserveSystem.

Abstract This paper describes the finished goods inventory behavior of more than 700 U.S. manufacturing firms between 1985-93 using a new Census Bureau longitudinal data base. Three key results emerge. First, there is a broad mix of production-smoothingand production-bunchingfirms, with about two-fifths smoothingproduction. Second, firm-level inventory adjustment speeds are about an order of magnitude larger than aggregate adjustment speeds due to econometric aggregation bias. Finally,accountingfortimevariationintheinventoryadjustmentspeedduetofluctuationsin firm sizeimprovesthefit ofatraditionalaggregateinventorymodelbyone-fifth. JEL Classifications: E22,D21,C23,C43 Keywords: Inventoryinvestment,productionsmoothing,adjustmentspeed,aggregationbias

Introduction Linear-quadratic (LQ) inventory models in the spirit of Holt et al (1960) have not fared well empirically. As Blinder and Maccini (1991) point out, there exists a tension between microeconomic and macroeconomic views of inventory behavior. Microeconomic theory suggests that firms should use inventories as buffer stocks, allowingproductionto be smoothedrelativeto fluctuatingdemand. Aggregatedata,however,indicatethatproductionismorevariablethansalesand henceinventoriesappeartohaveanacceleratoreffectthatgeneratescycles. Notsurprisingly,then, econometricinventorymodelsbasedonmicroeconomictheoryandappliedtoaggregatedatahave failedmiserably,producingahostofpuzzlingresults. Most previous attempts to reconcile this tension rest on one of three explanations. One explanation is that the microeconomic theory is wrong and the model must be modified to generate more variable production. Stockout avoidance, increasing returns to scale and (S,s) ordering policies are the leading alternatives offered in the literature. Another explanation is that the standard Department of Commerce inventorydata contain measurement error that leads to faultyinference about inventory behavior. A third explanation is that econometric inference is faulty, for example with respect to the estimationof adjustment speeds and of structural (Euler equation) models. Unfortunately,however,noneoftheseexplanationshassucceeded inresolvingthetension. Afourthexplanation—aggregation—hasbeenrelativelyunexplored. However,recentresearch hints that aggregation across agents may be responsible, at least in part, for the tension between microeconomictheoryandmacroeconomicevidence. Forexample,Hunt(1981),Blinder(1986b), Seitz(1993),andLovell(1993)providesomeincompleteindicationsthataggregationacrossfirms maybiasadjustmentspeedestimatesdownward.1 Also,Krane(1994)andLai(1991)showtheoretically how aggregate data can exhibit production bunching even though firms smoothproduction. Evidencefromdisaggregatedindustrieswithphysicalunitsdata,suchasGhali(1987),Fair(1989), and Krane and Braun (1991), provides some evidence this hypothesis may be true. Each of these studies,however,has beenseverelylimitedbyalackofbroad-basedfirm-leveldata. This paper provides new evidence on firm-level inventory behavior and examines the hypothesis that aggregation effects are responsible for the poor performance of applied LQ inventory models. The evidence comes from a new longitudinal data base developed by the Census Bu- 1Adjustmentspeedsmaybebiaseddownwardforotherreasonsaswell. ChristianoandEichenbaum(1987)cite temporalaggregation,Irvine(1988)citesmisspecification,andBivin(1989)citesproductdifferentiationandmarket spillovers.Thispaperdoesnotconsidertheseinterestingreasons. 1

reau called the M3 Longitudinal Research Database (M3LRD). As the name suggests, this data base is closely related to the Census Bureau's well-known LRD except that the M3LRD includes a more limited range of data (the Manufacturers' Shipments, Inventories and Orders survey) and primarilycoversanarbitrarysampleofcompaniesratherthanaprobabilitysampleofplants.2 The M3LRD contains monthlydata on shipments (henceforth, sales), stage-of-fabrication inventories, andordersforabout4,300companiesand8,200divisionsinU.S.manufacturingduringtheperiod 1985-93. Intermsofnumberoffirms,industrialcoverage,periodicity,disaggregationofinventory types, and data reliability, the M3LRD is the most comprehensive source of microeconomic data oninventorybehavioravailableintheUnitedStates. In this first study to exploit the M3LRD, the focus is on estimating traditional LQ inventory models at the firm level and on the relationship between the firm-level and aggregate results.3 The investigation centers on a panel of more than 700 continuously operating firms, for which individual time-series—not cross-section—regression models are estimated. Since the LQ model primarily is suited for inventories of finished goods and since most empirical applications have focusedontheseinventories,thisstudyconcentratesonfinishedgoodsinventoriesaswell. Atthemicroeconomiclevel,twokeyquestionsfromtheinventoryliteratureareaddressed: (1) do firms smooth production?; and (2) do traditional LQ models fit firm-level inventorydata well? Acentralaspectofthesecondquestioniswhetherornotfirmsadjusttheirinventorystockstotheir targetlevelsatplausiblerates. Answerstothesequestionsaboutmicroeconomicbehaviorarethen comparedwithanswersobtainedfromaggregatedata. Thegoalofthiscomparisonistodetermine the extent to which aggregation may adversely affect inference about the suitabilityof traditional inventorymodels. Figure 1 summarizes the paper's two main results for all manufacturing firms. First, firms exhibit a wide array of production smoothing and nonsmoothingbehavior. The upper panel plots distributions of firm-level ratios of production variance to sales variance. Ratios of less than 1.0 traditionally are interpreted as production smooothing behavior, so about two-fifths of all firms smooth production. But the median variance ratios (1.06 for companies and 1.04 for divisions) are consistent with ratios constructed from aggregate data reported in previous studies, which 2ThenameM3originatesfromthealphanumericCensuslabelofthesurveyformsenttothecompanies. 3Additionalstudiesin progressexploittheM3LRDto examineother questionsandissues aboutmicroeconomic inventorybehavior.Schuh(1992)testsforevidenceofnonconvexityininventorybehaviorbyestimatinganadjustment hazardmodelofthetypedevelopedbyCaballeroandEngle(1993). MacciniandSchuh(1995)examinetheroleof financialmarketconditionsonstage-of-fabricationinventorybehavior. 2

041 021 001 08 06 04 02 0 Variance(Production)/Variance(Sales) 041 021 001 08 06 04 02 0 Figure 1 Summary of Main Results Firm-Level Production Variance Ratios 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 seinapmoC 003 052 002 051 001 05 0 snoisiviD Companies Divisions 004 003 002 001 0 Median Number of Months 004 003 002 001 0 Duration of Firm-Level Inventory Gaps (I - I*) 0 1 2 3 4 5 6 7 8 9 10 seinapmoC 0001 008 006 004 002 0 snoisiviD Divisions CompaniesM3LRD BEA Companies 2.06 2.47 4.00 6.50 Divisions 3

typically exceed 1.0 a bit as well. Thus, on average, a strict production smoothing inventory model is inconsistentwiththefirm-level dataon thisscore as well. Nevertheless,the distributions generally are consistent with a standard LQ inventory model that incorporates both production smoothing and stockout avoidance motives. Further, evidence presented later shows that firms smoothproductionmoreatseasonal frequencies thannonseasonalfrequencies. The second result is that firms eliminate the gap between actual and target inventory stocks much more quickly than is apparent from aggregate data. The lower panel plots distributions of the duration of firm-level inventory gaps, defined as the difference between actual ( I ) and target ( I (cid:3) )inventorystocks,wherethetargetisderivedfromatypicalstockadjustmentinventorymodel. The bars represent the median number of months that each firm experiences consecutive positive or consecutive negative gaps ( I (cid:0) I (cid:3) )—that is, how long it usually takes for firms to elimate inventorygaps. The verticallines comparemeans of thesedistributionswithmediandurationsfor the aggregate M3LRD data and for the published aggregate data from the Bureau of Economic Analysis(BEA). Inventorygapsclearly persistlongeronaverage inthetwoaggregatedataseries. The BEA data used in most previous applied inventory studies imply that the representative firm experiences inventory gaps about three times longer than the median firm. This finding seems to explainthepuzzlepointedoutbyFeldsteinandAuerbach(1976)thatinventoryadjustmentspeeds are implausiblyslow. A host of more detailed results are presented throughout the paper, but some general implications for inventory theory are worth emphasizing. First, the broad mix of production smoothing and nonsmoothingfirms argues against idea that “bad” data lead to the rejection of the basic production smoothing model. Instead, it appears that inventory theory must be able to encompass both types of behavior. Second, except for more plausible adjustment speed estimates, traditional LQinventorymodelsfare nobettereconometricallywithfirm-leveldatathanwithaggregatedata. In fact, these models only account for a small fraction of firm-level inventory variation. Third, the vast heterogeneity in variance ratios and adjustment speeds generally is not well-explainedby observable firm characteristics such as industry and size. Together, these latter two implications suggestthereremainsconsiderableroomforimprovedunderstandingofinventoryfluctuations. The paper proceeds as follows. Section 1 lays out the standard LQ inventory model and the econometricvariants ofit usedin thisstudy. Section2 introducesand describes thenewM3LRD. Section 3 presents evidence and results from the firm-level data. Section 4 presents evidence and results from the aggregate data, and compares them with the firm-level results. Section 5 4

summarizesandconcludes. 1 The Model The econometric models used in this study are derived from a simple version of the standard LQ model for finished goods inventories (see Blinder and Maccini (1991) or West (1993) for surveys of the inventory literature). The representative firm chooses the inventory level to minimize the present discountedvalueofcostssubject toaproductionidentity. Formally,themodelis: m I f i +t n j g E 0 8 < : 1 = j X 0 (cid:12) j [ C Q ( Q +t j ) + C I ( I +t j ; S +t j ) ] 9 = ; (1) s.t. I +t j = I +t j (cid:0) 1 + Q +t j (cid:0) S +t j (2) C Q ( Q +t j ) = ( (cid:14) = 2 ) Q 2 +t j (3) C I ( I +t j ; S +t j ) = ( (cid:30) = 2 ) ( I +t j (cid:0) ! S +t j ) 2 (4) given I 0 and the S t process.4 The notation is: I t is real end-of-period finished goods inventories; S t is real sales; Q t is real production; (cid:12) is the constant discount factor; (cid:1) is the first difference operator; E t = E [ : j (cid:10) t ] isthemathematicalexpectationsoperator; and (cid:10) t is thefirm's information set,whichincludesallvariablesdated t (cid:0) 1 andearlierplusanyknownat t . Structural parameters (cid:14) , (cid:30) , and ! are assumed to be positive constants. For simplicity, production adjustment costs, ( (cid:11) = 2 ) ( (cid:1) Q t ) 2 , are assumed to be zero even thoughsome studies — Blanchard (1983) and Fuhrer, Moore, and Schuh (1995) — have found marginally significant evidence for them in the data. Sales are assumed to be exogenous, as in most applied inventory research, so S t 62 (cid:10) t . The latter assumptionimpliesthat I t isabufferstockbecausethefirmcannotalteritsproductionplanduring theperiod t . The model's structural parameters determine the relative variabilityof productionand sales in the model. The sign of parameter (cid:14) determines the sign of the marginal cost of production, hence the extent to which there is a production smoothing motive due to convex costs. If ! = 0 and allotherparameters are positive,thefirm's optimalinventorydecisionrulewillleadtoproduction smoothing(i.e.,productionvariancelessthansalesvariance). But if ! > 0 ,asadvocatedbyKahn 4Constants, linear termsand white noise shocksare omitted for simplicity herebut includedas necessary in the econometricworkreportedinsections3and4. 5

(1987), the firm has a stockout avoidance motive and the decision rule may lead to production bunching (i.e., production variance greater than sales variance). Given ! > 0 , the smoothing or bunching of production depends on relative magnitudes of (cid:14) and (cid:30) (see Krane (1994) on this point).5 Themodelissolvedintheusualmanner. TheEulerequationforperiod t is E t f (cid:14) Q t + (cid:30) ( I t (cid:0) ! S t ) (cid:0) (cid:12) (cid:14) Q +t 1 g = 0 ; (5) which shows that the firm chooses I t to balance the cost of producing a unit of output today and storingitoneperiodagainstthecostofproducingtheunitofoutputnextperiod. FollowingEichenbaum (1989), substitute for production and re-express equation (5) as the second-order stochastic differenceequation E t n ( 1 (cid:0) (cid:21) L ) ( 1 (cid:0) ( (cid:21) (cid:12) ) (cid:0) 1 L ) I +t 1 o = E t f (cid:0) S +t 1 + ( (cid:20) = (cid:12) ) S t g (6) where (cid:21) and ( (cid:21) (cid:12) ) (cid:0) 1 aretherootsofthecharacteristicequation L 2 + L + (cid:12) (cid:0) 1 , = (cid:0) [ 1 + (cid:12) + ( (cid:30) = (cid:14) ) ] , and (cid:20) = 1 (cid:0) ( (cid:30) ! = (cid:14) ) . The parametric assumptions imply that (cid:20) (cid:20) 1 and stability requires that j (cid:21) j < 1 . Equation(6)iscloselyrelatedtothetraditionalstockadjustmentmodelofLovell(1961),which has beenusedinmanypreviousstudies. Thesolutiontoequation(6)isthedecisionrule I t = (cid:21) I t(cid:0) 1 (cid:0) S t + ( 1 (cid:0) (cid:20) (cid:21) ) E t S t + ( 1 (cid:0) (cid:20) (cid:21) ) 1 =i X 1 ( (cid:21) (cid:12) ) i E t S +t i (7) which is expressed in terms of actual rather than expected inventories.6 A simple version of the stockadjustmentmodelis (cid:1) I t = (cid:22) ( I (cid:3) t (cid:0) I t(cid:0) 1 ) (cid:0) (cid:25) ( S t (cid:0) E t S t ) : (8) Ifthetarget inventorystockis I (cid:3) t = (cid:23) E t S t ,thenthemodelbecomes I t = ( 1 (cid:0) (cid:22) ) I t(cid:0) 1 (cid:0) (cid:25) S t + ( (cid:22) (cid:23) + (cid:25) ) E t S t : (9) 5Thetimingofsalesinthetargetstockspecification, C I ( (cid:1)) ,variesacrossmodelsintheliteraturebetween S t and S +t 1 . Giventhecrudenatureoftheapproximationitishardtoarguestrenuouslyforonespecificationortheother. I find S t moreappropriateforapproximatingstockoutcostsinperiod t asopposedtoperiod t + 1 ,butthechoiceisnot crucialtotheeconometricresults. 6Theexpectedinventorypolicybecomestheactualinventorypolicyasfollows. Taketheexpectationoftheinventorylawofmotion,equation2: E t I t (cid:0) I (cid:0)t 1 = Q t (cid:0) E t S t ( Q t isknownatthebeginningofperiod t ). Thenrewrite thelawofmotionas (cid:1) I t = ( Q t (cid:0) E t S t ) + ( E t S t (cid:0) S t ) andsubstitute E t I t (cid:0) I (cid:0)t 1 for ( Q t (cid:0) E t S t ) inthelawof motion.Finally,substitutethesolutiontoequation(6)for E t I t intothelawofmotionandrearrange. 6

Equations7and9areobservationallyequivalentif (cid:25) = 1 andeither ! = 0 or E t S +t i = 0 8 i (cid:21) 1 . Theserestrictionsfurtherimplythat (cid:21) = ( 1 (cid:0) (cid:22) ) and ( 1 (cid:0) (cid:20) (cid:21) ) = ( (cid:22) (cid:23) + (cid:25) ) .7 A third,and related,versionof thisinventorymodel arises whenthedata are nonstationary. In theeventthattheinventoryandsalesdataareintegratedoforderone(i.e., I t (cid:24) I(1)and S t (cid:24) I(1)), andtheirlinearcombination z t = I t (cid:0) I (cid:3) t = I t (cid:0) A S t (10) is integrated of order zero (i.e., z t (cid:24) I(0)), then inventories and sales are cointegrated. Nickell (1985)showsthatthiskindofmodelcanbewrittenas anerror correctionequation: (cid:1) I t = (cid:11) (cid:0) (cid:13) z t(cid:0) 1 + L =i X I 1 (cid:18) I i (cid:1) I t(cid:0) i + L =i X S 1 (cid:18) S i (cid:1) S t(cid:0) i + (cid:17) t (11) For the model described in equations (1) through (4), L I = 0 ; if the model includes adjustment costs due to changing production, then L I > 0 . Equation (11) admits any exogenous univariate salesprocess,whichinturndetermines L S . Unlikethedecisionruleandstockadjustmentversions ofthemodel,theerrorcorrectionmodeldoesnotrequirespecificationofasalesprocesstoestimate the parameters. In this study equation (11) is used primarily as an alternative method to estimate theadjustmentspeed (cid:13) ,whichisequivalentto (cid:22) and ( 1 (cid:0) (cid:21) ) . Ideally, it would be preferable to obtain reliable firm-level estimates of (cid:14) , (cid:30) , and ! . Unfortunately, the inventory literature is filled with evidence that such estimates are not readily forthcoming. West and Wilcox (1994) and Fuhrer, Moore, and Schuh (1995) demonstrate that traditional generalized methodof moments(GMM)estimationof such parameters from equations like (5) tends to produce biased and insignificant estimates. Gregory, Pagan, and Smith (1993) show that it is possible to identify the structural parameters (except, perhaps, (cid:12) ). However, identification requires a parametric specification of sales, which must be strictly exogenous, and definitive knowledgeoftheorderofintegrationofsales(specifically,I(0)versusI(1)). Fuhreretalarguethat maximum likelihoodestimationof inventorydecision rules is a promisingapproach to estimating thestructuralparameters,buttherequisitecomputerequipmenttoconductsuchestimationwasnot availableat theCensusBureau at thetimeofthisstudy. Instead,theeconometricestimationfocusesonusingthethreemodelstoestimatetheinventory speedofadjustment(i.e., ( 1 (cid:0) (cid:21) ) , (cid:22) ,and (cid:13) )andaddressingthewell-knownpuzzleintheinventory 7In many econometricapplications, the inventorystock adjustmentmodelincludesthe real interest rate (cost of capital)asadeterminantof I (cid:3) t . Consequently,therealrateisincludedintheeconometricestimationreportedlaterin thispaper. 7

literature that adjustment speed estimates are implausibly small. Typical adjustment speed estimates usingmonthlyaggregatedata are around 0.1,implyingthat the representativefirm can take more than a year to adjust its actual inventory stock to the target level. Feldstein and Auerbach (1976)argued that sincethe entiremanufacturinginventorystockamounts toless than onemonth of aggregateproduction,firms shouldbe able toadjust theirstocks muchfaster than is impliedby estimatedadjustmentspeeds. MacciniandRossana(1984),Blinder(1986b),NguyenandAndrews (1988),andEichenbaum(1989)haveattemptedtoresolvethepuzzlewithlimitedornosuccess.8 2 The M3 Longitudinal Research Database 2.1 Composition and Structure The M3 Longitudinal Research Database (M3LRD) is a new panel data base containingmonthly, seasonally unadjusted data on the domestic shipments, inventories, and orders of U.S. manufacturing firms (manufacturing operations only). In 1985, the Census Bureau began building the M3LRD by linking the firm-level data from the Manufacturers' Shipments, Inventories, and Orders (M3) survey. The main economic variables in the M3LRD are the dollar values of sales, stage-of-fabrication inventory stocks (finished goods, work-in-process, and materials and supplies on a current-cost basis) and new and unfilled orders. This study primarily uses two variables, the value of sales ( V S ), and finished goods inventories ( F G ), plus the production identity ( Q = V S + (cid:1) F G ). See the data appendix for a complete list of variables. Data are available throughAugust1993,andupdateddatabecomeavailableperiodically. Two types of firm characteristics are available in the M3LRD: industry and geography. Industrial composition is determined by an industry category variable, which is based on a special Census classification system that includes about 80 categories of combined four-digit SIC manufacturingindustries.9 Censusassignsafirmtotheindustrycategoryinwhichitmakestheplurality of its sales. The M3LRDincludes firms from all 20 two-digitSIC manufacturingindustries. Firm nameandaddress are alsoincludedintheM3LRD,buttheaddress isonlyforfirms' headquarters andthusdoesnotprovideanygeographicinformationabout itsproductionsites(plants). The primary reporting unit in the M3LRD is the company (enterprise), which may include 8Eichenbaum(1989)getsthehighestadjustmentspeedestimates,usingunobservableautoregressivecostshocks. However,section3notesthattherearesomedoubtsabouttherobustnessofhisparticulareconometrictechnique. 9SeeAppendixBofBureauoftheCensus(1987)fordetailsabouttheindustrycategories. 8

one or more plants (establishments). 10 In contrast, the primary reporting unit in the LRD is the plant. Thus,althoughtheM3LRDandLRDaredrawnfrom thesamemanufacturinguniverse,the M3LRD is based on more aggregated reporting units and—due to different sampling schemes— includes a different mix of plants. However, Census obtains disaggregated reports for most large, diversified companies to improve data precision. Typically, the disaggregated report corresponds toadivisionwithinthecompanyoperatinginrelativelymorehomogenousindustrialareasthanthe companyasawhole. Butthecompaniesthemselvesdeterminetheexactcorporateentityreporting dataandtheentitycanvaryacrosscompanies. Fortheperiod1985-93,theM3LRDcontainsabout 8,200 disaggregated reporting units (henceforward referred to as divisions) belonging to about 4,300 companies. To maximize the degree of disaggregation, this study exploits both companylevel and division-level data; for ease of exposition, both are referred to as the “firm” and the distinctionismadeonlywhennecessary forclarity.11 2.2 Sample Selection The M3 survey is the flagship of the Census Bureau's Current Industrial Reports program and a supplement to the Bureau's Census of Manufacturers (CM) and Annual Survey of Manufacturers (ASM) surveys.12 The M3 survey includes “most manufacturing companies with $500 million or more in annual shipments [sales]. Selected smaller companies are included to strengthen the sample coverage in individual industry categories” (Bureau of the Census (1992), p. VII). Thus, unlike the CM and ASM, the M3 survey is not a probability sample and therefore cannot produce unbiased universe estimates.13 Furthermore, firms may not appear continuously in the M3 survey because it is voluntary and the arbitrary sampling strategy for smaller firms changes over time. Consequently, the M3LRD does not permit accurate identification of firm entry and exit or corporateorganizationaldynamics(e.g.,mergers,acquisitions,etc.). Nevertheless,theM3LRDoffersanopportunitytoanalyzetheinventorybehaviorofhundreds 10Acompany,orenterprise,isdefinedasfollows:“Fordefinitionsofplants(establishments)andcompanies(enterprises),seeBureauoftheCensus(1979,p. 12). 11Thisusageassumesthatdivisionsareindependentcostminimizers,whichmaybeincorrect.Aggregatingdivisionlevel inventory decision rules may not—perhaps likely will not—produce the company-leveldecision rule if there are important economic interactions among divisions within the company. These interactions are interesting and potentiallyimportant,butpostponedforfuturestudy. 12FormorecompletedetailsontheM3surveyanddata,seeBureauoftheCensus(1992). 13Instead,Censususesthelink-relative(LR)growth-ratemethodtoobtaintotalmanufacturingdatathatcorrespond toCMandASMestimates. Seethedataappendixfordetails. 9

of manufacturingcompanies, and thousands of divisions,over nearly a decade-long period. From the pool of all M3 firms, this study concentrates on an M3LRD aggregation panel of 734 companies and 2,332 divisions.14 Because the goal is to estimate firm-specific time-series econometric models, the aim in selecting the aggregation panel was to obtain a subset of M3 firms containing thehighestqualitydataandcoveringthelongestpossiblesample. To be selected for the M3LRD aggregation panel, a company or division was required to satisfy certain criteria. The criteria included: (1) a sufficiently long sample period for all data; (2) a high percentage of reported, rather than imputedor missing,data observations; (3) a low percentage of measurement error detected from visual inspection, violation of data identities, and other miscellaneous editing procedures; and (4) limitations on outliers. In addition to the sample selection procedures, some firms' data required relativelyminoreditingfor obviousand correctable measurementerror. See thedataappendixforfurtherdetails. Obviously, the arbitrary sample selection procedure may reduce the representativeness of the M3LRD, though it could cause it to be more representative. It turns out that the only significant differencebetweenthetotalM3LRDandtheaggregationpanelisthat theaveragefirm islargerin theaggregationpanel(measuredbysales). Thereasonisthat largerfirms tendtoreportdatamore oftenandmoreaccurately,inlargepart becausetheCensusBureaumakesamoreconcertedeffort toobtainaccurate datafrom thesefirms. 2.3 Data Adjustments Several adjustments were made to the data to make them as consistent as possible with the published aggregate M3 data. The firm-level sales and inventory data are deflated with industry-level deflators. Sales are adjusted for monthly variation in trading days. No adjustment was necessary for LIFO compositionof inventories,but someinventorydata were adjusted for a shift inthe Census inventory reporting procedure in 1987. Sales and inventories were detrended and deseasonalized in accordance with the tradition of the literature.15 See the data appendix for further 14Tobeclear,the2,332divisionsdonotallbelongtooneofthe734companiesnor,conversely,arealldivisionsof eachofthe734companiesincludedinthe2,332divisions. Eachcompanyanddivisionenterstheaggregationpanel bysatisfyingthecriteriadescribednext. 15Themainqualitativeresultsinthepaperarethesameifthedataarefirst-differenceddataratherthandetrended. Theerror-correctionmodelprovidesevidencefromanalternativewayofdealingwithtrends. However,theissueof integrationatthefirmlevelhasnotbeenthoroughlyexaminedyetintheliterature,eithertheoreticallyorempirically, andthustheappropriatetreatmentisambiguous. ForthefirmsintheM3aggregationpanel,manydataseriesexhibit 10

detailsaboutdataadjustments. 2.4 General Characteristics and Statistics Table 1 provides information about the industrial composition, size, average inventory holdings, and corporate organization of the firms in the M3 aggregation panel. The panel is almost evenly split between nondurable goods and durable goods firms in terms of number of companies and manufacturingshares,thoughdisaggregationintodivisionsismoreprevalentamongdurablegoods firms (3.2 divisionsper companyversus 2.1).16 Finishedgoods ( F G ) inventoriesare more importantfornondurablegoodsfirms,wheretheygrewatanearly4percentannualrate,thanfordurable goodsfirms,wheregrowthwasflat. More relevant for inventory theory is the industrial disaggregation by production technology. Aboutone-quarterofthefirmsinthepaneloperateinproduction-to-stock(PS)industries,i.e.,they do not backlog orders. The traditional LQ inventory model is best suited to these kinds of firms. A key advantage of the M3LRD is that it permits disaggregation of the remaining productionto-order (PO) firms into those that report unfilled orders (PO-Y) and those that do not (PO-N). The latter firms, which account for almost three-fifths of the PO firms, may behave more like PS firms despite operating in PO industries.17 Like nondurable goods firms, PS firms tend to rely relatively more on F G in terms of growth rates, shares, and inventory-to-sales ratios. PS firms alsosawsalesgrowabout50percentfasterthanPOfirms. ParticularlynotablearethePO-Nfirms which,althoughaccountingforonly6percentofallsales,hadthehighestsalesgrowthrateofany industrialcategory. Despite the sampling bias toward larger firms, the panel nevertheless includes a significant number of smaller firms. About 60 percent of all firms have fewer than $250 million annual average sales, a common cutoff for the definition of small in the literature on financial market effects on real activity. About 15 percent are just plain small, with average sales of less than $25 positivetrends,manynegative,andmanynotrendatall. Inlightofthisambiguity,Ielectedtofollowthetraditional detrendingprocedureappliedtotheaggregatedata. 16Althoughitispossibletoexaminemoredisaggregatedindustries,suchastwo-digitindustries,therearenotenough firmsinallindustriestoproduceconsistent,reliableresults. Schuh(1992)containstheresultsbytwo-digitindustry fortheinterestedreader. 17PO-Nfirmsareidentifiedbecausetheydidnotreportunfilledorders. Althoughtheyareassumedto produceto stock,itispossiblethattheyholdunfilledordersbutdonotreportthem. 11

c P C l a I n d T A n L $ $ $ M M o 0 0 0 0 M N u O M R e C D a C r o s s b C c T O - N d S e S f S g S T a b l e M 3 L R D F i r m C l a s s i (cid:12) c a t i o n s N u m b e r C o m p s s a n i e s c u s t r y o t a l m a n u f a c t u r i n g ( T M ) 7 3 4 d N o n d u r a b l e g o o d s ( N ) 3 3 5 e D u r a b l e g o o d s ( D ) 3 9 9 f P r o d u c t i o n - t o - s t o c k ( P S ) 2 0 6 g P r o d u c t i o n - t o - o r d e r ( P O ) 5 2 8 W i t h o u t b a c k o r d e r s ( P O - N ) 3 0 6 W i t h b a c k o r d e r s ( P O - Y ) 2 2 2 n u a l a v e r a g e V S s i z e , 1 9 8 6 - 8 7 ( $ 1 9 8 7 e s s t h a n $ 2 5 m i l l i o n 1 1 1 2 5 - 1 0 0 m i l l i o n 1 9 1 1 0 0 - 2 5 0 m i l l i o n 1 3 5 2 5 0 - 1 ,0 0 0 m i l l i o n 1 5 9 o r e t h a n $ 1 b i l l i o n 1 3 9 n t h l y a v e r a g e i n v e n t o r y - t o - s h i p m e n t .0 0 - 0 .2 5 1 0 1 .2 5 - 0 .5 0 1 1 1 .5 0 - 0 .7 5 1 2 2 .7 5 - 1 .2 5 1 7 6 o r e t h a n 1 .2 5 1 6 8 m b e r o f e s t a b l i s h m e n t s i n t h e c o m p a n e 5 0 o r e t h a n o n e 6 8 4 p o r t i n g u n i t t y p e o m p a n y 3 6 7 i v i s i o n 3 6 7 o m p a n y c o u n t s fo r s u b c la s s e s w ill n o t s u m t o o v e r c la s s e s . o m p o n e n t s h a r e s m a y n o t s u m t o 1 0 0 d u e t o h e in d u s t r ia l a g g r e g a t io n id e n t it ie s a r e : ( 1 ) T . I C in d u s t r ie s 2 0 - 2 3 , 2 6 - 3 1 . I C in d u s t r ie s 2 4 - 2 5 , 3 2 - 3 9 . I C in d u s t r ie s 2 0 - 2 1 , 2 3 , 2 8 - 3 0 . I C in d u s t r ie s 2 2 , 2 4 - 2 7 , 3 1 - 3 9 . 1 a n d S u m m a a o f : D i v - T M s h a i s i o n s V S 2 3 3 2 1 0 0 7 1 1 5 3 1 2 5 4 4 7 4 7 3 4 2 1 4 9 2 5 8 3 4 9 6 1 1 4 3 5 3 ) 4 9 8 0 6 8 9 2 4 3 8 4 3 9 5 1 3 1 5 8 8 1 s r a t i o ( F G / V 3 0 6 1 2 3 2 6 2 1 2 6 9 2 2 4 5 1 2 9 6 5 2 1 4 n y 6 5 0 2 2 6 7 1 0 0 3 6 7 1 2 1 9 6 5 8 8 t h e t o t a l if r e p o r o u n d in g e r r o r s . M = N + D = P r y S T i m r e ( % F G 1 S ) 1 r t in g S + t e ) 0 5 4 4 5 5 1 7 1 1 3 3 0 1 8 u P a S b 0 4 6 2 8 6 2 0 2 4 5 9 2 0 9 8 2 0 0 2 8 n O t i s t i c s e r i e s M e a n G r o w t h V S .2 0 .1 8 .2 4 .2 6 .1 7 .3 9 .1 4 .0 6 .1 1 .2 5 .2 0 .2 0 .2 1 0 .3 3 .2 6 .2 4 .2 9 .2 0 .2 6 .2 0 it s w it h in c ; ( 2 ) P O = ( 1 9 R a t F G o m p P O 8 6 - 9 e ( % ) .1 7 .3 1 0 .3 4 0 .1 0 .0 3 .1 1 .1 3 .1 8 .2 1 .1 6 (cid:0) .4 4 .0 4 .1 5 .2 1 .2 3 .1 1 .1 7 .2 1 .1 6 a n ie s - Y + 2 ) F L G e v / 1 e V .7 .7 .7 .7 .7 .7 .7 .7 .8 .8 .8 .7 .1 .3 .6 .9 .6 .5 .7 .7 .7 l S 5 6 3 5 4 4 4 9 1 6 3 3 2 6 4 7 6 7 5 5 5 12

million. Still, the vast bulk of sales and inventories are in the largest firms.18 Unless meaningful differences in inventory behavior due to size occur at some size much smaller than $25 million, theM3LRDshouldbecapableofidentifyingtheimpactofsize. Middle-sizedfirmsarethefastest growing, in terms of both sales and inventories, and have the highest inventory-to-sales ratios. Interestingly,the smallest firms have the lowest sales growthrate. For the remainder of the paper, firms areassignedtosmall(averageannualsales lessthan$250million)andlargesizeclasses.19 Firm-level average inventory-to-sales ratios exhibit considerable heterogeneity, running from zero to more than 1.25. A majority of of firms hold less than one-month's supply of inventories, as measured by the average monthly inventory-to-sales ratio. Interestingly, 56 companies (7.6 percent)and270divisions(11.6percent)reportholdingno F G inventoriesatall. Firmsinthetwo lowestratioclassestendtohaverelativelylowsalesandinventorygrowthrates. Inventory-to-sales ratios tend to be inversely related to firm size (i.e., sales share per firm falls monotonically with ratioclass). The vast majority of firms in the data base own multiple establishments (plants)—well over 90 percent of companies and divisions. In terms of reporting units, half of the companies report company-level data and half report data by division. Companies reporting division-level data account for 84 percent of all divisions in the data base, and the average company reports data for 5.4 divisions. Single-establishment firms and company reporting unit firms account for very little of total sales and inventories. Because the traditional LQ inventory models do not capture these aspects ofcorporateorganization,nofurtheranalysisisconductedalongthesedimensions. 2.5 Comparison of M3LRD and BEA Aggregate data InlightofsampleselectionconcernsabouttheM3surveyingeneral,andtheM3LRDaggregation panelinparticular,itisimportanttoascertainhowcloselytheaggregateM3LRDaggregationpanel datacorrespondtothepublishedaggregateBEAdata.20 Figure2demonstratesthatthetimeseries 18Ofcourse,thepanel'ssizedistributionisfarfromrepresentativeofthesizedistributionoffirmsintheuniverse. Thepointhereis thatthe biastowardlargerfirmsdoesnotprecludeasignificantnumberofsufficientlysmallfirms fromenteringthepanel. 19Foramoredetailedinvestigationoftheimpactofsizeoninventorybehavior,particularlywithrespecttofinancial marketconditions,seeMacciniandSchuh(1995). 20FollowingReaganandSheehan(1985),theconstantdollar($1987)BEAdatawerereseasonalizedusingtheratio of seasonally adjusted to unadjusted nominal data then deseasonalized in a manner analogousto the M3LRD data. TheBEAdataarebasedonthenominalM3datapublishedbyCensus,andthustheBEAdataincorporate(potentially 13

selaS AEB DRL3M 042 25 DRL3M AEB 032 05 022 84 012 64 44 002 3991 2991 1991 0991 9891 8891 7891 6891 0099..00 == nnooiittaalleerrrrooCC 7891$ snoilliB 7891$ snoilliB noitcudorP AEB DRL3M 042 25 032 05 022 84 012 64 44 002 3991 2991 1991 0991 9891 8891 7891 6891 9988..00 == nnooiittaalleerrrrooCC 7891$ snoilliB 7891$ snoilliB htworG yrotnevnI GF )etar launna( AEB DRL3M 08 08 06 06 04 04 02 02 0 0 02- 02- 04- 04- 3991 2991 1991 0991 9891 8891 7891 6891 5566..00 == nnooiittaalleerrrrooCC 7891$ snoilliB 7891$ snoilliB oitar selas-ot-yrotnevnI GF AEB DRL3M 86.0 67.0 66.0 47.0 46.0 27.0 26.0 07.0 06.0 86.0 85.0 66.0 65.0 46.0 3991 2991 1991 0991 9891 8891 7891 6891 2277..00 == nnooiittaalleerrrrooCC 7891$ snoilliB 7891$ snoilliB 2 erugiF )detsujdA yllanosaeS( ataD etagerggA gnirutcafunaM latoT 14

properties of the M3LRD aggregation panel data (henceforth, simply “M3LRD”) and BEA data are reassuringlysimilar. Althoughthe M3LRD data onlyaccount for slightlyless thanone-fourth of the BEA data, seasonally adjusted data from the two aggregates are highly correlated (.90 for salesand.89forproduction).21 Thecorrelationisevenstrongerforseasonallyunadjusteddata(not shown). Correspondence between inventory series is less robust, but correlations of growth rates (.68) and inventory-to-sales ratios (.72) still significantly positive. Note that the average M3LRD inventory-to-salesratioisslightlyhigherthantheBEAratio(.68versus.60). In sum,theM3LRD aggregatedataappeartobeareasonablyrepresentativesubsampleoftheBEAdata. 2.6 M3LRD Versus Other Disaggregated Inventory Data To close this first glance at the M3LRD, it is instructive to compare the M3LRD with other disaggregated data bases containing inventory data. The comparison illustrates the advantages and disadvantages of M3LRD and, in summarizing results from other data bases, provides a useful benchmarkforevaluatingtheresultsinthispaper. Table2listsanddescribestheprimarydisaggregated inventorydata bases cited in the literature. Two general impressions emerge from the table. First, no singledata base exhibitsall of the most desirable characteristics. Second, the M3LRD is as close as any data base—and perhaps is the closest—toexhibitingthe most advantageous set of characteristics.22 AdvantagesoftheM3LRDaredisaggregation(offirmandinventorytype)andfrequency. Only the LRD provides more disaggregated data, but its data are only available annually, which is too low frequency given the extremely short-run nature of inventory decisions. Only three data bases (M3LRD, LRD, and Compustat) offer complete stage-of-fabrication disaggregation, and of these adverse)effectsoflink-relativeestimationofaggregatedata,whichpermitsfluctuationsinsamplecomposition,and dataimputedbyCensusformissingobservations. 21Aggregate V S for all firms in the M3LRD equals roughly half of total manufacturing V S reported by BEA. Thus,thesampleselectionrestrictionsreducecoveragebymorethan50percent.Nevertheless,theaggregateM3LRD dataarebroadlyconsistentwiththeBEAdata. Thepercentageoftotalmanufacturing V S coverageinthecomplete M3LRD varies from 20 to 90 percent across two-digit SIC industries. Not surprisingly, the correlations between M3LRDandBEAdataaggregatesatmoredisaggregatedlevels,suchastwo-digitindustries,canbemuchlower. 22TheM3LRDisasuccessortothequarterlyDepartmentofCommerceManufactures' InventoryandSalesExpectationsSurveyused by Hirsch and Lovell(1969)to estimate reduced-forminventoryequationsat the industrylevel andforan83-firmsubset.Someofthesedatawerequalitativeratherthanquantitative.Thestudyconcludedthatthere islittleevidenceforproductionsmoothing,butshowedthatsmallerfirmstendtohavehigheradjustmentspeedsand moreflexibleproductionplans. 15

C Q G A g d T a b l e 2 C o m p a r i s o n o f D i s a g g r e g a t e d I n v e n t o r y D a t a B a s e s U n i t o f F r e q a b D a t a B a s e A n a l y s i s C o v e r a g e u e n c y S a m p l e L R D P l a n t P r o b a b i l i t y s a m p l e o f a b o u t A 1 9 7 2 - 9 0 5 0 ,0 0 0 - 7 0 ,0 0 0 m a n u f a c t u r i n g (cid:12) r m s M 3 L R D D i v i s i o n A r b i t r a r y s a m p l e o f a b o u t M 1 9 8 5 - 9 3 1 ,7 0 0 m a n u f a c t u r i n g (cid:12) r m s W a r d ’s D i v i s i o n D o m e s t i c a n d t r a n s p l a n t e d M 1 9 3 8 - 9 5 f o r e i g n a u t o m a k e r s e C o m p u s t a t C o m p a n y A l l p u b l i c a l l y t r a d e d (cid:12) r m s A 1 9 5 8 - 9 4 C o m p u s t a t C o m p a n y A l l p u b l i c a l l y t r a d e d (cid:12) r m s Q 1 9 5 8 - 9 4 Q F R C o m p a n y P r o b a b i l i t y s a m p l e o f a b o u t Q 1 9 7 7 - 9 1 3 0 ,0 0 0 m a n u f a c t u r i n g , m i n i n g , a n d t r a d e (cid:12) r m s . I F O C o m p a n y A r b i t r a r y s a m p l e o f 4 ,0 0 0 M 1 9 7 5 - 8 6 m a n u f a c t u r i n g (cid:12) r m s C e m e n t D i s t r i c t A r b i t r a r y s a m p l e o f 1 9 P o r t - M 1 9 5 0 - 6 0 l a n d c e m e n t i n d u s t r y p r o d u c t i o n d i s t r i c t s w i t h c o n t i n u o u s d a t a S C B I n d u s t r y M i s c e l l a n e o u s M v a r i e s s u b s e t s o f 4 - d i g i t S I C m a n u f a c t u r i n g i n d u s t r i e s F a i r I n d u s t r y S e v e n 3 - a n d 4 - d i g i t S I C m a n - M v a r i e s u f a c t u r i n g i n d u s t r i e s a L R D = L o n g it u d in a l R e s e a r c h D a t a b a s e ( C e n s u s B u r e a u ) ; W a r d ’s = W a r d ’s A u t o m o m p u s t a t = S t a n d a r d & P o o r ’s C o m p u s t a t S e r v ic e s , I n c . ( S e c u r it ie s a n d E x c h a n F R = Q u a r t e r ly F in a n c ia l R e p o r t ( C e n s u s B u r e a u ) ; I F O = I F O I n s t it u t e f (cid:127)u r W ir t s c h a ft s fo e r m a n y ) ; C e m e n t = M o h e b G h a li ( 1 9 8 7 ) d a t a b a s e ; S C B = S u r v e y o f C u r r e n t B u s in e s s ( B u n a ly s is ) ; F a ir = R a y F a ir ( 1 9 8 9 ) d a t a b a s e . b A = a n n u a l; Q = q u a r t e r ly ; M = m o n t h ly . c F = (cid:12) n is h e d g o o d s ; S O F = a ll s t a g e - o f- fa b r ic a t io n ( m a t e r ia ls a n d s u p p lie s , w o r k - in - p r o o o d s ) ; T = t o t a l ( s u m o f a ll S O F s t o c k s ) . d $ = d o lla r v a lu e ( c u r r e n t - c o s t ) ; P = p h y s ic a l q u a n t it ie s ; C = c a t e g o r ic a l ( q u a lit a t iv e ) r e s p e A c c o r d in g t o H u n t ( 1 9 8 1 ) , t h e S O F d a t a a r e o n ly a v a ila b le b e g in n in g in 1 9 6 9 t h o u g h t a t a g o b a c k t o 1 9 5 8 . S t o c k D a t a c d T y p e T y p e S O F $ S O F $ F P S O F $ T $ T $ F C F P F P F P o t iv e Y e a r b o o k ; g e C o m m is s io n ) ; r s c h u n g ( M u n ic h , r e a u o f E c o n o m ic c e s s , a n d (cid:12) n is h e d o n s e s . h e t o t a l in v e n t o r y 16

onlyM3LRDoffers itat high(monthly)frequency. Wards,theonlyotherdivision-leveldatabase, ishighlyspecialized(autoindustry)andonlycontainsfinishedgoodsinventories.23 Potential disadvantages of the M3LRD are scope, sample, and data type. Unlikethe LRD and QFR, the M3LRD is not based on a probability sample and, unlike the QFR and Compustat, the M3LRD covers only manufacturing. Still, the M3LRD coverage within manufacturing is broad and it includes some very small nonpublic firms that Compustat does not. Although approaching a decade, the M3LRD's sample period is on the short end. Nevertheless, new data continue to become available periodically. Finally,if critics such as Foss, Fromm, and Rottenberg (1980) and Miron and Zeldes (1989) are correct, the M3LRD's dollar-value data are subject to serious measurement error relativeto physical units data.24 But evidence reported later in this paper indicates that the dollar-value data may not be so bad after all, particularly for the purpose of quantifying theeffectsofaggregationwithinadatatype. Furthermore,theM3LRDdataislesssubjectivethan thecategorical-responsedata,whichmeasuretheopinionsof firms' managers about,forexample, whetherinventoriesare toolow,toohigh,orjustright. Two general findings in the literature pertain to the alternative disaggregated inventory data bases. First,studiesusingphysicalunitsdatatendtofindmoreevidencethatfirmssmoothproduction. Ghali (1987), Harris (1988), Fair (1989), and Krane and Braun (1991) report considerably lower ratios of production variance to sales variance than do studies using the Commerce data— i.e., more evidence of production smoothing. This result could, however, be misleading if the narrow industries under study are more likely than others to smooth production or if aggregation across firms tendstobias thevarianceratiosfortheseindustriesdownward. Asecondgeneral resultisthatdisaggregatedinventorydatatendtoproducehigheradjustment speed estimates. Using BEA data, Blinder (1986b) observed that estimates for nondurable and durable goods manufacturing aggregates are generally lower than for two-digit industries within the aggregate groupings. Using categorical response data, Seitz (1993) found that the aggregate adjustment speed estimate is biased down from two-thirds to one-fourth after aggregating across 23Tobemoreaccurate,theWard'sdataareactuallyavailableatthemoredisaggregatedlevelofvehiclemake(e.g., Taurus,Accord,etc.). However,product-linedisaggregationinamulti-productenvironmentdoesnotconformwellto theconceptofafirm,whereacentralizedeconomicagentisassumedtobemakingdecisionsaboutallchoicevariables. Blanchard(1983)useddivision-levelWard'sdata. 24Toinvestigatethisissuemorecarefully,Icomparedthedollar-valueandphysical-unitsdataforthemotorvehicle assemblyindustry.M3LRDproduction(salesplusfinishedgoodsinventoryinvestment)isveryhighlycorrelatedwith dataonfinalproductionofassembledunitsfromtheMotorVehicleManufacturers' Association,bothatthefirm-and industry-level.Confidentialitylawsprohibitmorespecificassessmentsandevidenceonthiscorrelation. 17

more than 500 German manufacturing firms. Using Compustat data, Hunt (1981) found scattered evidence in a few industries that aggregation biases adjustment speed estimates downward, but Haltiwanger and Robinson (1987) obtained extremely low estimates from pooled time-series cross-section estimation. And Harris (1988), using industry-level physical units data, did not find anysystematicbias. However,noneofthesestudiesarecapableofproducingthebreadthofquantitativeevidenceavailablefrom theM3LRD. 3 Firm-level Evidence and Results This section provides evidence on firm-level production variance ratios and regression estimates of linear-quadratic inventory models for the M3LRD aggregation panel. All results include firms that reported holding no finished goods inventories. Henceforth, notation for inventories, sales, andproductionwillbeas follows: I = F G , S = V S ,and Q = V S + (cid:1) F G = S + (cid:1) I . 3.1 Variance Ratios Aggregate BEA manufacturing data, including total manufacturing and two-digit SIC industrylevel data, suggest that firms tend to bunch production. That is, for BEA data the variance of production tends to exceed the variance of sales, expressed as V a r ( Q ) = V a r ( S ) > 1 (see Blinder and Maccini (1991), and references therein, for evidence). West's (1986) variance bounds test with BEA data also strongly rejects the hypothesis of production smoothing. On the other hand, physical units data in more disaggregated industries tend to exhibit production smoothing, with variance ratios often less than 0.9 (see Ghali (1987), Krane and Braun (1991), and Fair (1989)). There is little or no evidence, however, on firm-level variance ratios measured with data of any kind. Akeycontributionofthisstudy,then,istoprovidesomeevidenceontheextentofproduction smoothingatthefirm-level. Figure 3 plots firm-level variance ratio distributions for all companies and divisions in the M3LRDaggregationpanel.25 Thetailcellsineachdistributionincludeallratiosgreaterinabsolute 25Resultsarepresentedonlyfor V a r ( Q ) = V a r ( S ) andnotfortheratiowiththevalueaddeddefinitionofproduction, V a r ( Q S U M ) = V a r ( S ) ,where Q S U M = Q + (cid:1) W P and W P isthework-in-processinventorystock.Qualitatively, thedistributionofvarianceratioswith Q S U M isthesameaswith Q ,althoughthemeanandvariancearehigherwith Q S U M .Work-in-processinventoriesarelesspreciselymeasured,ingeneral,andmeasurementerrortendstoincrease thevarianceratios. Tomyknowledge,therearenotheoreticalmodelsthatpredictwhetherthevarianceratiosshould 18

051 001 05 0 latoT 051 001 05 0 3 erugiF soitaR ecnairaV noitcudorP leveL-mriF )noitaived dradnats = DS ,naidem = M( 60.1 = MC )C( seinapmoC 46. = DSC )D( snoisiviD 40.1 = MD 71.1 = DSD 2 8.1 6.1 4.1 2.1 1 8.0 6.0 4.0 2.0 0 005 004 003 002 001 0 snoisiviD seinapmoC 051 001 05 0 lanosaeS 051 001 05 0 99. = MC 87. = DSC 10.1 = MD 82.1 = DSD 2 8.1 6.1 4.1 2.1 1 8.0 6.0 4.0 2.0 0 005 004 003 002 001 0 snoisiviD seinapmoC 051 001 05 0 lanosaesnoN 051 001 05 0 51.1 = MC 19. = DSC 70.1 = MD 65.1 = DSD 2 8.1 6.1 4.1 2.1 1 8.0 6.0 4.0 2.0 0 005 004 003 002 001 0 snoisiviD seinapmoC 051 001 05 0 naideM ssalC sunim latoT 00. = MC 36. = DSC 00. = MD 41.1 = DSD 1 8.0 6.0 4.0 2.0 0 2.0- 4.0- 6.0- 8.0- 1- 005 004 003 002 001 0 snoisiviD seinapmoC 051 001 05 0 naideM ssalC sunim lanosaeS 00. = MC 87. = DSC 00. = MD 72.1 = DSD 1 8.0 6.0 4.0 2.0 0 2.0- 4.0- 6.0- 8.0- 1- 005 004 003 002 001 0 snoisiviD seinapmoC 051 001 05 0 naideM ssalC sunim lanosaesnoN 00. = MC 98. = DSC 00. = MD 35.1 = DSD 1 8.0 6.0 4.0 2.0 0 2.0- 4.0- 6.0- 8.0- 1- 005 004 003 002 001 0 snoisiviD seinapmoC 051 001 05 0 etagerggA ssalC sunim latoT 10.- = MC 57. = DSC 31.- = MD 03.1 = DSD 1 8.0 6.0 4.0 2.0 0 2.0- 4.0- 6.0- 8.0- 1- 005 004 003 002 001 0 snoisiviD seinapmoC 051 001 05 0 etagerggA ssalC sunim lanosaeS 80. = MC 28. = DSC 00. = MD 23.1 = DSD 1 8.0 6.0 4.0 2.0 0 2.0- 4.0- 6.0- 8.0- 1- 005 004 003 002 001 0 snoisiviD seinapmoC 051 001 05 0 etagerggA ssalC sunim lanosaesnoN 90.- = MC 40.1 = DSC 02.- = MD 97.1 = DSD 1 8.0 6.0 4.0 2.0 0 2.0- 4.0- 6.0- 8.0- 1- 005 004 003 002 001 0 snoisiviD seinapmoC 19

value than the endpoint values. The figure contains three columns: the total ratio (first column), andadecompositionintoseasonal(secondcolumn)andnonseasonal(thirdcolumn)components.26 The figure reveals a broad mix of production smoothing and bunching behavior at the firm level. Abouttwo-fifthsofallfirmssmoothproduction(ratio < 1)intotal(upperleftpanel),though the majority do not and the central tendency is for mild production bunching. The distribution is about the same in production-to-stock industries as in production-to-order industries, as well as in more disaggregated industries. The diversity of variance ratios also supports the basic LQ inventorymodelofsection1,whichallowsforavarietyofsmoothingandnonsmoothingbehavior byincorporatingastockoutavoidancemotive(model parameter ! > 0 ).27 Viewing across the top row, one can see that there is more evidence of production smoothing at seasonal than nonseasonal frequencies for companies, though less so for divisions. Whereas only about two-fifths of all companies smooth production in total, about half smooth production at seasonal frequencies. Conversely, only about one-fourth smooth production at nonseasonal frequencies. For divisions, the fraction of smoothing ratios is about 45 percent for both total and seasonalratios,andabout40percentfornonseasonalratios. Withregardtocompanies,thisfinding supportsthehypothesesandconclusionsofMironandZeldes(1988)andKraneandBraun(1991) using more aggregate data. In particular, to the extent that seasonal fluctuations are deterministic whilecyclicalfluctationsarestochastic,rationalfirmsshouldbeabletosmooththroughtheformer better than the latter. Interestingly, though, the dispersion of both the seasonal and nonseasonal distributions is higher than for the total. This finding suggests that most firms tend to exhibit substantiallydifferentseasonal(smoothing)andnonseasonal(bunching)varianceratios. The substantial heterogeneity of firm-level variance ratios raises an important question: what accounts for dispersion in variance ratios? To answer this question, the data were divided into 40 classes based on two observable firm characteristics — industry and size (20 two-digit SIC industries and two sizes, large and small). Each firm-level ratio then was deviated around the industry-size median ratio (shown in the second row of Figure 3) and the industry-size ratio calbehigherorlowerfor Q S U M thanfor Q . 26The decomposition follows the method in Krane and Braun (1991). The seasonal component is obtained by runningaregressionwithseasonalandtimedummiesontheproductionandsalesdataandusingthefitted seasonal part. Theresidualfromthisregressionisthenonseasonalcomponent. Thesampleperiodis1985:1through1993:8, dependingonavailabilityoffirm-leveldata. 27Fuhreretal(1995)reportthatthefeasibleparameterspaceforabenchmarkLQinventorymodelusingaggregate BEA nondurablegoodsdata producesvarianceratios between about0.5 and 1.5. It seems plausible that firm-level dataeasilycouldproduceawiderrangeofvarianceratios,asisobserved. 20

culated with aggregated data (third row). The standard deviations of the distributions show that observable characteristics account for little of the heterogeneity in variance ratios. The mediandeviation ratios have marginally smaller variance at best, and the aggregate-deviation ratios are all larger. On a size-weighted basis (not shown), it is clear that both tails of the distributions are populatedprimarilybysmallerfirms. Lastly, it is interesting to examine the M3LRD firm-level variance ratios in the M3 industries that correspondto the four-digit industries examinedbyFair (1989)and Krane and Braun (1991). Atissueiswhetherthereisevidencethatfirmsintheseindustriestendtosmoothproductionmore than other manufacturing firms. The M3LRD aggregation panel contains 52 companies and 98 divisions in these industries.28 The median total variance ratio in these industries is 1.1 for both companies and divisions; the median seasonal variance ratio is .95 for companies and 1.1 for divisions. Thus there does not appear to be a tendency for firms in these industries to smooth productioningeneral or relativetoother manufacturingfirms. To theextent that M3LRDfirms in these industries are representative, this result lends somesupport to the contentionthat the dollarvalueM3dataare “bad”relativetothephysical-unitsdataduetomeasurementerror. 3.2 Econometric Results This section reports regression results from firm-level estimation of the three econometric inventory models described in section 1: the stock adjustment model, equation (9), the error correction model,equation(11), and theEuler equationmodel, equation(6). For each model, theeconometric specification is designed to follow as closely as possible the standard practice in the inventory literature. 3.2.1 Specifications Allthreeeconometricmodelssharethefollowingspecifications. Regressionsarerunforeachfirm individually; these are not panel data regressions so there are no parametric restrictions between firms.29 The sample period for each firm is the maximum number of observations available over 28Theindustriesare: cigarettesandcigars(category21Awith fourfour-digitSICindustries),tires(category30A withonefour-digitindustry);cement(category32Cwith22four-digitindustries);andcopper,lead,andzincrefining (category33Cwith17four-digitindustries). 29Inprinciple,thisregressionstrategycouldbenefitfromapplyingtheseeminglyunrelatedregression(SUR)techniqueto the individualfirm-levelregressionsif there exists exploitablecovarianceamongfirm residuals. Given the 21

the period 1985:01 to 1993:08, minus relevant leads and lags. Data are in logs, deseasonalized, and,exceptfortheerror-correctionmodel,detrended. Theeconometricstockadjustmentmodelforfirm k is I k t = (cid:22) k (cid:23) 0 k + ( 1 (cid:0) (cid:22) k ) I k ;t(cid:0) 1 (cid:0) (cid:25) k S k t + ( (cid:22) k (cid:23) 1 k + (cid:25) k ) E k t S k t + (cid:22) k (cid:23) 2 k E k t r it + (cid:15) k t (12) where (cid:15) k t is the usual random normal OLS regression error. Equation(12) includes a real interest rate, r it , defined as the difference between the nominal three-month commercial paper rate and expected industry (subscript i ) output price inflation. The real rate term enters via a more general inventorytarget, I (cid:3) t = (cid:23) 0 + (cid:23) 1 E t S t + (cid:23) 2 E t r t ,whichiscommonintheinventoryliterature. Thesign of (cid:23) 2 shouldbe negative. Expected sales and the expectedreal rate are replaced inthe regressions withfittedvaluesfromindependentlyestimatedAR(3) models,asis usuallydone.30 Theeconometricerrorcorrectionmodelforfirm k is (cid:1) I k t = (cid:11) k (cid:0) (cid:13) k ( I k ;t(cid:0) 1 (cid:0) ’ S k S k ;t(cid:0) 1 (cid:0) ’ r k r k ;t(cid:0) 1 ) + (cid:18) I k (cid:1) I k ;t(cid:0) 1 + (cid:18) S k (cid:1) S k ;t(cid:0) 1 + (cid:18) r i (cid:1) r i;t(cid:0) 1 + (cid:17) t (13) wheretheindustryrealinterestratehasbeenaddedtothecointegratingrelationship. Theequation also includes seasonal dummies and the inventory valuation dummy. Theoretically, equation (13) isvalidonlyforfirmsforwhich I , S ,and r areI(1)andthelinearcombinationofthesevariablesis I(0). StandardaugmentedDickey-Fuller(ADF)unitroottestswereperformedon I , S , r ,andtheir linear combination to test these conditions. Only firms that pass all the criteria at the 10 percent level were included in the error correction model estimation (288 companies and 632 divisions qualified).31 largenumberoftotalfirms(especiallydivisions),SURisnotfeasiblefortheentiresample. However,Iexploredthe effectsofSURestimationon25companiesinthestone,clay,andglassindustry(SIC32)andfoundlittleimpacton eitherthepointestimatesorstandarderrors.Thefirm-leveldifferencesbetweenthesingle-equationandSURstandard errorsfortheadjustmentspeedparameterhadamedianofzeroanda90-10decilerangeof (cid:0) :0 3 to.03. Ingeneral, thereisverylittlecorrelationbetweenresidualsamongfirms,evenwithinindustries. 30In principle,thestandarderrorsforequation(12)shouldbecorrectedforthetwo-step estimationasin Murphy and Topel (1985), for example. However, in earlier work (Schuh (1992)), I found that the adjustment made little difference.Breusch(1978)andGodfrey(1978)Lagrangemultipliertestsindicatedlittleornoserialcorrelationinthe residualssonocorrectionsweremade.Thus,theadjustmentspeedestimatesarenotsubjecttotheupwardbiasdueto multipleequilibriasometimesencounteredinthesemodels(see,forexample,Blinder(1986b)andHallandRossana (1991)). 31Theoretically,therealrateprobablyshouldbeI(0),butempiricallyonecannotrejectthehypothesisthatitisI(1). Becausetheconductofmonetarypolicymayinducenonstationarityintheshort-termrealrate,Iincludedit. However, 22

TheeconometricEulerequationforfirm k is I k + ;t 1 (cid:0) [ (cid:21) k + ( (cid:21) k (cid:12) ) (cid:0) 1 ] I k t + (cid:12) (cid:0) 1 I k ;t(cid:0) 1 + S k + ;t 1 (cid:0) (cid:20) k (cid:12) (cid:0) 1 S k t + c k = (cid:24) k + ;t 1 (14) where c k is a constant and (cid:24) k + ;t 1 includes the expectational error, which is MA(1) because (cid:10) k t S k t 62 , plus any white noise cost shocks in the model. Discount factor (cid:12) is preset to 0.995. The instrument set is Z k t = [ 1 ; I k ;t(cid:0) 1 ; I k ;t(cid:0) 2 ; S k ;t(cid:0) 1 ; S k ;t(cid:0) 2 ] , which is consistent with the standard assumptions of exogenous and unknown sales. Parameters (cid:21) k and (cid:20) k are estimated with Hansen's (1982) GMM estimator using the Newey and West (1987) weight matrix with lags equal to one. The J statistic from the test of overidentifying restrictions is distributed (cid:31) 2 with three degrees of freedom. This GMM implementation is virtually identical the analogous model in Eichenbaum (1989). Formanyfirms,theGMMcriterionfunctionisveryflat overthestablerangefor (cid:21) k (i.e.,0-1). In these cases, the actual estimate of (cid:21) k is often greater than one (unstable) and the standard errors are extremelylarge. Toovercomethisproblem,IfollowEichenbaum(1989)andDurlaufand Maccini (1992)andmultiplyequation(14) by ( 1 (cid:0) (cid:21) k ) (cid:0) 1 . Althoughintheorythistransformation islegitimatebecauseitdoesn'taffectthemomentcondition,inpracticeitaffectssmallsampleestimatesbychangingthelocationandincreasingthecurvatureofthecriterionfunction. Inparticular, it tends to significantlyreduce estimates of (cid:21) k (thereby increasingtheadjustmentspeed) andtheir standarderrors.32 3.3 Regression Estimates Table 3 reports the results of the firm-level regression estimation for each of the three models at the company level (upper half) and division level (lower half). The first three columns report 10th, 50th (median), and 90th percentiles of the distributions of parameters and regression diagnostic statistics. The next two columns indicate the percentage of parameter estimates that are theparameterestimatesoftheerrorcorrectionmodelarenotparticularlysensitivetotheinclusionofrealrateexcept thattheadjustmentspeed( (cid:13) )isslightlylargerwiththerealrate(medianfirm-leveladjustmentspeedsof.30withthe realrateand.25without). 32Ididsomerandomexperimentationwiththiseconometrictechniquebytransformingtheequationby ( 1 (cid:0) (cid:21) k ) (cid:0) (cid:26) for (cid:26) = f :5 ; 1 ; 2 ; :::; 8 g ,sinceanypositiverealvalueof (cid:26) is legitimate. Theestimatesof (cid:21) k (theadjustmentspeed) weremonotonicallydecreasing(increasing)in (cid:26) ,thoughatadecreasingrate. Thusitappearsthatthistransformation permits generation of an arbitrarily large estimate of the adjustment speed. This issue requires a more systematic MonteCarloinvestigation. 23

positive or negative and significantly different from zero at the 10 percent level. The final two columns measure the degree to which observable firm-level heterogeneity accounts for variance intheparameter estimates. “Median”indicatesthepercentage ofvarianceattributabletovariation in median firm-level parameter estimates across 40 industry-size classes. “Other” indicates the percentageattributabletoidiosyncraticvariationwithinindustry-sizeclasses.33 TheoverallimpressionconveyedbyTable3isthatthefirm-leveldatagenerallydonotsupport the linear-quadratic inventory model, regardless of the econometric methodology employed. In most respects, these firm-level estimates are consistent with prior results from regressions with aggregate data. On one hand, the adjustment speed estimate is highly significant and the correct sign and magnitude. But on the other hand, most parameter estimates are insignificantlydifferent from zero,thewrongsign,orboth. Specifically: (cid:15) Stock adjustment model—The median buffer stock parameter ( (cid:25) ) is zero instead of one. Roughly half of the estimates of (cid:25) , (cid:23) 1 , (cid:23) 2 are the wrong sign, and estimates of all three are aboutequallylikelytobesignificantlynegativeassignificantlypositive. (cid:15) Error correction model—The lagged difference coefficients ( (cid:18) 's) are about evenly distributed around zero. The estimates are significantly different from zero less than half the time, but about twice as likely to be significantly positive rather than negative. In this unrestricted form, the coefficients don't have much of a structural interpretation other than a significant (cid:18) I indicatingthepresence ofadjustmentcosts. (cid:15) Euler equation model—The (cid:20) parameter is less than 1.0 for most firms, as predicted, but notforallfirms. Inaddition,lessthantwo-fifthsofthe (cid:20) estimatesaresignificantlydifferent fromzero. Despitethesegenerallyunsupportiveresults,thetablerevealsthreenotabledifferencesbetween thefirm-levelresults andpreviousaggregateresults: 1. Adjustment speeds—Firm-level adjustment speed estimates are considerably larger than most estimates reported from regressions with monthly aggregate data. Figure 4 plots the 33Asinthestandardvarianceformula,thereisacovariance-typetermbutitisgenerallysmallandcanbeinferred fromthetable. Theactualformulasareasfollows.Let (cid:12) denotethemedianparameterestimateforallfirmsandNthe totalnumberoffirms. Subscriptsareasfollows: i denotesindustries, j denotessizes,and k denotesfirms. Thenthe “median”columnis ( N (cid:0) 1 ) (cid:0) 1 P 2 0 =i 1 P 2 j = 1 ( (cid:12) ij (cid:0) (cid:12) ) 2 ,andthe“other”columnis ( N (cid:0) 1 ) (cid:0) 1 P N k = 1 ( (cid:12) ij k (cid:0) (cid:12) ij ) 2 . 24

d f S o C D e n r e e F i M o d e l o m p a n S A E C M E u l e r i v i s i o n S A E C M E u l e r a S A d e n o t e s t h e t e s t s o f t e x t f o r r m - L a i e s : s : o t e s s E u l e r (cid:12) r s t - o m o r e e v e l I n (cid:22) (cid:25) 1 (cid:23) 2 (cid:23) 2 R S C ( 1 ) (cid:13) I (cid:18) S (cid:18) r (cid:18) 2 R S C ( 1 ) (cid:0) (cid:21) 1 (cid:20) 2 (cid:31) (cid:22) (cid:25) 1 (cid:23) 2 (cid:23) 2 R S C ( 1 ) (cid:13) I (cid:18) S (cid:18) r (cid:18) 2 R S C ( 1 ) (cid:0) (cid:21) 1 (cid:20) 2 (cid:31) t o c k a d j u e q u a t i o n r d e r s e r i a d e t a i l s . v (cid:0) (cid:0) (cid:0) (cid:0) s t m l e P (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) m o c n t o P 1 0 . 1 3 . 3 0 2 . 2 0 . 0 3 0 . 2 3 . 1 2 : 2 5 : 2 8 : 0 2 . 1 0 . 2 8 . 3 1 1 . 2 3 . 1 7 . 1 3 . 5 1 3 . 2 8 . 0 3 0 . 2 6 . 1 1 : 2 5 : 3 0 : 0 2 . 1 1 . 2 6 . 3 3 1 . 3 5 . 1 7 e n t d e l . o r r e T a b r y M o e r c e n t i l e P 5 0 . 3 2 (cid:0) . 0 1 . 2 0 . 0 0 . 0 2 . 7 8 . 3 0 (cid:0) : 0 3 (cid:0) : 0 2 0 . 2 3 . 8 1 . 4 8 . 4 7 . 7 2 . 4 0 (cid:0) . 0 2 . 0 9 0 . 0 2 . 7 8 . 3 2 (cid:0) : 0 3 (cid:0) : 0 3 0 . 2 6 . 8 1 . 5 0 . 5 0 . 7 2 , E C M d S C ( 1 ) a n l a t i o n a n l e d s e d d e P 3 1 1 1 1 3 1 1 1 n (cid:31) o 3 l P 9 0 . 7 2 . 5 1 . 0 8 . 0 3 . 0 9 . 0 0 . 6 3 . 2 1 . 2 4 . 0 2 . 4 4 . 0 0 . 6 9 . 5 4 . 9 7 . 0 3 . 8 6 . 9 0 . 0 3 . 2 7 . 0 0 . 7 9 . 2 7 . 2 4 . 0 3 . 5 0 . 0 0 . 7 0 . 7 0 . 9 7 o t e s 2 s t v e r i a r a m e % S i g n i (cid:12) + 9 9 2 7 4 6 3 3 9 4 1 4 1 1 1 3 8 9 3 5 9 3 2 4 4 0 2 8 9 3 1 3 9 1 1 8 6 3 3 e r r o r c o a t i s t i c s r d e n t i f y i n t e c a (cid:0) r r e p g r n 2 3 3 2 2 1 3 3 3 2 2 1 e o r E t 0 9 6 3 0 6 3 5 1 4 4 2 7 2 0 7 6 7 0 4 c t i r t e s t o t r s n h i t A M e c t i m a t e % V a c c o u n t e d i a n 1 0 0 0 4 3 0 1 0 0 0 0 0 0 2 1 0 0 0 0 m o d e l , p r o b a b i o n s , r e s r i a n c e e d f o r O t h a n d E i l i t y v a s p e c t i v b e 1 1 1 u l u e y : r 0 0 9 7 9 8 0 0 8 8 9 1 9 8 9 2 9 4 9 9 9 9 9 4 9 5 0 0 9 5 9 5 9 7 9 7 9 6 9 8 l e r e s l y . 25

4.0 3.0 2.0 1.0 0.0 m Companies seinapmoC fo noitcarF 4.0 3.0 2.0 1.0 0.0 Figure 4 Firm-Level Inventory Adjustment Speed Estimates Significant Insignificant 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4.0 3.0 2.0 1.0 0.0 m Divisions snoisiviD fo noitcarF 4.0 3.0 2.0 1.0 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4.0 3.0 2.0 1.0 0.0 g Companies seinapmoC fo noitcarF 4.0 3.0 2.0 1.0 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4.0 3.0 2.0 1.0 0.0 g Divisions snoisiviD fo noitcarF 4.0 3.0 2.0 1.0 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4.0 3.0 2.0 1.0 0.0 1-l Companies seinapmoC fo noitcarF 4.0 3.0 2.0 1.0 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4.0 3.0 2.0 1.0 0.0 1-l Divisions snoisiviD fo noitcarF 4.0 3.0 2.0 1.0 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 26

weighted distributions of adjustment speed estimates for further inspection. The median firm-level adjustment speeds for the stock adjustment and error correction models run from .30 to .40. Compare these with estimates in Blinder (1986b, Table 1) of .06 and .14 for durable and nondurable goods industries, respectively. The median firm-level adjustment speeds for the Euler equation model are .48 and .50; Eichenbaum (1989, Table 1) reports estimates of .30 and .39 for nondurable goods industries.34 The adjustment speed estimates are slightly higher for divisions than companies in all three models, supporting Blinder's assessment from two-digit industries that “aggregation seems to bias the estimated speed of adjustment downward” (p. 359). Also, for (cid:22) and (cid:13) , disaggregation from companies to divisions leads the adjustment speed distributions to move from a skewed (e.g., (cid:31) 2 ) shape closertoauniformshape. 2. Serial correlation—Firm-level residuals exhibit very little serial correlation, unlike residuals from regressions using aggregate data. The table reports p-values for: (1) tests of the nullhypothesisofnoserial correlationagainstthealternativeoffirst-orderserialcorrelation (denoted by SC(1)); and (2) tests of overidentifying restrictions (denoted by (cid:31) 2 ). There is virtuallynoevidence of serial correlationin the stockadjustmentand error correction models. In contrast, the inventory literature has devoted considerable space to debating how the treatment of serial correlationinregressions withaggregatedata can radically affect regression results. Similarly,there is virtually no evidence against the overidentifyingrestrictions in the Euler equation model. In contrast, most GMM estimates of Euler equations produce highly autocorrelated residuals and rejections of the restrictions (see, for example, Eichenbaum(1989)andKashyapandWilcox(1993)). 3. Model fit—Firm-level models do not account for much of the variation in the inventory data. The median R 2 in the stock adjustment model is close to zero; the median in the error correction model is considerably higher, but still only about one-fourth. Given that the 34Two robustness issues regarding the firm-level Euler equation estimates are worth noting. First, if there is no buffer stock motive ( S t is known and in the instrument set) then the adjustment speed distributions have the same centraltendencies. However,the distributionsareconsiderablyless disperseandthe overidentifyingrestrictionsare rejectedmorefrequently. Second, whentheparametrictransformation ( 1 (cid:0) (cid:21) k ) (cid:0) 1 is notmadeto equation(14)the distributionof (cid:21) k changesmarkedly—themedianestimateis0.63andaboutone-fourthofallestimatesaregreater thanorequalto1.0(i.e., theadjustmentspeedsaremuchsmaller). Theseissuesapplyequallyto thefirm-leveland aggregatedata. 27

stock adjustment model is approximately a restricted version of the error correction model, thisresult weighs againsttherestrictionsof thestockadjustmentmodel. Incontrast,regressionswithaggregatedata usuallygenerate R 2 ofone-halfor moreormorefor bothtypes of models. It seems reasonable to infer, however, that this result is directly related to the other two. Thetablealsoportraysconsiderableheterogeneityinthefirm-levelresults,buttheheterogeneity is essentially totally unexplained. For example, the 90-10 decile band of adjustment speed estimates encompasses most of the stable region. Likewise, all other estimates exhibit a broad mixof negativeandpositivevalues,many(but not most)of whichare significantlydifferent from zero. Whatcanexplainthisheterogeneity? Thelasttwocolumnsofthetableindicatethatsubstantiallyless than10 percent of thedispersion,and usuallynoneof it,can beattributedtoobservable characteristics (industryandsize). Instead,virtuallyall thedispersionisidiosyncratic. Combiningthesefindings,oneisleftwiththefollowingdepictionofmicroeconomicinventory behavior. Although important theoretically, standard accelerator (expected demand) and cost of capital (real interest rate) effects explain very little of the variation in firm-level inventory investment. Observable firm characteristics cannot explain the idiosyncratic component of firm-level inventoryinvestment. In addition,the idiosyncraticcomponent of firm-level inventoryinvestment is not very persistent, which is consistent with the finding that firms do not experience inventory gapsforabnormallylongperiodsoftime. Several factors could potentially explain the large and transitory idiosyncratic component of firm-levelinventoryinvestment. Measurementerrorisanobviouscandidate,giventheheterogeneity and lack of persistence. But this explanation is, of course, unsatisfying. Misspecification due to missing variables is an obvious economic explanation, though it is not immediately clear what firm-specific variables are missing. Some low-frequency factors, such as inventory-saving capital and production techniques, are possibilities, but these cannot account for the high-frequency volatilityof the idiosyncraticcomponent. Also, because the inventorytarget depends on expected variables, it is possible that simple univariate time-series projection methods do not adequately account for all information available to firms which may lead them to very different sales plans, for example. Finally, at the firm level there may be functional nonlinearities or discontinuities at work, such as an (S,s) rule. In any case, the results point to much room for improvement and furtherresearch. 28

4 Aggregate Evidence and Results This section provides evidence on aggregate production variance ratios and regression estimates oflinear-quadraticinventorymodelsforabalanced-panelsubsetoftheM3LRDaggregationpanel (613 companies and 903 divisions). The balanced panel includes all firms from the M3LRD aggregationpanel withcontinuousdata overthe period 1986:01through1992:12. A balanced panel is necessary for examining the link between firm-level and aggregate inventory behavior because neitherthedatanorthemodelframeworkadequatelyaccount forfirm entryandexit. In this section, three aggregate concepts arise. First is MW, which pertains to the weighted firm-level mean value where the weight accounts for variation in average firm size. Second is MA, which pertains to the aggregate data (simple summation across all firms) obtained from the balanced-panel subset of the M3LRD aggregation panel. Third is BEA, which pertains to the aggregate data published by the Bureau of Economic Analysis (the data used in most previous inventorystudies). 4.1 Representative Agent Issues Before turningtotheaggregate results,it is necessary to address the representativeagent assumption. In particular, the question arises: can aggregate inventory behavior be modeled adequately using a representative agent hypothesis? This hypothesis has come under increasing fire recently. Forexample,Geweke(1985)andKirman(1992),argue strenuouslythatthereisnoguaranteethat a representative agent exists even in theory, much less in practice. In the inventory literature—as in most of macroeconomics—relatively little attention has been given to the representative agent hypothesis. ButtheM3LRDprovidesanopportunitytoassessexistingassumptionsandtocompile evidenceonthehypothesis. The remainder of the paper investigates this issue by adopting Theil's (1954, 1971) approach to aggregation analysis. The Theil approach begins with the assumption that there exists a true microeconomic model for each of the N firms in the economy. Then, using the simple linear aggregator function, one can derive the implied“true” aggregate model. The representative agent hypothesis,then,isthattheaggregatemodelaccuratelyreflectsthetrueaggregateoftheunderlying microeconomicrelations. It turns out that linearity of the microeconomic relations is critical to having a hope of evaluating the representative agent hypothesis,so the linear-quadratic inventorymodel qualifies. How- 29

ever, even with linear microeconomic relations, one of the following two conditions is necessary for the existence of a representative agent model: (1) all data (e.g., inventories and sales) are the same across firms; or (2) all micro parameters are the same across firms. West (1986, p. 383), for example, adopts the latter approach by assuming that the structural parameters of his inventory model are identical across firms. Clearly, however, previous sections have provided ample evidenceagainstbothhomogeneityassumptionsbydocumentingtremendousheterogeneityinthe data. Ramey (1989, p. 345) adopts another approach to the aggregation problem by assuming: (1) firm-specific components of observed and unobserved variables are independent across firms and time; (2) firms face identical prices, wages, and interest rates; and (3) cross-sectional output variance is constant. The extensive unexplained heterogeneity in the firm-level data and the lack of correlation among firm-level data series might be interpreted as evidence in favor of (1). However, Abbott (1987) and Davis and Haltiwanger (1991) provide strong evidence against (2) for manufacturingprices andwages,respectively. In addition, Figure 5 shows evidence against Ramey's point (3). The two panels in the first column plot the time-series of the first four cross-section moments of detrended firm-level production. Clearly, the cross-section mean of production can vary more than 10 percent over the business cycle (1990-91), and even more for the other moments. Note, especially, that the crosssection variance of production rises substantially and persistently during the 1990-91 recession much the same way job reallocation (a measure of dispersion) does, as reported by Davis and Haltiwanger (1992). The two panels in the second columndemonstrate the same principle for the momentsoftheequilibriumerrorfrom thecointegratingequation. 4.2 Variance Ratios This section provides evidence on the relationship between firm-level and aggregate production variance ratios. At issue is whether the production variance properties of the aggregate data are representativeof the firm-level evidence discussedpreviously. Put another way: is it possiblethat thefirm-leveldataexhibitmoreproductionsmoothingthanisobservedintheaggregatedata? The first four columns of Table 4 report the evidence on four aggregate measures of total, seasonal,andnonseasonalproductionvarianceratios (i.e., V a r ( Q ) = V a r ( S ) ). Comparingthefirst two columns with the next two, it is clear that the firm-level data do not reveal more production smoothing behavior than the aggregate data. The MA ratios (column 3) generally are about the 30

5 erugiF lenaP noitagerggA DRL3M eht ni stnemoM noitceS-ssorC rorrE muirbiliuqE gnitargetnioC eht fo noitaiveD dradnatS dna naeM noitcudorP leveL-mriF fo noitaiveD dradnatS dna naeM veD dtS naeM veD dtS naeM 05.0 60.0 05.0 51.1 naeM naeM 54.0 veD dtS 40.0 54.0 veD dtS 01.1 04.0 04.0 20.0 53.0 50.1 53.0 00.0 03.0 03.0 20.0- 52.0 00.1 52.0 40.0- 02.0 3991 2991 1991 0991 9891 8891 7891 6891 3991 2991 1991 0991 9891 8891 7891 6891 rorrE muirbiliuqE gnitargetnioC eht fo sisotruK dna wekS noitcudorP leveL-mriF fo sisotruK dna wekS sisotruK wekS sisotruK wekS 021 4 052 41 wekS wekS sisotruK sisotruK 001 2 21 002 01 08 0 051 8 06 2- 6 001 04 4- 4 02 6- 05 2 0 8- 0 0 3991 2991 1991 0991 9891 8891 7891 6891 3991 2991 1991 0991 9891 8891 7891 6891 31

N e c M d s A g g r e g a t e V a I n d u s t r y M e T o t a l : T M N D P S P O P O - N P O - Y S L S e a s o n a l : T M N D P S P O P O - N P O - Y S L N o n s e a s o n a l : T M N D P S P O P O - N P O - Y S L O T E S : T h e t a b l e r e p o r t s t h s t i m a t e d o v e r t h e p e r i o d 1 9 8 6 o n s t r u c t e d w i t h t h e f o l l o w i n g W | w e i g h t e d s u m o f (cid:12) r m a t a ; B | B E A r e s e a s o n a l i z e e a s o n a l , a n d n o n s e a s o n a l . T a r i a n c e s V a r i a n d i a n M W 1 . 0 5 1 . 3 1 . 0 4 1 . 5 1 . 0 7 1 . 2 1 . 0 8 1 . 7 1 . 0 4 1 . 2 1 . 0 5 1 . 2 1 . 0 3 1 . 2 1 . 0 4 1 . 3 1 . 0 8 1 . 3 1 . 0 0 1 . 4 1 . 0 0 1 . 7 1 . 0 0 1 . 1 1 . 0 0 2 . 3 1 . 0 0 1 . 1 1 . 0 0 1 . 1 0 . 9 9 1 . 1 1 . 0 0 1 . 5 0 . 9 9 1 . 3 1 . 1 5 1 . 5 1 . 1 2 1 . 6 1 . 1 7 1 . 4 1 . 1 8 1 . 6 1 . 1 5 1 . 4 1 . 1 7 1 . 5 1 . 1 0 1 . 4 1 . 1 2 1 . 5 1 . 2 1 1 . 4 e v a r i a n c e : 0 1 t o 1 9 9 2 d a t a : M e d - l e v e l r a t i o d a g g r e g a t b l e 4 o f P r o d u c t i o c e R a t i o s M A B E A 9 1 . 0 8 1 . 4 3 4 1 . 1 8 0 . 7 6 6 1 . 0 2 2 . 5 0 6 1 . 2 6 0 . 6 6 5 1 . 0 4 2 . 1 6 8 1 . 1 7 2 0 . 9 1 9 1 . 0 2 7 1 . 2 2 4 0 . 9 1 0 . 9 1 9 0 . 9 8 0 . 6 6 5 0 . 8 6 1 . 3 0 0 1 . 3 3 0 . 6 4 3 0 . 8 2 1 . 1 8 4 0 . 9 4 1 0 . 7 3 0 0 . 8 3 3 1 . 1 6 1 1 . 2 2 3 . 9 1 0 1 . 3 8 1 . 3 8 4 1 . 1 3 8 . 0 8 3 1 . 2 2 0 . 9 9 7 1 . 2 2 5 . 9 4 1 1 . 3 1 3 1 . 1 0 3 1 . 2 0 9 1 . 2 5 Q r a t i o s , V a r ( ) / : 1 2 . C o l u m n h e a i a n = (cid:12) r m - l e v e l s ; M A | M 3 L R e d a t a . S e e t h e n V a d i n m e D t e a n d S V C o m p a S 6 4 . 7 2 4 . 5 4 0 . 2 1 3 . 9 5 0 . 9 2 4 . 8 2 6 . 0 4 4 . 4 2 0 . 4 2 7 . 4 1 1 . 6 1 5 . 8 5 . 1 2 2 . 3 9 . 2 1 3 . 1 2 1 . 3 6 . 0 3 7 . 4 1 2 . 9 2 4 . 5 8 . 8 2 8 . 6 1 5 . 7 1 2 . 9 2 3 . 0 1 4 . 4 S r ( ) , a n g s i n d i c d i a n f r o b a l a n c e x t f o r d a l e s a r i a n c e S u m s n i e s D i v i s i o n s Q S Q 7 7 . 5 1 2 0 . 5 1 5 3 3 3 . 3 3 2 . 8 3 7 4 4 . 3 5 3 . 2 7 6 2 1 . 1 2 6 . 4 2 9 5 6 . 4 5 9 . 6 8 4 3 0 . 9 8 . 7 1 7 2 5 . 5 5 0 . 8 6 7 4 9 . 6 8 3 . 0 1 0 0 2 7 . 9 3 7 . 5 5 3 2 5 . 7 4 8 . 9 4 8 1 1 . 7 1 3 . 5 1 2 1 4 . 0 1 7 . 2 2 0 6 . 9 9 . 7 8 1 8 . 8 2 0 . 9 2 4 8 . 6 4 . 2 5 1 0 . 1 1 6 . 7 1 8 1 8 . 5 3 5 . 1 3 2 7 . 1 1 3 . 8 1 6 5 1 . 8 7 3 . 7 1 0 5 2 1 . 5 2 0 . 5 2 6 3 0 . 2 3 6 . 9 5 6 1 4 . 1 1 7 . 9 2 1 3 7 . 6 3 9 . 5 6 1 2 2 . 3 4 . 7 1 1 1 5 . 4 3 4 . 9 4 9 3 1 . 0 4 8 . 9 6 8 2 0 . 8 2 4 . 8 3 7 d v a r i a n c e s o f Q , a t e v a r i a n c e r a t i o s m b a l a n c e d p a n e l ; d p a n e l a g g r e g a t e e (cid:12) n i t i o n s o f t o t a l , . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7 7 7 7 0 7 1 0 4 2 3 6 0 5 5 2 2 7 2 6 9 0 4 5 0 7 32

samemagnitude,oronlyslightlylarger, thanthemedianfirm-levelratios(column1). But theMA ratios are notably smaller than the MW ratios (column2). Apparently, firms with higher variance ratiosaredisproportionatelylarge. Forallmanufacturing,theBEAdataexhibitconsiderablymore productionbunching,butthecross-sectionalevidenceisdisparate(perhapsduetosampleselection or small sample problems). The evidence on total ratios versus the seasonal and nonseasonal componentsisessentiallythesameasatthefirmlevel. Insum,itappearsthataggregationactually tendstoreducethedegreeofproductionbunchingintheaggregatedata. Recently, Krane (1994) and Lai (1991) have hypothesized that production bunching may be observedintheaggregatedataeventhoughall(ormost)firmssmoothproduction. Theirargument is clear from the aggregation formula for the difference between production variance and sales variance: V a r ( Q ) (cid:0) V a r ( S ) = k X K = 1 V a r ( q k ) (cid:0) k X K = 1 V a r ( s k ) + 2 2 4 =i6 X j j X =6 i C o v ( q i ; q j ) (cid:0) C o v ( s i ; s j ) 3 5 (15) where lowercase letters pertain to firm-level data and uppercase to aggregate data. Suppose each firm smooths production, i.e., that the sum of firms' production variance is less than the sum of firms' sales variance. Then aggregate production variance would exceed aggregate sales variance onlyifthesumofcovarianceamongfirms' productionsufficientlyexceededthesumofcovariance among firms' sales. Whether this covariance condition is true depends largely on the underlying parameters of the cost functions C Q ( (cid:1) ) and C I ( (cid:1) ) . Krane shows that the condition is true when marginalstockoutcosts(afunctionof (cid:30) and ! )sufficientlydominatemarginalholdingcosts(which implicitlyenter themodel throughconstants). Lai shows that theconditionis truewhen shocksto firms' marketshares are sufficientlymorevariablethancommon(aggregate)shocks. The remaining four columns of Table 4 report evidence on this hypothesis for companies and divisions in the M3LRD balanced aggregation panel. The data speak strongly against the Krane andLaihypotheses: forthevastmajorityofindustriesforalltypesofratios,thesumofproduction variance exceeds the sum of sales variance. Thus, there is not a significant difference between thecovariancesamongfirm productionandfirm salestogenerateamisleadingaggregatevariance ratio. Infact,thecorrelationamongfirm-levelproductionandfirm-levelsalesvariablesisgenerally closetozero. 33

4.3 Econometric Results and Aggregation Bias Thissectionprovidesevidenceontherelationshipbetweenfirm-levelandaggregateparameterestimatesfromthethreeinventoryeconometricmodels. Atissueiswhethertheaggregateparameter estimates are representative of the firm-level parameter estimates. That is, we want to determine whether or not there is evidence in favor of the representative agent hypothesis for the standard inventorymodels. Forlinearmodelsliketheinventorymodels,Theil's(1954)analogyprincipleprovidesaframeworkforanalyzingtheeffectsofaggregationontheparametersofmacroeconometricmodels. For a general linear model, the framework is as follows. Let uppercase variables denote aggregates (simple linear sums) and lowercase variables denote firm-level variables. The firm-level models are y k t = x k t (cid:12) k + (cid:15) k t k = 1 ; : : : ; K (16) where x k t and (cid:12) k are J-dimensional vectors of predetermined variables and time-invariant firmlevelparameters,respectively. Thustheaggregatemodelis Y t (cid:17) k X K = 1 y k t = k X K = 1 ( x k t (cid:12) k + (cid:15) k t ) : (17) Relation(17)impliesthat Y t = k X K = 1 2 4 j X J = 1 ( w j k t (cid:12) j k ) X j t + (cid:15) k t 3 5 (18) where w j k t = ( x j k t = X j t ) , and X j t and x j k t are (T (cid:2) 1) matrices from the j th columns of X t and x k t . Therefore each aggregate parameter, (cid:12) j t = P K k = 1 w j k t (cid:12) j k , is the size-weighted sum of firmlevel parameters. Although the firm-level parameters (cid:12) j k are time-invariant (by assumption), the “true”aggregateparameters (cid:12) j t are time-varying. The“true”aggregateparametersbecometime-invariantonlyifone(orboth)oftwoconditions is true: 1) all of the weights are time-invariant (i.e., w j k t = w j k ), which implies that firm size is time-invariant; or 2) the firm-level parameters are the same for all firms (i.e., (cid:12) j k = (cid:12) j 8 k ). If oneoftheserestrictionsholds,thentheaggregatemodelwillbe“representative”oftheunderlying firm-levelmodels. Otherwise,afixed-parameter(FP)aggregatemodelwillbemisspecifiedrelative tothe“true”time-varying-parameter(TVP)aggregatemodel. On one hand, the FP aggregate model may provide misleading information about firm-level behavioriftheregressionestimatesofafixedaggregateparameter( (cid:12) )arenotrepresentativeofthe 34

underlying firm-level parameters ( (cid:12) k ). A simple measure of the “true” fixed aggregate parameter is a weighted sum (average) of the firm-level parameters: (cid:12) = P K k = 1 w k (cid:12) k , where w k is the time series mean of the true weight w k t . The difference between these twofixed aggregate parameters, (cid:12) (cid:0) (cid:12) , is a measure of econometric aggregation bias.35 Past research has shown that aggregation bias can be as large as an order of magnitude (ten times) or more—a finding confirmed in this paperfortheinventorymodel.36 On the other hand, however, (cid:12) is not necessarily the value of (cid:12) that minimizes the sum of squaredresidualsinanaggregateOLSregression. AsGrunfeldandGriliches(1960)pointout,the FP aggregate model may very well be preferred to the firm-level models for two related reasons. First,theFPaggregatemodelislikelytoobtainamuchhigher R 2 thanarethefirm-levelmodels— a finding also confirmed in this paper. Second, the forecasting performance of the FP aggregate modelmayexceedthat ofthecollectiveforecasts ofthefirm-levelmodels. Inlightofsubstantialeconometricaggregationbiasbutpoorfitoffirm-levelinventorymodels, the questionarises as to whether thedisaggregateddata offer anyimprovementoverFP aggregate models in explaining inventory behavior. To answer the question, this paper compares the FP aggregatemodel withthe TVP model,which allowsfor heterogeneityinfirm-level fixed parameters and timeseries fluctuationinthe size distributionof firms. At issueis whetherthe TVP aggregate modelcanfit theaggregatedatasubstantiallybetterthantheFP aggregatemodel.37 The remainder of this section shows that severe aggregation bias arises in standard aggregate inventory models. Because the inventory models do not perform well, at either the firm level (as shown earlier) or at the aggregate level (as shown in the literature), the focus is restricted to the one parameter that is estimated precisely—the adjustment speed. In particular, Table 5 reports adjustmentspeedestimates( (cid:22) , (cid:13) ,and ( 1 (cid:0) (cid:21) ) ) for: (1)the weightedsum offirm-level parameters (MW column); (2) the FP aggregate model using aggregate M3LRD data (MA column); and (3) 35SeeTheil(1954,1972)foraderivationoftheaggregationbiasandademonstrationthat E [(cid:12) ] 6= (cid:12) . 36See Boot and de Wit (1960) and Sasaki (1978) for exampleswith capital investmentstock adjustmentmodels, Gupta(1971)andLee,Pesaran,andPierse(1991)forexampleswithlabordemandequations,andBils(1985)foran examplewithawageequation. 37ATVPmodelfitstheaggregatedatabetteressentiallybydefinition,ofcourse,becauseaFPmodelisjustaspecial caseofaTVPmodel. Insomesense,itmightbemoreappropriatetocomparetheaggregationTVPmodelwithother TVPmodelssuchasthoseobtainingtimevariationfrommodelnonlinearities,variationincross-sectiondistributions, etc. But given thatthe firm-levelmodelsareactually FP models, this exerciseprovidesrelevantinformationon the extenttowhichaccountingforfirm-levelsizevariationcanaccountforunexplainedvariationinaggregateinventory investment. 35

A g g r e g a t (cid:22) I n d u s t r y D M W C M W (cid:3) (cid:3) : : T M 4 5 3 7 ( .1 5 ) ( .0 8 ) : : N 4 3 4 0 ( .1 6 ) ( .0 8 ) (cid:3) (cid:3) : : D 5 0 3 4 ( .1 4 ) ( .0 7 ) : : P S 4 0 3 9 ( .1 6 ) ( .0 8 ) (cid:3) (cid:3) : : P O 5 0 3 6 ( .1 4 ) ( .0 8 ) (cid:3) (cid:3) : : P O - N 5 0 3 6 ( .1 5 ) ( .0 8 ) (cid:3) (cid:3) : : P O - Y 5 0 3 5 ( .1 4 ) ( .0 8 ) (cid:3) (cid:3) : : S 5 5 3 8 ( .2 5 ) ( .0 8 ) (cid:3) (cid:3) : : L 4 3 3 5 ( .1 2 ) ( .0 8 ) N O T E S : R e g r e s s io n s a r e e s t im a e r r o r s a p p e a r in p a r e n t h e s e s . C w e ig h t e d s u m o f d iv is io n - le v e l p d a t a fr o m t h e M 3 L R D b a la n c e d P O - Y , S , a n d L in d u s t r y c a t e g in d ic a t e s t h a t t h e p a r a m e t e r e s t p e r c e n t le v e l. S e e t h e t e x t fo r m e I n v e n t o r y M A B (cid:3) : : 2 7 0 5 ( .0 8 ) ( .0 5 ) (cid:3) : : 3 8 2 2 ( .1 0 ) ( .0 6 ) (cid:3) : : 2 3 0 6 ( .0 7 ) ( .0 4 ) (cid:3) : : 4 0 1 9 ( .1 0 ) ( .0 6 ) (cid:3) : : 3 2 0 6 ( .0 8 ) ( .0 4 ) :3 3 ( .0 8 ) :5 5 ( .1 9 ) :2 6 ( .0 9 ) :2 9 ( .0 8 ) t e d o v e r t h e p e r o lu m n s in d ic a t e a r a m e t e r s ; C M W a g g r e g a t io n p a n o r ie s ) . A ll d a t a im a t e is s ig n i(cid:12) c a o r e d e t a ils . T a b l e 5 A d j u s t m e n t S p e e d E s t i m a t e s (cid:0) (cid:13) (cid:21) 1 D M W C M W M A B D M W C M W M A B (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) : : : : : : : : 3 7 3 2 1 8 0 2 5 0 4 8 5 6 3 5 ( .1 1 ) ( .0 6 ) ( .0 6 ) ( .0 2 ) ( .2 3 ) ( .2 2 ) ( .1 7 ) ( .1 4 ) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) : : : : : : : : 4 1 3 6 1 1 0 2 5 0 4 9 5 6 3 5 ( .1 3 ) ( .0 6 ) ( .0 7 ) ( .0 3 ) ( .2 3 ) ( .2 1 ) ( .1 1 ) ( .3 6 ) (cid:3) (cid:3) (cid:3) (cid:3) : : : : : : : : 3 5 2 9 2 6 0 5 5 0 4 7 4 8 3 5 ( .1 0 ) ( .0 6 ) ( .0 7 ) ( .0 3 ) ( .2 3 ) ( .2 2 ) ( .2 1 ) ( .1 3 ) (cid:3) (cid:3) (cid:3) (cid:3) : : : : : : : : 4 2 3 5 1 2 0 4 5 0 5 0 5 5 3 4 ( .1 4 ) ( .0 7 ) ( .0 7 ) ( .0 3 ) ( .2 4 ) ( .2 3 ) ( .1 3 ) ( .3 8 ) (cid:3) (cid:3) (cid:3) (cid:3) : : : : : : : : 3 5 3 0 2 7 0 4 5 0 4 7 4 4 3 5 ( .1 0 ) ( .0 6 ) ( .0 7 ) ( .0 3 ) ( .2 3 ) ( .2 1 ) ( .2 3 ) ( .1 3 ) (cid:3) (cid:3) : : : : : : 3 3 2 9 2 8 5 2 4 7 4 9 ( .0 7 ) ( .0 6 ) ( .0 7 ) ( .2 2 ) ( .2 0 ) ( .1 9 ) (cid:3) (cid:3) (cid:3) : : : : : : 3 7 3 2 1 5 4 9 4 7 4 2 ( .1 0 ) ( .0 5 ) ( .0 5 ) ( .2 3 ) ( .2 2 ) ( .1 1 ) (cid:3) (cid:3) (cid:3) : : : : : : 4 1 3 4 0 1 5 1 4 8 3 8 ( .1 2 ) ( .0 6 ) ( .0 3 ) ( .2 5 ) ( .2 0 ) ( .2 7 ) (cid:3) (cid:3) (cid:3) : : : : : : 3 6 2 9 2 0 4 9 4 7 6 0 ( .1 1 ) ( .0 6 ) ( .0 6 ) ( .2 2 ) ( .2 3 ) ( .1 8 ) io d 1 9 8 6 :0 1 t o 1 9 9 2 :1 2 , le s s a p p r o p r ia t e la g s . A s y m p t o t ic s t a n d a r d p a r a m e t e r e s t im a t e s a s s o c ia t e d w it h t h e fo llo w in g d a t a : D M W | | w e ig h t e d s u m o f c o m p a n y - le v e l p a r a m e t e r s ; M A | a g g r e g a t e e l; B | B E A r e s e a s o n a liz e d a g g r e g a t e d a t a ( n o t a v a ila b le fo r P O - N , w e r e d e t r e n d e d a n d d e s e a s o n a liz e d u s in g s e a s o n a l d u m m ie s . A * n t ly d i(cid:11) e r e n t fr o m t h e e s t im a t e in t h e c o lu m n t o t h e r ig h t a t t h e 1 0 36

the FP aggregate model usingaggregate Commerce Department data (BEA).38 The BEA estimate cannot be compared strictly with the MW and MA estimates because the M3LRD aggregation panelrepresentsonlyasubsetoftheBEAdata. However,theBEAestimategivessomeindication of the effects of higher degrees of aggregation and the effects of the sampling scheme used to producethedata(see thedataappendixonthislatterpoint). Table 5 illustrates the most striking result of the paper: aggregation tends to bias adjustment speeds estimates downward. For the stock adjustment and error correction models, the adjustment speed estimates ( (cid:22) and (cid:13) ) in nearly all industry classes decline monotonically as the level of aggregation increases from divisions (DMW) to companies (CMW) to aggregate (MA and BEA).39 Further, the bias is most often statistically significant and large. For total manufacturing, the division-levelweighted mean adjustment speed is 67 percent and 105 percent larger than the aggregate (MA) estimate in the two models, respectively. Aggregation bias is evident in the Euler equation adjustment speed estimates ( 1 (cid:0) (cid:21) ), but considerably smaller and less significant. However, given the sensitivity of the Euler equation estimation technique to parametric transformations,it isunclearhowseriouslytotaketheresults. Two other important characteristics stand out in Table 5. First, there is some variation in the extentofaggregationbiasacrosstypesoffirms. Aggregationbiastendstobesmallerinproductionto-stock(PS)andnondurablegoodsindustries—forwhichthestandardinventorymodelsaremore applicable—than in production-to-order (PO) and durable goods industries. Perhaps aggregation intheselatterindustriescancelsoutsomemisspecificationbiasessuchasfailingtoincludeunfilled orders inthemodel. A second important characteristic is that the BEA adjustment speed estimates are markedly lower than any other estimates. For most industry categories, the decline in the BEA estimate relativetotheMAestimateis greater thanthedeclinedue toexplicitaggregationbias (MW-MA). Apparently,theadditional aggregationand thesamplingmethodologytogetherreduce adjustment speeds even more. Thus the BEA adjustment speeds, which are comparable to most estimates 38Theeconometricspecificationsfortheaggregatemodelsareidenticaltothosedescribedforthefirm-levelmodels insection3. 39TheMAestimatesof (cid:13) arenotstrictly comparablewiththe DMWandCMW estimates. Thereasonis thatthe MW estimates include only the firms which exhibited cointegration between inventories and sales, while the MA estimateis forall companiesin theM3LRDbalancedaggregationpanel. MA-typeestimatesfromdataaggregating only over the subset of firms with cointegrated inventoriesand sales are not uniformly smaller than the DMW and CMWestimates. IchosetoreporttheestimatesthiswaybecauseIbelievethatthesmallersampleadverselyaffects theaggregatemorethanitdoesthecentraltendencyofthefirm-levelparameters. 37

reportedintheinventoryliterature,are theleastrepresentativeoftheunderlyingfirm-leveladjustmentspeeds.40 What are the economic implications of this finding that aggregation biases adjustment speeds downward? The result seems to resolve the long-standing puzzle in the literature that adjustment speeds are implausibly slow (Feldstein and Auerbach (1976)). At the aggregate level, the BEA data imply that the representative firm reduces its inventory gap by a small fraction each month. But the firm-level M3LRD data reveal that the average firm actually eliminates close to half of its gap—a much more plausible rate of adjustment. In fact, with even more disaggregated data (e.g., at the plant level) we should expect to find even higher adjustment speeds—perhaps well above one-half. More generally, this finding provides evidence that persistence in aggregate data may not be due to adjustment costs, learning, “stickiness”, incomplete markets, and other theoretical hypothesesadvancedinmanymacroeconomicmodels. The aggregation bias also suggests that there may be econometric gains to exploiting disaggregated data in aggregate models. Clearly, it is not theoretically or computationally feasible to buildandmaintainseparatemodelsforeachagent intheeconomy. But itispossibletoinvestigate whethertheheterogeneityinfirm-levelparametersandtime-variationinfirmsizecanbeexploited toimprovetheaggregateinventorymodel. Figure 6 demonstrates how time variation in the “true” aggregate adjustment speed parameter canimpacttheaggregateinventorymodel. Theupperpanelplotsthetimeseriesof“true”aggregate adjustmentspeedsforeachofthethreeinventorymodels. Forexample,thestockadjustmentmodel estimate( (cid:22) ) of the “true” aggregate adjustmentspeed (based on divisions)has a time series range ofabout.015(ora littlemorethan3percent). Althoughthetimevariationissmall,theimpactonthefitofthestockadjustmentmodelislarge. The lower panel shows the residuals from the FP stock adjustment model using MA data and the TVP model with (cid:22) varying only.41 The figure shows that the TVP model fits the data better than the FP model: the R 2 rises from .74 in the FP model to .88 in the TVP model—an improvement 40Inmostotherrespects,theaggregateMAandBEAestimatesconformwellwiththeestimatestypicallyreported intheliterature. Forexample,the R 2 estimatesforthestockadjustmentanderrorcorrectionmodelsaremuchhigher thanforthefirm-levelregressions,typicallywellabove.5. Estimatesotherthantheadjustmentspeedparametersare ofteninsignificantorthewrongsignorboth. Surprisingly,though,thereislittleevidenceofserialcorrelationinthe aggregateregressionsforthissampleperiod. 41For themost part, allowing other parametersto varydoesn'tchangetheresults much largelybecausetheother parametersare insignificant. However, allowingparametersthatare occasionallythewrong sign to vary, especially therealinterestrateparameter (cid:23) 2 ,canactuallyworsenthefitofthemodel. 38

Figure 6 Time-Varying Aggregate Parameter Model ’True’ Time-Varying Aggregate Adjustment Speed Parameters 0.015 0.010 0.005 0.000 -0.005 1986 1987 1988 1989 1990 1991 1992 1993 naeM morf noitaiveD m l g Inventory Stock Adjustment Model Residuals 0.03 0.02 0.01 0.00 -0.01 -0.02 1986 1987 1988 1989 1990 1991 1992 1993 )etaR( dnerT morf noitaiveD FP: R2=.74 TVP(m ): R2=.88 39

of 19 percent. Note that this improvement is about the same magnitude as the improvement in fit reported by Caballero, Engel, and Haltiwanger (1994) for an aggregate employment model that incorporatescross-sectionmomentstoaccount forfirm-levelnonlinearities. 5 Summary and Conclusions This paper informs us of three new facts about inventory behavior. First, there is a broad mix of production smoothing and production bunching bunching firms. Second, firm-level inventory adjustment speeds are much higher than has been inferred previously from aggregate adjustment speed estimates. Finally, although inventory models generally perform no better with firm-level datathanwithaggregatedata,accountingforvariationinthesizedistributionoffirmscandramaticallyimprovethefit ofaggregateinventorymodels. What are theimplicationsofthesenewfacts? Followinginasummarylist: 1. Adjustment speeds—Given that firm-level adjustment speeds appear to be more plausible thanpreviouslythought,anewpuzzlearises: howdowereconcileslowaggregateadjustment speeds with fast firm-level adjustment speeds? One possible answer is that the appropriate periodicity of interpretation is different for aggregate and firm-level adjustment speeds. Firm-leveladjustmentspeedsprobablyshouldbeinterpretedatcalendarfrequency. Withthe averageM3LRDinventorystockequal toabout three-quarters ofa monthofsales,itshould not take a rational, optimizingfirm many months to eliminate an inventory gap. But aggregate adjustment speeds probably should be interpreted at business cycle frequency, where the typical gap lasts about as long as the recession. Thus, the aggregate adjustment speed reveals howfast the aggregateeconomyeliminates theaggregate inventorygap. Slow stock adjustmentarises naturallyfrom theheterogeneityoffirm-level adjustmentspeeds (someof which are quite slow) if demand shocks hit all firms simultaneously. But slow stock adjustment canbe exacerbatedif thereis variationinthetimingofthedemandshockacross firms as well. 2. Persistence—The finding on adjustment speed bias may pertain to other macroeconomic variables as well. To explainthe extreme persistence observed in most real macroeconomic data, economists have resorted to explanations such as adjustment costs, price “stickiness” andwagecontracts,imperfectorincompletemarkets,limitedinformation,less-than-rational 40

expectation formation, strategic behavior, and fixed costs. But the results in this paper support: (1)Trivedi's(1985)contentionthataggregatepersistenceintheformoflongdistributed lagsis simplya consequenceofaggregationacross firms withshortdistributedlags;and(2) Stoker's(1986)findingthat“distributionaleffectscanbemistakenlyinterpretedasevidence of dynamic effects”. Thus, a problem of interpretation arises. Consider an aggregate inventorymodelthatincludesafixed-parametercostofchangingoutputtoaccountforpersistence inthedata. Typically,thisterm isinterpretedas somesortofphysicaladjustmentcosts. But theresultsinthispapersuggestthattheinterpretationofthisspecificationisincorrectontwo grounds: (1) it is unrepresentativeof firm-level behavior; and (2) it misses time variationin persistencethat occurs duetofluctuationsinfirm size. 3. Convexityandaggregation—Theresultsinthispapershowthataccountingforaggregation over convex firm-level behavior can produce dramatic improvements in the fit of aggregate models equal to the improvements obtained by aggregating over nonconvex firm-level behavior. Of course, this finding does not rule out the existence or importance of nonconvex behavior. But it does indicate that disaggregating only to the firm level, where behavior is relativelyconvex,rather than to theplant level,where behavior is relativelynonconvex,can yieldsignificantgainsformacroeconomicanalysis. 4. Room for improvement—The poor overall performance of the standard inventory models atthefirm-levelsuggeststhatthereisstillmuchroomforimprovementindevelopingapplied inventorymodels. Whileit is importantto understandwhyaggregateadjustmentspeeds are soslow,thisstudyseemstoruleoutaggregationbias asthereasonfortheoverallpoorfit of appliedinventorymodels. 5. Production smoothing—The heterogeneity in production variance ratios suggest that future inventory modeling efforts should allow flexibly for both production smoothing and bunchingbehavioratthefirmlevel. TheM3LRDdatademonstratethatthereisasubstantial fraction of firms with smoothing behavior. While there may be some measurement problems in this data—perhaps even enough to cause the average variance ratio to exceed one insteadofbeinglessthanone—thereistoomuchheterogeneitytoforceaninventorymodel tonecessarilyproduceonebehaviorortheother. 6. Disaggregateddata—Thisstudydemonstratesthevalueofworkingwithdisaggregateddata byshowingtwothings. First,it showsthatthedatareveal howanimportantfeature offirm- 41

level behavior (the inventory adjustment speed) had been misinferred from past research using aggregate data. Second, it shows that the data provide an opportunity to account for heterogeneityandtimevariationinsizethatcanmarkedlyimproveasimpleaggregatemodel relatively easily. These achievements motivate expanding the availability of disaggregated data. 42

A Data Appendix This data appendix provides more detailed information about the Census Bureau's M3LRD and howitwas usedinthestudyreportedinthispaper. A.1 Contents Table A.1 lists the main variables in the M3LRD and their definitions. There are three main categoriesofvariables: information,economic,andreportingstatus. Theinformationvariablespertain primarilyto the name, address, and Census identification numbers of each firm's central administrativeoffice,butthesenamesandaddressesarenotnecessarilyassociatedwithphysicalproduction sites. Two other information variables tell about the firm's accounting period and, relatedly, it's monthlytrading-day variationin the economic data. Finally, there is a sample weight for the limitednumberof firms that were involvedinhistoricaltestingofrandom samplingtechniques(most firms haveaweightof1.0). In addition to the two main variables used in this study ( V S and F G ), the M3LRD includes economic variables containing information on other stage-of-fabrication inventories—materials and supplies and work-in-process—and on new and unfilled orders. The total inventory and new ordersidentitiesprovideanopportunitytocheckfordatameasurementerror,asdoestheproduction identity (to some extent). The percentage of nonzero values of the LIFO variables is less than 3 percent. The last group of variables are reporting statuses. These useful variables provide information aboutM3LRDdatareliabilitybyindicatingwhetherthedataarereportedbythefirm ornot. Ifso, theyshowhow the data were reported; if not, theyshow howCensus “filled in” the data. There is one reporting status variable associated with each economic variable and with each versionof the data(describednext). A.2 Data Versions The M3LRD contains three data versions: REPORTED, FINAL, and BENCH. Each version corresponds to a different stage of the data collection, editing, and publicationprocess. REPORTED is data reported by firms in the M3 survey and included in the preliminary published M3 report. FINALisREPORTEDdataeditedbyCensusanalystsinpreparationforfinalpublication,pluslate 43

T A D I I N O R S T W F L L M N N Q T U V W A C D E G H I P R h V D D R A T E D N D A M E I D P T A T E D F T G I F O R E S S O D O R I O S P , K , M a O n e e s e r e a E r p T a b l e A . 1 r i a b l e s i n t h e M 3 L o n g i t u d i n a l R e s e a r c h I n f o r m a t i o n V a r i a b l e s S S C e n t r a l o (cid:14) c e a d d r e s s o f r e p o r t i n g u n i t o r c o M o n t h a n d y e a r C e n s u s r e p o r t i n g u n i t i d e n t i (cid:12) c a t i o n n u m b e r p a n y a (cid:14) l i a t i o n M 3 i n d u s t r y c a t e g o r y N a m e o f r e p o r t i n g u n i t ’ s c o m p a n y O l d I D f r o m p r e c e d i n g p a r e n t c o m p a n y ( i f a A c c o u n t i n g r e p o r t i n g p e r i o d S t a t e a s s o c i a t e d w i t h A D D R E S S T r a d i n g d a y f a c t o r S a m p l e w e i g h t E c o n o m i c V a r i a b l e s F i n i s h e d g o o d s i n v e n t o r i e s P o r t i o n o f T I v a l u e d o n a L I F O b a s i s L I F O a d j u s t m e n t o r r e s e r v e M a t e r i a l s a n d s u p p l i e s i n v e n t o r i e s N e w o r d e r s , d e r i v e d ( V S + (cid:1) U O ) N e w o r d e r s , r e p o r t e d P r o d u c t i o n ( V S + (cid:1) F G ) T o t a l i n v e n t o r i e s ( M S + W P + F G ) U n (cid:12) l l e d o r d e r s V a l u e o f s h i p m e n t s W o r k - i n - p r o c e s s i n v e n t o r i e s R e p o r t i n g S t a t u s e s f o r E c o n o m i c V a r i a b C o r r e c t i o n o f r e p o r t e d d a t a b y C e n s u s a n a l y C o r r e c t i o n o f r e p o r t e d d a t a b y c o m p a n y D e r i v e d b y c o m p u t e r e d i t A n a l y s t e s t i m a t e w i t h n o r e p o r t e d d a t a a v a i E s t i m a t e d b y c o m p a n y E s t i m a t e d f r o m c o m p a n y ’ s h i s t o r i c a l d a t a I m p u t e d f r o m i n d u s t r y ’ s h i s t o r i c a l d a t a R e p o r t e d d a t a r e c e i v e d b y p h o n e R e p o r t e d d a t a r e c e i v e d b y s u r v e y e p o r t i n g s t a t u s v a r i a b l e i s a s s o c i a t e d w i t h e a c h o f t h e 1 o r t i n g s t a t u s e s a r e d e s i g n a t e d R F I , R L I F O , e t c . D a t a B m p a n y i n d i c a t i n p p l i c a b l e a l e s s t l a b l e 0 e c o n o m a g ) i c s c v e o a m r i a b l e s . 44

reported data. Analyst edits include imputation, deletion of non-manufacturing data, reclassification of stage-of-fabrication (SOF) inventory data, etc. BENCH is FINAL data benchmarked by Census analyststotheAnnualSurveyofManufactures (ASM)data,plusotherlatereporteddata. Data versions may differ significantly, but only for certain rare observations. The differences, which occur only in the FINAL and BENCH versions, signify time-series structural breaks in the data caused by corporate reorganization (e.g. a merger), reporting procedure changes (e.g. redefinition of company divisions), and data redefinitions (e.g. changes in a firm's assessment of what is work-in-process inventoryversus finished goods inventory). In the eventof a structural break,Censuscreatestwodifferenttimeseries: onethatreflectsthepropermonth-to-monthgrowth rate of the data based on the old structure and one on the new structure. Consequently, there are two versions of each of the FINAL and BENCH data series: FINAL1, FINAL2, BENCH1, and BENCH2 data series (1 and 2 denote the series on a pre- and post-structural break basis, respectively). Because considerable skill, time, and knowledge about specific industries and firms are required to edit the data versions, I chose to work with the BENCH data. It would be interesting to re-estimate the inventory models using the REPORTED data to discover what affect, if any, the Census editing process has on the parameter estimates. However, it turns out that there is very littledifferenceamongtheversionsanyway. Lessthan.5percentoftheREPORTED,FINAL,and BENCH data differ within any particular month, and less than .02 percent of the BENCH1 and BENCH2 data differ. Where the BENCH versions differed, the BENCH1 version was selected becauseBENCH1 representstheactual datavalue. A.3 Sample selection and restriction The entire M3LRD contains more than 222,000 monthly observations for more than 8,200 divisions of more than 4,300 companies. A subset of the M3LRD sample was selected for this study thatwouldprovidereliabledatawithwhichtotesttheLQinventorymodel. Thefollowingcriteria wereusedtoselect andrestrictthesample: (cid:15) Sample size—Only firms with sufficiently long continuous data samples were included. Sales( V S )datawererequiredtobenonzerofor90outof104possiblemonths. Thiscriterion ensures sufficient observations to maintain acceptable degrees of freedom for econometric estimation. 45

(cid:15) Reported data—Only firms with high percentages of reported, rather than imputed, data were included. For each firm and each economic variable I calculated the time-series percentageofmonthlyobservationsof“reported”dataasthoseobservationswithreportingstatusesA,C,G,K,P,andR.Allotherobservationswereconsiderednotreported,orimputed. Sales and total inventory data ( V S and T I ) were required to be reported for 90 percent or more of all time-series observations. Stage-of-fabrication (SOF) inventory data ( F G , W P , and M S ) wererequiredtobereportedfor70percentormoreofall timeseries observations because the SOF data have lower average rates of reported data. This criterion reduces the extentofmeasurementerrorinthedataattributabletoimputation. (cid:15) Dataidentities—Onlyfirmswithlimitedviolationofdataidentitieswereincluded. Thetwo main identities are total inventories ( T I ) and derived new orders ( N O D ). Firms with data violating these identities for a majority of observations were eliminated.42 (Some of these identities were used to correct data as well; see below.) This criterion also limits the extent ofmeasurementerror. (cid:15) Size distribution—Only firms that fit smoothly into the size distribution were included. BecausetheM3sampleisunrepresentative,andbecausesampleselectionproceduresfurther reduce the sample, the possibility arises of very large outlier firms dominating the results. Consequently,thelargestfirminthesamplewasprohibitedfromhavingannualaveragesales morethantwiceas largeas thenextlargest firm. This criterionpreventstheskewnessofthe sizedistributionfrom distortingtheresultsgiventhenonrepresentativenessofthesample.43 42Usingidentitiestojudgethedatacanbeproblematicbecausethedataarebook-value(dollar)data. Forexample, althoughthetotalinventoryidentityholdsintermsofphysicalquantities,theequationmaynotholdindollartermsif accountingortaxprovisionsallowcompaniestodelayorspeeduptherecordingofinventoryinvestment.Nevertheless, mostidentitiesaresatisfiedbythereporteddata—infact,Censusforcesthenewordersidentity—sotheidentities seemtobeareasonablequalitycheck. 43Tobemoreconcrete,thecompositionoftheM3LRDaggregationpanelissuchthatitispossibletohaveonefirm withmoresalesthanthesumofthesalesofallotherfirmsinthepanel. Consequently,thebehaviorofonefirmcan largelydeterminetheoutcomeoftheaggregateM3LRDanalysis.Becausethereisonlyonefirm,thereislittlechance that it is exactly representativeof all very large firms. Thus, although it would be preferable to include very large firms,insmallsamplesselectionbiascanbesevere. 46

A.4 Data Editing and Imputation Evenafterpassingthesesampleselectioncriteria,somefirmsstillrequiredsomeeditingandimputationofasmallfractionoftheirdataobservations. Obviousdataerrors,suchasrandomrounding errors, were cleaned via visual inspection of time series plots of each data item. Data imputed by the Census Bureau using industry average growth rates were replaced with data imputed using firm-specific time-series methods described below. Other editing and imputationis due to the occurrence ofzero,missing,andidentity-violatingdata. Sales ( V S ) data are considered by Census to be the most reliable data variable and were addressed first. The need for editingand imputationwas minimal because the vast majorityof sales data is nonzero, nonmissing,and reported. In the few cases where they are not, the data were imputedusingtheSASsoftwarePROCEXPANDfunction,whichfitsacubicsplinetothefirm-level data. Totalinventory( T I )dataareconsideredthesecondmostreliabledataandwereaddressednext. Fortherelativelyfewobservationswherethetotalinventorydatarequiredimputation,fittedvalues oftotalinventorywere obtainedfromtheregression T I k t = (cid:12) k V S k t + 1 =i X 2 1 (cid:13) k i S D ik t + (cid:14) k 1 T + (cid:14) k 2 T 2 + (cid:15) k t (19) where k indexesfirms, S D indicatesmonthlyseasonaldummies, T isalineartimetrend,andonly reported V S observationswere included. Use of this regressionwas predicated on the theory that thereisa stable,long-runrelationshipbetweensales andtotalinventories. Stage-of-fabrication (SOF) inventory data are considered notably less reliable than the sales and total inventory data and were addressed last. Some firms' SOF data also exhibit occasional compositionalchangesapparentlyrelatedtochangesindefinition. Forexample,firmsmaychange theirassessmentofwhatisamaterialinventoryversusawork-in-processinventory. Whereobvious definitional changes could be identified, the stocks were redefined to be consistent over time (the stockwith datafor the majorityof observationswas selected). Then, theremainingzero, missing, imputed,andidentity-violatingdatawereobtainedfrom thefittedvaluesoftheregression I k t = (cid:12) I k T I k t + (cid:13) V S k t + 1 =i X 1 1 (cid:13) k i S D ik t + (cid:14) k 1 T + (cid:14) k 2 T 2 + (cid:15) I k t (20) where I = f M S ; W P ; F G g and only reported SOF data observations where the T I identity is satisfiedwereincluded. The (cid:12) I parametersrepresenttheaverageproportionof I in T I ,conditional onsales,andsumtoone. 47

A.5 Trading Day Adjustments The sales ( V S ) data were trading-day adjusted using the Census adjustment factor. Trading-day adjustment is necessary because accounting data collected on a monthly calendar basis are not comparable between months throughout the year for two reasons. First, the number of calendar and work days vary across months. Second, accounting periods vary across firms. Since time periodsintheLQinventorymodelimplicitlyareassumedtobeidentical,thedatamonthsmustbe standardized. Three types of accounting periods are reported in the M3LRD data: 4-4-5 periods (two 4-week months and one 5-week month per quarter); 13-4 periods (13 4-week months per year), andcalendar monthperiods. Althoughmanydifferenttradingdayadjustmenttechniqueshavebeenusedinempiricalwork, the Census Bureau has used some form of the following technique since the 1960s. Weekdays receiveaproductionweightofoneandweekenddaysaweightofone-half;thus,therearesixproductiondaysperweek. Astandardnumberofmonthlytradingdaysisobtainedfromthefollowing formula: T r a d i n g d a y s = m o n t h = 6 (cid:2) ( 3 6 1 5 2 : 2 5 = 7 ) (cid:25) 2 6 : 0 8 9 : (21) Consequently,thetradingdayfactorfor4-4-5reporters inmonth t is T D F t ( 4 (cid:0) 4 (cid:0) 5 ) (cid:25) 2 6 6 : 0 X 8 9 (22) for X = f 4 ; 5 g . Asimilarfactor is usedforthe13-4reporters. Thefactor forcalendar month(M) reporters is T D F t ( M ) (cid:25) W D 2 t 6 + : 0 0 8 : 9 5 W E t (23) where W D t is thenumberofweekdaysand W E t thenumberofweekenddays inmonth t . Thetradingdayfactorisappliedmultiplicativelytoflowvariables,suchas V S ,butnottostock variables, such as the various inventory types. Although the LQ inventory model is specified in termsofinventorystocks,itimplicitlydeterminesthechangeininventories(inventoryinvestment), which is also a flow. Consequently, it is logical to adjust the inventory stocks so that inventory investment(aflow)istrading-dayadjustedinamannerconsistentwiththetrading-dayadjustment of sales. Trading-day adjustment factors for stock variables, denoted T D F (cid:3) , can be derivedfrom thetrading-dayfactorforflowvariables, T D F ,usingtheequation T D F (cid:3) I t = T D F I t + ( T D F (cid:3) I ;t(cid:0) 1 (cid:0) T D F I t ) ( I t(cid:0) 1 = I t ) (24) 48

where I = f M S ; W P ; F G ; T I g denotes the inventory type and t > 0 . By assumption, 1 : 0 T D F (cid:3) = at t = 0 . Experimentation with stock trading-day factors revealed they had no substantive or systematic effect on the construction of aggregate data, on the basic time-series patterns of the data,oronfirm-leveloraggregateregressionresults,andthustheywerenotused. A.6 Other Data Adjustments AnumberofotheradjustmentsweremadetotherawM3LRDdata: (cid:15) Deflation: Firm-level price data were not available, soall nominal data were deflated at the firm level with highly disaggregated fixed-weight industry-level price deflators (1987=100) fromtheBureauofEconomicAnalysis(BEA),asdescribedinHinrichsandEckman(1981). Sales ( V S ) deflators are at the M3 industry category level and all inventory deflators are at the two-digit SIC industry level. Although the deflators are not firm-specific, the breadth of product diversification in many of the large M3 companies increases the suitability of industry-leveldeflators. (cid:15) LIFO inventory adjustment: No adjustment was made for differences in inventory valuation methodology, specifically differences between LIFO and non-LIFO methods. Since January of 1987, Census has collected all nominal inventory data on a current-cost, or pre- LIFO, basis, which eliminates the need for a LIFO adjustment. During 1985-86, the extent of LIFO inventory accounting reported in the M3LRD is less than 3 percent of all observations during that period. Further, the usefulness of LIFO data is limitedbecause most firms usingLIFOusuallyonlymaketheappropriatecalculationsannuallyorquarterlyratherthan monthly. (cid:15) Detrending and deseasonalizing: Because the M3LRD contains time-series data, nonstationarity is an issue. Following inventory literature tradition, e.g. Blinder (1986a), the data weredetrendedanddeseasonalizedat thefirm levelwiththefollowingregression: l o g ( X k t ) = 1 =i X 2 1 (cid:12) k i S D ik t + (cid:13) 1 k T + (cid:13) 2 k T 2 + (cid:14) D 8 7 t + (cid:15) k t (25) where X = f V S ; T I g and D 8 7 is a dummy variable for potential intercept shift due to valuation changes (one in months from January, 1987, forward and zero prior to that). The D 8 7 adjustment is only made where (cid:14) is positive and significant (because the switch from 49

LIFO to current cost will increase the nominal value of inventories), and where the firm experience abnormally large growth rates in all inventory stocks (but not sales) in early 1987. A.7 M3 Sampling Methodology and Sample Selection TheCensusBureauconductsthreemaineconomicsurveysofmanufacturing: (1)thequinquennial CensusofManufactures(CM),whichcoverstheuniverseofmanufacturingplantsandcompanies; (2) the Annual Survey of Manufactures (ASM), which covers a probability sample of plants and companies; and (3) the monthly Manufacturers' Shipments, Inventories, and Orders (M3) survey, whichcoversanon-probabilitysampleofabout1,700companies. Thus,theCMandASMsurveys produceunbiasedestimatesofmanufacturingdatalevels,buttheM3surveydoes not. Census designs the M3 sample as follows. It attempts to include all companies with 1,000 or more employees plus about 60 percent of an arbitrarily selected sample of companies with 100 to 1,000 employees. The 1,000 employee cutoff roughly corresponds to $500 million in annual sales and pertains to about 550 companies. The original M3samplingframe, establishedin 1962, includedasampleofcompanieswithfewerthan100employees. Censusdroppedthesmallcompanies in1963becausetheresponserate was toolow. Sincethen,resamplingoccurs occasionallyto improvecoverageincertainindustriesbutCensusmakesnosystematicefforttoupdatethesample. See BureauoftheCensus (1992)formoredetailsaboutthesurvey. Without probability-based sampling weights, the M3 data levels cannot provide unbiased estimates of the universe. However, there is some chance that the growth rate of the M3 data is an unbiased estimate of the universe growth rate. Thus, the Census Bureau constructs a link-relative (L)growth-rateestimatoroftheuniverseinventorystock, I = P K k = 1 I k t ,whichis I L m y = m Y t = 1 " I I M m M m (cid:0) y 1 ;y # I A y (cid:0) 1 (26) where subscript m denotes month and y denotes year; I M m y is the aggregate of all M3 companies; and I A y isthe(unbiased)aggregateinventorystockforyear u fromtheASM.44 Unfortunately,even thisaggregateestimatorisnotrepresentativebecauselargefirms tendtohavemuchloweraverage growth rates than small firms (see Dunne, Roberts, and Samuelson (1989), for example). Wakim 44TheDepartmentofLabor,BureauofLaborStatistics,usesasimilartechniquetoconstructmonthlyemployment estimatesfromitsestablishmentsurvey. 50

(1986) found some evidence that the M3 data tended to underestimate the ASM growth rates, primarily because the M3 data was not representative for about half of the industries examined. However, the Bureau's M3 Branch later found that the M3 level estimate for total manufacturing actually overestimatedthe ASM estimates,whichthemselves badlyunderstated the CM levels(in 1982).45 AsecondM3sampleselectionshortcomingtakenupinthisstudyisnonresponse. Becausethe M3 survey is voluntary, important differences may arise between responding and nonresponding firms. For example, if economically distressed firms stop reporting data (to cut costs) but prosperous firms do not, then the M3 link-relative growth rate would likely be biased. Furthermore, firmsmaytemporarilystopreportingbutnotbedroppedfromthesample. Fornonreportingfirms, Censusimputestheirdatausingindustry-levelgrowthrates. Totheextentthatthereisconsiderable heterogeneity among firms, random imputation with industry growth rates is likely to impart bias in the firm-level data as well as in the link-relative estimator.46 To avoid this potential bias, this study uses a panel of M3 firms with only minimally imputed data and re-imputes the data with firm-levelstatisticalmodels. 45Althoughthe ASMproduces, in principle, unbiasedestimatesof theuniverse, it too hasdifficultieswith biases enteringbetweencensuses(seeDavis,Haltiwanger,andSchuh(1990)foradescriptionofthisbenchmarkingproblem). Inanycase,thisstudycannotcorrectfor,orstudytheeffectsof,thissampleselectionproblem. 46Onlyfirmswithreporteddataforatleasttwoofthethreemostrecentmonthsareallowedtoenterintothelinkrelativecalculation. 51

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