Hodrick Prescott 1997 Postwar U.S. Business Cycles an Empirical Investigation

Postwar U.S.Business Cycles: An Empirical Investigation Author(s): Robert J. Hodrick and Edward C. Prescott Source: Journal of Money, Credit and Banking, Vol. 29, No. 1 (Feb., 1997), pp. 1-16 Published by: Blackwell Publishing Stable URL: http://www.jstor.org/stable/2953682 Accessed: 10/07/2009 11:45 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. Blackwell Publishing is collaborating with JSTOR to digitize, preserve and extend access to Journal of Money, Credit and Banking. http://www.jstor.org As Lucas(1981)hasemphasized.Both are research associates of the National Bureau of EconomicResearch. HODRICK is NomuraProfessorof InternationalFinance at the GraduateSchool of Business.We documentthe natureof the comovements of the cyclical componentsof a varietyof macroeconomictime series. Journal of MoneyvCreditvand Banking. Chari. We find that these comovements are very differentthan the correspondingcomovements of the slowly varying trendcomponents. No. economy. ColumbiaUniversity. BusinessCycles: An EmpiricalInvestigation We propose a procedurefor representinga time series as the sum of a smoothly varying trendcomponentand a cyclical component.Priorto Keynes'General Theory. combinedwiththe attemptto reconcilethe observationswith an equilibrium theory. THE PURPOSE OFTHISARTICLE is to documentsomefeatures of aggregateeconomicfluctuations sometimesreferredto as businesscycles. PRESCOTT PostwarU.In particular. and JohnH. KennethJ. Nelson. ROBERT J.was regardedas themainoutstanding challengeof economicresearch. we apologize to the many authorswho have used the Hodrick-Prescottfilter and studied its propertiesin the interveningeighteen years since its original development.The fluctuations studiedarethosethataretoo rapidto be accountedfor by slowlychangingdemographicandtechnologicalfactorsandchangesin the stocksof capitalthatproduce seculargrowthin outputpercapita.aggregateeconomicvariablesin capitalisteconomiesexperiencerepeatedfluctuations abouttheirlong-termgrowthpaths. 29. Since we did not updatethe citations.S. This paperis substantiallythe same as our 1981 workingpaper.EDWARD C.The only majorchange to the paperis the additionof an Appendixof Tablesthatmirrorour originalsand containdataending in 1993. We also thankthe WhartonEconomic ForecastingAssociates for providingthe data. Sargent.ROBERT J. 1 (February1997) Copyright1997 by The Ohio State University Press . Vol.V. The investigationuses quarterly datafromthe postwarU. PRESCOTT is Regents'Professorat the University of Minnesotaand Advisorto the FederalReserveBank of Minneapolis. the studyof theserapidfluctuations. Lars Peter Hansen. Wood for comments.we thankRobertAvery.S. Singleton. We also acknowledge helpful comments by the participantsat the 1979 SummerWarwickWorkshopon Expectationand the money workshops at the Universities of Chicago and Virginia and at Carnegie-MbllonUniversity. V. Thomas J. CharlesR.AlthoughtheKeynesianRevSupport of the National Science Foundationis acknowledged. HODRICK EDWARD C. Themaintained hypothesis.andlittlechangein thehoursof employment per capitaor household.l Ourstatistical approachdoes not utilizestandardtime seriesanalysis.consumption. andSingleton(1980). thefailureof theKeynesianTheoryin the 1970shas causedmanyeconomiststo wantto returnto the studyof business cycles as equilibrium phenomena. b).Sims(1980. is thatthegrowthcomponent of aggregateeconomictimeseriesvariessmoothlyover time.the correlation withcyclicaloutputis evennegative. Lucas(1980)intexprets theworkof Mitchell(1913)in a similarlight.a.NelsonandPlosser(1980).Neftci(1978).In thecase of the cyclicalcapital stocks in both durableand nondurablemanufacturing industries. Section2 presentsour findingsregardingthe comovementsof theseserieswith the cyclicalcomponentof realGNP.We do think our approachdocumentssome interesting regularities.basedupongrowththeoryconsiderations.as well as anexamination of thecyclicalcomponentsof prices.In theirsearchfor an equilibrium modelof the businesscycle.2 Ourview is thatno one approach dominatesall the othersandthatit is best to examinethe datafroma numberof differentperspectives.Thethesisof thispaperis thatthesearchfor an equilibrium modelof the businesscycle is only beginningandthatstudyingthe comovementsof aggregateeconomicvariablesusingan efficient.Growthis characterized by roughly proportional growthin (percapita)output. usingalternative methods.andnominalandrealmoneybalances.Sargent andSims(1977). moderneconomistshavebeen guidedby the insightsof Mitchell (1913)andotherswhohaveusedtechniquesof analysisthatweredevelopedpriorto thedevelopment of moderncomputers. Severalresearchers.easilyreplicable techniquethatincorporates our priorknowledgeaboutthe economywill provide insights into the featuresof the economy that an equilibriumtheory should incorporate. 2.AND BANKING olutionredirected effortawayfromthisquestionto theone of determining the level of outputat a pointin timein disequilibrium.Weproceedin a more cautiousmannerthatrequiresonly priorknowledgethatcanbe supported by economictheory.Ourpriorknowledgeconcerningtheprocessesgenerating thedatais notof thevarietythatpermitsusto specify a probability modelas required forapplication of thatanalysis.investment. capitalstock andproductivity (outputperhour). 1.Cyclicalconsumption variesonly one-halfandinvestmentthreetimesas muchas doescyclicaloutput.Anotherdifferenceis in thevariability of componentsof aggregatedemand.thecyclicalvariations in outputariseprincipally as the resultof changesin cyclical hoursof employmentand not as the resultof changesin cyclicalproductivity or capitalstocks.In contrast. ExamplesincludeLitterman andSargent(1979).haveaddedandareaddingto our knowledgeof aggregateeconomicfluctuations. CREDIT. Thisstudyshouldbe viewedas documenting somesystematicdeviationsfromthe restrictions uponobservations impliedby neoclassicalgrowththeory. Wefindthatthe natureof thecomovements of thecyclicalcomponents of macroeconomictimeseriesareverydifferentfromthecomovements of theslowlyvarying componentsof the corresponding variables. . The sensein whichit variessmoothlyis madeexplicitin section1.2 : MONEY.Section3 examinesthe serialcorrelation propertiesof a numberof the series. interestrates. estimatingthe cyclicalcomponentwouldbe an exercisein moderntimeseriesanalysis.thiscomponenthas alreadybeenremovedby thosepreparing the dataseries.There it is calledtheWhittaker-Henderson TypeA method(Whittaker 1923)of graduating or smoothingmortalityexperiencesin constructing mortalitytables. Ourconceptualframeworkis thata given time seriesYtis the sumof a growth componentgt anda cyclicalcomponentct: Yt=gtict fort= 1.thereis also a seasonalcomponent.gt mustbe arbitrarilynear some constant. .Bt. Theparameter Ais a positivenumberwhichpenalizesvariability in the growthcomponentseries.Theseconsiderations leadto the followingprogramming problemfordetermining the growthcomponents: T T Min { E ct2+ A E [(gt-gt-l)-(gt-l-gt-2)]2 {g.If our priorknowledgeweresufficientlystrongso thatwe couldmodelthegrowthcomponentas a deterministic component. Denison1974).ROBERTJ. . amongothers. 3.}. particularly in theactuarial sciences.Fora sufficientlylargeA.T. attheoptimumall thegt 1 . WethankPaulMilgromforbringingto ourattention thattheprocedure we employedhasbeenlong usedin actuarial science.Thelargerthevalueof A.Growththeoryaccounting(cf.closelyrelatedmethodsweredevelopedby theItalianastronomer Schiaparelli in 1867andin theballisticliterature in the earlyfortiesby.=-1 t=1 t=1 ) (2) wherect = Yt.computingthe cyclicalcomponentwouldbejusta matterof calculatingthedifferencebetweentheobservedvalue andthe growthcomponent. (1) Ourmeasureof the smoothnessof the {gt} pathis the sum of the squaresof its seconddifference. DECOMPOSITION PROCEDURE The observedtime seriesareviewedas the sumof cyclicalandgrowthcomponents.theiraverageis nearzero.3 As pointedout in Stigler's(1978)historicalreviewpaper.the smootheris the solutionseries.This impliesthatthelimitof solutionsto program(2) as Aapproaches infinityis theleast squaresfit of a lineartimetrendmodel.vonNeuman. Ourpriorknowledgeis not of this variety. If growthaccountingprovidedestimatesof the growthcomponentwitherrors thatweresmallrelativeto the cyclicalcomponent. . Ourmethodhasa longhistoryof use.plus a stochasticprocessandthe cyclicalcomponentas someotherstochasticprocess. . HODRICKAND EDWARDC.Band thereforethe gt arbitrarily neargO+ .Actually.gt.butas the dataareseasonally adjusted. .so thesepowerfulmethodsarenot applicable.Ourpriorknowledgeis thatthe growthcomponentvaries"smoothly" overtime.The ct aredeviationsfromgt andourconceptualframework is thatoverlongtimeperiods.Themethodis still in use.possiblyconditionalon exogenousdata.in spite of its considerable success. PRESCOU : 3 1.is farfromadequateforprovidingsuchnumbers. andin the subsequent periodit wasevensmaller.even aftercorrectingfor cyclicalfactors. CREDIT.are in naturallogarithmsso thechangein thegrowthcomponent. which are treatedas unknownparameters with diffuse priors.4 : MONEY.g_i. efficientKalmanfiltering techniquescanbe usedto computethesegt. one need not inverta (T + 2) by (T + 2) matrix(T is the numberof observations in the sample)as wouldbe necessaryif one solvedthelinearfirst-order conditionsof program(2) to determinethe gt. normalvariableswithmeanszeroandvariancescr2 andcr2 (whichthey arenot).5 Thesenumberschangelittleif Ais reducedby a factorof four 4. how sensitivearethe resultsto the valueof A thatis selected?To explorethis issue. corresponds to a growth rate.41 percent.Table1 containsthe (sample)standard deviationsandautocorrelations of cyclicalrealGNPfor the selectedvaluesof the smoothingparameter as well as statisticsto testforthepresenceof a unitrootin the cyclicalcomponents.in the 1968-73 period. giventhe observations.gt .Thisled us to select < = 5/(1/8) = 40 or A = 1.the conditionalexpectationof thegt.Toproceedas if it werewouldresultin errorsin modelingthegrowthcomponentandtheseerrorsarelikelyto be nontrivial relativeto thecyclicalcomponent. gOand gO.ANDBANKING Valueof the SmoothnessParameter The dataanalyzed..1950-79. given the observedy.Partof thesechangescanbe accountedfor by a changingcapital-labor ratioandchangingcompositionof the laborforce.as is a one-eighthof 1 percentchangein the growthratein a quarter. Ourpriorview is thata 5 percentcyclicalcomponentis moderately large.. The conditionalexpectationof the g. variousothervaluesof A weretried.the annualgrowthratewas4. 5. Forthisreason.g_ 1 are the generalizedleast squaresestimates. the level of the cyclical component. This minimizationhas two elements.600as a valueforthesmoothingparameter.20 percent. The Kalmansmoothingtechnique(see Pagan 1980) was used to compute efficiently the conditionalexpectationsof the g. The tests for the presence of a unit root are augmentedDickey-Fullertests in which the change in the cyclical componentis regressedon a constant. If T is large.with the exceptionof the interestrates. in the 1953-68 period. for t 2 1 are linearfunctions of these parametersand the observations.61 percent. The posteriormeans of gOand gO.gt_l.4By exploitingthe recursivestructure. as shownby McCarthy.only 1.this is importantbecauseinvertinglargematricesis costly and therecan be numerical roundingproblemswhenimplemented on computers. wouldbe the solutionto program(2) when< = (rl/Cr2 As thisprobability modelhasa statespacerepresentation.an infinitevalueforthe smoothnessparameter was notselected. If the cyclicalcomponentsand the seconddifferencesof the growthcomponentswere identicallyandindependently distributed. a sizableandvariableunexplained componentremains.But. Thefollowingprobability modelis usefulforbringingto bearpriorknowledgein the selectionof the smoothingparameterA.The assumptions thatthe growthratehas been constantoverourthirty-year sampleperiod. The largestmatrixthatis inverted usingthe Kalmanfilteringcomputational approachis 2 by 2. is not tenable. The growthrateof labor'sproductivity has variedconsiderably overthisperiod (see McCarthy1978).Kalmanfilteringcanbe performedwithcomputerpackagesthatarewidelyavailable.2.and six lags of the .In the 1947-53 period. Oneissue is. CORRELATIONS SAMPLE PERIOD: A = 400 Standard A = 1600 1..theannual rateof changeof the growthcomponentvariedbetween2.02 .44 -.20 -.80% A = infinity 2..39 ...20 ..1.52 .Themaximum growthrateoccurredin 1964.48 .73 . alongwiththe cyclicalcomponent.thestandard deviation increasesandthereis greaterpersistence.20 .It is noteworthy thatonlytheresultsforthelineardetrending violatethe assumptionthat no unit root is giving rise to nonstationarity in the cyclical component.41 .44 -.Wecautionagainstinterpreting the cyclicalcharacteristic of the differenceas a cycle of long duration.80 1 Order -.44 Order 9 Order 10 Order Order Order Order Unit-Root .If thesamplesize wereinfinite.1-1979. Withourprocedure foridentifyingthegrowthcomponent(A= 1.11 .41 .61 . t-statistic for the coefKcient of (5 percent) -2.Theaverage growthrateoverthe periodwas 3.36 - Test - 5.withthe resultsbeingverydifferent for A = oo.84 . E (3) i=l whereT is thelengthof thesampleperiod.4.65 .56% Deviations GNPFOR OF CYCLICAL DIFFERENT VALUES OF 1950. the presence of a unit component root is more in the cyclical negative than component if the the critical value .00 .Thedifferencesbetweenourcyclical componentsandthoseobtainedwithperfectsmoothing(A = oo)aredepictedin Figure 1.89 (4) Yjti _00 on the or -3.15 .relativeto the variationin the cyclicalcomponent.31 -.57 ..03% 3.3 and4.The smoothnessof the variationin this difference. foreachseriesj T gjt = WitYji..withthe minimaoccurringin 1957andin 1974.94 .84 .44 .3.withanotherpeakof 4.31 .17 .indicatesthatthe smoothingparameter chosenis reasonable. PRESCOTT : 5 TABLE 1 STANDARD THE DEVIATION SMOOTHING AND SERIAL PARAMETER.41 .400. HODRICKAND EDWARDC.12% Autocorrelations Order 2 3 4 .15 to 400 or increasedby a factorof fourto 6.32 Order 6 7 8 . Thesametransformation was usedfor all series:thatis.32 .4 percentin 1950.87 .38 .50 (l One rejects level of the cyclical percent).27 . As Aincreases.25 .01 -.2 A = 6400 1.42 .27 -.30 .14 Order 5 -.4 percent.600).9 percentover the sampleperiod.47 .it would notbe necessaryto indexthesecoefficientsby t and 00 gjt E = i= change Wl in the cyclical component.57 .ROBERTJ...38 -.Suchpatternscan appearas artifactsof thedataanalysisprocedure. 6.l < m.} sequence belongs to the space for which 0 E . \ | 2 o 0|41 \t-01 AW W A T\ l As -1 IV.Whenusingthe statisticsreportedhere to examinethe validityof a modelof the cyclicalfluctuations of an artificialeconomy.The resultsdo indicatethat thereis considerablepersistencein the rapidlyvaryingcomponentof output. CREDIT. where wiw= 0. The abovemakesit clearthatthe dataarebeingfiltered.11168i)] (s) for i 2 0 andwi = w-i forl < o.S. the model'soutputseriesshouldbe decomposedpreciselyas was thedataforthe U.the serialcorrelationof the rapidlyvaryingcomponentof the model'saggregateoutputseriesshouldbe comparedto thesenumbers.8941l'llyj. We requirethat the {y.AND BANKING cydical GNP (x=O) .=-r . -2 -3 cycilcal GNP (X=1600) -4 -5 1950 1952 1954 1956 1958 1960 19621 1964 1966 YEAR 1968 1970 1972 1974 t976 1978 FIG. farfromeithertheendor thebeginningof the sample. may not exist. 1.Theadvantage of usingthe exactsolutionis thatobservations nearthebeginningandtheendof thesampleperiod arenotlost.As any filteraltersthe serialcorrelationpropertiesof the data. See Miller (1946) for a derivation.Thatis.055833sin(0..8941i [0. sequence when the sample is infinite. so ourmethodis approximately a two-waymoving averagewithweightssubjectto a dampedharmonic. the gj. / \ rcyclical GNP (1=1600) (AX :. Otherwise. .the wTarenearwt_i.the reportedserialcorrelations shouldbe interpretedwith caution.056168cos(0.6 : MONEY.11168i) + 0. There are certainimplicit restrictionson the y.6 Fort. Thisis a checkforthe stabilityof the measuresovertime. VARIABILITYAND COVARIABILITYOF THE SERIES The componentsbeingstudiedarethe cyclicalcomponentsandsubsequently all referencesto a seriesrelateto its cyclicalcomponent.Theinvestmentcomponents.The samplestandard deviationsof a seriesis ourmeasureof a series'svariability.even if it were. .Wedo thinkit is important thatall seriesbe filteredusingthe sameparameter A. theR-squared forthe regression 2 cjt= aj + E i= -2 >iGNPt_i (6) for eachseriesj was computed. WechosethismeasureratherthanapplyingsomeF-testfortworeasons. AggregateDemand Components Thefirstset of variablesstudiedaretherealaggregatedemandcomponents.8 percentvalueforrealoutput.consumption of nondurables andstateandlocal government purchasesof goodsandservices.Covariabilities of consumption andinvestmentwithoutputaremuchstrongerthanthecovariability of government expenditures withoutput.It is a numberbetweenzero and one.Therefore.butleador lag realoutput.Only then. andthecorrelation of a series withrealGNPis ourmeasureof a series'scovariability.but the relativemagnitudesof fluctuations of the serieschangelittle.Witha largerA.600arereported. A variablemightbe stronglyassociatedwithrealoutput. would the model's statisticsand those reportedhere be comparable. Thesemeasuresarecomputedfor the firsthalf andthe secondhalfof the sample. PRESCOTT : 7 economy. with one indicatingthatthe best-fitequationis preciselythe samein the firstandsecondhalvesof the sample. Theseriesthatvarytheleastareconsumption of services. Theratioof the explainedsumof the squaresforthisregressionto the explained sum of squaresfor the regressionwhenthe coefficientsare not constrained to be equalin thefirstandthesecondhalvesof thesampleis ourmeasureof stability.we do notthinkthe assumption of uncorrelated residualsis maintainable.includingconsumerdurableexpenditures.The resultsaresummarized in Tables2 and3.as a secondmeasureof the strengthof associationwithrealoutput.in the subsequent analysisonlythe statisticsfor A = 1.as well as for the entire sample. HODRICKAND EDWARDC.ROBERTJ.are aboutthreetimes as variableas output. As the comovementresultswere not particularly sensitiveto the value of the smoothingparameter A selected. the amplitudes of fluctuations arelarger. 2. it is very difficultto deducethe magnitudeof the instabilityfromthe reportedteststatistic.Eachof thesehasstandard deviationless thanthe 1.First. Second. 5 4.698 .7 22.642 .500 .877 .8 26.509 .+ Stability Measure .123 .2 8.468 .377 -.4 14.Theseresultsare summarized betweenhoursandoutput.968 .129 .6 3.2 4.574 .8 6.7 1.152 .829 .5 11.631 .In addiThereis a strongandstablepositiverelationship to thevariabilityin output.0 3.8 8.6 1.2 SAMPLE PERIOD: R2 for Regression Correlationwith Real Output Squared Total Consumption Services Nondurables Durables Total Invest.554 .329 .441 .3 5.266 -.3 First Half 1.367 .922 .602 .8 1.298 .424 .1-1979.454 .3 5.503 .809 .415 .707 .884 .6 10.589 .0 6.029 cJ.917 .6 5.808 .8 11.353 .512 .441 .575 .2 4.125 .4 5. Fixed Residential Nonresidential Equipment Structures Total Government Federal State and Local 1.4 .the tionof output.5 8.9 12.739 .436 .760 .7 1.9 5.7 4.546 .8 4.6 Second Half 1.7 6. thevariabilityin hoursis comparable andoutputis weakandunstablewith poraneousassociationbetweenproductivity deviabeingmuchsmallerthanthe standard the standarddeviationof productivity to notethatwhenleadandlag GNPsareincluded.2 5.378 .884 .0 5.777 .552 .615 .8 Factors of Production andproducThe secondset of variablesconsideredarethe factorsof production in Tables4 and5. Fixed Residential Nonresidential Equipment Structures Total Government Federal State and Local .190 .4 9.509 .It is interesting TABLE 3 OFASSOCIATION WITHGNP ANDMEASURE STRENGTH DEMAND COMPONENTS: AGGREGATE OFSTABILITY 1950.GNP.7 1.834 .095 .225 . = aJ + 2 E fij.4 5.1 4.714 .482 .0 opferReenalt Correlations with Real Output in Percents First Half Whole .408 Second Half GNP .2 Average Standard Deviations Whole Real GNP Total Consumption Services Nondurables Durables Total Invest.8 : MONEY. COMPONENTS: STANDARD DEVIATIONS AND CORRELATIONS WITHGNP 1-1979.684 .262 . tivity whichis outputper hour.7 1.831 .6 1.3 .1 10.Thecontemtion.067 .9 5.071 .258 .7 26.620 .5 4. CREDIT.2 1.1 2.2 .131 61.714 .9 1.510 .908 .170 .298 .637 .785 .781 .119 .AND BANKING 2 TABLE AGGREGATE SAMPLE DEMAND PERIOD: 1950. andtheircorrelations the otherhand.622.0 Inventory Capital Stock Durables Capital Stock Nondurables Hours Work .800 .235 .7 1.8 .600 .838 .728 . goodsindustries.areless Capitalstocks.935 .l First Half Second Half Stocks Employees 1.513 .2 First Half Correlations with Real Output in Percents Whole Second Half 1.Further.453.The in the firstandsecondhalvesof the sample. Correlations betweennominalmoney.4 2.withthe differencesin the correlations instabilityovertimein theserelaexceptionof nominalM1.361 . the increasein theR-squared Monetary Variables Resultsforthe finalset of variablesarepresentedin Tables6 and7.bothin durablegoodsandnondurable stocks.Thisis indicatedby from.309 -.andrealmoneywithGNPareall positive.236 .2 R2 for Regression Correlation with Real Output Squared Capital Inventory Stock Durables Capital Stock Nondurables Hours Week Employees Average 2 fij.453 .7 2.686 -.5 Week .2 .954 .824 .700 .853 .044 .The correlationships.297 .056 .2 1.010 to .672 .velocity.+I t=-2 Stability Measure Stocks Capital Work 2 cjt = aj + Product of Labor .622 .7 1.0 . PRESCOTT : 9 4 TABLE OF PRODUCTION: FACTORS SAMPLE PERIOD: STANDARD DEVIATIONS AND Standard Deviations Whole Real GNP Capital WITHGNP CORRELATIONS 19SO.828 .801 .740 .GNP.8 1.A similarconclusionholdsfor the short-terrn tionsof GNPwiththepricevariablesarepositivein the firsthalfof the sampleand TABLE 5 OF PRODUCTION: FACTORS SAMPLE PERIOD: 1950.210 -.782 .4 .Inventory with to output.773 .4 1.231 associationbetween GNP and productivityincreases dramaticallywith the R-squared increasingfrom.773 . STRENGTH OF ASSOCIATION WITH GNP AND MEASURE OF STABILITY 1-1979.100 .0 1.7 1. suggestconsiderable interestrate.6 1.havea variabilitycomparable outputarepositive.5 1.274 -.178 -.7 2.129 .on variablethanrealoutputandnegativelyassociatedwithoutput.507 -.257 to .854 .257 .the strengthof associationof inventorieswithGNPincreaseswhenlag andleadGNPsareincludedin theregression.1-1979.0 1.732 .9 1.6 Productivity 1.896 .1 .010 .185 .820 .831 -. HODRICKAND EDWARDC.0 1.ROBERTJ. 9 1.738 .1 .037 .3 Correlationswith Real Output Second Half Whole First Half Second Half 1.376 .3 1.280 .661 . SERIAL CORRELATIONPROPERTIESOF DATA SERIES A sixth-order autoregressive processwas fit to a numberof the serieswhichdisplayedreasonablestablecomovementswithrealoutput.567 .6 1.7 .Figure2 presentsplotsof the unitimpulseresponsefunctionsforGNPandnineotherseriesfortheestimated TABLE 7 MONEYANDPECEVARIABLES: STRENGTH OFASSOCIATION WITHGNP ANDMEASURE OFSTABILITY SAMPLE PERIOD: 1950.2 2. .529 .490 .187 .223 814 799 19 1. 3.10 : MONEY.675 .010 .2 1.381 .684 .260 .319 ..316 . .408 .3 negativein thesecondhalfwiththecorrelation fortheentireperiodbeingsmalland negative.0 1.261 .06 1.9 1.330 .0 1.1 1. .0 1.378 . ANDBANKING TABLE 6 MONETARY ANDPRICEVARIABLES: STANDARD DEVIATIONS ANDCORRELATIONS WITHGNP SAMPLE PERIOD: 1950.865 1.255 175 .818 221 .565 .131 .4 .281 .828 510 .9 27 .175 .8 2.2 22 CoITelationwith Real Output Squared for Regession CJf= atJ + 2 E ISJIGNPr+I t= -2 Stability Measure M1 Nominal Value ve OClty Real Value M2 .8 1.230 .445 .1-1979.415 .24 .0 1..506 . > omlna va ue Velocity Real Value InterestRates Short Long Price Index GNP Deflator CPI .481 *057 . 1 1.5 .7 1.649 .495 .2 su Whole Real GNP M1 Nominal Value Velocity Real Value M2 Nominal Velocity Real Value InterestRates Short Long Price Indexes GNP Deflator CPI ndardDeviations in Percents First Half 1.CRED1T.432 .437 .428 .665 .640 .614 . .079 .650 .1-1979.3 1.480 .748 .9 .378 .0 1.371 .749 .4 1.239 -.801 .724 .678 .8 2. * . uz o o o - * WORKWEEK \ o o - o r uz . . EMPLOYEE t o ./ .2. .NTORY INVE \ o \ /- - UZ cr. ' '-o b 31 20 110 ' Po T o - of M1 X d CM w | \ 210 ' s o °. Unit Impulse Response Function 20 3|0 . I o CJURABLE CAPITAL \ -- UZ - 20 30 \ UZ o o o 1'0 20 r\ - 0 l / \ CM 3'0 b ' l l 1o - \ - - a \ CM o _ cM - VELOCI5M1 \ 2 - D o | ) 30 0 | § 10 1 FIG._ - o o - a CSJJ o CONSUMPTION \ @ GNP o o \ o V-- CM 310 20 11O 20 20 10 N - FIXEDINVESTMENT \ t j° o HOUR \ \ o / \ o 1b 110 1 5 3 T w 1'o l X X | | | | do 3l0 - 0- uz o . 1A.6. The interestrateseriesare fromthe FederalReserveBulletinand are constructedfromthe monthlyseriesin Tables1. The patternsfor conis in the sumptionandinvestmentaresimilarexceptthatthe peakfor consumption andeachof its threecomponents(not initialperiod.The functionfor consumption pictured)aresimilarto the one for the aggregate. Treasurybill rate.4.All labordatain TablesA. One must take care in interpretingthe responsepattern. be the innovationsand ct Ct= i Zia.parameter0.33 and 1.The short-term ForecastingAssociationQuarterly taxablethree-monthU.6 andA.7 alsocomefromCitibase.35.ANDBANKING 12 : MONEY.Themonetaryvariableshaveverydifferentresponsepatterns. Others are the spectrum.Dataforthepriceseriesin TablesA.indicatingsepropertiesverydifferentthanthoseof realoutput.Two moving averageprocesses can be observationallyequivalent (same autocovariancesfunction) yet have very differentresponsepatterns.exceptfor the greater amplitude. the long-termbonds. It is just one way to representthe serial correlationpropertiesof a covariance stationarystochastic process.publishedby wereobtainedfromtheBusinessCycleIndicators of Commerce. 1-A.S.8. and the long-terminterestrate.1to 1993.39 in periodeight.5 andA.7 containdatafrom 1947.Surveyof Current Businessas anstockdatain TablesA.Themaximumamplitudeof theresponseis muchgreater. RealM1 andReal M2 HistoricalDiskette. function.being2.7ThefunctionforGNPincreasesinitiallyto a peakof 1. andoccursslightlyovera yearsubsequent (notpictured)is simigoodsindustries patternforthecapitalstockin thenondurable larthoughthemaximumamplitudeis smaller. rialcorrelation Thereis a dramaticdifferencein theresponsepatternforthecapitalstockin durable goodsindustries. Letting a.beto the unitimpulse.is verysimilarto the patternfor GNP.Weusedquarterly ital stocksto constructquarterlyseries.S.6 come fromCitibase.Thecapital Business. All datafor TablesA.S.3 comefromthe NationalIncomeandProductAccounts:HistoricalNIPAQuarof Commerce. equals the value of the unit responsefunction in period i.S.The ing about3.CRED1T.U. Department terlyData. yieldon U._i i=o be the invertiblemoving averagerepresentation.Nominalserieswerecalculatedby multiplying the U. Department by the GNPdeflator.6 comefromthe Surveyof Current investmentseriesfromtheNIPAwiththeannualcapnualseries. APPENDIX All the datafromthe originalpaperwereobtainedfromthe WhartonEconomic interestratewas the DataBank. 7. however. Forbothcapitalstocksthe peaksin the unitresponsefunctionarein periodfive. Government TablesA.Theaveragework-weekpattern.begins to declineimmediatelyand the periodof dampedoscillationis shorter. the autoregressive representation. The patternfor totalhoursandthe numberof employees.5 andA.and the autocovariancefunction. We chose the invertiblerepresentationbecause it is unique.15 autoregressive in periodone andhas a minimumof -. . 453 .39 -.047 Cj.52 .571 .121 .731 .73 .012 .30 -.8 3.544 .796 .873 .470 . = 0tj + 2 22 jiGNP.348 .637 .9o .64 .798 .4 4.74 .397 .86 .209 ..2 0. Fixed Residential Nonresidential Equipment Structures Total Government Federal State and Local .732 .517 .540 .659 .200 .557 .52 VALUES 44.707 .05 -. CORRELATIONS SAMPLE PERIOD: OF CYCLICAL GNPFOR 1947.396 .662 .469 .1 6.16 .4 5.9 0.9 5.0 21.715 .8 1.02 -.685 ..511 .37 -.19 26 28 98 -5.40 -.469 .8 5.5 9.4 A = 400 Autocorrelations Order 1 Order2 Order3 Order4 OrderS Order6 Order7 Order 8 Order9 Order 10 Unit-Root Test TABLE A = 1600 1.536 .112 .81 .2 1.820 .1-1993.6 1.34 09 .6 5.27 -.528 -.69 .8 1.872 .96 .500 .745 .1 .9 6.6 6.5 5.8 TABLE A3 AGGREGATE DEMAND COMPONENTS: STRENGTH OFASSOCIATION WITHGNP ANDMEASURE OFSTABILITY SAMPLE PERIOD: 1947..86 .436 .1 4.39 .1-1993.47 -2.787 .2 5.955 .9 1.746 .91 A2 AGGREGATE DEMAND COMPONENTS: STANDARD DEVIATIONS AND CORRELATIONS WITH SAMPLE PERIOD: 1947.infinity 2.520 .216 .8 3.462 .702 .7 31.669 .1 4.6 6.7 1.34 .91 .929 .03 -.229 .1 10.164 .719 .015 61.7 10.482 .8 0.14% .43 -.224 .6 5.500 .792 .47% DIFFERENT A = 6400 1.213 .9 1.457 .1 6.5 6.94% .40 -.871 .9 1.927 .220 .+ Stability Measure .827 .32 -.5 1.2 5.35 -6.80 .4 9.580 .58 .80% A = .810 .4 12.324 .123 .9 15.808 .512 .875 .558 .TABLEA1 STANDARD DEVIATION OF THE SMOOTHING Standard Deviations AND SERIAL PARAMETER.18 .2 24.40 . Fixed Residential Nonresidential Equipment Structures Total Government Federal State and Local GNP Average opferReenalt Correlationswith Real Output Whole First Half Second Half Whole First Half Second Half GNP 1.7 1.515 .53 .350 .1 4.22 -.4 0.4 R2 for Regression Correlationwith Real Output Squared Total Consumption Services Nondurables Durables Total Invest.0 5.3 5.5 10.5 1.53 .6 10.21 -.63 .080 .755 .8 1.4 StandardDeviations in Percents RealGNP Total Consumption Services Nondurable Durables Total Invest. 1-1993. 436 1.8 2.764 .6 0.032 .247 -.GNP.7 2.7 1.5 0.1 2.8 2.404 ..1 1. 1 .828 .9 1.0 1.7 3.619 .779 .547 .0 1.510 .239 .778 .057 TABLE A6 WITH GNP ANDCORRELATIONS DEVIATIONS STANDARD MONETARY ANDPRICEVARIABLES: 1947.368 .992 .8 2.358 -.672 ..324 .8 1.874 .151 .994 .778 .337 .883 .8 1.911 .357 .680 .9 1. .1 1.585 -.6 1..228 . + ' Squared Capital Stocks Inventory Capital Stock Durables Capital Stock Nondurables Hours Work Week Employees Average Productof Labor S J t=-2 .260 .1 993.356 .967 .1 3.2 0.801 .7 .1-1993.4 withRealOutput Correlations SecondHalf Whole FirstHalf Standard Deviations in Percents FirstHalf SecondHalf Whole RealGNP CapitalStocks Inventory CapitalStockDurables CapitalStockNondurables Hours WorkWeek Employees Productivity TABLE 1.327 .1 1.2 2.49l 1.260 .9 2.8 1.+I Stability Measure ' .728 .156 -.0 1.542 .9 .360 AS FACTORS OF PRODUCTION: STRENGTH OF ASSOCIATION WITH SAMPLE PERIOD: 1947.465 .858 .3 2.8 1.222 .4 GNPAND MEASURE OF STABILITY R2 for Regression CoITelationwith Real Output c.6 2.635 -.328 .003 .783 .125 .1 1.6 1.510 .933 .319 .7 1.335 .347 .685 .058 .1 1.4 1.104 .4 .1 0.5 1.4 1.2 1.8 wole - with RealOuWut colTelations Second Hif FirstHif - 2.8 1.324 .0 1.0 1.020 1.373 .8 1.4 2.2 1.8 .6 0.318 .151 .8 .5 0.1-1993.0 1.sx.4 SAMPLE PERIOD: Real GNP M1 Nominal Value Velocity Real Value M2 Nominal Velocity Real Value InterestRates Short Long Price Indexes GNP Deflator CPI Deviations inPercents Standard Second Hif Whole FirstHif 1.850 .869 ..989 .387 .021 .TABLEA4 WITHGNP FACTORSOF PRODUCTION:STANDARDDEVIATIONSAND CORRELATIONS SAMPLEPERIOD:1947.055 .808 .5 0.5 2.4 1.6 2.219 .0 1.475 .860 .605 . 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