CHRISTOPHER
A. SIMS
Universityof Minnesota
Output
Labor
in
and
Input
Manufacturing
THE RELATION between output and labor input in manufacturing is important in quantitative analysis of economic fluctuations. Empirical work
on this topic generally supports the conclusion that labor inputs respond
with a delay, and not in full proportion, to changes in output. Thus, variations in output are accompanied by corresponding variations in average
labor productivity. This phenomenon is something of a paradox, for shortrun increasing returns to labor, or SRIRL, are difficult to rationalize if the
sector being explained is assumed to operate on a static production function
in which labor is the most variable factor. One need not make this assumption, and a variety of plausible deviations from it, which can explain the
qualitative empirical results, have been advanced. It also seems possible
that some of the apparent SRIRL reported in previous studies reflects
statistical bias in estimating the labor-output relation.
Note: Researchresults describedin this paper were obtainedwith financialsupport
of the National ScienceFoundationgrantGS 36838.Some supportwas suppliedalso by
the Cowles Foundation.
695
696
Brookings Papers on Economic Activity, 3:1974
Most previous regression studies of this relation have used quarterly
data,1 and the estimates have often suggested that most of the response of
labor input comes in the same quarter in which output changes. When an
estimated lag distribution has such a form, the discrete estimates may not
be a reliable guide to the underlying dynamics.2
Also, most previous studies have estimated single-equation distributed
lag regressions, treating labor input as the dependent variable. The justification for doing so appears to be that output changes are thought Ito
be causally prior to employment changes in the firm-levelbehavioral mechanisms most often invoked to explain SRIRL. Since these mechanisms do
not generate a theory of the error term in the aggregate labor-output relation, they do not justify making labor the dependent variable in a regression.3 Indeed, if some of the error in the relation resides in output, inappropriate regressions of labor on output might imply spuriously large estimates
of SRIRL.
The statistical procedures of this paper sharpen estimates of the laboroutput relation by using monthly data, by estimating regressions with both
output and labor as the dependent variable (testing the hypothesis that
right-hand variables are exogenous), and by using only weak maintained
hypotheses about the forms of estimated lag distributions.
This paper shrinks the paradox of SRIRL in two ways. First, the empirical results show that the response of manhours of production workers in
manufacturing industries to a change in output is essentially complete
within six months (which was implicit in the results of earlier studies), and
that the total response is fully proportionate (which was not). It should be
noted that the empirical work in this paper is confined to production workers and in no way examines the existence of SRIRL for other workers,
whose employment is presumably less variable.
Second, the theoretical discussion shows that, once the formation of expectations is treated realistically, a standard dynamic theory of decisionmaking under uncertainty does not imply that the sum of coefficients in
1. An importantexceptionis Ray C. Fair, The Short-RunDemandfor Workersand
Hours (Amsterdam:North-Holland, 1969). Fair uses monthly data, as does this study,
but works at a finerlevel of disaggregation.
2. For discussionof the effects of temporalaggregationon distributedlag relations,
see ChristopherA. Sims, "Discrete Approximationsto Continuous Time Distributed
Lags in Econometrics,"Econometrica,Vol. 39 (May 1971), pp. 545-63.
3. Correspondingly,the fact that in the theory of consumerbehaviorpriceis causally
prior to quantityin the consumer'sdecision does not imply that regressionsof market
quantityon price can be interpretedas demandcurves.
ChristopherA. Sims
697
estimated lag distributions of labor on output representsthe static optimum
response of labor to output. Thus even where the total response of labor to
output is less than fully proportionate, a nonconcave static production
function is not implied.
The SRIRL Paradox
It is mildly paradoxical that fluctuations in output should tend to induce
less than proportionate fluctuations in labor inputs. Presumably, most
other inputs are less flexible than labor and therefore vary less as output
changes. If a production function relating output to inputs shows constant
returns to scale, percentage changes in output are a weighted average, with
positive weights summing to one, of percentage changes in inputs. But the
point of the empirical SRIRL phenomenon is precisely that output fluctuates proportionately more than does labor, which is only one of the inputs
to production. Since there is no strong empirical evidence of aggregate increasing returns to scale in the United States, and increasing returns are
difficult to reconcile with the standard competitive model of microeconomics, SRIRL is a paradox.
If inventories exist, output measured by deflated sales obviously can vary
more than actual production over short periods. Even if labor input responded immediatelyand more than proportionately to production changes,
it might then respond with a lag to "output" changes, in a pure timing
effect. If production of finished goods were measured directly, similar pure
timing effects might arise if there were inventories of goods-in-process or
postponable maintenance tasks in the production process. But lags due to
pure timing effects cannot explain the lack of proportionality in the eventual total response of labor input to properly measured output.
The more widely accepted explanations of SRIRL rest on the cost of
adjusting labor input. In one version, part of the work force is regarded as
fixed over the relevant time horizon, either by contract or by the need to
man the fixed capital stock regardless of output. Alternatively, the work
force is treated as homogeneous, but changing it rapidly is assumed to
incur costs.4
4. Two pieces of work based on theory of this type are Thomas A. Wilson and Otto
Eckstein, "Short-RunProductivityBehaviorin U.S. Manufacturing,"Review of EconomicsandStatistics,Vol. 46 (February1964),pp. 41-54; and M. IshaqNadiriand Sherwin Rosen, A DisequilibriumModel of Demandfor Factors of Production(Columbia
UniversityPressfor the National Bureauof Economic Research, 1974).
BrookingsPaperson EconomicActivity,3:1974
698
Clearlythere are plausibleexplanationsof labor-outputrelationsthat
wouldexplainSRIRL over varyingtime horizons.Carefulstatisticalestimationis neededto evaluatetheirquantitativeimportanceand the implied
behavioralstructure.The followingsectionspresenta discussionof statistical issues and of the relationbetweenestimationand structureas these
applyto the analysisof the labor-outputrelation.Quantitativeresultsare
presentedand discussedin subsequentsections.
of the Regressions
StructuralInterpretation
Supposethat in everymonth, t, each firm in the industryhas an ideal
levelof labor,L*(t).Thisidealleveldependsonly on variablesthe firmcannot control,suchas marketinputand outputprices,and possiblyshiftsin
the firm'sown short-rundemandcurve.Wereit not for the costsassociated
with adjustingactuallabor,L(t), they would alwaysequalL*(t).To illustratehow suchcosts couldgive riseto a distributedlag relationof L to L*,
assumethat, in any period,t, the tradeoffbetweenthe costs of deviating
from L* and the costs of rapidlychangingL is determinedby the cost
function5
C(t) = a[L(t)
-
L*(t)]2 + b[L(t)
L(t
-
-
1)]2.
If the firmchoosesL(t) at time t to minimizeexpecteddiscountedcosts,
Et ,
C(t + s)R8],
whereR is the discountfactor,it can be shownthatthe firm'sbehaviorwill
be describedby
(1)
L(t)
-
AL(t
-
1) = (1
-
A)(1
-
B)
E2B8EtL*(t + s),
8=0
whereEt denotes the expectationbased on informationavailableup to
5. It would be more naturalto formulatethe dynamicoptimizationproblemin continuous time. A short appendixin which this is done and in which the implicationsof
rational expectationsand the existence of inventoriesis exploredis availablefrom the
author on request.It would also be naturalto have L and L* as vectors of factor inputs,
and a and b as matrices.This procedurewould lead to a set of distributedlag relations,
one for eachfactorinput, but the qualitativeconclusionsof the analysiswouldremainthe
same.
Christopher
A. Sims
699
time t and A and B are parametersthat dependon a, b, and R.6 In the
theoreticalmodelsused to justifypreviouseconometricwork on the relation of laborand output,it has beenassumedthatexpectationsarestaticthat Et[L*(t+ s)] = L*(t) for all s and all t. This implies that the right-hand
side of (1) is simply(1 - A)L*(t),so that the distributedlag relationbetweenL and L* is givenby
(2)
L(t)
-
AL(t
-
1) = (1
-
A)L*(t).
Thisis a Koycklag distributionwhosecoefficientssumto one. Whensome
exogenousinfluencechangesL*,the lag distributionimpliesthateventually
L will adjustin full proportionto the change.
But now supposethat in fact L*(t)containsa trendcomponent,l*(t),
that, over the relevanttime horizon,can be predictedwithouterror;and
anothercomponent,L*(t)- l*(t), that has expectedvaluezerobut is serially correlated.A forecastthatputsL*permanentlyat its currentlevelthen
makesno sense.Instead,it will be naturalto forecasta more or less rapid
convergenceof L* to its trendline l*-for example,
Et[L*(t+ s)] = l*(t + s) + Gs[L*(t)-
(t)],
where G is a constant between 0 and 1 that governs how quickly L*
approachesits trend.7If in additionP is an exponentialfunctionof time,
(1) impliesthe followingequationinsteadof (2):
(3)
L(t)
-
AL(t
-
1) = (1
-
A)(1
-
B)(1
-
BG)-lL*(t) + T(t),
whereT is an exponentialfunctionof time.Now the lag distributionimplies
less than fully proportionatetotal responseof L to L*. This is intuitively
reasonable,since now no level of L* is expectedto persistindefinitely.8
6. This relation follows from the principleof first-periodcertaintyequivalence.See
H. Theil,EconomicForecastsandPolicy(Amsterdam:North-Holland,1958),pp. 507-14;
and Herbert A. Simon, "Dynamic Programmingunder Uncertaintywith a Quadratic
CriterionFunction," Econometrica,Vol. 24 (January1956), pp. 74-81. Application of
this principleto a dynamictheory of the firm is not new; see, for example,CharlesC.
Holt and others,PlanningProduction,Inventories,and WorkForce(Prentice-Hall,1960).
7. Such a forecast would actually minimize the variance of forecast error, e, if
L*(t)-L*(t)
=
G[L*(t - 1)
- L*(t-
1)] + e(t), where e is serially independent, and
the informationavailableat time t was equivalentto that containedin currentand past
values of L*.
8. This resultarisesin essentiallythe same way in whichan economycharacterizedby
rationalexpectationsmay displaya distributedlag relationof unemploymentto the rate
of changeof pricesin whichthe sum of coefficientsis nonzero,as explainedby RobertE.
Lucas, Jr., "EconometricTesting of the Natural Rate Hypothesis," in Otto Eckstein
700
BrookingsPaperson EconomicActivity,3:1974
To generatean industry-levelrelationrequiressummingacrossfirms.In
this process,many exogenousinfluenceson L* that are importantat the
firmlevel will becometrivial.Furthermore,some variablesthat mightnot
plausiblybe assumedexogenousat the firmlevelmaynaturallybe regarded
so whenmeasuredas industryaggregates.This seemsa particularlyimportant point for output,but appliesalso to input and outputpricesif firms
have some short-runflexibilityin the relationof these prices to market
averages.
In this paper,industryoutputis the sole explicitlymeasuredexogenous
influenceon L*. In principle,others,particularlyinputand outputprices,
should be included.If the estimatedregressionsare to be interpretedas
reflectingforecastingand adjustmentprocesseslike those that lead to (3),
one hopes that the omittedexogenousinfluencesare well measuredby the
exponentialtrendthat entersall the estimatedrelations,and that forecasts
of industryoutputareno betterthanthose obtainablefromits ownhistory.
If this pair of assumptionsis not at least approximatelycorrect,the estimatedregressionsare likelyto fail exogeneitytests.
Besidesthe negativeconclusionthat the sum of coefficientson current
and laggedL* need not be one, modelslike that discussedhereyieldpositive conclusionsthat may help in interpretingresults.The parametersA
and B tendto zero as b/a tendsto zero-that is, as the costs of adjustment
diminishrelativeto the costs of deviatingfrom L*. Thus if Et[L*(t+ s)]
is close to L*(t) for a short period into the future,as seems reasonable,
equation(2) will becomenearlyaccurate,with A nearzero, as adjustment
costs go to zero: whenadjustmentcosts aresmall,adjustmentwillbe rapid
andnearlyfullyproportionate.Similarly,the smalleris G the closerto one
willbe the sumof coefficientson laggedL* in (3); in industrieswheredeviations of outputfrom trendare short-lived,the degreeof SRIRL,as measuredby the total responseof laborto output,shouldbe large.Note that
in two industrieswith the samecost function,and thus the sameA and B,
the total responseof L to L* could still differbecauseof differencesin G,
or more generallyin the serialcorrelationpropertiesof the outputseries:
interindustrydifferencesin the size of total responseof L to L* need not
implydifferencesin the speed of the response.
(ed.), The Econometricsof Price Determination(Board of Governors of the Federal
ReserveSystem, 1972),p. 57. The behavioralinsightembodiedin the resultis similarto
the distinctionbetween "permanent"and "transitory"income in Milton Friedman,A
Theoryof the ConsumptionFunction(PrincetonUniversityPress, 1957).
A. Sims
Christopher
701
TemporalAggregation
In this study, temporalaggregationcannot be dealt with in a footnote
devotedto the peculiaritiesof the data.The empiricalresultsand the analysis presentedbelow suggestthat it could be the source of a substantial
portionof the observedresidualsin output-laborrelations.Furthermore,
the methodof temporalaggregationfor U.S. data on laborinputand output is a likely sourceof certainsystematicbiases.
Data on employmentand averageweeklyhoursof productionworkers,
on whichthe "labor"variablesof this study and many othersare based,
apply to the payrollperiodthat includesthe 12th of the month, and are
thusmonthlysamplingsof weeklyaverages.9Data on productionand sales
in manufacturing,on the otherhand, are monthlysamplingsof monthly
averages.Relativeto outputdata, then,labordata are less smoothed,and
are shiftedback in time by about one-tenthof a month.
The estimatedlag distributionin a regressionof output on labor will
have its meanreducedby 0.1 monthby the time shift in the data and will
includea bias towardlocal smoothnessgeneratedby the tendencyof the
lag distributionto correctfor the greatersmoothnessof the outputseries.
On the assumptionthat labor data are exactly and always 0.25-month
averagescentered40 percentof the way throughthe monthwhilethe output data are full-monthaverages,methodssimilarto those I have used
of these biases.10Thus suppose
elsewhereallow an exact characterization
that the true relationbetweenoutputand labor in continuoustime is the
identityY = L. Then,assumingY andL in fact changeonly slowly,11the
firstrow of the tablebelow givesthe lag distributionfor L on Y estimated
in a largesampleof the discretedata, and the next two rowsshow, respectively,the separateeffectsof the time shift and of the greatersmoothness
of observedY. The coefficientsshownfor negativet applyto futurevalues
of L. The truelag distributionwouldhave weightconcentratedentirelyat
9. Most manufacturingfirmsare on a weekly payroll period.
10. In "Discrete Approximations";and ChristopherA. Sims, "ApproximateSpecification in DistributedLag Models," in InternationalStatisticalInstitute,Proceedings
of the 38th Session, 1971 (Bulletin of the InternationalStatistical Institute, Vol. 44,
August 1971), Pt. 1, pp. 285-94.
11. Technically,L and Y are assumedto be first-orderMarkov processeswith very
small parameters.
Brookings Papers on Economic Activity, 3:1974
702
zero; the biased lag distributions show weight spilling over to adjacent
coefficients.
t (month)
Source of lag
distribution
-1
0
1
Combined effects
0.18
0.75
0.075
Shift only
Aggregationonly
0.10
0.125
0.90
0.75
0
0.125
Note also that the sum of coefficients in the first row of the table is not
exactly one. This is not rounding error, but an intrinsic feature of the
analysis. In contrast to the unit averaging of data on both sides of a distributed lag equation,12the asymmetric time aggregation of labor and output data destroys any exact connection between long-run effects in discrete
and continuous data. One can show that so long as weekly L is quite
smooth, the bias from this source in sums of coefficients from distributed
lag regressions of Y on L will be small.
Distributed lag regressions of labor on output will have their mean lag
increased by 0.1 month by the time shift in the data and will include a bias
toward local fluctuations generated by the tendency of the estimated lag
distribution to correct for the greater smoothness of the output series. The
above example for regressions of output on labor can be computed for
regressions of labor on output, but such computations are somewhat less
helpful here. The limiting forms of the lag distributions of labor on output
will in general be complicated, reflecting the tendency to "unsmooth" Y;
and the nature of the complications will depend on the unobservable weekly
serial correlation patterns in output. Also, the sum of coefficients in this
lag distribution will in general show greater bias than the other for a given
degree of smoothness in the data. The following tabulation gives an example
of the lag distribution for output on labor when the true relation is identity,
on the same assumptions underlying the previous tabulation.
t (month)
-5
-4
Coefficients 0.00 0.00
-3
-2
-I
-0.01
0.02
-0.09
0
1
2
3
1.00 0.01 0.00 0.00
12. In which case the sum of the discretelag distributionis the integralof the continuouslag distribution.
ChristopherA. Sims
703
All thesesourcesof distortionfromtimeaggregationcauselittledifficulty
if the truelag distributionhas a largemeanandis itselfverysmooth.If the
meanlag is severalmonths,0.1 month of bias one way or the otheris not
important.
Whenthe adjustmentof labor to outputis rapid,as appearsto be the
case with manhoursin responseto productionor sales in the empirical
work reportedbelow, the foregoinganalysissuggeststhat the regressions
of output on labor are betterindicatorsof the true relationthan are the
regressionsof labor on output.On the otherhand, when the adjustment
involvesa long, smoothlag distribution,as appearsto be the case in some
of the empiricalworkwith employmentand production,time aggregation
will producelittle distortionin a regressionof labor on output. Since in
this case the correspondinglag distributionof outputon laborwill involve
rapidoscillationsor derivatives,time aggregationis likelyto producerelatively greaterdistortions.
Many of the estimatedlag distributionsof output on labor discussed
belowindicatethat a substantialproportionof the adjustmentoccurswithin the contemporaneousmonth, and standarderrorson them are small
enoughthat the spuriouscoefficienton futurelaborin the lag distributions
listed above would be statisticallysignificant.Therefore,the estimated
first futurecoefficientin lag regressionsof output on labor is treatedas
part of the effectof currentand laggedlabor on output.13
While the biases discussedhere would be less apparentwith quarterly
data, they would not necessarilybe less misleading.The assumptionof a
locallyverysmoothoutputvariablemakesthe bias in the reportedsum of
the coefficientsfor the regressionof laboron outputoptimisticallysmallabout7 percent.That bias would not be any smallerwith quarterlydata,
thoughthe suspiciousoscillationsin the coefficientsof the lag distribution
woulddisappear.
Sourcesof EquationError
Althoughthe theoriesof maximizingfirmbehaviorconsideredthus far
includestochasticelements,they deriveexact relationsbetweenlabor and
13. This is a reasonableprocedureexcept in the case (which never arises here) in
whichthe estimatedfirstfuturecoefficientis significantlylargerthan could be accounted
for by this kind of leakage of contemporaneouseffects.
704
BrookingsPaperson EconomicActivity,3:1974
output. Least-squares estimates of the distributed lag regressions considered in the previous section would provide consistent estimates of the equations biased by time aggregations, as discussed there. In at least some of
the estimated equations, however, other sources of residual error affect
results.
In empirical estimates using output as the dependent variable, results are
similar using either deflated sales or the Federal Reserve industrial production index to measure output. When labor is the dependent variable in the
regression, however, results with the two output variables are quite different. Labor is estimated as a two-sided, smooth distributed lag function of
deflated sales, which does not fit into any natural behavioral story. One
possible explanation is some source of pure measurement error in deflated
sales that is substantially greater than any pure measurement error in the
production index.
The sales data are compiled from a 5,000-firm subsample of the 65,000firm sample used to generate the annual survey of manufactures. This is
a substantially smaller sample than that used to obtain the employment
data which, according to the same source, covers two-thirds of all employees in manufacturing.'4 The sampling reliability of the production
index varies by industry, but if it approaches that for employment, pure
sampling error is at least a candidate for explaining the results.
Suppose that in the discrete data L = Y for the data without sampling
error,but that Y*is observed, where Y* = Y + e and e is sampling error.15
If e is serially uncorrelated, Y is first-orderMarkov with parameter 0.9 in
monthly discrete time, and e has one-third the variance of Yt - 0.9 Yt-1,
then the form of the distributed lag regression of L on observed Y will be
as follows:16
14. U.S. Bureauof Economic Analysis, 1973 BusinessStatistics, explanatorynote 2
for p. 26 and note 1 for p. 73, respectively.
15. Since the sampleof firmsused is presumablynot redrawneach month, thereis no
strong presumptionthat e is seriallyuncorrelated.
16. If the true lag distributionin discretetime is somethingmore complicatedthan
the identity (a unit coefficientat zero), then the effect of the samplingerror will be to
convolutethe true lag distributionwith the lag distributiondisplayedin the text.
The lag distributionwas obtainedunderthe assumptionsabout the serialcorrelation
propertiesof Y and e, the samplingerror,given in the text. The projectionof L = Y on
Y* = Y + e is given by H(L)H(L-1)Y*, where H(L) = g(L)/[g(L) + h(L)] and Y =
g(L)u, e = h(L)v,where u and v are independent"white noises," and H, g, and h are
polynomialsin the lag operatorL.
705
Christopher
A. Sims
t (month)
Coefficient
-3
-2
-J
0
1
2
3
0.00
0.03
0.13
0.67
0.13
0.03
0.00
Generally, as in this example, the effects of measurement error on an identity lag distribution will be symmetric about zero; and when the sampling
error shows less positive serial correlation than the true component, the
bias will be toward a smoothed lag distribution.
Equation error can arise from other sources, though none of those discussed below appears to be essential in explaining the observed results in
this paper. (None can be decisively rejected, either, for that matter.)
Producers might make mistakes. If the output variable is exogenous at
the industry level, errors made by firms must show up entirely in labor and
be independent of output. If these errors had substantial variance, the
effect would be similar to the effect of sampling error, but now in the Y on
L rather than the L on Y distribution. Thus, where there is little evidence of
two-sided lag distributions of Y on L, this sort of optimization error is
probably not very important.
Since output and labor data are always aggregated over industries and
firms with different normal ratios of output to labor, and since the composition of the aggregates shifts over time, aggregation is undoubtedly a
source of residual error in the output-labor relations. How it should affect
the relations is unclear, a priori. As with any other general source of specification error, a substantial error of this kind is likely to distort the onesided form of a causal dynamic relation. Thus, the large number of acceptably one-sided relations found here may suggest that aggregation error is
not a serious problem.
Fair, and Nadiri and Rosen, have explained regressions of labor on output in which coefficients on future output were significant by arguing that
firms have knowledge of future output.17Hirsch and Lovell have suggested
that future output might be known well enough to make actual output a
good measure of expectations.'8 In regressions of labor on current and
past output, forecasting by firms that is better than that obtainable from
linear combinations of current and past output or labor data would cer17. Fair, Short-Run Demand for Workers and Hlours; Nadiri and Rosen, Disequilibrium Model of Demand.
18. Albert A. Hirsch and Michael C. Lovell, Sales Anticipations and Inventory Be-
havior(Wiley, 1969).
706
Brookings Papers on Economic Activity, 3:1974
tainly be a source of equation error. However, where significant coefficients
on future output do appear in the estimates reportedbelow, it will be argued
that the prescience of firms is not a convincing explanation.
Finally, the equations actually estimated are all log-linear, and are therefore only approximations to the linear relations that follow from the
quadratic-linear theory discussed earlier. Since, in some industries, the
estimated equations include data affected by strikes or by very sharp seasonals, the log-linear approximation could be a substantial source of specification error, but whether any systematic bias is likely to result from this
source of error is difficult to determine a priori.
SEASONALITY
The theory discussed thus far does not in principle distinguish seasonal
from nonseasonal movements in output and labor. In the empirical work
the regressions have not been required to fit across seasonal and nonseasonal variation. The deterministic component of seasonal movementswhich can be captured in ordinary seasonal dummy variables-ought to
be removed from the distributed lag regressions. The data are corrected for
workdays per month on the output side, but not for variations in numbers
of workdays due to major holidays, which are regarded as seasonal movements. Because of the asymmetric timing of the labor and output data, the
result is that variations in output per month due to holidays may not be
reflected in the unadjusted labor data, depending on what week of the
month the holiday falls in. One working day is about 4 or 5 percent of a
working month, and the residual standard error in the estimated equations
is 1 percent to 2 percent. Thus, a regular tendency for Christmas to fall
outside the sampled payroll period but for President's Day to fall within it
would worsen the fit of the regression equation across seasonal frequencies.
Since the location of the sampled pay period within the month, as well as
that of some major holidays, shifts from year to year, not all of the distortion in the labor-output relation arising from major holidays will be confined to the deterministic component of the seasonal.
Even in the absence of distortion in the underlying relation except at the
deterministic seasonal frequencies, equations of the form estimated in this
paper could still run into problems arising from the attempts of firms to
anticipate slowly evolving seasonal movements in output.
To capture the likely form of firms' projections of seasonal movements
ChristopherA. Sims
707
in output would require estimation of lag distributions considerably longer
than those reported below. But to do this appropriately, allowing a flexible
form to the seasonal components of the lag distribution, is beyond the
scope of this paper.
THE DATA
Estimates were made with data for all manufacturing, for durables and
nondurables, and for three two-digit level industries for which physical
production measures are available: primary metals, apparel, and paper.19
In all industries, two labor variables were considered-manhours and
employment of production workers. The Federal Reserve index of industrial production was an output variable in all industries. In addition, for
total manufacturing, shipments deflated by the wholesale price index for
manufacturingwas used as an alternative output variable (called "sales").20
All data were collected in seasonally unadjusted form. The statistical procedures used included a kind of seasonal adjustment as one step of the
analysis.
For data up to 1963, a substantial part of the Federal Reserve index for
total manufacturing estimates "production" from manhour data. In these
estimates, a judgmental allowance was made for the possibility that SRIRL
exists; but the allowances appear to be small and rare enough that the
main effect of using manhour data would be to bias regression estimates
toward an identity relation between the labor and production series.
Fortunately, cross-checks on results using other data are available.
In this study, such independent cross-checks are provided by the sales
19. Productiondata for the petroleumindustryare directlymeasuredin the Federal
ReserveBoardindex. However,initial estimatesfor this paperturnedup a residualof 10
standarddeviationsfor the January1969 observation,correspondingto a short strike,
whose timingwas such that it had maximumeffecton the payrollreportingperiodof the
Bureauof Labor Statistics.New results, using a strike dummyvariable,eliminatedthe
January1969 problembut turned up two new outliers of 3.5 standarddeviationseach.
Apparently,output-laborrelations in the petroleumindustryhave highly non-normal
residuals,so that least-squaresmethods probablyare not applicable.
20. At an earlierstage of the research,experimentswere made with sales corrected
for changesin inventoriesof finishedgoods as an output variable.Results differedlittle
from those with sales itself, and since such data are availablefor only a restrictedperiod,
the experimentswere not pursued.
21. The officialdescriptionof the seriesis given in Board of Governorsof the Federal
Reserve System, Industrial Production, 1971 Edition (1972).
luo
DrooKings
rapers
on zconomic
ACtivity,
J;IYi/q
data,whicharecollectedindependentlyof the payrolldata,and by the use
for whichFRB outputis measureddirectlyfrom
of two-digitsubindustries
productionvolumes,at least in the most recentyears.As an addedcheck,
regressionsweretestedfor temporalhomogeneity,with the samplesplit at
March1961.The last half of the samplethuscontainsmuchless interpolation frommanhourdatathandoes the earlierhalf. If bias from this source
wereserious,one would expect a substantialshift in coefficientsbetween
the two componentsof the samplefor total manufacturing,durables,and
nondurables;and in fact such shiftsdo occur.The shift is marginallysignificantat the 10 percentconfidencelevel in total manufacturingand significantat the 5 percentconfidencelevel in durablesand nondurables,for
regressionsof outputon manhours.In all casesthe shift is in the expected
direction,showinga declinein the sum of coefficientson contemporaneous
and firstleadingvaluesof manhoursbetweenthe subperiods.The decline
is from 1.29to 1.13for durables,from 1.20to 0.97for totalmanufacturing,
and from 0.73 to 0.62 for nondurables.Somebias in the resultsfor aggregatemanufacturing
seemslikelyfromthissource,therefore,butits absolute
magnitudeis moderate.22
Thebasicdatausedcoverthe period1947-73.Twoyearsat the beginning
of the sampleandone at the endarelost to leadsandlagsin the regressions.
In addition,two months at the beginningare lost to prefilteringand an
additionaltwelvemonthsare droppedat each end of the seriesbecauseof
the unreliabilityof deseasonalizationat the ends of the series.Thus the
actualregressionscoverthe sampleperiodMarch1950throughDecember
1971.
StatisticalProcedures
The basic statisticalmodel of this studymakesthe logarithmof the dependentvariablea linearfunctionof twelvefuturevalues,the currentvalue,
and twenty-fourpast valuesof the logarithmof the independentvariable,
plus a constantand a trendterm,with a residualerrorindependentof the
regressors.Theseassumptionsimposeonly weakrestrictionson the struc22. An interestingfact noted too late for adequateinvestigationis that the last-half
results for all three aggregatesfit the hypothesis that all adjustmentof manhours to
output occurs within the contemporaneousmonth. The significantlags in adjustmentin
the full-sampleregressionarise solely from the pre-1963portion of the sample.
A. Sims
Christopher
709
ture generatingthe time serieson output and labor. For example,if the
data were generatedby a two-equationsystem and involvedexogenous
variablesnot included in these regressions,under general assumptions
there would still be two-sideddistributedlag regressionsof output and
labor on each other(involvingpast and futurecoefficients)with residuals
independentof the right-handvariable.Thoughin this case neitherregression wouldin generalbe a structuralequation,statisticalinferenceabout
the coefficientswould in itself be accurate.In particular,a test of the
hypothesisthatfuturecoefficientsarezero,whichcan be regardedas a test
of the hypothesisthat the equationis in fact a structuralequation,will be
valid.The basic statisticalmodel wouldbe substantiallymisspecifiedonly
if the covariancepropertiesof the variablesweretemporallyunstableor if
the restrictionof twelveleadsandtwenty-fourlags was seriouslymistaken.
In the centralset of estimates,a proceduredescribedand justifiedat
length elsewhereis applied.23The aim is to correctfor serialcorrelation
in the residualswithoutany strongpriorrestrictionon the parametricform
of the serialcorrelation,and also to eliminatefrom the datavariationin a
band about the seasonalfrequencies.The latterpart of the procedureis a
aimedat allowingfor possibleeffects
form of seasonal"overadjustment,"
on the data of a slowlyevolvingseasonalpatternof "noise,"or errorsin
variables.24
The theoreticalmodelsdiscussedabovegenerateexactdynamicrelations
betweenlabor and currentand past output.If, as is quite possible,those
23. ChristopherA. Sims, "Seasonalityin Regression,"Journalof the AmericanStatistical Association,Vol. 69 (September1974), pp. 618-26.
24. Logarithmsof the basic data are taken, and each seriesis prefilteredthroughthe
filter (1 - .9L)2,where L is the lag operator. Mean, linear trend, and a deterministic
(OLS) regression.The resulting
seasonal patternare removedby ordinary-least-squares
series are Fourier-transformed, using 768 points over the (-x,
sr) interval and fifteen
Fouriertransformordinatesare set to zero in a band centeredat each of the eleven seasonal frequencies.(The width of the band is thus approximately7r/26.)The data are then
and the basic regressionestimatedover the sample shortinverse-Fourier-transformed
ened as indicated above. The residuals from this first estimate of the regressionare
Fourier-transformed,the spectral density estimated using a smoothing window triangular with base 26 ordinates(about 7r/15,or 11 harmonicfrequenciesat the base;
effectivedegreesof freedomabout 18). The smoothedestimatesat the edges of the seasonal bands are correctedfor the downwardbias due to the erasureof variancein the
data, with seasonalbands erased,are
seasonal bands. The originalFourier-transformed
dividedby the squareroot of the estimatedresidualspectraldensityand inverse-Fouriertransformed.Then the final, efficient,regressionestimatesare obtainedby OLS. Appendix Table A-1 presentsresults of tests of the hypothesisthat regressionsfit seasonally.
710
Brookings Papers on Economic Activity, 3:1974
dynamicrelationshaveone-sidedinverses,theycan be writtenequallywell
as relationsof outputto currentand pastlabor.Sincethe relationsderived
in the theoryareexact,the theorysuggests,if anything,that regressionsin
eitherdirectionwill recoverthe true dynamics,even though in a certain
intuitivesensethe theorytakes industryoutputas exogenousto decisionmakers.Many possiblesourcesof significantcoefficientson futurevalues
of the independentvariablein two-sidedregressionshave been suggested
earlierin this paper.However,with the exceptionof a fairlysmallsignificant positive coefficienton the first future value of labor in explaining
output,25findingsignificantcoefficientson futurevaluesof the independent
variablewould suggestthat the estimatesare seriouslybiased or that the
centraleconomicmodelsof thispaperarenot usefulwaysto rationalizethe
results.Hence, a criticalspecificationtest for each of the regressionsconcernswhetherthe twelvefuturecoefficients(or eleven,excludingthe first
futurecoefficient,with labor on the right-handside) are zero. If the behavioralrelationof interestinvolvesno nonzerofuture coefficients,this
procedureconstitutesa test for the null hypothesisthat the independent
variableis statisticallyexogenous.Only in this case can the estimatedlag
distributionsbe consideredestimatesof the behavioralrelation.26
Results
MANHOURS
Tables 1 and 2 presentestimatesof lag distributionsfrom regressions
of manhoursand output on each other. Table 3 presentsthe cumulative
relationsbetweenoutputand laborobtainedby summingtheselaggedcoefficientsovervaryingintervals.The resultspresentedarefor regressionsin
25. Which,recall, might be expectedto arise from the special form of time aggregation in manufacturinglabor and output data.
26. This was pointed out in ChristopherA. Sims, "Money, Income, and Causality,"
AmericanzEconomicReview,Vol. 62 (September1972),pp. 540-52. The test is also a test
of the null hypothesisthat the independentvariable"causes"the dependentvariablein
the sensedefinedby C. W. J. Grangerin "InvestigatingCausalRelationsby Econometric
Models and Cross-SpectralMethods," Economerrica,Vol. 37 (July 1969), pp. 424-38.
Granger'sterminologyis a usefulverbalshorthandand a guideto intuitionif one already
graspswhat statisticalexogeneitymeans.However,the word "cause"has such a variety
of meaningsthat to attemptto reachan understandingof statisticalexogeneityby translating it into Granger'scausal terminologycan be counterproductive.
ChristopherA. Sims
711
Table 1. Lag Distributionsand SummaryStatistics, Regressions of
Manhours on Output, Selected ManufacturingIndustries,Sample
Period March 1950-December 1971
Lag
Totalmanufacturing
(month)
and
Index of
summary industrial
Durable
goods
statistic production Salesa
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
E13-24
...
...
...
...
...
...
...
...
...
...
...
-0.006
0.698
0.132
-0.022
0.030
0.044
0.020
0.062
-0.025
0.021
-0.043
0.126
-0.085
-0.047
0.029
Standard
errorof
Z13-24 0.056
Standard
errorof
coefficient
0.037
Largest
Smallest 0.032
0.8869
R2
Nondurable
goods
Paper
Primary
metals Apparel
-0.099
0.044
-0.016
0.015
-0.002
0.039
-0.031
0.012
0.044
-0.007
0.130
0.114
0.347
0.129
0.099
0.071
0.096
0.036
0.052
0.020
0.034
0.019
0.053
-0.012
-0.091
-0.080
...
...
...
...
...
...
...
...
...
...
...
-0.032
0.740
0.062
-0.008
0.028
0.048
0.031
0.036
-0.024
0.010
-0.024
0.095
-0.072
0.015
-0.021
...
...
...
...
...
...
...
...
...
...
...
0.039
0.551
0.202
0.075
0.050
0.064
0.011
0.092
-0.040
-0.017
-0.032
0.114
-0.040
-0.023
0.045
...
...
...
...
...
...
...
...
...
...
...
0.092
0.267
0.137
0.077
0.021
0.009
0.011
-0.021
0.024
0.025
0.004
-0.026
-0.026
0.017
0.003
...
...
...
...
...
...
...
...
...
...
...
0.006
0.644
0.006
0.058
-0.013
0.037
0.035
0.021
0.008
0.012
0.010
0.021
0.018
0.000
0.019
...
...
...
...
...
...
...
0.099
0.312
0.165
0.101
0.048
0.049
0.042
0.136
-0.002
-0.046
-0.050
0.053
0.042
-0.061
-0.075
0.126
0.048
0.093
0.024
0.063
0.119
0.039
0.036
0.5895
0.027
0.024
0.9139
0.052
0.047
0.7958
0.033
0.028
0.6049
0.017
0.016
0.9371
0.042
0.040
0.6077
...
...
...
Sources: Author's regressions,discussed in the text, using information describedin the data section of the
text.
a. Shipments deflated by the wholesale price index for manufacturing.
712
Brookings Papers on Economic Activity, 3:1974
Table 2. Lag Distributionsand Summary Statistics, Regressions of
Output on Manhours, Selected ManufacturingIndustries,Sample Period
March 1950-December 1971
Lag
Total manufacturing
(month)
and
Index of
summary industrial
Durable
statistic production Salesa
goods
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
E13-24
0.167
0.949
0.037
0.049
-0.071
-0.070
-0.040
-0.056
0.028
-0.024
0.003
-0.035
0.050
-0.041
-0.049
Standard
errorof
E13-24 0.064
Standard
errorof
coefficient
0.051
Largest
Smallest 0.044
R2
0.8761
Nondurable
goods
Paper
Primary
metals Apparel
0.046
0.198
0.519
1.376
0.079
0.053
0.018
-0.111
0.014 -0.176
-0.105
0.096
-0.056
-0.027
-0.173
-0.024
-0.020
-0.025
0.024
-0.019
-0.024
-0.009
-0.101
-0.023
0.155
-0.049
-0.153
-0.002
-0.327
-0.028
0.116
0.890
0.100
0.094
-0.278
-0.029
-0.009
-0.147
0.164
-0.098
-0.078
-0.026
0.178
0.031
-0.102
0.144
1.075
0.029
0.005
-0.046
-0.100
-0.046
-0.033
0.029
0.007
-0.004
-0.025
0.036
-0.071
0.025
0.219
0.583
0.032
0.036
-0.059
-0.041
0.003
-0.057
0.018
-0.064
0.035
-0.046
0.112
-0.102
-0.266
0.342
0.713
0.042
0.042
0.111
0.082
-0.308
-0.009
-0.040
-0.035
0.104
0.261
-0.171
-0.357
-0.307
0.116
0.050
0.124
0.188
0.130
0.254
0.112
0.092
0.6461
0.040
0.036
0.9161
0.067
0.060
0.6216
0.125
0.118
0.6044
0.034
0.031
0.9334
0.111
0.088
0.4060
Sources: See Table 1.
a. Shipments deflated by the wholesale price index for manufacturing.
whichcoefficientson lags -2 through -12 (the second throughtwelfth
futurecoefficients)wereconstrainedto be zero, exceptwherethe null hypothesisthat thesecoefficientswerezero was rejected(for outputon manhours)or the null hypothesisthat all twelvefuturecoefficientswere zero
was rejected(for manhourson output).As appendixTable A-2 demonstrates,in a majorityof the industriesthe exogeneitytest rejectsneither
directionof regression.For the paperindustryandfor total manufacturing
withdeflatedsalesas the outputvariable,the twelvefuturecoefficientsare
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714
Brookings Papers on Economic Activity, 3:1974
significantlydifferentfrom zero when outputis the independentvariable.
In no casearethe elevenfuturecoefficientssignificantlydifferentfromzero
in regressionsof outputon manhours(thoughthe resultsfor nondurables
aresuspecton this score).Thoughthe firstfuturecoefficientin theseregressionsis stronglysignificantin severalindustries,the coefficientis neverany
largerthancan be accountedfor by time-aggregation
bias of the type discussedabove.
Except where the exogeneitytest rejects one directionof regression,
resultsarebroadlyconsistentundereitherdirection.Somedegreeof SRIRL,
for at least a shorttime interval,seemslikelyin all the industriesreported
in Table 3 exceptthe regressionsof outputon manhoursfor appareland
nondurablegoods. (In assessingthis conclusion,rememberthat the zeroorderresultsof output on manhoursare slightlybiased towardzero by
aggregationover time.) In all cases except outputon manhoursfor paper
and apparelthe cumulativeresponseover six monthsis estimatedwith a
band
standarderrorof 0.11 or less, and in all cases a one-standard-error
bands about
about the six-monthresponseoverlapsone-standard-error
the twelve-andtwenty-fourmonthresponses.AppendixTableA-3 demonstratesthat the large zero-ordercoefficientsin these lag distributionsare
not the whole story. But thereis little evidencethat the dynamicsrequire
more than six months to work themselvesout; and by the end of six
months,thereis muchless evidenceof SRIRLin the regressionresults.
Only for primarymetals do regressionsin both directionsoffer firm
evidencethat a less than proportionalresponseof manhoursto output
persiststhroughthe wholesix-monthadjustmentperiodfor eitherdirection
of regression.For durables,the regressionof manhourson output shows
significant,thoughsmall,SRIRLat six and twelvemonths,but the regression in the other direction,which also passes an exogeneitytest, shows
none at all. SRIRLof this magnitudecouldeasilyresultentirelyfrom temporalaggregationbias. On the otherhand,becauseof the increasingstandard errors on the longer-runresponses,even for primarymetals the
resultscannotruleout constantor diminishingreturnsto laborover twelve
or twenty-fourmonths.27
For apparel,the regressionof output on manhoursshows significant
27. At a late stage of the work for this paperI found one verylarge residual(around
4.5 standarderrorsfor August 1959)for primarymetalsthat probablydeservesthe same
treatmentgiventhe even more implausibleJanuary1969petroleumresidual.Thus all the
resultsfor primarymetals should be treatedwith some skepticism.
ChristopherA. Sims
715
decreasingreturnsto labor at six, twelve,and twenty-fourmonths.However,in both directionsof regression,the standarderrorsfor apparelare
the largestfor all the industries,and the F-test resultsfor exogeneityin
Table A-3 are marginal.Probablyneitherdirectionof regressionfor this
industrygivesreliableresults,andthe exogeneitytestshavebeenpassedby
virtueof the poor fit of the equations.For the paperindustry,the six-,
responsesof manhoursto output are very
twelve-,and twenty-four-month
significantlyless than one, but this regressionfails to pass the exogeneity
test due to the presenceof a firstfuturecoefficient34 percentas large as
the zero-ordercoefficient(Table1).Theregressionfor outputon manhours
in the industrydoes not revealSRIRL at six months.
EMPLOYMENT
Tables4 through6, correspondingto Tables 1 through3 in the outputmanhoursregressions,show that the initial responseof employmentto
output,over the firstmonthor two, is much smallerthan the responseof
manhours.As appendixTable A-4 shows, in threeindustries,regressions
of outputon employmentfail the exogeneitytest. Standarderrorson coefficientsaremuchlargerin regressionsof outputon employmentthanthey
are in those of outputon manhours.
The tendencyfor the regressionsof output on employmentto perform
poorlyhas two naturalexplanations.First, since the ratio of employment
to outputcan undoubtedlybe variedin the shortrun withless inefficiency
than would accompanya similarvariationin the ratio of manhoursto
output, employmentmight be relativelymore affectedby variablesnot
accountedfor in the model.Second,becauseemploymentrespondsslowly
and smoothlyto outputchanges,it mightbe impossibleto determinecurrent outputfrom currentand past employment.28
bandsaboutthesix-monthresponsesoverAs before,one-standard-error
bands about twelve- and twenty-four-monthrelap one-standard-error
sponsesexceptfor the regressionof total manufacturingemploymenton
28. Technically,the lag distributionof employmenton outputmightnot be invertible.
If, as is true with many of the lag distributionsfor manhourson output, the zero-order
coefficientis largerin absolutevaluethan the sum of the absolutevaluesof the remaining
coefficients,the lag distributionautomaticallyhas a one-sidedinverse.Exceptfor durable
goods and primarymetals, the estimatedregressionsof employmenton output come
nowherenear meetingthis sufficientcondition for invertibility.
Brookings Papers on Economic Activity, 3:1974
716
Table 4. Lag Distributionsand SummaryStatistics, Regressions of
Employmenton Output, Selected ManufacturingIndustries,Sample
Period March 1950-December 1971
Lag
Totalmanufacturing
(month)
and
Index of
Durable
summary industrial
statistic production Salesa
goods
Nondurable
goods
Paper
Primary
metals Apparel
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
-12
-11
-10
...
...
...
-0.048
0.003
-0.005
...
...
...
...
...
-9
...
-0.001
...
-0.008
0.027
-0.006
-0.005
-0.008
0.015
0.063
0.087
0.287
0.074
0.121
0.062
0.082
0.052
0.036
0.044
0.015
0.047
0.032
0.022
-0.019
0.025
...
...
...
...
...
...
...
-0.008
0.570
0.029
0.071
0.010
0.044
0.075
0.025
-0.023
0.029
-0.017
0.087
-0.048
0.036
0.021
...
...
...
...
...
...
...
0.029
0.224
0.192
0.092
0.021
0.077
0.060
0.018
0.013
0.002
-0.024
0.081
0.016
0.006
-0.024
...
...
...
...
...
...
...
0.030
0.124
0.085
0.102
0.011
0.034
0.024
-0.022
0.044
0.023
0.008
-0.009
-0.001
0.038
-0.017
-0.023
0.620
-0.021
0.078
-0.030
0.056
0.033
0.047
0.008
0.025
0.029
0.030
0.032
0.013
0.120
0.075
0.158
0.142
0.069
0.007
0.056
0.046
0.054
0.032
0.014
0.012
0.025
0.037
-0.005
-0.043
0.087
0.033
0.070
0.055
0.095
0.076
0.029
0.026
0.7008
0.032
0.028
0.9192
0.024
0.023
0.7364
0.019
0.017
0.5357
0.023
0.020
0.8936
0.020
0.018
0.5984
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
Z13-24
...
...
...
...
...
...
...
-0.003
0.486
0.084
0.087
0.019
0.036
0.081
-0.003
-0.003
0.042
-0.022
0.121
-0.068
0.020
0.050
Standard
errorof
,13-24 0.041
Standard
error of
coefficient
0.034
Largest
Smallest 0.028
0.8949
R2
Sources: See Table 1.
a. Shipments deflated by the wholesale price index for manufacturing.
717
ChristopherA. Sims
Table 5. Lag Distributionsand Summary Statistics, Regressions of
Output on Employment,Selected ManufacturingIndustries,Sample
Period March 1950-December 1971
Totalmanufacturing
Lag
(month)
and
Index of
Durable
summary industrial
goods
statistic production Salesa
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
E13-24
...
...
...
...
...
...
...
...
...
...
...
0.257
1.178
0.151
-0.182
-0.108
-0.087
-0.217
0.034
0.085
-0.053
-0.111
0.012
-0.042
-0.091
-0.139
Standard
error of
,13-24 0.122
Standard
errorof
coefficient
0.076
Largest
Smallest 0.067
R2
0.7933
...
...
...
...
...
...
...
..
...
...
...
...
...
...
...
...
...
...
...
...
...
...
0.062
0.226
1.393
1.201
0.071
0.173
-0.132
-0.140
-0.189
-0.085
-0.239
-0.101
-0.117
-0.168
0.028 -0.022
0.124
0.063
-0.094
-0.023
-0.125
-0.033
-0.107
-0.004
0.217 -0.003
0.065 -0.121
-0.181
-0.014
Nondurable
goods
Paper
Primary
metals Apparel
-0.107
-0.003
0.290
-0.334
0.095
-0.073
-0.141
0.210
0.271
-0.106
0.195
0.599
0.775
-0.171
-0.100
-0.139
0.009
-0.129
-0.361
0.077
0.133
0.032
0.002
-0.020
-0.107
-0.478
-0.301
0.181
-0.282
-0.107
0.381
0.198
-0.516
0.091
0.645
-0.401
0.827
0.483
0.388
-0.055
0.093
-0.167
0.278
-0.567
-0.368
-0.102
0.431
-0.330
0.439
-0.143
-1.005
-0.327
-0.051
...
...
0.034
-0.023
...
0.060
...
0.029
...
...
-0.049
...
0.048
0.038
...
-0.089
...
0.106
...
0.175
...
0.493
0.079
1.006
1.333
0.177
0.274
-0.376
-0.118
-0.368
-0.004
0.402
-0.143
-0.383
-0.082
-0.340
-0.046
0.213
-0.035
-0.215
-0.079
-0.029
-0.027
0.124
0.019
-0.297
-0.093
-0.038
-0.089
-0.200
-0.172
0.139
0.090
0.329
0.734
0.266
0.333
0.141
0.118
0.6665
0.061
0.053
0.8652
0.151
0.111
0.6694
0.299
0.234
0.4877
0.063
0.049
0.9036
0.210
0.184
0.5517
Sources: See Table 1.
a. Shipments deflated by the wholesale price index for manufacturing.
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ChristopherA. Sims
719
FRB production.Nonetheless,the persistentrisein total responsefromthe
sixththroughthe twelfthmonth in all regressionsof employmenton output makesit likely,if not proven,that some adjustmentof employmentto
outputpersistsbeyondsix months.Changesin total responsebetweenthe
twelfthand twenty-fourthmonthsin these regressionsare all insignificant
and small,and the signs are inconsistent.It seemsreasonableto conclude
that the adjustmentof employmentto outputis virtuallycompletewithin
a year.
In contrastto the manhoursresults,the evidencefor less than proportionalresponseof employmentto outputoversix monthsis strongin every
industryin the regressionof employmenton output.This patternpersists
in most industriesfor the twelve-monthresponseas well. However,while
the six-monthresponsesareclearlysmallerfor employmentthan manhours
in eachindustryexceptprimarymetals,the differencesin the twelve-month
responsesof the two labor variablesare probablynot statisticallysignificant.
Themorerapidandthoroughgoingresponseof manhoursthanof employment to outputchangessupportsthe presumptionthat hours per worker
aremorefreelyvariablethanis the numberof workers.In fact, becauseof
the logarithmicform of the regressions,the differencein coefficientsbetweenthe equationsfor manhourson outputand for employmenton outputgivesestimatesof the regressionof hourspermanon output.For all but
the suspectresultsfor primarymetals,thelargestcoefficientin theseimplicit
regressionsis in the contemporaneous
monthand is positive.In the industriesfor whichregressionsof employmenton outputshow SRIRLpersisting even overtwelveandtwenty-fourmonthswhilethe regressionsof manhourson outputdo not, sumsof the coefficientsof theseimplicithours-permanequationswillbe positive,implyingthatpartof the effectof outputon
hoursis "permanent."
Findingsof OtherStudies
The principalfindingof this paper-that manhoursadjustaboutin proportionto an output change within six months-can be comparedwith
someof the previousempiricalworkon short-rundemandfor labor.Other
studiessupportthe presentfindingthat the lag in adjustmentof manhours
720
Brookings Papers on Economic Activity, 3:1974
to output is short; but most do not agree that the adjustmentis fully
proportionate.
Wilsonand Ecksteinstudiedmanhoursof productionworkersfor total
Theirspeedof adjustmentturnsout to varywitha dating
manufacturing.29
variable,which implies an elasticityof manhourswith respectto output
aftersix monthsof 1.06whentime is set at 1948and of 0.82 whentime is
set at 1961.Takinga rough average,their six-monthelasticityis 0.94.30
Their long-runelasticityis 1.15 in 1948 and 0.87 in 1961, with a rough
averageof 1.01. Nadiri and Rosen offer estimatesfor all manufacturing
productionworkers,using deflatedsales to measureoutput.3'After six
months,theirestimatedelasticityof manhourswithrespectto salesis 0.89,
indicatinga rapidand almostproportionalresponse.However,they estimate a long, continuing,adjustmentthat eventuallyreducesthe long-run
elasticityto about0.6. F. P. R. Brechling,studyingBritishmanufacturing
andapparentlyusingtotalmanhoursratherthanproductionworkersalone,
findsa six-monthelasticityof 0.46,witha long-runelasticityof 0.48.32Thus
the Wilson and Ecksteinresultsagree with the findingin this paperthat
SRIRLhas vanishedaftersix monthsand that reactionis completewithin
that time, whileNadiriand Rosen and Brechlingfinda long-runelasticity
well below that foundin this paper.All threestudiesagree,however,that
the six-monthelasticityis almostas large as, or largerthan, the long-run
elasticity.
29. "Short-RunProductivityBehaviorin Manufacturing."
30. In an attempt to make quarterlylag distributionscomparableto the monthly
resultsof this paper,the total responsethroughthe sixth month is measuredas the sum
of coefficientson the contemporaneousquarter,the firstlaggedquarter,and two-thirdsof
the second lagged quarter.The numbersdiscussedin this paragraphwere obtained by
adding the Wilson-Ecksteinequation (b) of Table 1 to their equation from Table 3
(addingthe straight-timehoursequationto the overtimehoursequation)and multiplying
throughby Ct, the capacityoutputvariableat time t. The resultingequationmakesmanhours a distributedlag function of output plus a capacity effect. The coefficientsin the
equationare themselvesfunctions of t, which is zero in 1948:4 and increasesby one in
each quarterthereafter.I have taken the "long-runeffect" of a change in output to be
that occurringwhenall laggedvaluesof outputhavereachedthe new level and are equal,
assumingoutput equal to capacity at the startingpoint.
31. DisequilibriumModel of Demand.The numbers that follow in this paragraph
were derivedfrom the Nadiri-RosenTable 4.1, p. 59, by the methods given in their section Dl, pp. 73-75, and in particularequation (4.3), p. 73. Separateresults for log of
hours per man and log of employmentwere added to obtain implied results for total
manhours.
32. "The Relationshipbetween Output and Employmentin British Manufacturing
Industries,"Reviewof EconomicStudies,Vol. 32 (July 1965), p. 213, equation(Avi).
ChristopherA. Sims
721
Evidenceon Forecastingby Firms
The significantlag in reactionof laborto outputin all industriesin the
face of insignificantvalues for future output in most industriessuggests
thatfirmforecastsof aggregateindustryoutputarenot substantiallybetter
than forecastsfrom its currentand past values. If firms could forecast
better,and if lags in adjustmentof laborare generatedby costs of adjustment,thenfirmsshouldbe ableto profitby makinglabordependon future
output.Also, sincecosts of adjustmentarepresumablygreaterfor employment than for manhours,the importanceof future output should be
greaterwhenemploymentis the dependentvariableif firmpresciencewere
the source of significantfuture coefficients.Neither the resultsfor total
withsalesthe outputvariablenor thosefor the paperindusmanufacturing
try fit this pattern,thoughthe apparelindustrydoes.
Finally, if firms know future output, it would be pure chance if the
resultingtwo-sidedregressionrelationfor laboron outputhad a one-sided
inverse,makingoutputa functionof currentand past laboronly. In every
case in whichfutureoutputenterssignificantly,the regressionin the other
directionpassesa test for one-sidedness.33
Recent ProductivityBehavior
hasbeenlow relativeto its predictedvalue
Productivityin manufacturing
throughoutthe 1973-74 period. This can be seen from Figure 1, which
shows the actualand predictedvaluesof manufacturingmanhoursalong
with the errorsof the predictions-the differencebetweenpredictedand
actual-from an equationsimilarto that presentedearlier.34The values
33. Here again, apparelis a partialexceptionin that the regressionof output on employment,while not significantlyin conflict with the hypothesisof a nonzerofirstfuture
coefficientarisingfrom time aggregation,has an implausiblylarge point estimatefor the
first futurecoefficientif time aggregationis the explanation.
1
34. The equationused to generatethis section was estimatedfrom the post-1963part
of the sample period only, since a significantchange in the equation'sparametershad
been detectedat 1963.Also, lags 13 through24 weredroppedfrom the lag distributionin
the estimation.The frequency-domainserial-correlationcorrectionusedin the estimation
techniquefor this paperproducesresidualsthat do not have the usual interpretationas
conditionalforecasterrors.In effect, the correctionfor serialcorrelationduringestima-
722
Brookings Papers on Economic Activity, 3:1974
given in the figureare percentagedeviationsfrom trendand measurethe
percentageerrorsin an impliedpredictionof productivity.
Two kinds of residualsare shown in the figure.The firstis simplythe
differencebetweenpredictedand actual manhoursfor each month. The
correctionestimatedin fitting
secondtakesaccountof the serial-correlation
the equationand uses it in adjustingeach month'spredictionfor the error
in the previousprediction.The uncorrectedresidualsshow the shortfallin
the level of productivity(the excessin the level of manhoursoverthe predictedlevel)risingfromabout2 percentat the startof the periodto about
4 percentat the end of the period,with variationsduringthe intervening
months. The errorsare exceptionallylarge in December 1973 and May
1974;but residualsof this size are not uniquein the periodfor whichthe
equationwas fit. The residualscorrectedfor serialcorrelationindicatethat
whateverproductivitymysteryexistedin this periodwas not growingpersistently.The large residualfor December 1973, when actual manhours
held steadywhilepredictedmanhoursdeclinedsharply,is abouttwo standarddeviations.In monthlydata, such a residualshouldoccuronce every
two years.Anotherlargeobservationis the residualfor May 1974,which
is about 2.3 standarderrors,but it is adjacentto a residualof minus 1.8
standarderrors.
Betweenthe summermonthsof 1973and the end of the data periodin
May 1974,the productivityshortfallwidenedsubstantially,althoughthe
reversalsin the seriallycorrelatedresidualswarnagainstinterpretingthis
as a new patternin the dynamicbehaviorof outputper manhour.Rather,
it seemsmorenaturalto attributethis productivitybehaviorto the unusual
difficultiesof makingshort-termdemandforecastsin this period.A full
documentationof just how irregularthis periodhas been would requirea
careful comparisonwith past periods of decline in output. But it does
tion uses a filter involving leads as well as lags. To get residuals correctedfor serial
correlationthat can be interpretedas conditionalforecasterrors,the followingprocedure
was used. Logarithmsof both variableswereregressedon lineartrendand seasonaldummies over 1947-73. Residuals of the FRB index from this regression were then fed
throughthe lag distributionestimatedby methods of this paperfrom post-1963data to
generatepredicteddeviationsfrom trend for the labor variable(manhoursor employment). Residuals from the regressionof labor on trend and seasonals are plotted in
Figure 1 as the true values. The differencesbetweenthese true values and the predicted
values were taken as the raw residuals,and an autoregressionwas fit to these residuals
over 1963-74, usinglags 1 through12, 24, and 36. The residualsfrom this autoregression
weresubtractedfrom the true values to generatethe predictedvalues of labor corrected
for serialcorrelation.
Figure 1. Productivity,Measured by Manhours per Manufacturing
ProductionWorker, Actual and Predicted, Monthly,
January 1973-May 1974
Percent
6
Actual<
5
4
3
Predicted
-1
*
4
Ii
%
%~~~~~~~~-
I
3
3
,,/^#
^-
,/'
,-~~~Error
''_-~~~~"
2I 1 F ,,'
'000,~
,*
J
F
M
A
M
J J
1973
/ i\\,
1'10~/
|
/\
~ ~ ~ ~ ~~~'
L-~
I
\
?~
'i
'%
\''
A
Serially correctederror
S o
N
D
J
F
\
i_
M A M
1974
Sources: Author's simulations, using the Federal Reserve Board's index'of industrial production as the
output measure. The sample period begins after 1963.
724
Brookings Papers on Economic Activity, 3:1974
appearthatthe substantialand unusualdropin productivityfor the aggregate economythat has been observedbetweenlate 1973and mid-197435
had a parallelin the productivityof productionworkersin manufacturing.
Conclusions
The responseof labor-taken here as productionworkersin manufacturing-to changesin outputappearsto be completewithinone year;the
adjustmentof manhours,as opposed to employment,is probablycompletedwithinsix months.The responseof employmentappearsto be less
than fully proportional,even when it is complete-the phenomenonof
short-runincreasingreturnsto labor-in degreesthat vary substantially
across industries.There is no noticeabletendencyfor a high degree of
SRIRLto be associatedwithan especiallydelayedcompletionof response.
Theoreticalsuggestionsmade above showed that SRIRL that does not
dwindleaway in estimated"long-runresponses"could arise with firms
optimizingunderuncertainty,even thoughcost and productionfunctions
havethe usualconvexityproperties.On the otherhand,in no industryexcept primarymetals is the estimatedresponseof manhoursinconsistent
withthe absenceof SRIRLaftersix months.
Theresultsof this paperindicatethat single-equationrelationsof output
are capableof generatingresultswith a behavto laborin manufacturing
ioral interpretation
(as opposedto purelystatisticalforecastingrelations),
and that exogeneitytests are useful for identifyingestimatedregressions
for whichstructuralinterpretationis hazardous.
In the threecases (all with employmentas the labor variable)in which
regressionsof output on labor are rejectedby the exogeneitytests, the
estimatedone-sidedlag distributionsall haveunusualandimplausibleestimatesof longer-runresponses.Implausibleestimatesarenot presentin the
regressionsin the other direction,which pass the test. For manhoursin
total manufacturing,resultswith the two outputmeasures-sales and the
FRB productionindex-are more consistentin regressionsof output on
manhours,whichpass the exogeneitytest, than in the regressionsof manhourson output,whichfail it. Whenboth directionsof regressionpassthe
35. See ArthurM. Okun, "Unemploymentand Output in 1974,"BrookingsPapers
on Economic Activity (2:9174), Table 2, last column, p. 498.
Christopher
A. Sims
725
test, in most cases they give roughlyconsistentpicturesof the dynamics
betweenlaborand output.36
The findingspresentedhere do not have directimplicationsfor rulesof
thumbfor relatingeconomy-wideemploymentto outputin businesscycle
analysisand forecasting.Most workersare not productionworkers,and
almostanyothercategoryof employmentseemslikelyto adjustlessrapidly,
less strongly,andless consistentlyto movementsin output.The regressions
displayedshouldnot be expectedto applyto broaderaggregates,and the
single-equationmethodsused here very likely would not succeedin such
applications.This paperhas focusedon productionworkersin manufacturingpreciselybecausethe labor demandfunctionsfor them arelikelyto
be welldefinedandbecauseSRIRL,if it exists,wouldbe most paradoxical
for this type of labor.
On the otherhand, the resultsdo have some implicationsfor business
cycle analysisfor policymakingpurposes.Labordemandregressionshave
universallyincludedsome sort of trendingvariable-for example,a polynomialin t, capitalstock, a brokenline interpolatedamongselectedpeak
yearsof outputor outputper worker,"normalhours,"or the labor force.
The theoreticaland empiricalresultsof this papersuggestthat attributing
causalsignificanceto the estimatedcoefficientsof thesetrendingvariables
is a mistake.Supposethe trendtermis "capacity."Soligo estimatesa distributedlag regressionof employmenton outputwhoselong-runelasticity
is 0.49,andconcludesthat "thelong-runproductivitychangeof .51percent
is a measureof the increasein efficiencyresultingfrommovingto a higher
degreeof capacityutilization."37This paper has shown that, within the
same industry,employmentdemandfunctionsare likely to have smaller
long-runelasticitiesthanmanhourdemandfunctions,andthat both elasticities can be less than one in the absence of nonconvexityin the static
productionfunction.In effect,trendingvariableswill pick up some of the
explanatorypowerthat shouldbe associatedwith the anticipatedpath of
output.An actualchangein capacityutilizationthat did not bearthe usual
relationto a changein the ratioof outputto its trendvaluewouldnot have
the effectspredictedby the "capacity"coefficientin the regression.
36. In assessingthis statementrecallthat, becauseof time aggregation,the veryshortrun dynamicsare expectednot to agree across the two directionsof regression.
37. Ronald Soligo, "TheShort-RunRelationshipbetweenEmploymentand Output,"
YaleEconomicEssays, Vol. 6 (Spring 1966), pp. 161-215 (the quotation is on p. 191).
Brookings Papers on Economic Activity, 3:1974
726
APPENDIX
Reportsof Statistical Tests
TABLES
A-1 throughA-4 presentthe resultsof statisticaltests for various
hypothesesdiscussedin the text relatedto the regressionsbetweenoutput
and manhours,and betweenoutputand employment,that are reportedin
text Tables 1, 2, 4, and 5.
Table A-1. F-Test for Null Hypothesis That Manhours-Output
ManufacturingRegressions Fit Seasonally
Total manufacturing
NonIndex of
Primary
industrial
Durable durable
goods Paper metals Apparel
Regressiona production Salesb goods
Manhourson
output
Output on
manhours
1.38
8.48
1.40
1.57
1.24
1.55
1.62
1.70
1.26
1.79
1.81
1.36
1.54
1.49
Source: The regression to be tested was estimated by the method described in note 24 above, except that
the setting to zero of components of the Fourier transforms in the seasonal bands was omitted. Then,
without repeating the correction for serial correlation, the Fourier-transformeddata were set to zero in the
seasonal bands and the final-stage OLS regression repeated. If interpreted in the frequency domain, this
procedureis equivalent in large samples to omitting the frequency-domain "observations"at those harmonic
frequencies that lie in the seasonal bands. Since the regressionsare fit to 262 observations, the seasonal bands
of the width used contain about forty-seven harmonic frequencies in addition to the eleven exact seasonal
frequencies. The second regression, then, is in effect fitted to a sample with forty-seven fewer degrees of
freedom, and the F(47,1763 statistics can be thought of as testing the significance of a hypothetical group of
forty-seven dummy variables accounting exactly for the variance in those forty-seven observations.
a. Fo.lo(47,176) - 1.38; Fo.o5(47,176) 1.45; Fo.oi(47,176) 1.66.
b. Shipments deflated by the wholesale price index for manufacturing.
727
ChristopherA. Sims
Table A-2. F-Test for Null Hypothesis That Future CoefficientsAre
Zero, Manhours-OutputRegressions, Selected ManufacturingIndustries
Regressiona
Outputon manhours
Manhourson output
Industry
Total manufacturing
Index of industrial
production
Salesb
Paper
Primarymetals
Apparel
Durables
Nondurables
11 future
coefficients
12 future
coefficients
11 future
coefficients
12 future
coefficients
1.001
2.391
1.404
0.966
1.151
0.891
0.370
0.921
2.944
1.936
0.896
1.486
0.957
0.393
1.308
0.667
1.012
1.343
1.444
0.867
1.804
2.434
0.741
1.634
1.373
1.666
2.041
2.758
Sources: Author's regressions, discussed in the text. The coefficients on lags -2 through -12 or on lags
-1 through -12 (see Tables 1 and 2) were constrained to be zero under the null hypothesis.
a. Fo.o5(12,165)- 1.81; Fo.oi(12,165) 2.29; Fo.o5(11,165) 1.84; Fo.oi(11,165) ; 2.39.
b. Shipments deflated by the wholesale price index for manufacturing.
Table A-3. F-Test for Null Hypothesis That Lags 2 through24 Have
Zero Coefficientsin Manhours-OutputRegressions Which Include
Lags -1 through24, Selected ManufacturingIndustries
Total manufacturing
NonIndex of
Durable durable
industrial
Primary
goods Paper metals Apparel
Regressiona production Salesb goods
Manhourson
output
Outputon
manhours
2.90
2.45
3.12
2.53
1.67
1.67
3.27
2.23
2.40
2.64
1.64
2.02
1.68
1.50
Sources: Author's regressions, discussed in the text. See Tables 1 and 2 for the lag distributions.
a. Fo.o5(23,176)
1.59; Fo.ol(23,176)
;
2.02.
b. Shipments deflated by the wholesale price index for manufacturing.
Brookings Papers on Economic Activity, 3:1974
728
Table A-4. F-Test for Null Hypothesis That Future Coefficients Are Zero,
Employment-OutputRegressions, Selected ManufacturingIndustries
Regressiona
Employmentonioutput
Industry
Total manufacturing
Index of industrial
production
Salesb
Durable goods
Nondurable goods
Paper
Primary metals
Apparel
11 future
coefficients
0.68
0.99
0.60
0.50
1.14
1.61
1.02
12 future
coefficients
Outputon employment
11 future
coefficients
0.63
1.89
0.56
0.43
1.24
1.72
2.10
12 future
coefficients
1.57
0.79
1.09
1.97
2.40
2.37
1.02
2.64
0.74
2.35
4.41
2.73
2.42
1.54
Sources:,Author's regressions, discussed in the text. The coefficients on lags -2 through -12 or on lags
- 1 through -12 (see Tables 4 and 5) were constrained to be zero under the null hypothesis.
a. Fo.o0(12,165)
1.81; Fo.ol(12,165) 2.29; Fo.o0(11,165) 1.84; Fo.oi(11,165) 2.39.
b. Shipments deflated by the wholesale price index for manufacturing.
Commentsand
Discussion
MichaelC. Lovell: ChristopherSims presentsa fresh analysisof a much
studiedphenomenon:short-runincreasingreturnsto labor, or SRIRL,
which describesthe tendencyfor fluctuationsin output to be associated
with less than proportionatefluctuationsin labor inputs. He arguesthat
the existingliteraturedoes not rule out the possibilitythat the size and
durationof SRIRL that are found so regularlyin the empiricalwork in
this areaare substantiallybiasedby errorsin variables.Thushis approach
model
resemblesMiltonFriedman'suse of the statisticalerrors-in-variable
in his studyof the consumptionfunction.Friedmanpostulateda trueproportionalityrelationshipbetween consumptionand income and argued
that the observedshort-runrelationshipbetweenobservedconsumption
and income was too flat because they contain errors of observationtransientconsumptionand transientincome.' In the same vein, Sims
arguesthat SRIRLmay be a statisticalartifact.
Sims postulatesa much simplereconomicmodel than is customaryin
the workof otherinvestigators.Themodelproposedby Holt and his associates in 1960 invoked costs of adjustment(hiring, training,and firing
costs)to explainwhy the workforce of a cost-minimizingfirmwouldnot
fluctuateas violentlyas output.2And a cadreof subsequentworkersused
modelsof varyingdegreesof sophisticationin theirempiricalwork. Some
modelsexplicitlyincorporatedsuch variablesas the capital stock, liquid
assets,and inventories;some made outputa decisionvariabledetermined
on the basis of anticipatedpriceand factorcost. Simsis probablycorrect
1. This interpretationis convenientlysummarizedin J. Johnston,EconometricMethods (lst ed., McGraw-Hill,1963), note 1, pp. 148-49.
2. Charles C. Holt and others, PlanningProduction,Inventories,and Work Force
(Prentice-Hall,1960).
729
730
Brookings Papers on Economic Activity, 3:1974
in complainingaboutthe lack of attentionto measurementerror.WhileI
regardhis treatmentof employmentas proportionalto currentand lagged
output-trend corrected-a grossoversimplification,
his attemptto reconcile observedfact with that simplemodel is intriguing.
Near the beginningof his paperSimsmentionsthat one of his aimsis to
use monthlyratherthanquarterlydata.In principle,this practiceincreases
the numberof observationsthree-fold.Ray Fair, who also used monthly
data,had warnedof potentialproblemsbecausethe employmentdatarefer
to a particularsampleweek whilethe outputand shipmentsdata are, for
the most part, based on the whole month.3With a clever examplepresentedin his first two text tables, Sims demonstratesthat these temporal
shift and aggregationeffectsmay introducea psuedo-distributed
lag relationshipand artificialSRIRLwhenin fact laboris proportionalto current
output.
Seasonalityis anothercontroversialproblem.Fair had criticizedthe use
of seasonallyadjusteddata in earlierstudies, arguingthat the seasonal
movementin sales should be capturedby seasonalmovementin output.
In contrast,Simsfeelsthatthe monthlydatashouldbe seasonallyadjusted,
andhe utilizesa sophisticatedprocedureto do so. Onecost is a loss of observations:his techniquecurtailshis sampleperiodfromthe basic 1947-73
span to March 1950 through December 1971. He does reveal in Table
seasonaladjustmentflunksthe F-test in nine of
A-1 that dummy-variable
fourteencases.My guessis that a somewhatmoreelaborateset of dummy
variables,allowingfor a movingseasonal,mighthave done the trick.But
a more sophisticatedapproachwould be to rely on the economicarguments advancedby Modiglianiand his co-authorslong ago. First of all,
Modiglianiand Sauerlenderarguedthat firmsconfrontedwith markedly
seasonalsalesarelikelyto have a planninghorizonof a year;4hence Sims
mighthave truncatedhis lead and lag distributionsat twelveratherthan
twenty-fourmonths. Second, Modiglianiand Sauerlenderargued-in a
departurefrom Fair-that the coefficientsin the productionscheduling
model are themselvessubjectto seasonalvariation-that is, the response
to anticipatedsalesand to previouserrorswill dependupon how close the
Demandfor WorkersandHours(Amsterdam:North3. Ray C. Fair, TheShiort-Run
Holland, 1969).
4. Franco Modiglianiand OwenH. Sauerlender,"EconomicExpectationsand Plans
of Firms in Relation to Short-TermForecasting"in Short-TermEconomicForecasting,
Conferenceon Research in Income and Wealth (Princeton University Press for the
National Bureauof Economic Research, 1955).
ChristopherA. Sims
731
firmis to the seasonalpeak or troughin sales. This notion suggeststhat
separateregressionsfor each season might be useful,possiblywith some
sort of a dummyto fudgefor variationsin the numberof workingdaysin
each month, followed by the customaryF-test to determinewhether,in
fact, poolingover seasonsis appropriate.
Simsdiscussesothertypesof measurementerror.For example,he points
out that the sampleused in estimatingsales is much smallerthan that for
employmentdata. And a problemmay arise because employmentdata
wereusedin the constructionof the FRB industrialproductionindex.One
mightpresumethat the presenceof substantialmeasurementerrorin both
variableswould mean that the standardregressionmodel is not appropriate; Sims runs things both ways, but in a majorityof industrieshis
exogeneitytest rejectsneither directionof regression.He suggests that
either directionwill uncover the true dynamics;presumedly,errors of
measurementin both variablesmean that either direction will involve
bias, althoughnot necessarilyof sizablemagnitude.
I think that Sims is to be congratulatedfor skillfullyexecutinga wellconceivedprojecton an intriguingquestion.It certainlyis usefulto know
that the manhourinputgenerallyadjustsin proportionto outputchanges
within six months. My major reservationconcernsthe appropriatemix
betweeneconometricversuseconomicsophistication.Sims workswith an
exceedinglysimplemodel and does not find it helpfulto distinguishbetweenshipmentsand output.In contrastto Solowand Soligo,5he does not
incorporatethe capitalstock in the regressions,hoping that this variable
can be nettedout by trend.WhileSimsdebateswithhimselfaboutwhether
outputor laboris exogenous,Schrammworkedwitha model in whichboth
variablesare endogenous.6I preferthe more elaboratemodelsbecauseof
their richereconomiccontent,for the same reasonthat I preferthe lifecycle to the permanent-income
hypothesis.Sims'extremelysimplemodel
carrieshim far once errorsof observationand aggregationare giventheir
due. And, in the currentstate of the economy,it is indeedinterestingto
5. Robert M. Solow, "Technical Progress, Capital Formation, and Economic
Growth," in American Economic Association, Papers and Proceedings of the Seventyfourth Annual Meeting, 1961 (American Economic Review, Vol. 52, May 1962), pp. 76-86;
Ronald Soligo, "The Short-RunRelationshipbetweenEmploymentand Output," Yale
EconomicEssays, Vol. 6 (Spring1966), pp. 161-215.
6. R. Schramm, "The Influence of Relative Prices, Production Conditions and
Adjustment Costs on Investment Behaviour," Review of Economic Studies, Vol. 37 (July
1970),pp. 361-76.
732
Brookings Papers on Economic Activity, 3:1974
learnthat whileno substantiallong-runproductivitychangeswill be associated with changesin output,the short-runcushioningeffect of SRIRL
will not persistlong in a periodof anticipatedstagnation.
RobertM. Solow: Themostimportantresultsof thispaperaresummarized
in Table3. That table shows an almostproportionalcumulativeresponse
of manhoursof productionworkersto output,withthe strongestresponse
in the contemporaneous
monthand the full responsecompletedwithinsix
to ninemonths.Thereis evidencefor this findingin all industriesreported
exceptpaperand possiblyprimarymetals.Such resultsare of interestfor
two mainreasons:theycan guidethe interpretation
of monthlyunemploymentstatisticsand they can help in analyzingthe short-termemploymentoutputrelationin macromodels.
If the manhoursfor all workerswerelike those of productionworkersin
then Sims'resultswould supporta statementthat the admanufacturing,
justmentof manhoursto outputtakes only about two quarters-indeed,
thatthe responseof employmentto outputis about80 percentcompletein
this time.As Simspointsout, however,only a smallfractionof all workers
are manufacturingproductionworkers;so more insight into the rest of
employmentis requiredbefore resultslike his can be used to interpret
economy-widedevelopments.But his projectionsfor 1973and early 1974
do coincide with the economy-widefinding reportedby Okun (BPEA,
2:1974) that the response of unemploymentto output was unusually
delayedin this period.
On the analyticissue,a lot of the paradoxof short-runincreasingreturns
to labor in macromodels has evaporatedin this paper.First, Sims estimates nearlyconstantreturnsto labor. The differencesbetweenhis estimatesandunityaremostlystatisticallyinsignificant.Evenif a puristwould
preferdiminishingreturnsto labor,it hardlymatters.
Second,thereareoverheadworkers,evenamongthosewho areclassified
Theirexistenceis not
in the data as productionworkersin manufacturing.
explicitlyallowedfor in the estimationprocess,so it makesthe findingof
constant,or even increasing,returnsto labornot surprisingat all.
A thirdreasonwhya findingof constantreturnsto laborin the shortrun
does not disturbme is that short-runvariationsin otherinputs,especially
the servicesof plantandequipment,arenot directlyobserved,so theycannot be held fixed. If, for instance,the propermodel is fixed proportions
betweenthe servicesof plantand equipmentand of productionworkersin
the shortrun,thenthe movementobservedis reallyone alongan expansion
ChristopherA. Sims
733
path,andnot in one variablealone.The cumulativeelasticityis to be interpretedas short-runreturnsto scaleratherthan to laboralone.In that case
unitywouldbe a very cozy value,not at all hard to accept.
Finally, Sims makes an importantanalyticpoint in discussingrational
expectationsthat I thinkcan be mademoregeneral.Suppose,for instance,
that labor requirementsare proportionalto output statisticallyso that
proportionalityis reallythe rule. If employersknow that sales will be a
randomvariable,fluctuatingarounda constantmean value, and if it is
muchcheaperto producea constantmean value of outputand meet fluctuationsin salesfrom inventoriesratherthan by variationsin production,
then labor employmentand manhourswill be constant.A regressionof
employmenton sales will show a zero sum of coefficients.This is fundamentallymeasurementerror.If one could measureoutput correctly,one
wouldfindlaborproportionalto output.I thinkthat Simsunderestimated
the importanceof this resultby tyingit too closelyto rationalexpectations.
Much the same thing would follow if employersthoughtincorrectlythat
sales wouldfluctuateas a seriallyuncorrelateddisturbancearounda constant mean value. All one needs is that kind of belief about the behavior
of output.
I havequestionsabouta coupleof pointsconcerningthe results.In most
of the industries,especiallyin appareland paper,the regressionof output
on manhoursappearsinconsistentwith the regressionof manhourson
output.In exact terms,if the cumulativeresponsein one case is less than
one, the responsein the other case ought to be greaterthan one. If that
relationbetween the two directionsis not necessarilytrue in statistical
terms,the reasonoughtto be explained.Furthermore,I wouldlike to have
the trendtermsfromtheseregressionsto see whetherthey are plausiblein
sign and in magnitude.How one interpretsthe resultsdependsupon the
plausibilityof these trendterms.
It is cheapto inventdifficultthingsfor peopleto do. However,I suspect
thereis an asymmetricalcyclicalresponseof manhoursto output.It would
be useful if Sims could allow for differentresponsesdependingupon
whetheroutputis risingor falling.
GeneralDiscussion
FrancoModiglianirelatedSims'resultsto his own view that most productivefactors are approximatelyfixed in the short run, suggestingcon-
734
Brookings Papers on Economic Activity, 3:1974
stant returnsto labor as a first approximation.The existence of some
overheadlabor-or of labor used for a time on overheadjobs, such as
paintingwalls-tilts the balancetowardincreasingreturns;whilethe mixture of vintagesin the capitalstock with whichlaborworkstilts it toward
decreasingreturns.Whicheffectdominatesat a particulartime is hardto
predict, at least until very high levels of utilizationare reached; but
Modiglianiwas not surprisedat Sims' findingof approximatelyconstant
returnson balance.
Otherpanelmembersdiscussedaspectsof the productionand employment decisionthat might affectcyclicalproductivity.ArthurOkun asked
why fluctuationsin inventoriesratherthan labor input shouldnot be expected to accommodatemost of the short-runvariationin demand.A
model that allowedfor this effectmight producedifferentelasticityestimates.Daniel Brillnoted, however,that holdinginventorieslays substantial carryingcosts on the firm. R. A. Gordon shared Robert Solow's
concernthat the ups and downsin economicactivityshouldbe separated.
He noted that the costs of hiringand layingoff workerswouldaffectproductivityas much as the presenceof overheadlabor would, and these
turnovercosts wouldbe asymmetricbetweenrisingand fallingperiodsof
the businesscycle.
The inadequaciesof the FRB productionindexesas measuresof output
drewsome comment.Okunand RobertJ. Gordonnoted the inaccuracies
associatedwithincludingin productiononlyfinishedgoods andnot goodsin-process.For example,a truckenginewouldnot be countedin the FRB
indexif it wereproducedby GeneralMotors,but wouldbe if it wereproducedby an enginecompany.Otherspointedto the use of electricityinput
to measureoutputfor someindustriesas well as the use of laborinputthat
Sims had discussed.LawrenceKlein arguedthat, at the industrylevel,
gross output would be a betterindex than value added since it could be
measuredmoreaccuratelyand avoidedthe necessarilyuncertainestimates
of inputs.
Klein and CharlesHolt secondedMichael Lovell's interestin a fuller
modelto supportthe empiricalwork.Simsrepliedthat the singleequation
he estimatedcould have been derivedfrom a completemodel with many
inputs,and cited the work of Nadiri and Rosen which did preciselythat.
Klein remainedunpersuadedthat the estimationscould be interpretedas
anythingmore than correlationsbetweenthe labor and outputvariables.
They appearedto be neither structuralproductionfunction relations-
Christopher
A. Sims
735
which would requiretechnicalinformationabout productionlags-nor
structurallabor demandfunctions-which would requireothervariables,
suchas relativeprices.Simsrespondedthat productionfunctionandlabor
demandrelationsneed not be different,althoughthe hiringresponseto
currentandfutureoutputthathe modelleddidexplicitlymakehis equation
one describinglabordemand.The exogeneitytest indicatedthat the equationswerecapturingthe dynamicsbetweenlaborandoutput,althoughthey
did not offera particularstructuralinterpretationof the relation.Simsdid
not put relativefactorpricesin his modelbecausethey changeslowlyand,
on evidenceof workby others,enterthe estimateswiththe wrongsign.Holt
and Saul Hymansobjectedthat, while includingrelativefactor pricesin
the equationposedproblems,omittingthem,or any otherrelevantvariable
suchas labormarketconditions,cast doubton the estimatesof the sum of
the coefficientson output.
R. J. Gordondescribedhis own labordemandequation,whichis similar
to Sims' except that it relatestotal privatenonfarmmanhoursto private
output. The sum of the coefficientson output is between0.65 and 0.85,
somewhatlowerthan Sims'estimatespresumablybecauseof the presence
of more overheadlabor. This equation,which has sufficientcoverageto
say somethingmeaningfulaboutglobalproductivity,was2.3 percentbelow
its forecastlevel for the third quarterof 1974.A slightlylargershortfall
was experiencedin 1969,and a smaller,althoughstill distinct,shortfallin
1956.These were all periodsof stagnantor decliningoutput.But similar
errorsare not observedfor otherperiodsof decline.