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FRS
Chicago
#94-25

Overtime, Effort and the Propagation
of Business Cycle Shocks

George J. Hall

Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
December (W P -94-25)

O ve rtim e , effort and the p ro p a g a tio n o f business
cycle shocks
George J. Hall*
Department of Economics
University of Chicago
December 1994

A b str a c t

This paper presents and estimates a variant of Hansen and Sargent’s (1988) real
business cycle model with straight time and overtime. The model presented has only
one latent variable, the state of technology, yet it does as good a job propagating and
magnifying shocks as labor hoarding models which incorporate unobserved effort.
This paper also finds that the implied effort series of labor hoarding models displays
a high coherence with U.S. overtime data at business cycle frequencies. This supports
the view that effort is procyclical.

Introduction

This paper estimates a dynamic general equilibrium real business cycle model of the U.S.
economy incorporating straight time and overtime. This model is a hybrid of Hansen and
Sargent’s (1988) model with straight time and overtime and Burnside, Eichenbaum and
Rebelo’s (1993) labor hoarding model. This model is studied along with an estimated
version of Burnside, Eichenbaum and Rebelo’s model. The two models are analyzed to
answer two questions. First, how do different assumptions about laoor market rigidities
effect the real business cycle model’s ability to propagate and magnify shocks? Second,
does the unobservable time series effort implied by the labor hoarding models make sense?
*1 thank John Cochrane, Martin Eichenbaum, Charles Evans, Lars Hansen, Robert Lucas, David Mar
shall, Ellen McGrattan, Edward Prescott Jr., Argia Sbordone, Thomas Tallarini and Mark Watson for
helpful conversations. I thank Thomas Sargent for his encouragement, support and comments. I gratefully
acknowledge research support from the Federal Reserve Bank of Chicago. All errors are mine.

A well-known and often repeated criticism of standard real business cycle (RBC) models
is their dependence on implausibly large aggregate technology shocks to account for the
variability in output . 1 Indeed Cochrane (1994) concludes “we haven’t found large, identi
fiable, exogenous shocks to account for the bulk of output fluctuations.” Another way of
phrasing this criticism is that standard real business cycle models possess^ weak propagation
mechanisms. That is, if RBC models could plausibly magnify and spread over time the
effect of the shocks, these models would be less dependent on large aggregate shocks to
produce sufficiently volatile time series. Consequently a challenge to business cycle theo
rists is to construct models that amplify shocks and realistically propagate shocks over the
business cycle.
This challenge has not gone unanswered; Burnside and Eichenbaum (1994) studies two
potential channels of propagation: time-varying effort and time-varying capital utilization.
Their model has the ability to substantially magnify and propagate shocks. Motivated by
Burnside and Eichenbaum’s success, the current paper puts capital utilization aside for
the moment and focuses attention on the implications for shock amplification and prop
agation inherent in the time-varying effort assumption. Consequently the relevant model
is Burnside, Eichenbaum and Rebelo’s (1993) labor hoarding model although the issues
addressed are from Burnside and Eichenbaum (1994). The current paper studies in detail
the propagation mechanism of the labor hoarding channel and proposes a complementary
channel - the differentiation between straight time and overtime.
The model presented here builds on the model of Hansen and Sargent. Like Hansen
and Sargent this model incorporates a version of Lucas’s (1970) instantaneous production
function in which straight time and overtime are not perfect substitutes. However, it differs
from Hansen and Sargent’s model in two ways. First, a government sector and a government
shock are added to the model. Second, firms must commit to the number of workers they
will employ before observing any shocks to the economy; once the shocks are observed,
firms can adjust the number of employees working overtime. In other words, the cost that
firms face in adjusting the contemporaneous number of workers after observing the shocks
is infinite.
The model presented can also be viewed as a variation on the model of Burnside,
1See for example Summers (1986), McCallum (1989) and Eichenbaum (1991).

Eichenbaum and Rebelo (BER ) . 2 In both models, firms must commit to the number of
workers employed before observing any shocks to the economy. In BER’s model, firms can
adjust the work effort after observing the shocks. In the model presented here, firms can
adjust the number of persons working overtime after observing the shocks. The advantage
to the model presented here is that unlike effort which cannot be measured, overtime
employment is observed and measured.
Using economically interpretable parameter values obtained by maximum likelihood,
the model presented here has the ability to magnify and propagate shocks over the busi
ness cycle as well as BER’s labor hoarding model. This success is achieved under the
constraint that all variables in this model, with the exception of technology, are observed.
Moreover, the model does as well as BER’s model in matching the first- and second-moment
properties of the data. This result demonstrates that theorists can construct models which
embody quantitatively important propagation and amplification mechanisms without using
the unobservable variable effort.
This paper also analyzes the relationship between effort and overtime. It finds that
the time series on effort implied by Burnside, Eichenbaum and Rebelo’s labor hoarding
model displays a high coherence with U.S. overtime employment data at business cycle
frequencies. The result is consistent with the belief that overtime and effort move together
over the cycle.
The remainder of this paper is organized as follows. In the second section the model is
presented. In the third section the log LQ approximation technique for solving the model
is described, and the method for estimating the parameter values of the model is discussed.
The data are reported in the fourth section. In the fifth section, empirical results are
calculated and assessed. In the final section some concluding remarks are made.

T h e

e c o n o m y

This section presents a variant of Hansen and Sargent’s (1988) straight time and over
time model modified to incorporate a government sector and the precommitment of total
employment.
2Other models in which employment must be set before observing the shocks include Christiano (1988)
and Sbordone (1993).

Consider an economy with a continuum on [0,1] of identical infinitely lived agents who
have preferences over consumption of a single nondurable good, C*, and leisure,
maximize:

L t.

They

O
O

(1)

E ' j j r p t [ \ o g C t + v \ o g L t],

t=o

where

is their subjective discount factor and u is strictly greater than zero.

Agents are endowed with

units of time each period which can be divided between

labor and leisure; consequently 0 <
of three values. Let

and

shift’ respectively. Let 0 <
<

T
T

Furthermore

is restricted to take only one

be the lengths of a ‘straight time shift’ and an ‘overtime

hi < h\

hi < T.

Therefore:

if the agent is unemployed
if the agent works only the straight time shift
if the agent works both the straight time and overtime shifts

—h i
—h i —h 2

Proceeding as in Hansen (1985) and Rogerson (1988) lotteries are employed to convexify
the commodity space. Assume 7rjt and 7r2( are the probability of working just a straight time
shift and the probability of working both straight time and overtime shifts, respectively.
Hence 1 — flu — 7r2t is the probability of being unemployed. Agents choose over labor
probabilities to maximize their expected single period utility:
7 lt[logCf
r
+vlog(T-/i1)]+7r2t[log<7t
+vlog(r-/i1-/i2)]+(l-7rit-7r2 )[logC,+t;log(T)]. (2)
t
t

Define
and

equals

N 2t

to be the fraction of agents who work both shifts (overtime employment)

to be the fraction of agents who work the straight shift (total employment). Hence
ttu

+ 7r2t,

N 2t

equals 7r2t and the agent’s utility function, ( 1 ), can be written:

O
O
E j 2 P > g C t - a i(N u -

N 2 t) -

a 2N 2t -

a„(l -

N u )},

(3)

(=0

where ao = —
vlog(T), ai = —
ulog(T —h i ) and

= —
ulog(T —h i —h 2). This preference

specification is from Hansen and Sargent.
Aggregate output,

Q t,

is produced by an instantaneous Cobb-Douglas production tech

nology such that at any instant, r, during period

Q (t

the rate of output per unit time is

+ r) = exp(A 1+,)(7 ,+T * ? + “W1+t.
"

where
7

K t+ T

is the capital stock, At+r is the state of technology, and

N t+T

is the labor force;

represents the growth rate of exogenous labor-augmenting technological progress. It is

assumed that technology and capital are constant throughout the period. Hence aggregate
output produced over the two shifts during period
Qt

is:

= exp(At)(7t)a ^ 1_“ [ ^ i ^ + W J .

Labor’s share, a , is restricted such that 0 <

< 1.

Total output is allocated each period to private consumption,
tion,

and investment,

G t,

(4)

%government consump
,

It:

(5)

C t + G t + It < Q t.

Productive capital depreciates each period at the rate 0 < < < 1, so:
5
(6)

K t + i = ( \ - 6 ) K t + I t.

Technology has the following law of motion
^i+l = /fx +
where

and

px^t

(7)

& \W \t+ U

is a sequence of i.i.d. normally distributed random variables

with mean zero and variance one.
Government consumption evolves according to
(8 )

G t = ^ e x p ig t),

such that

follows the first order autoregression
0t+l =

where

pg <

Pg + Pg9t

<7gWgt+1 ,

(9)

i. {to5 f} is a sequence of i.i.d. normally distributed random variables with

mean zero and variance one and is orthogonal to innovations in technology.
The timing of this economy differs from the one presented in Hansen and Sargent.
In this model
Ku+1

N u

must be chosen before, instead of after, (A*, G t ) are known;

N 2t

and

are chosen after observing the shocks. Formally, let the initial information set,

be generated by the set of initial conditions, {A0, Go, K

I q,

iV10}. Let the information set

be generated by {Ao, G o ,

Mo} and

Ko,

measurable functions of {Ao, G o ,

Ko,

{ ( w \ s , w gs) : s

M o} U {(in\s, w g3)

1 , 2 ,...£},

= 1,2 ,

consists of all

Hence {«;*«, u> ^ 0>
st}

is a conditionally homoskedastic martingale difference sequence adapted to the sequence
of information sets,

the model presented here, the set {M+i, Mt+i» Mt} is

restricted to reside in the space of measurable functions of
processes {M+i,Mt+i> M»}tSo resides in the space
L\

{yt

: y t is in

for

I t.

That is, the set of stochastic

given by:

= 0 , 1 , 2 ,..., and

E Y ^L 0^ y ] \ I o

< oo}.

(1 0 )

This condition requires that decisions made at time t depend only on information available
at time

If markets are complete the decentralized competitive equilibrium corresponds to the
solution of a social planning problem. In this case, the social planning problem is to choose
a set of stochastic processes
initial conditions {Ao, G o ,
the space

{ K t+ u

Ko,

M«+1 , M t } ^

maximize (3) subject to (4) - (9) given

Mo}- Moreover {M+i, Mt+ 1 , M t}^o

required to reside in

defined by (10). This definition of the social planning problem completes the

description of the model; this model is referred to as the labor precommitment model.
For comparison purposes, a version of Burnside, Eichenbaum and Rebelo’s labor hoard
ing model is presented. In their model, firms commit to the number of workers employed
before observing any shocks to the economy; after observing the shocks, firms can adjust
the work effort of their employees. Higher work effort increases output but lowers the
agents’ utility. Formally, in the labor hoarding model the social planner chooses a set of
stochastic processes

{K t+ i,

Mt+i, e* } ^ . 0 t° maximize

£ £*[Mt[log C t + u log(r -

i -

e t h i)}

- MO [log C t + u log(T)]]

(1 1 )

t=0

subject to
Qt

= exp(At)(7 ‘)“M1

and (5) - (9) where {M+i, Mt+i, e* } “

0 ls

(12)

a h i [ e t N l t ]a ,

iQ the sPace To given by (10)- In this model

is the fraction of agents who work a single shift of length

h i.

Two additional variables are

introduced in BER’s model. The date t level of employee work effort is denoted by e t;

is a

fixed cost of going to work. The only difference between this model and the one presented

in B E R is th at th e production function in (12) is instantaneous. T h is m odel is referred to
as th e labor hoarding m odel.
D u e to th e nonlin earity o f both social planner problem s, it is not p ossib le to obtain an
a n alytic solu tion to eith er of th ese m odels. H ence C h ristiano’s (1988) logarith m ic m odifi
cation of K ydlan d and P r e sc o tt’s (1982) linear-quadratic approxim ation is used to obtain
an approxim ate solu tion to th e social planner’s problem . In th e n ex t section this approx
im ation procedure is discu ssed, and th e estim a tio n techn iqu e of th e param eter values is
presented.

C o m p u t i n g

a n

equilibrium

In order to solve th e social planning problem s discussed above, th e follow ing procedure is
em p loyed. For each m od el, th e econ om y is transform ed into one w hich possesses a steadystate; then a log-linear-quadratic approxim ation of th e transform ed econ om y is m ade. D e
cision rules for th e log-linear-quadratic econom y are com p uted. W ith th ese decision rules
in hand, th e param eter values of th e approxim ate econom y are estim a ted using m axim um
likelihood. U sin g th e ob tain ed param eter values and decision rules, equilibrium allocation s
are com p u ted . To keep th e presen tation sim p le only th e labor p recom m itm en t m od el is
discussed; how ever th e analogous steps are used to solve th e labor hoarding m odel as w ell.
In th e labor p recom m itm en t m odel th e endogenous variables do not converge to a
n on -stoch astic stead y sta te but to a stead y sta te grow th path. Therefore th e econom y is
transform ed so th at it has a stead y sta te. D efine th e following:
kt

= lo g (/C ,/7 f)

ct = l o g i C t h * )

= lo g ( /t/

qt = \ o g { Q t h l )

£)

= log ( N u )

n 2t = log { N u )

For th e rem ainder of th e paper, lower case letters d en ote logged and detrended variables.
T h e social p lan n er’s problem can b e w ritten:
OO

m a x £ ^ ^ [ c ( - a i( e x p ( n it) - e x p (n 2t)) - a 2 e x p (n 2t) - a0( l - e x p (n at)) -f
t=o

(13)

su b ject to:

e x p (c t) + exp(fft) +

exp(fcm ) - (1 - 6) exp ( k t ) = ex p (g t)

(14)

exp(<7t) = exp(A t) e x p ( k t ) 1 "[/ii exp (n i*)“ + h 2 e x p ^ t ) " ]

(15)

and (7 )-(9 ). Since th e last term in the planner’s o b jective function is beyond th e planner’s
control it is dropped for th e rem ainder of th e analysis.
T h e m odel is solved using th e approxim ation techn iqu e em ployed by C hristiano (1988).
D efine th e sta te vector, x t = (1, At , g t , k t , n u ), th e control vector, u { = (n )t+ i, n 2 t, A i), and
;(+
th e shock vector, w t = ( w \ t , w gt). Let 8 = ( a , ( 3 , 6 , i , p \ , p g , p \ , p g , h i , h 2, T , v ,

Vg) be a

vector of param eters. Finally, let z t = [xj, u't].
T h e approxim ation technique proceeds as follows. F irst, su b stitu te (14) and (15) into
(13)

so th e u tility function is now a nonlinear function of the param eters and the sta te

and control variables. Call this function r ( z t , 0 ) . H ence the social planning problem can be
w ritten as:
OO

m ax E ^ 2 P ‘r ( z „ 0 )
(=0
su b ject to:

1
-W i
9 t+ i

M A Px
=

■

ftn+i

n 2t

&<+i

nit

0
0

9t
kt

k t+ i

’

0
0

w\t+i
w gt i
+

T h is constraint can be w ritten shorthand as:
x t+ i = A x t + B u t 4- C w t + i.

(16)

Second, com p u te th e stead y sta te values of the sta te and control variables; call this vector
z. Third, take a second order Taylor series approxim ation to r { z t , 8 ) around th e steady
sta te to form a m atrix M . such that:
d ri^ Y

M . — e ( r ( z , 8)

1 f.drfa 0Y
^
dz

d r{z,8)
dz

-e — ez

dz
f d 2r ( z , 8)
dz2

, d 2r ( z , 8 )

+ 2Z
d z ^ z)e +
d 2r ( z , 8 ) _ ,
d 2r ( z , 8 )
ze' +
;
)
dz2
dz2

where e is a selector m atrix - a vector of zeros w ith a one in th e elem en t corresponding to
the constant term . N o tice th at M . can be written:

5
W

Fourth and finally, substitute

z \M .z t

for r(zt, 9 ) . Hence the social planner’s problem is now

approximated by the following linear-quadratic problem.
O
O

max E q

ft1

xt
ut

' 5 W'
W' Q

Xt
ut

subject to (16). This approximated problem is a discounted optimal linear regulator prob
lem, and its solution is obtained in a straightforward manner.
The solution to the above optimal linear regulator problem yields the following time
invariant decision rules:
u( =
The matrix

is the solution to following matrix Riccati equation:
Jr = ( Q

for

(17)

—T x t .

+ / 3 B ' V B ) - 1( 0 B ' V A

(18)

+ W')

satisfying
V

0 A 'V A

- (W +

0 A 'V B )(Q

p B ' V B ) - x{ 0 B ' V A

(19)

W ).

Substituting (17) into (16) yields a closed form of the state space representation.
= A 0xt + Cwt+i

where
Let

A 0
yt

(20)

—B T .

denote a vector of observer variables:

y t = [qt <h i t 9 t k t h t n \ t Ti2 t}'-

Let

denote

the corresponding vector of untransformed (upper case) observer variables. All observer
variables are written as linear functions of the state variables.
Vt = Q x t
.

The matrix

is formed by stacking the coefficients of the linear equations of the ob

server variables as functions of the state variables. Total hours,
exp(ht) =

(21)

hi N u

h t,

is defined such that

Since output, consumption, investment and hours are

nonlinear functions of the state variables, Taylor approximations are used. The remaining
four variables in

are either state or control variables.

In this model there are two corner solutions that must be avoided to ensure the accuracy
of the log-linear-quadratic approximation. When the shocks to the economy are higher than
anticipated, firms increase the number of persons working overtime. However firms cannot
employ more overtime workers than straight time workers. Likewise, when the shocks
to the economy are lower than anticipated, firms reduce the number of persons working
overtime; however firms cannot reduce the number of persons working overtime below zero.
Parameter values must be set such that the constraint 0 < 7r2i < 7rlt never binds.
If measurement error is added to equation (21), then the linear system (20) and (21) can
be used to compute a Gaussian likelihood function. Economically sensible parameter values
for the vector

that maximize the likelihood function can then be extracted from the data.

This estimation procedure is discussed in more detail in Anderson, Hansen, McGrattan
and Sargent (1994).
Assume the vector j/t is observed with measurement error. Replace (21) with:
(22)

Vt = Q x t + v t

where

is measurement error such that:

Evt

v t = T>vt~ \

Evtvs =
7Z

f % if t=s
| 0 if t 7 ^ s
0 Vt > s .

is a diagonal matrix. Hence the state space system can be written as:
£«+i

and

A 0xt T Cwt+i

Q xt

y = y t+ i — V y t

Defining

and

Evtv's =

Assume

T>Vt - \ + 7] f

Q A 0 — T>Q

allows the state space system to be represented

by:

£t+i —

A 0x t

C w t +1

Q x t + Q C w t+1 +

v t +1 -

Define the following: let
X;

let

denote the linear least squares projection of

onto

and E denote the steady-state ‘Kalman gain’ and ‘state-covariance m atrix’ of the

time invariant Kalman filter. Therefore,
=

E =

(c c 'Q '

A 0Z A 0
’

A 0w ' ) { Q Y . g '
CC -

{C C 'Q '

and E must satisfy:

g c c 'g y 1

A 0Z G ' ) ( G X G ’ + n

+ G C C ' G Y X( G ^ A ! 0

Q C C ') .

Applying the time invariant Kalman filter to the above state space representation attains
the corresponding innovations representation:
£ 4+1 =
Vt

where

ut -

G C C 'G .

j/t+i -

E [ y t+ i \ y t ,

J/i, xx], x t

A 0i t
G it

(23)

JCut

(24)

= E [ x t \ y t , . . . , y u x x]

Hence the Gaussian log-likelihood function for

and

{ y t } J =x

E u tu't

= fi =

conditioned on

G W ' + Tl +
xx

is given

by:
log L(0) = - ( T - 1 ) log 2* —i ( T - 1) log |H| - i
Z

i z

u 'fl-'u , -

Z t=l

i x ; log |diag(V,)-1 1 (25)
.
1

4=1

The final term is necessary since the model’s implications are for transformed (logged and
detrended) data .3 The parameter estimates that maximize log L ( 0 ) are reported in section
5. Using these parameter values decision rules can be computed.
4

D a t a

In this section the data set constructed for this analysis is discussed. The data are quarterly,
aggregate data of the United States for the sample 1955:Q1 to 1992:Q4.4
The capital stock series,

K t,

is the net stock in 1987 dollars of fixed private capital plus

the net stock of durable goods. These series are reported in U.S. Department of Commerce
(1993) and are updated in the

S u r v e y o f C u r r e n t B u sin ess.

To convert these annual series

to quarterly series a linear interpolation is used. Since the data are reported as end-of-year
stocks, the year

observations are used as the first quarter of year t +

observations.

3See Anderson, Hansen, McGrattan and Sargent (1994).
4Unless otherwise stated, the data were obtained from the Federal Reserve Bank of Chicago’s database.
Data on total employment and overtime employment from 1955 to 1976 were provided by Gary Hansen.

Investment, 7(, is private fixed investment plus the personal consumption expenditure
on durable goods. Private consumption,

C t,

is the sum of the personal consumption expen

ditures on nondurable goods and services and the imputed service flow from the stock of
consumer durables.

is government purchases. The imputed service flow from the stock

of consumer durables is a quarterly seasonally adjusted series from the MPS database
documented by Bray ton and Mauskopf (1985). The other series are quarterly seasonally
adjusted data from the National Income and Product Accounts reported in 1987 dollars.
Output,

Q t,

is the sum of

C t, G t,

and

It.

This measure of output differs from the official

GDP measure in that it excludes net exports and the change in business inventories but
includes the service flow from the stock of consumer durables.
Total employment,

N \t,

is the number of employed persons at work in nonagricultural

industries who worked 35 hours and over a week; overtime employment, A^t, is the number
of employed persons at work in nonagricultural industries who worked 41 hours and over a
week. Total hours,

H t,

is the number of persons employed at work 35 hours and over a week

in nonagricultural industries multiplied by the average weekly hours worked in nonagricul
tural industries. All these series are from the Bureau of Labor Statistics’ household survey
and reported in

E m p lo y m e n t a n d E a rn in g s.

These series are adjusted for holidays in the

BLS’s reference week using OLS and seasonally adjusted using EZ-X1 1 . Since the labor
series are monthly, each quarterly observation is the average of the three corresponding
monthly observations.
In order to make these data consistent with the lower case variables in the theoretical
model, all data are converted to per capita terms using the civilian, noninstitutional pop
ulation, 16 years and older. All the series of flow variables are converted to per quarter
rates. The series

Q t , C t , It, G t

and

K t

are divided by 7 *. Logarithms of each series are

taken.

E m p i r i c a l results

This section assesses the quantitative implications of the two models. First, the parameter
values are computed by maximum likelihood. Second, the ability of the models, evaluated
at the computed parameter values, to fit the time-series properties of the data is measured.

Third, the propagation mechanisms embodied within each model are studied along the
dimensions discussed in Burnside and Eichenbaum (1994). Fourth, the relationship between
effort and overtime is studied.

5.1

P a r a m e t e r values

This subsection presents the parameter estimates that maximize the likelihood function (25)
for both the labor precommitment model and the labor hoarding model. The estimated
models are identical to the theoretical models described above with one exception; in the
estimated models, government,
series. Its trend is denoted

g t,

j g.

is allowed to grow at a different rate than the other

To facilitate comparisons across the two models, three

parameters are fixed prior to estimation;

is set to 1.0044,

is set to 1.0007 and

to 1369 hours per quarter. The value for 7 is chosen by separately regressing

qt ,

is set

ct, i t and

on a constant and a linear time trend. The coefficients on the time trend are constrained to
be equal across all four regressions. Likewise 7 g is computed by regressing
and a linear time trend. The value for the time endowment,

on a constant

corresponds to 15 hours

per day.
The innovations,

u t,

used in the likelihood function are constructed recursively using

(23) and (24). The time-invariant Kalman filter is employed with
steady state values. The initial value of the state vector,
of technology and the beginning of sample values for

27,

g, k

and fi set to their

is set to the steady state level
and ni. Data over the entire

sample are used in constructing the innovations; however to mitigate the influence of the
choice of initial conditions, the first two elements of the innovation series are not used in
computing the likelihood function.
Table 1 presents the point estimates of the preference and technology parameters for
the labor hoarding and the labor precommitment models. Standard errors are also reported
for all the estimates with the exception of 6 ;

is free during estimation but fixed when

computing standard errors. This was done to avoid a singularity in the information matrix
when computing standard errors.
The parameter values for the labor hoarding model are virtually identical to those found
by Burnside, Eichenbaum and Rebelo. Most of the point estimates are within one standard
error of BER’s point estimates. For example, in the labor hoarding model, the estimate of

Table 1
Preference and technology parameter estim ates

a
0
6
PX
Pg
Px
Pg
hr
h2
V
t
<*\
°9

7.004
93.56
0.0057
0.0089

5.706

Labor precommit
Value
S.E.
0.027
0.689
0.988
0.0225
0 .0 1 1 2
-0.0149
0.0887
0.0859
0.973
0.005
0.987
0 .0 1 2
451.1
48.5
158.9
47.5
6.670
0.609
O
O
oO
C

Parameter

Labor hoarding
Value
S.E.
0.656
0.093
0 .0 1 2
0.983
0.0225
-0.0145 0.0309
0.0886 0.0858
0.006
0.979
0.987
0 .0 1 2
511.4
1 0 .8

1 1 0 .2 0

0.0006
0.0014

0.0060
0.0092

0.0004
0.0013

labor’s share in the production function is 0.656; BER estimate labor’s share to be 0.654.5
The estimate for p \ in the labor hoarding model is 0.979 while BER compute 0.982. Likewise
the labor hoarding estimate for

is 0.987; this is identical to BER’s estimate. Recall that

the labor hoarding model presented here is a slight variation of Burnside, Eichenbaum and
Rebelo’s model. BER compute their parameter estimates using GMM on a different data
set. This replication of their results suggests that their selection of parameter values is
robust. There is good reason to be confident that these parameter values are reasonable
and are not dependent on specifics in the data set. Moreover it is reassuring that applying
GMM and maximum likelihood to this model yields similar parameter values.
In the labor hoarding model three parameters are estimated that are fixed by BER.
First is the discount factor,

(3.

The estimate of (3 is lower than usually assumed. The point

estimate, 0.983, implies an annual risk-free interest rate of 8.7%. Nevertheless the point
estimate is less than one and within one standard error of the values usually assumed.
Second is the shift length,

h \.

BER fix the shift length so steady state effort is one. In

5T h e B E R p a ra m e te r estim a te s are for th eir L abor H oarding I - O veridentified m odel which are presented
on page 256 of th eir pap er.

Table 2
M e a s u r e m e n t error p a r a m e t e r e s t i m a t e s
series ordered: q t , c*, i t , 9 t ,
Labor hoarding
Parameter Value
S.E.
D ( l,l)
0
©(2 , 2 )
0.999
2>(3,3)
0.980
0 .0 1 1
P(4,4)
0.024
0.980
2 ?(6 , 6 )
0.550
0.157
©(7,7)
0.576
0.166
V (8 ,S )

k t, h t , n u , n2t
Labor recom m it
Value
S.E.
0

0.999
0.977
0.983
0.757
0.821
0.946

0.008
0.024
0.204
0.170
0.037

> ( i,o
> ( 2,2 )

0.0036

0.0006

0.0033

0.0005

> ( 3 ,3 )

0.0109

0.0014

0.0104

0.0014

> ( 4 ,4 )

0.0064

0 .0 0 2 0

0.0064

0.0017

> ( 5 ,5 )

> ( 6,6)

0 .0 2 1 2

0.0049

0.0208

0.0065

> ( 7 ,7 )

0.0185

0.0047

0.0179

0.0052

0.0209

0 .0 0 2 0

> ( 8,8)

the model presented here, the point estimate is economically interpretable

(h t

= 511.4)

and corresponds to 39.3 hours per week. The third parameter is the fixed cost to working,
£. BER state that their results are insensitive to choices of £ between 20 and 120. These
results support that statement; the point estimate for £ is 93.56 with a standard error of
110.20.

The parameter estimates for the labor precommitment model are similar to those in
the labor hoarding model. The estimate of labor’s share is 0.689, a slightly higher number
than often used. The point estimate of /? is 0.988 - the same number King, Plosser and
Rebelo (1988) use. Both models estimate
parameters

and

to be 0.0225. The point estimates for the shift

are 451.1 and 158.9, respectively. These point estimates correspond

to a straight time shift of 34.7 hours a week and an overtime shift of 12.2 hours a week.
Since the definition of total employment excludes persons working less than 35 hours a

week, the point estimate of

is little troubling. Not too much should be made of the

point estimate however; its standard error is large, 48.5.
In general the standard errors on the preference and technology parameters of the
two models are of the same order of magnitude as those reported in McGrattan (1994).
Interestingly, the standard errors reported here suggest that both models have difficulty
matching the hours and employment series. On the one hand, the labor hoarding model
provides a tight parameter estimate of the shift length; the standard error on

1 0 .8 .

On the other hand, there are very large standard errors associated with the preference
parameters v and £. In contrast, applying maximum likelihood to the labor precommitment
model yields estimates for the shift lengths with large standard errors but a tight estimate
on

The parameter values that govern the measurement error process are presented in table
2.

Recall

is the variance-covariance matrix for the innovations to the measurement error

vector and is assumed to be diagonal. To ensure that estimated variances are positive, the
square root of the diagonal terms of

are estimated and reported. Several parameters in

the measurement error process are fixed prior to estimation.
At first the matrix

was assumed to be diagonal; however to avoid estimating a unit

root in the autoregressive coefficient for the measurement error of capital, the following
assumption from Christiano (1988) is made. It is assumed that the original capital stock is
measured without error, but investment is measured with error; therefore the measurement
error on the capital stock is a weighted sum of the past measurement errors on investment.
The law of motion for the measurement error on capital is constructed as follows. Let
it

be the model’s one-step-ahead forecasts of

v*t

where

and

are the time

and it, respectively. Then

and

-f- uf and

measurement errors on capital and investment,

respectively. Take the exponential of both sides of each equation, solve for
substitute into equation (6 ) and solve for u*+1 as a function of u* and

v\.

K t

and

It,

Taking a Taylor

approximation yields7
1-5

7 - 1+ < i
5
7
7
Unfortunately this assumption causes the likelihood function to reach a constrained maxi
J+i
t

-wf+

mum when the autoregressive coefficient of the measurement error process for consumption,
T > (2 ,2 ),

is one. Therefore

T>{ 2 , 2 )

is set to 0.999. Though this assumption “trades” one
16

Output

Consumption

Figure 1 : U.S. data (solid), fitted series (dotted), and predicted series (dashed) for output
and consumption. Since output is assumed to be observed without measurement error the
fitted and predicted series for output coincide.
unit root for another it improves the value of the likelihood.
Finally, output is assumed to be measured without error; so "D(l,l) is set to 0. For
numerical reasons, the variances of the innovations to the measurement error on output
and capital (72.(1,1) and 72.(5,5)) are set to

x 10-1° rather than exactly zero.

The computed point estimates of the parameters in both models are reasonable. More
over the standard errors are small enough for most parameters that the entire two standard
error confidence interval contains economically interpretable numbers. This result is in
contrast to some previous studies which had trouble computing economically interpretable
parameter estimates using maximum likelihood. For example Altug (1989) and Christiano
(1988) must fix f3 to avoid a point estimate greater than one. These results, along with stud
ies such as McGrattan (1994), demonstrate the usefulness of applying maximum likelihood
to linear-quadratic models to obtain economically interesting parameter values.

5.2

Diagnostics

This subsection assesses the performance of the two models evaluated at the parameter
estimates computed above.
To visually assess the performance of the labor precommitment model, the actual data,
the fitted series and the predicted series are graphed for the sample period 1955:Q3 to

Government

Figure 2 : U.S. data (solid), fitted series (dotted) and predicted series (dashed) for invest
ment and government
1992:Q4 in figures 1-4. The f-fl observation of the fitted series is E[yt+i\yt, yt-i, ...,yt,x i] =
Qxt+ Vxit Therefore the innovations, ut, are the vertical difference between the solid line
.

(the U.S. data) and the dotted line (the fitted series). The dashed line is the predicted
series; its date

observation is @E[xt\yt, yt-\, •••, J/t, #i] = Qxt. Recall that the actual data

for output, consumption, investment, government and capital are divided by 7 * (except for
government which is divided by 7 *) and logarithms are taken. Hours and employment are
assumed stationary; only logs of these series are taken. The corresponding graphs for the
labor hoarding model are not presented; they are qualitatively similar though not identical
to the graphs shown.6
The fitted series are the model’s one-step-ahead forecasts, incorporating measurement
error, of the data. In contrast the predicted series ignores the measurement error and
therefore displays just the contribution of the theory. Consequently, in figures 1-4 the
vertical distance between the dashed line (the predicted series) and the dotted line (the
fitted series) is the contribution of the measurement error. Since output is assumed to be
measured without error the fitted series and the predicted series coincide (see figure 1 ).
Not surprisingly, the one-step-ahead forecasts (the fitted series) of the labor precom
mitment model matches the data very well. In figures 1-3 it is clear that the fitted series do
an excellent job matching output, consumption, investment, government and capital; the
6The corresponding graphs for the labor hoarding model are available from the author.

Capital

Hour*

Figure 3 : U.S. data (solid), fitted series (dotted) and predicted series (dashed) for capital
and hours
fitted series have some difficultly matching the hours and employment data (see figure 4).
In particular the model has trouble forecasting (even one step ahead) the large rise in hours
and employment that occurred in the late 1980s. Unfortunately much of this excellent fit
is due to the contribution of the measurement error.
The theory does an excellent job matching the output data. Unfortunately, the theory
dramatically underpredicts the consumption series (see figure 1 ). Though the predicted
series for consumption (the dashed line) and the U.S. consumption data (the solid line)
appear to move in tandem, the predicted series is consistently below the actual series. This
is not surprising since the autoregressive coefficient on the measurement error process for
consumption was set to 0.999. Figures 2 and 3 illustrate that the theory also has some
trouble reconciling the investment and capital data. For the investment series, the theory
overpredicts the actual data before 1972; it underpredicts the data after 1972. Conse
quently, the predicted series for capital is consistently above the actual data before 1984
and consistently below after 1984. As with most business cycle models, the theory predicts
hours and employment to be much less volatile than the actual data (see figures 3 and 4).
Even with the total employment precommitment constraint in this model, the model still
predicts a much smoother overtime employment series than is observed in the data.
One caveat is worthy of note. It can be shown that the innovations to the capital stock
series are highly serially correlated. More specifically, the innovations are on average zero

Total Employment

Overtime Employment

Figure 4: U.S. data (solid), fitted series (dotted) and predicted series (dashed) for total
and overtime employment
but consistently above zero before 1970 and consistently below zero after 1970. Obviously
either the law of motion for capital, equation (6 ), is misspecified or the assumption that the
measurement error on capital is a weighted sum of the measurement error of investment is
a poor one. Nevertheless the innovations to capital are quite small; the largest in absolute
value is less than 0.004 whereas the mean of the capital stock is about 10.35. Consequently
there is reason to believe this error is not serious.
At first glance it appears that in figures 1-4 the fitted series (the dotted line) leads the
U.S. data (the solid line). Upon closer inspection one can see that the fitted series tends to
be below the actual data when the data are increasing and above the data when the data
are decreasing. Moreover the largest innovations occur at turning points in the data. To
see why this happens, assume the state vector
yt

does not equal its steady state value where
E [ y t+ i\y t ,y t- i,

xi]

y ss

is at its steady state value
=

Q x ss.

x aa.

Assume

Then:

G x t +

V y t

{G A > -

( I - V ) y aa + V y t .

V G ) x as

V yt

The above algebra demonstrates that the fitted value of yt+1 is a weighted sum of the
period

realization of

and an updating vector. The larger the persistence in the mea

surement error, the more weight is given to the realization of y t . Note that for investment,

Table 3
F r a c t i o n of v a r i a n c e e x p l a i n e d b y t h e o r y
series

Labor Hoarding
Labor Precommitment

1.00 0.22
1.00 0.22

0.65
0.73

0.75
0.73

0.50
0.58

0.44
0.34

0.50
0.35

0.12

consumption and government the autocorrelation coefficient for the measurement process
exceeds 0.97 (see table 2). It is clear that the measurement error process in this model has
important implications for the model’s performance.
To isolate and evaluate numerically the performance of the theory without the mea
surement error, the unconditional variances of the fitted time series are decomposed into
the fraction due to the theory and the fraction due to the measurement error. That is,
var(y) =
where var(x) =

•A ?C C 'A *

Q vax{x)G '

and var(u) =

+ var(u)
V ^T Z IA .

Table 3 reports the fraction of

the total unconditional variance explained by the theory for each series for both models.
This statistic is a

measure of the theory for each variable in the models.

Table 3 indicates the importance of the parameter estimates for the

matrix. Holding

all other things constant, the larger the persistence in the measurement error process, the
smaller the fraction of total variance explained by the theory. Since output is assumed
to be measured without error, none of its variance is explained by the measurement er
ror. However for consumption, the theory in both models explains only 1/5 of the total
variance of the fitted series. This result makes sense since

T>( 2,2)

is set to 0.999. The

labor precommitment model explains about 3/4 of the total variance of investment and
government - but only 1 /3 of the total variance in hours. Disappointingly, the labor pre
commitment model only explains 1/8 of the variance in overtime employment. The labor
hoarding model does not do much better; its theory explains about 2/3 of the variance
in investment,

3 /4

of the variance of government consumption and

1 /2

of the variance in

total employment. This measure of fit further confirms the visual assessment of figures 1-4;
the labor precommitment model (as well as the labor hoarding model) does a much bet-

Table 4
M e a n s a n d s t a n d a r d d e v i ations of t h e p r e d i c t e d a n d U. S . t i m e series
Labor hoarding
Mean Std. dev.

series
output, q
consumption, c
investment, i
government, g
capital, k
hours, h
total employment, ni
overtime employment,

8.347
7.743
6.783
6.938
10.397
5.330
-0.907

0.039
0.043
0.075
0.056
0.042
0.023
0.023

n-i

Labor precommit
Mean Std. dev.

United States
Mean Std. dev.

8.328
7.718
6.783
6.916
10.397
5.329
-0.909
-1.875

8^331
7.818
6.747
6.967
10.347
5.341
-0.896
-1.898

0.038
0.039
0.081
0.058
0.041
0.023
0.023
0.024

0.048
0.028
0.078
0.070
0.030
0.048
0.038
0.098

ter job explaining investment than consumption. Furthermore both models have difficulty
matching the variability in the hours and the employment series.
A more conventional metric to measure the performance of the two models is to compare
the models’ first- and second-moment properties to the moments in the data. Table 4
reports the unconditional means and standard deviations of the models’ predicted time
series and the actual U.S. data. The unconditional means of the predicted series are just
their steady state values,

y ss.

Their unconditional standard deviations are the square root of

the diagonal elements of the matrix

C j v a .r ( x ) ( y .

None of the moments reported incorporate

the effects of measurement error. The reported moments for the U.S. data are the sample
moments from 1955:Q3 to 1992:Q4 for the logged and detrended (lower case) time series.
Table 4 shows that both models imply very similar first- and second-moment properties.
In general the likelihood function chooses parameter values such that both models do
extremely well matching the means of each series (with the exception of consumption).
Both models do less well matching the standard deviations. Since output is assumed to
be measured without error, it is no surprise that both models match its mean; however,
both models underpredict its standard deviation. For example, the labor precommitment
model explains about (0.0382/0.0482) x 100 = 63% of the variance of output. As one would
expect from studying the previous figures, both models do a better job matching the mean

T a b le 5. C o rre la tio n o f aplt w ith qt+j
-4
-2
-1
-3
0
1

Labor hoarding
Labor precommitment
U.S. Data

0.78
0.73
0.58

0.80
0.75
0.60

0.82
0.78
0.62

0.84 0 . 8 6
0.81 0.82
0.63 0.65

0.85
0.82
0.65

0.85 0.85
0.82 0.82
0.64v 0.65

0.84
0.82
0.65

and standard deviation of investment than consumption, and both models underpredict
the volatility of hours and employment. It is difficult to use the implied second-moment
properties of the two models to differentiate between them.
Both models capture the procylicality of the average product of labor. The average
product of labor is defined as

a p l t = <lt — h t-

The correlation of the

apl

with

is presented

in table 5. Both models overestimate the correlation or coherence of the average product
of labor with output. Nevertheless both models imply that the correlations of the
the t + 1,...

+ 4 leads of q t are greater than the correlations with the t —1,...

a p lt

with

t —4

lags.

This is consistent with the observation that the average product of labor is procyclical and
falls at the end of an expansion.

5.3

P r o p a g a t i o n a n d magnification of shocks

Using the parameters for preferences and technologies given by the maximum likelihood
estimates, this subsection assesses the ability of the labor precommitment model and the
labor hoarding model to propagate and magnify shocks over the business cycle. Following
Burnside and Eichenbaum (1994), this analysis is executed along three dimensions. First,
for each model, the parameter values computed by maximum likelihood and the impulse
response functions are used to infer the strength of the model’s magnification mechanisms.
Second, each model’s ability to account for the observed autocorrelation of output growth
is evaluated. Finally, each model’s implied spectrum of output growth is reported and
compared to the corresponding spectrum in the data.
To analyze how each model propagates and magnifies shocks it is useful to once again
consider the parameter estimates. The bottom two rows of table 1 report the parameter
estimates for

<T\

and

crg ,

the standard deviation of the technology innovations and govern23

T a b l e 6. M a g n i f i c a t i o n of s h o c k s
Labor hoarding
0.0277

std(A)
std(?)/std(A)

Labor precommit
0.0260

1.42

1.48

ment innovations, respectively. For both models, the standard deviation of the technology
innovations is estimated to be about 0.006; likewise, for both models, the standard devi
ation of the government innovations is estimated to be about 0.009. Both models match
the data with shocks of the same volatility.
In table 4 one can see that both models imply an unconditional standard deviation of
output of about 0.038. Table

reports the unconditional standard deviation of the state

of technology, std(A), for each model. For the labor precommitment model it is 0.0260,
while for the labor hoarding model it is 0.0277. One measure of amplification is just the
unconditional standard deviation of output divided by the unconditional standard deviation
of A. This measure is reported in the second row of table 6 . It implies that both models
possess similarly strong magnification mechanisms. The magnification mechanism within
the labor hoarding model leads to a 42% increase in the volatility of output; for the labor
precommitment model the increase is 48%. This strength of the magnification mechanisms
embedded in these two models is a success for both models.
These two models magnify shocks by about the same amount. This is surprising con
sidering the different technologies the two models use to convert labor into goods. To
understand the dynamic properties of the two models, it is useful to consider the impulse
response functions of the linear system described by (2 0 ) and (2 1 ) for the two models.
These impulse responses are plotted in figures 5 and 6 . No measurement error is assumed
for any of the variables in the models. These figures plot responses to a positive unit shock
to A.
Both models produce “hump-shaped” impulse response functions. To understand how
the social planner responds to shocks in these two models look at figure 5. When a positive
technology shock occurs in the labor hoarding model, the social planner increases effort

Figure 5: Left: Labor hoarding model - impulse responses from a unit shock to technology.
Top - output; bottom - employment (solid) and efFort (dashed).
Figure 6 : Right: Labor precommitment model - impulse responses from a unit shock to
technology. Top - output; bottom - total employment (solid) and overtime employment
(dashed).
immediately but must keep employment fixed for one period. In the period after the shock,
the social planner increases employment such that efFort returns to its steady state value.
This rigidity in employment causes output and investment to deviate from its non-stochastic
steady state value more in the second period after the shock than in the initial period.
In a similar fashion, when a positive technology shock occurs in the labor precommit
ment model, the social planner increases overtime employment immediately but must keep
total employment fixed for one period. This is illustrated in figure 6 . In the second period
the social planner adjusts total employment. Unlike efFort in the labor hoarding model,
however, overtime does not return to its steady state level in the second period. Instead,
the ratio of total employment to overtime employment returns to its steady state level in
the second period. This is an important implication of the labor precommitment model.
Consider again tables 3 and 4. The most dramatic weakness of the labor precommitment
models is its inability to explain the volatility of overtime employment. From table 3,
one can see that the theory explains only

12%

of the unconditional variance in the fitted

overtime series. Moreover, in table 4, it is clear that the overtime employment data are
over twice as volatile as total employment data; however the model predicts overtime

Autocorrelations

T a b le 7. A u to c o rre la tio n of o u tp u t g ro w th
2
4
7
1
3
5
6

Labor hoarding
Labor precommitment
U.S. data

0.17
0.30
0.42

-0 . 0 2
-0.03
0.28

-0 . 0 2
-0.03
0.17

-0 . 0 2
-0.03
0.07

-0 . 0 2
-0.03
0.04

-0 . 0 2
-0.03
0.06

-0 . 0 2
-0.03
-0.03

-0 . 0 2
-0 . 0 2
-0 . 1 0

-0 . 0 2
-0 . 0 2
-0.04

-0 . 0 1
-0 . 0 2
-0 . 0 2

employment and total employment to have almost the same standard deviations. 7 Since
the ratio of total employment and overtime employment returns to its steady state level
in the second period following the shock, it is no surprise that the labor precommitment
model has difficulty matching the relative volatility of total and overtime employment.
The differences in the labor market response to a technology shock within each model
causes differences in the magnification and propagation properties of each model. For the
labor hoarding model, a 1 % shock to technology increases output 1 .2 1 % in the initial period;
by the second period output is 1.40% higher than its non-stochastic steady state. For the
labor precommitment model, a 1 % shock to technology increases output only 1.09% in
the initial period; but in the second period output is 1.48% higher than its non-stochastic
steady state. The relative increase in output from the first to the second period is much
larger in the labor precommitment model than in the labor hoarding model. This implies
that it is more efficient for the social planner to increase effort in the labor hoarding model
than overtime in the labor precommitment model.
A related way to measure the propagation mechanisms is to analyze the autocorrelation
of output growth. Cogley and Nason (1993) demonstrates that while output growth in the
data displays considerable autocorrelation, standard real business cycle models predict
this autocorrelation to be white noise. Moreover Watson (1993) shows that standard real
business models cannot explain the peak of the spectrum of output growth at business
cycle frequencies. Table 7 presents the two models’ implications for the autocorrelation of
output growth. Output growth is defined to be log Q t —log Q t - \

= qt — q t - i

+ log 7 -

7It is worth noting that adding an adjustment cost term for changes in total employment to the model can
lead to the implication that overtime employment is over twice as volatile as total employment. Adjustment
costs are not in the labor precommitment to simplify the contrast between the two models’ implications.
Results with adjustments cost are available from the author.

Figure 7: The spectrum of output growth in the labor hoarding model (left), the labor
precommitment model (center) and the data (right).
Consider table 7. Both models imply a positive correlation at lag l .8 The intuition
behind this result is illustrated by the impulse response functions (figure 5-6). After a
positive shock in the labor precommitment model (labor hoarding model) the social planner
immediately increases overtime (effort) and thus output but cannot immediately increase
total employment until the next period. This first increase in output due to the increase in
overtime (effort) predicts the increase in output in the following period from the increase
in total employment.
It is straightforward to consider each model’s implication on output growth in the
frequency domain. In figure 7 the spectrum of output growth is plotted for each model.
The spectrum of output growth from the data is also plotted. Both models imply spectra of
the same shape. Both the labor hoarding and the labor precommitment models succeed in
magnifying the shocks (which are white noise) such that a hump in the spectrum at business
cycle frequencies is produced. This hump is more pronounced for the labor precommitment
model than for the labor hoarding model. Although it is clear that both models dramatically
overestimate the variance of output growth at high frequencies, this hump in the spectrum
is a success for both models.
The analysis in this subsection demonstrates that the labor precommitment model does
as well as the labor hoarding model in propagating and amplifying shocks.
8If adjustment costs to changing total employment are incorporated, both models can im ply positive
correlation coefficients for lags greater than one. Results of the labor precommitment model with quadratic
adjustment costs to total employment are available from the author.

5.4

O v e r t i m e a n d effort

The purpose of this subsection is to use the overtime employment data to assess whether
the predicted effort series from the labor hoarding model is reasonable.
The intuition motivating the labor hoarding model and the labor precommitment model
is the same. It is costly to adjust total employment quickly; therefore, after observing
a shock to the economy firms first adjust worker effort (in the labor hoarding model)
or overtime employment (in the labor precommitment model). Clearly firms make their
contemporaneous labor decisions along more than just the one dimension each of these
models assumes; thus these models are not mutually exclusive. However since the labor
hoarding model makes use of an unobservable time series, effort, it is more difficult to
evaluate its implications.
Just as RBC theorists ask of their models, “what is the implied state of technology
series measuring?” one can ask, “what is the implied effort series in the labor hoarding
model measuring?” A useful way to try to answer this question is to find instruments
that are expected to co-move with effort. If these instruments exhibit the same time-series
properties that the labor hoarding model predicts for effort, then one can develop increased
confidence that effort is moving in the direction implied by the model.
One of the advantages of using the Kalman filter to estimate the labor hoarding model is
that the predicted series on the logarithm of effort, log et, can be constructed by multiplying
the matrix of time-invariant decision rule coefficients, —
T

and the constructed

series.

Consequently one can ask: is this predicted effort series reasonable? does it behave like
the overtime employment series? Recall that the overtime employment series, U2 j, is not
used in constructing or estimating the labor hoarding model. Moreover one might think
that when agents are working harder they are also working more overtime. Consequently
overtime employment is a suitable instrumental variable for effort.
To study the relationship between the predicted effort series, loget, from the labor
hoarding model and the actual U.S overtime employment series, the two time series are
plotted in figure 8 . In this figure both series are normalized to have a mean of zero and
a variance of one. At first glance the two series appear to have little relationship to each
other; the correlation coefficient for the two series is just 0.26. However a regression of the

2
8

Figure 8 : Scaled overtime employment
(solid) and effort (dotted)

Figure 9: The coherence between overtime employment and effort

first difference (growth rate) of overtime employment on the first difference (growth rate)
of the predicted effort series yields:9
Aloge*

0.026

A n2<
.

(0.011)

[2.32]
The significant positive regression coefficient implies that in periods in which overtime
employment increases (or decreases) so does predicted effort.
Moreover the predicted effort series and the actual overtime employment series move
together over the business cycle. In figure 9 the coherence between the two series is plot
ted. This figure indicates that at business cycle frequencies, the variance of the overtime
employment series explains between 1/3 and 2/3 of the variance of the predicted effort
series. Indeed the coherence between the two series peaks at the frequency corresponding
to a periodicity between four and five years. At high frequencies the two series display a
low coherence. Interestingly, but not surprisingly, the pairwise coherence of the two series
at the seasonal frequency (period equals four quarters) is less than 0 .0 1 .
The results from this subsection suggest that the predicted effort series of the labor
hoarding model makes sense. Although the effort series displays much larger volatility at
high frequencies than the observed overtime employment series, the growth rates of the two
9The standard error is in parentheses, and the t-statistic is in braces.

series are correlated. Moreover the two series display a high pairwise coherence at business
cycle frequencies. These results are consistent with the belief that overtime and effort move
together over the cycle.

Conclusion

The labor hoarding and labor precommitment models are not mutual exclusive mod
els. Clearly firms make labor decisions along both the effort margin and the straight
time/overtime margin. In this sense these are complementary models - not competing
models. But since both models ignore one of these margins, both models overstate the role
of the other.
The purpose of this paper has been to assess the implications of assuming that firms
contemporaneously adjust the labor input along either the effort margin or the overtime
margin. To that end, this paper demonstrates three things. First, it illustrates the useful
ness of applying maximum likelihood to compute reasonable parameter values for linearquadratic models. Second, it demonstrates that the standard real business cycle model can
be modified to embody quantitatively important propagation mechanisms without employ
ing additional unobserved variables. Third, it finds that the effort series implied by the
labor hoarding model moves with overtime employment over the cycle.
One interpretation of these results is as follows. Theorists need not incorporate timevarying effort into their models in order to construct models with quantitatively important
propagation mechanisms. However if they chose to employ time-varying effort, there is good
reason to believe that the predicted effort series from their model is reasonable. Moreover,
since the predicted effort series and the U.S. overtime data display a large coherence at
business cycle frequencies, there is good reason to believe that effort is procyclical.

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