Download NBER WORKING PAPER SERIES OF EARNINGS AND HOURS CHANGES

NBER WORKING PAPER SERIES
ON THE COVAR lANCE STRUCTURE
OF EARNINGS AND HOURS CHANGES
John M. Abowd
David Card
Working Paper No. 1832
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
February 1986
We are grateful to Gary Chamberlain, Zvi Griliches, and Whitney
Newey for comments and suggestions. An earlier version of this
paper was circulated under the title "The Covariance Structure of
Earnings and Hours Changes in Three Panel Data Sets." We thank the
Department of Economics at the Massachusetts Institute of
Technology for their hospitality while this paper was completed.
The research reported here is part of the NBER's research program
in Labor Studies. Any opinions expressed are those of the authors
and not those of the National Bureau of Economic Research.
ER brking Paper #1832
Fdruary 1986
On the Covariance Stricture of Earnings and HOUrS Changes
ABSTRPCT
This paper presents an empirical analysis of changes in individual
earnings and hours ov&r time. Using longitudinal data from three panel
surveys, we catalogue the main features of the covariance structure of
changes
in
earnings and hours. We then present an interpretation of
these features in terms of both a life—cycle labor supply model and a
fixed—wage labor contract nDdel. Our major findings are:
(1) there is
a remarkable similarity in the covariance structure of earnings and
hours changes across the three surveys; and (2) apart from simple
measurement error, the major component of variance in earnings and
hours affects earnings and hours equi—proportionately.
John M. Abowd
Department of Economics
MIT
Cambridge, MA 02139
(617) 253—1526
David Card
Department of Economies
Princeton University
Princeton, NJ 08544
(609) 452—4045
—1—
Recent empirical studies of longitudinal earnings and hours data
focus on the contemporaneous correlation between hours of work and
average hourly earnings.L" This focus arises naturally from the life—
labor supply model, which explains movements in hours over time in
cycle
terms of changes in the value of work and unanticipated changes in
wealth)' Nonetheless, there is widespread agreement that most of the
observed variation in hours over time is not explained by contemporaneous
movements in wages)'
correlation
On one hand, the cross—sectional
between percentage changes in annual hours and percentage
changes in average hourly earnings is apparently dominated by measure-
ment error.' On the other hand, state—of—the—art estimates of the
life—cycle labor supply model yield small and often statistically
insignificant elasticities between hours variation and wage movements.-"
In
this paper, we present a more general analysis of the relation
between movements in earnings and movements in hours over time.' Using
longitudinal
data from three panel surveys, we catalogue the main
features of the covariance structure of earnings and hours. We then
present an interpretation of these features in terms of both a life—
cycle
labor supply model and a model of fixed—wage labor contracts. Our
major findings are:
(1) there is remarkable similarity in the
covariance structure of earnings and hours changes across the three data
sources and (2) apart from simple measurement error, the major component
of variance in earnings and hours affects earnings and hours equi—
proportionately.
In
the first section of the paper, we describe the covariance
—2—
matrix of changes in annual earnings and annual hours of work for male
household heads in three panel surveys:
the Panel Study of Income
Dynamics (PSID); the National Longitudinal Survey of Older Men (NLS);
and the control group of the Seattle and Denver Income Maintenance
Experiment (SINE/DIME). Using method—of—moments estimation techniques,
we test for parsimonious representations of the data from all three sur-
veys. We find that changes in earnings and changes in hours are
uncorrelated at lags in excess of two years in the PSID and the NLS, and
at lags in excess of one year in the SINE/DIME. We also find evidence
of nonstationarity In the covariances of earnings and hours from all
three data sources.
In the second section of the paper, we present a life cycle labor
supply model and derive its Implications for the data described in
Section I. We also discuss the implications of a labor contracting
model in which hours of work are varied by employers with no
corresponding change in average hourly earnings.
In the third section of the paper, we present estimation results
for a simple two—factor model of earnings and hours generation. Th.e
model for earnings and hours consists of a systematic component
(reflecting individual productivity in the labor supply model or changes
in labor demand in the alternative model) and an unsystematic or
measurement error component. The estimation results reveal that both
cnponents are important in the data, and that the systematic component
generally influences earnings and hours equi—proportionare].y. This
leads us to conclude that most observed changes In earnings and hours
—3—
that are not attributable to measurement error occur at fixed average
hrly wage rates.
I. Data Description
The
longitudinal earnings and hours data in this paper are drawn
from the Panel Study of Income Dynamics, the National Longitudinal
Survey of Older Men, and the nonexperimental families in the Seattle and
Denver Income Maintenance Experiment. From the PSID, we have drawn
1448 male household heads whose records indicate nonzero earnings and
hairs in each year from 1969 to 1979 (the
of the survey). A brief summary
contained
of
third
through thirteenth waves
the characteristics of the sample is
in Table 1. We have included only those male household heads
who were between 21 and 64 years of age in each of the sample years.
Average annual hours (at all jobs) are more or less constant throughout
the eleven—year period,
while average hourly earnings (adjusted for
grow erratically. Our data set includes the Survey of
inflation)
Economic Opportunity subsample of the PSID and consequently overrepre—
sents nonwhite and
eleven
proportionately more low income and nonwhite households from
sample.
Fran
1318
low income households. The requirement of
continuous years of earnings and hours data, on the other hand,
eliminates
the
relatively
the National Longitudinal Survey of Older Men we have drawn
men who where less than 65 years old in 1975 and who reported non-
zero earnings and hours in each of the survey years 1966, 1967, 1969,
1971, 1973 and l975.' Table 2 summarizes the characteristics
NLS
of the
sample. Average annual hours for this older sample of men decline
—4—
throughout the nine—year sample period, particularly between 1973 and
1975. In interpreting the NLS data, it is important to keep in mind
that the later waves of the survey were administered biennally. As a
consequence, the changes in earnings and hours from 1969 to 1975 refer
to changes in annual totals over two—year intervals.
From the Seattle and Denver Income Maintenance Experiment we have
drawn
a sample of 560 male household heads who enrolled in the control
group
of the experiment.-" These male heads were between 21 and 64 years
of age during the first four years of the experiment, and reported nonzero earnings and hours in each of the first eight experimental half—
year periods. We excluded a small number of heads who reported more
than 2500 hours of work in any one of the six month periods. The size
of the SINE/DINE sample reflects the relatively small initial survey and
the reduction in sample size associated with the requirement of eight
periods of labor market data.
The demographic characteristics of the SINE/DIME sample and time
series information on earnings and hours are recorded in Table 3. Labor
market information in the income experiment was collected every four
months and aggregated into six—month data by the experiment's contrac-
tors. Since enrollment dates differ by household, the experimental
periods correspond to different periods of calendar tine for different
households. The first experimentalperiod, for example, contains data
from calendar periods between early 1971 and late 1972.
Like the other two samples the SINE/DIME control group over repre-
sents low—income and nonwhite households. Average hours of work
— 5—
(recorded at an annual rate in Table 3) were essentially constant
throughout
earnings
the first six
of the SIME/DB4E
periods
of the expe rinnnt.
control
Hours and real
group fell significantly in the eighth
and ninth experimental periods. These periods contain labor market
information
hours in
from Late 1974 arid
early
1975. changes in mean earnings and
the SIME/DIME sample are therefore consistent with
observed in the other two
changes
samples.
The complete covariance matrices oF changes in the logarithms of
annual earnings and annual hours for the three samples are recorded in
Tables 4, 5, and 6.
In order to partially control for differences in
labor force experience
in
the three samples, we have computrl LiLen
covariances using the residuals from unrestricted multivariate
regressions of changes in earnings and hours on time period indicator
variables and
potential
characteristics
experience (age rnLrtus
education
minus 5). The
of the data are not signt[icantly affected by removing
the predicted effects of experience, since the explanatory power of the
experience regressions is negligible in all these data sets. The
covariances of the changes in earnings and hours with potential
experience are recorded in the final rows of the Tables.
To control for the fact that the SIME/DIME data are drawn from different calendar periods, depending on the date of assignment into the
experiment, changes in earnings and hours from the SIME/DIME were
regression adjusted using potential experience and a series of indicator
variables for month—of—assignment into the experiment.
In none of the
sixteen regressions for the eight changes in earnings and
hours
were
—6--
these
indicator variables jointly statistically significant at conven-
tional significance levels, however.
Table 4 contains the covariances of changes in the logarithms of
annual earnings and annual hours for our sample of household heads from
the PSID. The upper left hand triangle of the table presents the
covariances of changes in earnings and their associated standard errors.
The lower right hand triangle of the table presents the covariances of
changes in annual hours. The lower left hand block contains the cross—
covariances
between changes in earnings and hours.
The cross—sectional variation in percentage changes in annual
earnings
and hours is large: the standard deviation of the change in
the logarithm of earnings is at least 35 percent, while the standard
deviation of the change in the logarithm of annual hours is at least 25
percent. The variances and covariances of changes in annual earnings
and hours also vary over time. Cross—sectional dispersion in earnings
and hours growth
is relatively small in 1972—73 and 1973—74,
and rela-
tively large in 1975—76. In contrast to aggregate time series data,
consecutive changes in individual earnings and hours are strongly nega-
tively correlated. The first—order serial correlation coefficients of
the changes in earnings and hours range from —.25 to — .45.
The first—
order serial cross—correlations of earnings and hours are also negative,
although smaller in absolute value (between —.15 and —.25) than the
corresponding autocorrelations.
The contemporaneous covariances of changes in earnings and hours
for household heads in the PSID are significantly positive, although too
—7—
small to generate a positive correlation between changes in hours and
changes
hourly
in average hourly earnings. Since the logarithm of average
earnings is the difference in the logarithms of annual earnings
and annual hairs, the covariance between changes in annual hours and
changes in average hourly earnings is just the difference between the
covariance
in hours.
of changes in earnings and hours and the variance of changes
For all ten changes in Table 4 this difference is negative
and well determined.
A final important feature of the covariance matrix in Table 4 is
the absence of any large or statistically significant autocovariances at
lags greater than two years. Year—to—year changes in earnings and hours
in the p510 are apparently well—represented as a nonstationary bivariate
second—order moving average process.
Table 5 presents the covariance matrix of five experience—adjusted
changes in earnings and hours from the NLS. Overall, the data are very
similar to the corresponding data from the PSID, although there is more
evidence of nonstationarity in the NLS data. On
the second—order autocovariances
the
other hand, none of
or cross—covariances in
Table 5
are
large or statistically significant, so that the NLS data may perhaps be
adequately suinnarized as a nonstationary bivariate first—order moving
average (MA(l)) process. If changes in annual earnings and hours
between conseaitive years are represented by a second—order moving
average process (NA(2)) then changes in hours and earnings at two—year
gaps are represented by a first—order moving average.2' Thus the data
in Tables 4 and 5 are potentially consistent with the se underlying
model of earnings and hours generation.
—8—
Table 6 presents the covariances of experience—adjusted changes in
earnings and hours between consecutive six—month intervals in the
SINE/DIME survey. Again, these covariances are similar to the covari—
ances of the PSID and NLS samples. There is less evidence of nonsta—
tionarity in the SINE/DINE than in the other two surveys, although this
may reflect the fact that changes in earnings and hours in the SINE/DIME
survey are averaged over several different calendar periods. The first—
order autocorrelations of earnings and hours are similar in all three
surveys. The covariances between contemporaneous changes in earnings
and hours, on the other hand, are relatively higher in the SINE/DINE.
This implies that the simple regression coefficient of changes in the
logarithxo of annual hours on changes in the logarithm of average hourly
earnings is smaller in absolute value in the SINE/DINE (—.17) than in
the other surveys (—.31 in the PSID and —.32 in the NLS). None of the
third—order autocovariances or cross—covariances is statistically
significant in Table 6, suggesting that semi—annual changes in log
earnings and log hours in the SINE/DIME are close to a bivariate MA(2)
process. Since a bivariate MA(2) representation of semi—annual changes
in earnings and hours implies only a first—order moving average representation of annual changes, the SINE/DIME data exhibit lower order
serial correlation than the PSID data.
Tables 7, 8, and 9 present formal test statistics for several
restrictive models of the covariance matrices of earnings and hours from
the three samples. Details of the method—of—moments estimation frame—
—9—
work that we use to obtain these test statistics are presented in
Appendix A. Table 7 presents the test results for the PSID sample. The
first rw of the Table reports the goodness—of—fit statistic for a
nonstationary bivariate MA(2) representation of the PSID data. This is
a test that the third and higher—order autocovariances and cross—
covariances
of
in Table 4-are
jointly equal to zero. The probability value
the test statistic is about six percent. Given the absence of
any
large or individually significant covariances of third order or higher,
we conclude that a second order moving average provides an adequate
representation of changes in earnings and hours in the PSID sample.
The next three rows of Table 7 present goodness—of—fit statistics
associated
with further restrictions on the general MA(2) model. These
tests are performed on the subset of autocovariances and cross—
covariances up to second order, utilizing the sample variance matrix of
the selected covariances..!Q! The first test is for a symmetric MA(2)
model of earnings and hours. This model imposes symmetry on the block
of
cros s—covariances in the lower left—hand corner of Table 4.
statistic
has
The test
a marginal significance level of 2 percent, indicating
some evidence against the symmetry hypothesis. The next test statistic
reported in Table 7 is a test for an MA(l) representation of earnings
and hours changes in the P519. Against the MA(2) alternative, this is a
test that the second—order autocovariances and cross—covariances
Table
in
4 are jointly equal to zero. Again, the test statistic indicates
some evidence against the null hypothesis. The final row of Table 7
presents the test statistic for the hypothesis that the covariance
—10—
structure of earnings and hours changes is time—stationary (i.e. ,
that
all 10 variances of earnings are equal, that all 10 variances of hours
are equal, etc.). This hypothesis is strongly rejected by the data.
Tables 8 and 9 present an identical sequence of test statistics for
the NLS and SIME/DIME samples.
In both samples, the data are consistent
with a bivariate MA(2) representation of earnings and hours changes. In
the NLS, the second—order covariances and cross—covariances of earnings
and hours are also approximately equal to zero. Like the PSUD, the
SIME/DThIE and NLS covariances exhibit significant nonstationarity. The
SIME/DIME and NLS covariance matrices are more nearly consistent with
the symmetry restriction than the PSID covariance matrix.
In light of the data presented in Tables 4—6, and the test results
in Tables 7—9, we conclude that changes in individual earnings and hours
are adequately summarized as a nonstationary bivariate second—order
moving average. This description holds for annual data (PSID), semiannual data (SIME/DIME), and annual data at two year gaps (NLS). More
restrictive models of the covariance structure of earnings and hours
changes are not generally valid.
In particular, nonstationarity is an
important feature of the covariances of earnings and hours from all
three panel surveys.
II. Alternative Models of the Covariance Structure of Earnings
and Hours thanges
In this section we present a simplified life—cycle labor supply
model and derive its implications for the covariance matrix of changes
in
earnings and
hours in panel data. We also present an alternative
—11—
of earnings and hours determination that corresponds to a labor
model
contract with fixed average hourly earnings and employer—determined
hours of
work. We discuss the econometric identification of these
models
in the presence of survey measurement error and outline an esti—
ination
strategy that decomposes changes in earnings and hours into two
components: a systematic component, reflecting individual productivity
(in the labor supply model) or employer demand (in the contract nvdel);
and a measurement error component.
A. A Life—Cycle Labor Supply Model.
The starting point for our model of life—cycle labor supply is an
additively separable utility function of the form
T
(1)
[log c1+. —
a.+.
tqhere T is the time horizon of individual I ,
is a fixed discount
factor, c1+. represents the consumption of individual i in period
represents a preference—shift variable,
n
is
a strictly
positive parameter, and h.÷. represents the hours of work of indivi-
dual i in period t+j .
supply and
consumption
We
assume that individuals choose their labor
in each period in order to maximize the expected
value of (1), given the Information available in that period
to
and subject
the following equation describing the evolution of their nominal
assets:
(2)
A
it-I-i
=(pt 9
t
it
it hit —p tc.it -4-A)(l+R).
—12—
In this equation
Alt represents the nominal value of assets held by
individual i in period t
Rt represents the nominal interest rate,
represents the price level of consumption goods, and
Pt
repre-
sents the real wage rate available to individual I in period
t
Maximization of the expected value of (1) subject to the wealth
constraint (2) implies the Bellman equation:
l+ii
(3)
where
max
Vi(A.)
log c1 —
alt
i-:hi
h.11 + $E
denotes the expectation operator conditional on information
available to individual i In period t
The first—order conditions for consumption and labor supply in
period t are
(4a)
—
Sp(l + R) EV!1(A) 0
(4b) it
a th1
—
80
p (1 + R ) E
it t
t
V'
(A
t it+l it+1 )
0
In addition, the derivatives of the value function V. (A )
it it
follow
the
stochastic difference equation:
(5)
Alt E V(A1) = 8(1 + R) E A11
There is no exact solution to equations (4) and (5) in general. We
proceed by using a well—known approximation to equation (5); namely
—13—
logA = log
(6)
loX1 +
S +Rt + Et
d.
where d1 is the error of approximation and is assumed to be time sta-
The labor supply equation can be derived from (4b)
tionary.
definition
it =
where
related
tion
of
of A1 in (5):
log hi =
(7)
using the
—n
n log
+
ii log
Pt +
p log
it
+
According to this equation, hours of work are
log alt .
to the contemporaneous wage rate by the intertemporal subs titu—
wealth
Hours
n
elasticity
,
and the
of work also depend on the marginal utility
preference parameter
Making use of the approximation in equation (6), equation (7) can
be differenced to yield the following description of the change in the
logarithn of hours:
(8)
A log hi log hj — log
—
p A log 0ir +
+
where p
n(log
Alt —
r1)
Ei log A.) +
A
is the discount rate defined by 5 E 11(1
— A
log Pt
real Interest rate defined by r
it
+p) and r is the
Since labor earn-
ings are the product of the wage rate 8it and hours of work hit
the labor supply model implies a similar specification for the change in
the logarithm of earnings (g):
(9)
A log gif E
A
log hj +
A
=
(l+p)A
+
ii (log X1 —
log
log O
+
fl(P-r1)
Etl log Ai)
+
Ai
—14—
Changes in earnings and hours share common components associated with
the
revision of the marginal utility of incane; the change in tastes for
leisure; and the difference between the discount rate and real
rate. In addition, an individual—specific wage component
interest
enters the equations for earnings and hours changes with relative factor
loadings of (1-i-n) and n , respectively.
To complete this model of earnings and hours, we need to specify a
stochastic process for the wage rate. We assume that individual wages
contain a permanent component, an aggregate component, an experience—
related component, and an idiosyncratic component. Specifically, we
a Ssume:
(10)
log
e1 = a1 +
+ 11x1
+
2
+
is an
economy—wide time effect, x is the number of years the individual
where a1 is an individual—specific permanent wage effect,
has been
This
in the labor force, and
is an idiosyncratic wage shock.
specification explicitly rules out both
individual—specific respon-
ses to the aggregate shock and individual—specific experience effects.
We defer a detailed discussion of these assumptions to Appendix B.
Using equation (10), the change in the wage rate in period t may
be expressed as:
(11) A log
where c
—
=
6t—l +
+
yx0
—
+
Az
÷ 2y2t ,
force experience at calendar date t = 0.
y
,
and
is labor
—15—
Substituting equation (11) into equations (8) and (9) and appending
a pair of measurement errors to log g and log hj yields a
theoretical model for the bivariate stochastic process governing
observed changes in individual earnings and hours
(l2a) A log
=
(l-1-n)c +n(—r1)
+
(l+n)yx10
(l2b) A log hj =
+
(l+)
—
Ai ÷ Mit,
flK +
+ rryx0 + nAzi
where 6it E log
+
Az
+161
+
+ Av,
Etl log Alt
uli E measurement error in observing log gft , and
measurement error in observing log hit
E
The implications
earnings
of equations (12) for
the covariance structure of
and hours depend on the serial correlation properties of the
productivity shock z1 ,
and
income
the
taste shock
, the measurement errors
the unexpected change in the marginal utility of
assume that tastes for leisure contain a permanent
vi •, and
•
We
component, a hoimgeneous quadratic age effect, and an individual—and—
period—specific component. These assumptions imply that the error com-
ponent Mit associated with the change in tastes for leisure can be
written as
(13)
=
+ 2x0
+
Avi
4
—16—
where Uft
represents
We assume that
an individual— and period—specific taste shock.
is serially uncorrelated with a constant variance.
The change in tastes for leisure is therefore a first—order moving
average with a serial correlation coefficient of —.50.
The measurement errors u and v'
have a special structure.
They are intended to represent systematic and random deviations of the
survey measures of earnings and hours from their theoretical analogues.
To capture these ideas, we assume that
(14)
rut1
tii
[v7tJ
'2iJ
+
[cut
Lc2it
where
and •2i represent individual— and survey—specific permanent response biases, and
and 21 represent transitory
measurement errors.
We assume that c1
and
are serially
uncorrelated with an arbitrary time—stationary covariance matrix. Since
the changes in the measurement errors of earnings and hours reflect only
the changes in the transitory components of these errors, and since the
transitory
measurinent errors are serially uncorrelated, Au* and v*
are first—order moving averages with serial correlation coefficients
equal to —
.50.
Notice that we are unable to separately identify the
variance contribution of the random shock to preferences and the
variance contribution of the transitory measurement errors, since we
have assumed that they have the same serial correlation properties, and
we have not
restricted the correlation between the measurement errors.
—17—
We therefore combine these components into an unsystematic component of
the changes in earnings and hours:
r.it11
1 rAU
it
(15))
+it11
i r
lit
it
L itJ Ltt v.J L"21t ÷ Avit
+
which is a stationary bivariate MA(l) process with an arbitrary
contemporaneous covariance and with first—order autocorrelations equal
to —.50.
On the basis of the evidence that changes in earnings and hours are
adequately represented as a bivariate MA(2) process, we assume that
changes in individual productivity shocks in all three surveys follow a
nonstationary second order moving average. The parameters of this process include the variance of the change in individual productivity
shocks in period t (var Az.) , the covariance of changes in individual
productivity shocks in periods t and t—l (cov(Az. ,
Azj)),
and
the covariance of changes in individual productivity shocks in periods
it
t and t—2 (cov(Az,
Az
)).
The final corrrponent of variance in equations (12a) and (12b) is
the unexpected change in the logarithm of the marginal utility of
wealth.
In general,
is a function of the unanticipated component
of the individual productivity shock Az11 and the unanticipated corn—
—18--
ponent of the aggregate productivity shock. Consider the linear projection (across individuals at a point in time t):
(16)
Since
+
bjAz — Eti
Azj)
+
the forecast error of AZft is serially uncorrelated, the serial
correlation propertiesmf c1 depend only on the serial correlation
properties of the projection error e This error depends on both
individual and
aggregate
productivity shocks, since individuals can vary
in their response to an idiosyncratic shock and in their response to an
aggregate shock. For simplicity, we assume that the projection errors
are serially uncorrelated (in the sense that plim
This
assumption is satisfied if
ei e11 =
0
)
depends only on individual produc-
tivity shocks, or equivalently, if unanticipated aggregate shocks
generate a homogeneous shift in the marginal utility of income that is
incorporated into the cons tant term
in (16)
We are now in a position to catalogue the implicationsof equations
(l2a) and (l2b) for the covariance structure of earnings and hours in
panel data. We note that in the absence of prior information these
equations place no restrictions on the mean changes in earnings and
hours observed in a cross—section)-! Depending on the specification of
life—cycle preferences for leisure, however, equations (12) may restrict
the
regression
coefficients of changes in earnings and hours on poten-
tial experience. Equation (iZa) implies that the regression coefficient
of the change in earnings on potential experience is
(l-e-)y +
—19—
while (l2b) implies that the regression coefficient of the change in
hours on potential experience is
+
If
=
life—cycle preferences exhibit no systematic curvature, then
0,
and all the curvature in life—cycle hours is due to curvature in wages.
In that case, an estimate of
ii
is
available frcm the ratio of the
covariance of the change in earnings with experience to the covariance
of the change in hours with experience. This is precisely the instru-
mental variables estimate of i
fran
the labor supply equation (l2b),
using potential experience as an instrument for wages.--'
Table
10 presents the average covariances of changes in earnings
and hours with potential experience from the three data sets.
important to keep
earnings
in
ft is
mind that the timing intervals of the changes
in
and hours are different in the three surveys. If the
underlying parameters were the same in all three samples, then one would
expect
to observe covarlances in the SINE/DIME roughly one—half as large
as those in the PSID, and
covariances in the NLS roughly twice as large
as those in th PSID.!±" The first row of the Table presents the
average covariances of the change in earnings with potential experience.
In all three data sets this covariance is negative, although in the NLS
the average covariance is relatively small. The second row of the Table
presents the average covariances of changes in hours with experience.
In the PSID and SINE/DIME samples the covariance of hours changes with
experience is smaller in
absolute
value than the covariance of earnings
changes with experience. In the NLS sample, however, the covariance of
—20—
hours
changes with experience is larger in absolute value than the
covariance of earnings changes with experience. Under the assumption
that labor supply preferences exhibit no systematic curvature, the
instrumental variables estimate of n
is
ChI(Cg_Ch)
where
and
Cg represent the average covariances of experience with hours and earnings changes, respectively. The instrumental variables estimates of i
for
n
the PSID and SIME/DIME are therefore positive, while the estimate of
for
the NLS is negative and significantly different from zero.!i
The fourth r of Table 10 contains the goodness—of—fit statistics
associated with the restriction that the covariances of earnings and
hours with potential experience are stable over time.
The hypothesis of
stationarity is marginally accepted in the PSID and SIME/DIME samples,
and strongly rejected in the NLS sample. In view of this finding, for
the
remainder of our empirical analysis we leave the regression coefficients of the changes in earnings and hours on potential experience
unrestricted, and concentrate on the covariance structure of the
experience—adjusted changes.
Denote the residuals of individual changes in earnings and hours
from a multivariate regression on potential experience and time—specific
means by A log
and A log
, respectively. Equations (12a) and
(12b) imply
(17a) A log it = (lfn)Azi
+
—
÷
Au.
and
(17b) A log hit =
Experience
+
—
+ Av1
adjusted changes in earnings and hours consist of three
components: a productivity component (Azit) that enters earnings and
—21—
hours with coefficients (l+) and n , respectively;
a serially
uncorrelated component associated with the unanticipated change in the
marginal utility of income; and a restricted bivariate MA(l) component
reflecting survey measurement error and random shocks to preferences.
Table 11 summarizes the implications of equations (l7a) and (17b)
for the covariance structure of experience—adjusted changes in earnings
and
hours.
Measurement error components contribute negative first—order
autocovariances in direct proportion to their variance contribution.
Unanticipated changes in the marginal utility of income contribute a
time—specific component to the variances and
covariance
of earnings and
hwrs, but do not contribute to the autocovariances. Individual—
specific productivity shocks contribute a time—varying variance and
covariance component, and represent the only source of second—order
autocovariance. As indicated in the table, the labor supply model im-
symmetry on the cross—covariances of earnings and hours——a
poses
restriction that is satisfied in the NLS and STtW/DThIE samples but
marginally rejected in the PSID sample. The labor supply model also
restricts
the four second—order autocovariances in each year to have the
same sign. Inspection of Tables 4, 5 and 6 suggests that this restriction is typically satisfied, althcxigh the second—order covariances are
estimated with relative imprecision.
It is clear from Table 11 that the labor supply elasticity
ii
is
generally overidentified. First, ratios of the second—order autoco—
variances provide estinates of l+/ , the relative factor loading of
productivity
shocks in earnings as compared to hours. Second, if the
—22--
first—order covariances vary over time, then the non—stationary com-
ponents of the first—order autocovariance matrix are restricted by the
Let
ratio l+n/n .
c'(t) denote the estimate of the first—order
autocovariance matrix of earnings and hours in petiod t.
Equations
(17a) and (17b) imply
(1+1)2 n(l+n)
(18) c'(t) —
C'(s)
=
{cov(Azi,Azii)
—
cov(AzjAzi
[
2
p
Provided that the first—order autocovariances of the productivity shock
are not equal, equation (18) Identifies the relative contribution of
productivity shocks in earnings as
individual
provides
thus
In
(17a)
compared to hours, and
an estimate of the labor supply elasticity i
the stochastic model of labor supply represented by equations
and (l7b),
variances
the zero—order covariances of earnings and hours (the
of earnings and
and the contemporaneous covariance) are
hours
unrestricted. This is a consequence of the fact that there are three
free paraneters (var (Azit)
var it' and
coy (A z., €ft2) asso-
ciated with the three zero—order covariances in each time period. In a
perfect foresight model in which the marginal utility of income is
constant, on the other hand, c is zero in every period and the labor
supply model restricts the zero—order covariances in the same manner as
the
first—order autocovariances.
Even in a perfect foresight model, however, the covariances of the
measurement error component of earnings and hours are not separately
identifiable fran the variances and first—order autocovariances of the
—23—
productivity
variance
shodc)-" It is therefore impossible to provide a unique
deccmposition of the changes in earnings and hours into systema-
tic and unsystematic components. One solution is to impose a prior
estimate of the correlation coefficient between the measurement errors
in earnings and hours. The estimate of (l-t-)/ is invariant to the
choice of correlation coefficients. In this paper we do not compute
variance
decompositions of the earnings and hours data.
B. A Fixed Wage Contract Model.
The institutional structure of many
employment situations is not
easily reconciled with a model of the labor market in which employees
unilaterally determine their hours of work subject to their current
wage rate. In contrast, a wide variety of contractual models emphasize
fixity in the labor market and resulting discrepancies between observed
average hourly earnings and the marginal rate of substitution of consurnp—
tion for leisure)2_" A prototypical model is one outlined by Abowd and
Ashenfelter (1981). In their model, employers offer
job packages con-
sisting of a fixed average hourly wage rate and a probability distribu-
tion over hours. Equilibrium in the labor market is obtained by equating
expected
the
utilities of job packages.
In any particular realization of
random demand shodc driving employers' demands for hours, however,
employees may prefer to work more
contract.
or less than required in their
The implied correlation
earnings and
between changes in average hourly
changes in labor supply is zero: earnings move in direct
proportion to employer—determined hours with no corresponding change in
wage rates.
—24—
To derive the implications of this model for the covariances of
earnings and hours in panel data, assume that the wage rate received by
individual i in period t contains only a permanent component, a
deterministic experience component, and a period—specific component:
(19) log 9it =
a. +
+ y1xft
+
where, as before, x1 indicates the potential experience of i in
period t .
consist
Assume that hours of work of individual i in period t
of a permanent component '
,a
deterministic experience com-
and a stochastic component
ponent, a period—specific effect Pt
reflecting employer demand for hours:
(20)
+
log hi =
+
-
+
-Yx + yit
Adding measurement errors u and vj to observed earnings and
hours, respectively, this fixed—wage contract model implies that the
residuals
of individual changes in earnings and hours from a regression
on potential experience are given by:
(2la)
log g
(21b) A
log
=
AYit +
and
As it
hit = Ayft
happens,
guishable fran
+
this simple fixed wage contract model is indistin-
a labor supply model in which the ratio (l+)/ is
unity, or equivalently, in which the elasticity of labor supply is
arbitrarily large. To see this, let
—25—
p =
(l+n)/n
= nAzi
,
c=
— e)
and rewrite (l7a) and (17b) as
(22a)
A log
=
i.i Az'
÷
+
and
(22b) A log hj =
If
p = 1
,
var(Ay1) =
Az +
÷
(22a) and (22b) are identical to (2la) and (Zlb) with
var(Azr)
÷ var(ct) + 2 coy (Azt, e) and
coy (Ay, Ay) = coy (Azr. Az) for
•
It
is therefore
impossible to reject the labor supply model in favor of the fixed
wage contract model. If estimates of p in equation (22a) are
close to unity, however, our interpretation of the data is not that
small changes in wage rates are necessarily driving large changes
in hours, but rather that hours and earnings are moving proportionately
at constant average hourly wage rates.--'
III. Estimates of a Two—Factor Model of Earnings and Hours.
In this section we present estimates of the two—factor model of
earnings and hours generation derived fran the theoretical models in the
previous section. We report estimation results for two versions of
the labor supply model: a perfect foresight version that ignores
changes in the marginal utility of income; and a stochastic version that
allas for unanticipated changes in the marginal utility of income. For
each version of the model, and each of the three panel surveys, we pre—
—26—
sent
sumnnry
statistics
for the goodness—of—fit of the model, as well as
estimates of the relative contribution of productivity shocks in
earnings as compared to hours (the parameter p in equation (22a)), and
the implied elasticity of intertemporal substitution.
The estimates are obtained by method—of—moments techniquS, using
as data the covariances of experience—adjusted changes in earnings and
hairs. On the basis of the preliminary data analysis in Section I, we
fit the model to the contemporaneous covariances of earnings and hours
and their first—
and second—order
autocovariances, ignoring the higher—
order autocovariances.
Two
alternative
method—of--moments estimates are presented. The
first, whidi we call the equally weighted minimum distance (EW?e) estimator, minimizes the quadratic form
[m —
f(b)J'
[m —
f(b)]
where m represents the vector of adjusted covariances to be fit by the
model, b represents the vector of parameters of the model, and f(b)
represents the vector of predicted values of the fitted moments, con-
ditional on the parameter vector b •
The second, which we call the.
optimal minimum distance (Oft)) estimator, minimizes the quadratic form
[m -
f(b)J' V[m - f(b)]
where V denotes the sample variance matrix of the data vector
in.
Chamberlain (1984) shows that the latter estimator minimizes the asymptotic covariance matrix of the estimated parameters among the class of
minimum distance estimators.
Table 12 sunmarizes the structural estimation results for the PSID.
—27—
Columns (1) and (2) report equally weighted minimum distance and
optimal
minimum distance estimates, respectively, for the perfect foresight
model, while columns (3) and (4) report EWFID and 01W
estimates of the
uncertainty model. The individual productivity process
(Az1) is
estimated as an arbitrary nonstationary MA(2) process. In the PSID,
this introduces 27 parameters: 10 variances, 9 first—order covariances,
and 8 second—order covariances. The measurement error/preference shock
component of variance introduces 3 additional parameters (the variance
of the measurement error component in earnings, the variance of the
measurement error component in hours, and the correlation of the two
measurement errors). The final structural parameter is the relative
contrihitlon of the productivity component to earnings
(1').
In the
uncertainty specification of the model there are 20 additional para-
meters corresponding to the additional unrestricted zero—order covar—
lances. Since the measurement error components and the productivity
components are not separately identifiable, for purposes of estimation
we have arbitrarily fixed the correlation of the measurement errors in
and hours. The goodness—of—fit of the models and the estimates
earnings
of u
are invariant to the choice of normalizing assumptions.
first r of Table 12 shows the goodness—of—fit of the structural models relative to an unrestricted model (98 unrestricted
The
moments). The test statistics in the first two columns of the table
suggest
that the perfect foresight model is not supported by the data.
The test statistics in the last two columns of the Table, on the other
hand, indicate that the uncertainty model is consistent with the data at
—28—
conventional significance levels. The second row of the table reports
All
the estimated values of the productivity contribution parameter i.t .
of these estimates are within sampling error of one for the PSID. Even
though the perfect foresight model does not fit well, the estimates of
p from this model are essentially the same as the estimates of
i.t
from
the uncertainty model. As row 3 of Table 12 shows the implied estimates
of the labor supply elasticity n
are
extremely variable and impre—
19/
cisely estimated.—
The finding that productivity shocks influence earnings and hours
proportionately
suggests that systematic changes in individual hours
occur at fixed hourly wage rates. There are a variety of possible
interpretations of this finding. One possibility is that changes in
hcnrs
are determined by employer preferences, and are uncorrelated with
changes in average hourly earnings. A second possibility is that the
estimate of p
is biased downward by changes in the marginal utility of
that enter earnings and hours proportionately. If we
income
incorrectly fit a perfect foresight model, assuming eft = 0
in
estimate of p. ,
equation (l7a) is biased toward one.
,
then the
If cross—
sectional variability in e is the source of downward bias in the
perfect foresight estimate of p , however, then we would expect the
estimate of p
to increase between the perfect foresight and uncer-
tainty specifications of the labor supply model
Since the perfect
foresight and uncertainty versions of the labor supply model yield very
similar
estimates of
utility,
of income
ji
, we conclude that changes in the marginal
are not a likely source of bias in the estimates of
—29--
p
.
Year—to—year changes in earnings and hours are consistent with
either a large substitution elasticity, or alternatively with employer
determined hours at fixed average hourly wage rates.
The results in Table 12 nonetheless reveal a problem for the fixed—
wage interpretation of the estimates of p .
Under a stochastic labor
supply model, the contemporaneous covariances of earnings and hours are
not necessarily consistent with a two—factor model. Under the simple
contract model outlined in Section 118, on the other hand, the zero—
order covariances of earnings and hours are consistent with the same
two—factor model as the higher—order autocovariances. The
underlying
goodness—of--fit statistics for the two—factor model when the contemporaneous
covariances are restricted (in columns (I) and (2)) and
unrestricted
(in columns (3) and (4)) suggest that this is not the case.
Whether the contanporaneous covariances of earnings and hours are
restricted by the model or not, the estimate of p and the
corresponding goodness—of—fit statistics are very similar between the
OMD and EWMD estimation methods. Differences between the OMD and EWMD
parameter estimates have the interpretation of a specification test
(Hausman (1978)). Under the hypothesis that the model is correctly specified, both the EWHD and OMD estimates are consistent, and the OND
estimates are efficient. It is straightforward to show that the
covariance
parameter
matrix of the vector of differences between the EWMD and 01W
estimates is equal to the difference between the OMD
parameter
covariance matrix and the EWMO parameter covariance matrix.--' The test
statistics associated with the difference between the 01W and EWMD esti—
—30—
mates of p are reported in the fourth row
of
the Table. For both the
perfect foresight and uncertainty versions of the labor supply model,
the OMD and EWMD estimates of j
are
not significantly different.
Table 13 summarizes the estimation results for the NLS. In fitting
the labor supply model to the NLS, the covariances of the underlying
productivity shock (Az.) contribute 12 free parameters (five variances, four first—order autocovariances, and three second—order
autocovariances). Columns (1) and (2) of the table present estimation
results for the perfect foresight version of the labor supply model.
Columns (3) and (4) contain the results for the stochastic labor supply
model, which adds 10 extra parameters corresponding to the additional
unrestricted zero—order covariances.
The equally weighted minimum distance estimates in columns (1) and
(3) suggest that the parameter i
is
greater than unity, but not
significantly so. The associated estimates of
ii
are large and Impre-
cise. The goodness—of—fit of the perfect foresight model is poor: the
chi—squared test statistic is 124.1 with 28 degrees of freedom. The fit
of the stochastic labor supply model, on the other hand, is considerably
better. The optimal minimum distance estimates in columns (2) and (4)
of Table 13 provide about the same goodness—of—fit as the corresponding
EWMD estimates, but yield very different estimates of the labor
supply elasticity. The (ThD estimates of the perfect foresight model, in
particular,
earnings
indicate a remarkably stronger productivity component in
as compared to hours, and a relatively small estimate of i
The OMD estimates of the stochastic labor supply model
are closer to the
—31—
corresponding equally weighted minimum distance estimates, but yield a
negative point estimate of
ii.
Our interpretation of the difference in the estimates in columns (1)
and (2) of Table 12 is that the perfect foresight labor supply model provides a poor fit to the data, and as a consequence the parameter estimates are sensitive to the relative weight assigned to the deviations of
particular moments from their predicted values. The fourth row of Table
4 presents specification tests based on the difference between the EWND
estimates of p and the corresponding OND estimates. For the perfect
foresight model the statistic cannot be computed because the estimated
standard error under OMD is larger than under EWMD.?' In any case, the
goodness—of—fit statistics providefairly strong evidence against the
perfect foresight labor supply model. For the uncertainty model, on the
other hand, the difference in the EWMD and OMP estimates of p is .71,
with a standard error of .51. This result and the goodness—of—fit statistics in the third and fourth columns of the Table suggest that the
stochastic labor supply model provides a reasonable description of the
data.
Table 14 contains goodness—of—fit statistics and parameter esti-
mates obtained from the SIME/DIME sample. The conclusions from this
sample are essentially the same as the conclusions from the other data
sets. The perfect foresight labor supply model fits relatively poorly.
The alternative model that frees up the zero—order covariances of the
data is only marginally rejected. The estimates of p are within
sampling error of unity in all cases. In the SIME/DII{E data, as in the
PSID data, the specification tests based on the difference between the
—32—
EWMD and aMP
estimates of p are not significant at conventional signi-
ficance levels.
IV. Conclusions
We have presented evidence on the autocovariance structure of ear-
nings and hours changes in three panel data sets: the PSID, which
measures hours and earnings annually; the NLS Survey of Older Men, which
measures annual hours and earnings at two year intervals; and the
SIME/DIME, which measures semi—annual earnings and hours. In spite of
these timing differences, and some differences across the surveys in the
questionnaire used to measure hours and earnings, the autocovariance
structure of all three data sets is remarkably similar. Individual
changes in earnings and hours have strong negative first order auto—
correlation, weak second order autocorrelation, and negligible higher
order autocorrelations. Contemporaneous changes in earnings and hours
are positively correlated in all three data sets. This cross—covariance
is too small, however, to generate a position covariance between changes
in hours and changes in average hourly earnings in any of the data sets.
The sparse autocovariance structure of earnings and hours changes
makes it difficult to identify models that include general specifications of measurement error, tastes, and
two
economic components. We develop
models of hours and earnings that distinguish between an unsystema-
tic component (attributable to measurement error and/or taste changes),
and a systematic component. In the first model, the systematic component is interpreted as an underlying shock to individual productivity
—33—
that
influences wages directly and hours indirectly through an intertem—
poral substitution effect. In the second model, the systematic component is interpreted as a shock to employer demand for hours that
influences
results
earnings and hours equi—proportionately. Our empirical
suggest
that the systematic component of variance effects ear-
nings and hours equi—pifoportionately in all three panel data sets.
Apart from the zero—order covariances of earnings and hours (the
variances of earnings and hours and their contemporaneous covariance), a
simple two—factor model provides an adequate description of the
covariance structure of earnings and hours changes in all three data
sets. A two—factor model is apparently too restrictive as a model of
the complete covariance matrix of earnings and hours changes. In the
context of a labor supply model, this finding is interpreted as evidence
against perfect—foresight. In the context of a fixed—wage contract
model the interpretation of this finding is unclear. More work is
required to distinguish between the two models, and to develop alternative models of the covariance structure of earnings and hours.
—34—
Footnotes
See for
example Ashenfelter and Ham (1977), MaCurdy (1981),
Altonji (1984), and the surveys by Killingsworth (1983) and Pencavel
(1985).
VThe life—cycle labor supply model is described in Ghez and Becker
(1975) and Heckman (1976).
Browning, Deaton and Irish (1985) provide a
useful summary of the theory of consumer behavior over time.
1"Pencavej (1985, p. 151) concludes that
.
. . the
focus of most
economists' research [on labor supply] has been on a behavioral response
for men appears to be of a relatively small order of magnitude.
that
'Altonji
(1984) compares the correlation between changes in hours
and changes in average hourly earnings (about — .35 in his sample and in
the PSID and NLS data sets used in this paper) with the correlation between changes in hours and changes in reported wage rates of hourly rated
workers (about .01 in Altonji's sample). Altonji interpretes this difference as evidence that masurement error in hours induces a strong
negative correlation between changes
in hours and changes in average
hourly earnings.
'HaCurdy
(1981) and Altonji (1984) both arrive at estimates of the
intertemporal substitution elasticity in the neighborhood of .10 — .40.
-"Our
empirical
analysis is closely related to work by Hause (1980)
and MaCurdy (1982). Hause and MaCurdy both model the serial correlation
—35—
structure of earnings in longitudinal data. We extend the analysis to
the bivariate process of earnings and hours.
1The NLS surveys
survey
were administered in July through October of the
years, and asked questions on earnings and hours in the previous
twelve months. Unlike the PSID survey, the NLS collects no auxilliary
information
on overtime hours or hours of work on secondary jobs. For
this and perhaps other reasons, a large fraction (30 percent) of NLS
respondents report exactly 2,000 hours per year.
!"our SINE/DIME data is drawn from the so—called Work Impact File
assembled by SRI from the underlying survey data. The SIME/Dfl1E survey
contains detailed questions on overtime and secondary jobs, and is
conceptually more similar to the PSID survey than the NLS survey.
Denote the change in earnings between period t—l and period t
.2!
by
Ax .
If
Ax is MA(2), then Ax
example, where
between
= s
+
b11 ÷ b2e
is serially uncorrelated. The change in
period t—2 and period t is x —
x2 =
x2
=
Ax
for
2
earnings
+
+
(l+bi)e 1 + (b1 + b2)e2 +
, but not with
Notice that x — x2 is correlated with x2 —
Therefore, x —
x4 —
x6 (or Ax4).
!Q!There are at least two alternative ways to test restrictions on
subsets of the covariances in Tables 4, 5, and 6, depending on whether we
incorporate the restrictions that the third— and higher—order covariances
are jointly equal to zero when
we
test the properties of the lower—order
—36—
covariances. Our procedure is to ignore the higher—order covariances in
estimating and testing restrictions on the lower—order covariances.
i2/The error
aggregate
e
it
may
be serially correlated, for example, if
shocks have a systematically larger effect on some
individuals' marginal utility of income than others'. In this case,
however, we would expect to observe nonvanishing covariances between
changes in earnings and hours from years with large aggregate shocks,
irrespective of the gap between these years. There is no evidence of
this phenomenon in Tables 4—6.
il/There are two time—varying components of the mean changes in
earnings and hours in each year: the shift in the aggregate productivity
shock
;
and the average revision in the marginal utility of
income
!'MaCurdy (1981) and Altonji (forthcoming) present instrumental
variables estimators of the intertemporal labor supply
elasticity, using
polynomials of age and experience as instruments.
---"Write equation (l2a) as
Alog gft =
where
+
io
÷
represents the regression coefficient of the change in ear-
nings on experience,
is a period—specific constant, and
includes the stochastic components of earnings changes.
If time is
measured in years, then the change in annual earnings over the two year
interval from t—l to t+l is Alog g.1 ÷ Alog
=
1t+l
+
+
—37—
28x1 +
+ 6.
.
Thus the covariances of changes in earnings and
hours with potential experience should be roughly twice as large in the
NLS as the PSID. The comparison between changes at six month intervals
and changes at annual intervals is similar.
instrumental variables estimate of n
for
the PSID is
substantially larger than the estimates reported by MaCurdy (1981) or
Altonji (forthcoming). Both authors use overidentified estimators that
make use of several instrumental variables for wage changes. Since our
primary interest is not in the instrumental variables estimates themselves, we have not explored the estimates of
ii
when other exogenous
variables are used as instruments for changes in wages.
is straightforward to show that each of the variances of the
productivity shock can be increased by a fixed factor, the measurement
error
variances can be decreased, and the first—order autocovariances of
the productivity shock can be increased in such a way as to hold
constant
the predicted covariance matrix of earnings and hours.
!ZiHart (1983) and Rosen (1985) survey most of the theoretical
literature on labor contracts.
!JIn other work (Abowd and Card (1984)) we have considered the
implications of labor contracts that smooth individual earnings.
Suppose, for example, that observed earnings in period t represent a
geometric average of earnings in the absence of contracts (g°) and
desired consumption in t (c) , which we assume is constant. Then
log g = (1—y) log g° + y log c , where
'y'
represents the extent of
—38—
earnings stabilization. Taking first differences and using the notation
of equation (22a), log
=
ILAz*.
+
E*i +
p = (1—y) (l+n)/n and Az*. and C*1
, where
are suitably defined. In the
presence of earnings smoothing contracts, the parameter
may fall
i'
below unity.
12!The labor supply elasticity n
U by the formula
ii
= (p—l)1
is
obtained from the estimate of
Standard errors for
.
in
Ti
Tables 12,
13, and 14 are calculated by the delta—method.
derive the covariance matrix of the difference between the OMD
estimate b° and the EWND estimate be
,
note that under the hypothesis
that the model is correct,
/N (b —
b*)
a
/nbe - b*) -
—1 —l
—1
/r(F'V F) FTV (m_f(b*))
/F(F'F)1
F!(m_f(b*))
where V is the covariance matrix of the vector of moments
is the predicted vector of moments given the true value b*
sample size, and F represents the Jacobian matrix of f
b* •
,
in
,
, f(b)
N
is the
evaluated
at
These equations follow from the first order conditions for
and be ,
respectively,
and a series of regularity assumptions. Using
the fact that V is the variance matrix of (m_f(b*))
,
it
is straight-
forward to derive the asymptotic variance matrix of the difference
(ho_be).
li/The
only when
variance of the OND parameter estimate is necessarily smaller
both variances are estimated at the same parameter values.
—39—
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Temporary Layoffs and Compensating Wage Differentials." Studies
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—40—
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1013—1030.
Hausnan, J.A. "Specification Tests in Econometrics." Econometrica, 46
(September 1978): 1251—1272.
Heckman, James J. "A Life—Cycle Model of Earnings, Learning and
Consumption."
Journal of Political Economy, 84 (August 1976):
811—544.
Killingsworth,
Mark. Labor Supply.
New York:
Cambridge Univ. Press,
1983.
MaCurdy, Thomas E. "An Empirical Model of Labor Supply in a Life—Cycle
Setting." Journal of Political Economy, 89 (December 1981):
1059—1085.
__________________
Structure
"The Use of Time Series Processes to Model the Error
of Earnings in a Longitudinal Data Analysis." Journal of
Econometrics,
Newey, Whitney IC.
18 (Special Issue 1982): 83—114.
"Generalized Method of Moments Specification Tests."
Unpublished manuscript, Princeton University, 1985.
Pencavel, John. "Labor Supply of Men: A Survey." In The Handbook of
Labor
Economics, ed. by Orley Ashenfelter and Richard Layard.
Forthcoming, 1985.
Rosen, Sherwin, "Implicit Contracts". Journal of Economic Literature,
23 (September 1985): 1144—1176.
—41—
Appendix A
In this appendix we summarize the estimation and -inference proce-
-
dures used throughcxit the paper. The basic unit of data for each individual in a particular data set is the vector of experience—adjusted
changes in earnings and hours. If we denote these by A log
and
A log hj , then the data vector is
A log
V
Ji a
A log
A
log ll
A 1g "iT
which has dimension 2T , where T is the number of changes observed in
the data set.
Let "
let C=-t —
=
4f
'—'
these changes.. Estimates of C
represent
the mean vector of changes and
represent the covariance matrix of
are presented for the three data sets
in Tables 4, 5, and 6.
The
models we estimate are models for C.
Let m represent a
vector whose elements are the distinct elements of C •
Since C is
symntric, there are only 2T(2T-i-l)/2 elements in in •
In
the PSID,
this corresponds to 210 elements; in the NLS, 55 elements, and itt
SIME/DD4E, 136. Conformably with in,
let
m represent the distinct
elements of the individual cross—products matrix
in
=
E
in
the
(—)' . Then
—42—
The variance matrix V of the vector of covariance elements is
estimated by computing the cross—sectional variance of m1 in the usual
manner:
—in)'
v=- (m —in) (m
A typical elanent of V is V
cov(A
in
log ,
A
cov(m
m)
,
If
= cov(A
log g1
log hj.) and
,
A log
then
=
j [(A
— A
(A log
where
A log j1
A
log it —
represents
of earnings in period t •
— A
log
—
log is—k — A log
—
log XA log
log ')(A
m]
x
the sample average change in the logariti-in
Let
Q
represent
the matrix of fourth
moments of
Q=ji mimi'
Q
and
V
are related by V = Q — in in'
Under fairly general conditions, independence of the
y1 implies
that the sample mean of
hF Cm — ii)
where p
m has an asymptotic normal distribution:
N(O ,
V*)
is the population value of m (i.e. , the true covariance
matrix of earnings and hours changes) and V* is the population value
of V.
Consider a model for the vector of covariance elements that depends
on a lcwyer—djmensional parameter vector b
,
say
m =
f(b).
Several
—43—
estimators
are available for b: among these are the optimal minimum
distance (OIID) estimator b° , which minimizes (m—f(b)) V1 (m—f(b)),
and
the
equally weighted minimum distance (EWMD) (or least squares)
estimator
former
be , which minimizes (m—f(b) ) '(m—f(b)).
Optlinality of the
estimator is discussed in Chamberlain (1984).
For the OMU estimator, inference is based on the fact that under the
hypothesis of a correct
N •
(in
—
specification, the minimized
f(b°fl'
V1 (m —
f(b°))
has an asymptotic chi—squared distribution with
to
quadratic form
degrees of freedom equal
the difference between the dimension of m and the rank of the
Jacobian
evaluated at b* ,
matrix F(b)
the
true value of b.
(See Chamberlain (1984)).
If the model for in
has
= f(b)
=
H
in which
the special form
=
L°i
only the last k elements of
straightforward
N•
in
are restricted to zero, it is
to show that the minimized quadratic form reduces to
—l
in2
V22
m2
where V22 is the block of V
corresponding
to the elements in
that are restricted to zero. This is the chi—squared test for
in2
=0
in
—44--
For an estimator bA that minimizes an arbitrary quadratic form
(m —
f(bfl'
ACm —
f(b))
where A is a positive definite matrix, Newey (1985) shows that the
quadratic form
(in
- f(b R (mS- f(bA))
has an asymptotic chi—squared distribution. Here, C is a generalized
inverse of the matrix R = pVp'
,
where
— F(F'F)1
p =
F'A , and
F
represents the Jacobian matrix of f evaluated at hA. Newey (1985)
suggests a generalized inverse of R of the form S(S'RS)1 S , where
S is a selection matrix of rank equal to the difference between the
dimension of
in
(the number of movements fit) and the rank of F. In
general, for a nonlinear model, the value of the quadratic form depends
the exact generalized inverse selected for R. In case of a linear
on
model, however,
the value of the quadratic form is invariant, and in the
simple case of testing the restrictions in2
restrictions on m1 ,
equal N
the
0
,
with
no additional
value of the quadratic form can be shown to
in2' V221 in2. Tests of zero restrictions are invariant to
the choice of OMD or any arbitrary minimum distance estimator.
Our procedure is to first test for zero restrictions and then to
work with the nonzero covariances of earnings and hours. In the notation of the previous discussion, once we have accepted the hypothesis
= 0
V11
,
we estimate ndels for
m1 and use
the covariances matrix
for inference. This procedure is not fully efficient, since
better estimates of the unrestricted elements of m can be formed by
taking into
those
in in2)
account the fact that certain other elements of at (i.e.
are zero.
—45—
Appendix B
In this appendix, we comnnt on the assumption underlying equation
(10) of the text that aggregate shocks and experience effects enter
homogeneously into individual
productivity. More
generally, consider
the alternative specification:
(ba) log
a1 + r16
+
+2x +
z
Here, we have permitted the aggregate productivity shock
individual
ficient
that
r1
to affect
productivity through an individual—specific response coefCross—section
dispersion in r
reflects
the possibility
individuals have permanent cyclical attributes: individuals with
higher values of r1 are more responsive to aggregate shocks. We have
also permitted the linear experience coefficient
to have an indi-
vidual component of variance.
Equation (l0a) implies that the first difference of wages can be
written as:
(ha) A log
where
K
= rA6
—
=
K+
+ 2y2t
+ Y2x0
,
and
+
(r —r) Aót + Az1
r is the average value of
Individual heterogeniety in the experience slope and the response to
aggregate shocks introduces two additional components of variance into
the first differences of earnings and hours. Both of these components
contribute to covariances between lagged changes in earnings and hours.
First consider the component of variance associated with (r — r)
If
this component is large, then changes in earnings and hours in years
with a negative product [vity shock should be more highly
correlated with
—46—
changes in other years with negative shocks, and less correlated with
changes in years with aggregate productivity improvements. By most
measures aggregate productivity fell with business cycle recessions in
1969—70, 1973—74, and 1980—81. In the PSID data, however, covariances
of earnings and hours changes between these years are very similar to
covariances
between arbitrary pairs of years at the same gap. We there-
fore conclude that the component of variance due to individual—specific
responses to aggregate shocks is small.
Next, consider the component of variance in (ha) associated with
cross—section dispersion in earnings growth rates. This component
contrihites
a fixed positive element to earnings and hjrs autocovar—
iances and cross—covariances of all orders. In
the data, autocovariances
of third order and higher are all negligible. Again, we conclude that
cross—section
dispersion in experience growth rates contrihites a
relatively small component of variance to wage growth.
Table 1
Characteristics of the PSID Sample of Male Household Heads
Average
Hourly
Earnings
Average
Annual
Hours
1969
1970
1971
1972
1973
1974
1975
3.62
3.71
3.85
4.00
4.13
4.10
4.02
2308
2276
2266
2302
2324
2246
2220
1976
1977
1978
1979
4.19
4.26
4.26
4.25
2231
2236
2244
at
Year—
1. Annual Hours and Average
Hourly Earnings
(1967 dollars)
Change
2.
Changes in Log Real Annual
Earnings and Log Annual
Hours (x 100)
.
Change
Earnings
in
.
1969—70
1970—71
19 71—72
1972—73
1973—74
1974—75
1975—76
1976—77
1977—78
1978—79
2186
Change
in Hours
2.5
3.0
—0.8
-0.3
6.9
4.7
2.0
1.9
—5.5
—4.2
—4.1
—2.4
4.1
2.5
0.2
—5.5
0.6
0.3
0.5
—4.2
3. Demographic Characteristics
Average Age in 1969
Percent Nonwhite
Average Potential Experience in 1969
4. Sample Size
35.8
27.3
18.9
1448-"
NOTES: a/Data are for the calendar years listed.
.1Eight outliers with reported average hourly earnings greater
than $100/hour (1967 dollars) were excluded.
Table 2
Characteristics of the NLS Sample of Older Men
Year
1. Annual Hours and Average
Hourly Earnings
(1967 dollars)
1966
1967
1969
1971
1973
1975
.
Change
2. Changes in Log Real Annual
Earnings and Log Annual
Hours (x 100)
Average
Hourly
Earnings
3.50
3.46
3.55
3.66
3.63
3.50
Change
in Earnings
1966—67
1967-69
1969—71
4.5
4.0
1971—73
1973—75
—0.2
—16.8
3.1
Average
Annual
Hours
2209
2190
2190
2161
2160
2003
Change
in Hours
0.0
0.0
—0.1
—1.5
—11.6
3. Demographic Characteristics
Average Age in 1969
49.2
Percent Nonwhite
29.8
Average P.otential Experience in 1969
34.4
4. Sample Size
1318
NOTE: Data are for twelve—month periods preceeding the interview date.
Changes in earnings and hours over the two year intervals are not
at annual rates.
Table 3
Characteristics of the SINE/DIME Sample
of
Male—Reacts of Dual—Headed Households
Experimental
Period
1. Mm..] Hours and Average
Hourly Earnings
(1971 dollars)
i
3.47
2093
3.53
3
4
3.63
3.73
3.83
3.88
3.88
3.85
3.88
2098
2087
2117
2135
7
8
9
Change
Earnings and Log Semi—Annual
Hours (z 100)
Average
Annual
Hours
2
5
6
2. Changes in Log Real Semi—Annual
Average
Hourly
Earnings
Change
in Earninjs
2104
2131
2074
2059
Change
Hours
1—2
1.2
2-3
—1 .3
2.8
5.8
1.6
4.0
5—6
6—7
7—8
—1.3
0.7
—3.5
0.6
8—9
—3.0
—2.7
3—4
4—5
3.9
3. Demographic Characteristics
Average Age at start of experiment
34.7
Percent Nonwhite
48.8
Average Potential Experience at start
18.2
of experiemnt
4. Sample Size
NOTE: Data
560
are for six—month periods following assignment into the
income experiment. The changes in earnings and hours between
consecutive six—month intervals are not at annual rates.
73—06
Alog
A L.a a
76—79
70—71
Lloj
Ii
.00)
.003
(.004)
2).
)0006
(.939)
—.506
(.00))
.001
79-79
l0.Abo(h
(.004)
—.002
.004
(dl?)
.!10
.000
(.003)
(.975)
4900
.004
(.906)
(.006)
(.004)
(.006)
—.006
—-009
ILAljh
77—79
76—77
1.61096
75—76
17.81n9k
(.006)
.006
76—75
6.61096
65.
—.00)
(.006)
(-00))
.006
(.004)
(.0(0)
-.031
.07)
(.01)5
(.010)
—.9)1
(.006)
6
Alo
73—li
.003
(.00))
—
—-001
-
—.001
(-007)
(.011)
—.022
(.016)
.067
.000
.136
NOTflt
(.1 (7)
323
-.00]
(.003)
.00)
(.00))
(.00))
—.001
—.002
.001
(.005)
(.099)
.224
-009
(.119)
—.01!
I92
(-III)
(.006)
—.006
(.005)
—.091
(.021)
—.046
.106
(.021)
(.013)
—.0)1
(.007)
—.008
(.003)
—.001
(.006)
—.1)0
(.117)
(.00))
—.004
033
(.013)
4967
(.027)
(.010)
—.067
.002
(.009)
(.003)
—.0)0
(4905)
.007
—.003
(.003)
(.964)
.209
.003
(4906)
.088
(.020)
.017
(.014)
-00]
(.010)
(.009)
—.011
.001
(.004)
—.00'
(.005)
—.00!
(-005)
.005
(.004)
.001
(.005)
(.006)
-.009
(.004)
—.001
.046
(.013)
—
.191
(.0?))
(9)
*009 a
7779
(.005)
—.004
.003
(.060)
(-0)9)
—.064
.996
(.034)
76—77
AIg
(9)
(.006)
.006
.00'
(.006)
(.019)
.0l3
(.009)
—.087
(.032)
-
75—76
âIog
(7)
(.006)
—.002
490?
(.003)
(.006)
—.006
(OIl)
—.068
.071
(.093)
(.079)
—.008
—.003
(.004)
—.009
(.006)
(.0052
.00)
(.005)
—.001
.000
(.006)
.006
(.007)
—OIl
(.007)
(.01))
—.088
.111
((II)
AThg g
76—75
(6)
(.11])
.217
.075
(.016)
(-060)
—.006
.009
(.009)
(.0063
—.007
(.006)
4906
.001
(.906)
(.004)
.00)
(.00))
.001
(4906)
—.001
(.004)
—.007
.17'
(.0)0)
79.79
Abs
0)0)
of:
—.311
(.118)
(.005)
—.00!
(.005)
—.001
.005
(.00))
(.003)
—.002
(.006)
—.006
.006
(.006)
.000
(.00))
—.002
(.007)
—.060
(.0(0)
(.013)
.1)8
(II)
Slog 6
69—70
C.art.nt.
.115
—.123
(.096)
(-006)
.00)
000
(.006)
(.004)
—.002
.000
(.006)
.006
(.006)
(.004)
—.006
(.00))
-M05
(.010)
—.068
(.015)
Ii
(12)
&I
70-71
ii, p.rev,the.ea).a'
.119
—.953
(.096)
(.006)
—-006
.002
(.006)
(.006)
—.001
.00)
(.003)
(.003)
—.005
(.003)
—.090
—.016
(.090)
900r7 of
the
an.
t
oeot
.atrtx.
an given potential eocpertence and year
onple fourth
tonal
are ba.ed on the
coodi
of
ohaerva
—-III
(402)
002
(.006)
(.003)
—-003
(.006)
—.010
(.01])
—.058
.110
(.013)
(16)
6Io 6
767)
tb,.
and
the
scsi.) 'curs
re calculated
See Table I for
(.07))
—.05)
(.00))
430!
(-004)
—.001
0.00,0
—.001
(.006)
—.006
(.005)
—.010
.073
(.001)
73—16
(I))
61086
.Vcovartance of potential labor force exper(ence with the change (9 log earnings (column. (1)—fib))
ch.nge (9 log annual hours (column. (0l)—(20)h
i/standard error.
a
sfter.dJoetbOg (or to..
log
(.091)
.076
(-001)
—.001
(-006)
—.006
.008
(.005)
(.003)
—.006
(.003)
—.00'
(.001)
—.016
.090
(.0(9)
71—71
(14)
hog h
71—11
(II)
61086
(.015)
Il
-The data ste change. to log annual e.rnlng. (A log g I, roes .nol cclii.,. (1)—flu)) end
(A log I. to row! .nd cclii.,,. (tl)—(20)) 4) Iferenced between the year. indlc.ted. Ca
0_Ott)
4906
(.001)
—450
—490?
-003
(.004)
—.007
.000
(.00))
(.004)
—.001
(.007)
—.016
.036
(.007)
(.003)
.00)
(.004)
—.00)
(.0(1)
.002
(.001)
—.001
.000
(-(06)
(.006)
—.00!
(.006)
—.OIO
(.006)
—.001
(.015)
—.050
(.016)
.129
9
73—76
Ito)
0)
(.003)
.005
(.005)
(.006)
—.00)
(490))
(.006)
—.009
.006
(.003)
.039
(.010)
(.00))
.000
—.0)!
(.006)
(4911)
—.021
(.015)
—4922
(.010)
(.007)
.079
—.00)
(.011)
.001
(.004)
.000
(.00))
.006
(.007)
.011
(.009)
.000
(.00!)
(.006)
—.711
(.007)
—.036
(.014)
—.026
(.007)
—.001
.001
(.005)
(.009)
(-007)
(.006)
—.003
.006
(.007)
-00]
.009
(.009)
.000
(.0070
6l
73-34
II
79—02
I3A0o)t
II.
69—70
II. 60056
In.
77—79
9.ilogg
76—77
9.460I
.010
(.008)
(.005)
(.005)
(.006)
(.0076
—.010
—.002
—.007
—.007
.00)
(.005)
(.005)
—.007
(.0")
—.041
.000
).h!o.e
75—76
.170
(.0:9)
(.005)
(.00))
.003
.003
(.006)
76—75
6. Alog
L
(.009)
—.002
.000
(.0060
72—73
70—7)
6.hlogg
.959
—.066
(.093)
—.097
(III)
—.116
.161
(.011)
(.019)
72—72
71—71
70.71
(.007)
3. 6)0(6
70—It
1. how
69—10
1. Aba
ilo g
660!.
aloe!
(4)
g
69—70
6t
(3)
(2)
II)
error.
to
" 165
4
N81fl 1969—1970
(Itanitard
rot £010
Y.bla
.070
CIII)
-.003
(.005)
(.006)
—.006
(.0)8)
—06A
.167
(On)
73—76
(II)
loa I,
(.60)1
I9'
(.00))
—.00)
(.015)
—493)
483
(.026)
76—77
(08)
Mo 6
—.151
(.121)
(.008)
—.02!
.1)4
(.020)
7779
(I!)
t08 It
(.129)
—.29!
.016
(.019)
II
79—79
(20)
6lfl
vitb:
66—67
1. Alog g
Table 5
NLS
(3)
(4)
(5)
Alog h
66—67
(6)
Alog h
67—69
(7)
8log h
69—71
(8)
Alog h
71—73
(9)
(.039)
.132
(.040)
.178
(.024)
.330
(.036)
69—71
3. Alog
Data l966—75'
Covariances of Experience—Adjusted Changes in
Earniis and Hours:
(standard errors in parentheses)
(2)
Alog g
Covariance of:
(1)
Alog g
71—73
—.068
(.038)
—.040
(.008)
—.073
(.018)
7375
Alog g
69—71
—.032
(.010)
.000
(.004)
—.011
(.010)
(.006)
(.007)
—.017
.013
—.026
.198
—.225
.
blog g
67—69
—.000
(.004)
.007
—.003
(.004)
h
Experience1
73—75
67—69
7. Alog
11.
(.086)
(10)
—.036
(.012)
—.038
(.008)
(.015)
—.028
(.007)
.066
(.011)
—.050
(.010)
(.014)
—.291
(.078)
.235
(.028)
7375
A1.og h
—.002
(.003)
.007
(.005)
.011
(.005)
—.158
(.04!)
.081
—.001
(.003)
—.013
(.006)
.052
(.036)
,
.005
(.005)
.000
(.048)
.091
—.018
(.049)
.114
(.023)
8log g
66—67
.
—.005
(.004)
(.011))
.002
(.003)
.001
(.005)
(.005)
.109
(.016)
.004
(.006)
.002
(.005)
—.004
(.004)
.002
(.006)
(.009)
.133
4. tlog g
71—73
.038
(.012)
.016
(.004)
.015
(.004)
.060
(.014)
(.027)
2.
5. Alog g
73—75
—.007
(.004)
—.007
(.004)
-.007
(.005)
—.034
(.010)
tb8 g
6. Alog h
66—67
.002
(.003)
—.001
(.004)
—.006
(.009)
—.160
(.056)
67—69
8. Alog h
69—71
.001
(.003)
.009
(.009)
.016
(.057)
g
9. Alog h
11—13
.002
(.007)
.078
(.063)
.005
10. Alog h
.062
(.051)
NOTES
-11The data represent
in
changes
and
log
earnings
hours
over
log
the
intervals
indicated, adjusted for the
conditional mean given potential experience and year of observation.
See Table 2 for means.
21Covariance of potential labor market experience with the
change in log earnings (colunns (l)—(5)) and the
change in log hours (columns (6)—(10)).
with:
in
(5)
Table 6
Change
in parentheses)
/
Semi—Annual Data (roe. SINE/DIME1
Covariances of Experience—Adjusted
F.arninga and Hours
(standard errors
(10)
Period 2
.153
Slog h
(.028)
(.024)
.130
.119
.130
(3)
—.050
—.052
(.027)
(.025)
(.026)
cnarinc. of
(7)
(8)
(9)
Slog g
Slog g
Slog I'
Period I
Period 7 PerIod 8
(.018)
(.018)
(.020)
—.050
—.038
(.038)
.140
.134
6
(6)
Slog g
—.022
.006
(.012)
.171
(.032)
—.006
(.025)
(.026)
(2)
Slog g
—.046
(.014)
(.010)
.009
—.076
(.023)
—.086
(.024)
(.007)
(.017)
—.038
PerIod5 Period
(.022)
—.003
(.008)
.010
(.012)
.019
(.017)
.006
.002
4
(4)
Slog g
—.007
(.008)
(.009)
—.01]
—.013
(.010)
.000
(.012)
.008
(.008)
(.010)
Period
(.013)
—.001
.005
(.013)
.000
(.007)
(.014)
.009
Slog g
Slog g
PerIod 2 Period 3
—.016
(.007)
—.006
(.010)
—.013
(.012)
(.007)
—.002
1
(1)
Slog g
Period
.155
(.014)
.004
(.013)
.015
(.015)
(.012)
—.001
—.DLl
(.030)
.002
(.013)
.000
(.010)
—.007
—.022
.011
(.008)
1. Slog g
Period I
—.005
6. Slog g
Period 6 (.008)
—.036
(.012)
(.013)
g
PerIod
4. Slog
Period
4
I
Period 5
5. Slog g
7.
Period
Slog h
(.010)
(.013)
(.013)
—.005
(.017)
—.042
(.022)
.120
.119
(12)
Slog h
(.008)
Period 4
(.009)
(.023)
3
(.017)
(.017)
—.052
(11)
Alog h
PerIod
(.022)
—.020
.145
.010
(.010)
(.018)
—.036
—.049
2
—.006
(.008)
.130
.119
(.017)
2. Slog g
Period
.133
(.026)
(.026)
II.
3
Slog g
Period 7
—.045
—.046
3. Slog g
8. Slog g
Period 8
.001
(.021)
(.011)
(.01])
.217
9. Slog h
2
(.014)
(.011)
—.011
.005
ID. Slog Ii
Period
3
.005
(13)
Slog ii
Period S
(14)
Slogh
.100
.111
(.013)
—.040
(.016)
(.023)
(.010)
(.013)
(.0L2)
—.036
—.009
.010
.004
.005
(.009)
(.008)
(.009)
—.003
—.010
—.041
(.007)
(.008)
.116
—.003
.000
—.055
(.009)
(.006)
—.017
.001
.006
—.014
.003
(.012)
(.008)
—.031
.020
.100
(.010)
(.0'S)
(.015)
(.012)
(.021)
—.071
(.024)
.113
—.042
(.019)
(.024)
.010
—.040
(.017)
(.015)
(.012)
(.013)
.007
—.006
(.009)
(.008)
(.013)
—.011
—.002
(.008)
.001
(.006)
(.007)
.004
(.008)
(IS)
Slog
(16)
Slog h
Period
.157
.002
—.069
—.077
.000
.155
—.009
—.077
—.003
.008
(.03!)
—.010
(.031)
.008
.006
of
.208
(.011)
(.022)
.065
(.022)
.017
(.009)
.008
(.014)
(.010)
(.00!)
(.015)
i.lel)
.002
(.011)
(.010)
.174
.002
(.013)
(.006)
(.187)
.009
(.013)
(.018)
—.056
.000
(.023)
(.234)
.004
(.214)
—.346
(.185)
—.126
.198
.061
(.211)
—.244
(.16])
.039
—.087
(.214)
(.022)
(.017)
.014
(.012)
.010
.002
.00!
(.013)
(.151)
.139
(.031)
(.013)
.016
(.016)
100
(.138)
(.024)
—.00!
(:169)
—
235
(:165)
—
(.160)
(.011)
(.009)
(.010)
007
(:195)
—.168
(.010)
(.007)
— 127
(:230)
—.005
156
(.195)
—
—.040
—.004
(.010)
(.010)
.001
Period
12. Slog Ii
8
7
6
S
PerIod 4
13. Slog h
Period
Period
14. Slog K
15. Slog h
Period
—
Period
Ex e
16. Slog h
17
NOTES:
experience and period—specific shocks by regreeeing changes in log earnings and
i/The data have been adjusted for conditional means given potential
log hours on potential experience and dummy variablea for the date of entry into the experimental group. The data are aligned in experimental tine:
Period 1 refers to the first sixth—month nerind after enrollment into the control group. See Table I for means.
in log hours (columns
earnings (columns (l)—(8)) nnd the change
potential labor market experience with thearechange in log
control group.
adjusted for date of entry into the experimental
Experience and the change in log e.nrnlnp.e and hours
covariancem
(9)—(16)).
Vpartial
Table 7
Summary
of Estimated Bivariate MA(2) Representation
of Experience—Adjusted Changes in Earnings and Hours:
Annual
Data
from the PSID
1. Goodness—of—fit statistic — MA(2)
(probability value — 112 degrees of freedom)
137.19
2. Test statistic for symmetric MA(2)'
31.02
(.020)
(probability value — 17 degrees of freedom)
(.062).
3. Test statistic for MA(l)--"
(probability value — 32 degrees of freedom)
(.020)
4. Test statistic for stationary MA(2)1'
(probability value — 87 degrees of freedom)
143.69
(.000)
50.39.
NOTE: "Against the alternative hypothesis of a nonstationary
bivariate MA(2).
Table 8
Summary of Estimated Bivariate MA(2) Representation
of Experience—Adjusted Changes in Earnings and Hours:
Annual Data from NLS
1. Goodness—of—fit statistic — MA(2)
(probability value — 12 degrees of freedom)
12.89
(.377)
2. Test statistic for symmetric MA(2)uI'
(probability value — 7 degrees of freedom)
(.047)
3. Test statistic for MA(l)"
14.47
14.64
(probability value — 12 degrees of freedom)
(.263)
4. Test statistic for stationary MA(2)11
(probability value — 21 degrees of freedom)
68.14
(.000)
NOTE: !iAgainst the alternative hypothesis of a nonstationary
bivariate MA(2).
Table 9
Summary of Estimated Bivariate MA(2) Representation
of Experience—Adjusted Changes in Earnings and Hours:
Semi—Annual Data from SIME/DIME
1. Goodness—of—fit statistic — MA(2)
(probability value — 60 degrees of freedom)
2. Test statistic for symmetric tfA(2)L"
(probability value — 2 degrees of freedon)
74.98
(.092)
5.90
(.434)
39.95
3. Test statistic for MA(1)(probability value — 24 degrees of freedom)
(.022)
4. Test statistic for stationary MA(2)"
(probability value — 65 degrees of freedom)
100.70
(.000)
NOTE: VAgainst the alternative hypothesis of a nonstationary
bivariate MA(2).
Average Covariance of Change
in Earnings with
Experience
Average Covariance of
Change in Hours
with Experience
(C1)
Implied Estimate of
Substitution
Elasticity (ii)
Goodness—of—fit Statistic
(probability value)
[degrees of freedom]
2.
3.
4.
NOTE:
PSID
NLS, and SIME/DIME
[14]
[8]
[18]
1"Changes in earnings and hours between 1966 and 1967 are expressed at two—year
rates for comparability with later changes.
20.78
(.107)
.56
(.47)
40.96
(.000)
(.61)
(.04)
—.05
(.04)
—.13
25.80
(.104)
—1.62
(.44)
(.02)
1.52
—.09
(.02)
(.02)
(.03)
—.13
—.03
NLE4"
SIME/DIME
(six—month changes)
(annual changes
over two—year gap)
—.21
(annual changes)
1.
(Cg)
PSID,
(standard errors in parentheses)
with Potential Experience:
Covariances of Changes in Log Earnings and Changes in Log Hours
Table 10
Table 11
Implied Covariances of Experience —
Adjusted
Changes in
Earnings and Hours: Labor Supply Model
1. var (A log gft)
+
2. var (A log hit)
2n(l+n) coy (Azj, jt + 202
var (Azi) +
+ 21,2 coy
3. coy (A log
var
(1+1)2 var (Azj) +
A log
—
4.
coy (A log
A
5.
coy (A log
A log 1itl
6.
coy (A log
log 11)
A
(cit)
(Azi, e) +
log hii),
202
var
n(1+n) var (Azi) +
+
-
var
n(l+2n) coy (Azj. 12
+
2
(l+ri) coy (Azj, Azii) —
(cit
2paa
2
coy (Azj, Azit_1) —
n(l+n) coy (Azi, Azi1) —
uv0u0v
coy (A log it' A log j1)
7. coy (A log
A log gj_)
8. coy (A log hi A log h2)
9. coy (A log
A
log hi2),
(1+1)2 coy (Azt, Azi_2)
coy (Azj, Az.2)
n(l+n) coy (Azi, Az12)
coy (A log hi A log
Notation: A log g and A log h refer to the residuals of the changes in
the logarithm of earnings and hours from regressions on potential experience.
represents the variance of the measurement error of earnings, 2 represents the variance of the measurenent error of hours, and uy represents their
correlation.
Goodness—of—Fit Statistic
(probability value)
[degrees of freedoml
NOTES:
is
(0.772)
0.29
8,83
(2.63)
11.30
(14.59)
a
unit contribution to hours.
fit to autocovariances of changes in earnings and hours up to second order.
i
(4.81)
7.27
(0.11)
(0.03)
(0.09)
1.68
(0.093)
—19.11
(28.45)
0.95
(0.08)
[48]
[48]
1.09
[68]
[681
61.38
(.027)
Optimal
Minimum Distance
62.26
(.026)
1.11
261.12
(.000)
261.15
(.000)
Equally Weighted
Minimum Distance
uncertainty Model
(Zero—Order Covariances Unrestricted)
1.14
Optimal
Minimum Distance
Equally Weighted
Minimum Distance
VProductivity component is normalized to have
A"Model
ON!)
estimate of
(probability value)
vs.
4. Specification test of EWMT)
Substitution Elasticity
(standard error)
3. Implied Intertemporal
2/
Component to Earnings(p)—
(standard error)
MA(2) Productivity
2. Relative Contribution of
1.
-
Perfect
Annual Data from
Estimated Two—Factor Model of Changes in
Foresight Model
Covariances
Restricted)
(Zero—Order
-
of
Earnings and Hours:
Summary
Table 12
estimate
of p
of
———
5.86
(4.11)
3/
n/a—
0.15
(0.06)
7.71
(2.79)
———
(1.53)
1.61
1.56
(0.59)
[161
[28]
1.17
31.38
(0.026)
1.38
(0.167)
—6.80
(13.45)
0.85
(0.29)
116]
31.21
(0.027)
Optimal
Minimum Distance
order.
Equally Weighted
Minimum Distance
Uncertainty Model
(Zero—Order Covariances Unrestricted)
94.49
(0.000)
Optimal
Minimum Distance
(0.12)
[281
124.13
(0.000)
Equally Weighted
Minimum_Distance
Perfect Foresight Model
(Zero—Order Covariances Restricted)
and Hours: Annual Data front NLSU
Estimated Two—Factor Model of Changes in
Earnings
Sumtnary
Table 13
'Model is fit to autocovariances of changes in earnings and hours up to second
VProductivity component is normalized to have a unit contribution to hours.
--Test statistic cannot he computed.
(probability value)
OMI)
Specification test of EWMD
vs.
NOTES:
4.
Implied Intertemporal
Substitution Elasticity
(standard error)
Relative Contribution of
MA(2) Productivity
2/
Component to Earnings(p)—(standard error)
2.
3.
Goodness—of—Fit Statistic
(probability value)
[degrees of freedom]
1.
"C
Goodness—of—Fit Statistic
(probability value)
[degrees of freedom]
is
227.19
(0.000)
[52]
229.36
(0.000)
[52]
Changes in
.
(0.124)
1.54
(7.23)
16.67
———
0.002
(0.998)
123.00
(438.74)
(0.03)
(0.07)
125.00
(1140.63)
1.01
1361
1361
1.01
(0.009)
56.98
(0.014)
58.86
Optimal
Minimum Distance
Equally Weighted
Minimum Distance
Uncertainty Model
(Zero—Order Covariances Unrestricted)
fit to autocovariances of changes in earnings and hours up to secondorder.
———
—55.55
(178.98)
1.06
(0.03)
Optimal
Minimum Distance
Equally Weighted
Minimum Distance
Perfect Foresight Model
(Zero—order Covariances Restricted)
0.98
(0.06)
of
Semi—Annual Data from SIME/DINEII'
Estimated Two—Factor Model
FProductiviry component is normalized to have a unit contribution to hours.
!"Model
Specification test of EWMD
vs. OMD estimate of p
(probability value)
4.
NOTES:
Implied Intertemporal
Substitution Elasticity
(standard error)
3.
MA(2) Productivity
2/
Component to Earnings(p)—
(standard error)
2. Relative Contribution of
1.
of
Earnings and Hours:
Summary
Table 14