Estimation and Testing of the Union Wage Effect Using Panel Data
Author(s): George Jakubson
Source: The Review of Economic Studies, Vol. 58, No. 5 (Oct., 1991), pp. 971-991
Published by: Oxford University Press
Stable URL: http://www.jstor.org/stable/2297947
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Review of Economic Studies (1991) 58, 971-991 0034-6527/91/00560971 $02.00
? 1991 The Review of Economic Studies Limited
Estimation and Testing of
the Union Wage Effect
Using Panel Data
GEORGE JAKUBSON
Cornell University
First version received October 1986; final version accepted September 1990 (Eds.)
We present estimates of the union wage effect controlling for unmeasured individual effects,
and subject the conventional fixed-effects model to specification tests. For PSID men the union
wage effect is 5-8% after controlling for person effects, as opposed to 20% in cross-section.
Omnibus tests based on an unrestricted reduced form and instrumental variables tests based on
differencing are consistent with conventional models. Tests based on comparing those who enter
and leave union coverage provide evidence against the usual model. We find evidence for
interactions between union status and other variables even after controlling for person effects.
1. INTRODUCTION
Whether unions truly raise wages is a longstanding question in labour economics. A
considerable body of empirical work, based on cross-section data, suggests that the union
wage premium is on the order of 20% (Lewis (1986)). This large rent for union workers
may seem implausibly large, and several economists have questioned whether cross-section
studies estimate a pure union effect (see Lewis (1986) for a review). They argue that the
natural employer response to higher union wages is to raise hiring standards.' If employers
react in this manner, and if they observe more about workers than does the researcher,
then union workers will have higher unmeasured (by the researcher) productivity
attributes. The "union wage premium" contains a return on these unmeasured attributes
and the estimated union wage effect is biased upwards.
Two approaches to this omitted-variables bias have appeared in the literature. The
first accounts for the endogeneity of the union variable using either an instrumental
variables technique or a selectivity adjustment (Heckman (1979)). This approach is
intuitively appealing since it is based on the notion that an individual obtains a union
job when a weighted sum of his observed and unobserved characteristics crosses a
threshold. The main drawback is the need to make arbitrary distributional and/or
exclusion assumptions.
The use of panel data allows one to avoid those assumptions. If the unobserved
productivity differences are constant over time one can control for them by estimating a
fixed-effects model. One can use standard specification tests (e.g. Hausman (1978)) to
investigate whether previous studies, which do not control for unobserved productivity
differences, have produced biased results.
1. See Pettengill (1980) and comments by Pencavel (1981) for a particularly careful development of this
argument.
971
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972 REVIEW OF ECONOMIC STUDIES
The use of panel data and fixed-effects models is an important advance in empirical
economics. However, one weakness in current empirical practice is the fact that the
fixed-effects model is usually the most general model considered. In fact, panel data
contain more information than that required to identify a fixed-effects model. Thus, the
appropriateness of the fixed-effects model itself can be investigated by testing the overidentifying
restrictions it places on the reduced form. This is not commonly done in the
empirical literature.
The purpose of this paper is two-fold. The first is to compare the estimates of the
union wage premium when controlling for unmeasured productivity differences to those
obtained when one does not control for such differences. As part of the comparison, the
results of a formal test for the absence of person effects which are correlated with the
union variable are presented.
The second goal of this paper is to test the appropriateness of the standard personeffect
model. Four types of tests are considered. The first is an omnibus test based on
the following idea (Chamberlain (1982)): Consider the wage equation from a cross-section
model which has been augmented by a person-specific effect. If this person effect is
correlated with union status as a point in time, then, in general, it will be correlated
with union status at all points in time. Values of time-varying variables from other
periods enter the equation for period t only through their correlation with the person
effect, and enter the current period equation in a manner that is constant over time.
This produces testable restrictions on an unrestricted reduced form that relates values
of time-varying variables to the value of the wage in each period. Tests based on
these restrictions are carried out both under the assumption of homoscedasticity and
of arbitrary heteroscedasticity.
This omnibus test has the potential to detect misspecification in many directions. Its
generality comes at the cost of potentially low power in detecting misspecification in
particular directions. We also consider two narrower tests of the person-effect model. A
person-specific wage model with time-constant coefficients implies a regression equation
in differences. The difference equation implies restrictions on the coefficients relating the
current wage to the lagged wage and to current and lagged values of the explanatory
variables. Testing these restrictions in an instrumental variables framework leads to a
narrower test of the person-effect model.
The second narrow test considers misspecification in directions not considered by
the omnibus test. The test is obtained by noting that the difference equation implies that
changes in the explanatory variables should have the same effect (in absolute value) on
wage rates whether they are increases or decreases. The change in union status is
decomposed into a variable denoting entering union status and one denoting leaving
union status. If the standard person-effect model is correct, these two variables will have
equal but opposite effects on the change in wage. This specification adds a cross-product
term to the difference equation, and therefore is sensitive to misspecifications that will
not be picked up by the omnibus test or the instrumental-variables approach. Both sets
of narrow tests will be performed under both variance assumptions.
The last test considers a different type of misspecification. In the conventional model
the wage effects of all other explanatory variables are the same for union and non-union
workers. Here we test that hypothesis while allowing for a person effect which is correlated
with union status. For numerical reasons (see below) this test is only carried out under
the homoscedasticity assumption.
The outline of the paper is as follows: Estimation methods and testing procedures are
presented in Section 2. The data are described in Section 3, and the empirical results are
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JAKUBSON THE UNION WAGE EFFECT 973
discussed in Section 4. The null hypothesis of no person effect correlated with the union
variable is strongly rejected, Allowing for such a person effect lowers the estimate of the
union wage premium from 20% to 5-8%. The standard person-effect model is not rejected
by the omnibus test nor by the first narrow test. However, the second narrow test
(decomposing the change in union status) does reject the model. The last test for interaction
effects finds evidence for the hypothesis that the union wage effect is different for different
workers. Non-white workers, younger workers, and workers with less schooling have
larger wage premia. These results suggest the importance of using narrower, but potentially
more powerful, tests as well as omnibus tests to assess model specification. Section 5
concludes the paper.
One final note of caution. This paper presents a battery of tests which are clearly
not independent. Hence the nominal size of any particular test is not the true size. For
purposes of exposition, we refer to a "rejection" when the nominal significance level is
greater than 0-05. However, we would then expect 1 test in 20 to "reject" even if all the
null hypotheses were correct. The reader is invited to combine his or her own priors with
the sample evidence presented here. In particular, one should not overly focus attention
on any particular "rejection".
2. MODEL TESTS
2.1. Basic specification
Consider estimating a standard human capital wage equation (Mincer (1974)) using panel
data. Specifically, assume that N (large) individuals (indexed by i) are observed at T
(small and fixed) time points (indexed by t). The equation for individual i's log wage
in period t takes the form
Yit= tf+f3txit + it i=I,...,N, t=1,..., T. (2.1)
The individual's characteristics in period t have been divided into a q-vector of observed
characteristics f which are fixed over time (e.g. race) and a k-vector of time-varying
characteristics xit. For this study, the most important element of xit is the union dum
variable which is coded one if the wage in period t is covered by a collective bargaining
agreement and zero otherwise. Assume that the disturbance term Eit takes the following
form:
Eit = ytci + uit. (2.2)
The general model is then
Yit = ft f + fXait + ytCi + uit. (2.3)
We assume that uit is uncorrelated with ci, f, and xi = (xi,, .. ., xiT)'. The term ci is
the unobserved person effect (unmeasured productivity). We cannot assume that ci is
uncorrelated with f and xit. In particular, if ci contains productivity differences which
are observed by employers (though not by the researcher), then one expects a positive
covariance between ci and the union dummy, as employers facing higher union wages
respond by hiring more productive workers. The inability to control for ci in a cross
section leads to an upward bias in the cross-section estimate of /t.
The conventional person-effect model in the literature imposes restrictions on (2.3).
The yt's are assumed equal in all periods and the 83t's are assumed constant and equal
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974 REVIEW OF ECONOMIC STUDIES
to p. That model (normalizing the common y to unity) is
Yit = 4tf + 8'xit + ci + uit. (2.4)
The common ,3 is the "union effect" on wages.
Panel data may allow us to control for unmeasured productivity differences (c). For
example, in the conventional model with two years of data, differencing the data (or
equivalently, adding dummy variables for each person) allows consistent estimation of
(4/2-q/1) and p (as N -oo for fixed T).
The consistency of these estimates depends on the assumption that the true model
is given by (2.4). Most previous work has treated (2.4) as the most general model
considered, and has focussed on the bias (if any) which arises from ignoring unmeasured
productivity differences. The focus here will be two-fold. In addition to asking what
bias results from ignoring c, we will also be concerned with the appropriateness of the
specification (2.4). That model places testable restrictions on the process generating wage
rates which we investigate below.
We begin with an omnibus test in the next section, and then proceed to narrower,
but potentially more powerful tests. In the empirical work we present two sets of results
for each test. First, we treat the linear equations as conditional expectation functions,
and use the conventional covariance matrix estimators as weighting matrices in estimation
and testing. Second, we treat the linear equations as projections, on the grounds that the
unknown conditional expectation functions may be nonlinear. (In particular, we know
nothing of the form of E(c Ix).) If they are, the disturbances will be arbitrarily heteroscedastic,
since the variance of the approximation error will depend on the values of the
explanatory variables. In this case we use the methods of Chamberlain (1982) and White
(1982) to construct covariance matrix estimators which allow for arbitrary heteroscedasticity.
Lastly, we construct the test of the hypothesis that other variables have the same
effects for union and nonunion workers.
2.2. Omnibus test
Assume for simplicity that, conditional on ci, wages depend only on union status, so that
xit is a scalar and f drops out of (2.3). (This assumption is relaxed in the empirical work
below.) Then (2.3) becomes
Yit = ,txit + ytci + Uit. (2.5)
The bias arises because ci is correlated with xit. If ci is correlated with x,,, in general it
will be correlated with xis, s = 1, . . ., T. Formalize this notion as
Ci = AlXil + * ** + ATXiT + ei = Ak xi + ei, (2.6)
where A = (A1,.. ., AT)' is the vector of partial correlation coefficients and ei is a distur
ance which is orthogonal to xi by construction. Substitute for ci in (2.5) to obtain an
estimating equation based on observables:
yit= =3txit + ytA'xi + (ytei + Uit). (2.7)
A convenient scale normalization is yi = 1. For each period, (2.7) im
yit onto all observed values xi with restrictions on the coefficients over time. The
unrestricted reduced form of the model is
yi = xIxi + ei, (2.8)
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JAKUBSON THE UNION WAGE EFFECT 975
where Yi = (Yil,... , YiT)'- II is the T x T matrix of projection coefficients, and ei is the
T-vector of disturbances. The model in (2.7) implies the nonlinear restrictions
H = diag {138, ... .,T} + YA', (2.9)
where y = (yl,..., YT)'. That is, lTts =13t + ytAt for s = t, and ytAs for s $ t.
Equation (2.9) suggests a method for estimating the parameters f3, A, and y and
testing the implied restrictions. First estimate the reduced form I. Then use a method
of moments estimator to obtain ,3, A, and y using (2.9). This amounts to restricted GLS.
Finally, test the validity of the person effect model by testing the overidentifying restrictions.
The test is an omnibus test in that rejection does not imply a specific alternative,
since the test is against an unrestricted reduced form. The test of the null hypothesis that
the unobserved person effect is uncorrelated with union status is a test of A =0.
Note that in the usual person effect model the restrictions are linear. There (2.4)
implies that
I = /3I + lA'. (2.10)
where 1 is a T-vector of ones. In contrast, (2.9) allows departures in two economically
interesting directions. If macroeconomic conditions affect the extent to which employers
can select high productivity workers or the return to this productivity we would expect
yt $ 1. (Note that if yt $ 1 the usual difference estimator is inconsistent.) Alternatively,
if wage bargains are affected by the business cycle then ,3 may vary over time.
2.3. Instrumental variables test for misspecification in a particular direction
The omnibus test above may not be very powerful, particularly if the II matrix is imprecisely
estimated. We now consider an alternative approach which has a natural solution via
the use of instrumental variables. In essence, the test of whether the conventional
person-effect model (2.4) is valid is a test of whether a differencing procedure is valid.
This test has the advantage of providing linear, as opposed to nonlinear, restrictions and
should be easier to implement.
To ease the exposition, retain the assumption that xit is a scalar and there are no f
terms. We suppress the person-subscript i for notational clarity. In equation (2.5) solve
for c in the equation for Yt-i and substitute this expression into the equation for Yt to obtain
Yt = f3txt + (Yt/ Yt-X)Yt-i - (Pt-iYt/ Yt-J)Xt- + [ut - (Yt/ Yt-) Ut-i]. (2.11)
To set scale let yi = 1. This equation must be estimated using instrumental variables (o
a similar technique) since Yt-i is correlated with ut-1. Begin with an overparameterized
version of (2.11) and estimate
Yt = (Pl tXt + (P2tXt-1 + (P3tYt-1 + 77t, t = 2,..., T, (2.12)
where (p, t = 36t, (P2t = - (8t-l yt/yt-1), and (p3t = (yt/yt-1). Estimate this system usin
stage least squares (3SLS) where all values of x are used as instrumental variables for
the lagged y term. At this point we test the over-identifying restrictions of the simultaneous
equations model (i.e. the exclusion restrictions).
The next step is to test the y's. If they are not all equal (to 1), then there is no clean
interpretation of the regression which uses the change in y as the dependent variable.
The test of whether the coefficients on the lagged y terms above are unity is equivalent
to testing yt = 1 in the omnibus test framework.2 While in the omnibus test framework
2. If the asymptotics involved T as well as N we would be testing a unit root, with the concomitant
problems. Since the asymptotics here only involve N, there is no such problem.
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976 REVIEW OF ECONOMIC STUDIES
we required nonlinear methods to estimate the yt's here we can use linear methods to
estimate ratios of the yt's. The use of the change in y as the dependent variable is justified
if we fail to reject this hypothesis.
The conventional person effect model (2.5) further implies that the coefficients on
current and lagged x are equal in absolute value but opposite in sign,
Yt -=83xt -f3xt1 + Yt-i + [ut- ut-l]. (2.13)
This is simply the difference equation described above. If the yt's are all unity but the
183 's vary over time, then the coefficient on lagged y will be unity, but the coefficients on
current and lagged x will not be equal but opposite.
2.4. A nonlinear specification: decomposing the change in x
The third test changes the basic estimating equation to include cross-products in the
union variables. If xit never changes it is perfectly collinear with ci and we cannot est
,l. The union variable is discrete and can change in only two ways. An uncovered
might become covered by a collective bargaining agreement, or a covered worker may
no longer be covered. We examine the possibility that the two events do not have the
same impact on wage rates.
Decompose the change in x into the following:
ENTERt = 1 if xt = 1 and xt-, = 0; 0 otherwise. (2.14a)
LEAVEt = 1 if xt = 0 and xt-, = 1; 0 otherwise. (2.14b)
STAYt = 1 if xt = 1 and xt- = 1; 0 otherwise. (2.14c)
The fourth possibility, that the individual was never covered by a union contract, is
absorbed into the intercept. The equation of interest is then
Ayt = alt ENTERt + a2t LEAVEt + a3t STAYt + A/Lit. (2.15)
where Ayt = Yt -Yt-i and Aut is defined similarly. An equivalent specification is
Ayt = 7r1xt-1 + r2Xt + r3Xtxt-1 + Aut, (2.16)
where a, = r2, a2 = r1, and a3 = rl + r2+ r3 . If the specification
the restriction that the coefficients are constant over time)., then the restrictions are
alt =-a2t =Pl3 t = 2,...,I T. (2.17a)
a3t-O, t=2,..., T (2.17b)
Note that this test could be used with a continuous x variable as well by separating the
change in x into increases and decreases. Then test whether the coefficient on the increase
in x is equal in absolute value but opposite in sign to the coefficient on the decrease in x.3
2.5. Do the effects of other regressors vary with union status?
The last test is whether the effects of other regressors vary with union status. If c is
correlated with union status, then it is also correlated with interactions of union status
3. In principle one could begin with an unrestricted reduced-form coefficient matrix H1, as in the omnibus
test. This would entail entering all possible interactions of the five union variables freely into each of the five
unrestricted cross-section equations, as opposed to just entering the levels and estimate 153 (31 interaction
times 5 equations) free coefficients for the portion of the 1H matrix corresponding to the union variables. This
procedure is not feasible with the data used here. The empirical work below does not implement this test until
after examining the omnibus test and the instrumental-varibles test. These tests suggest that the decomposition
proposed is a natural direction in which to proceed.
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JAKUBSON THE UNION WAGE EFFECT 977
and other regressors. We use the methods of sections 2-3 and 2-4 here, adding interaction
terms to the model while also allowing them to be correlated with c.
3. DATA
The data are a sample of male heads of household drawn from the Panel Study of Income
Dynamics (PSID) from the years 1976 though 1980. We included men who were part of
the representative portion of the PSID, not disabled, between the ages of 20 and 60 in
1976, either black or white, and worked for money income in each of the years 1976
through 1980. We excluded those who were self-employed, had labour income from more
than one source, or were paid by some method other than wage or salary, in order to
construct a clean wage variable. These selection criteria produced a sample of 588 men.
Twenty per cent of them experienced a change in union status over the sample period.
A full description of the data appears in an appendix (available on request). The
variables used in the analysis are the following:
LW,: logarithm of the hourly wage rate (in cents) in year t.
U': 1 if the wage in year t was set by collective bargaining; 0 otherwise.
SM': 1 if living in an SMSA in year t; 0 otherwise.
AGE: age in years in 1976.
SCHL: years of schooling.
RACE: 1 if black, 0 if white.
SOUTH1: 1 if living in the South in 1976; 0 otherwise.
Appendix Table Al contains descriptive statistics.
4. EMPIRICAL RESULTS
4.1. Simple cross section
Table 1 displays the results treating each year of the panel as a separate cross-section
and fitting the least-squares projection of LW,t onto fJ and xi,. These estimates provide
the benchmark against which we compare the panel estimates below, and permit us to
determine the extent of the bias in the cross-section estimate which ignore the person
effect c,.
TABLE 1
Simple cross-section OLS regressions of log wage on current union status
(N = 588) (standard errors in parentheses)
Ul U2 U3 U4 U5
LWI 0163
(0-025)
LW2 0-198
(0 024)
LW3 0-177
(0 025)
LW4 0-171
(0 027)
LW5 0-184
(0 026)
Note: All equations also include age, age-squared, schooling, schooling-squared, age x schooling, race, Southl,
current SMSA, and a constant.
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978 REVIEW OF ECONOMIC STUDIES
We display only the coefficients associated with the union variables in all tables.
(Complete results are available on request.) The time notations one to five correspond
to the years 1976 to 1980. The other variables in the equations are a full quadratic in
age and schooling, race, South 1, SMt, and a constant.4
Tlhe union coefficients range from 0-163 to 0-198, with a simple mean of 0-179. The
cross section estimate of the union wage premium is 20% (exp {0 179} -1 = 0196),
comparable to those of most cross section studies. If the sorting story were correct,
unmeasured productivity is positively correlated with union status, and we expect the
panel estimate to be smaller.
4.2. Omnibus test: restricted GLS
The first step is estimating the unrestricted reduced form by projecting each (log) wage
onf and all leads and lags of x. Appendix Table A2 presents the portion of the unrestricted
II corresponding to the union variables. If leads and lags of union status are unimportant
in wage determination the off-diagonal elements of II should be zero.
TIhe reduced form disturbances in (2.8) are likely to be serially correlated even if the
u's are i.i.d. since eit = Yti + uit. If the u's were i.i.d. one could impose the factor structure
on the disturbance covariance matrix when estimating II, yielding greater precision in
the estimates. Alternatively, one could allow the u's to be freely correlated and therefore
allow the e's to be freely correlated over time. This approach prevents misspecification
of the disturbance covariance structure from generating inconsistencies in the slope
coefficient estimates.
Our focus is the slope coefficients. Since the cross-section dimension (N) is large
and the time-series dimension (T) is small, and since the asymptotics rely on large N
for fixed T, the latter approach is safer, and is the one adopted in all the empirical work
below. This approach is in the same spirit as allowing for arbitrary heteroscedasticity in
the reduced-form disturbances.
Table 2 displays the results of imposing restrictions on II using minimum-distance
techniques. The first three panels of the table use the conventional (homoscedastic)
covariance matrix of II as the norm and the second three use the covariance-matrix
estimator which allows for arbitrary heteroscedasticity. The first panel imposes the
restrictions of the omnibus test for the presence of a person effect (2.9). This is the most
general model considered. There is no evidence against the restrictions (X2(11)= 923).
The estimated union effects (,3's) range from 0065 to 0 093 and the 's range from 0-852
to 1-474. Similar results obtain when we allow for arbitrary heteroscedasticity in the
fourth panel. The 13's range from 0031 to 0K119 and the 9's range from 0-847 to 1-334.
In both cases there is no evidence against the restrictions (X2(1 1) =923 under homoscedasticity
and 11-57 with heteroscedasticity).
The second and fifth panels display the results for a standard person-effect model
(2 10) in which the y's are all unity and the 13's are all equal. The estimated union effect
is 0084 (with a standard error of 0015) under homoscedasticity, about half the average
cross-section estimate from Table 1. With heteroscedasticity the estimate is somewhat
smaller, 0 055 (0019), about one fourth the average cross-section estimate. In both cases
the models are consistent with the reduced form (X2(19) = 1546 under homoscedasticity
and 20 12 with heteroscedasticity). The test statistic for the incremental restrictions is
4. If the region variable varies over time there is not enough variation to estimate the GLS model with
heteroscedasticity. The homoscedastic results are not sensitive to the treatment of the region variable or the
SMSA variable. In the empirical work the SMSA coefficients vary freely.
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JAKUBSON THE UNION WAGE EFFECT 979
TABLE 2
Omnibus test for the presence of c based on the estimated H from Table A2: restricted GLS
(standard errors in parentheses)
Conventional GLS: Homoscedasticity Extended GLS: Arbitrary Heteroscedasticity
2.1. Restrictions from (2.9): X2(11) =9-23 2.4. Restrictions from (2.9): X2( 1) = 11-57
13 P2 P3 P4 P5 P1 P2 P3 P4 P5
0-081 0-093 0 074 0-065 0-087 0 074 0-119 0-050 0-031 0-062
(0-028) (0.028) (0-033) (0-021) (0-035) (0-031) (0-031) (0-040) (0-022) (0.027)
Yi Y2 Y3 Y4 Y5 Yi Y2 Y3 Y4 Y5
1-000 0-852 1*334 1*474 1-221 1*000 0-847 1-223 1-334 1-117
(0-210) (0-252) (0.357) (0-191) (0-200) (0-192) (0.298) (0-187)
Al A2 A3 A4 A5
0-018 0-022 0'045 0-012 0-018 0-028 0-027 0-058 -0-010 0 043
(0.047) (0-050) (0.056) (0-044) (0.047) (0.047) (0.041) (0-051) (0-050) (0-041)
2.2. Restrictions from (2.10): X2(19) = 15-46 2.5. Restrictions from (2.10): X2(19) = 20-12
P 1P
0-084 0-055
(0-015) (0-019)
A1 A2 - A3 A4 A5 Al A2 A3 A4 A5
0-016 0-024 0-045 0.011 0-019 0-026 0-028 0-052 -0-007 0-040
(0-049) (0-054) (0-054) (0-049) (0-49) (0-047) (0.044) (0-047) (0-045) (0-042)
2.3. A = 0: X2(24) = 30 54 2.6. A = 0: X2(24) = 42-58
0-111 0-109
(0-013) (0-015)
the difference between the x2 statistics, and shows no evidence against the restrictions
fA =,f and ,= 1.
The third and sixth panels display the results for the conventional random-effects
model (Balestra and Nerlove (1966), Maddala (1971)) which assumes c is uncorrelated
with union status (A = 0). In this model the cross-section estimates in Table 1 are consistent
though not efficient. We obtain virtually identical estimates in both cases: 3 = 0 111
(0.013) under homoscedasticity and 0K109 (0.015) with heteroscedasticity. The estimated
union effect is the optimal (matrix-) weighted average of the cross-section estimates under
the null hypothesis A = 0, though it is considerably smaller in magnitude than the crosssection
estimates.
There is strong evidence against the hypothesis A = 0. The incremental x2(5) statistics,
against the model in (2.10), take the values 15 08 (homoscedastic) and 22X46 (heteroscedastic).
A Hausman-type test of the null hypothesis that the 8's are equal yields a X
statistic of 13X02 (homoscedastic) and 21X44 (heteroscedastic).
4.3. A test of differencing: instrumental variables
We now implement the instrumental-variables procedure from Section 2.3 which provides
both a test of the person-effect model as well as a clean estimate of the union effect. We
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980 REVIEW OF ECONOMIC STUDIES
use the equation for the previous year's wage to solve for c, and substitute into the
equation for current wage to get an equation giving the current wage as a function of the
lagged wage and current and lagged union status in equation (2.11). Consider the
unrestricted version of this system in equation (2.12). If the coefficients are constant over
time, and if the standard person-effect model is correct, then the coefficient on the lagged
wage is unity and the coefficients on current and lagged union status are equal in absolute
value but opposite in sign. This is the model of equation (2.13), which is essentially a
difference equation. This specification is equivalent to the standard person-effect model
of equation (2.5).
Estimation of (2.12) must account for the correlation between the lagged wage and
the disturbance term. Three-stage least-squares (3SLS) procedures are used, treating the
lagged wage as endogenous, and using the union status at other points in time as
instrumental variables.5 There are two versions of 3SLS, corresponding to the two versi
of GLS used in the omnibus test. Conventional 3SLS (83) assumes homoscedascitity in
the disturbances. "Extended 3SLS" allows for arbitrary heteroscedasticity. There are
two (asymptotically equivalent) computational methods for extended 3SLS. The first
(6G3) uses the residuals from conventional two-stage least-squares (2SLS) in order t
compute the weight matrix used in the minimization problem. The second (G3) uses
the residuals from heteroscedastic 2SLS to compute the weight matrix. (For details, see
Chamberlain (1982).)
The unrestricted 3SLS estimates are contained in Appendix Table A3. They show
no evidence against the 20 overidentifying restrictions. The X2(20) statistics take the
values 16-98, 13-85, and 12-62 for 83, SG3, and 8G3, respectively. There is general
agreement among the three estimators on the signs and orders of magnitude of the
estimated coefficients. Hence, in the rest of this section we will maintain the hypothesis
that the overidentifying restrictions are correct. To test against the unrestricted reduced
form one would add the chi-squares above to those of the incremental tests. In all cases
below for which the restrictions are not rejected, the test against an unrestricted reduced
form also does not reject.
If there is a time-invariant person effect in the wage equations, then the coefficient
on the lagged wage should be unity in each of the four equations. Imposing only the
restriction that '3p(= Yt!/ Yt-) from equation (2.12) is unity for each t results in incremental
x2(4) statistics of 4-86, 4-62, and 5 79 for A3, 8G3, and 3, respectively (table not shown)
Therefore there is a clean interpretation to a regression with the wage change as the
dependent variable.
We impose restrictions on the unrestricted 3SLS coefficients in a series of steps.
These results appear in Table 3. Table 3.1 displays the results of imposing the restriction
that the 3SLS coefficients are constant over time, that is, that 5Pt, P2t, and 9p3, in equation
(2.12) are equal to pl, P2, and 03, respectively. There is no evidence against these
restrictions. The incremental Y2(9) statistics take the values 11 98, 12 15, and 13-17 for
83, SG3, and 8G3, respectively. The coefficient on the lagged wage is approximately unity,
and the coefficients on current and lagged union are close in absolute value.
Table 3.2 presents the results for the difference equation (2.13) which imposes these
restrictions formally. The added restrictions here are that the coefficient on the lagged
wage is unity and the coefficients on current and lagged union status are equal in magnitude
5. The varibles in the f vector are allowed to enter freely into each of the equations. The equations also
contain current and lagged SMSA variables, and the SMSA variables from other points in time are used as
instrumental variables. All results presented here and below are insensitive to the treatment of the SMSA
variables.
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JAKUBSON THE UNION WAGE EFFECT 981
TABLE 3
Restricted 3SLS estimates: restrict coefficients from Table A3
(standard errors in parentheses)
3.1. Coefficients constant over timeA A A
63 803 0G3
LWt-I 1-100 1P053 1*058
(0-092) (0 075) (0.066)
Ut 0-077 0-049 0-044
(0-015) (0.019) (0-018)
Ut_l -0-092 -0-056 -0 055
(0.019) (0-021) (0.021)
x2(9) 11 -98 12- 15 13- 17
3.2. Change model
83 8G3 BG3
U 0-081 0 050 0-046
(0-014) (0-018) (0-018)
x2(11) 13-96 13-46 14-00
Note: Column headings: 83 = conventional 3SLS (homoscedasticity).
CG3 = extended 3SLS (heteroscedasticity) with
residuals from conventional 2SLS used as norm. 8G3*
extended 3SLS with residuals from extended 2SLS as norm.
but opposite in sign. The test against the 3SLS model appears at the bottom of the table,
so the incremental X2(2) statistics take the values 1-98, 1X31, and 0-83 for 83, SG3, and
8*3, respectively. The estimated union coefficient is 0-081 (0.014) for the model which
assumes homoscedasticity (83) which is very similar to the GLS estimate in Table 2.2 of
0-084 (0.015). Allowing for arbitrary heteroscedasticity yields estimates of the union
effect of 0-050 (0.018), and 03046 (0.018) for 8G3 and 83, respectively. These are similar
to the restricted GLS estimate in Table 2-5 of 0 055 (0.019).
These results suggest that the usual fixed-effects approach to this problem is valid.
The fixed-effects estimator uses the deviations from individual-specific time-means, and
regresses the deviations in the wage on the deviations in the union variable. For these
data, this procedure results in an estimated union coefficient of 0-083, with a standard
error of 0-020, which is similar to the others derived under the homoscedasticity
6
assumption.
4.4. Entrants versus leavers: a nonlinear specification in x
The results thus far indicate that the data are consistent with a model with an unobserved
person-specific effect which is correlated with the union status variable. Further, there is
no evidence against the standard person-effect model (equation (2.4)) in which a regression
of the change in wage on the change in union status yields a clean estimate of the union
effect, purged of the omitted variables bias which affects the cross-section estimate. This
6. These equations also take deviations of other time-varying variables. We imposed the restriction that
the coefficients on the time-invariant explanatory variables were constant, so that they dropped out of the
equation. The estimate shown is from an equation in which the SMSA variable was taken as deviations from
time-means.
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982 REVIEW OF ECONOMIC STUDIES
section presents the results of a narrower test of the specification, detailed above in
Section 2.4, in which we decompose the change in union status into indicators for entering
covered status (ENTER) and for leaving it (LEAVE), and ask whether their effects on
the wage change are symmetric.
We employ the same general strategy as above. Appendix Table A4 displays the
estimates of the wage change equations (2.15) for years 2 to 5. These equations allow
current and lagged SMSA to enter freely into each change equation. (The results are not
sensitive to the treatment of the SMSA variable.) The point estimates suggest a positive
union wage premium. The coefficients on ENTER are positive, while those on LEAVE
are negative. The coefficients on STAY are not significantly different from zero in three
of the four equations.
Table 4 displays the results of imposing restrictions on the wage change coefficients.
There are two columns. The first displays the results assuming homoscedasticity, while
the second displays the results when we allow for arbitrary heteroscedasticity.
We impose the restriction that the wage change coefficients are constant over time
in the first panel of the table. This is the restriction that a,,, a21, and a31 in equati
(2.15) are equal to al, a2, and a3 respectively. There is no evidence against the restrictions.
The x2(9) statistics take the values 13O00 and 10O00 for the conventional GLS and the
extended GLS, respectively. The coefficient on STAY is not different from zero, either
statistically or substantively, in either case. The second panel of the table constrains the
coefficient on STAY (a3) to zero. There is no evidence against the restriction, and the
coefficients on ENTER and LEAVE do not change. Both conventional and extended
GLS yield similar estimates of the coefficients on ENTER (a,) and LEAVE (a2). The
TABLE 4
Restricted wage change equations: impose restrictions on coefficients in
Table A4
(standard errors in parentheses)
Conventional GLS Extended GLS
4.1. Coefficients constants over time
ENTER 0-117 0*107
(0-107) (0-029)
LEAVE -0-037 -0-026
(0-017) (0 025)
STAY 0-001 0-001
(0.005) (0?005)
X 2(9) 13-00 10*00
4.2. Add restriction that coefficient on STAY is 0
ENTER 0-116 0-106
(0-017) (0-029)
LEAVE -0-037 -0-027
(0-017) (0.025)
x2(10) 13-02 10-07
4.3. Add equal but opposite restriction
UNION 0-076 0-060
(0-013) (0-020)
x2(11) 27-74 14-87
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JAKUBSON THE UNION WAGE EFFECT 983
ENTER coefficients are 0-116 (0.017) and 0-106 (0.029), respectively, while the LEAVE
coefficients are -0-037 (0-017) and -0-027 (0-025), respectively.
The conventional person-effect models implicitly constrain the coefficients on ENTER
and LEAVE to be equal in magnitude but opposite in sign. Table 4.3 presents the results
of constraining a, = -a2- Maintaining the hypothesis that the coefficients are constan
over time and that a3 = 0, there is evidence against the equal but opposite restriction.
The incremental Y2(1) statistics take the values 14X72 and 4-80 for the conventional and
the extended GLS, respectively. Maintaining the hypothesis that the restriction is correct,
the estimated union coefficients are 0-076 (0-013) and 0-060 (0.020), respectively, which
are very similar to the results from the omnibus test and the instrumental variables
procedures.
There is some disagreement between the conventional and the extended GLS. The
chi-square statistic at the bottom of Table 4.3 is a test of the restricted model against the
unrestricted alternative in Appendix Table A4. The conventional GLS shows clear
evidence against the model (X2(11)= 27 74), while the extended GLS does not (Y2( 1)=
14. 87). The sum of the ENTER and LEAVE coefficients (a, + a2) in the second panel
of the table is 0 079 (0-021) for conventional GLS and 0-079 (0.036) for extended GLS.
We conclude that the bulk of the evidence is against the restriction.
4.5. Testing for interactions
The final specification test asks whether the effects of the other regressors varies with
union status, so that the appropriate specification would include interactions between
union status and those variables. To be concrete, suppose that the effect of race varies
with union status. The appropriate specification then includes union and union x race.
If union is correlated with the unobserved c, so is union x race. We utilize both the GLS
approach of Section 2.3 and the instrumental variables approach of Section 2.4 to address
this question. We add interactions between union status at each point in time and the
time-invariant regressors (age and its square, schooling and its square age x schooling,
race and south) to the model. These models are only estimated under the homoscedasticity
assumption since the covariance matrix under the heteroscedasticity assumption was
numerically singular, ruling out minimum distance tests.
Table 5 displays the results of these experiments. The first two columns are GLS
results using the method of Section 2.3. The unrestricted reduced form (coefficients not
shown) contains interactions of all union variables with the time-invariant regressors in
addition to all levels of the union and SMSA variables. In the first column we display
the f3 coefficients from a model comparable to Table 2.2 in which all interaction terms
are set to zero and the fl, A structure is imposed on the union coefficients. There is
evidence against the model (X2(194) = 316-5) though ,3 of 0-083 (0-013) is quite close to
the results in Table 2.2 of 0-084 (0-015). In the second column we display the 8's when
we impose the f3, A structure on the interaction terms as well. There is evidence against
the restrictions, as compared to the unrestriced reduced form (X2(152) = 244X7). The 's
are estimated quite imprecisely, with the exception of the union x race interaction.
The third column displays the "3's using the instrumental variables approach of
Section 2.4. The exclusion restrictions from the "unrestricted" 3SLS model are consistent
with the data (y2(92) = 111 1, table not shown). The incremental restrictions of the change
model are not consistent with the data (X2(60) = 155-5), however. Similarly to the GLS
results, the f3 's are estimated rather imprecisely.
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984 REVIEW OF ECONOMIC STUDIES
TABLE 5
Interactions Between Union Status and Other Explanatory Variables
(standard errors in parentheses)
(1) (2) (3)
U 0-083 0-277 -0-092
(0-013) (0.346) (0.355)
U x age -1-198 -0-765
(1-132) (1-150)
Uxage2 0 073 0-077
(0-113) (0-113)
Uxschool 0-184 0-663
(0-361) (0.367)
U x school2 -0-169 -0-311
(0.119) (0-118)
U x age x school 0-299 -0.101
(0 466) (0.474)
U x race (0-078 0-066
(0 030) (0.030)
U x southl -0 004 0-021
(0.003) (0.032)
X2 316.5 244-7 155-5
df 194 152 60
Notes:
(1) GLS model; zero-out all interaction terms. fl, A on U.
(2) GLS model. fl, A structure on U and interaction terms.
(3) 3SLS model restricted to be a change model.
x2 is against an unrestricted reduced form for columns (1) and (2). x2 is
against the "unrestricted" 3SLS model for column (3). The "unrestricted"
3SLS model had X2(92) = 111 1.
We calculate the implied union wage effect for five different data configurations below.
In all five we set age and schooling to their mean values in the sample. In the first we
set race and south to their sample means. In the second we set race and south to zero,
and in the third and fourth we alternately set race and south to one, and in the fifth we
set both to one. The implied union wage effects are:
Data configuration GLS IV
mean age, school, race, south. 0X075 0X074
mean age, school. race = 0. south = 0. 0 050 0 040
mean age, school. race= 0. south = 1. 0X046 0X061
mean age, school. race = 1. south = 0. 0 128 0 106
mean age, school. race = 1. south = 1. 0-124 0-127
The results in the first row are similar to the GLS and instrumental variables results
(under homoscedasticity) in Tables 2 and 3. This is to be expected, since the single ,3 in
the models above refers to the union wage effect for a randomly chosen individual. We
find that the union wage premium is higher for non-whites. At the mean age, the union
effect is higher for those with lower levels of schooling. At the mean schooling level, the
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JAKUBSON THE UNION WAGE EFFECT 985
union effect is higher for younger workers. These are consistent with the literature (Lewis
(1986)).
5. DISCUSSION
5.1. Size of the union wage effect
This paper has two goals. The first is to provide estimates of the union wage effect which
are free of an omitted variables bias due to the presence of an unmeasured person-specific
effect which is correlated with the union variable. That person effect may be thought of
as unmeasured productivity characteristics which are observed by the employer but not
by the researcher. A rational employer might respond to higher union wages by raising
hiring standards, leading to a positive covariance between the unmeasured productivity
characteristics and the union status variable. That covariance will impart an upward bias
to cross section estimates of the union wage effect.7
Cross-section estimates of the union wage effect are on the order of 20%. When we
allow for a person effect which is correlated with the union variable, the estimate drops
to the 5-9% range. Table 6 presents a summary of the various estimates of the union
wage effect. One-half to three-fourths of the union effect estimated in the cross section
may be attributed to the existence of a person effect. There is evidence that the union
effect varies among different types of workers, even after controlling for unobserved
individual-specific effects. Younger workers, less educated workers, and non-whites gain
more from union coverage, even after controlling for c.
TABLE 6
Summary of estimated union coefficients
(standard errors in parentheses)
Homoscedastic Heteroscedastic
OLS-GLS 83 OLS-GLS 8G3 SG3
1. Simple cross-sectiona 0179
2. Fixed effectsb 0-080
(0-020)
3. Omnibus testc 0-084 0055
(0-015) (0-019)
4. Instrumental variablesd 0-081 0.050 0-046
(0.014) (0-018) (0-018)
5. Wage changee 0-076 0-060
(0-013) (0-020)
6. Interactions at meansf 0-075 0-074
Notes: a Table 1. b From text. c Table 2. d Table 3 e Table 4. Wage change model is rejected by the data.
f From text, derived from Table 5.
Both the point estimates of the union effect and the size of the reduction as compared
to the cross-section estimate are similar to those found by Chamberlain (1982) using NLS
data for young men. His sample contained a larger number of men, though only three
7. Note that this is not the only motivating story for the person effect. If more able workers choose union
jobs, the empirical implications would be the same.
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986 REVIEW OF ECONOMIC STUDIES
years of data each. The tests of the person-effect model become sharper with more years
of data, since there are considerably more restrictions.8 The results are also similar to
those of Brown (1980) and others who have used these techniques (see Lewis).
5.2. Appropriateness of the usual person-effect model
The second goal of this paper is to test the appropriateness of the person-effect model
most commonly employed in empirical economics. Four tests are suggested. The first is
an omnibus test for the presence of a person effect which is based on restricting the way
past and future union variables enter the equation for the current wage. The second is
a narrower test which uses instrumental-variables techniques to overparameterize and
then test a model of differencing. The third introduces a nonlinear specification of the
union varibles, and decomposes the change in union status into two components. It then
tests whether the two components have symmetric effects on the wage change. The fourth
is a test of whether interaction effects belong in the model.
The first two tests provide no evidence against the usual person-effect model. The
estimated coefficients are similar across the tests, and tests against unrestricted reduced
forms fail to reject. When the nonlinearity in the explanatory variables is introduced,
however, there is evidence against the usual model. There is also evidence in favour of
the hypothesis that the effects of other regressors vary with union status. For these reaso
the estimated union effects should be interpreted cautiously.
5.3. Comparison to conventional IV solutions
The conventional solution in the case of a suspected endogenous explanatory variable is
the use of instrumental variables. For purposes of comparison we estimated two sets of
IV models, treating the union status variable as endogenous. In the first set we use a
linear predictor for union and in the second we use the predicted probability from a
probit model. The auxiliary equation (not shown) contains a third-order polynomial in
age and schooling, all SMSA variables, race, south and interactions of age and schooling
with race and south.
The results using a linear predictor appear in columns (1)-(3) of Table 7. In the
first column we use single-equation methods, while in the second column we treat the
wage equations as a system, allowing the union coefficients to vary over time. In the
third column we constrain the union coefficients to be equal over time in the wage system.
The estimated 8's are significantly larger here than in the models above. The single
equation results are larger than the cross-section estimates. When system methods are
used the 8's drop on the order of I, and when constrained to be equal the estimate is
similar to the cross-section average.
The results are basically similar when we use a probit predictor of union status in
columns (4)-(6). The single-equation estimates are larger than those with the linear
predictor, while the system estimates are smaller. The constrained estimate in column
(6) is smaller than that obtained with the linear predictor, but is still much closer to the
cross-section estimate than to the panel-data estimates above.
8. Consider the omnibus tests. In testing (2.9) against an unrestricted reduced form, there are T2- 3 T
restrictions. This test is not feasible when T = 3. In testing (2.10) against an unrestricted reduced form, there
are T2- T - 1 restrictions. This leads to 5 restrictions when T = 3 as compared to 19 restrictions when T = 5.
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JAKUBSON THE UNION WAGE EFFECT 987
TABLE 7
Conventional Instrumental Variables Results Treating Union as Endogenous
(Standard errors in parentheses)
(1) (2) (3) (4) (5) (6)
Union 0-214 0-164
(0.089) (0-078)
U1 0-328 0-238 0-376 0-208
(0.156) (0.124) (0.147) (0.110)
U2 0-315 0 186 0-343 0-160
(0-134) (0100) (0-135) (0-092)
U3 0 358 0 243 0 392 0 190
(0.144) (0.110) (0-139) (0.096)
U4 0-290 0.188 0-327 0-147
(0 147) (0.116) (0.145) (0.104)
U5 0.281 0-246 0-273 0-142
(0.142) (0.109) (0.136) (0.096)
Notes: (1) Linear IV, single equation methods (2) Linear IV, system method. (3) Linear, IV,
system method, coefficient constrained equal across years. (4) Probit IV, single equation methods.
(5) Probit IV, system method. (6) Probit IV, system method, coefficient constrained equal across
years.
Instrumental variables are a full third-order polynomial in age and schooling and interaction of
age and schooling with race, region, and SMSA.
Why are these estimates so different from the panel-data estimates? In the panel
models above we do not assume that the time-invariant variables, for example, age, are
uncorrelated with the unobserved c. We simply cannot separately identify the "structural"
effect of age and its correlation with c. If age is correlated with c, then the instrumental
variables used here are not valid instruments, and the resulting parameter estimates are
inconsistent. The difference between the conventional IV results and those of the panel
models suggests that this is the case. In general, then, it will be difficult to obtain consistent
estimates of the union wage effect using instrumental-variables methods (at least in the
cross section).
5.4. Caveats
The panel-data methods employed here require that the sample contain individuals wh
union status changes over time. There are two potential problems in basing the estimate
of the union wage effect on changers. The first is measurement error, and the second is
the implicit assumption that union status changes are random.
It has been argued (e.g. Mincer (1983), Freeman (1984), and Chowdhury and Nickell
(1985)) that many of the ".observed" union status changes are not true changes, but
merely the result of measurement error. Mincer argues that observed changes in union
status which occur without a job change are likely to be spurious.9 Freeman uses outside
9. We attempted to address this problem with these data by constructing a dummy variable indicating a
job change (JOBCHG) and redoing the analysis of Section 4.3 using ENTER, LEAVE, STAY, their interactions
with JOBCHG and JOBCHG. The interactions of the status change variables with the job change dummy
should yield better estimates of the union effect if this kind of measurement error were a problem. Unfortunately,
the cell sizes were very small and the estimates unstable. The pooling hypothesis that the coefficients are
constant over time was strongly rejected. We have little confidence in the results and do not display them here.
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988 REVIEW OF ECONOMIC STUDIES
information (gathered apart from the household survey which generates the individual
data) from the employer and concludes that there is measurement error in the union
status variable. Estimates of the union wage effect using only the sample for which worker
and firm responses agree are larger than the usual within-sample estimates. Chowdhury
and Nickell parameterize both the measurement-error process and the process generating
the true union variables. They use instrumental-variables procedures and undo the fixed- T
bias in the resulting within estimator. They too find that allowing for measurement error
leads to much larger estimates of the union wage effect.
In other work with these data (Jakubson (1986)) we examine the possibility that
measurement error is driving these results. There a model is derived which nests both
the measurement-error models of Freeman and of Chowdhury and Nickell as well as the
person-effect model as special cases. That model allows one to determine the extent to
which unobserved heterogeneity, as opposed to measurement error, is responsible for the
difference between the panel and cross-section estimates. The results there suggest that
while there is evidence of measurement error in the union status variable, adding measurement
error to a model with person effects results in only a small increase in the estimate
of /3 while adding a person effect to a model of measurement error results in a large
decrease in the estimate of 86. Hence most of the difference is ascribable to unobserved
heterogeneity, which is the focus of this paper.
The second potential problem is the implicit assumption that union status changes
are random. Changes in status generally require job changes, since certification and
de-certification elections are infrequent events. If job changes depend on the wage, as
in a matching model (Jovanovic (1979), Flinn (1982)), then the change in union status
is endogenous and the estimates here are subject to the usual simultaneous-equations bias.
This argument suggests that the exogeneity of union status, conditional on c, may
not hold. To address this point, we also implemented the instrumental-variables procedure
in Section 2.3 using only past values as instruments. The results were quite similar to
those presented above. The evidence, combined with the failure to reject the overidentifying
restrictions of the 3SLS model suggests (Appendix Table A3) that forcing all the
endogeneity to appear through c is not a bad approximation in these data.'?
Finally, there is an issue of interpretation. According to the employer selection story,
employers see a queue of applicants for union jobs, and pick those with high values of
c (unmeasured productivity characteristics). If the union wage effect were the same for
all people, then eliminating c from the wage equation would lead to a clean estimate of
that effect.
However, if the union effect, say b, varies among people, then we would expect that
only .those people with high values of b would enter the queue for union jobs. The
estimates of the union effect above, then, do not estimate E(b), but rather E(b I b > b*),
where b* is the minimum value of b for which an individual joins the queue. The estimates
at best refer only to those who would be better off in a union.
10. We also took another approach to this issue. To the extent that union status changes are upgrading,
either because a low skill worker moves up into a union slot, or because a union worker is promoed to a
supervisory position, we might expect more random changes among younger workers than among older workers.
We re-estimated the models of Section 4.3 on the subsample of those 30 years of age or younger in 1976
(N = 220). The results were virtually identical to those of Section 4.3. This suggests either that non-random
changes in union status are not a problem, or, more likely, that non-random changes are not confined to older
workers.
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JAKUBSON THE UNION WAGE EFFECT 989
APPENDIX
TABLE Al
Means and standard deviations
Variable Mean Standard deviation
LWl 6-235 0-356
LW2 6-334 0-351
LW3 6-435 0-363
LW4 6-516 0-374
LW5 6-627 0370
AGE/100 0-364 0-112
AGESQ/1000 1-447 0-887
SCHL/10 1150 0-282
SCHLSQ/100 1-403 0 625
AGExSCHL/1000 0-408 0 142
RACE 0-352 0-478
SOUTH 1 0-464 0-499
SMi 0 675 0-469
SM2 0-672 0-470
SM3 0-673 0-469
SM4 0-682 0-466
SM5 0-687 0-464
U1 0-427 0-495
U2 0-432 0-496
U3 0-434 0-496
U4 0-427 0-495
U5 0-417 0-494
TABLE A2
Unrestricted cross-section GLS regressions of log wage on all values of union status,
(standard errors in parentheses)a
U1 U2 U3 U4 U5
LW1 0-090 -0-004 0-076 0-043 -0-022
(0-055) (0-060) (0-060) (0-055) (0-054)
(0-055) (0.060) (0-060) (0-058) (0.050)
LW2 0-020 0-125 0-049 0 017 0-005
(0-052) (0.057) (0-057) (0-052) (0-052)
(0.055) (0.051) (0-056) (0-052) (0-050)
LW3 0-025 0-055 0-080 -0-013 0-057
(0-055) (0-059) (0-060) (0-054) (0.054)
(0-051) (0.053) (0-058) (0-054) (0.050)
LW4 0-022 0-021 0-056 0-071 0-026
(0-058) (0.063) (0-063) (0-057) (0.057)
(0-051) (0.054) (0-056) (0-054) (0-052)
LW5 0-009 0.011 0 035 -0-005 0-142
(0-056) (0.061) (0-062) (0-056) (0.055)
(0.057) (0-060) (0-064) (0.057) (0-055)
Notes: All equations also include age, age-squared, schooling, schooling-squared, age x schooling,
race, Southl, all values of SMSA and a constant.
a The first standard error shown is calculated assuming homoscedasticity. The second allows for
arbitrary heteroscedasticity.
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990 REVIEW OF ECONOMIC STUDIES
TABLE A3
Unrestricted 3SLS: Current wage on lagged wage and current and lagged union status
(standard errors in parentheses)
83 8G3 8G3* 83 0G3 303*
Year 2: t = 2 Year 3: t = 3
LW_,- 0-832 0-822 0 845 LW_,- 1-562 1-342 1-339
(0.250) (0.204) (0-221) (0-311) (0-173) (0-152)
U, 0-122 0-122 0-119 U, 0-024 0-016 0-019
(0-027) (0.033) (0-037) (0.035) (0.035) (0.032)
Ut-I -0-061 -0-063 -0-064 Ut-1 -0-161 -0-108 -0- 112
(0.036) (0.035) (0-041) (0.056) (0.049) (0.043)
Year 4: t=4 Year 5: t=5
LW,-, 1.163 1.091 1.063 LW_,- 1-041 1-037 1-038
(0-198) (0-148) (0-139) (0-189) (0-186) (0-176)
U, 0-060 0-029 0 028 U, 0 095 0-071 0-066
(0.029) (0.032) (0.029) (0.030) (0.038) (0.037)
Ul-_ -0-951 -0 049 -0-042 Ut-1 -0-101 -0 075 -0-071
(0.038) (0.034) (0-031) (0.038) (0.038) (0.037)
x2(19) 16-98 13-85 12-62
Notes: (1) All equations also include age, age-squared, schooling, schooling-squared, agexschooling, race,
Southl, current and lagged SMSA, and a constant. (2) Lagged wage endogenous. All values of union, SMSA,
and f used as instruments. (3) Chi-square statistic is a test of the exclusion restrictions. (4) Column headings:
83 = conventional 3SLS (homoscedasticity). 8G3 = extended 3SLS (heteroscedasticity), residuals from conventional
2SLS as norm. 8G3* = extended 3SLS (heteroscedasticity), residuals from extended 2SLS as norm.
TABLE A4
Unrestricted wage change equations
(standard errors in parentheses)
Dependent Variables A LW = L Wt-L WALW2
ALW3 ALW4 ALW5
ENTER 0-111 0-092 0-096 0-177
(0.036) (0.032) (0-035) (0-039)
(0.063) (0.053) (0.055) (0.075)
LEAVE -0-080 -0-013 -0-032 -0-036
(0.038) (0-032) (0-032) (0.036)
(0-054) (0.046) (0.063) (0-057)
STAY 0 035 -0-018 -0-001 0.001
(0-015) (0-014) (0-018) (0-018)
(0-014) (0-015) (0-016) (0-017)
Notes: (1) First standard error calculated under homoscedasticity. Second
standard error calculated with arbitrary heteroscedasticity. (2) Equations
also include age, age-squared, schooling, schooling-squared, age x
schooling, race, Southl, current and lagged SMSA, and a constant.
Acknowledgement. This work owes a great deal to advice and instruction from Gary Chamberlain, Glen
Cain, and Arthur Goldberger. I have also benefited from comments from Richard Blundell and David Lillien,
and from participants at workshops at Cornell University, Princeton University, SUNY-Binghamton, and the
University of Western Ontario. Special thanks to John Ham for extensive comments on earlier drafts. The
comments of Charles Bean and three anonymous referees have greatly improved the paper. I retain sole
responsibility for any errors which remain. This work was partially funded by SSRC Grant SS-55-82-07 and
by NSF Grant SES-8016383.
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JAKUBSON THE UNION WAGE EFFECT 991
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