The Joint Determination of Union Status and Union Wage Effects: Some Tests of
Alternative Models
Chris Robinson
The Journal of Political Economy, Vol. 97, No. 3. (Jun., 1989), pp. 639-667.
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The Joint Determination of Union Status
and Union Wage Effects: Some Tests
of Alternative Models
-
-
Chris Robinson
Universzty of Western Orltario
The problems of estimation and interpretation of union wage differentials are examined. T h e properties of cross-section and longitudinal estimators are compared. Estimates are presented and those in
the literature summarized. Conflicting results are obtained. Longitudinal estimators typically produce results smaller than those of OLS,
while cross-section methods (instrumental variables or inverse Mills
ratio) raise the estimate. T h e paper offers a reconciliation of these
results. It supports a more optimistic conclusion than that reached in
reviews by Freeman and Lewis, who argued that little has been
learned from attempts to deal with the endogeneity issue. Comparisons between estimators are used to throw light on the process governing union status and to suggest interpretations of "union differentials" consistent with the current evidence.
I. Introduction
In recent discussions of the estimation of "union differentials" there
appears to be a quite general agreement that union status is not exogI wish to thank without implicating Robin Carter, James Heckman, Peter Kuhn,
Glenn MacDonald, Aman Ullah, and Mike Veall fbr helpful discussions on the issues
addressed in this paper. Helpful comments on an earlier draft were received from
workshop participants at the Universities of Chicago, Guelph, Western Ontario, and
Wilfrid Laurier. An anonymous referee provided many useful suggestions that are
reflected in the present version. Finally, thanks are also due to Paul Rilstone for making
available his program for nonparametricjoint density estimation. Financial support for
this research was provided by the Social Science and Humanities Research Council of
Canada and by the Center for the Study of the Economy and the State. University of
Chicago, through a grant from the Lilly Endowment.
[ J o u m l of Poltt~ralE ror~omj,1989, vol 9 i , no 31
0 1989 b) T h e University of Chlcago. All rights reser\ed 0022-5808!89/9i03-0007$01 50
J O U K N $1. OF POLITIC ,$L ECONC)M'I'
640
enous (Freeman 1984; Robinson and .I'onies 1984; Duncan and Leigh
1985; Le~vis1986). Given that some endogenous process determines
union status, further questions have naturally arisen, along lvith discussions about ~ v h a tunion differentials measure in the endogenous
union status context and h o ~ vthey should be interpreted. Unfortunately, there has been much less agreemerit over these issues. Follo~ving a variety of attempts to "deal with" the endogeneity, several authors, notably Freeman and Xledoff (1982) arid Le~vis(1986), have
asked ~vhetheranything has been learned f'rom these exercises. 'I'he
different estimates of union differentials obtained from these attempts have been compared. 'I'lie pessimistic conclusion that has usually been d r a ~ v nis that there is no discernible pattern to the estimates-many of ~vliichare considered unreasonably high or lo~vand that therefore no improvement in our understanding of the
union differential problem has been made (1,elvis 1986, pp. 58-59).
A subgroup of these estimates that appeared to hold out great
promise and that dici offer more "stable" estimates Ivere those obtained from longituciinal estimators. Panel estimates of union differentials have been presented by hlincer (1981), C:hamberlain (1982),
Jakubson (1984), and others. Typically the longitudinal estimates are
much smaller than the comparable cr-oss-section estimates. Freeman
(1984) argues that these results are d u e to the differential effects of
both measurement error- and selectivity effects across the two types o f
data sets. In particular, either singly or (less restrictively) together,
measurement error and selectivity problems bias longitudinal estimates do~vn~var-d.
Some evidence is provided of severe measurement error problenls
~vithlongitudinal (relative to cross-section) data that, Freeman argues,
considerably reduce their value. Indeed, put into context, Freeman
(1984) and Freelnari ancl lledoff (1982) argue that longitudinal estimates are a yet further manifestation of'the great disparity in union
differential estimates that occur ~ v h e r eany attempt is made to deal
lvith the union endogeneity problem. I'hese estimates are thus characterized as being "less useful" than the "more stable" estimates obtained from ordinary least squares (OLS) cross-section estimates. AAt
best the longitudinal estimates may serve as a lolver bound on the
"true" differential (Freeman 1984).
T h e present paper offers evidence that supports a more optimistic
view than that of Freeman and l,e\vis. In Section 11, a tlvo-period
model of unionization is specified as a frarne~vork~vithin~vhichboth
cr-oss-section and longitudinal estinlators may be compared. Since
both interpretation and properties of the estimators depencl crucially
on the ~inderlyirigendogenous process, a detailed, explicit characterization is provided for the stochastic structure. T ~ v obasic sources of
UNION W A G E EFFECT'S
641
endogeneity are identified: the behavior of the agents (~vorkersor
firms) and measurement error. Three alternative approaches to solving these pi-oblems are considered: control function methods (e.g.,
inverse Mills ratio), instrumental variables, and longitudinal differencing. T h e different properties of these estimators under various
assumptions are used to thro\v light on the process governing union
status.
Section I11 conducts exogerieity tests and presents both "corrected"
and "uncoi-rected" estimates of the union differential from both
cross-section and longitudinal data sets. Exogeneity of union status is
generally rejected. T h e union rvage differential estimates provide
substantial evidence of a definite pattern to biases induced both by
selection problems and by measurement error. T h e test results and
the pattern of differentials are compared ~viththe existing literature
in Section IV. It is argued that the pessimistic \-ie\v of Freeman and
Le~vis,that \ve have little to learn from alternative (to OLS) estimates
of union differentials, may be unjustified.
Section V asks the follo~vingquestions: What kind of interpretation
is necessary for the reconciliation of the results of the different estimators! Does it require the imposition of ~vhatniight be regarded as
unr-easonable assumptions governing the under-lying str-ucture? Does
it imply other conclusions that are at variance lvith our observations?
In Section VI, comparison of alternative estirriators designed to deal
~viththe endogeneity issue is used to test hypotheses about the union
status process. A simple "fixed effects" model (of union status) is
contrasted ~vitha "1.andom effects" model. Comparison betlveen instrumental variables estimators and contr-ol function estimators suggests that the simpler fixed effects rnodel may be a good approximation for the union probleni. Some conclusions are presented in
Section VII.
11. A Framework for Union Wage Differential
Estimates
T h e purpose of this section is to extend the cross-section nod el of
Robinson and Tomes (1984) to t ~ v operiods, keeping the specification
as simple as possible. This cross-section model was estimated on 1979
Canadian data that contained high-quality Lvage data for hourly paid
workers. second year of data (1981) is now available. 'The primary
advantages of the additional year of data are the larger sample size
and the possibility of testing a Lvider variety of hypotheses regarding
union status. T h e simple t~vo-periodmodel of this section is designed
to capture only a very limited extension of the one-period setting.
Wages in each sector (union [('I or nonunion [N]) in each period
642
JOURNAL O F POLITICAL ECONOMY
are as specified in Robinson and Tomes (1984):
where the superscript t indicates period t values; Xj is a vector of
exogenous regressors.
T h e precise nature of the disturbances E:., and E,;., determines the
difference in the properties of the estimators considered below and
'
I An explicit characterization that would generate (1)-(4) is as follows. Let workers
have observable inputs ( f I , ) that they cotivert into output ( 0 , )in the union and nonunion sectors. according to individual functions that may be approximated t)y indi\idual proportionality factors:
where C ,
I if worker I is in the union sector. Variation across individuals in
=
+,.,(or
+,-,) could simply model the unobser\able fraction of individual 1 ' 5 total skills. For a
given individual I , I$, ., and +,-,ma) differ to allow for the utilization of diff'erent t)pes 01,
skills or the operation of different production processes in the two sectors. Payment to
each worker is. in general. some function o f output that ma) differ by sector; assume
that these may be approximated as
I-', (0,)= RI.+,.,fI, it C',
= 1,
T h e log wage (payment) form is then sirnpl)
where y:'
ET,, = In
In M',.,
=
y:
+
In W,-,
=
y,*
+ X,y
X,y
+ E:.,,
+ ec,,
In K,. + y,,, y: = In K,- + y,,. X , y = C:= I ylX,,. It1 H , = y,, + I:_,ylX,,,
,, and ,E:
= In +.,-,. Assume that X, and the +'s are independent. Then
=
+,
E(ln
+, lH)
+, ) = €7-
=
E(ln
=
E(ln I$,-) = e:
and
E(In +.,-/fI)
In general, ET and E-:
rameteri7ing. we get
will be nonzero so that y:.
and y:
= y,.
In K',.,
+ X , y + E,.,.
In W,.,
+
=
y,-
X,y
+
are not identified. Kepa-
t,-,.
where y , = y: + E:., y, = y:. + E:, E ( . , = e:,
e:, and E,-, = E:, - E:.. Thedifference,
in terms o f structural parameters, that unions make to observed earnings comes from
two sources. First, there is a difference in the payment functic~n(y?. y:.) associated
with union "rents"; second, there may be difference5 in the production processes,
implying differences in the relationship between observed Inputs and output (E:. - E: ),
associated with union productivit) effects. T h e "union effect" that ma) be estimated
from the log wage equations is
-
-
7,.
-
7,-
=
(y:
-
yX-) + (e:.
-
E:.).
Since, in general, e? Z E:., the separate "rent" and "productivity" effects cannot be
identified, only an overall union effrct.
U N I O N \VAC;E: E:FEFCTS
(343
the interpretation o f t h e estiniates obtained. Since two periods of data
are available, the disturbances may usefully be decomposed into tirneinvariant components and remainders:
T h e time-invariant components, O 1 . and 0,,-,, niay be thought of as
aspects of omitted skills that persist for a given ~vorkerover time. T h e
remainders r({ and T/,,, then represent time-varying omitted chax-acteristics or, more generally, aspects of the production process o r payments process that may be represented as raridom drawings for each
individual and that may differ over- time.' If the vectors y,\. and y l .
differ only in their first elements, (1) and (2) may he combined to
form the staridard OL,S esti~natirlgequation for the union problem,
that is, an earnings equation that includes a dumtny mr-iable for
union status:"
'I'he standard "union effect" estimates (Freeman 1982; L,e\vis 1986)
are OL,S estimates of the parameter (yl - y,\) in equation (3). If
union status, l':, is uncorrelated with the disturbance \':, this pariinleter is consistently estimated by OLS and represents the earnings difference any given ~vorkerobtains in the union sector relative to the
nonunion sector. '1-he problem ~ v i t l itreating (3) as a standard cross
section that [nay be estimated by OLS is that union status ia likely to be
correlated lvith the disturbance. T h e r e are t ~ v osources of this correlation. T h e first source is the correlation induced by the behavior of
some of the agents in the model. For example, there is the argument
that en~ployerschoose, frotn some union pool, those indivitluals tvith
higher (unobserved) aljility (C),,,, 0[ ). 'I'here is also the nlore general
selection argument that the ~vorkersknow (:it least part o f ) the disturbance iind select thenlselves into their most p r e k r r e d sector taking
this into account. T h e second source is the correliitioti iliduced bv
For example, o u t p u t at the incli\idual le\el nra\ I)e \to( h,~\tic.. ~ n dthe ~ratllr-eo t tllc
stochastic- proce5s rna\ ditfer tn sector hut l)e \t;~hleo i e i time. T h r 111(11\idl1al
tlr:~\\.i~rg\
\vill diff'cr acro5s i ~ i d i \ i d u , ~ l wctors,
s,
ntid t i ~ n r .
Tests o n the clata 5ets ernplovecl in Sec. 111 hrlo\\ cannot jiener,iIl\ reject ecll~;ll~t\
of the earnings function slope coefficients. '
644
JOUKKAI. OF POLITICAL. ECONO>I.I1
errors in the measurement of union status. Freeman (1984) has argued that measurement error is snlall enough to be ignored in crosssection estimators. Nevertheless, OI,S estimation of (3) for individual
years o r as a pooled cross sectionlti~neseries ~vouldstill be inconsistent hecause of the first source of correlation incluced by the choice
behavior of Lvorkers o r firms.
T h e possibility of a more coniplex error structure than that assumed for consistency of OLS raises both estiniatiorl and identification problems. Without further restrictions on V:, eq~latiorl(3) represents a random effects model of union status on earnings. A4
~vorker'searnings in the union sector differ fi-om ~ v h a they
t
~vouldbe
in the nonunion sector by t ~ v oseparate terms in (5).'I'11e first is (yl. y,,), ~vhichis common to all ~vorkers;the second is (E:. - EL,),representing a random effect of union status. If this random component of
the ~ l n i o neffect is suppressed by imposing E:. = E;, for all i , then a
fixed effects model of union status results:
In the fixed effects model, the parameter (y,. - y,.) represents the
union effect, as in the 0 1 , s studies, though the parameters \\.ill not be
consistently estimated by OLS if cov(i', E , \ ) # 0. In the random effects
model, the randoni component complicates the identification of the
union effect. T h e parameter (y,. - y,.) represents the difference in
earnings bet\\,een union and nonunion sectors ~ v h e n~vor-kersa re randonily assigned to sectors, as in the fixed effects nod el. Ho~vever,
contrary to the fixed effects model, (y,. - y v ) does not represent the
difference in earnings expected if a union ~vorker(i.e., one having
been allocated to the union sector by the union status determination
process) Lvas transferred to the nonunion sector. Under the OLS assumptions, o r mor-e generally under fixed effects assuniptions, any
experiment that moves a ~vorkerbet~veensectors results in a fixed
change in the ~vorker'searnings given by (yl- - y,.). Under ranctorn
effects, union Lvorker i ~ n o v e dto the nonunion sector Lvill experience
the earnings change (y,. - y,,.) + (E, - E,.). T h e expected change is
(Y,, - Y . ~ )+ E[(E[.,- E,\.)/(', = 11. LVith data o n earnings alone (i.e.,
in the absence of, say, a 1-egressor in the process determining union
status), all workers have the same \.slue of E[(E,.,- E \ - ) I L', = 11. AS a
result, (yl - y,,) cannot be separately identified. This is apparent if
equation ( 5 ) is reparameterized to
615
UNION W A G E E F F E C T S
Equation (7) is econometrically equivalent to the fixed effects model
(6) but identifies a different union effect.' T h e effect ( y , . - y,.)* does
not have the interpretation of the union differential resulting from
random assignment.
Alterncrtzve Estzmators of C'nzon Effects
T h e problem of union status endogeneity, cov(C':, 1':) # 0, has resulted in several alternative approaches to estimating ( y l- - y v ) . Two
alternative approaches to the problem for single cross sections are
instruniental variables (I\') and control function methods. Consider
first the I\' method. This requires the existence of instruments for C':
that are uncorrelated r\.ith the disturbance in ( 5 ) .Since in general the
disturbance contains C':, E:.,, and E:, some special assumptions are
required.
example is given in Duncan and Leigh (198.5). T h e
validity of the assumptions necessary for the 'I.\ method to be consistent depends on the underlying process generating union status. This
issue is discussed below.
T h e alternative class of estimators considered is the control function type, of ~vhichthe inverse hlills (IhI) ratio method has been most
widely used. In this case, union status occurs rvhen 1: > 0, where I: is
the net-of-"costs" ( C : ) rvage difference from choosing the union sector:
I," = 111 W:.,
-
In W!v, - C:
=
Z:*i
+
E:,
c: = s:s +
Given joint normality of E:, E:.,, and
follorving modification to (5):
E:
E:,,
E:
=
E:.,
- E:
- E:
.
+ E(.~[(c;
=
E{('j[(e;,
=
0.
-
-
e;.,~
-
e',,)
-
(8)
(9)
( l ) , ('L), ( 8 ) , and (9) yield the
* 111 the fixed effects model ( 6 )the distur-bance iu e',,. H \ assumptiorr, Ee:
the disturbance is I.':*, which also has rer-(1 mean:
El.':* = Ee;,
,,
E ( e ; , - c;
E ( E ; , - e',.,i(lr:
=
=
0. Ill ( 7 )
= 11
l]}Pr(( , = 1 )
See Heckrnan a n d Robb (198.3) for- m o r e detailed diucussionu in the context of estin~atirig effects of tr-aining programs.
J O U R S A L OF POLITICAL ECOh'Ohll
and A:., anti A(, are the relevant inverse hlills ratios. T h e tfisturbarice in
this case al~vayshas a zero mean and is uncorrelated ~vithC':; hence n o
instrument is necessary. This method will produce consistent estimates fhr any type of assumption governing the disturbances, O's, q's,
and E(.'s provideti that there is a regressor in the selection rule and
that the conditionai ciisturbances k1(8[. + q:.)I l', = 1 and E(O,v, +
q\- ) I I', = 0 can be consistently estimated.
T h e availability of longitudinal data for equation (5) suggests an
alternative to the cross-section approach to the union endogeneity
probleni, by exploiting the tirne invariance of the person-specific 8's.
Unfortunately, the longitudi~laldifferencing approach rvill not ~ v o r k
in general. T h e person-specific effect is not elimi~latedbecause of the
differences no-oss sectors. In adtfition, the different q's across sectors
each period result in 1,'; and c',' both appearing in the error term. T h e
standard longitudinal approach has been to impose 8[. = O.y and
= T( , t = 1, 2 (Freeman 1983, pp. 3-4, eqq. 1-3). I n that case
the (individual) mean difference form of ( 5 ) is
rvhere =
xi. This 12.ithin-groupregression, ~ v h e r ethe "group"
is the data for both time periods for- the same individual, rvill provide
unbiased estimates if I', and the q's are uncorrelated. that is, if the
permanent components (0's) are the o~liysource o f the encioge~leity
problenl.
..lltr? ~ i a t ~ .Ilocicl,
~rp
of tkc Df~tfv?)iinatlo~i
of ( 'nzon Strrtuc
T h e ~alidityof the assumptions necessary for consistency for the estimators considered above depends on the nature of' the process b!,
tvhich individuals become union members and the error rvith rvhich
union status is measur-ed. Unfortunately there is little agreement in
the current literature on the union process. hlost of the discussions,
ho~vever,involve sonle form of choice by rvorkers or- firms based on
t3oth obser-vable and unobservable factors (Lee 1978; Abolzd and Fart3er 1982; Farber 19830; Robinson and Tomes 1984). O n e general
point that emerges frorn these discussions is that rvhere workers know
some of their characteristics that the investigator does not, the most
sel-ere econometric problem arises. T h e corollar-y to this, pointed out
t x Heckman and Kobb (1985) in the training model context and
exploited earlier by Zellner, Kmenta, and Dreze (1966), is that the
econometric problems are often reduced if the agent's perfect knorcledge is replaced by uncertainty. ?'he agents then have to make their
decisions on. tor example. union status, not or1 the basis of unobserved (by the investigator) individual characteristics but rather- on
expectations the agents torm about horc rvell they would do in each
sector. Under appropriate assunlptions these expectations, conditioned on observables, tvill be the same across i~idividuals,thus removing the correlation b e t ~ c e e ~uriion
i
indistatus and ~~nobservable
victual characteristics present in the certaint) case. 1)iffel-e~lcesacross
estimators may theretore be used to construct tests o n alternative
hypotheses of' the union status process. '1.0 facilitate tlisc.ussion of the
t3asic issues, a simplified structure for union choice over ttvo periods is
specified and used to contrast the properties o f the alternative estimators.
Assume that two sectors are available to each rvorker, unio11 o r
nonunion, in each period and that the rvage rates in each sector are
given by (1) and (2). T h e r e is a cost attached to union status, including
psychic costs of' union membership, together ~ c i t hresources spent on
actually gaining membership anti any differences in expected \cork
duration o r rvorking conditions. Assurne that this cost, f'or simplicity,
may be expressed, relative to the nonunion tcage, 1)y C:, as specified in
(9). Consider first the case of per-f'ect certainty for the agents. 'I'he
individual's objecti1.e is to maximize net tvages over the ttvo periods.
111the simplified framercork of this section, this requires choosing the
sector that niaximizes net rvages in each period. Hence the rule tor
each period is
choose C':
=
1 i f f I: > 0, t
=
1, 2
Given this process, longitudinal differencing methods generally fail,
even in the absence of measurement error, because since I: depends
on the pil's, CT: and the q's rcill t) pically be correlated. 'll1e available
of the
instr-urnents for CT, are the Z, (or some knorcn transforrn;~tio~i
Z,). Consistency f'or the I\' estimation thus reyuir-es that (some transformation of) the Z, be uncorrelated ~ c i t h\', in the limit. ,\ detailed
discussion and precise conditions for consistency are available in
Heckman and Robt) (1983). Duncan and Leigh (1985) satisfy the consistenc), requirement by assumi~lgthat the joint distributions of (e( , e)
and ( E , ~ E, ) are the same. This implies that E;17,1Z,= E17, = 0. Since I',
= ev + I',(E[ - E , ~ , )a nd EE,,, = 0, EI', = 0 +L;Lr, (el - E , , ) = 0.
Thus union status and (eI. - E Y , ) are required to be uncorrelated.
This is guaranteed by the fixed effects (of union status) rnodel, rchich
sets el. = E.\. , for all i. More generally in the random effects frarne-
'j49
L'NION W A G E EFFECTS
tained by probit estimation o f t h e probability of union status. Consistency in this case will therefore fail if normality fails.
An obvious alternative to perfect certainty at the level of the agent
within the present framework is to assume that agents kno~vthe 0's
but that they have the same information as the investigator regarciing
the q's. In this case 0 may be thought of as omitted individual characteristics o f the same type as schooling and experience whose value in
each sector is common knowledge. T h e q's then represent individual
variation in earnings across sectors that arise from stochastic elements
in the sectors themselves on different realizations to individual character-istics whose value in each sector is uncertain, for example, the
ability to function well in a regulated union environment. Let the
criterion for choosing the union sector then be that the expected net
wage in each period is positive, that is,
choose C':
Given Eq:
=
Eq:,
Elf
=
=
=
0, 1
E(Z:n
=
1 iff El: > 0,
1 =
1, 2.
1, 2,
+ e:)
=
Zfn
+
0( - 0,
-
e: .
Thus e: reduces to
- O., , - c::~). T h u s L': will be correlated with ( O ( .
- O.v, - E::,),
but not with
o r q;.,, assuming that e:,, is uncorrelated
with the q's.
T h e longitudinal estimator remains inconsistent unless O1., = O,,
and C': is measured without error. If (01- - 0.\.,) # 0, this term remains
in the disturbance and is correlated with CT:. T h e I\' approach now
has to contend with the term (0(., - O.\.,) in the criterion function
rather than (E(., - e . ~ , )Suppose
.
that EL',(qc., - q.,,,) = 0 so that
E\~,Iz,= EC',(O1, - O.,z)
=
E(Oc IC',
=
l)Pr(CTl= 1, Z,)
Since indiviciuals know ( O r - O,v,) when the union decision is taken,
this will not, in general, be zero on the same grounds as before.
Special assumptions, analogous to those made in the perfect certainty
case, are thus required for consistency.
T h e Ihl approach would yield consistent estimators subject to the
normality assumption as before. T h e only change would be the interpretation of the parameter estimates for the coefficients on the inverse hlills ratios.
Finally, a more extreme form of depai-tur-e from perfect certainty
assumes that the agent knows only what the investigator knows, except, possibly, for ec, In that case the expected wage difference is El:
= Zfn + EeC,,and E, thus reduces to (at most) €I:,. In this case, C': thus
depends, given Z f , only o n ec,. Since (el., - E.,,~) is now eliminated from
650 J O U R N A L O F POLII'ICAL ECONOSlY
the criterion function, there is no reason to suppose a nonzero cot-re, E-Y,). T h e "special" assumptions required
lation between C', and ( E ~ in the previous section no longer appear restrictive, given the choice
model generating union status. From the econometric point of view,
this form of the random effects model presents estimation problen~s
equivalent to those of the fixed effects n~odel.'
T h e Ihl model remains consistent under any intormation assumptions, subject only to the distributional requirement. T h e imperfectforesight assumption reduces the problems for the longitudinal estimator. However, consistency still requires the difference in union
status to be uncorrelated with (rl' - -qj). Ho~vever,if union status is
generally uncorrelated lvit11 the ~ ' s the11
,
all the estimators in the
imperfect-foresight case, including OLS, will be consistent.' I'his
raises the question of the gain from longitudinal estimators in this
context. Consistency of the longitudinal estimator- requires stringent
assumptions. T h e only source of the endogeneity problem must be
time-invariant individual components that are the same across sectors. In that case, horvever, given obsei-ved characteristics, the union
status changes over time have to be random, tvhich is an unattractive
behavioral model for the evolution of union status.
Differences across estimators may be exploited to test alternative
hypotheses about the process generating union status. In the next
section, union differential esti~~lates
and exogeneity tests based o n the
alternative estimators are presented. I n Section I V , the results are
compared rvith the existing literature.
111. Empirical Results on the Alternative
Estimates of Union Wage Differentials and
Exogeneity Tests
T h e estinlates of union rvage differentials presented in this section are
based on the Canadian Quality of Life Sui-vey descl-ibed in Robinson
and To111es (1984). Table 1 presents summary measures of the data.
T h e alternative estimators ciiscussed in the previous section tvere used
to construct exogeneity tests and to obtain estimates of ( y l . - y.,.). T h e
' T h e hxed effects rnodel \ i ~ p p o s e se i , = E \ , \o that I., = E , , . T h e random etfect\
model u n d e r uncertaint> has I., = E , + C ' , ( E , , E \ . , ) b u t E C ' , ( t l
t.\ ) = 0 . so that the
additional tern1 C ' , ( t l , - E , , ) presents rlo further econometric problern tli,un tlie o n c
that was present with the flxed effects model (Heckrnan a n d Kobb 1085).
' T h e dihtur-ba~iceis [ ( H I + T; , ) l ' :+ ( 0 , ) + r(', ) ( I - (./)I. Since I./ a n d the 8's and r('\
a r e unco~-I-elated,
E{.} = 0 :
-
Hut
(
:' =
L':
since
C: takes o n the ~ a l u er e r o
01
-
o n e hence
DATADEFINITIONS,
MEANS,
A N D STANDARD
DEVIATIONS
1979 DATA
VARIABLE
NAME
Atlantic
Quebec
Ontario
Prairies
Yrssch
Expr
Tenure
Male
POW
Ln
u!
Union
Part time
Married
Public
Plantsize
DEFINITION
Dummy variable = 1 if
resident in Atlantic
province*
Dummy variable = 1 if
resident in Quebec
Dummy variable = 1 if
resident in Ontario
Dummy variable = 1 if
resident in Prairies
Years of schooling
Experience = age yrssch - 6
Number of years with
same firm
Dummy variable = 1
if male
Percentage of organized workers in respondent's industry
Satural logarithm of
respondent's hourl!
wage (1980 dollars)
Dummy variable = 1
if individual belongs
to a union
Dummy variable = 1 if
respondent works
fewer than 30 hours
per week
Dummy variable = 1
if respondent is
married
Dummy variable = 1 if
respondent works in
public sector
Mean size of plant
in individual's
industryt
Mean
Standard
Deviation
1981 DATA
Mean
Standard
Deviation
.063
(.074)
,320
(.288)
,407
(.436)
,104
(.093)
11.353
(1 1.249)
18.1 17
(19.237)
5.632
(6.622)
,508
(377)
38.459
(39.447)
1.877
(1.946)
,464
(.542)
,257
(.228)
,620
(.683)
,241
(.256)
27.109
(27.842)
NOTE.-Data for longitudinal sample are In parentheses
* Reference group 1s Br~tishLolulnbid
'Computed from respondents' answers to question on number of enrplo\ees ar r-esl~ondcnt'iplace of work (see
Roblnson and Tomec 1984). Units are 10 emploveer
632
.JOLIRiVAL O F P O L I T I C A L E(:ONOhlY
1 ABLE 2
EXOGE:NEIT
1. 7 EST S
FOR
UNIONS1'h.l'~'~
Test a n d D'ttn Set
Keault
.A. Tests Based on I\' versus OLS Corn1)arisons
M'u-Hausman test: F-test o n ,joint significarlce
of i~lstrurnentsin instru~llent-augmented
regression:
1079 cross section
198 1 cross section
Pooled 1979-8 1 cross sectiorl
F
F
F
=
=
=
1.883 vs. critical F ( 5 9 ) = 2.10
2.367 vs. critical F(.5R i = 2.10
3.623 vs. critical F ( 5 % ) = 2.10
B. Tests Based o n IRl Lersus OLS (:oniparisons
F-test o n joint signif~canceof the i n ~ e r s e
Xlills ratios:
1979 cross sectiori
198 1 croaa section
Pooled 19711-8 1 cross section
F
F
F
=
=
=
4.613 vs. critical F(.5'/;) = 2 9 9
4.162 vs. critical F(.5%)= 2.99
6.175 \s. critical F ( 5 R ) = 2.99
(;. Tests Based on Longitudinal Lersus Cross-Section Estimates
\Vu-Hausman test: t-test on the difference
in the union coefficient: witliin-group
estimator L s. NerloL e-Hausman-Ta)lor
estiniator:
1979-8 I longitud~nalsample
1 =
3.335
results of the exogeneity tests are presented in table 2. Panel A reports the results based on I\.' versus OLS con~parisons.T h e I\.' approach was used to construct a Wu-Hausman test similar to that employed in Duncan a n d Leigh (1985). Conditional 011 the observed
unio11 status, under the null hypothesis of'exogeneity of union status,
for a single year of data, OLS applied to the separate union and
nonunion sectors is unbiased and efficient. T h e r e are n o common
parameters to be estimated between the sectors. If, however, the
union and tlotlunion disturbance variarlces are equal-[u'(Ol~) +
u~(~;.=
) ] [u2(0.,,) + u2(~<-)]-then E V ~= [u2(0.,,) +
and
OLS applied to the pooled union and nonunion sector samples will be
unbiased and efficient.
T o test for exogeneity using a single cross section, the OLS estimates may be compared with the estimates obtained using an instrument for C':. 'The natural instruments are
'The probability was estimated using probit estimates for n and assuming no~.malitjfor E:. T h u s
CTiVIOiV WAGE EFFECTS
633
T h e term 0:
does not have to be a consistent estimator for (,':;Ahence
violation of the normality assumption here does not invalidate C': as an
instrument. Under the null hypothesis of exogeneity. I\' is consistent
but not efficient. Under the alternative hypothesis, OLS is inconsistent. T h e I\' estimator may o r may not be consistent, depending on
the assumptions on the nature of the disturbances. However, in general under the alternative. plim 4ol.s f plim
T h e tests reported in panel A utilize the "altel-native form" of
Hausman (1978). This consists of a test of the significance of auxiliary
regressors in the OLS estimation of' equation (5), perrrlitting interactions in the standard earnings function variables." T h e auxiliary
regressors are the vector of instruments for union status and its rele\;ant interactions. This test was performed for both 1979 and 1981
cross sections. Under the null hypothesis of exogeneity, the true
covariance matrix is consistently estimated by the standard OLS
covariance matrix, M-hich is used to construct the F-test on the
significance of the auxiliary regressors. These are reported in the first
two ~ O \ V Sof panel A. Tests were also performed on a sample that
pooled the independent observations from 1979 and 198 1. This consists of the 1979 cross section plus those individuals added to the
sample in 198 1. T h e covariance matrix in this case remains proportional to the identity matrix under the null hypothesis of exogeneity.
T h e F-test based o n it is reported in the third row in panel A. Exogeneity is rejected at the 5 percent level for 1981 and the pooled
sample. It is rejected at the 7 percent level for 1979.
Panel B reports the results for tests based on I M versus OLS comparisons. T h e null hypothesis of exogeneity is equivalent to the hypothesis that the inverse Mills ratio terms d o not enter the regressions. Under the null hypothesis of exogeneity, OLS consistently
estimates the covariance matrix for equation (8). This covariance matrix is used to construct the tests in panel B. ( T h e results are very
similar if the covariance matrix is corrected for the heteroscedasticity
and the use of estimated \fills ratios that would follow under the
alternative hypothesis of significant selection.) T h e F-tests were constructed on the joint significance of the inverse Mills ratio terms.
Exogeneity is always rejected in these tests.
Finally, panel C presents the test results based on the longitudinal
data. T h e presence of a n individual component, assumed constant
over time, creates a nonzero covariance in the disturbances for the
'
.As rloted in n. 3, zero slope interactio~~s
cannot generally be rejected. Howeler, the
pattern of slope coefficierlt differences between union arld nonunion sectors is consistent with that found in other studies. e.g., a lower schoolillg coefficient in the union
sector. 1 he model kcas theretore estimated pernlittillg slope illteractions for the standard earnings function variables.
same individual in different time periods. However, the pooled (2year) model may be consistently and efficiently estimated by OLS,
under the null hypothesis of exogeneity, provided that an appropriate weighting scheme is used to transform the covariance matrix to
one that is proportional to the identity matrix. T h e appropriate transformation for the case of OI., = O,,. and
= rl:., is quite straightf'orward and, as described in Nerlove (1971) and Hausman and Taylor
(1981), is equivalent to a differencing of the observations as follows:
In Wj
-
(1
-
-
w)ln 1VJ = [ciXj
-
(1
-
-
w)L',X,]yl,
where o = { ~ r " ( ~ ) / [ c ~+' ( ~202(0)])'
)
'. Equation (13) is a matrix
weighted average of the within-group regression ( 1 1) and the between-groups regression delirled by averaging (-5) across t for each
irldividual i :
In M7,= y,.
+
(yl
-
y,.)c
+ Fy+ c.
(13)
?'he longitudinal aspect of the sample permits a test of' independence of' union status from 0, under a maintained assumption of
independence of L:; and 77:. provided 0(., = O.,.,. Under the null hypotheses of exogeneity, OLS estimates of (13) will be consistent and
efficient."' ?'he OLS estimates of (1 1) will also be consistent, but since
they use none of the information from the between-groups regression, they will be inefficient. Thus a \Vu-Hausman test may be constructed f'i.on1 a compai-ison of' these estimators. 'I'he test reported
in panel (: was constructed from the within-group estimators and
the efficient (under exogeneity) weighted OLS (Nerlove-13ausmanTaylor) estimator. Again exogeneity is rejected.
''
I"
T h e e\timate requires a conslslent e5timate of
w.
'1-lii\ ol)t;~inedfrom the fact that
where SSK(LV(;) and SSRIBC;) are the surii\ of \cluared residuals froni the within-g~.oup and I)et\\een-groups regression, (Ilausnian ant1 Ta\Ior l!IHl). I I In general, 0, , f 0,. and ~i , # q; . In this caw tlie 1Vi1-Ilau\man tests nia\ \till I)e t ~ , ~ n \ l o r m a performed as indicated abo\e. 'The major dittere~iceis in the ,~ppropri;~te
tion for eq. (13) and the estin>ation ot w. An added cornplicatiou ctelns from ctle tact
that not all of the i~idi\idualsIn tlie 1979 ant1 1981 ianlple\ \\-ereinte~.vie\\edin botli
years. If the ditterent data aiailahilit) is randoni, then the 5aml)les ma) tle reatiilv
estimated together in a form analogous to eq. (13).
TABLE. :3
-
Data Set
Zero Slope
11iter;rctions
Earnirigs
Function Slope
Intcractions
A. 1979 cross section (.V = 6851:
O1.S
Iv
I hl
B. 1981 cross section (.V = 85.5):
OLS
I \.
IS1
C;. Pooled independent 197Cl-81 cross rectlons
= 1,220):
0I.S
Iv
I hi
D. Lotigitudi~lalsample. pooled 1979-8 I data
( S = 621):
OLS
I \'
I hL
E:. Lotigitutlinal s a ~ ~ i p l~ntli\idual-average
e.
data.
between-groups estimate5 (.V = 3 12):
0I.S
11'
I hi
F. Longitirdinal sample, mean-differencecl data.
within-group estimates (.V = 6001:
OLS
111table 3 the alternative estimators are used to contrast the alternative estimates of union wage differentials, calculated as exp[(?( ?,,)XI - 1. T w o fhrms of the basic model ( 5 ) are estimated. ?'he first
imposes ~ e r ionteractions between C': and all the elemenrs of' X: apart
from the constant; the second permits slope interactions in the earnings function (see n. 9).T h e first three panels, A-(:, report the results
of single-year cross sections individually and pooled. T h e ) show O12S
estimates o f approximately 'LO percent for all the cases. T h e I\.' estimators use as instruments the predicted values of union status obtained f.1-om maximum likelihood estimates of a probit model for
equation (8).12T h e IhI estimator is as described in the previous section. Both the I\.' and 111 estimates are much larger than the OLS
estimates. T h e estimates range fiom 27 percent to 33 percent. T h u s
" Instrurne~itstrom a lineal. prol)at)ilit\ n~odel5 x1-eal\o useti and procluced almost
identical results.
636
]OC'Kh'.AL OF 1'OI.ITICAL E C O N O M Y
attempts to correct for endogeneit? by either the I\.' or I M method
have the effect of raising the estimated union wage differential.
'I'he lo~lgitudinalresults are presented in panels D-F. T h e longitudinal estimator employed is the within-group regression defined by
(1 1). This estimator requires data on the same individuals in both the
I979 and 1981 cross sections. L~lfortunately,in the Quality of Life
data, not all individuals in the 1979 cross section appear in the 1981
cross section and vice versa. In order to produce an OLS cross-section
estimate from the same sample used to compute the within-group
estimate, both the 1979 and 1981 data sets were reduced to include
only individuals working both years. T h e result is termed the longitudinal sample.':' Because of the sr:laller sample size for this analysis,
only the model imposing zero slope interactions was estimated.
First, panel D uses both the 1979 and 1981 observatio~isfrom the
longitudi~lalsample and computes pooled estimates equivalent to
those in panel C. 'I'he observations in this sample are not in general
independent, but the coefficients are consistently estimated by OLS.
As in the full pooled cross-section sample, the corrected estimates are
approximately 30 percent larger than the OLS estimates, though the
general size of the differentials is snlaller.
In panel El, estimates based on the between-groups regressio~~
(14)
are presented. Inlposing O ( . , = O.,., and T i - , = T.;,. t = 1,2, as in the case
of the within-group estimator yields 1',= 0, + 7,.
T h e disturbance
thus retains all the potential problems encountered with the cross
sections. T h e results are therefore expected to be similar to those in
panel D, as in fact they are. These may be used as the cross-section
standard by rvhich to compare the longitudinal within-group estimates presented in panel F.
'I'he first row of panel F presents the estimated differential from
the standard within-group regression represented by e q u a t i o ~(1
~1).
This shows a union differential of 11.5 percent, substantially lower
than the 13.8 percent from the between-groups regression or the 13.7
percent from the OLS r e g r e s s i o ~on
~ the pooled longitudinal data.
There is thus an apparent conflict between the results obtained by the
alternative methods of addressing the endogeneity problem. Crosssection methods (I\' and IhI) result in upward adjustment; longitudinal (differencing) methods result in downward adjustment. ?'he
remaining estimates in panel F are discussed in Section \' below.
IV. Comparison with the Existing Literature
T h e rejection of exogeneity of union status is in general agreement
with results presented in the literature (see, e.g., Duncan and Leigh
13
Some o1)vious prol~lemsof selectiorl bias ma) arise because ot this restriction; these
are discussed below.
U K I O N %'AGE EFFECTS
TABLE 1
Data
Stud\
Reqult
A. Sample Selection Bias
KLS Older Men
SEO
PSID
Duncan and Leigh (1980)
Lee (1978)
Heckman and Neurna~ln
(1977)
Ro1)insoti and Tomes (1981)
SWH
Kumar and Stengos (1985)
LCWS
Simpson (1985)
NLS Yot~tigMen
Leigh (1980)
KLS Young Men
and Womeri
QOE
QOE
Blau and Kahn (1983)
Farber (19836)
Atrostic (1981)
CPS
Wessels (1981)
Grant, Swidinsky, and
X'ariderkamp ( 1 98.5)
Podgursky (1980)
Sul~stantialrise (all workers)
I'ery slight decline (operatives)
Moderate or substantial rise (all
workers)
Moderate risc (hourl) paid
workers)
Moderate or sul~stantialrise (all
workers)
Moderate or substantial rise (all
workers)
Moderate or sut)stantial rise (all
workers)
Moderate rise (all workers)
Moderate rise
Substantial rise (blue-collar,
rnanuhct~~ring)
Small rise
Substantial rise
No change or- substantial fall
B. Instrumental I'ariables arid Other Simultaneous Equation Rlethods
QoL
Farber (1980)
Duncan and Stafford
( 1 980)
Robinson and Tomes (1984)
QoL
Leigh (1978)
CPS
Podgul-sky (1980)
NLS Y o t ~ n gMen
PSID
Substantial rise (all workers)
\loderate rise (all workers)
Moderate rise (hourly paid
workers)
Rise (2oung men); fill (older
men) Highly ~ a r i a h l e NOTE.-NLS: Ndtlonal Longlrud~nalS u r \ e \ (Cnlted States), SEO S u r t r * of Fconomlc Opportun~tk1L n ~ t e d
States), PSID. Panel Stud) of Income D)narnlc\ (L'nlted Surer), QOI Quallt\ of Llfe Surte, iLanada). S\( tl
S u n e v of Work Histor) (Canada), LCWS. Labour Canada \cages Sur\e* ((,anadd), I'lC r>KD L n e n ~ p l o ~ r n r n t
Insurance C o m m ~ s s ~ odata
n ~ l t matched
h
Satwnal Re\enue Department data ((.anadd). QOF Qualll, of Frnplo\ment ( L n ~ t e dStates), CPS Current Populat~onSurbei ( L nlted States)
1985). Few investigators appear to doubt that union status is endoge~louslydetermined. There is, however, major disagreement about
what has been learned from attempts to deal with the endogeneity
problem.
It has bee11 forcefully argued by Freeman and Medof'f' (1982) and
Lewis (1986) that the 11' and IM methods yield no discernible pattern
relative to OLS. This may be true for data with any unit of observation, but for the large individual-level data sets there is considerable
uniformity. Table 4 displays the estimates from the studies using such
data sets listed in Freeman and Medoff (1982, table 2) and Lewis
J O L K N A I OF I'OI.ITIC A L E(,O~YO.M\
638 (1986, tables 4.1,4.2),together with some post-1982 studies, using the
I\.' or IS1 method. Except for Podgursky's (1980) \iork with the Current I'opulation Survey files, these studies provide substantial evidence of a corisistent rise in the union differential relative to OLS
estimates when the endogeneit) of' union status is addressed by the I\'
o r Ihl method. T h e estimates presented in Section I11 above for the
current data set reinforce this pattern fol-I\.' ant1 IhI estimators.
T h e longitudinal estimator approach to solving the union endogeneity problern has also exhibited considerable ~ ~ n i f o r m i tof'
y results.
Ho~vever,in this case it is to r r d z ~ cthe
~ estimated differential relative
to OLS I-ather than to increase it. Again, the results fhr- the current
data set presented above conform to this pattern. I'he reduction
found for the longitudinal estimate relative to OLS in table 3 is comparable to the magnitude reported in Freeman (198.1) and the studies
cited therein. Some investigators using longitudinal data have attributed this reduction to fixed effects of higher-quality workers being in
unions. However, this is not consistent with the cross-section studies
that deal with such effects b y the I\.' o r IhI method.
V. Reconciling the Results of Alternative
Approaches to the Union Endogeneity
Problem
T h e conflict reported in Section 111 and, as argued above, found
more generally in the literature was investigatetl further using the
longitudinal sample. If the longitudinal results are to be reconciled ~\.iththe cross-section corrections hi. endogeneit), tlifferenceinduced Ineasurenient error o r selection effects must reverse the direction of' endogeneit) bias from positive to negative. Freeman (1984)
has provided some evidence to suggest that while the number of
rnisclassified union members may he small enough to produce n o
discernible effect on cross-section estimates based on levels, it is large
enough to have a major impact on longitudinal estiniators based o n
differences because of the relatively small nunil)er of union status
changes.
If measurement error of the magnitude reported b) Freeman is
generally present, then even if' selection effects in cross sections result
in differentials larger than OLS differentials, sufficiently large difference-induced measurement error could produce an estimate lo\\-er
than OLS cross-section estimates. Further, even if selection effects
result in "corrected" estimates of ( y ( , - y,) based on cross-section or
lrz~rls-basedrstiniates being larger than 0 1 3 estimates, the same selec-
''
I 1
For- a n an;ll>sis of the PI-obleni of measurement rrl-01- i ~ cr lichotomous \alial~lcs.
Tee, e.g., Cochr-an il!If 8) and Xigrler- ( lLI'i3).
tion process may )ield dzffrrenced-based corrected estiniates of (yl y,.) that are biased downward arid may be lo~verthan O1,S estimates.
This issue is addressed in the last t ~ . orows of panel F in table 3.
T o examine tlie measurement error and selection issues iri the longitudinal estimates, the within-group regression as reestimated in
two forms. First, an instrument was used tor the change in union
status. T h e individual mean-differenced observations for union status
Fvere regressed o n the union status regressors in mean-differenced
form and tlie predicted values m-ere used as instruments. Second, the
regressors in the rvithin-group regression b.ere augmented b) the
inclusion of the inverse Slills I-atio terms. T h e IV estiniate is 22.7
percent and the IhI estiniate is 16.2 pel-cent. Both of these estimates
are in excess riot only o f t h e staridard within-group estimate but also
of the OLS estimates in panels D and E. T h e IV estimate is in fact very
similar to the IV estimates in panels D and E. T h e I'LI estimate is
lo\\.er than the IS1 estirrlate in panels D and E. T h u s there is evidence
in these data that measurement error o r selection effects result in a
downward bias in tlie longitudinal esti~nator.
T h e results above should be vie\\.ed with caution since the longitudinal data set of hourly paid workers is not large arid is sub-ject to
some potential attrition bias pr-oblems that, though perhaps
mitigated, are not necessarily solved by the inclusion of the conditional disturbance terrns in the analysis. Ho~+.ever,
they d o provide
some evidence fol- a reconciliation of cross-section and longitudinal
results. They are also in conformity rvitli a study by Chowdhury and
Nickell (1982), cited in Freenian (1984), \\.ho also found a substantial
increase in their longitudinal estimate ( f r o ~ n.10 to .30) when an instrument was used for union status. T h e results d o not support tlie
"hounding" results of Freeman. This assumes that agent-induced endogeneit) o r selection will bias OLS cross-section estimates upward,
\\.hich conforms rvith many researchers' priors, and quite persuasive
arguments have heen advanced in its favor. At the sirnplest theoretical level, these arguments are based on the notion that "better" workers \\.ill want to join unions. It might also be argued that the large
differentials i.eported in the corrected results suggest large rents that
should induce widespread and obvious queuing activity that, it is suggested, is not universally observed.
T h e results in table 3 d o sho~vlarge differentials and do suggest
that the corrected differentials are larger than the OLS differentials.
In the renniinder of this section a structure is outlined that is consistent with these results and that at the same tirne attempts to deal ~+.ith
tlie "bounding-result" arguments sketched above. It appears to he
quite generally agreed that union status is not exogenous. 111 that
case, what corrected estimates attempt to measure is the outcome of
660
J O U R N A L O F POI.ITICAI. E C O N O ~ I Y
an experiment conceptually different from what OLS estimates measure. In particular, corrected estimates measure the difference in
earnings (conditional on the exogenous variables) of individuals randornl) assigned to the union and nonunion sectors. T h e OLS estiniates measure the difference given the nonr-andom outcome of the
sector assignment process. T o simplify the comparison, consider an
economy of individuals with exogenously endowed human capital
that is unobserved. (Assunie no observed exogenous variables.) T h e
endowment of human capital permits certain tasks to be performed.
Let the nonunion sector be characterized by production processes in
rvhich individual outputs are readily observed, and one in which there
is scope for individual "initiative" to be productive. Let the rvage rate
then depend on the individual output: In
= E Z , Let the union
sector be characterized by production processes at the other extreme
rvhere (for simplicity) a teani structure for the process renders all
operatives o n the production line equally pr-oductive. Let the wage
I-ate in the union sector thus be the same for all: In W [ , ,= EL,, = EL..
Consider the union differential estimated by one of the methods (e.g.,
IM) that correct for the endogeneit) of union status. This differential
corresponds to that obtained from a randotn assignment to sector.
T h e expected value of this differential is
T h e differential obtained from OLS would be the difference between
the average wage of those actually in the union sector and the average
of those in the nonunion sector. Its expected value is
T h e expected difference betmeen the corrected estimator and OLS is
therefore
This will be positive, provided E(E.,.~C', = 0) > EE,\.. There are two
mechanisms that determine E(E,,.~C', = 0). T h e first is the choice by
Lvorkers of offering themselves to the union sector. This ensures that
E , ~ ,<
. This mechanism therefore tends to raise E(E,,.\ C', = 0) by
removing the upper tail off'(^.,.) from the union queue. T h e second
mechanism is the choice by eniployers of workers from the union
queue. Given the simplifying assumption that all workers are equally
productive in the union sector, so that employers would be indiffer-
C S I O K IVAGE EFFECTS
FIL. 1.-Urliorl
tents a n d differentials
ent about who works in the union sector, the first effect would determine the outcome. If nwrkers were randomly selectetl fronn the queue
o r those with the most to gain are most likely to be in the union, then
E ( E , ~ ?CT,
I = 0) > E E , ~ T
. h e outcome of this case is an estirrlated union
differential that is larger \\.hen endogeneity is accounted for than the
one that occurs with OLS. Even if employers choose only the ~ i o r k e r s
with the highest values of E . ~fl-om
,
the queue, it may still be the case that
E ( E , ~ ,CT,
I
= 0) > E E .because
~
of the removal of the upper tail off‘(^,,)
by the first mechanism.
Consider next the problem of rent size and queues. T h e differential estimated by correction methods is not an estimated rent for the
marginal individual. Suppose that in equilibrium E I - > E:, where E,: =
rnin(es, I C', = 0). T h e n this individual ~\.ouldgain E( - E,: by joining
the union sector, hut he is not permitted to d o so because of the queue
rationing process. This is also the maximutr~rent that could be earned
by any individual by assumption. H o ~ r e v e reven
,
this rnaxirnum rent
map actually be smaller than the corrected union differential, as illustrated in figure 1. If those with t h r largest gain enter "first," the
fraction unionized is F(E,$), \\.here F ( . ) is the cumulative distribution
function of'^.^. T h e union wage is E ~ . T. h u s the corrected differential
is length b; the OLS differential is length a , and the maximum rent is
c, where c < b. T h e average gain for those indi\,iduals rlot in the
662 JOURNAI
O F POI.ITl(,.AI.
ECONOhl\
union, but \\.ishing to join, is
This highly si~nplifiedexample illustrates that, given a nonhierand E , ~ the
,
corrected estimates of union
archical interpretation of
differentials may be higher than the 01,s estirnates. hloreover, the
concerns over apparently large rents to union rne~nbershipassociated
with the large differentials estimated by endogeneity-corrected methods may he unfounded since these differentials will not in general he
rents in the standard sense.
VI. Testing Alternative Hypotheses Regarding
the Union Process
T h e results of the earlier sections, along ~ i t hother results in the
literature, reject the hypothesis of exogeneity for union status. T h e
task remains of identifying the specific endogenous process. As
noted in Section 11, the IV and I M estimators are based on different
specifications of the underlying process. X comparison of these estimators may therefore provide evidence on some of these processes.
A Wu-Hausman test \%.asp e r h r m e d on a comparison of the union
coefficients estimated by IS1 and IV for the zero slope interactions
model. T h e results shotv n o significant difference in the 1x1 and IV
estimators. For 1979 the difference between the coefficients was .0150
with standard. error ,0605. For 1981 the difference was .012f ~ . i t h
standard err-or ,0682.
T h e similarity of'the IV and IM estimates suggests that the simpler
fixed effects (of union status) framework may suffice for the union
problem. T h e IV estimator is consistent in the fixed effects framework and also in the random effects frarne~%.ork
under uncertainty,
\\.hich yields the same econonletric properties as the fixed effects
framework. T h e IhI estimator is consistent under normality. Consistency of the IV estimator follows if \', satisfies
E1',IZ,
=
E ( E [, ( C ,= l)Pr(C, = 1, Z,)
Under normalit\ this become,
4
E;(E,\,IC',= O)Pr(C, = 0, Z,)
=
k.
(Xi3
UKION \VAGE EFFECTS
Thus, in general, E17,1Z,depends on Z, unless a(€(, E)
that case, howe\ er, k = 0 and
=
a ( € \ E, ) ; in
T h e select~onterms must therefore be opposlte In sign. T h ~ follor\.s
s
~mnied~atel\,
in the fixed effects fr'inieuork where E I = E \ for all I
T h e restriction k = 0 mav be tested b\ exnmirling the equl\ d e n t null
h\ pothes~c:
For 1979, D = ,2544 with standard error .07 17; for 1981, L) = ,2445
rvith standard error ,0566. Since this test is carried out under the
assunlption of union endogeneity, the standard errors used are those
obtained from the covariance matrix corrected tor heterogeneity and
estimated inverse Mills terms. Thus, for both 197'3 and 1981, k > 0.
Positive selection into both sectors, under no1-rnality, however, is not
compatible with EI;,IZ, = El', and hence is not con1p;itible with a
consistent I V estimator based o n Z,.
An alternative interpretation of the similarit) of the I V and IS1
estimates is that normality fails but nevertheless approximates the
true distribution. Under normality, the conditional distul-bances are
pr-oportional to the hazard f'(.)!F(.) or f ( . ) /[ 1 - F(.)].
Suppose instead that
E(E( I CT,
=
1, Z,)
1 C',
=
o, z,) =
=
aI
Pr(CT,
=
1 , Z,)
and
E(E\
a\
Pr(C',
=
0, Z,)'
'4s \\.ith f ( . ) ! F ( . ) , l!Pr(C', = 1, Z,) decreases as Z, is varied to increase
the probability of sanlple inclusion. If this holds,
so that I V will be consistent. For the control function approach, all
that is needed for consistency is a consistent estimate of Pr(CT, = 1 , Z,).
Since the distribution is unknown, this may be estimated nonparanletrically.
To examine this interpretation, nonparanietric estimates of Pr(Cr, =
1 , Z,) were obtained, and the selection terms [Pr(C', = I)]- and
[Pr(C', = O ) ] for union and nonunion workers, respectively, were
used for the control f ~ ~ n c t i oapproach
n
in place of the inverse YIills
ratio terms.'"he
results of' this analysis are presented in table 5 .
''
T h e nonp;iramrtric estimates rvere obtained trorn a progt-an1 \\-I-itcen),t Paul KIIstone that use5 a kernel eqtlmatlon approach described in Ullah a n d S i ~ ~ g(1989)
li
-
-
E c t ~ r n a t ~ oRlethod
n
1970 cross section:
Nonparanletric
control function
1x1
I \'
198 1 c l - o s section:
Nonpara11irt1-ic
contl-ol function
I \1
I\'
9\
k
4;
1 .04 13
1 04!)0
.02(i5
1 Oti78
...
1.0433
1 .(I027
...
.
.
...
...
1 .O<IXCI
.08.57
1. 12!)0
...
...
...
1.1520
-%-??
,3591
.2$11)3
,284.5
.33 8 .5
,2706
,2579
Under this interpretation, I\' estimation of equation ( 1 1) yields a
consistent estinlate of y.\. = (y.,, + k ) as the constant term. T h e control
fr~nctionyields an estimate of' y,,, as the constant term and k = a ( ,
q v as the sum of the coefficients o n the selection ternis in the union
arld nonunion sectors. T h e control function and IV estimates may
thus be compared for co~npatibilityin this regard, together with their
respecti\-e estimates of the rnairl parameter of' interest ( y ( . - y,,,).
T h e results in table 5 show that the new controi function estimates
are compatible with the IV estimates, both ~\.ithrespect to the estimate
o f t h e union dif'ferential (7,. - y,,,) and \iith respect to the predicted
difference in the constant terms. T h e esti~natesof k for both years are
positive. T h e difference between the IV constant, y;,, and the nonparametric control function constant, y,,,, should thus be positive.
This is true for both 1979 and 1518 1 data. For 1979 the estimate of k
accounts for one-half of the difference in the constant terms; for 198 1
the estimate of k accounts f i ~ four-fifths
r
of the constant term difference. For both years the estimates of the union differential (y,, - y,,,)
are close. Finally, conlparison with the IM control function estimator
indicates that the selection terrns computed f i x this esti~natorl\.ere
sufficiently close approxi~nationsto the alternative selection terms to
yield similar results for both (y,. - y , ) and y,\ . T h e positive values for
k are also consistent ~\.iththe tests for D > 0 above.
+
VII.
Conclusions
O n surveying the recent literature on union differentials, Lewis
(1986) reached the pessirllistic conclusion that, in essence, nothing
had heen learned about union effects on wage rates from the many
scridies designed to deal with the endogeneity of' union status. T h e
same conclusion is reached in Freeman and Medoff (1982). T h e basic
U N I O K WAGE EFFECTS
665
objection of these authors to the "other-than-OLS" studies is that they
exhibit no stable pattern, that they fluctuate ~\.ildly,and that therefore
they are unreliable. In this paper, existing evidence has been reviewed and new evidence provided that supports a much more positive view of what can be learned fronn the attempts to deal with union
endogeneity. With the large individual data sets, there is a strong
pattern to the "corrections" for endogeneity. T h e unconditional differential (for instance, that arising fronn workers randomly assigned
to sectors) estimated by correction methods is systematicall) larger
than the conditional (on union status) differentials estimated by OLS.
This pattern holds ~\.hetherthe attempt to deal with endogeneity uses
cross-section methods such as I V o r control function approaches o r
longitudinal estimators.
T h e results presented in Section 111 imply that the cross-section
OLS estimates are biased downward hy selectiorl effects. This conflicts
with the priors of man) researchers, on the basis of the notion that
"better" workers will be picked from a queue by unionized firms.
This, however, will result only in an upward bias for OLS under a
hierarchical notion of omitted "abilit)"; if this is abandoned in favor
of a comparative advantage specification, OLS may be biased downward. T h e idea that the corrected estimates are larger than 0 1 , s
estinnates is also considered a problem by many researchers because of
the large implied "rents." If union status I\.as exogenous, large estimated difftrentials would be very troublesome because of their association with rents in that setting. Large rents, especially in the absence
of obvious and widespread queuing activity, would clearly be difficult
to explain. However, if union status is endogenously determined, the
relation between the estinlated differentials and rents in the standard
sense is much more complex: in particular, a large differential does
not imply large rents.
Finally, the Canadian longitudinal data used in Section I11 were
also employed to construct extensive tests on the exogeneity of union
status. With exogeneity rejected, Section \.'I exploits the conflicting
properties of different estimators to examine hypotheses on the process governing union status. An insignificant difference bet~veenIV
and IM estimators suggests that simple "fixed effects" (of union
status) models may be good approximations for the union problenn.
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The Joint Determination of Union Status and Union Wage Effects: Some Tests of
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Chris Robinson
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[Footnotes]
6
The Endogeneity of Union Status: An Empirical Test
Gregory M. Duncan; Duane E. Leigh
Journal of Labor Economics, Vol. 3, No. 3. (Jul., 1985), pp. 385-402.
Stable URL:
http://links.jstor.org/sici?sici=0734-306X%28198507%293%3A3%3C385%3ATEOUSA%3E2.0.CO%3B2-I
10
Panel Data and Unobservable Individual Effects
Jerry A. Hausman; William E. Taylor
Econometrica, Vol. 49, No. 6. (Nov., 1981), pp. 1377-1398.
Stable URL:
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14
Errors of Measurement in Statistics
W. G. Cochran
Technometrics, Vol. 10, No. 4. (Nov., 1968), pp. 637-666.
Stable URL:
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References
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Errors of Measurement in Statistics
W. G. Cochran
Technometrics, Vol. 10, No. 4. (Nov., 1968), pp. 637-666.
Stable URL:
http://links.jstor.org/sici?sici=0040-1706%28196811%2910%3A4%3C637%3AEOMIS%3E2.0.CO%3B2-C
The Endogeneity of Union Status: An Empirical Test
Gregory M. Duncan; Duane E. Leigh
Journal of Labor Economics, Vol. 3, No. 3. (Jul., 1985), pp. 385-402.
Stable URL:
http://links.jstor.org/sici?sici=0734-306X%28198507%293%3A3%3C385%3ATEOUSA%3E2.0.CO%3B2-I
The Determination of the Union Status of Workers
Henry S. Farber
Econometrica, Vol. 51, No. 5. (Sep., 1983), pp. 1417-1437.
Stable URL:
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Longitudinal Analyses of the Effects of Trade Unions
Richard B. Freeman
Journal of Labor Economics, Vol. 2, No. 1. (Jan., 1984), pp. 1-26.
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Specification Tests in Econometrics
J. A. Hausman
Econometrica, Vol. 46, No. 6. (Nov., 1978), pp. 1251-1271.
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Panel Data and Unobservable Individual Effects
Jerry A. Hausman; William E. Taylor
Econometrica, Vol. 49, No. 6. (Nov., 1981), pp. 1377-1398.
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A Note on Error Components Models
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Econometrica, Vol. 39, No. 2. (Mar., 1971), pp. 383-396.
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Union Wage Differentials in the Public and Private Sectors: A Simultaneous Equations
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Chris Robinson; Nigel Tomes
Journal of Labor Economics, Vol. 2, No. 1. (Jan., 1984), pp. 106-127.
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