Full text of DOI:10.1111/1467-937X.00147

Review of Economic Studies (2000) 67, 609–633
0034-6527͞00͞00290609$02.00
2000 The Review of Economic Studies Limited
Differences in Wage Distributions
Between Canada and the United
States: An Application of a
Flexible Estimator of
Distribution Functions in the
Presence of Covariates
STEPHEN G. DONALD
Uniûersity of Texas
DAVID A. GREEN
Uniûersity of British Columbia
and
HARRY J. PAARSCH
Uniûersity of Iowa
First ûersion receiûed October 1995; Final ûersion accepted June 2000 (Eds.)
We construct a tractable, flexible-functional-form estimator of cumulative distribution func-
tions for non-negative random variables which admits large numbers of covariates. The estimator
adopts and extends techniques from the spell-duration literature for estimating hazard functions
to distribution functions for wages, earnings, and income. We apply these methods to investigate
sources of wage inequality for full-time male workers between Canada and the United States,
finding that the Canadian wage density has a thinner left tail because low-educated workers have
higher pay and a thinner right tail because of a lower proportion of highly-educated workers.
Unions appear to play a large role in these outcomes.
1. INTRODUCTION
Canada and the United States, in spite of broadly similar institutions as well as human
and physical resources, have very different levels of income inequality. Inequality in the
Canadian income distribution has been consistently below that in the American distri-
bution over the last 20 years and did not increase as rapidly in the 1980s. In part, this
difference is due to differences in the tax and transfer systems of the two countries. How-
ever, Canada also has a more equal distribution of male pre-tax annual earnings and,
during the 1980s, experienced a smaller increase in inequality in this distribution than the
United States.1 These facts suggest non-trivial differences in the labour markets of the
1. See Hanratty and Blank (1992) as well as Blackburn and Bloom (1993) on these points.
609

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REVIEW OF ECONOMIC STUDIES
two countries. Pinpointing the sources of the differences can provide insights into the
workings of labour markets in the two countries and the effects of differing labour-market
institutions. The substantial similarities between the two countries heighten the usefulness
of such a comparative exercise because differences attributed to a factor such as relative
patterns of unionization are more likely to be a reliable measure of union impacts than
one would obtain from a comparison of less similar countries. In this paper, we investigate
the sources of differences in the wage distributions of Canada and the United States by
decomposing differences in the entire male wage distributions for the two countries in
1989.
To understand the sources of differences in the wage distributions of the two count-
ries, we require a way of examining how differences in the distributions vary by subgroups
in the populations. A common approach to this problem is to evaluate scalar measures
of dispersion for various subgroups in the population. As discussed in DiNardo, Fortin,
and Lemieux (1996), scalar measures mask details in wage, earnings, and income distri-
butions that are needed to identify the effects of labour-market institutions such as mini-
mum wages. Furthermore, scalar measures place different inherent weights on different
parts of the distribution.2 To avoid this sort of masking and prejudging, one needs tech-
niques for estimating the entire distribution in the presence of covariate effects. Our
second goal in this paper is to construct a tractable and flexible estimator of conditional
density and cumulative distribution functions for non-negative random variables that
admits large numbers of either discrete or continuous covariates.
Our approach to conditional-density estimation is to use a hazard-function based
estimator to introduce covariates in a consistent manner, and then to make the necessary
transformations to obtain an estimate of the associated conditional density function.
Below, we argue that, given the large literature and previously-developed estimation rou-
tines for hazard-based duration models, this is likely to be a very tractable and accessible
approach for many empirical economists. We also demonstrate that our estimator pro-
vides very flexible density estimates, in the sense that it imposes a minimal number of
restrictions on the shape of the density for any value of the covariate vector, and that the
estimates have the desirable statistical property of integrating to one. Just as important,
generating standard errors for these estimates is also very straightforward since they are
based on the variance–covariance matrix of the parameter estimates derived from the
estimation of the hazard-function specification. These advantages come at the cost of
some difficulty in interpreting the estimated parameters of the model but, as we shall see
in the case of American and Canadian wages, the estimator is nonetheless a useful tool
for examining and decomposing wage, earnings, and income distributions in the presence
of covariates. Finally, it is worth noting that, while we emphasize the usefulness of the
estimator for examining the conditional densities of wages, earnings, and income, the
formulae derived here could be equally well used to derive the conditional density func-
tions and cumulative distribution functions (CDFs), along with associated standard
errors, for spell durations such as the duration of unemployment. In this sense, our work
can also be viewed as a direct extension of the spell-duration literature.
2. DENSITY ESTIMATION IN THE PRESENCE OF COVARIATES
We require an estimator of the density function that incorporates the effects of covariates
across the entire range of the dependent variable. We desire an estimator with three main
2. For example, comparisons of Gini coefficients place weight on differences in the middle part of the
distribution, while top͞bottom-decile differences place weight on differences in the tails.

DONALD, GREEN & PAARSCH
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611
properties: first, tractability in that it can be adapted to a large number of alternatives
and can be easily understood and implemented by researchers with standard training in
applied econometrics; second, flexibility in the way covariates affect the shape and
location of the density, so that the estimator imposes a minimal number of restrictions on
shape and location; and third, consistency, in that estimated probabilities associated with
any subset of the dependent variables range lie in the interval [0,1] and that the overall
density-function estimate integrates to one.
Our approach is to translate techniques developed for estimating spell-duration distri-
butions to the estimation of wage, earnings, and income distributions.The main building
block of this estimation approach is the hazard function. Consider a non-negative random
variable Y with associated probability density function f(y) and CDF F(y). The hazard
function h(y) is then defined by the conditional probability
f(y)
f(y)
h(y)G
≡
,
(2.1)
[1-F(y)] S(y)
where S(y) is the survivor function. One key result from the literature on spell-duration
estimation is that the conditional nature of h(y) makes it easy to introduce flexible func-
tions of the covariates and to entertain complex shapes for the hazard function, while still
meeting the consistency objectives set out above. From (2.1), one can see that the hazard
function is simply a transformation of the probability density function. Thus, flexible
estimates of the hazard function can be translated into flexible estimates of the density
function in a manner discussed in detail below. In fact, our direct interest is not in the
hazard-function estimates, but in the density-function and CDF estimates we can derive
from them. We make no attempt to interpret the hazard-function estimates themselves,
but treat the hazard function effectively as a flexible functional form that allows us to
generate tractable, flexible, and consistent estimates of the functions in which we are
primarily interested.
We introduce covariates using a proportional-hazards model. Specifically, the hazard
function for person i conditional on xi, a particular realization of the covariate vector Xi,
is
h( y͉xi)Gexp (xiα )h0(y),
(2.2)
where h0(y) is the baseline hazard function common to all individuals and α is a vector
of unknown parameters conformable with xi. Two important shortcomings of this specifi-
cation are the restrictions that individuals with very different covariate vectors have haz-
ard functions with the same basic shape and that any particular covariate shifts the entire
hazard function up or down relative to the baseline specification—strong restrictions
indeed. To weaken these restrictions, we partition the range of Y into P subintervals ΩpG
[ypL, ypU), where Ωp∩ΩqG0 for all p≠q with *PpG1 ΩpG[0, S) and allow the covariate
effects to vary over these subintervals. In particular, following Gritz and MaCurdy (1992),
we replace xiα in (2.2) with
P
g[x (Ω ),β]G
1( y Ωp)xi(Ωp)βp,
(2.3)
∑
i
p
pG1
where 1( y Ωp) is an indicator function of the event ( y Ωp), equalling one if Y is con-
tained in the set Ωp and zero otherwise, xi(Ωp) is a (1Bk) vector of covariates defined on
the set Ωp, βp is a (kB1) vector of unknown parameters, and β is the vector
(β1T, β2T, . . . , βPT)T, a (KB1) parameter vector with K equalling (PBk). Within this
specification, covariates can shift the hazard function up over some regions and down

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REVIEW OF ECONOMIC STUDIES
over others, providing the possibility of quite different shapes for the hazard function for
individuals with different covariate vectors. The covariates are written as a function of
Ωp, the relevant range of Y, to allow for covariates that vary with wages, earnings, or
income, such as the tax rate or a dummy variable indicating whether Y equals the value
of the relevant minimum wage. This is effectively the same as allowing for covariates that
vary with spell duration in the spell-duration literature.
For both tractability and flexibility reasons, we approximate h0(y) using a step func-
tion. In particular, we create a set of dummy variables corresponding to each of the
segments [yj, yjC1) for jG1, . . . , J where J is finite, and estimate a parameter associated
with each of these segments. The properties of a proportional-hazards model with this
form of baseline specification are discussed in Meyer (1990). The advantage of this
approach is that, with a sufficiently large value for J, it can capture complicated shapes
for the hazard and associated density functions, including spikes such as those induced
by minimum wages. It also makes the transformation from hazard to density estimates
very straightforward. The main disadvantage is that spikes induced by the tendency of
individuals to report certain round numbers such as $10 per hour for wages or $20,000
per annum for earnings are also evident in the final estimated density figures. If these
reporting spikes are not of interest in themselves, then they can be a distraction when
examining the overall shape of the density. Once the estimated density is calculated, the
importance of these spikes can be reduced using smoothing methods.
Our estimation approach is first to select the J ‘‘baseline segments’’ defined above.
We choose quite fine segments ($0.25 segments for wages in the estimation below) between
the 5-th and 95-th percentile of the overall wage distribution and broader segments in the
two tails. This is similar to adaptive-kernel estimation techniques that permit variable-
window widths to reduce the roughness of estimates in the tails of the density. Next, we
choose the P ‘‘covariate segments’’ across which the parameter vectors are allowed to
vary. The exact way in which we select J and P is discussed in Subsection 2.1. Given these
decisions, we maximize the likelihood function corresponding to data for N individuals.
The data are (Xi,Yi)NiG1 where Yi is the wage (or earnings) for individual i and Xi is a
(1B k) vector of covariates that may be continuous or discrete.
To construct the likelihood function, note that the probability of observing a wage
in the j*-th baseline segment can be expressed as
f( yj*͉x)G S( yj*-1͉x)h( yj*͉x).
It equals the probability of observing a wage at least as large as the upper limit of the
( j*A1)-st segment, the survivor function value at ( j*A1), times the probability of
observing a wage in the j*Ath segment conditional on it being at least as large as the
upper limit of the ( j*A1)-st segment, the hazard function value at yj*. If the wage value
is top-coded (i.e. if all we know is that the wage value is larger than some specified value
yJ), then the probability of observing that top-coded wage is the probability that the wage
is greater than or equal to the top-code value, S( yJ͉x). If one assumes an underlying
continous hazard function with the simple proportional-hazards form, then, following
Meyer (1990), one can show that the hazard for a discrete baseline segment is given by
P
h( y ͉x)G1Aexp −exp {γ C
1[Ωp(j*)]xβp} ,
∑
j*
j*
pG1
΂
΃
where
γ jGlog
h0(u)du ,
(2.4)
΄
Ύ
΅
J
j

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
613
and
J is the set of Y-values corresponding to the j-th baseline segment. As before, Ωp is
j
the set of Y values corresponding to the p-th covariate segment, but notice that we have
replaced the p subscript with p( j). We do this to signify that because the covariate seg-
ments consist of sets of baseline segments, as j changes, the value of p may also change.
The corresponding survivor function is given by
*
P
S( y ͉x)Gexp(− j A1 exp{γ C
1[Ωp( j)]xβp}),
i
∑
∑
j*
jG1
j
pG1
so the logarithm of the likelihood function is then
N
*
*
L
(β, γ )G
[1(YiFyJ) log {1Aexp [−exp (γ j Cg{Xi[Ωp( j )], β})]}
∑
i
i
iG1
*
j A1 exp (γ jCg{Xi[Ωp( j), β]})],
(2.5)
i
A
∑
jG1
where the dependent variable for individual i falls in the j*-th baseline segment, and Yi is
i
less than yJ if Yi is not right censored (i.e. not top-coded). The vector γ contains the J
baseline parameters, where the notation γ j denotes the element of γ corresponding to the
j-th baseline segment. For consistent estimates, we require that covariate values not
change within the baseline segments; they may, however, vary across baseline segments.3
For the j-th baseline segment, estimates of the hazard- and survivor-function value
ˆ
ˆ
can be obtained by replacing the β and γ with their estimated values β and γ obtained
from maximizing (2.5). An estimate of the density is then
ˆ
ˆ
ˆ
f( yj͉x)GS( yj͉x)AS( yjC1͉x).
We now wish to derive an expression for the variance-covariance matrix of the den-
sity estimates. Before doing so, we note that the step-function approach to specifying the
baseline hazard function involves estimating a large number of parameters corresponding
to the baseline segments. In principle, as the number of observations approaches infinity,
both J and K, and hence the number of associated coefficients, could also approach infin-
ity. This introduces standard incidental-parameter problems in deriving the variance-
covariance matrix of the estimated parameters. In the following derivations, we follow
related work (e.g. Heckman and Singer (1984)) by treating the number of baseline seg-
ments as fixed.4 Of course, the key issue is whether the derived formulae provide a good
approximation to the actual variance-covariance matrix in standard-sized samples. Below,
in Subsection 2.3, we provide some evidence for this point.
Let δ be the (KCJ) vector of unknown parameters (βT,γ T)T, consisting of the K
parameters in β and the J baseline parameters in γ . Denote the true value of δ by δ0 and
ˆ
let δ be the maximum-likelihood estimator (MLE) that maximizes (2.5). Our first assump-
ˆ
tion concerns the properties of the MLE δ.
ˆ
Assumption 1. Assume the model is described by (2.5) and the MLE δ has the
following properties:
p
ˆ
(i) δ → δ0 ;
p
ˆ
ˆ
1
(ii) N(δAδ0) →
(iii) a consistent estimator of
N [0, V (δ)];
ˆ
V
(δ) exists.
3. Note that to implement this estimator we require some right censoring. Thus, we typically artificially
censor our data at the 99-th percentile, acting as if for values above the 99-th percentile we only know that they
are at or above the 99-th percentile and assign the indicator function 1(YIFyj) to zero.
4. One might justify this approach by noting that a natural lower limit exists to how small baseline seg-
ments can become because earnings are not paid in fractions of cents. Thus, an upper bound exists on the
number of baseline-segment parameters to be estimated.

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REVIEW OF ECONOMIC STUDIES
This assumption can be verified using the standard conditions (such as those sufficient to
verify consistency and asymptotic normality of the MLE) discussed in Kalbfleisch and
ˆ
Prentice (1980, Section 4.6). The form of
V
(δ)is also given in Kalbfleisch and Prentice
(1980, Section 4.6), and consistent estimates can be obtained by replacing the unknown
ˆ
parameters appearing in
V
(δ) with consistent MLEs.
We now turn to deriving the limiting properties of the aforementioned density esti-
mator. Toward this end, we first define some notation. Let
∂
∆jG S( yj͉x),
∂δ
which is a (KCJ) column vector with elements
∂
∂β k
P
S( y ͉x)G−S( y ͉x) jA1 1[Ωp(i)] exp (γ ) exp {
1[Ωp(i)]xβp}xT,
∑
∑
j
j
iG1
i
pG1
and
−S( yj͉x)h( yl͉x) for lFj;
∂
S( yj͉x)G
Ά0
otherwise.
∂γ l
Also, let Θj denote (∆jA∆ jC1). The asymptotic properties of the estimators for the survivor
and density functions are given as follows:
Lemma 1. Giûen Assumption 1,
p
ˆ
S(yj͉x) → S( yj͉x),
d
[0, ∆Tj
V
(δ)∆j],
ˆ
ˆ
1
N[S( yj͉x)AS(yj͉x)] →
N
p
ˆ
f(yj͉x) → f( yj͉x),
d
ΘT
[0, Θj
Θ
ˆ
ˆ
1
N[ f(yj͉x)Af( yj͉x)] →
N
V
(δ) j].
The proof of this lemma involves straightforward applications of the continuous mapping
theorem and the delta method, and follows given Assumption 1(i) and the nature of the
ˆ
ˆ
dependence of S( yj͉x) on δ.
2.1. Discussion of the estimator’s general features
To understand better the nature of the density estimator, consider an example with no
covariates. In this case, the estimator is just the Kaplan–Meier estimator of the hazard
function at each baseline segment. For a given segment j, the Kaplan–Meier estimate
equals the number of observations with wage values in the baseline segment range dj
(called ‘‘failures’’ in the duration literature), divided by rj, the number of observations
with wage values in the j-th segment range or in segments with higher values (called the
‘‘risk set’’ in the duration literature). Using the estimated hazard function and the formu-
lae given above, one can show that the estimated mass in the j-th segment equals dj divided
by the total sample size N. This equals the sample histogram estimate for the j-th segment.
In the presence of covariates, we could estimate a histogram for every available value of
the covariate vector. With complete flexibility (i.e. with J equalling P and each possible

DONALD, GREEN & PAARSCH
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615
combination of covariate values represented by a dummy variable), implementing our
estimator is the same as estimating a separate histogram for each possible combination of
covariate values. We implement a restricted version in which the impact of covariates are
assumed to follow a particular (flexible) functional form. Thus, our estimator can be seen
as a transformation of the sample histogram where the transformation is used to ensure
that covariate effects are introduced in a consistent manner.
The fact that our estimator is strongly related to the histogram estimator is useful in
considering both its strengths and its weaknesses. Like the histogram, this estimator can
approximate any density-function shape and has no difficulty incorporating outliers or
top-coding; it goes beyond the histogram by incorporating covariate effects in a flexible
way. At the same time, it is quite tractable. In our experience, convergence for this likeli-
5
ˆ
hood function can be obtained easily and quickly. The transformations from δ to esti-
mates of the hazard function and then to the density functions described above are
straightforward.
The estimator also shares some of the disadvantages of histograms. One difficulty in
implementing these types of estimators is that the choice of bin size (or baseline segment
size, in our case) is somewhat subjective. This problem is amplified in our context by the
fact that we estimate parameters associated with hazard functions, but are ultimately
interested in the density function. One possible response to this problem is to use the
formulae for optimal bin size developed in the histogram literature; e.g., see Scott
(1979).We do not adopt this approach because we prefer to use varying segment widths
rather than the constant widths assumed in the optimal-bin-width literature and because
the introduction of covariates complicates choosing a segment size based on the range of
the unconditional distribution of the dependent variable. Deriving a formula for optimal
segment widths is beyond the scope of this paper, and the large numbers of observations
and parameters we use make techniques of cross-validation computationally prohibitive
at present. Instead, based on substantial Monte Carlo evidence, some of which is reported
below, and hands-on experience in applying the estimator to real data, we have developed
a procedure that provides a rough rule-of-thumb for selecting segment widths as follows:
first, regress the dependent variable, in our case the wage, on the full set of covariates
using least squares; second, recover the residuals from the first regression and regress the
squared residuals on the covariate vector; and third, use the estimates from the second
regression to form estimates of the conditional variance for a set of 10 to 20 common
covariate vector values. Apply a middle value from the range of calculated conditional
variances to Scott’s formula for the optimal bin width with Gaussian data
hoptG3·49sN−1͞3
where s equals the square root of the chosen middle conditional variance and N is the
total sample size. Employ this bin width between the 5-th and 95-th percentiles of the
unconditional wage distribution, but use wider segments in the tails. This approach sets
segment widths which are narrow enough to fit even relatively narrow conditonal density
functions and admits baseline segment parameters which are estimated using all of the
observations.
A second potential difficulty common to both histograms and our estimator is that
resulting density estimates can be distractingly ‘‘spikey.’’ For our estimator, this is a direct
5. The exception to this claim is that finding useful starting values for the parameters estimates sometimes
takes effort. A quick way to get starting values is to transform Kaplan-Meier estimates for the baseline segment
values and set all parameters for the covariates to zero.

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REVIEW OF ECONOMIC STUDIES
result of using a discrete baseline-hazard function and is another way in which our esti-
mator is similar to the sample histogram. One response is to use standard techniques to
smooth the estimated densities after they are calculated. Of course, the extent to which
one pursues such smoothing depends on the question being investigated. For example, in
work where minimum-wage effects are of interest, the researcher is most interested in the
location and the size of spikes in the wage distribution, so smoothing is unwarranted.
Finally, we note that the estimates of the hazard function for wages, earnings, or
income are difficult to interpret. For weekly unemployment-spell data, the hazard function
at week τ is the proportion of spells ‘‘at risk’’ to end in week τ (i.e. that have not ended
or been censored before τ) that actually end in week τ. In the earnings context, it is
unclear whether a $50,000-a-year job was ‘‘at risk’’ of being a $10,000-a-year job. Thus,
some standard interpretations from the duration literature do not carry over to this con-
text. However, the conditional probability given in (2.1) is still defined and maintains its
attractive properties as a basis for flexible density estimation in the presence of covariates.
In particular, software currently used to estimate duration models and the estimation
experience developed with such models can be directly translated to this context. Thus,
the estimator represents a trade-off: interpretability of parameters for tractability and
flexibility. The key disadvantage of poor interpretability of parameters is that those par-
ameters, and potential restrictions on them, are difficult to relate to economic theory.
Nonetheless, the estimator is useful for uncovering patterns in the data, as we demonstrate
in the empirical investigation below.
2.2. Comparisons to alternatiûe estimators
Of course, ours is not the first attempt to incorporate covariates in ways that affect more
than the first moment of a distribution. Approaches to this problem vary from fully para-
metric, where a specific functional form is assumed for the density of interest and the
associated parameters are made to be functions of covariates, to fully non-parametric,
where the data are divided into subsamples corresponding to all possible combinations of
covariate values and a density estimate is generated for each subsample using a non-
parametric approach such as kernel-density estimation. The parametric approach is trac-
table but risks distorting the information in the data by forcing them into an inappropriate
functional form. The non-parametric approach is flexible, but suffers from the ‘‘curse of
dimensionality’’ because as new covariate combinations are introduced, cell sizes quickly
become too small to provide a source of useful inference. It is also difficult to introduce
continuous covariates. For these reasons, researchers often turn to semi-parametric and
flexible-function-form approaches, trading some restrictions for ease of implementation.
A feasible alternative is quantile regression; e.g. see Buchinsky (1994). This approach
admits the estimation of the impact of covariates on any quantile of the distribution and,
given recent advances in software, can be implemented relatively easily, even with continu-
ous covariates. The key difficulty is a potential lack of consistency: the predicted value
of the p-th quantile for a given covariate vector could be larger than the predicted
( pC1)-th quantile value for the same covariate vector. This could be remedied by
imposing appropriate restrictions in the estimation, but then standard software packages
could no longer be used. We suggest that our approach, based on hazard function esti-
mation, is more familiar and thus more accessible to most empirical researchers.
One could also form a flexible estimate of the CDF using an ‘‘ordered-probit’’-type
specification. This would also require imposing restrictions to ensure one always generates

DONALD, GREEN & PAARSCH
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617
positive probabilities, especially with small categories and covariate effects varying over
the earnings range. Fortin and Lemieux (1995) have used such an estimator.
Another interesting approach is the weighted-kernel estimator introduced by
DiNardo, Fortin, and Lemieux (1996), hereafter DFL. The DFL approach focuses on
decomposing differences in the densities for two populations according to the effects of
specific covariates. To do this, they weight the kernel function for a particular individual
by the relative probability that a covariate of interest takes a specific value in the two
populations, conditional on the values of other covariates. Thus, in examining the impact
of declines in unionization on changes in the American wage distribution between 1979
and 1987, they create a ratio of the probabilities that an individual is unionized in 1979
and in 1987, given the individual’s covariate vector. This relative probability is then multi-
plied by the individual’s kernel-function values at each wage rate to create an estimate of
what the 1987 union-wage distribution would have been had the 1979 unionization pattern
been maintained.
DFL argue convincingly for the need to examine the entire distribution when con-
sidering explanations for changes in wage dispersion over time or across countries. We
build on their work; our approach has three advantages over the DFL approach. First,
the DFL technique is primarily a way of decomposing differences in distributions; it does
not provide a method of graphing density functions or CDFs for specific covariate vectors.
In contrast, our approach allows one both to graph such functions and to carry out the
same kind of decompositions as in DFL. Second, our approach is more tractable, using
techniques more familiar to most researchers. Third, DFL do not provide standard errors
for their density estimates, while in our approach standard errors are straightforward to
calculate.
2.3. Some Monte Carlo results on the performance of the hazard-based estimator
We now turn to implementing the hazard-based estimator using artificially-generated data.
We do this in part to provide evidence on how well the estimator fits known density
functions and in part to investigate the properties of the estimator as we vary its main
estimation choice parameters, the number of baseline segments and the number of covari-
ate segments. We generate data using a mixture of log-normal distributions as follows:
first,we draw 1000 pseudo-random observations from a standard-normal distribution.
Next, we draw 1000 pseudo-random observations from a uniform [0,1] distribution. Thus,
we have 1000 observations each with a standard-normal-variate value for the i-th obser-
vation, call it Ui, and a uniform-variate value for observation i, call it Xi. Without loss of
generality, we use the first 500 observations to create the value of the dependent variable
according to
YiGexp (0·5C0·5Ui),
while for the second 500 observations we create the value according to
YiGexp (0·5C0·5XiC0·5Ui).
Together, the 1000 observations on Y constitute a random variable which is distributed
as a fifty-fifty mixture of log-normal distributions. The advantage of using a mixture is
that it generates a random variable with a distribution the shape and location of which
changes considerably for different values of X. For values of X near zero, the conditional
distribution is very similar to the log-normal distribution. As X increases, the conditional

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REVIEW OF ECONOMIC STUDIES
distribution becomes increasingly bimodal. Fitting conditional distributions that vary this
much with the covariate value is a challenge for our and any other estimator.
Our approach to evaluating the hazard-based estimator is as follows. We first gener-
ated 100 samples of 1000 observations each in the manner described above. We then
selected a number of baseline segments J and a number of covariate segments P. We
maximized the logarithm of the likelihood function given in (2.5) for each of the 100
ˆ
samples separately and recovered the estimated parameter vector δ in each case. We then
set the covariate value to 0·7, a value that generates a visibly bimodal density function,
and created an estimate of the conditional density using the formulae presented above
ˆ
based on each of the one hundred δ vectors. We established a uniform grid with 50
intervals running from the 1-st percentile of the unconditional distribution of Y (approxi-
mately 1·35) to the 99-th percentile (approximately 2·80). For each of the 100 fitted density
functions, we took the difference between the fitted and true conditional densities at the
right endpoint of each of the 50 intervals just mentioned. Using these, we constructed the
mean-squared-errors for each of the 100 fitted densities. We then averaged the mean-
squared-errors over the samples to obtain an overall measure of closeness of fit corre-
sponding to the chosen values of J and P. We repeated this exercise for a range of values
for J and P. The resulting root-mean-squared-error values are reported in Table 1.6
TABLE 1
Root-mean-squared-error ûalues
Monte Carlo experiments of the hazard-based estimator
PG1
PG5
PG10
PG20
JG10
JG20
JG25
JG30
JG35
JG40
0·0587
0·0471
0·0456
0·0454
0·0458
0·0462
0·0499
0·0310
0·0302
0·0315
0·0322
0·0323
0·0499
0·0324
0·0305
0·0312
0·0314
0·0317
—
0·0328
0·0324
0·0329
0·0331
0·0334
The results in Table 1 indicate that, for any number of covariate segments, increasing
the number of baseline segments improves the fit of the estimated density up to a point;
past that point additional baseline segments erode the fit. This is the same trade-off wit-
nessed in choosing bin widths for histograms: with a few, broad segments it is impossible
to capture nuances in the shape of the density and the fit is poor, but with too many,
narrow segments the number of observations used to identify each baseline segment par-
ameter is small and excessive variability results. Increasing the number of covariate seg-
ments (moving across rows in the table) leads to improvements at first, particularly in
moving from a P of 1 to a P of 5; however, adding too many covariate segments worsens
the fit. The fact that the root-mean-squared-error value changes little between a P of 5
and a P of 20 suggests that the penalty for using too many covariate segments is not large.
The very poor relative fit of the cases with one covariate segment show that it is indeed
necessary to use something more complicated than the simple proportional-hazards model
6. In each case, we allowed one baseline segment in the left tail of the distribution covering the range of
Y up to 1·4 and baseline segments with left endpoints 2·65, 2·73, 2·8 and 2·9 in the right end of the distribution.
The baseline segments between 1·4 and 2·65 consist of (J-5) even-sized intervals. The right endpoints of the P
covariate segments are placed in each case at approximately P even quantile points from the unconditional
distribution of Y. Thus, when P equalled 10, we divided the covariate segments at the decile points. In each
case, we estimated using both the generated X variable and that variable squared. We have also evaluated the
root-mean-squared-errors at different values of the X vector with very similar results to those presented in
Table 1.

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
619
F
IGURE 1
Plots of mixture log-normal and fitted densities: 25 baseline segments, 5 covariate segments, XG0·7
when introducing covariates. The J-of-20, P-of-5 specification provides the minimum
value of the average root-mean-squared-error, though a range of similar values for J and
P appear to provide comparable fits.7
Visual depiction of the fit of the estimated densities is given in Figures 1, 2, and 3.
In Figure 1, we plot the true density when X is 0·7, and a set of fitted densities and
associated standard errors based on the estimations with J of 25 and P of 5, the preferred
specification based on Table 1. The line labelled median is the median of the 100 fitted
density values at each baseline segment. Similarly, the 5-th- and 95th-percentile lines corre-
spond to the 5-th and 95-th percentiles of the fitted-density values at each segment. We
also plot an estimate of the 95% confidence band around the median fitted line. The
confidence band equals the median fitted value at each baseline segment plus or minus
1·96 times an estimate of the standard error at that segment.8 In Figure 2, we recreate
Figure 1 for an X of 0·2.
Both Figures 1 and 2 indicate that the the hazard-based estimator fits the true density
very well. This is noteworthy since we have been able to fit very different shapes, associ-
ated with different values of the covariate, from the same estimations. It is also clear from
the figures that the estimated standard errors calculated from our formulae are close
approximations to the 5-th and 95-th percentiles of the distribution of the fitted density
values. Thus, both the density estimates and the estimated standard errors perform well
in this example. In contrast, in Figure 3 we recreate Figure 1 based on estimates of a J of
7. Using the rough rule-of-thumb methodology described in Subsection 21, the estimated conditional
standard deviation of Y conditional on X being 0·5 is 0·195, which implies an optimal segment width of approxi-
mately 0·06 from Scott’s formula. Even division of the range between the 5-th and 95-th percentiles of the
unconditional distribution of Y for the J-of-25, P-of-5 case implies segments with widths of 0·055.
8. To construct the standard errors, we calculated the standard errors at each baseline segment for each
of the 100 density-function estimates using the formulae presented in the text. The standard error for baseline
segment j used in the figure is the standard error for segment j associated with the fitted density chosen as the
median at that segment.

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REVIEW OF ECONOMIC STUDIES
F
IGURE 2
Plots of mixing log-normal and fitted densities: 25 baseline segments, 5 covariate segments, XG0·2
F
IGURE 3
Plots of mixture log-normal and fitted densities: 20 baseline segments, 1 covariate segment, XG0·7
20 and a P of 1. From Figure 3, one can see that insufficient flexibility, in the form of too
few baseline and covariate segments, leads to visibly poorer approximations to the true
density. Finally, in Figure 4 we plot the true and fitted cumulative distribution functions
for the J-of-25, P-of-5, and X-of-0·7 case. As one might expect from Figure 1, the median

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
621
F
IGURE 4
Plots of mixture log-normal and fitted distribution functions: 25 baseline segments, 5 covariate segments,
XG0·7
fitted CDF is very close to the true value, and the constructed confidence band is close to
the 5-th and 95-th percentiles of the distribution of fitted CDF values.
3. AN EMPIRICAL EXAMPLE: DIFFERENCES IN WAGE DISTRIBUTIONS
BETWEEN CANADA AND THE UNITED STATES
We now turn to our examination of differences between the 1989 American and Canadian
wage distributions. The Canadian data are from the 1989 Labour Market Actiûity Surûey
(LMAS). The American data are from the outgoing rotation group for the March 1989
Current Population Surûey (CPS). Both data sets are large samples of individuals contain-
ing information on income sources as well as individual demographic information such
as age and education. The two data sets are very similar in their target populations and
questions asked. Neither is strictly representative of the working population in the relevant
country, and assigned sampling weights from the surveys are used throughout our esti-
mation. In the American data, we obtain our measure of usual hourly wages by dividing
usual-weekly-wages by usual-hours-per-week. The LMAS questions respondents in Janu-
ary and February concerning up to five jobs held in the previous calendar year. The
information gathered includes union status, industry, occupation, usual hourly wage and
usual hours per week for each job. To match the American data, we use data from jobs
held in the first two weeks of March, 1989.9 The LMAS data are from the surveys taken
in January and February of 1990 and thus will include recall bias not present in the
American data. Otherwise, the two data sets are very comparable. We restrict the sample
to full-time (35 or more hours per week), non-self-employed, paid jobs held by males age
9. For individuals with multiple jobs in these weeks, we use the job with the highest number of hours per
week.

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REVIEW OF ECONOMIC STUDIES
20 to 64. We convert Canadian values to their American equivalents using the 1990 bilat-
eral Purchasing-Power-Parity (PPP) exchange rate for household consumption created by
the OECD; see OECD (1993). This exchange rate is translated to 1989 using relative
changes in the CPI for the two countries. The actual exchange rate we use is 1·257. While
the results presented here are based on this PPP exchange rate, we also generated another
set of results using the average spot-exchange rate for 1989, 1·180. We discuss differences
in results arising from differences in the exchange rates at the end of the paper.
3.1. Empirical results
In Figure 5, we plot the CDFs for American and Canadian male wages in 1989. It is
clear from the plots that the Canadian distribution is more equal. The means of the two
distributions are very similar, US$11.67 for Canada and US$12.27 for the United States.
Thus, the shapes and the locations of the distributions provide a picture that accords with
the mythologies of the two countries: the more egalitarian Canadian society provides
relatively better earnings for poorer workers, while the more incentive-oriented American
society provides better returns for those at the top of the distribution; see Lipset (1990).
Plausible explanations for the differences in the two distributions include higher
unionization rates and lower variability in school quality in Canada. Another possibility
is that the more-generous Canadian social safety net may mean both that individuals do
not have to accept low-paying, full-time jobs and that the associated higher taxes make
taking the risks leading to much higher earnings less attractive. Under these explanations,
one might expect the pattern depicted in Figure 5 to be repeated for different skill groups
in the populations. Alternatively, the distributions within skill groups may be similar
between the two countries, but the proportions of the workforce in the various skill groups
may differ. Evaluating whether Figure 5 predominantly reflects one of these alternatives
or some combination of the two provides insight into the sources of inequality in the two
F
IGURE 5
Cumulative distribution functions: hourly wages

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
623
countries and into who benefits most from the earnings structure in each. In the remainder
of this paper, we carry out such an evaluation.
We estimate the hazard-based, wage-generation model described in section 2 separ-
ately for the two samples. Using the rule of thumb described in Subsection 2.1, we chose
to use 94 baseline segments, with uniform segment widths between the 5-th and 95-th
percentiles of the unconditional wage distribution obtained by pooling the wage data from
the two countries and wider segments in the two tails. Based on the Monte Carlo results
in Table 1, combined with the fact that we have more observations than were used in
those experiments, we chose 10 covariate segments with dividing points at approximately
the deciles of the unconditional pooled-wage distribution. The covariate vector includes:
dummy variables corresponding to the education categories, less-than-high-school or
some-but-not-completed-high-school, some- or completed-post-secondary, and com-
pleted-university (the base category is completed-high-school);10 categorical age variables;
categorical industry variables; an indicator variable for visible minorities; an indicator
variable for union members; the percentage of workers unionized in the individual’s two-
digit industry; and the latter variable interacted with union status.11 Sample means for
the covariates are presented in Table 2. Note that estimates of the densities and the CDFs
are non-linear functions of the covariates and their associated coefficient estimates. Thus,
as in standard Probit estimation, marginal effects on the ordinates from varying covariates
are functions of values for all covariates. In such situations, presenting results using inter-
esting person-types greatly eases exposition and interpretation.
In Figures 6, 7, 8, and 9, we present plots of CDFs and the associated confidence
intervals for Canada and the United States for each of four person-types: not-completed-
high-school, age 20 to 24; completed-high-school, age 20 to 24; completed-university, age
25 to 34; and completed-university, age 45 to 54. The age categories are chosen to corre-
spond to low-experience workers in the first three cases and high-experience workers in
the fourth case. In each case, we use only white workers to avoid complications due to
racial differences between the two countries. The confidence intervals are formed as plus
or minus 2 times the estimated standard errors for the CDF estimates at each baseline
segment.12 The plots for low-experience, not-completed-high-school workers are striking:
the Canadian CDF for this worker type lies everywhere to the right of the American
CDF. The confidence bands for the two CDFs do not overlap, and the difference appears
economically substantial: the 80-th percentile of the Canadian distribution occurs at
US$9.38 compared to only US$7.13 in the American distribution. The CDFs for low-
experienced workers who have completed high school, plotted in Figure 7, also follow a
pattern with Canada dominating. In this case, however, the two CDFs are not point-wise
statistically significantly different below the 40-th percentile, and the difference thereafter
10. The some- or completed-post-secondary category includes those obtaining a post-secondary diploma
or certificate other than a B.A. and those who attended, but did not complete a post-secondary programme. The
completed-university category includes all workers with a B.A. or more. These groupings are directly reported in
the Canadian data. In the American data, the educational categories are defined as follows: less-than-high-
school contains workers with less than 12 years of school attended or 12 years attended but only 11 years
completed; high-school-completed contains those with exactly 12 years of education completed; post-secondary
contains those with 13 to 15 years of education attended or 16 years attended but only 15 years completed; and
completed-university contains those with 16 years attended and completed or more than 16 years attended.
Individuals obtaining a GED are not captured in this division for the United States.
11. Age is not a continuous variable in the LMAS. We use all the age categories available in the LMAS
and group the American data to make them comparable.
12. To form these plots, we draw the subsamples of individuals who are in the specified age and education
groups and who are white. We calculate fitted CDFs for each individual in the subsample using estimated
parameters from the hazard-based estimation and then calculate the average of these fitted CDFs.

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REVIEW OF ECONOMIC STUDIES
TABLE 2
Sample descriptiûe statistics
full-time males, 1989
Variable
Wage
Canada
11.67
Variable
Wage
United States
12.27
Age:
20-24
25-34
35-44
45-54 (Base)
55-64
Age:
20-24
25-34
35-44
45-54 (Base)
55-64
0·09
0·33
0·30
0·18
0·10
0·10
0·34
0·28
0·18
0·10
Education:
Education:
Less Than High School
High School (Base)
Some Post-Secondary
University
0·28
0·23
0·33
0·16
Less Than High School
High School (Base)
Some Post-Secondary
University
0·15
0·37
0·22
0·26
Union Member or Covered
by Collective Agreement
0·48
Union Member or Covered by
Collective Agreement
0·24
Visible Minority
0·04
Visible Minority
0·11
Occupation:
White-Collar
Occupation:
White-Collar
Professional
White-Collar Other
Blue-Collar (Base)
0·29
0·19
0·52
Professional
White-Collar Other
Blue-Collar (Base)
0·33
0·21
0·46
Industry:
Industry:
Primary and Construction
Manufacturing (Base)
Transportation
Trade
Service
Public Administration
0·11
0·30
0·13
0·15
0·21
0·10
Primary and Construction
Manufacturing (Base)
Transportation
Trade
Service
Public Administration
0·11
0·30
0·10
0·17
0·25
0·07
Sample Size
15,142
Sample Size
6,384
is much smaller than that observed in Figure 6. For high-school-educated workers, the
80-th percentile in the American distribution occurs at US$9.13 compared to US$10.30
in the Canadian. The superior CDF for less-than-high-school educated workers in Canada
combined with the more similar CDFs for high-school completers in the two countries
points to a smaller premium to completing high school in Canada. The difference between
the 80-th percentiles of the high-school completers and the non-completers wage distri-
butions is US$0.92 in Canada, but US$2.00 in the United States. Finally, for both high-
school completers and non-completers, the Canadian CDFs are flatter than those for the
American, indicating that there is greater inequality in Canadian wages for these skill
groups.13 The patterns in the CDFs for university-educated workers, shown in Figures 8
and 9, are very different from their less-educated counterparts. The CDFs for younger,
university-educated workers presented in Figure 8 are virtually coincident over much of
the wage range and are never point-wise statistically significantly different. For older
university-educated workers, the CDFs for the two countries are very similar and are not
point-wise statistically significantly different at any point up to the 60-th percentile. The
American distribution is superior in the upper tail, with point-wise statistically significant
13. It is possible that the greater variability in Canadian wages is due to recall bias that is surely absent
in the American data. However, if this were true then one might expect greater variability in the distributions
for more educated workers, which, as we shall see, is not true.

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
625
F
IGURE 6
Cumulative distribution function fitted plots: wages for less-than-high-school-educated workers, age 20-24
F
IGURE 7
Cumulative distribution function fitted plots: wages for high-school educated workers, age 20-24
differences between the two countries occurring between the 60-th and the 80-th
percentiles.
Overall, no evidence exists suggesting that Canadian institutions compress the wage
distributions of workers over all skills groups. Rather, these Figures suggest that low-
skilled Canadians have superior earnings, but face a more unequal distribution. The most

626
REVIEW OF ECONOMIC STUDIES
F
IGURE 8
Cumulative distribution function fitted plots: wages for university-educated workers, age 25-34
F
IGURE 9
Cumulative distribution function fitted plots: wages for university-educated workers, age 45–54
educated, in contrast, fare slightly better in the United States. Finally, the American densi-
ties appear less dispersed than the Canadian densities; the opposite of the mythical more-
egalitarian Canadian distribution.
The markedly superior CDF for the wages of high-school non-completers in Canada
versus the United States combined with the more similar CDFs for higher-skilled workers

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
627
indicates that a key to understanding why the low end of the American and Canadian
overall wage distributions are different is understanding why low-skilled American work-
ers are paid so much less than their Canadian counterparts. Differences in unionization
rates appear to be a likely place to start when trying to explain the wage differences of
low-paid workers in the two countries. Sample means presented in Table 2 indicate that
Canadian workers are twice as likely to be unionized as their American counterparts. In
Figures 10 and 11, we present the fitted wage densities for non-union and union less-than-
high-school educated workers, respectively.14 These densities are presented smoothed,
using a simple three-point moving average, to make comparisons of overall patterns in
the densities easier to spot. Non-smoothed plots (available on request) show no evidence
of substantial spikes in either the American or Canadian densities at minimum-wage
points. These results are consistent with the findings of DFL; ûiz. that the real minimum
wage declined enough in the 1980s in the United States to make it irrelevant as a determi-
nant of the male wage distribution. Thus, it appears that differences in wages for the low-
educated in the two countries cannot be attributed to differences in minimum wages.
F
IGURE 10
Fitted density plots: nonunion, less-than-high-school-education, age 20-24
Union wages have less dispersion than non-union wages in these data sets, as has
been found in studies such as Lemieux (1993). The variance in the logarithm of wages for
American males with less-than-high-school education in the manufacturing sector is 0·11
for union workers and 0·19 for non-union workers. However, once one draws the densi-
ties, as in Figures 10 and 11, one notes interesting and complex differences in union effects
for the two countries. In the American case, the non-union density is more skewed than
the union density, with more mass near $5 but also significant mass up to $10. The union
density is more symmetric and concentrated near $8. This result fits with the idea that
unions generate greater wage equality for a given skill type mainly by creating the largest
14. Apart from union status, the worker characteristics used in creating these plots are: white, age 20 to
24, less-than-high-school educated, in a blue-collar occupation in the manufacturing sector, and in an industry
with average union concentration for the manufacturing sector.

628
REVIEW OF ECONOMIC STUDIES
F
IGURE 11
Fitted density plots: union, less-than-high-school-education, age 20-24
wage increases for the least well-paid. In contrast, the Canadian non-union density
appears to have the same basic shape as the union density, but with more dispersion.
Comparing American and Canadian densities reveals that the Canadian non-union
density has more mass above $8 and less below $6 than its American counterpart, and
appears to be more symmetric. The American and Canadian union densities are much
more similar, with only a small superiority for the Canadian density. Thus, the question
of why the less-educated are paid more in Canada can, to some extent, be refined further:
Why do the least-educated non-union workers fare so much better in Canada? One poss-
ible answer is that the much higher unionization rate in Canada means more pervasive
and credible union-threat effects that bring benefits to the non-union distribution. This is
also a potential explanation for the more symmetric ‘‘union-like’’ Canadian non-union
wage density. In Figure 12, we present the non-union fitted densities for an American of
the same type as in the previous two Figures evaluated both with the union-concentration
variable set to its average for the manufacturing sector in the United States (0·28) and in
Canada (0.53). Increasing the concentration rate raises average earnings and makes the
non-union wage density more symmetric. A reduction in the mass up to $5 is offset largely
by an increase in the mass in the $6 to $8 range, with little effect at $9 or above.15 The
greater benefit for low-wage earners relative to medium- or high-wage earners makes the
density more symmetric, moving it toward the shape of the union density. Of course,
increasing unionization rates means moving workers from the non-union to the union
sector and these workers would not be selected at random. This implies additional changes
in density shapes that may not be fully captured in our results. Thus, a counter-factual
exercise of this sort does not provide a full picture of what the American non-union
density would look like with higher unionization rates. It does, however, provide a graphic
15. The extra symbols on the Canadian union-concentration line denote points at which points on the
two lines differ by more than two standard errors.

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
629
F
IGURE 12
Fitted density plots: U.S., nonunion with average and high union concentration
illustration of who in the non-union sector, at the margin, benefits from increased unioniz-
ation. The distribution of these benefits implies a movement of the American non-union
wage distribution toward the Canadian distribution as concentration rates increase.16
A
plot of the fitted CDF corresponding to an American facing Canadian concentration rates
in Figure 13 shows that differences in union-concentration rates over-explain the Canad-
ian-American difference in CDFs below $6 for this person type, but explain virtually none
of the difference above $8. In contrast, density plots for university-educated workers show
little impact of the worker being unionized or of changes in union-concentration rates.17
Based on both the direct and indirect effects of unionization on the wage density, it
is plausible that differences in unionization could play a large role in explaining differences
at the low end of the wage distribution between Canada and the United States. To pursue
this possibility, we construct a fitted CDF in which Americans face the wage structure in
ˆ
the United States (i.e., δ for the American sample is used), but the Canadian union struc-
ture. Thus, using the Canadian sample, we estimate a Probit on union status as a function
of all the covariates used in the wage estimation, apart from union status itself and the
union-concentration rate. Using parameters from this estimation, we form a probability
of being covered by a collective agreement for each member of the American sample using
their own covariate values, and then replace their union dummy variable with this fitted
probability. We also assign each worker in the American sample the Canadian union-
concentration rate for their two-digit industry. We then form a CDF estimate for each
American worker and then average at each baseline point to create the final fitted CDF.
16. To validate these results, we selected a subsample similar to the workers depicted in Figure 12: specifi-
cally, a subsample of American, non-union, blue-collar workers. Using this subsample, in a regression of the
logarithm of wages on the same set of regressors used in the hazard-based estimation, the estimated coefficient
on the union-concentration variable was 0·545, with a standard error of 0·084. This implies that an increase in
union concentration from 0·28 to 0·53 would raise average wages for these workers by approximately 14%.
17. These figures are available from the authors on request.

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REVIEW OF ECONOMIC STUDIES
F
IGURE 13
Cumulative distribution functions: hourly wages with U.S. given Canadian union structure
The resulting CDF is plotted in Figure 13 along with the simple wage CDFs for Canada
and the United States. Based on these estimates, differences in union structure over-
explain the gap in the CDFs for the two countries between $5 and $10 by 24%, and
explain 30% of the gap above $15. Assigning Americans the probability of being unionized
taken from the Canadian economy, but not the Canadian union-concentration rates,
yields a fitted CDF that explains approximately 76% of the CDF gap below $10 and 11%
above $15. In comparison, Lemieux (1993) found that differences in union status between
Canada and the United States explained approximately 40% of the difference in the vari-
ance in the logarithm of wages between the two countries.
Finally, in Figure 14 we present the fitted-wage CDF formed using the Canadian
sample, but the wage-structure parameter estimates from the United States. Comparing
this to the simple Canadian CDF shows the effect of differences in wage structure (i.e.
differences in covariate effects and the shape of the underlying baseline hazard) in
explaining differences in the distributions between the two countries. Comparing the fitted
CDF to the CDF of the United States shows the effect of differences in the distribution
of covariate characteristics, holding the wage structure constant. Based on these compari-
sons, differences in covariate distributions explain virtually all of the gap in the CDFs
between $5 and $10, and 80% of the gap above $15.18 Thus, compared to Figure 13,
moving beyond just giving American workers the Canadian union structure to giving
them the whole Canadian covariate distribution leads to a worsening across the whole
distribution. In a simulation in which American workers were given the Canadian union
structure as above and predicted industry and occupation values based on parameters
from a multinomial-Logit model of industry and occupation choice using Canadian data,
the percentage of the gap explained was virtually unchanged relative to the case where
18. We replicated the exercises depicted in Figures 13 and 14 using only white workers in both countries
and obtained extremely similar results. Thus, our conclusions are not confounded by different racial effects in
the two countries. The alternative versions of Figures 13 and 14 are available from the authors on request.

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
631
F
IGURE 14
Cumulative distribution functions: hourly wages with U.S. wage structure and Canadian covariates
they were given only the Canadian union structure. Thus, the worsening in the fitted
CDF in Figure 14 is due to the remaining covariates: the education variables. The overall
implication is that Canada has a superior wage distribution at the low end of the wage
range largely because of differences in unionization between the two countries. Lower
education levels in Canada partially offset this effect at the lower end and cause a signifi-
cant worsening at the upper end. It seems plausible, although there is no direct evidence
on this point in this paper, that the two outcomes are related: unions have a beneficial
impact in improving the wages of low-paid workers in both the union and non-union
sectors, but that this improvement may have a negative impact on educational attainment.
Seeing the small return to completing high school in Canada relative to the United States,
it is not surprising that fewer individuals complete high school in Canada. In all of this,
one cannot rule out the possibility that Canadian wages for the least-skilled are more
symmetrically distributed with a higher mean for a reason such as some other facet of
labour legislation and that unions find workers in this situation easier to organize. In this
case, the results above cannot be attributed to direct union actions, but still indicate that
the union͞non-union distinction is a useful starting place in searching for the root cause
of differences between the two countries. Addressing the selection issue directly is of clear
importance, but beyond the scope of this paper.
Several researchers have noted previously the dramatic decline in unionization in the
United States and have examined its impact on the rapid growth in wage inequality in
the United States. Card (1992), Freeman (1991), and DiNardo et al. (1996) all find that
declines in unionization account for about 20% of the increase in the variance of male
wages in the United States over the 1970s and 1980s. DiNardo and Lemieux (1997) find
that differences in unionization-rate changes between Canada and the United States over
the 1980s account for two-thirds of the differential in wage inequality growth between the
two countries in that period. Our results reveal that differences in unionization rates have
impacts primarily at the low end of the wage distribution. Further, we show that unions

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REVIEW OF ECONOMIC STUDIES
have an indirect impact on non-union wages, making the Canadian non-union distribution
less skewed, with more mass at higher wages relative to its American counterpart. By
drawing the entire wage distribution for various skill types, one can see that a key differ-
ence between the distributions in the two countries is the high concentration of American
non-union workers at wages below $6 per hour. This appears to be related to union-threat
effects and is an element of the differences between the two countries that is missed by
focussing on scalar measures such as the variance in the logarithm of wages.
The exchange rate is of central importance in many parts of the cross-country com-
parison carried out above. In the calculations above, we use a PPP exchange rate for
Canada and the United States. We have also generated all of our results using 1·18, the
average spot-exchange rate for 1989. In our view, the PPP rate is clearly superior since it
sets the comparisons in terms of goods that can be consumed. Using the spot exchange
rate yields three main results: first, the smaller return to completing high school in Canada
remains since this is based on comparisons within countries; second, the less-than-high-
school educated Canadian workers have an even more substantial advantage over their
American counterparts; and third, American and Canadian workers with higher education
face almost identical earnings and wage distributions. The main difference in interpret-
ation is that with the alternate exchange rate, the American dominance in the upper part
of the distribution is due almost entirely to differences in the educational composition of
the two work forces. By comparison, in the results above, the American dominance at the
upper end is due to both higher returns to education and a more educated workforce.
4. SUMMARY AND CONCLUSIONS
In this paper, we investigated the sources of differences in the wage distributions between
Canada and the United States in 1989. We found, perhaps not surprisingly, that the
Canadian wage distribution has less dispersion than the American, and a slightly lower
mean. Decomposing wage distributions by skill groups based on age and education, we
found that the pattern evident in the overall distribution is not present within skill groups.
Thus, Canadian institutions do not generate greater equality and lower mean earnings for
each skill group relative to their American counterparts. Rather, the effect seen in the
overall distribution arises from a combination of lower penalties to not completing high
school in Canada and a larger proportion of Canadian workers having low education.
Also, higher unionization rates in Canada appear to play a very large role in generating
superior wages for those at the lower end of the distribution in Canada relative to the
United States. Lower education levels among Canadian workers offset this union effect
at the low end of the distribution and lead to a substantially poorer Canadian distribution
at the upper end.
Reaching these conclusions depended crucially on using an estimator that permitted
estimating complex differences in the shapes of density functions for different values of a
covariate vector. We proposed and implemented a hazard-based estimator of wage, earn-
ings, and income densities in the presence of covariates. Because the estimator makes use
of techniques that are standard in the analysis of spell durations, it should be familiar to
many empirical researchers. The estimator allows one to examine the effects of covariates
of interest on the entire density function, while holding the effects of other covariates
constant. This provided important added information when studying the effects of insti-
tutions such as minimum wages and labour unions. The estimator is potentially useful in
other circumstances where studying the entire distribution of wages, earnings or income
is required.

DONALD, GREEN & PAARSCH
WAGE DISTRIBUTIONS
633
Acknowledgements. Green and Paarsch wish to thank the SSHRC of Canada for financial support. The
authors are also grateful to Charles M. Beach, Paul Beaudry, Michael I. Cragg, James B. Davies, Barton H.
Hamilton, James J. Heckman, Joel L. Horowitz, Thomas E. MaCurdy, Thomas A. Mroz, James L. Powell,
Jeffrey Smith, and Frank A. Wolak as well as two anonymous referees for useful comments and helpful sugges-
tions. Benoit Delage provided excellent research assistance.
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