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Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Nonparametric Estimation of Labor Supply and
Demand Factors
Tsunao Okumura
To cite this article: Tsunao Okumura (2011) Nonparametric Estimation of Labor Supply and
Demand Factors, Journal of Business & Economic Statistics, 29:1, 174-185, DOI: 10.1198/
jbes.2010.08068
To link to this article: http://dx.doi.org/10.1198/jbes.2010.08068

© 2011 American Statistical Association

Published online: 01 Jan 2012.

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Nonparametric Estimation of Labor Supply
and Demand Factors
Tsunao O KUMURA
International Graduate School of Social Sciences, Yokohama National University, Yokohama 240-8501, Japan
(okumura@ynu.ac.jp)

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This article derives sharp bounds on labor supply and demand shift variables within a nonparametric
simultaneous equations model using only observations of the intersection of upward sloping supply curves
and downward sloping demand curves. Furthermore, I demonstrate that these bounds tighten with the
imposition of plausible assumptions on the distribution of the disturbance terms. Using Katz and Murphy’s
(1992) panel data on wages and labor inputs, I estimate these bounds and assess the supply and demand

factors that determine changes within male–female wage differentials and the college wage premium.
KEY WORDS: College wage premium; Gender wage differentials; Panel data model; Partial identification; Sharp bound; Simultaneous equations.

1. INTRODUCTION
Simultaneous equations models of supply and demand provide a useful framework for the analysis of changes in wage
structure. The seminal works of Katz and Murphy (1992) and
Murphy and Welch (1992) demonstrated the utility of this
framework by assessing what supply and demand factors cause
changes in gender and educational wage differentials. However,
these works only observed the equilibrium wages and labor inputs (i.e., the intersections of the labor supply and demand functions) and hence failed to identify the supply and demand factors. Naturally, this raises to the question of what can be identified from the data alone.
This article estimates sharp bounds on supply and demand
shift variables within a nonparametric simultaneous equations
model. The analysis requires only observations of the intersections of the upward sloping supply and downward sloping demand curves (wages and labor inputs); it is not necessary to
specify supply and demand functional forms or the distribution of disturbances. Such an approach mitigates possible problems relating to model misspecification and nonidentification
by asking what can one infer about shift variables within a nonparametric simultaneous equations model using only weak and
credible assumptions.
To introduce the intuition behind the bounds, consider the
following model:

qit = fit (pit ) + μt + εit (supply),

(1)
qit = git (pit ) + νt + ξit (demand),
where i indexes the gender/educational/skill group, t indexes
time, qit is the growth rate of labor inputs, and pit is the growth
rate of real wages. μt denotes a supply shift variable and νt
is a demand shift variable, which are fixed time effects. I will
assume that fit (·) is an increasing function, git (·) is a decreasing function, and fit (pit ) = git (pit ) = qit for (qit , pit ), which is
known. εit and ξit are disturbances and their medians for i are
zero. Imagine a figure in which the horizontal axis represents
qit , the vertical axis represents pit , and (qit , pit ) is the origin.
In period t, when most of the cross-section observations are
found in the southeast region to the lower right of (qit , pit ), it

is likely that the supply shift variable μt is positive. Symmetrically, when most observations are found in the northwest region to the upper left of (qit , pit ), it is likely that μt is negative.
I use this intuition to demonstrate that the supply-side shift variable μt is identified within bounds. The case for the demandside shift variable νt is handled symmetrically.
These bounds can be narrowed by introducing additional assumptions on the distributions of the disturbances εit and ξit .
This article considers the following restrictions: (1) the disturbances are symmetric about zero and (2) the distribution of the
disturbances are known. Analyzing the data used by Katz and
Murphy (1992), I estimate supply and demand shift variables
for each of their gender and educational categories. From these

estimates of the sharp bounds, I discuss how the shifts in the
demand and supply factors caused changes within the male–
female wage differentials and the college wage premium within
the United States.
Katz and Murphy (1992) argued that when most of the
growth in labor inputs and real wages (qit , pit ) are found in the
southeast and northwest regions (e.g., in the 1970s), supply shift
is a relatively important component of the changes in wages
and labor inputs; however, when most of the growth is found
in the northeast and southwest regions (e.g., in the 1980s), demand shift is a relatively important component. Katz and Murphy (1992) examined these relationships by investigating the
signs of the inner products between changes in wages and labor inputs. In contrast, consider the following approach: (1) observe that if (qit , pit ) is in the southeast region to the lower right
of (qit + α, pit ), then this is a sufficient condition for μt + εit
to be greater than α (since supply curves are upward sloping)
and (2) observe that if (qit , pit ) is in the northeast, southeast,
and southwest regions as compared to (qit + α, pit ), then this
is a necessary condition for μt + εit to be greater than α. The
probability that μt + εit is greater than α is bounded above and
below by the probabilities of these two events. Therefore, the
distribution of μt + εit [which equals 1 − P(μt + εit > α)] is

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© 2011 American Statistical Association
Journal of Business & Economic Statistics
January 2011, Vol. 29, No. 1
DOI: 10.1198/jbes.2010.08068

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Okumura: Nonparametric Estimation of Labor Supply and Demand Factors

identified within bounds. By assuming the median of εit for i is
zero, these bounds will translate into restrictions on the supply
shift variable μt , which will be shown to be identified within
bounds. The symmetric result holds for the demand shift variable νt .
The econometric methodology of this article follows in the
tradition of Manski (1997) and Manski and Pepper (2000).
Under the weak assumption of monotone treatment response,
Manski (1997) derived sharp bounds on functionals of the distribution of treatment response that respect stochastic dominance. In particular, corollary M1.3 in Manski (1997) identified
the bounds on the quantiles of the treatment response function

y(t) as a special case. Though this article is similar to Manski (1997) in spirit, as both embrace inference under weak and
credible assumptions, in order to estimate labor supply and demand shift variables, I derive sharp bounds on the median of the
disturbances that generate the distribution of the outcome y(t).
There is a large and recent literature that examines the
identification of nonparametric simultaneous equations models
(see Matzkin 2007 for a survey). Within this literature, Brown
and Matzkin (1998), Ekeland, Heckman, and Nesheim (2004),
and Matzkin (2008) studied nonparametric identification of the
supply and demand system without assuming a triangular system. Their work took into consideration that both supply and
demand functions can arise from the aggregation of heterogeneous individual behavior, which corresponds to μt + εit and
νt + ξit within the current framework. Rigobon (2003) and Lewbel (2008) used disaggregate heterogeneous individual behavior
to examine the identification of parametric simultaneous equations models. Recently, there is a growing literature on the application of Manski’s (1994, 1997) and Manski and Pepper’s
(2000) bounds to labor markets. Blundell et al. (2007) estimated
the distribution of wages in the United Kingdom using these
bounds, using selection-into-work as a treatment variable.
This article is organized as follows. Section 2 discusses the
econometric model. Section 3 estimates the labor supply and
demand shift variables by employing panel data on labor inputs
and wages in the United States. Section 4 concludes.
2. SHARP BOUNDS ON THE MEDIAN OF

DISTURBANCES WITHIN THE SUPPLY
AND DEMAND FRAMEWORK
In preparation to estimate supply and demand shift variables
I begin by setting up a simultaneous equations model with two
endogenous variables and two disturbances. The objective is to
estimate the medians of the disturbances:

q = f (p) + u,
(2)
q = g(p) + v,
where p and q are endogenous variables and u and v are disturbances. The population is formalized as a measure space (J,
, P) of agents, with P a probability measure. Then P[(u, v),
(q′ , p′ )] gives the distribution of disturbances and realized variables. f (·) and g(·) are an increasing and a decreasing function
of p, respectively. p and u may be correlated, as p and v also
may be. The solution (q, p) of Equation (2) is unique, given
(u, v).

175

Figure 1. NE(α), SE(α), NW(α), and SW(α) represent the northeast, southeast, northwest, and southwest regions of the point,

(q + α, p), respectively, for some real number α and (q, p), where
f (p) = g(p) = q.

I take the existence of a known point (q, p) such that f (p) =
g(p) = q as prior knowledge. As Figure 1 illustrates, NE(α),
SE(α), NW(α), and SW(α) represent the northeast {(q, p) |
q > q + α, p > p}, southeast {(q, p) | q > q + α, p ≤ p}, northwest {(q, p) | q ≤ q + α, p > p}, and southwest {(q, p) | q ≤
q + α, p ≤ p} regions of (q + α, p), respectively, for some real
number α. Suppose that (q, p) is observed in SE(α). Since the
f function is upward sloping, u (measured at the q-axis intersected by the f function) is greater than α. In contrast, if (q, p)
is observed in NW(α), u is smaller than α. This relation implies
the following proposition.
Proposition 1. Suppose Equation (2). For any real number α,
(i)

P((q, p) ∈ NW(α))
≤ Fu (α) ≤ 1 − P((q, p) ∈ SE(α)),

(3)

(ii) P((q, p) ∈ SW(α))
≤ Fv (α) ≤ 1 − P((q, p) ∈ NE(α)),

(4)

where Fu and Fv are the cumulative distribution functions of
the disturbances u and v, respectively.
These bounds are sharp.
Proof. See Appendix A.
Proposition 1 shows that the distribution of the observations
(q, p) reveals the distributions of the disturbances u and v up to
bounds because both bounds in Equations (3) and (4) weakly
increase in α. These bounds hold for arbitrary correlations between u and p, as well as those between v and p. For any real
number α, the bounds on Fu (α) and Fv (α) are estimated by
using the analogy principle by replacing population quantities
with their empirical counterparts.
The quantiles of Fu and Fv are restricted by the quantiles of
the bound estimates on Fu and Fv , respectively. Specifically, the
medians of u and v, which are denoted by u and v, are bounded
by the medians of the bound estimates on Fu and Fv .

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Journal of Business & Economic Statistics, January 2011

Lemma 1. Define:


α1 = inf α | P[(q, p) ∈ NW(α)] ≥ 0.5 ,


α2 = inf α | 1 − P[(q, p) ∈ SE(α)] ≥ 0.5 ,


β1 = inf β | P[(q, p) ∈ SW(β)] ≥ 0.5 ,


β2 = inf β | 1 − P[(q, p) ∈ NE(β)] ≥ 0.5 ,

Then

sup [α2 (γ ) + α2 (1 − γ )]/2
γ ∈[0,1]

≤ u ≤ inf [α1 (γ ) + α1 (1 − γ )]/2,
γ ∈(0,1)

(8)

(5)

sup [β2 (γ ) + β2 (1 − γ )]/2
γ ∈[0,1]

u = min{u | Fu (u) ≥ 0.5},

≤ v ≤ inf [β1 (γ ) + β1 (1 − γ )]/2.
γ ∈(0,1)

v = min{v | Fv (v) ≥ 0.5}.
Then
α2 ≤ u ≤ α1 ,
(6)

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β2 ≤ v ≤ β1 .

Proof. See Appendix C.

These bounds are sharp. u and v are bounded on one side; they
are never bounded on both sides, however.
Proof. See Appendix B.

Lemma 3. Assume Assumption 2. Then



Fu−1 1 − P((q, p) ∈ SE(α))
sup α − 
α∈R

If more (less) than a half portion of (q, p)’s are distributed
in the region where p is greater than p, then the upper (lower)
bounds on u and the lower (upper) bounds on v exist; the opposite bounds do not exist, however.
Manski (1997) presented sharp bounds on the quantiles of
the weakly increasing function y(t) at some treatment t by observing the realized treatment and the realized outcome (corollary M1.3). Let us suppose that yj (·) is specified as fj (·) + uj (·),
and fj (t) is known; then, the sharp bounds on the median of
uj (t) are attained by the median of yj (t) − fj (t), the difference
between treatment response and the known value of fj (t).
Introducing certain assumptions narrows the sharp bounds
of the disturbances’ medians. The following two assumptions
about the distributions of u and v are considered.
Assumption 1. The distributions of u and v are symmetric
around the medians of u and v, respectively.
Assumption 2. 
Fu is defined as the distribution of u − u,
where u is the median of u. 
Fv is defined as the distribution
of v − v, where v is the median of v. Let us assume that (1) 
Fu
and 
Fv are known by an econometrician and (2) 
Fu and 
Fv are
strictly increasing functions.
The locations of individual supply and demand curves (measured at q when p = p) are characterized by supply and demand
disturbances. Assumption 1, therefore, supposes that individual curves are symmetrically located with respect to u and v.
Assumption 2 demonstrates that the distributions of individual
disturbances are known up to a location parameter (u or v), as a
semi-parametric approach assumes.
Lemma 2. Assume Assumption 1. Define:


α1 (γ ) = inf α | P[(q, p) ∈ NW(α)] ≥ γ ,


α2 (γ ) = inf α | 1 − P[(q, p) ∈ SE(α)] ≥ γ ,


β1 (γ ) = inf β | P[(q, p) ∈ SW(β)] ≥ γ ,


β2 (γ ) = inf β | 1 − P[(q, p) ∈ NE(β)] ≥ γ .

These bounds are sharp. These bounds are tighter than or equal
to those in Lemma 1. u and v are bounded on one side; they are
never bounded on both sides, however. Under Assumption 1,
the unbounded sides of u and v in Lemma 1 are not bounded.

(7)




Fu−1 P((q, p) ∈ NW(α)) ,
≤ u ≤ inf α − 
α∈R




Fv−1 1 − P((q, p) ∈ NE(β))
sup β − 

(9)

β∈R




Fv−1 P((q, p) ∈ SW(β)) .
≤ v ≤ inf β − 
β∈R

These bounds are sharp.

Proof. See Appendix D.
Assumption 1 has identifying power in that it can tighten the
bounds on u and v. Assumption 1, however, cannot identify the
unbounded sides of u and v in Lemma 1. In contrast, by imposing Assumption 2, u and v are bounded on both sides.
In Equation (2), I assume that the supply and demand functions are additively separable. By relaxing this assumption,
I can generalize the above results to functions that appear in
a nonseparable form in the simultaneous equations model:

q = f ∗ (p, u),
(10)
q = g∗ (p, u).
Let us assume that:
Assumption 3. f ∗ (p, u) strictly increases in u; g∗ (p, v) strictly increases in v.
Assumption 4. f ∗ (p, u) weakly increases in p for any u;
weakly decreases in p for any v.

g∗ (p, v)

Assumption 5. f ∗ (p, 0) = g∗ (p, 0) = q.
It should be noticed that p and u, as well as p and v, may
be correlated. Equation (2) is the case of Equation (10) under
Assumptions 3, 4, and 5.
Define NE∗ (α) = {(q, p) | q > f ∗ (p, α), p > p}; SE∗ (α) =
{(q, p) | q > f ∗ (p, α), p ≤ p}; NW∗ (α) = {(q, p) | q ≤ f ∗ (p, α),
p > p}; and SW∗ (α) = {(q, p) | q ≤ f ∗ (p, α), p ≤ p}.
The following sharp bounds on the distributions of u and v
are obtained.

Okumura: Nonparametric Estimation of Labor Supply and Demand Factors

Proposition 2. Suppose Equation (10) and Assumptions 3, 4,
and 5. For any real number α,
(i) P((q, p) ∈ NW∗ (α))
≤ Fu (α) ≤ 1 − P((q, p) ∈ SE∗ (α)),
(ii) P((q, p) ∈ SW∗ (α))

(11)

≤ Fv (α) ≤ 1 − P((q, p) ∈ NE∗ (α)),
where Fu and Fv are the cumulative distribution functions of
the disturbances u and v, respectively.
These bounds are sharp.

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Proof. See Appendix E.
3. ESTIMATION OF LABOR SUPPLY AND
DEMAND SHIFT VARIABLES
Equation (2) is modified using the following panel data
model:

qit = fit (pit ) + μt + εit ,
(12)
qit = git (pit ) + νt + ξit ,
where i is the gender/educational/skill group index and t is the
time index. qit and pit are the growth rates of the labor inputs
and real wages of the ith group at time t. (qit , pit ) is a normalization satisfying fit (pit ) = git (pit ) = qit . fit (·) and git (·) are increasing and decreasing functions, respectively. μt and νt are
supply and demand shift variables and unknown parameters
(fixed time effects) to be estimated. εit and ξit represent disturbances with supply and demand functions and the medians
of εit and ξit for i are zero. μt + εit and νt + ξit correspond to u
and v in Equation (2), and thus, μt and νt correspond to u and v.
This model posits that the growth rates of labor supply and
wages comove across groups because of the change in the supply and demand shift variables (μt and νt ). Yet, this comovement is not deterministic as independent movement is caused
by idiosyncratic supply and demand disturbances (εit and ξit ).
Supply and demand curves, fit (·) and git (·), do not need to be
specified and may differ across
both groups and time.

Let us assume that qit = T1 Tt=1 qit and pit = T1 Tt=1 pit and
that fit (pit ) = git (pit ) = qit . (T is the sample period. Alternative
normalizations (qit , pit ) are utilized for the estimation in Appendix F.) I utilize the same dataset as Katz and Murphy (1992)
and Murphy, Riddell, and Romer (1998). Using the March Current Population Survey, Katz and Murphy (1992) created the
mean of real weekly wages (in 1982 dollars) and labor inputs
in person hours (measured in efficiency units) for 64 groups by
gender, education, and skill for each year in the 1963 to 1987
period. To investigate factors of the wage differentials by gender and education, 64 groups of data are classified into (i) male
and female labor and (ii) high-school equivalents and college
equivalents. Lemmas 1, 2, and 3 are applied to the estimation
of μt and νt for each category, using analogous sample frequencies. (These groups are described in Appendix F.)
Figures 2 through 6 show estimates of the sharp bounds
on the labor supply and demand shift variables for all samples (Figure 2), male samples (Figure 3), female samples (Figure 4), high-school equivalents (Figure 5), and college equivalents (Figure 6). The solid lines represent the estimates of the

177

upper and lower bounds, the dotted lines represent the 10th and
90th bootstrap percentiles. In line with Lemmas 1, 2, and 3, I assume the distributions of εit and ξit : (i) the distributions have a
zero median for i [Figure 2(a)]; (ii) the distributions are symmetric around zero for i [Figure 2(b)]; and (iii) the distributions
are time-invariant and normal distributions over i, where their
means are zero and their variances are the middle of the minimum and maximum estimates of the variances, as elaborated in
Appendix F [Figures 2(c), 3, 4, 5, and 6]. As Lemmas 1 and 2
show, in cases (i) and (ii), only one side of the bound estimates
is bounded. Some bound estimates in case (ii) are tighter than
those in case (i). Specifically, in years when the observations
whose pit is greater (smaller) than pit are more than one-half of
all observations, the upper (lower) bound estimates on μ and the
lower (upper) bound estimates on ν exist. When the proportion
of these observations is large, there is a high possibility that the
bound estimates in case (ii) are tighter than those in case (i). In
case (iii), both bounds are estimated, and these bound estimates
are significantly tighter than those in cases (i) and (ii). Therefore, imposing the assumption that the distributions are normal
distributions has sufficient identification power to explain the
shifts in the supply and demand curves. To compare the findings with those of previous studies, I principally use the bound
estimates in case (iii) for interpretation.
The stylized facts regarding the between-group wage structure changes are: (1) the male and female wage differentials
were stable in the 1960s and 1970s, and decreased substantially in the 1980s and (2) the college wage premium rose in the
1960s, declined in the 1970s, and increased sharply in the 1980s
(refer to Katz and Murphy 1992; Murphy and Welch 1992; Blau
and Kahn 1997, 1999; and Katz and Autor 1999).
The male and female wage differentials (Figures 3, 4, and 7):
The supplies of male and female labor were stable. In the 1980s,
the demand for male (female) labor was lower (higher) than in
the 1960s. In the 1970s, the demand for male and female labor
fluctuated.
Figure 7 shows the bound estimates of the difference between
male and female labor shift variables. (The bound is explained
in Appendix F.) The averages of the bound estimates in the
1960s, 1970s, and 1980s are shown by dashed lines. The relative demand for male labor, as compared to that for female labor, has decreased, which caused the gender wage gap to shrink
in the 1980s.
The college wage premium (Figures 5, 6, and 8): The supplies
of high-school and college equivalents were stable, whereas the
latter slightly decreased in the 1980s. The demands for highschool and college equivalents fluctuated. In particular, the demand for college equivalents was lower in the 1970s than in
other periods and increased sharply after 1977, while the demand for high-school equivalents decreased after 1980.
Figure 8 graphs the bound estimates of the difference between shift variables of college and high-school equivalents.
The relative demand for college equivalents, as compared to
high-school equivalents, drifted downward during the 1970s.
This trend reversed sharply in the 1980s, however. In contrast
to the demand shifts, the relative supply of college equivalents,
as compared to high-school equivalents, increased slightly in
the 1970s and decreased in the 1980s. These shifts explain why
the college wage premium rapidly rose from 1980, following a
small decline in the 1970s.

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(a)

(b)

(c)

Figure 2. (a) Estimates of the bounds on shift variables for all samples assuming the median of the disturbances is zero. (b) Estimates of the
bounds on shift variables for all samples assuming the distributions of the disturbances are symmetric about zero. (c) Estimates of the bounds
on shift variables for all samples assuming the distributions of the disturbances are normal. The solid lines are the estimates of the bounds and
the dotted lines are the 10th and 90th bootstrap percentiles. [In (c), only 10th percentiles of the lower bounds and 90th percentiles of the upper
bounds are shown.] In (a) and (b), the opposite sides of the bounded sides of the estimates (the solid lines) are not bounded (∞ or −∞).

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Okumura: Nonparametric Estimation of Labor Supply and Demand Factors

179

Figure 3. Estimates of the bounds on shift variables for males assuming the distributions of the disturbances are normal. The solid lines are
the estimates of the bounds. The dotted lines are the 10th bootstrap percentiles of the lower bounds and the 90th bootstrap percentiles of the
upper bounds.

Sixty-four groups of data are also classified into four categories: male high-school equivalents, male college equivalents,
female high-school equivalents, and female college equivalents.
Using these categories, I estimate the supply and demand shift
variables. The estimation results show that in the 1980s, demand for both male and female high-school equivalents decreased; this demand, however, decreased faster for males than
for females. The supply of male high-school equivalents increased in the 1980s, whereas the supply of female high-school
equivalents decreased from the mid-1970s to the mid-1980s.
This finding accounts for why male and female wage differentials decreased more rapidly for high-school equivalents than
for college equivalents. (The estimation results are available
from the author upon request.)
These estimation results are consistent with the findings of
Katz and Murphy (1992), Murphy and Welch (1992), Blau and

Kahn (1997, 1999), and Katz and Autor (1999). In particular, Katz and Murphy (1992) and Katz and Autor (1999) suggested that skill-biased technological change, shifts in product
demand across industries, and rising international competition
increased demand for educated labor in the 1970s and 1980s.
Katz and Murphy (1992) illustrated the possibility that a decrease in the growth rate of the supply of college graduates
may help explain the increase in the college wage premium
in the 1980s. By estimating sharp bounds on the supply and
demand shift variables for gender and educational groups, this
article provides evidence supporting their findings. To further
study the causes of supply and demand shifts, the gender and
educational groups can be divided into industry and occupational groups. Utilizing this study’s approach, the within- and
between-industry shifts of the supply and demand functions can
be estimated.

Figure 4. Estimates of the bounds on shift variables for females assuming the distributions of the disturbances are normal. The solid lines
are the estimates of the bounds. The dotted lines are the 10th bootstrap percentiles of the lower bounds and the 90th bootstrap percentiles of the
upper bounds.

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Journal of Business & Economic Statistics, January 2011

Figure 5. Estimates of the bounds on shift variables for high school equivalents assuming the distributions of the disturbances are normal.
The solid lines are the estimates of the bounds. The dotted lines are the 10th bootstrap percentiles of the lower bounds and the 90th bootstrap
percentiles of the upper bounds.

4. CONCLUSION
This article presents a nonparametric econometric model that
identifies sharp bounds for the supply and demand shift variables in a simultaneous equations model by only using observations of the intersections of upward sloping supply and
downward sloping demand curves. To clarify the causes of the
changes in the wage differentials across gender and educational
groups, labor supply and demand factors are estimated using
panel data of wages and labor inputs. The estimation results
show that (1) the relative demand for female labor, as compared
to male labor, has increased in the 1980s and (2) the relative
demand for college equivalents, as compared to high-school
equivalents, slightly decreased in the 1970s and increased in
the 1980s.

Since the assumption for identification is much less restrictive than that of the existing parametric approach, the estimated
bounds are large. Additional assumptions on disturbances narrow the bounds. Two important research inquiries have yet to be
considered: how to select assumptions on disturbances and the
normalization point. These topics, however, are left for future
study.
APPENDIX A: PROOF OF PROPOSITION 1
Since f is monotone increasing in p, for any real number α,

(q, p) ∈ SE(α) ⇒ u > α,
(q, p) ∈ NW(α) ⇒ u ≤ α.
Thus,
P((q, p) ∈ SE(α)) ≤ P(u > α) ≤ 1 − P((q, p) ∈ NW(α)).

Figure 6. Estimates of the bounds on shift variables for college equivalents assuming the distributions of the disturbances are normal. The
solid lines are the estimates of the bounds. The dotted lines are the 10th bootstrap percentiles of the lower bounds and the 90th bootstrap
percentiles of the upper bounds.

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Okumura: Nonparametric Estimation of Labor Supply and Demand Factors

181

Figure 7. Estimates of the bounds on the difference between male and female shift variables assuming the distributions of the disturbances
are normal. The solid lines are the estimates of the bounds. The dashed lines are the averages of the bound estimates in the 1960s, 1970s, and
1980s.

These bounds are sharp since the empirical evidence and
prior information are consistent with the hypothesis P((q, p) ∈
SE(α)) = P(u > α) and also with the hypothesis 1 − P((q, p) ∈
NW(α)) = P(u > α). Hypothesis P((q, p) ∈ SE(α)) = P(u >
α) is consistent with the case in which all supply curves
traversing the observations in NE(α) have enough gentle
slopes for their u’s to be smaller than α, and all supply
curves traversing the observations in SW(α) have enough
steep slopes for their u’s to be smaller than α. The hypothesis 1 − P((q, p) ∈ NW(α)) = P(u > α) is consistent with the
case in which all supply curves traversing the observations in
NE(α) have enough steep slopes for their u’s to be greater
than α, and all supply curves traversing the observations in
SW(α) have enough gentle slopes for their u’s to be greater
than α.

Since P(u > α) = 1 − Fu (α), for any real number α,
P((q, p) ∈ NW(α)) ≤ Fu (α) ≤ 1 − P((q, p) ∈ SE(α)).
These bounds are sharp.
Since g is monotone decreasing in p, for any real number α

(q, p) ∈ NE(α) ⇒ v > α,
(q, p) ∈ SW(α) ⇒ v ≤ α.
Thus
P((q, p) ∈ NE(α)) ≤ P(v > α) ≤ 1 − P((q, p) ∈ SW(α)).
These bounds are sharp since empirical evidence and prior
information are consistent with the hypothesis P((q, p) ∈
NE(α)) = P(v > α) and also with the hypothesis 1 − P((q, p) ∈
SW(α)) = P(v > α). Hypothesis P((q, p) ∈ NE(α)) = P(v >

Figure 8. Estimates of the bounds on the difference between shift variables of college and high-school equivalents assuming the distributions
of the disturbances are normal. The solid lines are the estimates of the bounds. The dashed lines are the averages of the bound estimates in the
1960s, 1970s, and 1980s.

182

Journal of Business & Economic Statistics, January 2011

α) is consistent with the case in which all demand curves traversing the observations in SE(α) have enough gentle slopes
for their v’s to be smaller than α, and all demand curves traversing the observations in NW(α) have enough steep slopes
for their v’s to be smaller than α. The hypothesis 1 − P((q, p) ∈
SW(α)) = P(v > α) is consistent with the case in which all demand curves traversing the observations in SE(α) have enough
steep slopes for their v’s to be greater than or equal to α, and
all demand curves traversing the observations in NW(α) have
enough gentle slopes for their v’s to be greater than α.
Since P(v > α) = 1 − Fv (α), for any real number α,

However, for any a,
P[(q, p) ∈ NW(a)] = P[q ≤ q + a, p > p] ≤ P[p > p].
Hence, there does not exist a such that P[(q, p) ∈ NW(a)] ≥
0.5.
The set of {α | P[(q, p) ∈ NW(α)] ≥ 0.5} is empty. Therefore, by Equation (5), α1 = inf ∅ = ∞.
If α1 < ∞, P[(q, p) ∈ NW(α1 )] ≥ 0.5. For any a,
1 − P[(q, p) ∈ SE(a)] = 1 − P[q > q + a, p ≤ p]
≥ 1 − P[p ≤ p] = P[p > p]

P((q, p) ∈ SW(α)) ≤ Fv (α) ≤ 1 − P((q, p) ∈ NE(α)).

≥ P[q ≤ q + α1 , p > p]

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:00 11 January 2016

These bounds are sharp.

= P[(q, p) ∈ NW(α1 )] ≥ 0.5.

APPENDIX B: PROOF OF LEMMA 1
Equations (3) and (5) imply that

Hence, α2 = −∞. (The set of {α | 1 − P[(q, p) ∈ SE(α)] ≥
0.5} is not bounded below.) Therefore, if α2 > −∞, α1 = ∞,
whereas if α1 < ∞, α2 = −∞.
Similarly,

u ≤ α1 ,
since
0.5 ≤ P[(q, p) ∈ NW(α1 )] ≤ Fu (α1 ),

β2 ≤ v ≤ β1 .
This bound is sharp. v is bounded on one side; however, it is
never bounded on both sides.

0.5 ≤ Fu (u),
APPENDIX C: PROOF OF LEMMA 2

and
u = min{u | Fu (u) ≥ 0.5}.
Equations (3) and (5) imply that
α2 ≤ u,

Define the γ quantile of Fu as muγ . Equations (3) and (7) and
the definition of muγ imply that
muγ ≤ α1 (γ ),
since

since



γ ≤ P (q, p) ∈ NW(α1 (γ )) ≤ Fu [α1 (γ )],

0.5 ≤ Fu (u) ≤ 1 − P[(q, p) ∈ SE(u)],
0.5 ≤ 1 − P[(q, p) ∈ SE(α2 )],

γ ≤ Fu (muγ ),
and

and

Hence,



α2 = inf α | 1 − P[(q, p) ∈ SE(α)] ≥ 0.5 .

muγ = min{m | Fu (m) ≥ γ }.
Equations (3) and (7) and the definition of muγ imply that
α2 (γ ) ≤ muγ ,

α 2 ≤ u ≤ α1 .
This bound is sharp, since the bound in Equation (3) is sharp.
That is, empirical evidence and prior information are consistent
with the hypothesis u = α1 and also with the hypothesis u = α2 .
u is bounded on one side. For the sake of contradiction, suppose that α1 = ∞ and α2 = −∞. Since α1 = ∞, by Equation (5), the set of {α | P[(q, p) ∈ NW(α)] ≥ 0.5} is empty.
Therefore, for any α ∈ R, P[(q, p) ∈ NW(α)] < 0.5. Thus,
P[p > p] < 0.5. Since α2 = −∞, by Equation (5), for any
α ∈ R, 1 − P[(q, p) ∈ SE(α)] ≥ 0.5. Thus, P[p > p] ≥ 0.5. This
is a contradiction. Hence, u is bounded on one side.
u is never bounded on both sides. (The author thanks the referee for providing the following proof.) If α2 > −∞, there exists a ∈ (−∞, α2 ), such that 1 − P[(q, p) ∈ SE(a)] < 0.5 because of the definition of α2 . Therefore, if α2 > −∞, then
0.5 > 1 − P[(q, p) ∈ SE(a)] = 1 − P[q > q + a, p ≤ p]
≥ 1 − P[p ≤ p] = P[p > p].

since

and

Hence,

γ ≤ Fu (muγ ) ≤ 1 − P[(q, p) ∈ SE(muγ )],


γ ≤ 1 − P (q, p) ∈ SE(α2 (γ )) ,


α2 (γ ) = inf α | 1 − P[(q, p) ∈ SE(α)] ≥ γ .
α2 (γ ) ≤ muγ ≤ α1 (γ ).

(C.1)

Similarly, for the (1 − γ ) quantile of Fu
α2 (1 − γ ) ≤ mu1−γ ≤ α1 (1 − γ ).

(C.2)

Since the symmetry of the distribution of u around u implies
muγ − u = −(mu1−γ − u), Equations (C.1) and (C.2) imply that
[α2 (γ ) + α2 (1 − γ )]/2 ≤ u ≤ [α1 (γ ) + α1 (1 − γ )]/2.

(C.3)

Okumura: Nonparametric Estimation of Labor Supply and Demand Factors

Since the inequality in Equation (C.3) holds for any γ ∈ [0, 1],
sup [α2 (γ ) + α2 (1 − γ )]/2
γ ∈[0,1]

≤ u ≤ inf [α1 (γ ) + α1 (1 − γ )]/2.

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:00 11 January 2016

γ ∈(0,1)

[Note: α1 (0) = −∞.]
Let us show that these bounds are sharp. It suffices to
show that for any sample size n, there exist the following
two types of {ui }ni=1 . Type (C-1) {ui }ni=1 satisfy the conditions that (C-i) {ui }ni=1 realize u = supγ ∈[0,1] [α2 (γ ) + α2 (1 −
γ )]/2, (C-ii) {ui }ni=1 are symmetrically distributed around u,
(C-iii) the supply curve corresponding to ui traverses (qi , pi ),
and (C-iv) all of the supply curves are upward sloping, i.e., ui ≥
qi for pi ≤ p and ui ≤ qi for pi > p. Type (C-2) {ui }ni=1 satisfy
the conditions that (C-v) {ui }ni=1 realize u = infγ ∈(0,1) [α1 (γ ) +
α1 (1 − γ )]/2, (C-ii), (C-iii), and (C-iv).
Let us show that there exist {ui }ni=1 of type (C-1). Define q(i)
for i ≤ k1 , as the ith order statistics of {qj } whose p’s are greater
than p; where k1 is the number of the observations with p being
greater than p. Define q(i) for i = k1 + 1, k1 + 2, . . . , n as the
(i − k1 )th order statistics of {qj }, whose p’s are not greater than
p. Define ul(i) as the disturbances corresponding to q(i) . Define
uL = supγ ∈[0,1] [α2 (γ ) + α2 (1 − γ )]/2.
By Equation (7), q(i) = α2 (i/n) for i ≥ k1 + 1. By the definition of uL , for k1 + 1 ≤ i ≤ n,







i
i
uL ≥ α2
+ α2 1 −
2 = q(i) + q(n−i) /2.
n
n
Therefore, uL − q(i) ≥ q(n−i) − uL for k1 + 1 ≤ i ≤ n.
In the case that k1 < n/2; uL > −∞ since α2 (n/2) > −∞.
Let us take ul(i) satisfying (C-i) through (C-iv) in this case as
follows.
For k1 + 1 ≤ i < n/2, take ul(n−i) = q(n−i) (e.g., the supply
curves traversing the observations of q(n−i) are vertical). Then,
it is possible to take ul(i) for k1 + 1 ≤ i ≤ n/2, such that uL −
ul(i) = ul(n−i) − uL (ul(i) and ul(n−i) are symmetric around uL ) and
q(i) ≤ ul(i) (the supply curves traversing the observations of q(i)
are upward sloping).
For i ≤ k1 , if q(i) ’s are distributed as uL − q(i) ≥ q(n−i) − uL ,
take ul(i) = q(i) and ul(n−i) = 2uL − q(i) . Otherwise, take ul(i) =
2uL − q(n−i) and ul(n−i) = q(n−i) . Then, uL − ul(i) = ul(n−i) − uL ;
and q(i) ≥ ul(i) and q(n−i) ≤ ul(n−i) .
In the case that k1 ≥ n/2; uL = −∞ since α2 (i/n) = −∞ for
i < k1 . Let us take ul(i) satisfying (C-i) through (C-iv) in this
case as follows.
Take ul(i) = ql(i) for i ≥ k1 + 1; and ul(i) = −∞ for i ≤ k1 ,
which implies ul(i) ≤ q(i) . Then, [ul(i) + ul(n−i) ]/2 = −∞ = uL
for all i.
Consequently, this distribution of ui , which is symmetric
around u and for which the corresponding supply curves are
upward sloping, implies u = supγ ∈[0,1] [α2 (γ ) + α2 (1 − γ )]/2.
Similarly, it is shown that there exist {ui }ni=1 of type (C-2).
Therefore, in Equation (8) the bounds on u are sharp.
The bounds on u in Lemma 2 are tighter than or equal to
those in Lemma 1 because supγ ∈[0,1] [α2 (γ ) + α2 (1 − γ )]/2 ≥
α2 (0.5) = α2 and infγ ∈(0,1) [α1 (γ ) + α1 (1 − γ )]/2 ≤ α1 (0.5) =
α1 .

183

u is bounded on one side because u is bounded on one side
in Lemma 1 and the bounds on u in Lemma 2 are tighter than
or equal to those in Lemma 1.
u is never bounded on both sides. If α2 (γ ) > −∞, there exists a ∈ (−∞, α2 (γ )), such that 1 − P[(q, p) ∈ SE(a)] < γ because of the definition of α2 (γ ). Therefore, if α2 (γ ) > −∞,
then
γ > 1 − P[(q, p) ∈ SE(a)] = 1 − P[q > q + a, p ≤ p]
≥ 1 − P[p ≤ p] = P[p > p].
However, for any a,
P[(q, p) ∈ NW(a)] = P[q ≤ q + a, p > p] ≤ P[p > p].
Hence, there does not exist a, such that P[(q, p) ∈ NW(a)] ≥ γ .
The set of {α | P[(q, p) ∈ NW(α)] ≥ γ } is empty. Therefore, by
Equation (7), α1 (γ ) = inf ∅ = ∞.
If α1 (γ ) < ∞, P[(q, p) ∈ NW(α1 (γ ))] ≥ γ . For any a,
1 − P[(q, p) ∈ SE(a)] = 1 − P[q > q + a, p ≤ p]
≥ 1 − P[p ≤ p] = P[p > p]
≥ P[q ≤ q + α1 (γ ), p > p]


= P (q, p) ∈ NW(α1 (γ )) ≥ γ .

Hence, α2 (γ ) = −∞. (The set of {α | 1 − P[(q, p) ∈ SE(α)] ≥
γ } is not bounded below.)
Without loss of generality, assume γ ≤ 1 − γ . If
infγ ∈(0,1) [α1 (γ ) + α1 (1 − γ )]/2 < ∞, for γ h ≡
arg minγ ∈(0,1) [α1 (γ ) + α1 (1 − γ )]/2 (γ h ≤ 0.5), α1 (γ h ) <
∞ and α1 (1 − γ h ) < ∞. As shown, α2 (γ h ) = −∞ and
α2 (1 − γ h ) = −∞. Since 1 − P[(q, p) ∈ SE(α)] weakly increases in α, α2 (γ ) weakly increases in γ . Therefore, for
δ ≤ 1 − γ h , α2 (δ) = −∞. Since 0.5 ≤ 1 − γ h , for δ ∈ [0, 0.5],
α2 (δ) = −∞. Hence, supγ ∈[0,1] [α2 (γ ) + α2 (1 − γ )]/2 = −∞.
If supγ ∈[0,1] [α2 (γ ) + α2 (1 − γ )]/2 > −∞, for γ l ≡
arg maxγ ∈[0,1] [α2 (γ ) + α2 (1 − γ )]/2 (γ l ≤ 0.5), α2 (γ l ) >
−∞ and α2 (1 − γ l ) > −∞. As shown, α1 (γ l ) = ∞ and
α1 (1 − γ l ) = ∞. Since P[(q, p) ∈ NW(α)] weakly increases
in α, α1 (γ ) weakly increases in γ . Therefore, for δ ≥ γ l ,
α1 (δ) = ∞. Since γ l ≤ 0.5, for δ ∈ [0.5, 1], α1 (δ) = ∞.
Hence, infγ ∈(0,1) [α1 (γ ) + α1 (1 − γ )]/2 = ∞.
Let us show that under Assumption 1 the unbounded sides
of u in Lemma 1 are not bounded. Since u is bounded on one
side in Lemma 1, the opposite sides of the unbounded sides are
bounded in Lemma 1. These bounded sides are also bounded
in Lemma 2 because the bounded sides of u in Lemma 1 are
also bounded in Lemma 2. (This is because the bounds on u in
Lemma 2 are tighter than or equal to those in Lemma 1.) Since
u is never bounded on both sides in Lemma 2, the opposite sides
of these bounded sides are not bounded in Lemma 2. Therefore,
the unbounded sides in Lemma 1 are not bounded in Lemma 2.
Similarly for v, it is proven that
sup [β2 (γ ) + β2 (1 − γ )]/2
γ ∈[0,1]

≤ v ≤ inf [β1 (γ ) + β1 (1 − γ )]/2.
γ ∈(0,1)

These bounds are sharp, as the empirical evidence and
prior information are consistent with the hypothesis v =

184

Journal of Business & Economic Statistics, January 2011

supγ ∈[0,1] [β2 (γ ) + β2 (1 − γ )]/2 and also with the hypothesis v = infγ ∈(0,1) [β1 (γ ) + β1 (1 − γ )]/2. The bounds on v in
Lemma 2 are tighter than or equal to those in Lemma 1. v is
bounded on one side; it is never bounded on both sides, however. Under Assumption 1 the unbounded sides of v in Lemma 1
are not bounded. The proof is similar to that for u and is available from the author upon request.
APPENDIX D: PROOF OF LEMMA 3
Fu (α − u), Equation (3) implies that
As Fu (α) = 

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P((q, p) ∈ NW(α)) ≤ 
Fu (α − u) ≤ 1 − P((q, p) ∈ SE(α)).

Since 
Fu is known and strictly increases, by taking the inverse
of 
Fu


α −
Fu−1 1 − P((q, p) ∈ SE(α))


≤ u ≤ α −
Fu−1 P((q, p) ∈ NW(α)) .
This holds for any real number α;



sup α − 
Fu−1 1 − P((q, p) ∈ SE(α))
α∈R





Fu−1 P((q, p) ∈ NW(α)) .
≤ u ≤ inf α − 

Let us show that there exist {ui }ni=1 of type (D-2). Define
FL,u (α) as Fu (α) satisfying Fu (αH ) = P((q, p) ∈ NW(αH )),
where αH = arg infα {α − 
Fu−1 [P((q, p) ∈ NW(α))]}, i.e., FL,u is
−1
the lower bound of Fu . Take uh(i) = FL,u
(i/n) for i = 1, 2, . . . , n,
h
where u(i) is the disturbance corresponding to q(i) . Then, uh(i) attains uH ≡ infα {α − 
Fu−1 [P((q, p) ∈ NW(α))]}.
n
Similarly to {ui }i=1 of type (D-1), it is shown that q(i) ≥ uh(i)
for i ≤ k1 . Thus, the supply curves traversing uh(i) and the observations with q(i) for i ≤ k1 have positive slopes.
Since uL ≤ uH and 
Fu (α) is known, ul(i) ≤ uh(i) . Since q(i) ≥
uh(i) for i ≤ k1 , ul(i) ≤ uh(i) ≤ q(i) for i ≤ k1 . Therefore, the supply
curves traversing ul(i) and the observations with q(i) for i ≤ k1
have positive slopes.
Since q(i) ≤ ul(i) for k1 ≤ i ≤ n, q(i) ≤ ul(i) ≤ uh(i) for k1 ≤ i ≤
n. Therefore, the supply curves traversing uh(i) and the observations with q(i) for k1 ≤ i ≤ n have positive slopes.
Similarly, as Fv (β) = 
Fv is known and strictly
Fv (β − v) and 
increases, Equation (4) implies that



sup β − 
Fv−1 1 − P((q, p) ∈ NE(β))
β∈R

β∈R

α∈R

Let us show that this bound is sharp. It suffices to show
that for any n there exist the following two types of {ui }ni=1 .
Type (D-1) {ui }ni=1 satisfy the conditions that (D-i) {ui }ni=1 reFu−1 [1 − P((q, p) ∈ SE(α))]}, (D-ii) {ui }ni=1
alize u = supα {α − 
are distributed in quantiles of Fu , (D-iii) the supply curve corresponding to ui traverses (qi , pi ), and (D-iv) all of the supply curves are upward sloping, i.e., ui ≥ qi for pi ≤ p; and
ui ≤ qi for pi > p. Type (D-2) {ui }ni=1 satisfy the conditions that
(D-v) {ui }ni=1 realize u = infα {α − 
Fu−1 [P((q, p) ∈ NW(α))]},
(D-ii), (D-iii), and (D-iv).
Let us show that there exist {ui }ni=1 of type (D-1). Define
FH,u (α) as Fu (α) satisfying Fu (αL ) = 1 − P((q, p) ∈ SE(αL )),
where αL = arg supα {α − 
Fu−1 [1 − P((q, p) ∈ SE(α))]}, i.e.,
−1
FH,u is the upper bound of Fu . Take ul(i) = FH,u
(i/n) for i =
l
1, 2, . . . , n, where u(i) is the disturbance corresponding to q(i) .
(q(i) is defined in the proof of Lemma 2 in Appendix C.) Then,
ul(i) attains uL ≡ supα {α − 
Fu−1 [1 − P((q, p) ∈ SE(α))]}. (ul(i)
and uL are newly defined and different from those in Appendix C.)
1 − P((q, p) ∈ SE(q(i) )) = i/n for k1 < i ≤ n, where k1 is
defined in the proof of Lemma 2. Since FH,u (α) = 
Fu (α − uL ),
−1

(γ ) − uL . Therefore, for k1 < i ≤ n,
Fu−1 (γ ) = FH,u


 
uL ≥ q(i) − 
Fu−1 1 − P (q, p) ∈ SE q(i)






i
i
−1
Fu−1
= q(i) − FH,u
− uL
= q(i) − 
n
n
= q(i) − ul(i) + uL .

The first inequality holds because of the definition of uL . Hence,
q(i) ≤ ul(i) for k1 < i ≤ n. Thus, the supply curves traversing
ul(i) and the observations with q(i) for k1 < i ≤ n have positive
slopes. The positive slopes of the counterparts for i ≤ k1 will be
shown later.




Fv−1 P((q, p) ∈ SW(β)) .
≤ v ≤ inf β − 

This bound is sharp.

APPENDIX E: PROOF OF PROPOSITION 2
For (q, p) ∈ SE∗ (α); f ∗ (p, α) < q = f ∗ (p, u). Since Assumption 4 and p ≤ p, f ∗ (p, u) ≤ f ∗ (p, u). Therefore, f ∗ (p, α) <
f ∗ (p, u). By Assumption 3, u > α.
For (q, p) ∈ NW∗ (α); f ∗ (p, α) ≥ q = f ∗ (p, u). Since Assumption 4 and p > p, f ∗ (p, u) ≥ f ∗ (p, u). Therefore, f ∗ (p, α) ≥
f ∗ (p, u). By Assumption 3, u ≤ α.
Hence,

(q, p) ∈ SE∗ (α) ⇒ u > α,
(q, p) ∈ NW∗ (α) ⇒ u ≤ α.
Thus,
P((q, p) ∈ SE∗ (α)) ≤ P(u > α) ≤ 1 − P((q, p) ∈ NW∗ (α)).
These bounds are sharp since the empirical evidence and
prior information are consistent with the hypothesis P((q, p) ∈
SE∗ (α)) = P(u > α) and also with the hypothesis 1−P((q, p) ∈
NW∗ (α)) = P(u > α). Hypothesis P((q, p) ∈ SE∗ (α)) = P(u >
α) is consistent with the case in which any q = f ∗ (p, u) traversing the observations (q, p) in NE∗ (α) and SW∗ (α) satisfies
f ∗ (p, α) ≥ f ∗ (p, u). Because of Assumption 3, u ≤ α for u’s
which are associated with (q, p) in NE∗ (α) and SW∗ (α). Hypothesis 1 − P((q, p) ∈ NW∗ (α)) = P(u > α) is consistent with
the case in which any q = f ∗ (p, u) traversing the observations
(q, p) in NE∗ (α) and SW∗ (α) satisfies f ∗ (p, α) < f ∗ (p, u). Because of Assumption 3, u > α for u’s that are associated with
(q, p) in NE∗ (α) and SW∗ (α).
Since P(u > α) = 1 − Fu (α), for any real number α,
P((q, p) ∈ NW∗ (α)) ≤ Fu (α) ≤ 1 − P((q, p) ∈ SE∗ (α)).
These bounds are sharp.

Okumura: Nonparametric Estimation of Labor Supply and Demand Factors

Similarly,
∗

∗

P((q, p) ∈ SW (α)) ≤ Fv (α) ≤ 1 − P((q, p) ∈ NE (α)).

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:00 11 January 2016

These bounds are sharp.
APPENDIX F: DESCRIPTION OF DATA AND
ESTIMATION METHODS

(1) Normalization, (qit , pit ): For qit = T1 Tt=1 qit and pit =
1 T
t=1 pit ; if the time averages of μt , νt , εit and ξit are zero,
T
and the time average of fit (pit ) equals fit (pit ), then fit (pit ) =
git (pit ) = qit .
I also use (1) lagged values of the variables, qit = qit−1 and
pit = pit−1 and (2) the observation at some specified year (the
beginning, the middle, and the end of the sample periods),
qit = qis and pit = pis (s = 1, T/2, T). The interpretation of the
estimation results regarding the causes of gender and educational wage differentials does not change. Estimation results are
available from the author upon request.
(2) Group and category: The sample is divided into 64 groups
distinguished by two sex categories, four education categories
(8–11, 12, 13–15, and 16+ years of schooling), and eight experience categories (1–5, 6–10, 11–15, 16–20, 21–25, 26–30,
31–35, and 36–40 years).
High-school equivalents consist of those with 8–12 years of
schooling (high-school graduates and high-school dropouts).
College equivalents consist of those with 13–15 and 16+ years
of schooling (college graduates and those with some college).
(3) Variances of normal distributions in case (iii): Minimum variance is attained when the lengths of the estimated
bounds of the shift variables in some years are zero and
positive in other years. Since the length of the estimated
bounds increases as the variance increases, the maximum variance is attained when the length of the estimated bounds
is longest. This longest length of the estimated bounds of
the supply shift variables is the distance between the maximum q-values of the observations where p > p and the
minimum q-values of the observations where p ≤ p. The
counterpart of the demand shift variables is the distance
between the maximum q-values of the observations where
p ≤ p and the minimum q-values of the observations where
p > p.
(4) Bound estimates of the difference of shift variables between groups: In Figure 7, the lower bound estimates of the
difference between the male and female labor shift variables
are identified by subtracting the female upper bound estimates
from the male lower bound estimates. The upper bound estimates are calculated by subtracting the female lower bound estimates from the male upper bound estimates. These bound estimates are not sharp. The bound estimates of the difference
between shift variables of college and high-school equivalents
in Figure 8 are similarly derived.
ACKNOWLEDGMENTS
This is a revision of a chapter of my dissertation from Northwestern University. An earlier version of this article was entitled “Nonparametric Estimation of Supply and Demand Factors

185

with