Nonparametric Estimation of Labor Supply and Demand Factors

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1 Journal of Business & Economic Statistics ISSN: (Print) (Online) Journal homepage: 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, , DOI: / jbes To link to this article: American Statistical Association Published online: 01 Jan Submit your article to this journal Article views: 609 Citing articles: 2 View citing articles Full Terms & Conditions of access and use can be found at

2 Nonparametric Estimation of Labor Supply and Demand Factors Tsunao OKUMURA International Graduate School of Social Sciences, Yokohama National University, Yokohama , Japan 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 = f it (p it ) + μ t + ε it (supply), (1) q it = g it (p it ) + ν t + ξ it (demand), where i indexes the gender/educational/skill group, t indexes time, q it is the growth rate of labor inputs, and p it is the growth rate of real wages. μ t denotes a supply shift variable and ν t is a demand shift variable, which are fixed time effects. I will assume that f it ( ) is an increasing function, g it ( ) is a decreasing function, and f it (p it ) = g it (p it ) = q it for (q it, p it ), which is known. ε it and ξ it are disturbances and their medians for i are zero. Imagine a figure in which the horizontal axis represents q it, the vertical axis represents p it, and (q it, p it ) is the origin. In period t, when most of the cross-section observations are found in the southeast region to the lower right of (q it, p it ),it is likely that the supply shift variable μ t is positive. Symmetrically, when most observations are found in the northwest region to the upper left of (q it, p it ), it is likely that μ t is negative. I use this intuition to demonstrate that the supply-side shift variable μ t is identified within bounds. The case for the demandside shift variable ν t is handled symmetrically. These bounds can be narrowed by introducing additional assumptions on the distributions of the disturbances ε it and ξ it. 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 (q it, p it ) 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 (q it, p it ) is in the southeast region to the lower right of (q it + α, p it ), then this is a sufficient condition for μ t + ε it to be greater than α (since supply curves are upward sloping) and (2) observe that if (q it, p it ) is in the northeast, southeast, and southwest regions as compared to (q it + α, p it ), then this is a necessary condition for μ t + ε it to be greater than α. The probability that μ t + ε it is greater than α is bounded above and below by the probabilities of these two events. Therefore, the distribution of μ t + ε it [which equals 1 P(μ t + ε it >α)]is 2011 American Statistical Association Journal of Business & Economic Statistics January 2011, Vol. 29, No. 1 DOI: /jbes

3 Okumura: Nonparametric Estimation of Labor Supply and Demand Factors 175 identified within bounds. By assuming the median of ε it for i is zero, these bounds will translate into restrictions on the supply shift variable μ t, which will be shown to be identified within bounds. The symmetric result holds for the demand shift variable ν t. 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 μ t + ε it and ν t +ξ it 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). 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) (ii) P((q, p) NW(α)) F u (α) 1 P((q, p) SE(α)), (3) P((q, p) SW(α)) F v (α) 1 P((q, p) NE(α)), (4) where F u and F v are the cumulative distribution functions of the disturbances u and v, respectively. 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 F u (α) and F v (α) are estimated by using the analogy principle by replacing population quantities with their empirical counterparts. The quantiles of F u and F v are restricted by the quantiles of the bound estimates on F u and F v, 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 F u and F v.

4 176 Journal of Business & Economic Statistics, January 2011 Lemma 1. Define: Then α 1 = inf α P[(q, p) NW(α)] 0.5 }, α 2 = inf α 1 P[(q, p) SE(α)] 0.5 }, β 1 = inf β P[(q, p) SW(β)] 0.5 }, β 2 = inf β 1 P[(q, p) NE(β)] 0.5 }, u = minu F u (u) 0.5}, v = minv F v (v) 0.5}. α 2 u α 1, β 2 v β 1. u and v are bounded on one side; they are never bounded on both sides, however. Proof. See Appendix B. 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 y j ( ) is specified as f j ( ) + u j ( ), and f j (t) is known; then, the sharp bounds on the median of u j (t) are attained by the median of y j (t) f j (t), the difference between treatment response and the known value of f j (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. F u is defined as the distribution of u u, where u is the median of u. F v is defined as the distribution of v v, where v is the median of v. Let us assume that (1) F u and F v are known by an econometrician and (2) F u and F v 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: α 1 (γ ) = inf α P[(q, p) NW(α)] γ }, α 2 (γ ) = inf α 1 P[(q, p) SE(α)] γ }, β 1 (γ ) = inf β P[(q, p) SW(β)] γ }, β 2 (γ ) = inf β 1 P[(q, p) NE(β)] γ }. (5) (6) (7) Then sup [α 2 (γ ) + α 2 (1 γ)]/2 γ [0,1] u inf [α 1(γ ) + α 1 (1 γ)]/2, γ (0,1) sup [β 2 (γ ) + β 2 (1 γ)]/2 γ [0,1] v inf [β 1(γ ) + β 1 (1 γ)]/2. γ (0,1) 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. Proof. See Appendix C. Lemma 3. Assume Assumption 2. Then α F 1 [ ]} 1 P((q, p) SE(α)) sup α R sup β R u u inf α F 1 [ ]} u P((q, p) NW(α)), α R β F 1 [ ]} 1 P((q, p) NE(β)) v v inf β R Proof. See Appendix D. β F 1 [ ]} P((q, p) SW(β)). v 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), q = g (10) (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; g (p, v) weakly decreases in p for any v. Assumption 5. f (p, 0) = g (p, 0) = q. It should be noticed that p and u, aswellasp 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. (8) (9)

5 Okumura: Nonparametric Estimation of Labor Supply and Demand Factors 177 Proposition 2. Suppose Equation (10) and Assumptions 3, 4, and 5. For any real number α, (i) (ii) P((q, p) NW (α)) F u (α) 1 P((q, p) SE (α)), P((q, p) SW (α)) F v (α) 1 P((q, p) NE (α)), (11) where F u and F v are the cumulative distribution functions of the disturbances u and v, respectively. Proof. See Appendix E. 3. ESTIMATION OF LABOR SUPPLY AND DEMAND SHIFT VARIABLES Equation (2) is modified using the following panel data model: qit = f it (p it ) + μ t + ε it, (12) q it = g it (p it ) + ν t + ξ it, where i is the gender/educational/skill group index and t is the time index. q it and p it are the growth rates of the labor inputs and real wages of the ith group at time t. (q it, p it ) is a normalization satisfying f it (p it ) = g it (p it ) = q it. f it ( ) and g it ( ) are increasing and decreasing functions, respectively. μ t and ν t are supply and demand shift variables and unknown parameters (fixed time effects) to be estimated. ε it and ξ it represent disturbances with supply and demand functions and the medians of ε it and ξ it for i are zero. μ t + ε it and ν t + ξ it correspond to u and v in Equation (2), and thus, μ t and ν t 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 (μ t and ν t ). Yet, this comovement is not deterministic as independent movement is caused by idiosyncratic supply and demand disturbances (ε it and ξ it ). Supply and demand curves, f it ( ) and g it ( ), do not need to be specified and may differ across both groups and time. Let us assume that q it = 1 Tt=1 T q it and p it = 1 Tt=1 T p it and that f it (p it ) = g it (p it ) = q it.(t is the sample period. Alternative normalizations (q it, p it ) 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 μ t and ν t 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 upper and lower bounds, the dotted lines represent the 10th and 90th bootstrap percentiles. In line with Lemmas 1, 2, and 3,Iassume the distributions of ε it and ξ it : (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 p it is greater (smaller) than p it 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 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.

6 178 Journal of Business & Economic Statistics, January 2011 (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 ).

7 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.

8 180 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.

9 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 F u (α), for any real number α, P((q, p) NW(α)) F u (α) 1 P((q, p) SE(α)). 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.

10 182 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 F v (α), for any real number α, P((q, p) SW(α)) F v (α) 1 P((q, p) NE(α)). APPENDIX B: PROOF OF LEMMA 1 Equations (3) and (5) imply that since and u α 1, 0.5 P[(q, p) NW(α 1 )] F u (α 1 ), 0.5 F u (u), u = minu F u (u) 0.5}. Equations (3) and (5) imply that since and Hence, α 2 u, 0.5 F u (u) 1 P[(q, p) SE(u)], P[(q, p) SE(α 2 )], α 2 = inf α 1 P[(q, p) SE(α)] 0.5 }. α 2 u α 1. 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 = α 1 and also with the hypothesis u = α 2. u is bounded on one side. For the sake of contradiction, suppose that α 1 = and α 2 =. Since α 1 =, 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 α 2 =, 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 α 2 >, there exists a (,α 2 ), such that 1 P[(q, p) SE(a)] < 0.5 because of the definition of α 2. Therefore, if α 2 >, then 0.5 > 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)] 0.5. The set of α P[(q, p) NW(α)] 0.5} is empty. Therefore, by Equation (5), α 1 = inf =. If α 1 <, P[(q, p) NW(α 1 )] 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 q + α 1, p > p] = P[(q, p) NW(α 1 )] 0.5. Hence, α 2 =. (The set of α 1 P[(q, p) SE(α)] 0.5} is not bounded below.) Therefore, if α 2 >, α 1 =, whereas if α 1 <, α 2 =. Similarly, β 2 v β 1. This bound is sharp. v is bounded on one side; however, it is never bounded on both sides. APPENDIX C: PROOF OF LEMMA 2 Define the γ quantile of F u as m u γ. Equations (3) and (7) and the definition of m u γ imply that since and m u γ α 1(γ ), γ P [ (q, p) NW(α 1 (γ )) ] F u [α 1 (γ )], γ F u (m u γ ), m u γ = minm F u(m) γ }. Equations (3) and (7) and the definition of m u γ imply that since and Hence, α 2 (γ ) m u γ, γ F u (m u γ ) 1 P[(q, p) SE(mu γ )], γ 1 P [ (q, p) SE(α 2 (γ )) ], α 2 (γ ) = inf α 1 P[(q, p) SE(α)] γ }. α 2 (γ ) m u γ α 1(γ ). Similarly, for the (1 γ)quantile of F u α 2 (1 γ) m u 1 γ α 1(1 γ). (C.1) (C.2) Since the symmetry of the distribution of u around u implies m u γ u = (mu 1 γ u), Equations (C.1) and (C.2) imply that [α 2 (γ )+α 2 (1 γ)]/2 u [α 1 (γ )+α 1 (1 γ)]/2. (C.3)

11 Okumura: Nonparametric Estimation of Labor Supply and Demand Factors 183 Since the inequality in Equation (C.3) holds for any γ [0, 1], sup [α 2 (γ ) + α 2 (1 γ)]/2 γ [0,1] u inf [α 1(γ ) + α 1 (1 γ)]/2. γ (0,1) [Note: α 1 (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 u i } n i=1. Type (C-1) u i} n i=1 satisfy the conditions that (C-i) u i } n i=1 realize u = sup γ [0,1] [α 2(γ ) + α 2 (1 γ)]/2, (C-ii) u i } n i=1 are symmetrically distributed around u, (C-iii) the supply curve corresponding to u i traverses (q i, p i ), and (C-iv) all of the supply curves are upward sloping, i.e., u i q i for p i p and u i q i for p i > p. Type (C-2) u i } n i=1 satisfy the conditions that (C-v) u i } n i=1 realize u = inf γ (0,1)[α 1 (γ ) + α 1 (1 γ)]/2, (C-ii), (C-iii), and (C-iv). Let us show that there exist u i } n i=1 of type (C-1). Define q (i) for i k 1, as the ith order statistics of q j } whose p s are greater than p; where k 1 is the number of the observations with p being greater than p. Define q (i) for i = k 1 + 1, k 1 + 2,...,n as the (i k 1 )th order statistics of q j }, whose p s are not greater than p. Define u l (i) as the disturbances corresponding to q (i). Define u L = sup γ [0,1] [α 2 (γ ) + α 2 (1 γ)]/2. By Equation (7), q (i) = α 2 (i/n) for i k By the definition of u L, for k i n, [ ( ) ( i u L α 2 + α 2 1 i / 2 = n n)] [ ] q (i) + q (n i) /2. Therefore, u L q (i) q (n i) u L for k i n. In the case that k 1 < n/2; u L > since α 2 (n/2)>. Let us take u l (i) satisfying (C-i) through (C-iv) in this case as follows. For k i < n/2, take u l (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 u l (i) for k i n/2, such that u L u l (i) = ul (n i) u L (u l (i) and ul (n i) are symmetric around u L) and q (i) u l (i) (the supply curves traversing the observations of q (i) are upward sloping). For i k 1, if q (i) s are distributed as u L q (i) q (n i) u L, take u l (i) = q (i) and u l (n i) = 2u L q (i). Otherwise, take u l (i) = 2u L q (n i) and u l (n i) = q (n i). Then, u L u l (i) = ul (n i) u L; and q (i) u l (i) and q (n i) u l (n i). In the case that k 1 n/2; u L = since α 2 (i/n) = for i < k 1. Let us take u l (i) satisfying (C-i) through (C-iv) in this case as follows. Take u l (i) = ql (i) for i k 1 + 1; and u l (i) = for i k 1, which implies u l (i) q (i). Then, [u l (i) + ul (n i) ]/2 = =u L for all i. Consequently, this distribution of u i, which is symmetric around u and for which the corresponding supply curves are upward sloping, implies u = sup γ [0,1] [α 2 (γ ) + α 2 (1 γ)]/2. Similarly, it is shown that there exist u i } n i=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 sup γ [0,1] [α 2 (γ ) + α 2 (1 γ)]/2 α 2 (0.5) = α 2 and inf γ (0,1) [α 1 (γ ) + α 1 (1 γ)]/2 α 1 (0.5) = α 1. 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 α 2 (γ ) >, there exists a (,α 2 (γ )), such that 1 P[(q, p) SE(a)] <γ because of the definition of α 2 (γ ). Therefore, if α 2 (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), α 1 (γ ) = inf =. If α 1 (γ ) <, P[(q, p) NW(α 1 (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 + α 1 (γ ), p > p] = P [ (q, p) NW(α 1 (γ )) ] γ. Hence, α 2 (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 inf γ (0,1) [α 1 (γ ) + α 1 (1 γ)]/2 <, for γ h arg min γ (0,1) [α 1 (γ ) + α 1 (1 γ)]/2 (γ h 0.5), α 1 (γ h )< and α 1 (1 γ h )<. As shown, α 2 (γ h ) = and α 2 (1 γ h ) =. Since 1 P[(q, p) SE(α)] weakly increases in α, α 2 (γ ) weakly increases in γ. Therefore, for δ 1 γ h,α 2 (δ) =. Since γ h, for δ [0, 0.5], α 2 (δ) =. Hence, sup γ [0,1] [α 2 (γ ) + α 2 (1 γ)]/2 =. If sup γ [0,1] [α 2 (γ ) + α 2 (1 γ)]/2 >, for γ l arg max γ [0,1] [α 2 (γ ) + α 2 (1 γ)]/2 (γ l 0.5), α 2 (γ l )> and α 2 (1 γ l )>. As shown, α 1 (γ l ) = and α 1 (1 γ l ) =. Since P[(q, p) NW(α)] weakly increases in α, α 1 (γ ) weakly increases in γ. Therefore, for δ γ l, α 1 (δ) =. Since γ l 0.5, for δ [0.5, 1], α 1 (δ) =. Hence, inf γ (0,1) [α 1 (γ ) + α 1 (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 [β 2 (γ ) + β 2 (1 γ)]/2 γ [0,1] v inf [β 1(γ ) + β 1 (1 γ)]/2. γ (0,1) These bounds are sharp, as the empirical evidence and prior information are consistent with the hypothesis v =

12 184 Journal of Business & Economic Statistics, January 2011 sup γ [0,1] [β 2 (γ ) + β 2 (1 γ)]/2 and also with the hypothesis v = inf γ (0,1) [β 1 (γ ) + β 1 (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 As F u (α) = F u (α u), Equation (3)impliesthat P((q, p) NW(α)) F u (α u) 1 P((q, p) SE(α)). Since F u is known and strictly increases, by taking the inverse of F u α F u 1 [ ] 1 P((q, p) SE(α)) u α F u 1 [ ] P((q, p) NW(α)). This holds for any real number α; sup α R α F 1 [ ]} 1 P((q, p) SE(α)) u u inf α F 1 [ ]} u P((q, p) NW(α)). 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 u i } n i=1. Type (D-1) u i } n i=1 satisfy the conditions that (D-i) u i} n i=1 realize u = sup α α F 1 u [1 P((q, p) SE(α))]}, (D-ii) u i} n i=1 are distributed in quantiles of F u, (D-iii) the supply curve corresponding to u i traverses (q i, p i ), and (D-iv) all of the supply curves are upward sloping, i.e., u i q i for p i p; and u i q i for p i > p. Type (D-2) u i } n i=1 satisfy the conditions that (D-v) u i } n i=1 realize u = inf αα F 1 [P((q, p) NW(α))]}, u (D-ii), (D-iii), and (D-iv). Let us show that there exist u i } n i=1 of type (D-1). Define F H,u (α) as F u (α) satisfying F u (α L ) = 1 P((q, p) SE(α L )), where α L = arg sup α α F u 1 [1 P((q, p) SE(α))]}, i.e., F H,u is the upper bound of F u. Take u l (i) = F 1 H,u (i/n) for i = 1, 2,...,n, where u l (i) is the disturbance corresponding to q (i). (q (i) is defined in the proof of Lemma 2 in Appendix C.) Then, u l (i) attains u L sup α α F u 1[1 P((q, p) SE(α))]}. (ul (i) and u L are newly defined and different from those in Appendix C.) 1 P((q, p) SE(q (i) )) = i/n for k 1 < i n, where k 1 is defined in the proof of Lemma 2. Since F H,u (α) = F u (α u L ), F u 1 (γ ) = F 1 H,u (γ ) u L. Therefore, for k 1 < i n, u L q (i) F u 1 [ ( ( ))] 1 P (q, p) SE q(i) ( ) [ ( ) ] i i = q (i) F u 1 = q (i) FH,u 1 u L n = q (i) u l (i) + u L. The first inequality holds because of the definition of u L. Hence, q (i) u l (i) for k 1 < i n. Thus, the supply curves traversing u l (i) and the observations with q (i) for k 1 < i n have positive slopes. The positive slopes of the counterparts for i k 1 will be shown later. n Let us show that there exist u i } n i=1 of type (D-2). Define F L,u (α) as F u (α) satisfying F u (α H ) = P((q, p) NW(α H )), where α H = arg inf α α F u 1[P((q, p) NW(α))]}, i.e., F L,u is the lower bound of F u. Take u h (i) = F 1 L,u (i/n) for i = 1, 2,...,n, where u h (i) is the disturbance corresponding to q (i). Then, u h (i) attains u H inf α α F u 1 [P((q, p) NW(α))]}. Similarly to u i } n i=1 of type (D-1), it is shown that q (i) u h (i) for i k 1. Thus, the supply curves traversing u h (i) and the observations with q (i) for i k 1 have positive slopes. Since u L u H and F u (α) is known, u l (i) uh (i). Since q (i) u h (i) for i k 1, u l (i) uh (i) q (i) for i k 1. Therefore, the supply curves traversing u l (i) and the observations with q (i) for i k 1 have positive slopes. Since q (i) u l (i) for k 1 i n, q (i) u l (i) uh (i) for k 1 i n. Therefore, the supply curves traversing u h (i) and the observations with q (i) for k 1 i n have positive slopes. Similarly, as F v (β) = F v (β v) and F v is known and strictly increases, Equation (4) implies that β F 1 [ ]} 1 P((q, p) NE(β)) sup β R This bound is sharp. v v inf β F 1 [ ]} v P((q, p) SW(β)). β R 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, Thus, (q, p) NW (α) u α. 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 >αfor u s that are associated with (q, p) in NE (α) and SW (α). Since P(u >α)= 1 F u (α), for any real number α, P((q, p) NW (α)) F u (α) 1 P((q, p) SE (α)).

13 Okumura: Nonparametric Estimation of Labor Supply and Demand Factors 185 Similarly, P((q, p) SW (α)) F v (α) 1 P((q, p) NE (α)). APPENDIX F: DESCRIPTION OF DATA AND ESTIMATION METHODS (1) Normalization, (q it, p it ):Forq it = T 1 Tt=1 q it and p it = 1 Tt=1 T p it ; if the time averages of μ t, ν t, ε it and ξ it are zero, and the time average of f it (p it ) equals f it (p it ), then f it (p it ) = g it (p it ) = q it. I also use (1) lagged values of the variables, q it = q it 1 and p it = p it 1 and (2) the observation at some specified year (the beginning, the middle, and the end of the sample periods), q it = q is and p it = p is (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 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 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 with Applications to Labor and Macro Economics. The author thanks two anonymous referees, an associate editor, the editor Arthur Lewbel, Joseph Altonji, Gadi Barlevy, Lawrence Christiano, Jia-Young Fu, Roza Matzkin, Fumio Ohtake, Joon Y. Park, Christopher Taber, Elie Tamer, Emiko Usui, Randal Watson, Zhixiong Zeng, and especially Charles Manski for their helpful comments. The author thanks Lawrence Katz for sending his dataset. 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