metadata: authors: 0: author: Chiswick, Barry R.
docna: 0: docna: Interpreting the coefficient of schooling in
            the human capital earnings function
docty: Policy Research Working Paper
subtopic: Labor Markets,Skills Development and Labor Force Training,Disease Control & Prevention,Educational Sciences,Rural Labor Markets
teratopic: Poverty Reduction
abstracts: cdata!: The "human capital earnings
            function" (HCEF) has become a fundamental tool in
            research on earnings, wages, and incomes in industrial and
            developing economies. It is accepted procedure in litigation
            about earnings, such as cases involving the value of lost
            earnings due to injury, death, or discrimination. It is also
            often used to make educational policy decisions based on
            estimates of the rate of return from schooling. The HCEF
            relates the natural logarithm of earnings to investments in
            human capital measured in time, such as years of schooling
            and years of post-school work experience. Among its
            desirable features: 1) it is not an ad hoc specification; it
            is derived from an identity. So the coefficients of the
            equation have economic interpretations; 2) it uses data
            effieciently; 3) it is flexible, allowing for easy
            incorporations of variables appropriate for a particular
            study; and 4) the coefficients of the HCEF are devoid of
            units, facilitating comparisons across space (such as
            countries) or across time periods (such as decades). In
            estimating the rate of return from schooling, the
            coefficient of the schooling variable is often interpreted
            as the rate of return from schooling. This may be the
            correct interpretation but the author shows that in
            principle -and in many circumstances- it is not. He also
            discusses the effects on the coeffiecient of schooling of
            the treatment of the labor supply (weeks worked and hours
            worked per week) and other measures of labor market outcomes,
repnme: repnme: Interpreting the coefficient of schooling
            in the human capital earnings function
display_title: Interpreting the coefficient of
            schooling in the human capital earnings function
listing_relative_url: /research/1997/06/438486/interpreting-coefficient-schooling-human-capital-earnings-function
topicv3: Labor Markets m1326488 855,Education m1327048 470,Injury m1327524 1428,On-the-Job Training m1439215 ON--2823,Labor Supply m1504055 2751
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Poverty Reduction
origu: Off. of VP & Head of Network (HDNVP)
no_of_pages: 20
model_extractions: I POLICY RESEARCH WORKING PAPER 1790

# Summary findings

The "human capital earnings function" (HCEF) has become a fundamental tool in research on earnings, wages, and incomes in industrial and developing economies. It is accepted procedure in litigation about earnings, such as cases involving the value of lost earnings due to injury, death, or discrimination. It is also often used to make educational policy decisions based on estimates of the rate of return from schooling.

The HCEF relates the natural logarithm of earnings to investments in human capital measured in time, such as years of schooling and years of post-school work experience. Among its desirable features:

* It is not an ad hoc specification; it is derived from an identity. So the coefficients of the equation have economic interpretations.

* It uses data efficiently.

* It is flexible, allowing for easy incorporation of variables appropriate for a particular study.

> * And the coefficients of the HCEF are devoid of units, facilitating comparisons across space (such as countries) or across time periods (such as decades).

In estimating the rate of return from schooling, the coefficient of the schooling variable is often interpreted as the rate of return from schooling. This may be the correct interpretation but Chiswick shows that in principle - and in many circumstances - it is not. He also discusses the effects on the coefficient of schooling of the treatment of the labor supply (weeks worked and hours worked per week) and other measures of labor market outcomes, such as occupational status.

This paper-a product of the Human Development Department-is part of a larger effort in the department to understand the economics of education. Copies of the paper are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Promila Singh, room J3-227, telephone 202-458-5631, fax 202-522-3233, Internet address psinghl@worldbank.org. June 1997. (18 pages) _The Policy Researcb Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors, or the countries they represent._ Produced by the Policy Research Dissemination Center
Interpreting the Coefficient of Schooling in the Human Capital Earnings Function

**Barry** R. Chiswick
# **Interpreting the Coefficient of Schooling in the Human Capital Earnings Function**

# **Barry R. Chiswick**

# **I. Introduction**

The "human capital earnings function" (HCEF) has become a fundamental tool in research on earnings, wages and incomes in developed and developing economies.' It is an accepted procedure in litigation involving earnings, such as cases involving the value of lost earnings due to injury, death or discrimination (see, for example, Gastwirth, 1988 and Federal Judiciary Center, 1994). It is also frequently used to make educational policy decisions based on estimates of the rate of return from schooling (see, for example, Psacharopoulos and Mattson, 1996).

The basic feature of the HCEF is that it relates the natural logarithm of earnings to investments in human capital measured in time, such as years of schooling and years of postschool work experience. It has several desirable features:

- (1) It is not an ad hoc specification. It is derived from an identity. As a

result, the coefficients of the equation have economic interpretations.

* Research Professor, Department of Economics, University of Illinois at Chicago. This paper was written while Barry Chiswick was a Visiting Scholar in the Education Group, Human Development Department, World Bank. He appreciated the comments received on an earlier draft from Harry Patrinos, George Psacharopoulos and Jee-Peng Tan.

I The simple schooling version was first developed in Becker and Chiswick (1966), and extended to include on-the- job training in Mincer (1974).
- (2) Because of the positive skewness of earnings and the rise in earnings inequality as schooling level increases, by using the natural logarithm of earnings rather than earnings as the dependent variable the residual variance in the HCEF is less heteroskedastic and the distribution of the residuals is closer to normal.

- (3) It is an efficient user of data. Although data on earnings, years of schooling and years since leaving school are readily available, data on individual schooling costs are not readily available. The HCEF procedure involves converting a relationship between earnings and dollar investments in human capital to one between the natural logarithm of earnings and years of investment in schooling and training.

- (4) The HCEF is flexible, allowing for easy incorporation of additional variables appropriate for the particular purpose of the study.

- (5) Finally, the coefficients of the HCEF are devoid of units, facilitating comparisons across space (e.g., countries) or across time periods (e.g., decades).

One feature of the HCEF is its frequent use for estimating the rate of return from schooling. The coefficient of the schooling variable is often interpreted as the rate of return from schooling (see, for example, Psacharopoulos and Mattson 1996, Ram 1996, Rosen 1987, Willis 1986). While this may be the correct interpretation in some circumstances, it will be shown here that in principle, and in many circumstances, this is not the correct interpretation. This paper will also discuss the effects on the coefficient of schooling of the
**3**

treatment of labor supply (weeks worked and hours worked per week) and other measures of labor market outcomes, such as occupational status.

# II. **Derivation of the Coefficient**

Por simplicity of presentation, what follows in this section will ignore post-school investments in on-the-job training and other variables, and will focus exclusively on the schooling variable. Let: Eo = Earnings if there is no schooling, Et = Earnings received each year after obtaining t years of schooling, C, = Dollar amount of investments in year t of schooling, r,= Rate of return on investments in year t of schooling, K, = **Ct .** E, l = Investment in level of schooling t relative to a full-year's potential earnings if investments were not made in this level of schooling.

If there is one period of investment in schooling for the individual, earnings after schooling is completed are:

- (1) _El = Eo + r,C, =_ **_Eo + r,K 1EO=Eg(1 +r,K , )_**

For two periods:

- (2) _E2 = El +_ **_r2C2=E,_** _+_ **_r2K2E,_** _=_ **_E,(1 +r2K)=EO(I +rK,()( +r2K)_**

Using the principle of mathematical induction, s (3) **_Es = E._ I|** **_(1+r,kt) t-1_**

where S is the number of years of schooling completed. Taking natural logarithms,
4

**s**

- (4) _LnE, = LnE 0+F Ln(l+rtKt)_

If rtK, is small, we can apply the ruleregarding natural logarithms Ln (1 +e) = for small values of E.' Then, Separate values of rYK, can be estimated for each level of S, either individual years or groups of years (grade level). For simplicity of exposition, assume r, and K, do not vary with years of schooling (r0 = **rt** for all t and Ko = **Kt** for all t). Then,

- (6) _LnE, = LnE0 + (r0K.)S._

_2_ For alternative values of Ln(1 +E):

|Ln(1 +|**_e_**|
|---|---|
|.0100|.01015|
|.0500|.0513|
|.1000|.1052|
|.1500|.1618|
|.2000|.2214|
|.3000|.3500|
|.4000|.4918|
|.8000|1.2255|

The smaller is **e,** the closer is the approximation Ln(1 +e) **e**
5

# **III. Interpreting the Coefficient**

Adding a residual to equation (6) and estimating the regression equation, the coefficient of schooling is an estimate of the average percent increase in earnings per year of schooling.' Note that the coefficient of S is not the rate of return from schooling, but rather is rK. If the parameter K is known and the regression coefficient (b) is estimated, the rate of return from schooling is _r = b/K._ The parameter K = 1 if the investment in schooling equals the full-year potential earnings if there were no further investment. This assumption was made to simplify the exposition in the orginal formulation of the specification (Becker and Chiswick, 1966). This assumption is also made in later treatments (see, for example, Mincer 1974, Willis 1986, Rosen 1987, Psacharopoulos and Mattson 1996). In most estimates of the rate of return from schooling using the HCEF there is no acknowledgmen **t** hat this assumption is made; the coefficient of schooling is just accepted as the rate of return from schooling.

There are certain circumstances in which K= 1. Ct = El 1 can occur, for example, if there are no out-of-pocket costs (Cd = 0) and the foregone earnings or opportunity cost (Cf = Et,1<sup>) is a full year's earnings. It would also occur if, for example, opportunity costs were</sup> 75 percent of full-year potential earnings and it just so happened that direct costs were equivalent to 25 percent. However, K need not equal unity. Consider a case in which the direct costs of school are fully funded by the govermment, including books and school 3 Mincer (1974) shows that the coefficient of schooling (S) is biased downward if years of labor market experience is not included in the equation.
6 supplies. The student can work during school breaks, so the forgone earnings do not equal a full-year of potential earnings, but only 60 percent of a full-year. Then K = 0.60, and the rate of return is r = b/K = (1.67)b. A coefficient of schooling of .06 (or, six percent), which might seem low, would imply a rate of return of .10 or 10 percent.

Alternatively, consider a situation in which the student pays for tuition and all school fees and supplies, and other out-of-pocket expenditures related to schooling. Suppose the direct and foregone earnings equal 150 percent of potential earnings or K = 1.5. Then if b = .06, r = .06/1.5 = 0.04 or 4 percent. Very different interpretations emerge depending on the value of K.

The value of r or K for a country need not be constant across schooling levels. One may think of three levels of schooling, years of primary (P), years of middle or secondary (M), and years of tertiary or higher (H) education. Then,

# _(7)_ **_LnE._** _= LnE,+ (rpK)P +_ **_(r,,K)M_** _+_ **_(rhKh)H,_**

where S = P+M+H.

The K's may vary by level of schooling. If, as it does in many developing countries, secondary schooling involves tuition charges as well as foregone earnings, Km may exceed unity; if higher education involves subsidized tuition, fees and living expenses (i.e., lowering forgone earnings), **Kh** may be less than unity. Interpretations regarding the relative rates of return from different levels of schooling can be influenced by the estimated values of K.

An alternative procedure is to use dummy variables for each year of schooling or for each level of schooling. For example, we can define Dp, Dm and Dh as dummy variables that
7 are unity if the person has completed primary (p), secondary (m), and higher (h) education, respectively. Then the regression coefficient of the dummy variable D, is rjKjSj, where Sj is the number of years of schooling for education level D,.

Thus far, the discussion has been in terms of private costs and private benefits. The implied rate of return is a private rate of return, The HCEF procedure allows for the computation of the "social" rate of return (r*), defined as the rate of return based on total costs (private and social or public costs) and private benefits.' The regression coefficient is still b, but the interpretation is b = r*K*, where K* incorporates the social cost of the investment.

Consider the following scenario: Foregone earnings constitute 75 percent of potential full-year earnings, and there are no tuition charges, school fees or other out-of-pocket expenses paid by the individual, but the cost to the public not paid by the individual is 50 percent of a student's full-year potential earnings. The regression coefficient is 0.06. The private rate of return is: r = b/K = .06/.75 = 0.080, or 8.0 percent.

The social rate of return is: r' = b*/K* = .06/1.25 = 0.048, or 4.8 percent.

With an estimate of K and K*, which can be estimated on a group basis, not needing microlevel or individual data, both private and social rates of return can be computed.

4This concept of the social rate of return does not include what are referred to in the literature as the externalities from schooling.
**8**

Recall that if we do not make use of the approximation regarding the natural logarithm of a number close to one: Similarly, the social rate of return is, _r*_ **_=(e b_** 1) _K*_ Hence the typical assumption that the coefficient of schooling in the HCEF is the rate of return from schooling is not correct in principle, and is approximately correct only if K = 1 and the regression coefficient is a small number,

# **IV. Effects of other Explanatory Variables**

The HCEF is highly adaptable, and a variety of variables have been added to the righthand side. The most important and imaginative of these has been post-school labor market experience in Mincer's classic study (1974). Mincer shows that the coefficient of schooling
9 is biased downward if age rather than labor market experience is held constant.<sup>5</sup> Moreover, he develops the rationale for the now standard quadratic form for the experience variable.

Measures of employment (labor supply) can be incorporated into the analysis. Let Ea equal annual earnings and Eh equal hourly earnings, where W is weeks worked in the year and H is hours worked per week. Then, _E=Eh(W)Y,(H)Y2,_ where -y, i = 1,2, is the elasticity of earnings with respect to time worked. Then, (8) **_LnlEa_** _= LnEh + YA(LnW)_ + _Y2 (LnH),_ where LnEh, is replaced by the human capital earnings function variables, schooling, experience, etc. Suppose the elasticity of earnings with respect to time units worked is unity, that is, -yl and 72 are unity. Then, a 10 percent increase in time units worked (weeks or hours) increases annual earnings by 10 percent.

_5_ If LnY = bo + b, S +b2 T, where T equals Age-S-Z, that is, potential labor market experience is the number of years since leaving school (S+Z), where Z is the age at which school is started. Then LnY = bo + b1 S +b2 (A-S-Z) = (be- Zb2 ) + (b, - b2) S + b2 A. In some circumstances, especially in developing countries where schooling attainment is very low, potential labor market experience (T) should be measured a non-negative number which is as the lesser of: T = Age-S-Z, and T = Age - X, where X is the age of the onset of employment that provides labor market experience relevant for adult work. The level of potential labor market experience may also need to be adjusted for grade repetition.
10 Note, however, that labor supply decisions (time worked) are not exogenous with respect to wage rates. This raises issues of endogenous explanatory variables which are typically ignored. As a result, however, the elasticity of earnings with respect to time units worked may exceed unity. This arises if labor supply curves are upward rising. It also arises if data are available for weeks worked but not hours worked per week, and yet weeks worked and hours worked per week are positively correlated.

Alternatively, the elasticity of earnings with respect to time units worked may be less than unity. This arises if labor supply curves are backward bending (i.e., the income effect of a higher wage dominates the substitution effect) or if there is anticipated variable period (peak-load) employment. Seasonality of employment, for example, would result in higher weekly earnings, but lower annual earnings for those who work fewer weeks in the year if, say, there is unemployment in the off season which is compensated at a lower rate. A similar situation may arise for hours worked per week. Purely random measurement errors will tend to bias -y downward, resulting in a measured y less than unity even if the true population value is unity.

The expression in equation 8 permits the estimation of the elasticities of earnings with respect to time units worked, where the interesting null hypothesis is that -yl = Y2 = 1. Empirically, for developed countries the elasticity of earnings with respect to weeks worked is often close to unity, while the elasticity with respect to hours worked is substantially and significantly below unity.

This issue is particularly relevant because of a tendency to convert the dependent variable from annual earnings into hourly earnings. Then, transforming equation (8),
**11**

_(9) Ln( a) = LnE,+(yl-l)Ln(W)+(y 2_<sup>_-l)Ln(H)._</sup> However, when hourly earnings is the dependent variable, whether observed or constructed, variables for weeks worked per year and hours worked per week are generally not included in the regression equation. Then, to the extent that _-yj_ does not equal unity and the time worked variable is correlated with an included variable, the coefficient of the included variable is biased. For example, suppose _wyi_ < 1 and -y2 < 1 and that those with more schooling work more weeks in the year and more hours per week. If the dependent variable is computed hourly earnings (Ea/WH), the coefficient of schooling is biased downwards. This results in a downward biased estimate of the rate of return from schooling. Another variable that can bias the coefficient of schooling is an alternative measure of labor market success. Consider the effects of putting occupational dummy variables, an occupational prestige score, or a variable for living in a low-income area on the right hand side of the equation. Then the regression estimates the effect of schooling on earnings within an occupational or household income strata. The coefficient does not incorporate the effect of schooling in raising occupational level and household income. It is not that this approach is "wrong" but that it tends to result in incorrect interpretations. Consider the following equation:

# _(10) LnE = bo +b,'S+b2_<sup>_1_</sup> _OCC,_

where OCC = 1 for professionals and managers and OCC = 0 otherwise. The effect of schooling on earnings is:
12 (11) _8aLnE_ I _1_<sup>_aoCC_</sup> _as ~b+ 2 as_ The coefficient of schooling b,' is a downward biased estimate of the partial effect of schooling on earnings. It merely measures the average effect of schooling on earnings within occupational strata. The coefficient b2 ' is the effect of a higher occupational strata on earnings and _aocC_ is the effect of schooling on occupational status. It is not the _as_ coefficient of schooling, but rather _aLnE = b,_ **_+b2_ ..** that is the estimated value of _as as_ rK. Similarly, stratifying the data by measures of labor market success (e.g., within occupational categories) and computing regression equations within strata also biases downward the coefficient of schooling.

When interpreted correctly, the HCEF can be an invaluable tool for estimating the rate of return from schooling.