1 An introduction to Mincer Wage Regres- sions - IZA

1 An introduction to Mincer Wage Regressions
The wage regression, as developed by Mincer, dates back to 1958 and 1974.
Ben Porath (1967) is the first formal presentation.
the starting point of the human capital accumulation model is the link
between wages (observed) and the quantity of skills owned by an individual
(unobserved) in a competitive labor market. That is
where
• Wt is the market wage rate
Wt = Pt.Ht
• Pt. is the price of a unit of skills
• Ht is the total quantity of skills (Human capital)
• now, clearly
log Wt =logPt. +logHt
The major contribution of human capital theory is to develop a production
function approach to the specification of Ht(.) such that it may be
expressed as a function of observable variables (time allocation)
1.1 Mincer (1974): The Mincerian wage Regression
In his classical book, Mincer develops the famous Mincerian wage regression;
the statistical relationship between market wages, education and experience.
at this stage, the Mincer wage equation is solely descriptive and it is not
informative about the optimal quantity of schooling (it is typically developed
under the assumption of an exogenously determined rate of human capital
accumulation). Define
• Et as potential earnings at t (if working full time). Think of Et as
human capital.
• ct = kt · Et as investment in human capital (say training)
1
(1)
(2)

• kt is the amount of time (exogenous) devoted to training (kt =1when
in school)
• ρ t is the return to training (or schooling)
Et+1 = Et + ct · ρ t = Et(1 + kt · ρ t) (3)
• In general, starting from period 0 (earnings potential at 0 is E0)
t−1
Et =[
Y
(1 + kj · ρj)]E0 j=0
now, divide the horizon in 2 periods (school and post schooling training)
• ρ t = ρ s (in school)
• ρ t = ρ 0 (in training activities)
• clearly
t−1
Et =[
Es =[(1+ρ s) s · E0
Y
(1 + kj · ρ0)](1 + ρs) s · E0
j=s
• taking logs of potential earnings, and noting that both ρ0 and ρs are
small
Xt−1
(6)
ln Et =lnE0 + s · ρ s + ρ 0 ·
• the next steps are based on the willingness to introduce experience (X)
in the expression. The trick is to specify kj(X) by imposing an exogenous
rate of capital accumulation beyond school completion. Noting
that at t=T-s=X
j=s
ln(w(s, x)) = α0 + ρ s · S + β 0 · X + β 1 · X 2
which is the celebrated Mincer wage regression.
• note that, returning to (.), the intercept term is now the product of the
log skill price and the initial ability.
2
kj
(4)
(5)
(7)

• ρ s acts as the ”return to schooling”
• β 0/β 1 define the return to experience (and reflect concavity of the age
earnings profile when β 1 is negative)
• When (.) is augmented with race/ sex, then it may be used to study
”discrimination”
1.2 measuring the returns to schooling
1.2.1 background
Before describing the key aspects of the structural literature, it is useful to
survey some historical aspects. For the sake of the presentation, I consider
two distinct periods. First, as the literature devoted to the return to schooling
can hardly be distinguished from the Mincerian wage regression, I review
briefly the creation and the development of the Mincerian wage regression
model. A second period covers a large number of empirical studies that
used Instrumental Variable (IV) techniques to infer the returns to schooling.
These papers belong to an even larger set of empirical research using the
“Natural Experiment” approach.
The standard form of the Mincer wage regression is
log Wt = wt = β 0 + β 1 · Schoolingt + β 2 · expt + β 3 · exp 2 t +εt
where exp denotes accumulated labor market experience, β 0 is the sum of the
rental price of human capital (in log) and the level of ability, and where β 1,β 2
and β 3 are the parameters that represent the technology by which schooling
and labor market experience are transformed into skills.
For a fixed (known) level of ability, the term εt represents a pure random
shock affecting wages at a particular point in time. When (1) is fit
to a cross-section of heterogenous individuals, the error term will typically
be interpreted as unmeasured differences in innate ability. The assumptions
of linearity in the error term and in accumulated schooling are ad hoc assumptions
which are only based on statistical choices. As it will be clear
later, these assumptions are not innocuous. Finally, the quadratic term in
experience is meant to allow for the possible decline in post-schooling human
capital acquisition.
3
(8)

The term “ability” should be understood as a time invariant level of skills
that exists prior to the start of the human capital accumulation process and
that affects labor market wages, even after controlling for acquired human
capital. I refer to β 1 as the return to schooling. It designates the marginal
effect of schooling in percentage on log wages (as opposed to the internal rate
of return). If the effect of schooling is non-linear, the local return to schooling
refers to the percentage wage increase per additional year of schooling. 1 The
average return refers to the slope of the straight line between the intercept
and the expected log wage at a given number of years of schooling. Finally,
β 2 and β 3 are those that correspond to the return to experience.
While the human capital literature has been generalized so to incorporate
post schooling skill acquisition (say on-the-job training), it should be noted
that the Mincerian wage regression is a representation of the statistical relationship
between wages and experience (given schooling) for an exogenously
determined rate of on-the-job training. The Mincerian wage regression disregards
the endogeneity of post-schooling human capital accumulation and
treats schooling and training symmetrically. More precisely, the Mincerian
approach usually ignores the possibility that schooling may change the onthe-job
human capital accumulation process. 2 Statistically, these refinements
would either require the joint modeling of schooling and training or the nonseparability
between schooling and experience. As far as I know, these issues
have not been addressed extensively in the empirical literature but are likely
to be the object of research in a near future.
1.2.2 The instrumental Variable (IV) Literature
By the early 1970’s, the estimation of the returns to schooling using Mincerian
wage regressions had become one of the most widely analyzed topic in
applied econometrics. In a famous survey, Griliches (1977) pointed out several
econometric problems that arise in estimating the returns to schooling
and, in particular, those pertaining to the measurement of both schooling
and ability. Until then, substantial effort had been devoted to the estimation
of the return to schooling with control variables measuring (or proxies
of) unobserved ability. These measures are typically IQ tests or objects of a
1 The terms local and marginal returns may be used interchangeably.
2 In practice, this could arise if schooling affects traning opportunities or on-the-job
search outcomes. Some related issues are mentioned in surveys such as those written by
Willis (1986), Rosen (1984).
4

similar nature, such as the Armed Forces qualifications (AFQT) tests. Those
are most likely imperfect measures of labor market skills. Casual empiricism
would reveal that AFQT scores (available in the popular National Longitudinal
Survey of Youth) are more strongly correlated with schooling attainments
than they are with wages 3
More interestingly, Griliches recognized that the endogeneity of schooling
decisions, virtually ignored until then, was a serious issue which might
have prevented economists to uncover the true causal effect of education on
earnings. Subsequently, a wide range of empirical papers using instrumental
variable (IV) techniques have been published. A large segment of this literature
is based on “institutional features” of the education system. Card
(2001 and 2002) presents an extensive survey of this literature and discuss
the main conceptual issues within a unifying theoretical structure in which
individuals compare the benefits of schooling with the costs of schooling born
early in the life cycle. 4
At the econometric level, the main issue may be illustrated by the following
cross-sectional regression function
wi = β 0 + β 1 · Schoolingi + η i = β 0 + β 1 · Si + η i
Ignoring post-schooling labor market experience, it is clear that the discrepancy
between OLS and IV estimates is a reflection of the correlation between
schooling and unobserved ability (η i). A positive (negative) correlation is
associated with a positive (negative) ability bias. In the IV literature, the
ability bias is only indirectly investigated through the discrepancy between
IV and OLS estimates, assuming that the linear model is correct. As we will
see later, in structural models, the ability bias may be evaluated directly.
With estimates of the utility of attending school and market ability, it may
be obtained by simulation methods.
E( ˆ β ols) =β + cov(η iSi)
VarSi
(9)
(10)
if cov(ηiSi) > 0 ability bias (positive) (11)
if cov(ηiSi) > 0 ability bias (negative) (12)
3 See Belzil and Hansen (2003).
4 The model is intertemporal but is non-stochastic. It borrows from becker (1967).
5

To get around ability bias arguments, it is customary to estimate the
returns to schooling using IV methods. So let’s define an instrument Zi, such
that
The IV estimator is such that
Corr(Zi,Si) 6= 0 (13)
Cor(Ziηi) = 0
E( ˆ cov(log wi,Zi)
βIV )=
cov(SiZi) = β + cov(ηiZi) (14)
cov(SiZi)
inthecontextwhereSi (and Zi) is discrete, so di =1if individual i goes
to school and 0 if not, we obtain a Wald estimator
E( ˆ β IV )= E(log wi | Zi =1− E(log wi | Zi =0)
pr(di =1| Zi =1)− pr(di =1| Zi =0)
(15)
While a positive correlation between schooling and labor market ability
is usually expected, Card (2001, 2002) reports that a large number of studies
find that IV estimates exceed OLS estimates by a wide margin. 5
ˆβ IV > ˆ βols (16)
As of now, the IV literature retains two main explications for the high
estimates.
• First, the existence of significant measurement error in schooling may
cause the OLS to be severely biased and underestimate the true return.
My understanding of the literature is that the measurement error argument
often advanced is typically set within a classical measurement
error framework which ignores the correlation between schooling levels
and the measurement error itself and also ignores the discrete nature
of the schooling variable.
• Second, in presence of heterogeneity in the returns (say when β 1
varies across individuals), the IV estimate is inconsistent for the population
average and the resulting estimate may reflectthereturnsof
5 Estimates of the order of 15% per year of schooling are not uncommon.
6

a sub-population only. Indeed, there is a large econometric literature
concerned with the interpretation of IV estimates when the slopes are
individual specific. 6 That is
where
which implies that
log wi = β 0 + β i · Si + η i
β 1i = ¯ β 1 + η β
i
log wi = β 0 + ¯ β 1 · Si + η β
i · Si + η i = β 0 + ¯ β 1 · Si + η ∗ i
(17)
(18)
(19)
where
η ∗ i = η β
i · Si + ηi. (20)
Clearly, ˆ βIV is inconsistent for ¯ β1. why? Can Zi ⊥ η∗ i ? only if Si ⊥ η β
i . In
theory, optimal schooling (Si) is function of η β
i .
When the wage regression is given by (.). we refer to a correlated
random coefficient model (again randomness refers to the dispersion in
β1i so it is not random from the perspective of the agent.
The intuition for the failure of IV is simple; in such a framework, individual
reactions induced by Zi are heterogeneous, so the parameter arising
from IV estimation is only valid for those who changed status following the
experiment. So, the IV estimator is only valid for a sub-population (it is not
¯β 1)
1.2.3 Other Problems with the IV literature
Regardless of measurement error and neglected population heterogeneity, the
IV literature may be criticized for three main reasons.
• First, Instrumental variables (IV) techniques may be applied in a context
where the instrument is only weakly correlated with schooling
attainments (Staiger and Stock, 1997). Indeed, before the late 90’s,
most empirical researchers concentrated their efforts on finding an instrument
uncorrelated with neglected ability, but the power of the instrument
chosen was practically never investigated. In the presence of
6 For a statistial discussion, see Imbens and Angrist (1994).
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weak instruments, reported estimates may be at best imprecise and, at
worst, seriously biased (or inconsistent). As a consequence, the validity
of very high returns to schooling, reported in a simple regression framework,
should be seriously questioned. two cases are worth considered:
cov(SiZi) ≈ 0 (cov(η iZi =0)→ σ 2 (Z 0 D) −1 Z 0 Z(Z 0 D) −1 →∞ (21)
cov(SiZi) ≈ 0 (cov(ηiZi ≈ 0) → cov(ηiZi) →∞(bias) (22)
cov(SiZi)
• Second, when IV techniques are chosen, the log wage regression is usually
assumed to be linear in schooling and, perhaps, quadratic in experience.
However, there is no obvious reason to presume that the local
returns to schooling are independent of grade level. As individuals with
lower taste for schooling tend to stop school earlier, OLS (or IV) estimates
of the return to schooling, which impose equality between local
and average returns at all levels of schooling, will be strongly affected
by the relative frequencies of individuals with high and low taste for
schooling. 7 More precisely, if there are large differences in local returns
between various grade levels, the OLS estimate (measuring an average
log wage increment per year of schooling) will tend to be biased toward
the local returns at schooling attainments that are the most common
inthesampledata.
wi = β 0 + ϕ(Schoolingi)+η i
(23)
• Finally, the IV literature rarely addresses labor market experience per
se. This is surprising as labor market experience represents a key substitute
to schooling as a mean for enhancing life cycle wages. A survey
of the IV literature reveals that practically no paper presents a joint
estimation of the returns to schooling and experience. A quick glance
at the Mincerian wage regression (2) reveals that without controls for
individual differences in accumulated post-schooling human capital, it
is difficult to give a interpretation to the discrepancy between OLS and
7 This issue is sometimes referred to as the Discount Rate bias.
8

IV estimates. 8 To see this, return to the simple homogenous return
case β 1i = β 1 for everyone.
wi = β 0 + β 1 · Si + δ · PSHCi + η i = β 0 + β i · Si + η ∗ i
(24)
where PSHCi denotes all post schooling investment activities (training,
search...) and
η ∗ i = δ · PSHCi + η i. (25)
Now reconsider estimating (.), this implies that for Zi to be valid (if wages
are measured after entrance in the labor market), the following must be true
Zi ⊥ η ∗ i
(26)
but, in general, this cannot be true because Zi ⊥ PSHCi cannot be
true (think about season of birth, and what happens if you lose one year of
human capital accumulation). That if if, for instance, one individual loses
oneyearthenhe/shemayreactbyinvestinginpostschoolingtraining,search
activities and any other wage enhancing activities (Rosenzweig and Wolpin,
2000, Journal of Economic Literature).
8Many issues related to the endogeneity of work experience are discussed in Rosenzweig
and Wolpin (2000)
9