Earnings Functions, Whereby the Log of Earnings Is Regressed on Years

Earnings functions, whereby the log of earnings is regressed on years of education, years
of on the job training, and individual characteristics, have been studied for a variety of
reasons.Some studies have focused on the returns to education, others on discrimination,
union non-union differentials, etc.For all these studies, a major concern has been the fact
that ability should enter as a determinant of earnings, but that it is close to impossible to
measure and therefore represents an omitted variable.
Assume that the coefficient on years of education is the parameter of interest.Given that
education is positively correlated to ability, since, for example, more able students attract
scholarships and hence receive more years of education, the OLS estimator for the
returns to education could be upward biased.To overcome this problem, various authors have used instrumental variable estimation techniques.For each of the instruments
potential instruments listed below briefly discuss instrument validity.
(a)The individual's postal zip code.
Answer Instrumental validity has two components, instrument relevance $\left( \operatorname { corr } \left( Z _ { i } , X _ { i } \right) \neq 0 \right)$ , and instrument exogeneity $\left( \operatorname { corr } \left( Z _ { i } , u _ { i } \right) = 0 \right)$ . The individual's postal zip code will certainly be uncorrelated with the omitted variable, ability, even though some zip codes may attract more able individuals. However, this is an example of a weak instrument, since it is also uncorrelated with years of education. (b)The individual's IQ or testscore on a work related exam.
Answer: There is instrument relevance in this case, since, on average, individuals who
do well in intelligence scores or other work related test scores, will have more
years of education.Unfortunately there is bound to be a high correlation with
the omitted variable ability, since this is what these tests are supposed to
measure.
(c)Years of education for the individual's mother or father.
Answer: A non-zero correlation between the mother's or father's years of education and
the individual's years of education can be expected.Hence this is a relevant
instrument.However, it is not clear that the parent's years of education are
uncorrelated with parent's ability, which in turn, can be a major determinant of
the individual's ability.If this is the case, then years of education of the mother
or father is not a valid instrument.
(d)Number of siblings the individual has.