Palgrave Handbook of Econometrics: Applied Econometrics

John DiNardo 147

and Tobias (2008), which is interested in estimating the “effect” of the Body Mass
Index (BMI) on earnings.
Their point of departure is a two-equation model:

yi=f(si)+xiβ+ (^) i (3.16)
si=ziθ+ui, (3.17)
whereyiis log average hourly wages,xis a vector of demographic characteristics
(schooling, experience, and so on) thought to have an effect on wages,siis the
BMI of an individual, andf(·)is some continuous function ofswhich the authors
introduce to allow for the reasonable possibility that, if BMI has an effect on wages,
it is not necessarily linear.^64
The most obvious possible problem is “confounding” – the relationship we
observe between BMI and wages might merely represent the influence of other
omitted factors that are correlated with BMI: in their model this is represented
by a correlation between andu.^65 One such confounder they consider is “pref-
erences for long-term investments, which we mean to represent characteristics
that simultaneously impact decisions affecting both health and human capital
accumulation.”
One solution to this confounding problem is the identification of an “instru-
mental variable” that provides “exogenous” variation in BMI (that is, a variable
which is correlated with BMI but not correlated with the unobserved determi-
nants of wages). The authors discuss two possible instrumental variables for BMI

• mother’s BMI and father’s BMI – and argue for their validity in several ways,
  including references to other literature.
  In the case wheref(s)is linear ins, usual non-Bayesian practice is two-stage least
  squares or the method of instrumental variables. One test which sometimes seems
  to capture the notion of a“severe” test of the hypothesis that the instrumental
  variables are valid is an “overidentification test.” Specifically, if both instrumental
  variables are valid, the estimated effect of BMI should be similar whether mother’s
  BMI is used alone as an instrumental variable, father’s BMI is used alone, or both
  are used (Newey, 1985). If the test rejects, it is unclear how to proceed, but as the
  authors note:

From a theoretical perspective, however, it seems reasonable that the BMIs of

the parents are either jointly valid as instruments, or jointly invalid, thus poten-

tially calling into question what is actually learned from this procedure. On the

empirical side, however, the correlation between parental BMI was found to

be reasonably small (around .16), suggesting that something can be learned

from this exercise, and that its implementation is not obviously redundant or

“circular.”

The authors’ observation that the correlation between parental BMI is small seems,
to me, to suggest the importance of “severity.” Had the correlation been much
higher, one might have been tempted to conclude that the proposed test was
“obviously redundant” or “circular.”