158 Metastatistics for the Non-Bayesian Regression Runner
- Although this is an important focus of their paper, it is one that I will not focus on
since my interests in using this example lie elsewhere. To quote from their paper: “again,
it is important to recognize that many applied studies in the treatment-response litera-
ture, and to our knowledge all of those that have been conducted on this specific topic,
assume the relationship between the treatment variable and the outcome variable is lin-
ear (i.e.,f(s)=α 0 +α 1 s), and define the slope of this function as the causal effect of
interest. The assumption of linearity is likely made on computational considerations, as
IV [instrumental variables] is simple to use in this context.”
- Again because my interests lie elsewhere, one might not wish to stipulate that it is possible
to talk clearly about a “causal effect” of BMI on wages, because such an effect seems to pre-
suppose that, in whatever manner we “manipulate” an individual’s BMI, we would expect
that the “effect” of BMI on wages would be the same. However, it is possible to imagine
that the causal arrow runsfromBMItowages because (1) high BMI is equivalent to “bad
health” and “bad health” lowers wages, or (2) high BMI is equivalent to “unattractive,”
and employers discriminate against those who are “unattractive” for reasons possibly
unrelated to “productivity.” If the latter were true, a “successful” but “unhealthy diet”
that lowered BMI wouldraisewages; if the former were true, such a diet wouldlower
wages. See DiNardo (2007) for a discussion.
- It is also interesting to observe that this Bayesian “overidentification test” is arguably
better-suited to “severity” than recent non-Bayesian interpretations of such overidenti-
fication tests. In the linear instrumental variables model, for example, the failure of the
overidentification test has been recently reinterpretednotas a rejection of the premises of
the estimated model but as evidence of “treatment effect” heterogeneity (Angrist, 2004).
In this context, one possible cause (though not the only possible cause) of “treatment
effect heterogeneity” is that the true relationship between BMI and wages is, say, quadratic
but the investigator specifies a linear relationship. In such a case, one could no longer
ensure that the estimated relationship would be invariant to the choice of instrumental
variables even if the instrumental variables were “valid.” The informal test proposed by
Kline and Tobias (2008), however, easily accommodates such a situation since it allows
f(s)to be nonlinear without exhausting any overidentification.
- Some of this discussion draws from a brief discussion in unpublished lecture notes by
David Card, although for reasons of focus and brevity my account is not the same (Card,
2007).
- It should not be surprising that the term “structural model” encompasses a wide array
of activities which have very different emphases, including – to take just one example –
classic studies of demand systems, and so on (see, for example, Deaton and Muellbauer,
1980). Moreover, some work involving “structural estimation” occurs in studies that also
involve a design-based approach. For a simple illustration see DiNardo and Lemieux,
(1992, 2001). Consequently, I use the phrase “single strand” advisedly.
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