144 Metastatistics for the Non-Bayesian Regression Runner
whereUiis the union status indicator for workeri,wiis the wage of workeri,β
andγare parameters – the first two differ depending on whether the worker is
unionized or not –XandZare observed covariates, the are unobserved terms,
andψis the ratio of the return to unobserved time invariant individual specific
characteristics in the union to the non-union sector. The functionF(·)is some type
of cumulative density function and the elements ofZmay overlap withX.
The Bayesian analysis of Chib and Hamilton (2002) takes a variant of the above
model and is focused on how the effect of unions on wages varies acrossindividuals.
As has long been recognized, however, to a great extent unionization in the US
occurs at theestablishmentlevel (this is in contrast to unionization in Europe, which
frequently adheres to most workers in an industry). As Krashinsky (2004) observes,
this has meant that the aforementioned empirical work has been unable to rule out
the possibility of a “firm or enterprise specific fixed effect”: a worker’s union status
could merely be a marker, for instance, for the profitability or generosity of the
employer.^63 Put in other words, the list of “ceteris paribus” conditions considered
in previous research did not include “working at the same firm.”
As Freeman and Kleiner (1990) observe about the estimates of union wage
effects with individual data, the “treatment effect” of most interest comes from
an experiment on “firms” and not on “individuals”per se:
While it is common to think of selectivity bias in estimating the union wage
effect in terms of the difference between the union premium conditional on the
observed union (and nonunion) sample and the differential that would result
from random organization of a set of workers or establishments, we do not
believe that this is the most useful way to express the problem. What is relevant
is not what unionization would do to a randomly chosen establishment but
rather what it would do to establishments with a reasonable chance of being
unionized – to firms close to the margin of being organized rather than to the
average nonunion establishment.DiNardo and Lee (2004) use a regression discontinuity design, which, in their
context, provides a very good approximation to an RCT of the sort discussed in
the quote from Freeman and Kleiner (1990). Like previous work in this area, one of
the motivating ideas was to put the hypothesis “do unions raise wages” to a more
severetest, one that would allow for, among other things, a firm-specific effect.
This was possible in this research design since it used data on “firms.” The
research design essentially focused on comparing firms where the union “barely
won” to those who “barely lost.”
We can only be brief here, but the experiment is a “regression discontinuity
design” based on an aspect of (US) labor law. Workers most often become unionized
as the result of a highly regulated secret ballot. If more than 50% of the workers vote
for the union, the workers win collective bargaining rights. If 50% or fewer do so,
the workers do not win the right to collective bargaining. By comparing outcomes
for employers atfirmswhere unions barely won the election (for example, by one
vote) with those where the unions barely lost, one comes close to the idealized RCT.