involve a robust set of state- and year- fi xed effects, OLS is preferred; those fi xed ef- fects are useful in distinguishing whether or not the mean shifts at the discontinuity.
For censored outcomes, the use of traditional estimators with a global polynomial has been advised by Imbens and Wooldridge 2009. The tables report a fi rst- order
approximation of the marginal effects of the Tobit equation—⌽X
i
, the latent equa- tion estimate times the average fi tted probability that the observation is not censored.
Due to the nonlinear nature of these estimates, the t- statistics for the underlying pa- rameter estimates are reported for consistency, as they are for the linear probability
models.
I also provide graphical evidence of the discontinuities. In general, graphical anal- ysis can be carried out with the average value in “bins” according to the forcing vari-
able, as advised by Imbens and Lemieux 2008. Here, the forcing variable is the difference or distance in the graphs between the ratio of family income to Federal
poverty guideline and the relevant cutoff. The bins used here are as wide as 1 percent of the Federal poverty guideline. This corresponds to a 156- wide 1 percent bin for
a family of four in 1996. A fl exible fi tting of the bin averages is also provided in the form of a local linear smoother. The weighting function of the local linear smoother
uses a rectangular kernel, as advised by Lee and Lemieux 2010. The bandwidth of kernel is based upon the rule of thumb provided by Fan and Gijbels 1996, Sec-
tion 4.2.
5
B. Validity of the Design
In order to be valid, the regression discontinuity design needs to be robust to several key concerns. First, the forcing variable here, the fraction of family income to Federal
poverty guideline, is a choice variable for the household. In many RD design settings, the forcing variable, such as date of birth, is not a matter of individual choice. That
is suffi cient to satisfy concerns of “ducking” under the threshold. If you can exactly change your forcing variable to place yourself precisely below the threshold, then
receiving the treatment is partially a matter of self- selection and is not as good as randomly assigned.
However, just because the forcing variable is infl uenced by the choices of the in- dividuals does not mean that an RD design is invalid. Conceptually, this is related to
a key adverb used above: Can the forcing variable be manipulated to move precisely below the eligibility threshold? As discussed in Lee and Lemieux 2010, coarse ma-
nipulation does not invalidate a RD design. In this context, individuals can vary their wage earnings by varying their hours or insurance choices in order to manipulate the
numerator of the forcing variable. Alternatively, they could change their family struc- ture that is, have a new child, which would increase the family’s Federal poverty
guideline and increase the denominator.
The practical obstacles to precise manipulation should be clear. First, the household must know the administrative rules: the eligibility thresholds and the Federal poverty
guideline, both of which change over time. Second, the ability of a worker to precisely change his or her wages may be limited. If a worker changes insurance options in
5. In the online appendix, I vary the bandwidth; the variations considered do not materially change the fi nd- ings described below.
order to become eligible for public health insurance, the premiums are often large and predetermined subject to nondiscrimination policies. Similarly, it may be diffi cult for
an employee to add or drop work hours exactly the precise number of hours needed to get just below the threshold. Finally, manipulating household structure may be costly,
typically takes at least nine months, and adjusts the denominator of the forcing vari- able in a lumpy, discrete manner.
Moffi tt and Wolfe 1992 found that Medicaid generosity is negatively associated with labor supply. They measure labor supply as having a job, a rather coarse tool to
achieve the kind of income- manipulation required to invalidate the RD design. Yelow- itz 1995 and Ham and Shore- Sheppard 2005 both study the effects of the eligibility
thresholds themselves on labor supply that is, working or not. Yelowitz 1995 fi nds a relationship under more restrictive econometric assumptions, while Ham and Shore-
Sheppard 2005 fi nds that relaxing those restrictions diminishes the link between eli- gibility rules and labor supply. Similarly, Garthwaite, Gross, and Notowidigdo 2014
found evidence that public health insurance can be linked to decreases of labor supply on the extensive margin, in one particular recent example.
Hamersma 2013 studies the intensive margin that is, hours and wage growth. The author fi nds no evidence of “bunching” just below the eligibility threshold. The
author does fi nd some evidence of a broader, less- exact relationship between wages and eligibility thresholds. Again, while this broad relationship between the eligibility
rules and outcomes is of interest, the lack of a narrow and exact relationship supports the RD design requirement.
Validating a RD design with a forcing variable that can be manipulated need not be left to the imagination. McCrary 2008 suggests plotting the number of observations
in each bin to test for this kind of manipulation. Such a graph in Figure 4 suggests a lack of precise manipulation. The estimated log- difference in density fi nds a 1 percent
difference in smoothed density at the discontinuity with a standard error of fi ve times that value. Thus, the estimated change in density lacks both statistical and economic
signifi cance.
6
The weekly wages and hours of the parents near the eligibility thresholds reinforce this result. Using the regression framework, neither demonstrates evidence that house-
holds are manipulating their earnings or hours near the threshold. If they did, then we would expect an unusually small number of hours worked by households near
the eligibility threshold. Likewise, households that manipulate their weekly wage to become just- eligible would have unusually low weekly earnings near the threshold.
The regression estimates not reported are not statistically signifi cant, with t- statistics below one.
In the online appendix, I present evidence of manipulation using predetermined characteristics, using methods similar to Fang, Keane, and Silverman 2008. If there
is no manipulation of the forcing variable, then the frequency of predetermined char- acteristics around the threshold should be constant. The graphs suggest a modest
decrease in percentage black near the eligibility threshold. The estimates below are largely stable to the inclusion of demographic controls, including race.
6. This procedure does not account for the weighting and complex survey design of the data. A similar graph allowing for the weighting of the complex survey design shows a similar lack of ducking under the eligibility
threshold. The graph is available in the online appendix.
IV. Results
