V. Estimated Treatment Effects
A. Calculating Takeup, Private Coverage, and Crowd- Out Among Those Made Eligible by a Counterfactual Expansion in the Income Limits
Ignoring Selection Consider a nonmarginal change for example, a 10 percent increase in the Medicaid
income limits. Applying the new limits to the individuals in the data, we can observe who would become eligible and their characteristics. This assumes that families with
incomes above the new limits do not adjust their incomes to qualify for Medicaid. Thus when we ignore selection, the Medicaid takeup rate among all of the newly
eligible denoted by new=1 is given by
20a NEWATRE =
1 N
new new=1
∑
Pr pub
i
= 1| new
i
= 1 = 1
N
new new=1
∑
Pr pub
i
= 1 =
1 N
new new=1
∑
Pr X
i
ˆ + ε
i
0 = 1
N
new new=1
∑
[1 − ⌽
1
− X
i
ˆ ],
where N
new
refers to the number of newly eligible individuals.
18
Similarly, let new
k
= 1 denote individuals in a particular demographic group k who are made newly eligible,
and let N
new k
denote the number of these individuals. Then we can estimate the takeup rate for the newly eligible in group k when ignoring selection by calculating an analo-
gous equation to 20a for the subset of the newly eligible in group k: 20b
NEWATRE
k
= 1
N
new k
newk =1
∑
[1 − ⌽
1
− X
i
ˆ ].
The private insurance coverage rate among the newly eligible when we ignore se- lection is estimated by
21
NEWPRE = 1
N
new new=1
∑
Pr priv_elig
i
= 1| new
i
= 1 = 1
N
new new=1
∑
Pr priv_elig
i
= 1 =
1 N
new new=1
∑
[Pr X
i
ˆ␥
e
+ u
ei
0] = 1
N
new new=1
∑
[1 − ⌽
1
− X
i
ˆ␥
e
].
It is straightforward to estimate private insurance coverage for those newly eligible in group k as we did in moving from 20a to 20b, and we omit this equation to save
space. Next, for the newly eligible, the fraction that would have private insurance in the absence of eligibility when we ignore selection is given by
18. The conditioning on new = 1 in the fi rst equality refers to possible selection pertaining to the newly eligible, and the simplifi cation in the second equality follows because we are ignoring selection for now.
22 NEWPRNE =
1 N
new new=1
∑
Pr priv_nelig
i
= 1| new
i
= 1 =
1 N
new new=1
∑
Pr priv_nelig
i
= 1 = 1
N
new new=1
∑
Pr X
i
ˆ␥
ne
+ u
nei
= 1
N
new new=1
∑
[1 − ⌽
1
− X
i
ˆ␥
ne
]. The private coverage that group k would have in the absence of becoming newly
eligible is again a simple extension of Equation 22, and we omit the equation given space constraints. Finally, given the expression for the variance- covariance matrix
of the parameter estimates in Equation 19, it is straightforward to calculate standard errors for all treatment effects in this paper using the delta method, and we show an
example of this in the online Appendix.
Since they are counterfactuals, NEWATRE estimated average takeup rate among the newly eligible and NEWPRE estimated private insurance coverage for the newly
eligible cannot be directly observed in the data while NEWPRNE estimated private insurance coverage among the newly eligible in the absence of eligibility can be
directly measured in the data because children who would become eligible under our nonmarginal change are in fact currently observed to be ineligible. We defi ne crowd-
out among the newly eligible when we ignore selection as the private coverage rate the newly eligible would be predicted to have minus the private coverage rate they
would have if ineligible—NEWPRE minus NEWPRNE, or Equation 21 minus Equa- tion 22. This is a well- defi ned crowd- out measure, in contrast to many in the literature
where crowd- out is backed out from relative responses to eligibility for public and private coverage. Note that crowd- out among the newly eligible will depend on two
factors: the extent to which the benefi ts of private insurance exceed the costs of private insurance in the absence of eligibility, and the difference in the benefi ts and costs of
Medicaid insurance for these families. We would conjecture that newly eligible fami- lies are likely to value Medicaid insurance less than those currently eligible and may
dislike stigma more than the currently eligible; these factors will lead to crowd- out effects being smaller among the newly eligible than the currently eligible. However,
the newly eligible are more likely to be able to obtain a signifi cantly lower price for private insurance via employer sponsored insurance. Hence it is not clear which effect
will dominate and whether crowd- out will be higher or lower among newly eligible families than among the currently eligible.
19
The expressions for takeup, private coverage, and crowd- out for all currently eligible or the currently eligible in specifi c demographic groups—that is, average
treatment effects for the treated, are analogous to Equations 20a–22 when we ignore selection, and as noted above we provide them in the online Appendix. However, we
note that these estimates not only provide important policy parameters but also allow us to conduct a specifi cation check on our model by comparing the takeup and private
19. For example, consider a very poor family that effectively has a very low probability of purchasing private insurance in the absence of Medicaid. For this family there is essentially no private insurance for Medicaid
to crowd out.
coverage for eligible children and private coverage for ineligible children that are predicted from the model to the actual values observed in the sample. We can also do
this for a demographic group k.
B. Calculating Takeup and Crowd- Out Among Those Made Eligible by a Counterfactual Expansion in the Income Limits While Accounting
for Selection To calculate the policy effects while allowing for selection, we must differentiate be-
tween the values of the explanatory variables under the old and new income limits. For each individual let
Z
0i
denote the value of the previous explanatory variables and Z
1i
denote the value of the explanatory variables under the new income limits; note that Z
0i
and Z
1i
differ only in terms of Z
1i
having a higher value of FRACELIG
i
. To begin,
note that the probability that a child is newly eligible is given by 23
Prnew
i
= 1 = Pr−Z
1i
ˆ␦ e
i
≤ −Z
0i
ˆ␦] = ⌽
1
−Z
0i
ˆ␦ − ⌽
1
−Z
1i
ˆ␦. Now Medicaid takeup among the newly eligible is given by
24a NEWATRESEL =
1 N
new new=1
∑
Pr pub
i
= 1| newelig
i
= 1 =
1 N
new new=1
∑
Pr pub
i
= 1, newelig
i
= 1 Prnewelig
i
= 1 =
1 N
new new=1
∑
Pr X
i
ˆ + ε
i
0, −Z
1i
ˆ␦ e
i
≤ −Z
0i
ˆ␦ Pr
−Z
1i
ˆ␦ e
i
≤ −Z
0i
ˆ␦ =
1 N
new new=1
∑
⌽
2
X
i
ˆ ,
− Z
0i
ˆ␦, ˆ
34
− ⌽
2
X
i
ˆ ,
− Z
1i
ˆ␦, ˆ
34
⌽
1
−Z
0i
ˆ␦ − ⌽
1
−Z
1i
ˆ␦ ⎡
⎣ ⎢
⎤ ⎦
⎥ . We predict Medicaid takeup among the newly eligible in demographic group k as
24b NEWATRESEL
k
= 1
N
new k
newk =1
∑
⌽
2
X
i
ˆ ,
− Z
0i
ˆ␦, ˆ
ε,e
− ⌽
2
X
i
ˆ ,
− Z
1i
ˆ␦, ˆ
34
⌽
1
−Z
0i
ˆ␦ − ⌽
1
−Z
1i
ˆ␦ ⎡
⎣ ⎢
⎤ ⎦
⎥ . Similarly, private insurance coverage for the newly eligible is given by
25 NEWPRESEL =
1 N
new new=1
∑
Pr priv_elig
i
= 1| newelig
i
= 1 =
1 N
new new=1
∑
Pr priv_elig
i
= 1, newelig
i
= 1 Prnewelig
i
= 1 =
1 N
new new=1
∑
Pr X
i
ˆ␥
e
+ u
ei
0, −Z
1i
ˆ␦ e
i
≤ −Z
0i
ˆ␦ Pr
−Z
1i
ˆ␦ e
i
≤ −Z
0i
ˆ␦ =
1 N
new new=1
∑
⌽
2
X
i
ˆ␥
e
, −Z
0i
ˆ␦, ˆ
13
− ⌽
2
X
i
ˆ␥
e
, −Z
1i
ˆ␦, ˆ
14
⌽
1
−Z
0i
ˆ␦ − ⌽
1
−Z
1i
ˆ␦ ⎡
⎣ ⎢
⎤ ⎦
⎥ .
In the absence of becoming eligible, these individuals would have private coverage given by
26
NEWPRNESEL = 1
N
new
Pr priv_nelig
i
= 1| newelig
i
= 1
new=1
∑
= 1
N
new new=1
∑
Pr priv_nelig
i
= 1, newelig
i
= 1 Prnewelig
i
= 1 =
1 N
new
Pr X
i
ˆ␥
ne
+ u
nei
0, −Z
1i
ˆ␦ e
i
≤ −Z
0i
ˆ␦ Pr
−Z
1i
ˆ␦ e
i
≤ −Z
0i
ˆ␦
new=1
∑
= 1
N
new new=1
∑
⌽
2
X
i
ˆ␥
ne
, −Z
0i
ˆ␦, ˆ
23
− ⌽
2
X
i
ˆ␥
ne
, −Z
1i
ˆ␦, ˆ
24
⌽
1
−Z
0i
ˆ␦ − ⌽
1
−Z
1i
ˆ␦ ⎡
⎣⎢ ⎤
⎦⎥ .
Again note that one could also estimate NEWPRNESEL from the data. Thus crowd- out for the newly eligible while accounting for selection is given by the difference
NEWPRESEL minus NEWPRNESEL, or Equation 25 minus Equation 26. As before,
we can calculate the analogous effects including crowd- out for the currently eligible in the whole sample or in different demographic groups when we allow for selection;
the expressions for these calculations are available in the online Appendix.
VI. Comparison to Aakvik, Heckman, and Vytlacil 2005
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