Online Appendix for
Estimating Heterogeneous Treatment Effects of Medicaid Expansions on Takeup and Crowd-out*
John C. Ham
University of Maryland, IRP (Madison) and IZA
I. Serkan Ozbeklik
Claremont McKenna College
Lara Shore- Sheppard
Williams College and NBER
APPENDIX A – STANDARD ERROR CALCULATIONS
A.1 Linear Probability Model with Interactions (RCLPM)
Calculating the variance of the average take-up rate among the eligible (ATRE) in the RCLPM is
straightforward using the delta method since our parameter of interest is only a linear sum of regression
coefficients multiplied by the corresponding explanatory variables
 [ X ˆ
ATRE lpm 
i
elig i 1
 X iˆ1 ]/N e .
1
(A1)
Suppose that ˆ  {ˆ ,ˆ} .Thus the variance of the average take-up rate among the eligible (ATRE) can
be written as
 ATRE lpm
var ( ATRE lpm )  
ˆ


 ATRE lpm
ˆ
var
(

)


ˆ



,

(A2)
where
ATRE lpm

ˆ
X
elig 1
Ne
i
(A3)
.
The variance of the take-up response among the newly eligible families after the policy
expansion, TNEW lpm , as well as of all the private insurance coverage parameters, such as PITElpm ,
PITNElpm , and COElpm , can be calculated in an identical fashion using the observable characteristics of
the sample of the newly eligible families.
A2. Switching Probit Model (SPM)
A2.1. Average Take-up Response among the Eligible without Accounting for Unobservables
The marginal probability of participation, which is our average take-up rate as well as the take-up
response for the newly eligible population in our policy experiment assuming that the error term in the
index function for eligibility is random, is given by
1[ X i ˆ ] 
Xi 
 ( )d  .
i
(A4)
i

Our parameter of interest is
1
ATRE spm 
 [1-  (-X ˆ )] / N    ( X ˆ ) / N
1
elig 1
i
e
elig 1
1
i
e
(A5)
Then, the variance of ATREspm is
var( ATRE
spm
'
 1
)
 Ne
 1
1[ X i ˆ ] 
var ( ˆ ) 



ˆ
elig 1

 Ne
1[ X i ˆ ] 
.
ˆ
elig 1


(A6)
The derivative in (A6) can be written explicitly as
[ X i ˆ ] 1
1


N e elig 1 ˆ
Ne
 X ( X ˆ ) .
elig 1
i
(A7)
i
The variances of the take-up rates for those made eligible by the counterfactual policy experiment of
raising the 1995 Medicaid income limits are calculated in an analogous fashion.
A2.2. Average Take-up Response among the Eligible Accounting for Unobservables
Our parameter of interest is
ATRSELE spm 
1
Ne
 2 ( X i ˆ , Zi ˆe , ˆ ,e )

 ( Z ˆ )
elig 1
1
i
i
(A8)
e
where
 2 ( X i ˆ , Zi ˆe , ˆ ,e ) 
X i ˆ Zi ˆe
   ( , e , ˆ
2
i
i
,e
)de i d  i
and
 
1( Zi ˆe ) 
(A9)
Zi ˆe
 ( e )de .
i
i

In A9, 2 (.,., ˆ  ,e ) is the bivariate normal density,  (.) is the normal density, X i is the vector of
variables in the participation equation, Z i is the vector of variables in the eligibility equation, and ̂ and
ˆe are the vectors of coefficients in the participation and eligibility equations respectively.
The variance of ATRSELEspm can be written as follows.
ˆ  Var (ˆ ), and
Let   ATRSELE spm , ˆ '  ( ˆ ', ˆe' , ˆ' ,e ), 
  /   '  ((  /  )',(  / ˆe )',(  /  ,e )'). Then
2
ˆ   /   ',
Var( ATRSELEspm )  Var(ˆ )    /   ' 
(A10)
evaluated at   ˆ. The derivatives can explicitly be written as
 2 ( X i ˆ , Zi ˆe , ˆ  ,e )

1
ˆ
,


ˆ N e elig 1
 1( Zi ˆe )
(A11)
  2 ( X i ˆ , Zi ˆe , ˆ  ,e ) 
  ( Z ˆ ) 
 1( Zi ˆe )   1 i e   2 ( X i ˆ , Zi ˆe , ˆ  ,e )


ˆ

ˆe

1
  e





2
ˆ
 e N e elig 1
 ( Z ˆ )

1
i
e

(A12)
and
 2 ( X i ˆ , Zi ˆe , ˆ  ,e )
ˆ  ,e

1

,

ˆ  ,e N e elig 1
 1( Zi ˆe )
(A13)
where
 2 ( X i ˆ , Zi ˆe , ˆ  ,e )
 X i  ( X i ˆ )[( Zi ˆe  ˆ  ,e X i ˆ )/ 1  ˆ 2,e ] ,
ˆ
(A14)
 2 ( X i ˆ , Zi ˆe , ˆ ,e )
 Zi  ( Zi ˆe )[( X i ˆ  ˆ ,e Zi ˆe )/ 1  ˆ2,e ] ,
ˆ

(A15)
 2 ( X i ˆ , Zi ˆe , ˆ ,e )
 2 ( X i ˆ , Zi ˆe , ˆ ,e )
ˆ ,e
(A16)
e
1( Ziˆe )
 Zi( Zi ˆe ).
ˆ
and
(A17)
.
A2.3. Other Policy Effects
Standard error calculations for the other policy effects are contained in our online programs. For
other models, researchers can calculate standard errors using the appropriate analytical or numerical
derivatives and the delta method. Note that numerical derivatives can be easily constructed using
3
if y  f ( x 1 ,...., x j ,...., x K ) then
y / x j  [{ f ( x 1 ,...., x j   j ,...., x K )  f ( x 1 ,...., x j   j ,...., x K )}/ 2 j ],
where  j   x j if x j  0 and  j   if x j  0, where   0.001 or   0.0001.
4
(A18)
APPENDIX B
Table B1
Estimated Eligibility Coefficients for Medicaid Participation and Private Insurance Coverage
from Simultaneous Equation Linear Probability Models and Linear Probability Models with
Interactions
Eligibility
Medicaid
(1)
0.13***
(0.01)
Eligibility*Size of Household
-
Eligibility*White
-
Eligibility*Male
-
Eligibility*Two parents
-
Private
(2)
0.005
(0.01)
-
Medicaid
(3)
0.10
(0.12)
Private
(4)
-0.35**
(0.16)
0.03***
(0.01)
0.01
(0.01)
0.01
(0.02)
-0.01
(0.02)
-
-0.01
(0.01)
0.03**
(0.01)
-
-0.13***
(0.02)
0.01
(0.03)
-
Eligibility*Male head only
-
-
-0.11***
(0.04)
-0.09**
(0.04)
Eligibility*No earners
-
-
0.32***
(0.06)
-0.12
(0.078)
Eligibility*One earner
-
-
0.23 ***
(0.05)
-0.05
(0.07)
Eligibility*Two earners
-
-
0.12***
(0.04)
0.04
(0.07)
Eligibility*Highest earner’s age -
-
-0.00**
(0.00)
0.01**
(0.00)
Eligibility*Highest earner’s education
-
-0.02***
(0.00)
0.01
(0.00)
Notes: See notes to Table 2
5
Table B2
Estimated Coefficients for Medicaid Eligibility, Medicaid Participation and Private Insurance
Coverage from the Switching Probit Model
Medicaid
Eligibility
(1)
Medicaid
Participation
(2)
Private Coverage
When Eligible
(3)
Private Coverage
When Not Eligible
(4)
Size of Household
0.27***
(0.01)
0.15***
(0.01)
-0.13***
(0.01)
-0.12***
(0.01)
White
-0.29***
(0.02)
-0.34***
(0.03)
0.29***
(0.03)
0.29***
(0.02)
Male
-0.01
(0.01)
0.004
(0.01)
0.01
(0.01)
0.01
(0.01)
Two parents
-0.84***
(0.03)
-0.85***
(0.03)
0.57***
(0.03)
0.42***
(0.03)
Male head only
-0.24***
(0.04)
-0.77***
(0.06)
0.21***
(0.06)
0.17***
(0.04)
No earners
2.44***
(0.06)
1.86***
(0.10)
-1.89***
(0.11)
-1.79***
(0.05)
One earner
0.98***
(0.05)
0.73***
(0.10)
-0.71***
(0.11)
-0.60***
(0.03)
Two earners
0.10**
(0.05)
0.35***
(0.09)
-0.21*
(0.11)
-0.14***
(0.03)
Highest earner’s age
-0.03***
(0.00)
-0.02***
(0.00)
0.02***
(0.00)
0.01***
(0.00)
Highest earner’s education -0.15***
(0.00)
-0.07***
(0.01)
0.14***
(0.01)
0.16***
(0.00)
-
-
-
FRACELIG
0.48***
(0.01)
Notes: See notes to Table 2. Also, the estimates in column (1) are jointly estimated with the coefficients in column
(2). Medicaid eligibility estimates associated with column (3) and Medicaid eligibility estimates associated with
column (4) not shown but are very similar to those reported in column (1).
6
Table B3
Estimated Average Private Coverage Rates for Eligible Children
Group
All Population
Race
White
Non-White
Education of Highest Earner
High School Drop-out
High School Graduate
Some College
College Graduate
Family Structure
Female Head
Male Head
Two parents
Number of Earners
No earner
One earner
Two earners
More than two earners
(1)
RCLPM
(2)
SPM – Selection Ignored
(3)
(4)
SPM – Selection Accounted For
(5)
(6)
0.50***
(0.02)
0.44***
(0.01)
0.46***
(0.01)
0.44***
(0.01)
0.38***
(0.02)
0.34***
(0.03)
0.55***
(0.02)
0.38***
(0.03)
0.50***
(0.01)
0.31***
(0.01)
0.51***
(0.01)
0.33***
(0.01)
0.50***
(0.01)
0.30**
(0.01)
0.43***
(0.02)
*0.27***
(0.02)
0.39***
(0.03)
0.23***
(0.02)
0.34***
(0.03)
0.54***
(0.02)
0.61***
(0.02)
0.78***
(0.01)
0.25 ***
(0.01)
0.49***
(0.01)
0.58***
(0.01)
0.81***
(0.02)
0.26***
(0.01)
0.52***
(0.01)
0.61***
(0.01)
0.81***
(0.01)
0.26***
(0.01)
0.49***
0.01)
0.57***
(0.01)
0.79***
(0.01)
0.21***
(0.02)
0.43***
(0.02)
0.51***
(0.03)
0.70***
(0.03)
0.19***
(0.02)
0.37***
(0.03)
0.44***
(0.03)
0.64***
(0.04)
0.36***
(0.03)
0.58***
(0.02)
0.63***
(0.01)
0.27***
(0.001)
0.42***
(0.03)
0.62***
(0.01)
0.31***
(0.01)
0.51***
(0.01)
0.62***
(0.01)
0.27***
(0.01)
0.49***
(0.02).
0.61***
(0.01)
0.26***
(0.02)
0.42***
(0.03)
0.51***
(0.03)
0.21***
(0.02)
0.36***
(0.04)
0.47***
(0.04)
0.26***
(0.04)
0.65***
(0.01)
0.78***
(0.01)
0.78***
(0.01)
0.14***
(0.01)
0.62***
(0.01)
0.86***
(0.03)
0.88***
(0.07)
0.18***
(0.01)
0.64***
(0.01)
0.81***
(0.01)
0.81***
(0.01)
0.16 ***
(0.04)
0.62***
(0.01)
0.81***
(0.01)
0.85***
(0.02)
0.16***
(0.02)
0.53***
(0.03)
0.66***
(0.03)
0.68***
(0.03)
0.12***
(0.01)
0.47***
(0.04)
0.61***
(0.05)
0.69***
(0.05)
Notes: See notes to Table 2. Columns 1 and 3 are private insurance participation rates when a Medicaid eligible person becomes ineligible (ignoring his or her
unobservables) and columns 2 and 4 are private insurance participation rates when a Medicaid eligible person stays eligible (ignoring his or her unobservables). Column
5 presents private insurance participation rates when a Medicaid eligible person becomes ineligible (including the effect of unobservables) and column 6 presents
private insurance participation rates when a Medicaid eligible person stays eligible (including the effect of unobservables). Estimates in columns (3) and (5) are based
on maximizing (15) and estimates in columns (4) and (6) are based on maximizing (14). Estimates are based on parameter estimates for the entire sample and
characteristics of Medicaid eligible children from the SIPP 1995 panel.
7
Table B4
Counterfactual Policy Analysis: Estimated Private Coverage Rates for the Newly Eligible Population after Raising the Income
Limits by 10%
Group
All Population
Race
White
Non-White
Education of Highest Earner
High School Drop-out
High School Graduate
Some College
College Graduate
Family Structure
Female Head
Male Head
Two parents
Number of Earners
No earner
One earner
Two earners
More than two earners
(1)
RCLPM
SPM – Selection Ignored
(3)
(4)
(2)
SPM – Selection Accounted For
(5)
(6)
0.71***
(0.01)
0.74***
(0.03)
0.73***
(0.01)
0.71***
(0.01)
0.56***
(0.03)
0.49***
(0.05)
0.73***
(0.01)
0.61***
(0.02)
0.76***
(0.03)
0.65***
(0.03)
0.75***
(0.01)
0.63***
(0.01)
0.73***
(0.01)
0.62***
(0.01)
0.59***
(0.03)
0.45***
(0.03)
0.52***
(0.05)
0.39***
(0.05)
0.55***
(0.02)
0.72***
(0.01)
0.74***
(0.01)
0.89***
(0.01)
0.57***
(0.03)
0.74***
(0.03)
0.78***
(0.03)
0.99***
(0.04)
0.53***
(0.01)
0.75***
(0.01)
0.79***
(0.01)
0.92***
(0.005)
0.55***
(0.01)
0.72***
(0.01)
0.76***
(0.01)
0.90***
(0.01)
0.34***
(0.03)
0.58***
(0.03)
0.63***
(0.03)
0.79***
(0.03)
0.33***
(0.04)
0.51***
(0.05)
0.54***
(0.05)
0.70***
(0.05)
0.61***
(0.02)
0.72***
(0.02)
0.74***
(0.01)
0.59***
(0.03)
0.62***
(0.05)
0.80***
(0.03)
0.61***
(0.01)
0.69***
(0.02)
0.77***
(0.01)
0.58***
(0.01)
0.67***
(0.02)
0.76***
(0.01)
0.45***
(0.03)
0.49***
(0.04)
0.60***
(0.03)
0.37***
(0.04)
0.42***
(0.05)
0.55***
(0.05)
0.25***
(0.04)
0.70***
(0.01)
0.81***
(0.01)
0.85***
(0.02)
0.24***
(0.03)
0.71***
(0.03)
0 90***
(0.04)
0.99***
(0.08)
0.23***
(0.01)
0.72***
(0.01)
0.84***
(0.01)
0.92***
(0.01)
0.23***
(0.01)
0.69***
(0.01)
0.84***
(0.01)
0.93***
(0.02)
0.18***
(0.02)
0.55***
(0.03)
0.66***
(0.03)
0.74***
(0.03)
0.16***
(0.02)
0.47***
(0.05)
0.60***
(0.05)
0.72***
(0.06)
Notes: See notes to Tables 2 and B3. Estimates are based on parameter estimates for the entire sample and the characteristics of the children in the 1995 SIPP data
made newly eligible for Medicaid under our counterfactual policy experiment of raising the 1995 Medicaid income eligibility limits by 10%.
8