<p> Online Appendix for</p><p>Estimating Heterogeneous Treatment Effects of Medicaid Expansions on Take- up and Crowd-out*</p><p>John C. Ham</p><p>University of Maryland, IRP (Madison) and IZA</p><p>I. Serkan Ozbeklik</p><p>Claremont McKenna College</p><p>Lara Shore- Sheppard</p><p>Williams College and NBER APPENDIX A – STANDARD ERROR CALCULATIONS</p><p>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 </p><p> lpm ˆ ˆ A TRE=[ Xib1 + X i q 1 ]/ N e . eligi =1 (A1) Suppose that yˆ = { bˆ , q ˆ }.Thus the variance of the average take-up rate among the eligible (ATRE) can be written as </p><p> lpm lpm lpm 骣抖A TRE 骣 A TRE var( A TRE )= 琪 var (yˆ ) 琪 , 桫抖yˆ 桫 y ˆ (A2) where </p><p>X i A TRE lpm . = elig=1 yˆ N e (A3) The variance of the take-up response among the newly eligible families after the policy expansion,</p><p>TNEW lpm , as well as of all the private insurance coverage parameters, such as PITE lpm , PITNE lpm , and</p><p>COE lpm , can be calculated in an identical fashion using the observable characteristics of the sample of the newly eligible families. </p><p>A2. Switching Probit Model (SPM) A2.1. Average Take-up Response among the Eligible without Accounting for Unobservables</p><p>2 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 </p><p>X i m</p><p>F1[Ximˆ ] = f ( e i ) d e i . - (A4) Our parameter of interest is </p><p> spm A TRE=邋[1- F1 (- Ximˆ )]/ N e = F 1 ( X i m ˆ )/ N e elig=1 elig = 1 (A5) Then, the variance of A TRE spm is</p><p>' 轾1禙[Xmˆ ] 轾 1 禙 [ X m ˆ ] var(A TREspm )= 犏1i[ var (mˆ )] 犏 1 i . 邋 ˆ ˆ 臌Neelig=1抖m 臌 N e elig = 1 m (A6) The derivative in (A6) can be written explicitly as </p><p>1禙 [X i mˆ ] 1 邋 = Xif( X i mˆ ). Ne elig=1mˆ N e elig = 1 (A7) 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.</p><p>A2.2. Average Take-up Response among the Eligible Accounting for Unobservables Our parameter of interest is </p><p>1 F (Xmˆ , Z dˆ , rˆ ) A TRSELE spm = 2i i ee , e N ˆ e eligi =1 F1(Z id e ) (A8) where</p><p>3 ˆ Ximˆ Z i d e ˆ ˆ ˆ ˆ F2(Xim , Z i d e , re , e ) = 蝌 f 2 ( e i , e i , r e , e ) ded i e i and -� ˆ Zid e ˆ F1(Zid e ) = f ( e i ) de i . - (A9) ˆ In A9, f2 (.,., re ,e ) is the bivariate normal density, f(.) 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 mˆ</p><p>ˆ and de are the vectors of coefficients in the participation and eligibility equations respectively. </p><p>The variance of A TRSELE spm can be written as follows.</p><p> spm ˆ' ' ˆ Letv= A TRSELE, tˆ ' = ( m ˆ ', de , rˆe , e ),� V ar ( t ˆ ), and ˆ (抖v/ t) '= (( 抖 v / m )',( 抖 v / de )',( 抖 v / re , e )'). Then</p><p>Var(A TRSELE spm )= Var(tˆ ) = (抖 v / t) ' 宥ˆ ( v / t ) ', (A10) evaluated at t= tˆ. The derivatives can explicitly be written as</p><p>ˆ 禙 2(Ximˆ , Z i d e , rˆe , e ) v 1 mˆ = , ˆ ˆ m N e elig=1 F1(Zid e )</p><p>(A11)</p><p>骣禙 (Xmˆ , Z dˆ , rˆ ) 骣禙 (Z dˆ ) 琪 2i i ee , e F(Zdˆ ) -琪 1 i e F ( X mˆ , Z d ˆ , rˆ ) 琪 抖ˆ1i e ˆ 2 i i ee , e v 1 桫 de桫 d e (A12) = ˆ 2 d N e elig=1 ˆ e {F1(Z id e )} and</p><p>4 ˆ 禙 2(Ximˆ , Z i d e , rˆe , e ) v 1 rˆ = e ,e , ˆ ˆ re ,eN e elig=1 F1(Z id e ) (A13) where</p><p>禙 (Xmˆ , Z dˆ , rˆ ) 2i i ee , e =Xf( X mˆ ) F [( Z dˆ - rˆ X m ˆ )/ 1 - r ˆ 2 ], mˆ i i i ee, e i e , e (A14)</p><p>禙 (Xmˆ , Z dˆ , rˆ ) 2i i ee , e =Zf( Z dˆ ) F [( X mˆ - rˆ Z d ˆ )/ 1 - r ˆ 2 ], ˆ i i e ie, e i e e , e de (A15) ˆ ˆ ˆ 禙 2(Xim , Z i d e , re , e ) ˆ = f2(Xi mˆ , Z i d e , rˆe , e ) and rˆe ,e (A16)</p><p>禙 (Z dˆ ) 1 i e = Zf( Z dˆ ). . dˆ i i e (A17)</p><p>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</p><p> ify= f ( x1 ,...., xj ,...., x K ) then</p><p>抖y/ xj� [{ f ( x1 ,...., x j -e j ,...., x K ) - f ( x 1 ,...., x j e j ,...., x K )}/ 2 e j ], (A18) whereej= lx j if x j� 0 and e j = l if x j = 0, where l = 0.001or l 0.0001.</p><p>5 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</p><p>Medicaid Private Medicaid Private (1) (2) (3) (4) Eligibility 0.13*** 0.005 0.10 -0.35** (0.01) (0.01) (0.12) (0.16)</p><p>Eligibility*Size of Household - - 0.03*** 0.01 (0.01) (0.01)</p><p>Eligibility*White - - 0.01 -0.01 (0.02) (0.02)</p><p>Eligibility*Male - - -0.01 0.03** (0.01) (0.01)</p><p>Eligibility*Two parents - - -0.13*** 0.01 (0.02) (0.03)</p><p>Eligibility*Male head only - - -0.11*** -0.09** (0.04) (0.04)</p><p>Eligibility*No earners - - 0.32*** -0.12 (0.06) (0.078)</p><p>Eligibility*One earner - - 0.23 *** -0.05 (0.05) (0.07)</p><p>Eligibility*Two earners - - 0.12*** 0.04 (0.04) (0.07)</p><p>Eligibility*Highest earner’s age - - -0.00** 0.01** (0.00) (0.00)</p><p>Eligibility*Highest earner’s education - -0.02*** 0.01 (0.00) (0.00)</p><p>Notes: See notes to Table 2</p><p>6 7 Table B2 Estimated Coefficients for Medicaid Eligibility, Medicaid Participation and Private Insurance Coverage from the Switching Probit Model</p><p>Medicaid Medicaid Private Coverage Private Coverage Eligibility Participation When Eligible When Not Eligible (1) (2) (3) (4)</p><p>Size of Household 0.27*** 0.15*** -0.13*** -0.12*** (0.01) (0.01) (0.01)(0.01) </p><p>White -0.29*** -0.34*** 0.29*** 0.29*** (0.02) (0.03) (0.03) (0.02) Male -0.01 0.004 0.01 0.01 (0.01) (0.01) (0.01) (0.01) Two parents -0.84*** -0.85*** 0.57*** 0.42*** (0.03) (0.03) (0.03) (0.03) Male head only -0.24*** -0.77*** 0.21*** 0.17*** (0.04) (0.06) (0.06) (0.04) No earners 2.44*** 1.86*** -1.89*** -1.79*** (0.06) (0.10) (0.11) (0.05) One earner 0.98*** 0.73*** -0.71*** -0.60*** (0.05) (0.10) (0.11) (0.03) Two earners 0.10** 0.35*** -0.21* -0.14*** (0.05) (0.09) (0.11) (0.03) Highest earner’s age -0.03*** -0.02*** 0.02*** 0.01*** (0.00) (0.00) (0.00) (0.00) Highest earner’s education-0.15*** -0.07*** 0.14*** 0.16*** (0.00) (0.01) (0.01) (0.00) FRACELIG 0.48*** - - - (0.01)</p><p>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).</p><p>8 Table B3 Estimated Average Private Coverage Rates for Eligible Children</p><p>RCLPM SPM – Selection Ignored SPM – Selection Accounted For Group (1) (2) (3) (4) (5) (6) All Population 0.50*** 0.44*** 0.46*** 0.44*** 0.38*** 0.34*** (0.02) (0.01) (0.01) (0.01) (0.02) (0.03) Race White 0.55*** 0.50*** 0.51*** 0.50*** 0.43*** 0.39*** (0.02) (0.01) (0.01) (0.01) (0.02) (0.03) Non-White 0.38*** 0.31*** 0.33*** 0.30** *0.27*** 0.23*** (0.03) (0.01) (0.01) (0.01) (0.02) (0.02) Education of Highest Earner High School Drop-out 0.34*** 0.25 *** 0.26*** 0.26*** 0.21*** 0.19*** (0.03) (0.01) (0.01) (0.01) (0.02) (0.02) High School Graduate 0.54*** 0.49*** 0.52*** 0.49*** 0.43*** 0.37*** (0.02) (0.01) (0.01) 0.01) (0.02) (0.03) Some College 0.61*** 0.58*** 0.61*** 0.57*** 0.51*** 0.44*** (0.02) (0.01) (0.01) (0.01) (0.03) (0.03) College Graduate 0.78*** 0.81*** 0.81*** 0.79*** 0.70*** 0.64*** (0.01) (0.02) (0.01) (0.01) (0.03) (0.04) Family Structure Female Head 0.36*** 0.27*** 0.31*** 0.27*** 0.26*** 0.21*** (0.03) (0.001) (0.01) (0.01) (0.02) (0.02) Male Head 0.58*** 0.42*** 0.51*** 0.49*** 0.42*** 0.36*** (0.02) (0.03) (0.01) (0.02). (0.03) (0.04) Two parents 0.63*** 0.62*** 0.62*** 0.61*** 0.51*** 0.47*** (0.01) (0.01) (0.01) (0.01) (0.03) (0.04) Number of Earners No earner 0.26*** 0.14*** 0.18*** 0.16 *** 0.16*** 0.12*** (0.04) (0.01) (0.01) (0.04) (0.02) (0.01) One earner 0.65*** 0.62*** 0.64*** 0.62*** 0.53*** 0.47*** (0.01) (0.01) (0.01) (0.01) (0.03) (0.04) Two earners 0.78*** 0.86*** 0.81*** 0.81*** 0.66*** 0.61*** (0.01) (0.03) (0.01) (0.01) (0.03) (0.05) More than two earners 0.78*** 0.88*** 0.81*** 0.85*** 0.68*** 0.69*** (0.01) (0.07) (0.01) (0.02) (0.03) (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. </p><p>7 Table B4 Counterfactual Policy Analysis: Estimated Private Coverage Rates for the Newly Eligible Population after Raising the Income Limits by 10%</p><p>RCLPM SPM – Selection Ignored SPM – Selection Accounted For Group (1) (2) (3) (4) (5) (6) All Population 0.71*** 0.74*** 0.73*** 0.71*** 0.56*** 0.49*** (0.01) (0.03) (0.01) (0.01) (0.03) (0.05) Race White 0.73*** 0.76*** 0.75*** 0.73*** 0.59*** 0.52*** (0.01) (0.03) (0.01) (0.01) (0.03) (0.05) Non-White 0.61*** 0.65*** 0.63*** 0.62*** 0.45*** 0.39*** (0.02) (0.03) (0.01) (0.01) (0.03) (0.05) Education of Highest Earner High School Drop-out 0.55*** 0.57*** 0.53*** 0.55*** 0.34*** 0.33*** (0.02) (0.03) (0.01) (0.01) (0.03) (0.04) High School Graduate 0.72*** 0.74*** 0.75*** 0.72*** 0.58*** 0.51*** (0.01) (0.03) (0.01) (0.01) (0.03) (0.05) Some College 0.74*** 0.78*** 0.79*** 0.76*** 0.63*** 0.54*** (0.01) (0.03) (0.01) (0.01) (0.03) (0.05) College Graduate 0.89*** 0.99*** 0.92*** 0.90*** 0.79*** 0.70*** (0.01) (0.04) (0.005) (0.01) (0.03) (0.05) Family Structure Female Head 0.61*** 0.59*** 0.61*** 0.58*** 0.45*** 0.37*** (0.02) (0.03) (0.01) (0.01) (0.03) (0.04) Male Head 0.72*** 0.62*** 0.69*** 0.67*** 0.49*** 0.42*** (0.02) (0.05) (0.02) (0.02) (0.04) (0.05) Two parents 0.74*** 0.80*** 0.77*** 0.76*** 0.60*** 0.55*** (0.01) (0.03) (0.01) (0.01) (0.03) (0.05) Number of Earners No earner 0.25*** 0.24*** 0.23*** 0.23*** 0.18*** 0.16*** (0.04) (0.03) (0.01) (0.01) (0.02) (0.02) One earner 0.70*** 0.71*** 0.72*** 0.69*** 0.55*** 0.47*** (0.01) (0.03) (0.01) (0.01) (0.03) (0.05) Two earners 0.81*** 0 90*** 0.84*** 0.84*** 0.66*** 0.60*** (0.01) (0.04) (0.01) (0.01) (0.03) (0.05) More than two earners 0.85*** 0.99*** 0.92*** 0.93*** 0.74*** 0.72*** (0.02) (0.08) (0.01) (0.02) (0.03) (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%. </p><p>8</p>