Bayesian Analysis of the Ordered Probit Model withEndogenous Selection∗

Murat K. MunkinDepartment of Economics

531 Stokely Management CenterUniversity of Tennessee

Knoxville, TN 37919, U.S.A.Email: mmunkin@utk.edu

Pravin K. TrivediDepartment of Economics

Wylie Hall 105Indiana University

Bloomington, IN 47405, U.S.AEmail: trivedi@indiana.edu

February 2, 2007

Abstract

This paper presents a Bayesian analysis of an ordered probit model with endoge-nous selection. The model can be applied when analyzing ordered outcomes thatdepend on endogenous covariates that are discrete choice indicators modeled by amultinomial probit model. The model is illustrated by analyzing the effects of dif-ferent types of medical insurance plans on the level of hospital utilization, allowingfor potential endogeneity of insurance status. The estimation is performed usingthe Markov Chain Monte Carlo (MCMC) methods to approximate the posteriordistribution of the parameters in the model.

Key words: Treatment Effects; MCMC; Discontinuity Regression.∗We thank Jeff Racine for comments on an earlier version of the paper presented at the 2004 meetings

of the Southern Economic Association. In revising and rewriting the paper we have benefitted from thecomments of two anonymous referees, an Associate Editor, and Co-Editor John Geweke. However, weremain responsible for the current version.

1. Introduction

This paper develops an estimation method for the ordered probit model with endogenous

covariates, termed the ordered probit model with endogenous selection (OPES). Specif-

ically, we analyze the effect of endogenous multinomial choice indicators on an ordinal

dependent variable. Endogeneity is modeled using a correlated latent variable structure,

with multinomial choice represented by the multinomial probit model. Markov chain

Monte Carlo (MCMC) methods are then used to approximate the posterior distribution

of the parameters and treatment effects. The application of the model is illustrated by

analyzing the effects of different types of medical insurance plans on the level of hospital

care utilization by the US adult population.

The ordered probit (OP) model with exogenous covariates is well established in the

literature. Extending it to the case where some covariates are endogenous is empirically

useful. Then it can be applied also to models with count dependent variables whose

frequencies are restricted to just a few support points. Thus, the OPES model may

serve as an alternative to the existing count models with endogenous treatment.

Our model analyzes the effect of a set of endogenous choice indicators on a count

variable whose distribution displays a very large proportion of zeros. Specifically we

consider cases when even extensions of the Poisson model that allow for overdispersion

do not provide an adequate fit. Examples of such extensions include the negative bi-

nomial and the Poisson-lognormal mixture models (Munkin and Trivedi, 2003). There

are at least two empirical considerations which motivate this paper. First, using obser-

vational data we want to model an outcome (the biannual number of hospitalizations)

which is a count variable, but more than 80 percent of observations are zeros, and the

distribution has a short tail. Second, the outcome depends on some categorical dummy

variables (e.g., types of health insurance plans) which are potentially endogenous, i.e.,

jointly determined with the outcome variable. This is simply a particular case of an

often-encountered model in which some of the covariates are endogenous dummy vari-

ables. We develop a model that generalizes the OP model by including endogenous

choice variables among the covariates.

Our approach is Bayesian. The full model consists of an ordered probit equation and

2

a set of discrete choice equations. The interdependence between the OP and discrete

choice equations is modeled using a correlated latent variable structure. The defined

latent variables are made a part of the parameter set. Augmenting full conditional

densities with latent variables, following Tanner and Wong (1987) and others, simpli-

fies the MCMC algorithm. Our analysis is related to several previous contributions,

including Albert and Chib (1993), Cowles (1996), Chib and Hamilton (2000), Geweke,

Gowrisankaran, and Town (2003), Poirier and Tobias (2003), and Li and Tobias (2006).

Albert and Chib (1993) present a Bayesian treatment of the OP model using the Gibbs

sampler. However, the proposed Gibbs sampler mixes poorly in the case of many thresh-

old parameters and large samples. Geweke et al. (2003) analyze the endogenous binary

probit model (EBP) to study the quality of hospitals based on mortality rates in treating

pneumonia. In their analysis the patients self-select hospitals, so choices are endoge-

nous. Our model can be interpreted as an extension or synthesis of both the OP model

and the EBP model.

The rest of the paper is organized as follows. Section 2 describes the OPES model.

Section 3 presents the MCMC estimation algorithm for the model. Section 4 presents

an illustrative application using the Medical Expenditure Panel Survey (MEPS) data

on hospitalizations and health insurance. Section 5 concludes.

2. An Ordered Probit Model with Endogenous Selection

Assume that we observe N independent observations for individuals who choose the

treatment variable among J alternatives. Let di = (d1i, d2i, ..., dJ−1i) be binary ran-

dom variables for individual i (i = 1, ..., N) representing this choice (category J is the

baseline) such that dji = 1 if alternative j is chosen and dji = 0 otherwise. Define the

multinomial probit model using the multinomial latent variable structure which rep-

resents gains in utility received from the choices, relative to the utility received from

choosing alternative J . Let the (J − 1)× 1 random vector Zi be defined as

Zi =Wiα+ εi,

3

whereWi is a (J−1)× q matrix of exogenous regressors, α is a q×1 parameter vector,such that

dji =JYl=1

I[0,+∞) (Zji − Zli) , j = 1, ..., J,

where ZJi = 0 and I[0,+∞) is the indicator function for the set [0,+∞). The distributionof the error term εi is (J − 1)-variate normalN (0,Σ). For identification it is customary

to restrict the leading diagonal element of Σ to unity. We will impose identifying

restrictions after defining the entire model.

To model the ordered dependent variable we assume that there is another latent

variable Y ∗i that depends on the outcomes of di such that

Y ∗i = Xiβ + diρ+ ui,

where Xi is a 1 × p vector of exogenous regressors, β is p × 1 and ρ is (J − 1) × 1parameter vectors. Define Yi as

Yi =MXm=1

mI[τm−1,τm) (Y∗i ) ,

where τ0, τ1, ...,τM are threshold parameters and m = 1, ...,M . In our application Yi

is an ordered variable measuring the degree of medical service utilization. For identi-

fication, it is standard to set τ0 = −∞ and τM = ∞ and additionally restrict τ1 = 0.

Denote τ = (τ2, ..., τM−1). The choice of insurance is potentially endogenous to utiliza-

tion and this endogeneity is modeled through correlation between ui and εi. Assume

that they are jointly normally distributed such that cov(εi, ui) = δ with variance of ui

restricted for identification since Y ∗i is latent. Assume that V ar (ui) = 1 + δ0Σ−1δ.

Then ui|εi ∼ N¡δ0Σ−1εi, 1

¢.

We present our estimation strategy by first simplifying the exposition of the model

to be consistent with the application and reparameterizing Σ. In the application the

multinomial choice is among three alternatives so that J = 3. Let Zi =³Z1i, eZi´ such

that eZi = Z2i and use tilde to denote all parameters and variables related to eZi. DenoteV ar( eZi) = eσ22 where in fact eσ22 = σ22, cov( eZi, Z1i) = σ21 and restrict variance of Z1i

for identification such that V ar(ε1i) = 1 + σ221eσ−122 . Then ε1i|eεi ∼ N (σ21eσ−122 eεi, 1).4

Denote π0 = δ0Σ−1, π0 = (π1, eπ) (where π1 is 1 × 1 and eπ is 1 × 1) and eσ21 =σ21eσ−122 . There is a one-to-one correspondence between parameter sets (δ,Σ) and

(π1, eπ0, eσ21, eσ22). Then the model can be presented asY ∗i = Xiβ + diρ+ (Z1i −W1iα1)π1 + ( eZi − fWieα)eπ + ζi,

Z1i = W1iα1 + ( eZi − fWieα)eσ21 + ηi,eZi = fWieα+ eεi,such that ζi

ηieεi i.i.d.∼ N

03, 1 0 00 1 00 0 eσ22

.Let ∆i = (Xi,Wi, τ , β,ρ,π1, eπ,α1, eα, eσ21, eσ22). For each observation i the joint

density of the observable data and latent variables is

Pr³Y ∗i , Yi, Z1i, eZi,di|∆i

´= (2π)−3/2 eσ−1/222 exp

h−0.5eσ−122 ( eZi − fWieα)2i

× exph−0.5[Z1i −W1iα1 −

³ eZi − fWieα´ eσ21]2i× 3Xj=1

dji

3Yl=1

I[0,+∞) (Zji − Zli)

× exp·−0.5

³Y ∗i −Xiβ − diρ− (Z1i −W1iα1)π1 − ( eZi − fWieα)eπ´2¸

×"MXm=1

I{Yi=m}I[τm−1,τm) (Y∗i )

#.

The joint distribution of observable and latent variables for all observations is the

product of N such independent terms over i = 1, ...N . The posterior density is propor-

tional to the product of the prior density of the parameters and the joint distribution

of observables and included latent variables.

In order to identify causal effects of the endogenous treatment variables on the out-

come variable one needs exclusion restrictions, which arise if there are variables which

affect the insurance choices but not utilization. We discuss such restrictions at greater

length in the application section. However, there is a further identification issue in that

5

the multinomial probit structure is known to be difficult to identify in the absence of

additional restrictions. There are two potential sources of such restrictions, those that

arise from restricting elements of the covariance matrix and those that are generated

through restrictions on covariates which affect the utility levels of certain alternatives.

Keane (1992) shows, in the maximum likelihood framework, that without such exclusion

restrictions the estimation of the covariance parameters is tenuous. Because the identi-

fication problem also arises in the current Bayesian context, we propose such exclusion

restrictions in the application section. Further, we also study properties of the Markov

chain in our application under the no-exclusion-restriction specification.

The leading diagonal element σ11 of matrix

Σ =

·σ11 σ21σ21 σ22

¸is restricted for identification. However, the model is not formally identified if σ21 = 0

since in that case variance parameter σ22 must also be restricted. Bunch (1991) inves-

tigates estimability in the MNP model and recommends that both variance parameters

be restricted in which case the model is formally identified. We restrict parameter eσ22to unity. This restriction greatly improves the convergence properties of the resulting

Markov chain. When unrestricted even a visual examination of the chains for parameterseσ21, eσ22 in the application after considerably long runs indicates that convergence hasnot been achieved. This poor mixing is consistent with McCulloch, Polson, and Rossi

(2000) who have a similar reparameterization of the covariance matrix in their analysis

of the MNP model. They point out that more diffuse priors cause slower convergence

in their MCMC algorithm.

We select proper prior distributions for all parameters. The prior distributions

for parameters α1 and eα are normal N (α1,H−1α1) and N¡α,H−1α

¢, respectively, and

centered at zero vectors

α1 ∼ N (0, 10Ik1) , eα ∼ N ¡0, 10Iek¢ .

We also select proper normal priors centered at zero for parameters β and ρ

β ∼ N (0, 10Ip) , ρ ∼ N (0, 10I2) .

6

For the covariances among the errors εi in the MNP choice equations and the error ui

in the latent variable Y ∗i , we select prior distributions such that E(δ) = 0 and E(Σ)

is a diagonal matrix which means that E(σ21) = 0. The last assumption implies that

in the prior distributions parameters eσ21 and π are centered at zeros. We choose thepriors

π1 ∼ N (0,κ) , eπ ∼ N (0,κ) , eσ21 ∼ N (0,κ) .

We try two values κ = 1/2 and κ = 1/8 to evaluate the sensitivity of the tested null

hypothesis of no endogeneity to the choice of the priors. While parameters δ and σ21

must satisfy complicated restrictions such that Σ and the covariance matrix of vector

(ε0i, ui)0 be positive definite respectively, the new parameters π and eσ21 do not have

such restrictions.

The priors for the threshold parameters must respect the order restrictions placed

on them. It is easier to choose priors by reparameterizing these parameters first. The

prior distributions for the threshold parameters are specified in the next section.

3. MCMC algorithm

We block the parameter set ashZ1i, eZii , [Y ∗i , τ ] , α1, eα, [β,ρ,π1, eπ] , eσ21 and adopt a

hybrid Metropolis-Hastings/Gibbs algorithm. The steps of the MCMC algorithm are

the following:

1. The latent variable Z1i (i = 1, ...N) is conditionally independent with normal dis-

tribution Z1iiid∼ N (Z1i,H−11i ) where

H1i = 1 + π21,

Z1i = W1iα1

+H−11i

hπ1(Y

∗i −Xiβ − diρ− ( eZi − fWieα)eπ) + ( eZi − fWieα)eσ21i ,

and subject to

Z1i > max {Zli| l = 2, ..., J} if d1i = 1 andZ1i < max {Zli| l = 2, ..., J} if d1i = 0.

7

The latent vectors eZi (i = 1, ...N) are conditionally independent with the normaldistribution eZi iid∼ N (Zi,Hi−1) whereHi = eσ−122 + eπ2 + eσ221,Zi = fWieα

+Hi−1[eπ (Y ∗i −Xiβ − diρ− (Z1i −W1iα1)π1) + eσ21 (Z1i −W1iα1)] ,

and truncated such that

Zji > max {Zli| l = 1, ..., J, l 6= j} if dji = 1 andZji < max {Zli| l = 1, ..., J, l 6= j} if dji = 0.

We use the rejection sampling algorithm of Geweke (1991) to draw values from all

truncated normal distributions in our algorithm.

2. The full joint conditional density of block [Y ∗i , τ ] is

Pr¡Y ∗i , τ |Yi,Zi,di,∆i

¢=

NYi=1

"MXm=1

I{Yi=m}I[τm−1,τm) (Y∗i )

#

× exp·−0.5

³Y ∗i −Xiβ − diρ− (Z1i −W1iα1)π1 − ( eZi − fWieα)eπ´2¸ ,

which we write as

Pr¡Y ∗i , τ |Yi,Zi,di,∆i

¢= Pr

¡Y ∗i |τ , Yi,Zi,di,∆i

¢Pr¡τ |Yi,Zi,di,∆i

¢,

where ∆i includes the same parameters as ∆i except for τ . Latent variable Y ∗i is

N³Xiβ + diρ+ (Z1i −W1iα1)π1 + ( eZi − fWieα)eπ, 1´ and conditional on Yi = m

it is truncated on left by τm−1 and on the right by τm. The full conditional density

of vector τ = (τ2, ..., τM−1) is

NYi=1

"MXm=1

I{Yi=m} Pr¡τm−1 < Y ∗i < τm|∆i,di,Zi, Yi

¢#

8

where

Pr¡τm−1 < Y ∗i < τm|∆i,di,Zi, Yi

¢= (3.1)

Φ[τm − (Xiβ + diρ+ (Z1i −W1iα1)π1 + ( eZi − fWieα)eπ)]−Φ[τm−1 − (Xiβ + diρ+ (Z1i −W1iα1)π1 + ( eZi − fWieα)eπ)].

As the elements of vector (τ2, ..., τM−1) are ordered, the prior assigned to the

threshold parameters must be restricted. Instead, we follow Chib and Hamilton

(2000) and reparameterize them as

γ2 = log (τ2) , γj = log(τ j − τ j−1), 3 6 j < M − 1

and assign a normal prior N (γ0,Γ0) without any restrictions since elements ofvector γ =

¡γ2, ..., γM−1

¢do not have to be ordered. The full conditional for

vector γ is the product of the prior and the full conditional (3.1) after substituting

τ j =

jXk=2

exp (γk) .

This density is intractable and we utilize the Metropolis-Hastings algorithm to

sample from it, using t-distribution centered at the modal value of the full con-

ditional density for the proposal density. Note that it would be possible to avoid

the Metropolis-Hastings step with a reparameterization similar to Nandram and

Chen (1996) if the number of ordered categories did not exceed three. Let

bγ = argmax log p ¡γ|∆i,di,Zi, Yi¢,

andVbγ = −(Hbγ)−1 be the negative inverse of the Hessian of log p ¡γ|∆i,di,Zi, Yi¢

evaluated at the mode bγ. Choose the proposal distribution q (γ) = fT (γ|bγ,ϕVbγ , υ),a t-distribution with υ degrees of freedom and tuning parameter ϕ, an adjustable

constant selected to obtain reasonable acceptance rates. When a proposal value

γ∗ is drawn the chain moves to the proposal value with probability

Pr(γ,γ∗) = min

(p¡γ∗|∆i,di,Zi, Yi

¢q (γ)

p¡γ|∆i,di,Zi, Yi

¢q (γ∗)

, 1

).

9

If the proposal value is rejected then the next state of the chain is at the current

value γ.

3. Given the prior distribution of α1, N (α1,H−1α1 ), the full conditional distributionof α1 is α1 ∼ N (α1,H−1α1 ) where

Hα1 = Hα1 +NXi=1

W01i

¡1 + π21

¢W1i

α1 = H−1α1

"Hα1α1 +

NXi=1

nW0

1i

¡1 + π21

¢Z1i −W0

1i( eZi − fWieα)eσ12−W0

1iπ1(Y∗i −Xiβ − diρ− ( eZi − fWieα)eπ)oi .

4. Given the prior distribution of eα, N ¡α,H−1α

¢, the full conditional distribution ofeα is eα ∼ N (α,H−1α ) where

Hα = Hα +NXi=1

fW0i(eσ−122 + eπ2 + eσ221)fWi

α = H−1α [Hαα+

NXi=1

nfW0i(eσ−122 + eπ2 + eσ221) eZi

−fW0ieπ (Y ∗i −Xiβ − diρ− (Z1i −W1iα1)π1)

−fW0ieσ21 (Z1i −W1iα1)

oi.

5. Let Ci = (Xi,di, (Z1i −W1iα1) , ( eZi − fWieα)), χ0 = ¡β0,ρ0

¢, θ0 = (χ0,π1, eπ)

and specify prior distributions χ ∼ N (χ,H−1χ ) and π ∼ N¡π,H−1π

¢. The full

conditional distribution of θ is θ ∼ N (θ,H−1θ ) where

Hθ =

µHχ 00 Hπ

¶+

NXi=1

C0iCi

θ = H−1θ

"µHχχ

Hππ

¶+

NXi=1

C0iY∗i

#.

6. Given the prior distribution of eσ21, N ¡σ,H−1σ

¢, the full conditional distribution

10

of eσ21 is eσ21 ∼ N (σ,H−1σ ) whereHσ = Hσ +

NXi=1

( eZi − fWieα)2σ = H

−1σ

"Hσσ +

NXi=1

( eZi − fWieα) (Z1i −W1iα1)

#.

This concludes the MCMC algorithm.

4. Application

We study the effects of different types of medical insurance plans on hospital admission

rates of the US population aged between 55 and 75 years. Hospital utilization shows

large increases in admission rates at the age of 65, the age at which the majority of

individuals become eligible for coverage under the Medicare public insurance program

for the elderly. Hence in the literature the transition to age 65 has been treated as an

important source of variation in modeling the probability of Medicare coverage.

4.1. Modeling Considerations

The standard regression discontinuity approach assumes that measures of hospital uti-

lization would change smoothly with age in the absence of Medicare insurance and,

therefore, any change in utilization would occur due to changes in insurance status.

Following Lichtenberg (2002) and Decker, Dushi, and Deb (2005), it is argued that

there is no reason to believe that at age 65 individuals experience any structural breaks

in their health status. Although some measures of individual’s medical care use do show

a positive change at age 65, these movements are interpreted as reflecting the expression

of demand previously postponed by the uninsured individuals, and not as an indicator

of sharply deteriorating health. Hence the binary variable, DUM65, which indicates

whether an individual is older than or equal to 65 years of age, would affect utilization

only through the Medicare insurance status. In such a case DUM65 generates an ex-

clusion restriction; that is, under our specification DUM65 affects the insurance status,

but not utilization.

11

An example is the article by Card, Dobkin and Maestas (2004) which analyzes the

effect of medical insurance on utilization of medical services for the US subpopulation

aged 55-75 years in order to identify its causal effects. The article analyzes various

measures of demand for medical services, including hospital admission rates. As indi-

cated by Card et al. (2004), the study is subject to some limitations. First, it does not

distinguish among different types of medical insurance, but instead combines them in

a single insurance category. However, plan heterogeneity can potentially be a serious

issue. For example, if selection of insurance plans is partially driven by unobservable

attitude towards risk, with the privately insured being relatively more risk-averse and

the publicly insured being relatively more risk-inclined, then on average it could be dif-

ficult to first identify and then separate the selection effects if both plans are aggregated

into a single category. Second, the dependent variable in Card et al. (2004) is assumed

to be linear in covariates, a specification which would not accommodate nonlinearities,

such as variable marginal effects. Our specification allows for nonlinear relationships

between the dependent variable and covariates. Finally, the reference category in Card

et al. (2004) is the uninsured. Only a very small fraction of individuals is uninsured in

the over-65 group; in the MEPS sample less than one percent of the elderly is uninsured.

Potentially this makes it difficult to identify the coefficient of DUM65.

Our analysis is conditional on all individuals in the sample being insured. This solves

the problem of instability of parameter estimates arising from having very few uninsured

individuals aged over 65 years. Health economists generally expect that the treatment

effect of insurance relative to no insurance is positive. It seems more interesting to

identify signs and magnitudes of treatment effects for different insurance plans relative

to each other.

We consider only those individuals aged between 55 and 75 years who are insured,

either by Medicare only, or by a private insurance plan. Further, two types of private

insurance plans are considered: group and nongroup private plans. In order to have

a group plan an individual must be either employed or have it through an employed

spouse. This is referred to as an employer sponsored insurance (ESI) plan. A nongroup

plan must be purchased individually through a private insurance firm, and hence it is

referred to as an individually purchased insurance (IPI) plan. Group plans benefit from

12

a higher degree of risk pooling and typically involve lower premia than the correspond-

ing IPI plans. For the Medicare-eligible population, an ESI plan is a supplementary

insurance plan which may cover copayments for ambulatory visits and payments to-

wards prescription medications. An IPI plan may be a variant of a number of available

supplementary insurance plans. Some of these variants can be expected to have a higher

premium and less generous coverage. Goldman and Zissimopoulos (2003, p.198) show,

using 1998 Health and Retirement Survey data, that beneficiaries with ESI plans have

significantly lower average out-of-pocket expenditures than those with other supplemen-

tary plans. Atherly (2002, p.137) notes that in the past ESI plans have been a major

source of prescription drug coverage. Hence, it is possible that healthier individuals, or

those with a greater capacity for assuming health risks, would self-select into private

plans with lower coverage.

Individuals with Medicaid (public insurance for the low-income) are deleted from

the sample. To be eligible for Medicaid an individual must satisfy some strict income

criteria. In the health economics literature, Medicaid status is often assumed to be

exogenous (an important exception is related to nursing home care when some indi-

viduals “spend down” their assets to qualify for Medicaid which provides coverage of

long-term care). We follow Fang, Keane, and Silverman (2006) who also exclude the

Medicaid patients from their analysis of selection into Medigap private insurance. This

reduces heterogeneity in the sample since Medicaid patients are likely to be different

from the rest of the insured population. Thus, the choice of insurance is trinomial in

our application.

Our data set pools observations from the 1996-2003 waves of the Medical Expendi-

ture Panel Survey, a nationally representative survey of health care use, expenditure,

sources of payment and insurance coverage for the US civilian non-institutionalized

population. The MEPS sampling frame is a two-year overlapping panel, i.e., in each

calendar year after the first survey year, one sample is in its second year of responses

while another sample is in its first year of responses. To avoid panel and clustering

issues, we only use observations on the second round of the survey respondents in each

year with the dependent variable and income measuring the outcomes for the two year

period. We use only those individuals whose insurance status did not change during the

13

survey period. We set the nongroup insurance (IPI) plans as the baseline choice with

Medicare only (MEDICARE) and group private plans (ESI) as the choice categories for

insurance plans.

4.2. Covariates and Exclusion Restrictions

Table 1 gives summary statistics of all variables used in our analysis. The sample has

11,432 observations. Table 2 describes the distribution of the hospital visits (HOSPVIS)

up to cell 4. The table shows that the dependent variable has about 80% of the cases

with zero utilization; it seems unlikely that even mean-preserving transformations of

the Poisson model that allow for overdispersion will provide a satisfactory fit to the

data. Because it is not feasible to estimate threshold parameters for very sparse cells in

the right tail we combine them so that the last cell (> 4) has 1.36 percent of the wholesample (155 observations). HOSPVIS has five cells and three threshold parameter to

estimate. The mean and standard deviation (in parentheses) of the dependent variable

are 0.43 (0.87) for those on Medicare only, 0.27 (0.693) for those with group plans and

0.34 (0.79) for nongroup plans.

The covariate vector X consists of self-perceived health status variables VEGOOD,

GOOD, FAIR, POOR (excellent health is the omitted category), measures of chronic dis-

eases and physical limitation, CHRONIC and PHYSLIM, geographical variables NORE-

AST, MIDWEST, SOUTH and MSA, demographic variables BLACK, HISPANIC, FAM-

SIZE, FEMALE, MARRIED, EDUC, AGE, year dummies YEAR98-YEAR03 and eco-

nomic variable INCOME. The insurance variables are MEDICARE and ESI. The year

dummies are intended to capture the variations induced by trend-like changes affecting

the sample.

Vectors W1 and W2 include all variables included in X plus exclusion restrictions

to identify three covariance parameters. Exclusion restrictions are different for vectors

W1 and W2. Since latent variables Z1i and eZi measure gains in utility received fromMEDICARE and ESI choices, relative to the utility received from the baseline ISI

choice, we offer an explanation for why variables affect one utility level but not the

other. One of the exclusion restrictions included inW1 is DUM65, which was also used

by Card et al. (2004). The reasons why DUM65 should not directly affect utilization

14

have been given above. DUM65 indicates that the MEDICARE choice is available to

most individuals aged above 65 (One has to pay Medicare taxes for 40 quarters to be

eligible). However, DUM65 should not have a direct effect on the utility from the ESI

choice and, therefore, should not enterW2 since ESI is related to employment and there

is no evidence that the employment rates drop substantially at the age 65. In fact, in

our data the employment rate monotonically decreases with age.

Another restriction, entering vector W2 but not W1, is OFFER, an indicator of

whether the current employer offers a health insurance benefit. This variable should

affect the choice of insurance but does not have a direct impact on the utilization vari-

able. Some of those who are offered a group plan choose not to purchase it. Therefore,

being offered a group plan does not automatically make one insured with an ESI plan.

On the other hand, some of those with group insurance coverage receive it through a

spouse, and not as an employment benefit. Variable OFFER has a direct impact on

availability of employment-based private insurance but should not affect availability of

Medicare or nongroup plans.

Our final exclusion restriction is imposed through the variable SSIRATIO, which

enters both W1 and W2. It is defined as the share of the social security income re-

ceived for two years to the total individual’s income for that period. It is argued that

those individuals whose main source of income is social security are less likely to pur-

chase nongroup private insurance plans. That is, we assume that a high value of the

SSIRATIO, given income, reflects reduced affordability of such private insurance. Thus,

SSIRATIO should affect the utility level of the ISI choice and, since both Z1i and eZiare defined as measures of utility relative to the baseline choice it should be included

in both equations. Note also that SSIRATIO is an attribute of an individual, whereas

OFFER and DUM65 may be thought of as attributes of insurance plans. Hence it seems

correct that SSIRATIO enters both insurance equations, whereas OFFER and DUM65,

respectively, enter only one insurance equation.

If our data had information on insurance premiums for the considered plans then we

could use them as our exclusion restrictions. The actual premiums have direct impacts

on the choices. Given the limitation of our data we rely on proxy variables which are

correlated with the premiums. In general, the main difficulty which arises from the use

15

of proxy variables as exclusion restrictions is that there may be arguments that would

question their validity, in which case all variables excluded from the outcome equations

should enter the treatment equations through both W1 and W2. We estimate such

a specification of the model, study the properties of the resulting Markov chain, and

compare our findings with those of a similar investigation in the maximum likelihood

framework by Keane (1992).

4.3. Some Computing Issues

Our program has been fully tested with the joint distribution tests of posterior simulators

developed by Geweke (2004). We generate data with 10 observations and specify the

model such that there are three threshold parameters to estimate γ = (γ2, γ3, γ4) and

the dependent variable takes five values. The regressors are generated as

W1i = (1, w1i) and w1i ∼ N(0, 1);fWi = (1, ewi) and ewi ∼ N(0, 1);Xi = (1, xi) and xi ∼ N(0, 1),

and fixed. All the selected prior distributions are proper such as

α1 ∼ N (0, 0.25I2) , eα ∼ N (0, 0.25I2) ;

β ∼ N (0, 0.25I2) , ρ ∼ N (0, 0.25I2) ;

π1 ∼ N (0, 0.25) , eπ ∼ N (0, 0.25) ,

eσ21 ∼ N (0, 0.25) ;γ ∼ N (0, 0.25I3).

The tests are based on the 14 first moments and 105 second moments of the 14 para-

meters and 200,000 iterations. The algorithm passes the joint distribution tests.

4.4. Results

We estimate the OPES model and report posterior means and standard deviations of

parameters β, α1 and eα, except for the geographical and year dummies in Table 3 andparameters ρ, π, eσ21 and τ in Table 5. The reported results are for the set of priors

16

corresponding to κ = 1/2 and based on Markov chains run for 50,000 replications,

after discarding first 1000 draws of the burn-in-phase. We collect every 50th iteration,

discarding the rest. For the tuning parameter the value of ϕ = 2 is selected so that

the acceptance rates are equal to 0.25. The model has 87 parameters to estimate and

the autocorrelation functions for most of them die off after at most 5 lags. The slowest

convergence is observed for parameters ρ, π, eσ21 and τ for which serial correlation

is much more considerable. Table 4 reports the relative numerical efficiencies (RNE)

corresponding to them, which are between 0.028 and 0.09. However, all these parameters

pass the formal test of convergence based on Geweke (1992). The relative numerical

efficiencies for 12 more parameters are between 0.1 and 0.3 with the rest exceeding 0.3.

To facilitate discussion of the results we estimate and report in Table 3 the posterior

means and standard deviations of the average marginal effects for E (Y |X,d), Pr(d1 =1|W) and Pr(d2 = 1|W). The average is taken with respect to all sampled individuals

and the posterior distribution of the parameters. For the binary dummy variables the

marginal effects are calculated as the partial differences when the respective exogenous

variables change their values from 0 to 1.

The results for exogenous covariates are plausible. Health status indicators have

strong impacts on utilization as worsening health conditions increase the probability

of hospitalization. Age has a positive impact on probability of hospitalization. Being

female and living in a metro area have negative impacts. Income, race family size,

education have no impact. The variables that generate exclusion restrictions are strongly

correlated with the insurance choice variables. DUM65 has a strong positive impact

on the probability of being on Medicare only. OFFER has strong impact on being

privately insured. SSIRATIO has a strong negative impact on ESI and positive impact

on Medicare only.

We perform formal tests for null hypotheses that set the covariance parameters to

zero. Denote by M1 the unconstrained model specification and by M0 the constrained

model. Assume that modelsM1 andM0 employ the same priors for parameters common

to both models. Since models M1 and M0 are nested, we test the hypothesis using the

Savage-Dickey density ratio approach (Verdinelli and Wasserman, 1995) to calculate the

17

Bayes factor as

B01 =m(y|M0)

m(y|M1).

First we calculate the Bayes factor for H0 : π1 = eπ = 0 against the alternative that

leaves these parameters unconstrained. According to this approach the Bayes factor can

be calculated as

B01 =p(π∗1, eπ∗|y)p(π∗1, eπ∗) , (4.1)

where p(π1, eπ|y) is the joint posterior density and p(π1, eπ) is the prior for parametersπ1, eπ calculated for the unrestricted model at the point π∗1 = eπ∗ = 0. In general, lessinformative priors for these parameters would always favor the null hypothesis. This

motivates our choice of proper priors and the use of a sensitivity check. The posterior

means and standard deviations of parameters π1 and eπ are 0.092 (0.079) and 0.122(0.165) respectively. The calculated Bayes factor, 12.8 (6.6), does not provide strong

evidence to support or reject the null hypothesis. This conclusion is robust to both

specifications of the priors that correspond to κ = 1/2 and κ = 1/8, respectively.Similarly, we calculate the Bayes factor for H0 : eσ21 = 0 against the alternative that

leaves parameter eσ21 unconstrained. The estimated posterior distribution of the co-variance parameter eσ21 is centered at the posterior mean 1.011 with posterior standarddeviation 0.188. The estimated Bayes factor is 0.0 (0.0) and the null hypothesis is over-

whelmingly rejected. The positive sign of the covariance parameter indicates that the

common unobserved factors influencing the choices affect them in the same directions.

We also estimate the OPES model under the specification which imposes no choice

specific exclusion restrictions. The results are presented in Tables 4 and 5. It is interest-

ing to notice that variable DUM65 still has a strong positive impact on the probability

of being on Medicare only and has no impact on ESI which is consistent with our jus-

tification of the exclusion restrictions. Similarly, OFFER has a strong impact on being

privately insured and no impact on Medicare only. The Markov chain is constructed to

be of the same length as before but it has different convergence properties. The relative

numerical efficiencies in Table 5 indicate that the chain converges much more slowly.

Relative to the specification with choice specific exclusion restrictions the new posterior

standard deviations are larger in magnitude with that of parameters eσ21 increasing from18

0.188 to 0.425. This is consistent with findings of Keane (1992).

In addition, two competing specifications, denoted OP and EBP, respectively, are

estimated and compared with the OPES model. The posterior means, standard devia-

tions of parameters ρ, π, eσ21 and τ of the OP and EBP models are given in Table 5.The OP model ignores endogeneity of insurance status and, therefore, its estimates are

potentially subject to self-selection bias. The estimates indicate that insurance plans

have no strong impacts on utilization, 0.013 (0.058) and 0.061 (0.054). The estimated

Markov chains when endogeneity is ignored display almost no serial correlation and the

RNE values are high. The results are based on a shorter Markov chain run for 20,000

replications with every 20th iteration collected after discarding first 1000 draws. Table

2 reports the actual cell frequencies and those predicted by OPES and OP models.

Regardless of whether endogeneity is modeled both OPES and OP models give simi-

lar estimates of the cell frequencies. This result suggests that the main benefit of the

structural approach, which controls for selection on unobserved factors, is that the total

impact of insurance can be decomposed into selection and incentive components. The

incentive component can be interpreted as the pure effect of insurance.

The EBP model is applied to a binary dummy constructed as an indicator of zero and

positive hospital utilization. That is, the last (second) category includes all observations

of greater than or equal to one hospital visit. In this case there is no need to estimate

the threshold parameter. We impose the same choice specific exclusion restrictions as

those of the original OPES model specification. The reported RNE values are estimated

based on Markov chains of the same length as those for the OPES model. The Markov

chain of the EBP model displays very high serial correlations and the corresponding

RNE values are between 0.007 and 0.015. It is interesting to notice that the posterior

standard deviations for EBP model reported in Table 5 are substantially greater than

those of OPES model. This suggests a loss of precision or efficiency for the endogenous

parameters, as should be expected given limited information over cell frequencies that

the EBP model utilizes compared with that of the OPES model.

The coefficients of MEDICARE and ESI, −0.494 (0.361) and −0.333 (0.329), do notsuggest strong incentive effects. However, the signs of coefficients in the ordered probit

model do not necessarily coincide with those of the corresponding marginal or treatment

19

effects. Thus, treatment effects must be formally calculated to assess the direction and

magnitude of the incentives effects. We estimate the average treatment effect (ATE)

which is such a measure.

Denote ηi = (Zi,β,ρ,π,α, τ ) and define the expected utilization gain evaluated

at ηi for a randomly selected individual i between state j (dji = 1, j = 1, 2) and the

baseline choice (d3i = 1) as

E³Y ji − Y 3i |Xi,ηi

´=

MXm=1

m [Pr (Yi = m|dji = 1,ηi)− Pr (Yi = m|d3i = 1,ηi)] .

We calculate the ATE for the outcome variable as

E¡Y j − Y 3|X¢ = 1

N

NXi=1

Eηi|YhE³Y ji − Y 3i |Xi,ηi

´i,

where the expectation is taken with respect to the posterior distribution of the parame-

ters in the model and over N individuals.

The estimated ATEs are 0.096 (0.001) and 0.158 (0.002) for MEDICARE and ESI,

respectively, relative to nongroup private insurance status respectively. The actual

differences in utilization are 0.09 and −0.07 respectively. Thus, the ATE and the un-conditional difference in utilization for Medicare only and nongroup plans are very close

to each other, which indicates almost no selection effect between these plans. However,

the estimated ATE parameter for ESI plans indicates a strong selection effect with fa-

vorable selection into such plans. In other words, individuals with ESI plans on average

use hospital care less than their IPI counterparts, perhaps because they are healthier,

even though ESI plans may provide better coverage for care than the other alternative

plans. Favorable or advantageous selection is the converse of adverse selection. Ad-

verse selection implies that high risk individuals purchase higher insurance coverage.

Favorable selection has been discussed in the health insurance literature; see Fang et

al. (2006). One possible reason for it that has been offered is that those purchasing

supplementary insurance may be both the healthier and more risk averse individuals.

Risk aversion would increase the propensity for higher insurance coverage while being

healthy would contribute to lower utilization. It is also possible that ESI plans allow

substitution of other care for hospital care, e.g. nursing home care.

20

5. Conclusion

This paper proposes an OPES model to estimate the effect of endogenous treatment

variables on an ordinal dependent variable. The model is illustratively applied to ana-

lyze the effects of different types of insurance plans on hospital utilization allowing for

potential endogeneity of insurance status — a feature neglected by many previous studies.

In our illustration we find evidence that controlling for endogeneity is important.

21

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24

Table 1.Summary statistics

Mean St. dev.HOSPVIS Number of hospital admissions 0.303 0.734Insurance plan typesESI = 1 if plan is group private 0.766 0.423MEDICARE = 1 if Medicare only 0.159 0.366Demographic characteristicsFAMSIZE family size 2.166 1.035AGE age/10 6.389 0.615EDUC years of schooling 12.633 3.059INCOME $ income/1000 59.230 50.139FEMALE = 1 if female 0.526 0.499BLACK = 1 if black 0.107 0.310HISPANIC = 1 if hispanic 0.091 0.288MARRIED = 1 if married 0.716 0.451NOREAST = 1 if northeast 0.184 0.388MIDWEST = 1 if midwest 0.243 0.429SOUTH = 1 if south 0.376 0.484MSA = 1 if metropolitan statistical area 0.757 0.429Health characteristicsVEGOOD = 1 if very good health 0.316 0.465GOOD = 1 if good health 0.309 0.462FAIR = 1 if fair health 0.129 0.336POOR = 1 if poor health 0.044 0.206PHYSLIM = 1 if physical limitation 0.167 0.373CHRONIC = 1 if chronic conditions 0.718 0.450Instrumental VariablesDUM65 = 1 if AGE>6.5 0.447 0.497OFFER if the current employers offer insurance 0.325 0.468SSIRATIO = SSI/INCOME 0.239 0.310Year dummiesYEAR98 = 1 if year 1998 0.108 0.310YEAR99 = 1 if year 1999 0.104 0.306YEAR00 = 1 if year 2000 0.134 0.340YEAR01 = 1 if year 2001 0.103 0.304YEAR02 = 1 if year 2002 0.215 0.411YEAR03 = 1 if year 2003 0.146 0.353

25

Table 2.Utilization frequency

Actual PredictedCells Using OPES Using OP0 0.8062 0.8052 0.80541 0.1263 0.1268 0.12672 0.0395 0.0402 0.04013 0.0144 0.0145 0.0145>4 0.0136 0.0134 0.0134

Note: Ordered Probit (OP), Ordered Probit Model with Endogenous Selection (OPES)

26

Table 3.Posterior means and standard deviations of parameters β, α1 and eα and their marginaleffects. Ordered Probit Model with Endogenous Selection with choice specific exclusionrestrictions.

Coefficients Marginal EffectsHOSPVIS MEDICARE ESI HOSPVIS MEDICARE ESI

CONST -2.813 -1.003 1.880 − − −0.366 0.487 0.309

FAMSIZE -0.005 0.171 0.113 -0.002 0.026 0.0270.016 0.034 0.030 0.007 0.005 0.006

AGE 0.231 0.115 -0.263 0.100 0.017 -0.0630.037 0.075 0.046 0.016 0.011 0.009

EDUCYR 0.0005 -0.039 0.022 0.0002 -0.006 0.0050.0055 0.010 0.008 0.0024 0.0016 0.002

INCOME -0.0001 -0.0045 -0.0009 -0.0001 -0.0007 -0.00020.0004 0.0009 0.0005 0.0002 0.0001 0.0001

FEMALE -0.080 -0.219 -0.069 -0.035 -0.035 -0.0180.031 0.054 0.040 0.013 0.009 0.010

BLACK -0.046 0.696 0.540 -0.019 0.124 0.1100.049 0.110 0.095 0.021 0.023 0.017

HISPANIC -0.070 0.424 0.064 -0.029 0.072 0.0150.056 0.106 0.084 0.023 0.018 0.021

MARRIED -0.034 -0.351 0.179 -0.015 -0.057 0.0440.036 0.067 0.056 0.016 0.010 0.013

VEGOOD 0.169 0.123 0.104 0.077 0.019 0.0240.049 0.078 0.054 0.023 0.014 0.015

GOOD 0.414 0.185 0.064 0.194 0.029 0.0130.049 0.078 0.058 0.024 0.013 0.015

FAIR 0.785 0.447 0.208 0.452 0.075 0.0450.063 0.100 0.076 0.040 0.018 0.018

POOR 1.236 0.251 0.034 0.909 0.041 0.0060.081 0.136 0.106 0.074 0.025 0.025

MSA -0.089 0.150 0.203 -0.040 0.023 0.0500.034 0.060 0.045 0.016 0.008 0.011

PHYSLIM 0.329 0.177 0.057 0.159 0.029 0.0150.038 0.074 0.059 0.020 0.012 0.013

CHRONIC 0.518 0.046 0.141 0.190 0.007 0.0340.044 0.066 0.046 0.012 0.010 0.011

DUM65 − 0.646 − − 0.099 −0.076 0.011

OFFER − − 1.121 − − 0.2150.063 0.007

SSIRATIO − 0.309 -0.412 − 0.048 -0.0980.112 0.091 0.017 0.018

27

Table 4.Posterior means and standard deviations of parameters β, α1 and eα and their mar-ginal effects. Ordered Probit Model with Endogenous Selection without choice specificexclusion restrictions.

Coefficients Marginal EffectsHOSPVIS MEDICARE ESI HOSPVIS MEDICARE ESI

CONST -2.816 -0.940 2.035 − − −0.380 0.636 0.413

FAMSIZE -0.006 0.175 0.111 -0.003 0.026 0.0250.016 0.037 0.030 0.007 0.006 0.008

AGE 0.234 0.106 -0.285 0.101 0.015 -0.0670.037 0.095 0.069 0.016 0.012 0.016

EDUCYR 0.001 -0.045 0.021 0.0003 -0.007 0.0050.006 0.014 0.010 0.0024 0.002 0.002

INCOME -0.0001 -0.005 -0.0008 -0.0001 -0.0007 -0.00020.0004 0.001 0.0005 0.0002 0.0001 0.00013

FEMALE -0.083 -0.228 -0.069 -0.036 -0.034 -0.0160.031 0.066 0.045 0.013 0.008 0.011

BLACK -0.048 0.707 0.536 -0.020 0.121 0.1070.049 0.120 0.092 0.020 0.023 0.018

HISPANIC -0.071 0.448 0.061 -0.029 0.075 0.0150.056 0.124 0.091 0.023 0.019 0.020

MARRIED -0.033 -0.391 0.175 -0.015 -0.061 0.0420.036 0.089 0.066 0.016 0.012 0.015

VEGOOD 0.173 0.121 0.101 0.078 0.018 0.0220.050 0.082 0.056 0.023 0.012 0.013

GOOD 0.417 0.193 0.064 0.194 0.029 0.0140.052 0.082 0.056 0.024 0.012 0.012

FAIR 0.792 0.448 0.197 0.453 0.072 0.0430.070 0.111 0.075 0.042 0.017 0.014

POOR 1.249 0.257 0.029 0.913 0.040 0.0050.087 0.145 0.107 0.073 0.024 0.026

MSA -0.089 0.149 0.201 -0.039 0.021 0.0480.033 0.063 0.044 0.015 0.009 0.011

PHYSLIM 0.330 0.184 0.059 0.158 0.027 0.0130.040 0.081 0.059 0.020 0.011 0.013

CHRONIC 0.524 0.037 0.138 0.191 0.005 0.0330.047 0.063 0.043 0.012 0.009 0.011

DUM65 − 0.705 0.018 − 0.102 0.0050.139 0.081 0.015 0.019

OFFER − -0.266 1.045 − -0.037 0.1980.174 0.081 0.020 0.014

SSIRATIO − 0.326 -0.423 − 0.047 -0.0990.154 0.118 0.019 0.024

28

Table 5.Posterior means, standard deviations and RNE values of parameters ρ, π, eσ21 and τ

OPES OPES EBP OP(MNP restrictions) (no restrictions)Coefficients RNE Coefficients RNE Coefficients RNE Coefficients RNE

Insurance dummiesMEDICARE -0.494 0.040 -0.529 0.026 -1.536 0.007 0.013 0.334

0.361 0.381 0.953 0.058ESI -0.333 0.041 -0.362 0.022 -1.157 0.007 0.061 0.334

0.329 0.365 0.811 0.054

Covariance parameters between insurance and utilization equationsπ1 0.092 0.090 0.119 0.022 0.367 0.007 − −

0.079 0.112 0.279eπ 0.122 0.065 0.119 0.053 0.328 0.011 − −0.165 0.204 0.255

Threshold parametersτ2 0.748 0.085 0.753 0.028 − − 0.724 0.334

0.031 0.036 0.018τ3 1.251 0.072 1.259 0.027 − − 1.211 0.334

0.050 0.059 0.026τ4 1.612 0.077 1.624 0.028 − − 1.563 0.334

0.064 0.074 0.036

Covariance between insurance equationseσ21 1.011 0.028 1.055 0.010 0.876 0.015 − −0.188 0.425 0.201

Note: Ordered Probit (OP), Endogenous Binary Probit (EBP), Ordered Probit Modelwith Endogenous Selection (OPES), (MNP restrictions) indicates the OPES model withchoice specific exclusion restrictions, (no restrictions) indicates the OPES model withoutchoice specific exclusion restrictions

29