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Bayesian Analysis of the Ordered Probit Model with

Endogenous Selection∗

Murat K. Munkin

Department of Economics

531 Stokely Management Center

University of Tennessee

Knoxville, TN 37919, U.S.A.

Email: mmunkin@utk.edu

Pravin K. Trivedi

Department of Economics

Wylie Hall 105

Indiana University

Bloomington, IN 47405, U.S.A

Email: 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 that

depend on endogenous covariates that are discrete choice indicators modeled by a

multinomial probit model. The model is illustrated by analyzing the effects of dif-

ferent types of medical insurance plans on the level of hospital utilization, allowing

for potential endogeneity of insurance status. The estimation is performed using

the Markov Chain Monte Carlo (MCMC) methods to approximate the posterior

distribution 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 the

comments of two anonymous referees, an Associate Editor, and Co-Editor John Geweke. However, we

remain responsible for the current version.

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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

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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 Zibe defined as

Zi= Wiα + εi,

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where Wiis a (J −1)×q matrix of exogenous regressors, α is a q×1 parameter vector,

such that

JY

where ZJi= 0 and I[0,+∞)is the indicator function for the set [0,+∞). The distribution

of the error term εiis (J − 1)-variate normal N (0,Σ). For identification it is customary

to restrict the leading diagonal element of Σ to unity.

dji=

l=1

I[0,+∞)(Zji− Zli),j = 1,...,J,

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∗

ithat depends on the outcomes of disuch 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) × 1

parameter vectors. Define Yias

Yi=

M

X

m=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 uiand εi. Assume

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

restricted for identification since Y∗

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

iis latent. Assume that V ar(ui) = 1 + δ0Σ−1δ.

multinomial choice is among three alternatives so that J = 3. Let Zi=

thateZi= Z2iand use tilde to denote all parameters and variables related toeZi. Denote

for identification such that V ar(ε1i) = 1 + σ2

³

Z1i,eZi

´

such

V ar(eZi) = e σ22where in fact e σ22= σ22, cov(eZi,Z1i) = σ21and restrict variance of Z1i

21e σ−1

22. Then ε1i|e εi∼ N(σ21e σ−1

22e εi,1).

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Denote π0= δ0Σ−1, π0= (π1,e π) (where π1 is 1 × 1 and e π is 1 × 1) and e σ21 =

(π1,e π0,e σ21,e σ22). Then the model can be presented as

Y∗

i

=

Xiβ + diρ + (Z1i− W1iα1)π1+ (eZi−f

eZi

such that

e εi

density of the observable data and latent variables is

σ21e σ−1

22. There is a one-to-one correspondence between parameter sets (δ,Σ) and

Wie α)e π + ζi,

Z1i

=

W1iα1+ (eZi−f

Wie α)e σ21+ ηi,

=

f

Wie α +e εi,

ζi

ηi

i.i.d.

∼ N

03,

1

0

0

0

1

0

0

0

e σ22

.

Let ∆i= (Xi,Wi, τ, β,ρ,π1,e π,α1, e α,e σ21,e σ22). For each observation i the joint

³

×exp

j=1

l=1

·

"M

m=1

PrY∗

i,Yi,Z1i,eZi,di|∆i

´

= (2π)−3/2e σ−1/2

22

exp

h

−0.5e σ−1

22(eZi−f

Wie α)2i

h

−0.5[Z1i− W1iα1−

3Y

³

I{Yi=m}I[τm−1,τm)(Y∗

³eZi−f

Wie α

´

e σ21]2i

×

3

X

dji

I[0,+∞)(Zji− Zli)

×exp−0.5Y∗

i− Xiβ − diρ − (Z1i− W1iα1)π1− (eZi−f

i).

Wie α)e π

´2¸

×

X

#

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

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