Medicare Part D and the Use of Hospital Services

 
Medicare Part D and the Use of Hospital Services

                                          William J. Parish

                                    Department of Economics
                          The University of North Carolina at Greensboro

                                          August 21, 2014

                                              Abstract

    It has been hypothesized that government spending on Medicare Part D may offset government
spending on Parts A and B. A research literature exists exploring this hypothesis, but the set of
results produced vary substantially across studies. Furthermore, little is known about the extent
to which changes in beneficiary health may explain this hypothesis. In this paper I use panel data
covering the years 1996 through 2010 to estimate the effect of enrolling in Medicare Part D on
the use of hospital services among two types of Medicare beneficiaries: those with limited or no
prescription drug coverage prior to the implementation of Part D, and those with generous coverage
prior to the implementation of Part D. I also investigate the extent to which the enrollment effect
can be attributed to a change in the underlying health of Medicare beneficiaries. The econometric
model is a fixed effects Poisson specification. Among beneficiaries with limited or no prior drug
coverage I estimate that Part D reduced the number of overnight hospital stays by about 12%, and
among beneficiaries with generous prior drug coverage I estimate that Part D reduced the number
of hospital nights by about 21%. The estimated effect of Part D on overnight hospital stays for
those with generous prior drug coverage is positive, but small and statistically insignificant. The
estimated effect of Part D on hospital nights for those with limited prior drug coverage is negative,
but statistically insignificant. Results from testing for the presence of “health effects” suggest that
there may be substantial effects. These results are statistically imprecise, however.
Medicare Part D and the Use of Hospital Services

                                                William J. Parish
                                                August 21, 2014

1   Introduction

    Prescription medicines represent a clinically important component in the treatment of a variety

of illnesses. This is especially true among the chronically ill and elderly. Furthermore, drug spend-

ing accounts for a large proportion of total medical expenditures among both of these groups. In

the years preceding the implementation of Medicare Part D a substantial number of Medicare ben-

eficiaries were without prescription drug coverage. Safran et al. (2005) estimate that approximately

27% of seniors were without medication insurance in 2003, and that many of these uninsured se-

niors were poor or near-poor. Concerns regarding the health and financial well being of the un-, or

under-, insured provided impetus for the passage of the Medicare Prescription Drug, Improvement,

and Modernization Act of 2003, the legislative foundation for Medicare’s Part D program.

    In the health services research literature a hypothesis important to the evaluation of the Medi-

care Part D program has emerged known as the “cost-offset” hypothesis.1 This hypothesis proposes

that Part D may reduce non-drug medical expenditures, and thereby mitigate the costs associated

with the drug insurance program. Two mechanisms could explain this theory. First, drug insurance

could positively affect medication adherence and consequently decrease the likelihood of adverse

health events that would have otherwise led to an increased use of hospital or physician services.

Alternatively, medication therapies could be at least partially substitutable for hospital or physician

services. Accordingly, since drug insurance lowers the price of prescription medicines relative to the

price of other treatment modalities, standard economic theory predicts that agents will increase

their use of prescription drugs and decrease their use of non-drug health care services. Under this
    1
      See Briesacher et al., 2005; Sokol et al., 2005; Hsu et al., 2006; Shang and Goldman, 2007; Deb et al., 2009;
Stuart et al., 2009; Zhang et al., 2009; Chandra et al., 2010; Afendulis et al., 2011; Liu et al., 2011; McWilliams et
al., 2011; and Kaestner and Khan, 2012.
Parish

latter mechanism beneficiary health need not change.

       The cost-offset hypothesis is particularly relevant in assessing the costs and benefits associated

with Medicare Part D. In particular, it has distinct implications on both the cost and benefit side of

the assessment. It also suggests policy changes that could increase the efficiency of the Part D pro-

gram. On the cost side, if Part D insurance negatively impacts the use of other health care services,

then the calculated cost of the program should be adjusted to reflect the savings associated with a

diminished use of these health care services. On the benefit side, an inverse relationship between

Part D and the use of non-drug medical resources may indicate that Part D has an important effect

on the health of Medicare beneficiaries. Finally, Goldman and Phillipson (2007) demonstrate that

the degree of substitutability across multiple insured medical technologies is an important factor

in determining the optimal cost-sharing arrangement. Thus knowledge regarding the extent of the

substitutability of drugs and hospital resources may suggest program design changes that could

increase the cost-effectiveness of Medicare Part D.

       Previous studies have attempted to link prescription drug use or coverage with a change in

patient health,2 to link prescription drug insurance to changes in drug use or expenditure,3 and

to link prescription drug use or coverage with the use of other health care services.4 Among these

latter studies the evidence is inconclusive and more research is necessary to precisely determine

the magnitude of the relationship between Part D insurance and the use of non-drug medical care

(see Lau and Stubbings, 2012). Furthermore, no study has attempted to holistically investigate the

causal network between drug coverage, health, and the use of hospital services. I develop a testing

protocol based on nested econometric specifications that allows me to simultaneously estimate the

effect of Medicare Part D and to test the null hypothesis that health is not a mediator in the causal

   2
      See Lichtenberg, 1996 and 2001; and Heisler et al., 2004.
   3
      See Lichtenberg and Sun, 2007; Shea et al., 2007; Simoni-Wastila et al., 2008; Yin et al., 2008; Schneeweiss et
al., 2009; Zhang et al., 2009; Basu et al., 2010; Duggan and Morton, 2010; Polinski et al., 2010; Safran et al., 2010;
and Briesacher et al., 2011 for studies that have looked at changes in expenditures or uses of prescription drugs. See
also Piette et al., 2004; Goldman et al., 2004; Soumerai et al., 2006; Madden et al., 2008 and 2009; Polinski et al.,
2011; and Williams et al., 2013 for studies that have looked for changes in medication adherence.
    4
      See the references in footnote 1.

                                                          3
Parish

relationship between Part D and the use of hospital services. This is the primary contribution. A

secondary contribution is a focus on the previously uninsured. Despite the obvious policy relevance

of this group, surprisingly little research has directly analyzed the effect of Medicare Part D among

these beneficiaries. Briesacher et al. (2005), Zhang et al. (2009), and McWilliams et al. (2011)

are the only studies reviewed that provide point estimates associated with the previously un-, or

under-, insured. Moreover, among these studies, Briesacher et al.’s study is the only one that

focuses solely on drug coverage “gainers” versus “nevers,” and their study may be less relevant in

the post-Part D era. I estimate separate effect parameters for respondents that reported limited

or no drug coverage in all survey waves before 2006 and for respondents that reported generous

coverage in at least one survey wave before 2006.

2       Background

        Analyses of disease-specific study populations indicate a strong, inverse association between

prescription drug coverage (or use) and the use of hospital services. Sokol et al. (2005) estimated

the effect that medication adherence has on the probability of being hospitalized among samples of

patients with one of four health conditions.5 They found that medication adherence, derived from

data on prescription drug fills, strongly affects the probability of being hospitalized. For instance,

they estimated that diabetes patients with more than 80% adherence are 17 percentage points less

likely to be hospitalized than diabetes patients with less than 20% adherence. Cole et al. (2006)

estimated the effect of increasing prescription drug co-payments on the risk of hospitalization for

patients with chronic heart failure. They found that, depending on the medication regimen, the

probability of hospitalization increases between 6 and 9% for every $10 increase in drug co-payment.

Afendulis et al. (2011) analyzed state-level data on the number of hospitalizations associated with

eight illnesses.6 They compared the changes in the hospitalization rates associated with these
    5
    Diabetes, hypertension, hypercholesterolemia, and congestive heart failure.
    6
    Short-term complications of diabetes, uncontrolled diabetes, chronic obstructive pulmonary disorder, congestive
heart failure, angina, asthma, stroke, and acute myocardial infarction.

                                                        4
Parish

illnesses across states with “high” and “low” Part D take-up rates. They found that Part D is

associated with a reduction in the hospitalization rates associated with these illnesses by 4.1%, or

by 42,000 admissions. These studies do not straightforwardly generalize to the Medicare beneficiary

population because of the selected nature of the study populations used and because these analyses

are primarily descriptive.

   Studies that use proprietary insurance data are uniquely capable of identifying the role that

drug coverage versus drug use plays in the relationship between prescription drug insurance and the

use of physician or hospital services. Furthermore, these studies have clear sources of identifying

variation lending them increased causal interpretability. Hsu et al. (2006) analyzed the effect of

capping drug benefits at $1,000 per year. They report that capping benefits increased the number

of emergency department visits by 9%, and increased the number of non-elective hospitalizations

by 13%. Zhang et al. (2009) compared spending outcomes before and after the implementation of

Medicare Part D among a group of beneficiaries enrolled in a “large Pennsylvania insurer.” They

found that those individuals with limited drug coverage before Part D experienced offsetting de-

creases in non-drug medical spending. Chandra et al. (2010) used data on Medicare beneficiaries

receiving supplementary insurance through California Public Employees Retirement System. They

compared physician and hospital utilizations across beneficiaries that experienced drug co-payment

increases and those that did not. They concluded that cost-offsets are possible.

   Though the aforementioned studies suggest strong cost-offset effects associated with Part D,

concerns regarding the external validity of these results remain. Thus it is important to also con-

sider evidence produced by nationally representative studies. Shang and Goldman (2007) and Deb

et al. (2009) employed robust methodology and found evidence of a statistically significant rela-

tionship between prescription drug insurance and non-drug medical expenditure. However, these

studies produced radically different point estimates. Shang and Goldman estimated that for every

dollar spent on drug coverage Medicare saves approximately $2.06 on Part A and B expenditures.

Accordingly, their results imply that the net cost of Medicare Part D is negative. Deb et al., on the

                                                 5
Parish

other hand, found that the cost-offsets resulting from drug insurance are less than dollar-for-dollar,

implying a positive net cost of Part D. Kaestner and Khan (2012) conducted a descriptive analysis

that did not indicate any significant relationship between Part D and non-drug medical spending.

Liu et al. (2011) employed a differences-in-differences model with a small sample (N = 1, 105),

and estimated a small and statistically insignificant positive relationship between drug coverage

and emergency department and hospital use. Thus it is not clear from these latter studies whether

there is an effect associated with prescription drug insurance, and if so, how large that effect is.

   Two additional studies are very similar to mine: a study conducted by Briesacher et al. (2005)

and a study conducted by McWilliams et al. (2011). Briesacher et al. (2005) provide prospective

evidence of the effect of gaining Part D coverage on hospital and physician spending by comparing

drug coverage “gainers” with drug coverage “nevers” in the pre-Part D era. They used a longitudi-

nal data set covering years 1995 through 2000 constructed from the Medicare Current Beneficiary

Survey (MCBS). Their sample included respondents that either had no drug coverage between

1995 and 2000, or gained some drug coverage during this period and remained covered thereafter.

They estimated a fixed-effects differences-in-differences model, and find small and statistically in-

significant effects. McWilliams et al. (2011) used a panel data set covering years 2004 through

2007 constructed from the Health and Retirement Study (HRS) with linked Medicare claims data.

Their sample included respondents that had either generous or limited to no drug coverage before

2006. They investigated whether individuals with limited or no drug coverage before 2006 had a

greater reduction in health care expenditures in post-2006 quarters relative to those individuals

with generous drug coverage before 2006. Thus they did not actually analyze the effect of being

enrolled in a Part D plan. Rather, they estimated the effect of Part D being available. They found

that those with limited or no drug coverage before 2006 had a relative reduction in total non-drug

medical spending of $306 per quarter. The majority of this reduction was driven by changes in

inpatient and skilled nursing facility expenditures. I extend both of these analyses by analyzing an

extended panel data set, and by employing a more robust econometric methodology.

                                                  6
Parish

3       Methods

3.1      Data

        The Health and Retirement Study (HRS) is sponsored by the National Institute on Aging

(grant number NIA U01AG009740) and is conducted by the University of Michigan. The HRS

survey was first conducted in 1992, and has been conducted every two years thereafter. The HRS

is representative of the over-50 population of American residents. The Asset and Health Dynamics

Among the Oldest Old Study (AHEAD) is a companion study conducted by the University of

Michigan and sponsored by the same NIA grant. The AHEAD survey was administered separately

from the HRS survey until 1998 when the samples from the HRS and AHEAD were merged and

tracked thereafter as a part of the HRS. The RAND corporation has assembled a longitudinal file

containing data on all 30,671 observed individuals that have ever been interviewed between 1992

and 2010. The RAND HRS Data file is an easy to use longitudinal data set derived from raw HRS

data. It was developed at RAND with funding from the National Institute on Aging and the Social

Security Administration. The majority of the analysis variables I use were constructed using raw

HRS data or AHEAD data from the 1995 survey,7 but a handful of the control variables come from

the RAND longitudinal file.8

        The data used in this paper consist of a panel covering the years 1996 through 2010. 28,261

respondents were interviewed over this time period. The primary sample inclusion criterion is

enrollment in Medicare Parts A and B. Enrollment in both parts is a prerequisite for enrollment

in Medicare Part D. After omitting respondents that were ever not enrolled in Part A or Part B,

the sample included 21,922 individuals. The second inclusion criterion requires that a respondent

never skip an interview. Under this inclusion criterion individuals that died or were permanently
    7
    Approximately half of the respondents included in my 1996 cross-section come from the 1995 AHEAD survey.
    8
    Specifically, the following measures were derived from the RAND file: age, marital status, household income, and
geographic region.

                                                         7
Parish

dropped from the HRS study are however retained. Approximately 1,000 respondents included in

my final analysis sample died. The dependent variables derive from questions eliciting the number of

utilizations that occurred over the past two years or since the previous interview, whichever is longer.

Thus the second inclusion criterion ensures that the dependent variables measure utilizations over

the same length of time. After omitting respondents that did not meet the second criterion, the

sample included 17,974 individuals. The third inclusion criterion requires that respondents report

their Part D enrollment status in 2006 and 2008. 15,534 respondents met this, and the preceding,

inclusion criteria. At each survey wave the HRS instrument asks respondents to report whether

they have ever been diagnosed with several physical health conditions. Some records indicate that

a respondent has been diagnosed with one of these physical health conditions in an early wave,

but at a later wave has never been diagnosed with that condition. Since a secondary focus is on

the role that health plays in the relationship between Part D and the use of hospital services, I

omit respondents with these inconsistent records. After making this omission, the sample included

13,726 individuals. Finally, in addition to omitting observations with missing records for any of

the variables used in the regression models, the particular econometric model I use requires at least

two observations per respondent and at least one non-zero utilization outcome. After omitting

respondents that did not meet these criteria, the sample included 5,043 individuals. Of the 8,863

individuals lost in this final omission, 5,033 were lost because they only had one observation or

because they had all zero utilization outcomes.

3.2    Measures

      Two variables were derived to measure hospital utilizations. The first variable measures the

prospective number of overnight hospital stays occurring over a two-year period. The second vari-

able measures the prospective number of hospital nights occurring over this same length of time.

The econometric model is a panel data estimator. Therefore the primary regression controls consist

of a small set of time-varying variables. All time-invariant factors are omitted from the econometric

                                                   8
Parish

model along with the unobserved individual fixed effect. The primary regression controls include

age, household income, marital status, prescription drug coverage status at each survey period,

and a set of indicators for the type of health insurance contracts held. The econometric model

also includes region and year fixed effects. Table 1 provides descriptive statistics for the outcome

measures, and the primary regression controls. The table reports means and standard deviations

associated with the pooled data, and reports the within standard deviation. The within standard

deviation provides a measure of the variation that is used in the identification strategy employed

in the econometric model. As is typical, the within variation is substantially less than the between

variation. The loss of efficiency that results is counterbalanced by the ability to account for unob-

served individual effects.

   Several proxy measures were derived to characterize respondent’s health. In particular, mea-

sures associated with physical health conditions, measures indicating the respondent’s degree of

functional limitation, and measures indicating the respondent’s mental health status. The physi-

cal health conditions include hypertension, diabetes, cancer (excluding skin cancer), lung disease

(excluding asthma), stroke, arthritis, and heart disease. For each of these health conditions a set

of indicators were derived to differentiate between conditions that are in the patient’s history but

do not represent contemporaneous concerns and conditions that are in the patient’s history and

do represent contemporaneous concerns. For example, two indicators for hypertension were de-

rived. One indicates that the respondent has been diagnosed with hypertension and it is under

control. The other indicates that the respondent has been diagnosed with hypertension and it is

not under control. The base category for all the physical health conditions includes respondents

that have never been diagnosed with that particular illness. To measure functional limitations I

use an activities of daily living (ADL) score, and an instrumental activities of daily living (IADL)

score. The ADL score indicates whether the respondent is capable of basic daily tasks like bathing

and feeding oneself. The IADL score indicates whether the respondent is capable of performing

“higher order” daily tasks such as grocery shopping. To measure mental health status I use three

                                                 9
Parish

measures. The first two report whether the respondent has ever been diagnosed with a psychiatric

or emotional problem and they are currently receiving treatment, or the respondent has ever been

diagnosed with a psychiatric or emotional problem and they are not currently receiving treatment.

The third measure is the respondent’s Center for Epidemiological Studies of Depression (CESD)

score, which indicates the presence and intensity of depressive disorders. The CESD score is not,

however, intended for determining whether the respondent meets clinical definitions of depressive

disorders. Table 2 provides descriptive statistics for these measures.

3.3    Econometric Model

      To quantify the causal impact of Medicare Part D on the use of hospital services I estimate

incidence rate ratios (IRRs) associated with Part D enrollment. Though IRRs are standard post-

estimation effect parameters I derive them in terms of counterfactuals to describe the conditions

under which they can be interpreted causally. In particular, I demonstrate that they can be cast as

useful extensions of conventional average treatment effect (ATE) parameters, which conveniently

omit multiplicatively separable fixed effects in much the same way that standard ATEs omit addi-

tively separable fixed effects.9 The effect of Medicare Part D on the use of hospital services is likely

different among those beneficiaries that previously held prescription drug insurance than among

those beneficiaries that were without drug coverage prior to 2006. Accordingly, the two IRRs I

estimate quantify the impacts of Part D among beneficiaries with limited or no prior drug coverage

and among beneficiaries with generous prior drug coverage.

      Let y1a denote the hospital use outcome that would have occurred with Medicare Part D for

those respondents with limited or no prior drug coverage, and let y0a denote the hospital use out-

come that would have occurred without Medicare Part D for this same beneficiary group. Let

y1b and y0b be analogously defined for those beneficiaries with generous prior drug coverage. The

IRR associated with Part D for those beneficiaries with limited or no prior drug coverage can be
   9
     See Rubin (1974, 1977) for the prototypical definition. The discussion included here is heavily based on concepts
developed in Terza (2012a).

                                                         10
Parish

specified in terms of these counterfactuals as follows:

                 y1a
                    
        IRRa = E a .                                                                                               (1)
                 y0

Equation (1) represents the number of overnight hospital stays (or nights) that would have occurred

with Part D as a fraction of the number of overnight hospital stays (or nights) that would have

occurred without Part D. After subtracting 1 and multiplying by 100, (1) represents the amount

by which Part D reduces (or increases) hospital stays or nights in percentage terms. These ad-

justed IRRs are what I present throughout. The following represents this same quantity for those

respondents that had generous prior drug coverage:

                b  
                 y1
        IRRb = E b − 1 · 100.                                                                                      (2)
                 y0

      As with conventional ATE estimation the counterfactuals in (1) and (2) can be replaced with

regression predictions under an assumption of “comprehensive confounding” (see Terza, 2012a).

Intuitively, if one can assume that the conditioning vector used in the regression model is complete

in the sense that there are no omitted factors that are simultaneously related to treatment and the

outcome, then one can assume that treatment is conditionally exogenous. Let d1it denote a binary

variable that indicates if respondent i was enrolled in a Medicare Part D plan at period t, let d2it

denote a binary variable that indicates if respondent i had limited or no drug coverage in all periods

prior to period t, and let d3it denote the interaction d1it × d2it . The vector dit ≡ [d1it              d2it     d3it ]

encompasses the four distinct “treatment” states implicit in (1) and (2).10 With a comprehensive

vector of confounders, denoted by vit , the following equality holds:

               h                                       i
        y1a = E yit | d1it = 1, d2it = 1, d3it = 1, vit .                                                          (3)

 10
      Specifically, these “treatment” states are:
      1. Beneficiaries with limited or no prior drug coverage that enroll in a Medicare Part D plan
      2. Beneficiaries with limited or no prior drug coverage that do not enroll in a Medicare Part D plan
      3. Beneficiaries with generous drug coverage that enroll in a Medicare Part D plan
      4. Beneficiaries with generous drug coverage that do not enroll in a Medicare Part D plan

                                                          11
Parish

That is, with comprehensive confounding the counterfactual outcome, y1a , can be replaced with

the mean of yit conditional on the treatment state associated with y1a . Analogous equalities hold

for y1b , y0a , and y0b . Equation (3) is useful because the right-hand side can often be specified, and

consistently estimated, via standard regression-based methods. I assume that for all t = 1, 2, ..., T

       h                         i
      E yit | dit , ui , λt , xit = ui · exp (λt + dit αd + xit αx ) ,                              (4)

where ui denotes an unobserved individual fixed effect, λt denotes a period fixed effect, and xit

denotes a vector of time-varying regression controls. Table 1 lists the elements included in xit and

presents descriptive statistics. By making use of model (4) I implicitly assume that the individual

and period fixed effects combined with the set of regression controls provides a comprehensive set

of confounders. Accordingly, (3) and (4) suggest the use of the following estimate for y1a :

                                                        
      ŷ1a = ûi · exp λ̂t + α̂d1 + α̂d2 + α̂d3 + xit α̂x ,                                         (5)

where the “hats” denote estimates of the unknown parameters. Equations (3) and (4) suggest

analogous estimates for y1b , y0a , and y0b .

    Under the strict exogeneity assumption in (4) the parameters λ1 , ..., λT , αd , and αx can be

consistently estimated via the fixed effects Poisson (FEP) estimator developed by Hausman, Hall,

and Griliches (1984). The unobserved individual effects u1 , ..., uN cannot be consistently estimated

with a fixed, and small, number of time periods, but they are conditioned out of the likelihood

function and thus remain un-estimated. Consequently, the predictions in (5) and the analogous

predictions are infeasible. However, if one assumes that the unobserved individual effects u1 , ..., uN

are known, then a legitimate variant of (5) is

                                                       
      ŷ1a = ui · exp λ̂t + α̂d1 + α̂d2 + α̂d3 + xit α̂x .                                          (6)

                                                       12
Parish

Equation (6) and an analogous version of (6) for y0a suggests the following consistent estimator for

(1):
                                                                 
                 N X T  ui · exp λ̂t + α̂ + α̂ + α̂ + xit α̂x 
                                          d1     d2    d3
        ˆ a= 1
                X
       IRR                                                       − 1 · 100,                     (7)
             NT                ui · exp λ̂t + α̂d2 + xit α̂x    
                i=1 t=1

which greatly simplifies to

        ˆ a = {exp (α̂d1 + α̂d3 ) − 1} · 100.
       IRR                                                                                         (8)

Likewise, after simplification, the following is a consistent estimator for (2):

        ˆ b = {exp (α̂d1 ) − 1} · 100.
       IRR                                                                                         (9)

The asymptotic variances for the IRR estimators in (8) and (9) can be obtained via the general

method developed in Terza (2012b). The generic approach laid out therein is equivalent to the

delta method approximation in this case, however. I use the “nlcom” post-estimation command

available in Stata version 11 to calculate (8) and (9) and to conduct inference based on delta method

standard errors.

   The second empirical objective involves testing the null hypothesis that changes in health do

not explain the causal relationship between Part D and the use of hospital services. Let h∗it denote

a latent, continuous index of health for respondent i in period t. Under this null hypothesis, either

Part D has no effect on beneficiary health, or the effect on beneficiary health does not translate into

an indirect effect on the use of hospital services. To formulate a single test of this joint hypothesis

assume that the relationship between Part D insurance in the current period and beneficiary health

in the subsequent period is governed by the following linear projection:

       h∗i,t+1 = dit θ + υit ,                                                                    (10)

                                                  13
Parish

where E [dit υit ] = 0 and E [υit ] = 0 by definition. Equation (10) reflects a maintained assumption

that a respondent’s health in the current period is unrelated to factors in that period, but are

related to the previous period’s factors. Under this assumption contemporaneous health may be

an important confounding factor that should be included in model (4), but including prospective

health may bias the estimated impact of Part D on the use of hospital services. To see this write

the following variant of (4) in error-term form:

      yit = ui · exp λt + dit β d + xit β x + h∗i,t+1 δ + eit ,
                                                             
                                                                                                        (11)

          h                                            i
where E       exp (eit ) |ui , λt , dit , xit , h∗i,t+1 = 1. Now substitute equation (10) into (11). After

collecting terms the following obtains:

      yit = ui · exp (λt + dit {β d + θδ} + xit β x + {υit δ + eit }) .                                 (12)

                                                                                             
If αd ≡ β d + θδ, εit ≡ υit δ + eit , and it is assumed that E exp (εit ) |ui , λt , dit , xit = 1, then (12)

is equivalent to (4). Furthermore, the parameter vector, αd , consists of two components. The first

component is the “direct effect” of treatment, and the second component is the “indirect effect”

that treatment has through its effect on health. Model (11) provides a consistent estimator for β d

and δ only, which does not admit access to the total effect captured by αd .

   This discussion serves to suggest a simple test of the null hypothesis that the effect of Part

D is not explained by the effect that Part D has on beneficiary health. In particular, as stated

above this null hypothesis is a joint hypothesis that Part D either has no effect on health, or the

effect that Part D has on health does not translate into an effect on the use of hospital services.

Therefore, θδ = 0 under the null hypothesis, but θδ = 0 if and only if αd = β d . Accordingly,

one could estimate models (4) and (11), and conduct a “Hausman-type” test of the equality of the

parameter vectors αd and β d . This would require a measure of latent health, which is by definition

unobserved. Thus to operationalize this testing protocol I use a series of proxy variables for health

                                                      14
Parish

in place of h∗i,t+1 . Table 2 lists these variables and presents descriptive statistics. In Appendix A

I derive the variance-covariance matrix for a simultaneous estimator of (4) and (11), which can be

used to calculate a Wald statistic. The simultaneous estimator is equivalent to the estimators that

would be applied separately to equations (4) and (11), but the variance-covariance matrix that

results accounts for the covariance between the two estimators that exists due to the overlapping

of the data used with each estimator. In addition to testing the equality of αd and β d , I also test

the equality of the IRRs across the two specifications. In particular, IRRa is equivalent under both

models if and only if αd1 + αd3 = βd1 + βd3 , and IRRb is equivalent under both models if and only

if αd1 = βd1 .

4     Results

4.1    Descriptive Analysis

      Table 3 reports the frequency of overnight hospital stays across Part D abstainers and partici-

pants over time. Frequencies for those with limited or no prior drug coverage are presented in the

top panel, and frequencies for those with generous prior drug coverage are presented in the bottom

panel. The table only reports frequencies for the 2004-2006, 2006-2008, and 2008-2010 periods

to conserve space. The distributions at each period and among all subpopulations are typical of

hospitalization data in that a large proportion of the samples have zero hospitalizations, and a few

individuals have extreme observations. Among those respondents with limited or no prior drug

coverage the strongest evidence of a Part D enrollment effect occurs in the 2008-2010 period. In

this period Part D participants have more zero observations, and fewer observations at one through

four hospitalizations. These differences are not statistically significant, however. Among those with

generous prior drug coverage it is interesting to note that fewer Part D participants had zero hospi-

talizations in the 2006-2008 period (statistically significant at p < 0.05), and more observations at

one through four hospitalizations (the difference at two hospitalizations is statistically significant

                                                  15
Parish

at p < 0.05). This indicates a positive relationship between Part D and the use of hospital services

among those beneficiaries with generous prior drug coverage. This comparison is merely descrip-

tive, however, and holding confounding factors constant may alter the direction of the estimated

relationship.

   Table 4 reports the frequency of hospital nights across Part D abstainers and participants over

time. The table mirrors the organization of Table 3. There is no new information regarding the

proportion of zero observations, as these proportions are identical to those in Table 3. Inspecting

the frequencies over positive counts of hospital nights there is little evidence of an effect of Part D

as almost all counts occur with nearly the same frequency across within-period distributions.

   The test of the null hypothesis that changes in health do not explain the relationship between

Part D and the use of hospital services described above is a joint hypothesis test. In particular, this

hypothesis is true if Part D does not affect health and/or if the affect on health does not translate

into an impact on the use of hospital services. To provide some direct evidence on the affect that

Part D has on health in isolation of the relationship between Part D and the use of hospital services

I estimated logistic risk models for the risk of an improvement in health. The outcome of this model

was artificially generated following an approach described in Bound et al. (1999) and recently used

in a study by Coe and Zamarro (2011). Under this approach one estimates an ordered probit model

that uses subjective well-being as the dependent variable (i.e., an ordinal outcome recording self-

reported health as excellent, very good, good, etc.), and uses a set of objective health measures as

the right-hand side predictors. I used the health measures described in Table 2. Once the ordered

probit parameters are estimated, one obtains a measure of latent health as a linear prediction. I

then used this measure of latent health to create a variable that indicates if the respondent’s health

improved over time. I then estimated a logistic model with this generated improvement indicator

as the outcome, and estimated relative risks (RRs) associated with Part D following estimation.

   Table 5 reports the relative risks of having an improved health outcome in the subsequent period

for those respondents with limited or no prior drug coverage and those respondents with generous

                                                  16
Parish

prior drug coverage. Among those respondent with limited or no prior drug coverage, the relative

risk of having a health improvement is 1.16 (95% confidence interval: [1.02, 1.31]). Among those

respondents with generous prior drug coverage, the relative risk of having a health improvement

is 1.10 (95% confidence interval: [0.94, 1.25]). Thus Part D is associated with an improvement in

health, and the likelihood of an improvement is stronger among those with limited or no prior drug

coverage than among those with generous prior drug coverage.

4.2    Overnight Hospital Stays

      Table 6 presents the results of the fixed effects Poisson analysis of the relationship between

Medicare Part D, beneficiary health, and the use of hospital services. The first two columns

present specifications intended to estimate the causal impact of Part D on the number of overnight

hospital stays. The specification presented in column (1) does not include any observable health

measures, while the specification presented in column (2) does include contemporaneous health

measures. For beneficiaries with limited or no prior drug coverage, the estimated IRR indicates

that Part D insurance reduced the number of overnight hospital stays by about 12% (p < 0.10).

This estimate does not change appreciably after including health measures. For beneficiaries with

generous prior drug coverage the estimated IRRs indicate that Part D insurance increased the

number of overnight hospital stays between 1 and 2%. The estimated IRRs are not statistically

significant at any conventional level of significance, however.

4.3    Hospital Nights

      Table 6 also presents results pertaining to the number of hospital nights. Columns (4) and (5)

present specifications without and with health factors included, respectively. For those respondents

reporting limited or no prior drug coverage the estimated IRRs indicate that Part D insurance

reduced the number of hospital nights between 14 and 15%. These estimates are not statistically

significant at a 10% level of significance (p = 0.13), however. For those respondents reporting

                                                  17
Parish

generous prior drug coverage the estimated policy effects indicate that Part D insurance reduced

the number of hospital nights by about 21% (p < 0.05).

4.4    Health

      Columns (3) and (6) of Table 6 report the results obtained after adding prospective health

measures to the hospital stay and night models, respectively. In both columns inspection of the

IRRs suggests that health may play an important role in the relationship between Part D and the

use of hospital services. Among those with limited or no prior drug coverage the estimated IRRs

decrease (in absolute value) by more than 40%. Among those with generous prior drug coverage the

estimated IRRs for overnight hospital stays increases from between 1 and 2% to about 14%. The

estimated IRR for hospital nights among this group also decreased (in absolute value) by about

95%.

5     Conclusion

      The strongest estimated impact of Part D is that associated with the number of overnight hos-

pital stays among those beneficiaries that reported limited or no prior drug coverage. The average

number of hospital stays over all survey waves is 0.82 (see Table 1). Thus the estimated reduction in

the number of hospital stays for those with limited or no prior drug coverage is approximately 15%

of the average number of hospital stays. To put this estimate roughly into dollar terms, Pfuntner

et al. (2013) estimated that the average cost per hospital stay for an individual between the ages of

65 and 84 to be about $12,300 in a 2013 report. Using this estimate as the cost of a single overnight

hospital stay, the estimated effect of Part D suggests a reduction in hospital stay expenditures of

about $1,500 for every $12,300 that would have otherwise been spent. It is interesting that the

effect of Part D among those beneficiaries with generous prior drug coverage is small, but positive.

Little research has been conducted that separately analyzes these two groups. Accordingly, these

results suggest that future research should account for the possibility that Part D has a differential

                                                 18
Parish

impact across beneficiaries with limited or no prior drug coverage and beneficiaries with generous

prior drug coverage.

   The hospital night results broadly suggest that Part D has a much smaller impact on the

intensive margin. The point estimates for those with limited or no prior drug coverage only repre-

sent about 4% of the average number of hospital nights among these beneficiaries over all survey

waves. These point estimates are furthermore statistically insignificant. The estimated effect of

Part D on the number of hospital nights among those beneficiaries with generous prior drug cover-

age are particularly surprising. The estimated reduction only represents about 5% of the average

number of hospital nights for this group, however. The Kaiser Family Foundation (KFF) reports

that the average cost of a hospital night ranged from approximately $1,200 to $1,600 in 2006

(see http://kff.org/other/state-indicator/expenses-per-inpatient-day-by-ownership/). Thus the es-

timated reduction in hospital nights may be associated with as much as $336 in savings for every

$1,600 that would have otherwise been spent among those with generous prior drug coverage.

   Few studies have found convincing evidence supporting the conclusion that Part D positively

impacts the health of beneficiaries. The results in the FEP analysis strongly suggest the possibility

that Part D affects health and that this effect explains a substantial proportion of the estimated

effect of Part D on the number of hospital stays and nights. This suggests that an all-encompassing

evaluation of the Medicare Part D program should attempt to account for the important effect it

has on health.

                                                 19
Tables

         Table 1. Descriptive statistics for the hospitalization outcomes and primary
         regression controls used in the analysis of the relationship between Medicare Part
         D and hospitalizations (1996-2010).
                                                                     Mean
                                                                 (Overall SD)   Within SD
         Outcomes:
         Overnight Hospital Stays                                     0.82         1.06
                                                                     (1.37)
         Hospital Nights                                              4.20         9.93
                                                                    (12.35)
         Primary Regression Controls:
         Respondent's age                                            74.90         3.03
                                                                     (6.81)
         Respondent's total household income (in logs)               10.25         0.47
                                                                     (0.92)
         Respondent's marital status                                  0.57         0.19
                                                                     (0.49)
         Respondent has prescription drug coverage                    0.30         0.33
                                                                     (0.46)
         Respondent has a Medicare (or Medicaid) HMO                  0.21         0.23
                                                                     (0.41)
         Respondent is also enrolled in Medicaid                      0.07         0.13
                                                                     (0.26)
         Respondent is enrolled in a CHAMPUS plan                     0.05         0.14
                                                                     (0.22)
         Respondent is enrolled in a private insurance plan           0.65         0.28
                                                                     (0.48)
         Region of the Country:
         Respondent lives in the Northeast region                     0.16         0.05
                                                                     (0.36)
         Respondent lives in the Midwest region                       0.28         0.05
                                                                     (0.45)
         Respondent lives in the South region                         0.38         0.07
                                                                     (0.49)
         Observations:                                               19497
         Number of respondents:                                       5043
         Average panel length:                                       3.866
Table 2. Descriptive statistics for the set of observable health characteristics used in
the analysis of the relationship between Medicare Part D, beneficiary health, and
hospitalizations (1996-2010).
                                                              Mean
                                                           (Overall SD)      Within SD
Hypertension:
Condition is under control                                     0.61              0.21
                                                              (0.49)
Condition is not under control                                 0.02              0.11
                                                              (0.13)
Diabetes:
Condition is under control                                     0.19              0.15
                                                              (0.39)
Condition is not under control                                 0.01              0.08
                                                              (0.11)
Cancer (Excluding Skin Cancer):
In remission                                                   0.07              0.10
                                                              (0.26)
Not in remission                                               0.02              0.11
                                                              (0.15)
Lung Disease (Excluding Asthma):
Condition limits activity                                      0.05              0.12
                                                              (0.21)
Condition does not limit activity                              0.05              0.13
                                                              (0.22)
Stroke:
Problems persist                                               0.03              0.11
                                                              (0.18)
Problems do not persist                                        0.04              0.11
                                                              (0.20)
Arthritis:
Condition limits activity                                      0.29              0.28
                                                              (0.46)
Condition does not limit activity                              0.37              0.31
                                                              (0.48)
Heart Disease:
Myocardial infarction                                          0.03              0.14
                                                              (0.18)
Congestive heart failure                                       0.04              0.14
                                                              (0.20)
Angina with limited activity                                   0.04              0.13
                                                              (0.20)
Angina without limited activity                                0.05              0.16
                                                              (0.22)
Unspecified heart condition                                    0.18              0.23
                                                              (0.38)
Table continued on the next page…
Table 2 (Continued). Descriptive statistics for the set of observable health…
Functional Limitations:
Activities of daily living score                              0.30            0.47
                                                             (0.79)
Instrumental activities of daily living score                 0.08            0.24
                                                             (0.34)
Mental Health:
Contemporaneous psychiatric/emotional problems                0.04            0.11
                                                             (0.19)
History of psychiatric/emotional problems                     0.07            0.13
                                                             (0.26)
CESD score                                                    1.51            1.11
                                                             (1.88)
Observations:                                                19497
Number of respondents:                                        5043
Average panel length:                                         3.866
Table 3. Frequency table for the number of overnight hospital stays over time and
between Part D “abstainers” and “participants.”
              2004-2006                    2006-2008               2008-2010
        Abstainer Participant       Abstainer Participant   Abstainer Participant
                          Limited or No Prior Drug Coverage
0          209          512             277         447        153          370
          52.91       52.19           50.64        48.59      39.43       43.68
1          117          291             173         296        143          289
          29.62       29.66           31.63        32.17      36.86       34.12
2          45           109             60          108         56          107
          11.39       11.11           10.97        11.74      14.43       12.63
3          14           35              22          39          22          42
          3.54         3.57            4.02        4.24       5.67         4.96
4           7           18               4          19          8           14
          1.77         1.83            0.73       2.07**      2.06         1.65
5           2           10               6           6          3           13
          0.51         1.02            1.10        0.65       0.77         1.53
≥6          1            6               5           5          3           12
          0.25         0.61            0.91        0.54       0.77         1.42
Max:        8           12              12          10          10          20
N:         395          981             547         920        388          847
                             Generous Prior Drug Coverage
0          358          442             334         296        208          275
          53.19       51.04           51.86       44.92**     43.51       44.72
1          214          260             193         206        155          170
          31.80       30.02           29.97        31.26      32.43       27.64*
2          61           93              62          94          65          90
          9.06        10.74            9.63      14.26***     13.60       14.63
3          21           42              28          35          25          33
          3.12         4.85*           4.35        5.31       5.23         5.37
4           8           10              14          18          10          17
          1.19         1.15            2.17        2.73       2.09         2.76
5           8           10               3           5          6           17
          1.19         1.15            0.47        0.76       1.26         2.76*
≥6          3            9              10           5          9           13
          0.45         1.04            1.55        0.76       1.88         2.11
Max:        8           10              36          12          11          15
N:         673          866             644         659        478          615
Stars indicate statistical differences between “Abstainers” and “Participants”
*
  p < 0.10, ** p < 0.05, *** p < 0.01
Table 4. Frequency table for the number of hospital nights over time and between Part D
“abstainers” and “participants.”
               2004-2006                    2006-2008                   2008-2010
         Abstainer Participant       Abstainer Participant        Abstainer Participant
                           Limited or No Prior Drug Coverage
0           209          512            277          447            155          370
           52.91        52.19          50.64        48.59          39.95       43.68
1           34           81              54          88              47          90
           8.61         8.26            9.87        9.57           12.11       10.63
2           26           57              36          59              33          65
           6.58         5.81            6.58        6.41            8.51        7.67
3           21           56              36          57              32          55
           5.32         5.71            6.58        6.20            8.25        6.49
4           13           53              30          60              16          46
           3.29         5.40*           5.48        6.52            4.12        5.43
5           18           39              19          33              24          43
           4.56         3.98            3.47        3.59            6.19        5.08
≥6          74           183             95          176             81          178
           18.73        18.65          17.37        19.13          20.88       21.02
Max:        90           614            109          120            100          90
N:          395          981            547          920            388          847
                              Generous Prior Drug Coverage
0           358          443            336          297            209          276
           53.19        51.15          52.17       45.07**         43.72       44.88
1           53           58              68          54              42          55
           7.88         6.70           10.56        8.19            8.79        8.94
2           37           66              34          48              36          46
           5.50         7.62*           5.28        7.28            7.53        7.48
3           48           52              31          39              26          39
           7.13         6.00            4.81        5.92            5.44        6.34
4           23           38              26          34              25          18
           3.42         4.39            4.04        5.16            5.23        2.93*
5           28           36              23          26              18          24
           4.16         4.16            3.57        3.95            3.77        3.90
≥6          126          173            126          161            122          157
           18.72        19.98          19.57       24.43**         25.52       25.53
Max:        120          255            365          100             99          180
N:          673          866            644          659            478          615
Stars indicate statistical differences between “Abstainers” and “Participants”
*
  p < 0.10, ** p < 0.05, *** p < 0.01
Table 5. Relative risk of an improvement in latent health associated with Part D
enrollment.
                                           RR        95% Confidence Interval
Limited or No Prior Drug Coverage          1.16         1.02            1.31
Generous Prior Drug Coverage               1.10         0.94            1.25
The underlying probability model was a logit specification with a cluster-robust VCV estimator.
Standard errors for the RR estimates were obtained as described in Terza (2012b); see Appendix A.
Table 6. Results from the FEP analysis of the relationship between Part D, beneficiary health, and the
number of hospitalizations.
                                 Number of Overnight Stays                Number of Hospital Nights
                                 (1)           (2)          (3)           (4)       (5)         (6)
                                                          Part D Effects (%):
Limited Prior Coverage        -12.15*      -11.89*       -7.386         -14.12   -15.54       -8.181
                              (-1.89)       (-1.86)      (-0.80)        (-1.31)  (-1.50)     (-0.53)
Generous Prior Coverage        2.345         1.234        14.14        -20.92* -21.27**       -1.086
                               (0.28)        (0.15)       (1.22)        (-1.88)  (-1.96)     (-0.07)
                                                   Primary Regression Controls (%):
Age                             0.01          -3.24        5.03        73.44**   71.01**      6.47***
                               (0.00)       (-0.31)       (5.34)        (2.52)    (2.53)      (4.19)
Log(Household Income)          -4.14*        -4.43*        -1.71         -4.59    -5.55*       0.71
                              (-1.77)       (-1.87)      (-0.65)        (-1.40)  (-1.69)      (0.16)
Married                         0.35          0.81         9.94          -3.74     -0.99      -11.07
                               (0.05)        (0.12)       (1.14)        (-0.39)  (-0.10)     (-0.87)
Prescription Drug              -1.11          -1.30        2.15         -11.90   -11.89        -6.24
                              (-0.25)       (-0.30)       (0.37)        (-1.62)  (-1.59)     (-0.59)
Medicare HMO                   -2.79          -2.51        -5.63         -3.62     -2.77       -8.02
                              (-0.56)       (-0.50)      (-0.91)        (-0.43)  (-0.34)     (-0.87)
Medicaid                     -14.99**      -13.71**        -8.95        -14.66   -14.16        3.00
                              (-2.21)       (-2.04)      (-0.77)        (-1.28)  (-1.26)      (0.16)
CHAMPUS                         0.62          1.74         5.86          15.47    17.79        18.77
                               (0.08)        (0.23)       (0.60)        (1.05)    (1.26)      (1.12)
Supplementary Insurance        -0.47          -0.97        -0.45         4.63      3.25        -0.66
                              (-0.11)       (-0.23)      (-0.08)        (0.60)    (0.45)     (-0.07)
Health Characteristics?a         No            Yes          Yes           No        Yes         Yes
Prospective Health?b             No            No           Yes           No        No          Yes
Observations:                  19497        19497        11598          19497     19497       11598
Number of respondents:          5043          5043         3168          5043      5043        3168
Hypothesis 1 (p value):           -           3.59         2.09            -       0.86        2.47
                                  -          (0.31)       (0.55)           -      (0.84)      (0.48)
Hypothesis 2 (p value):           -           0.04         0.36            -       0.33        0.32
                                  -          (0.84)       (0.55)           -      (0.56)      (0.57)
Hypothesis 3 (p value):           -           0.67         2.08            -       0.02        2.37
                                  -          (0.41)       (0.15)           -      (0.90)      (0.12)
z statistics in parentheses; Standard errors are cluster robust
*
  p < 0.10, ** p < 0.05, *** p < 0.01
Hypothesis 1 tests the null hypothesis that the Part D coefficients are jointly equal to each other across models that add
health measures; hypothesis 2 tests the null hypothesis that IRRa is equivalent across models that add health measures; and
hypothesis 3 tests the null hypothesis that IRRb is equivalent across models that add health measures.
All models include region and period fixed effects.
To facilitate interpretation I present IRRs minus 1 and scaled up by a factor of 100.
a
  The set of health characteristics are jointly significant at p < 0.001 in all models.
b
  The set of prospective health characteristics are jointly significant at p < 0.001 in all models.
Parish

A     Details Regarding the Wald Test Used to Test for Health Effects

    The “Hausman-type” test mentioned in Section 3.3 requires that I estimate a joint variance-
covariance (VCV) matrix for the two estimators used as the basis of the test [i.e., models (4)
and (11)]. The basic approach I take is to conceptualize the independent estimators as solving
a “stacked” system of equations. From this stacked system of equations a standard sandwich
estimator for the joint VCV matrix is suggested. The covariance blocks of the resultant joint
VCV matrix account for correlation between the two models that exists because the same, or an
overlapping, sample
                 T is used by both models.
           T             T
    Let Θ = α          β    denote the stacked parameter vector associated with models (4) and
(11). Let x1it and x2it generically denote the vectors of regression controls used in (4) and (11),
respectively. Now, define the following two functions:
                 Ti
                                               !
                X              exp (x1it α)
      Q1i (α) =     yit ln PT                    ,
                               i
                t=1           r=1 exp (x1ir α)

and
                  Ti
                                                         !
                  X                   exp (x2it β)
      Q2i (β) =         yit ln     PTi                       ,
                  t=1                 r=1 exp (x2ir β)

where Ti denotes the maximum number of observations for individual i. Finally, let

      Qi (Θ) = Q1i (α) + Q2i (β).

According to Wooldridge (2010) the FEP estimator for Θ can be written as
                            n
                            X
      Θ̂FEP = argmax               Qi (Θ).                                                   (A.1)
                    Θ        i=1

Problem (A.1) is separable in α and β. To see this note that

      ∇Θ Qi = [∇α Q1i            ∇β Q2i ] ,                                                  (A.2)

where ∇κ Q denotes the gradient vector of Q with respect to κ, and has row dimensions equal to
one and column dimensions equal to the rows of κ. Therefore, the solution to (A.1) is given by the
solutions to the two systems of equations:
                  n
                  X
      G1 (α) =          ∇α Q1i = 0,                                                          (A.3)
                  i=1

and
                  n
                  X
      G2 (β) =          ∇β Q2i = 0.                                                          (A.4)
                  i=1

But (A.3) and (A.4) are the estimating equations associated with the FEP estimators for α and β
conducted in isolation of one another.

                                                                 27
Parish

    This discussion serves to point out that problem (A.1) is equivalent to estimating models (4)
and (11) in isolation. Furthermore, the estimator described by (A.1) has a well-known VCV matrix
estimator. Let
               n
               X
      G(Θ) =         ∇Θ Qi = 0.
               i=1

A robust estimator for the VCV of (A.1) is (see Wooldridge, 2010)
            ∂G(Θ)          −1 X n                                                      −1
      ˆ                                                 T         ∂G(Θ)
     Var Θ̂ =                          ∇Θ Qi |Θ=Θ̂ ∇Θ Qi Θ=Θ̂                                     .    (A.5)
                     ∂Θ Θ=Θ̂                                       ∂Θ                Θ=Θ̂
                                          i=1

The gradient of Q1i with respect to α is (see Cameron and Trivedi, 2005)
                Ti                                Ti
                       (                                )
               X                 exp (x1it α)    X
     ∇α Q1i =      x1it yit − PT                     yir .                                             (A.6)
                                  i
               t=1              r=1 exp (x1ir α) r=1

The gradient of Q2i with respect to β is analogous. Combining (A.6) and an analogous expression
for ∇α Q2i with (A.2) yields the gradient of Qi with respect to Θ. The Jacobian of G(Θ) is
                "                 #
                  ∂G1 (α)
     ∂G(Θ)          ∂α       0
              =            ∂G2 (β) ,
       ∂Θ            0       ∂β

where
                      n
                           (P                                                )
                              Ti   T
                                                       PTi    T        2
        ∂G(α)                 t=1 x1it x1it λ1it        t=1 x1it x1it λ1it
                 X
              =−   ȳi                             −           2
                                                                                 ,
         ∂α                        λ̄1i                      λ̄1i
                     i=1

where ȳi = t=1 yit , λ1it = exp (x1it α), and λ̄1i = Tt=1
             PTi                                      P i
                                                           exp (x1it α). The Jacobian of G2 (β) is
analogous.
   Under standard arguments the FEP estimator is asymptotically normal with variance approxi-
mated consistently by (A.5). Thus the Wald statistic
                   T h              i−1         
     W = RΘ̂FEP         RVarˆ Θ̂ RT           RΘ̂FEP                                         (A.7)

has a chi-squared distribution with k degrees of freedom, where k is equal to the number of rows
in the matrix R, or equivalently the number of linear restrictions tested. Accordingly, to test the
hypothesis that αd = β d define R1 as the 3 × (k1 + k2 ) matrix such that

      R1 Θ = αd − β d ,

and substitute R1 into (A.7). Likewise, let R2 and R3 denote matrices such that

      R2 Θ = αd1 + αd3 − βd1 − βd3

and

      R3 Θ = αd1 − βd1 ,

                                                        28
Parish

respectively. Substituting R2 and R3 yields test statistics to test the null hypotheses that IRRa
and IRRb are equivalent under models (4) and (11), respectively.

                                               29
Parish

                                           References

Afendulis, C. C., Y. He, et al. (2011). “The Impact of Medicare Part D on Hospitalization Rates.”
    Health Services Research, 46(4): 1022-1038.
Basu, A., W. Yin, et al. (2010). “Impact of Medicare Part D on MedicareMedicaid Dual-Eligible
    Beneficiaries’ Prescription Utilization and Expenditures.” Health Services Research, 45(1): 133-
    151.
Cameron, A. C., and P. K. Trivedi (2005): Microeconometrics: Methods and Applications, 1st Ed.
    New York, New York: Cambridge University Press.
Coe, N. B., and G. Zamarro (2011), “Retirement effects on health in Europe,” Journal of Health
    Economics, 30, 77-86.
Briesacher, B. A., B. Stuart, et al. (2005). “Medicare Beneficiaries and the Impact of Gaining
    Prescription Drug Coverage on Inpatient and Physician Spending.” Health Services Research,
    40(5p1): 1279-1296.
Briesacher, B. A., Y. Zhao, et al. (2011). “Medicare Part D and Changes in Prescription Drug Use
    and Cost Burden.” Medical Care, 49(9): 834-841.
Chandra, A., J. Gruber, et al. (2010). “Patient Cost-Sharing and Hospitalization Offsets in the
    Elderly.” The American Economic Review, 100(1): 193-213.
Bound J., M. Schoenbaum, T. R. Stinebrickner, and T. Waidmann (1999), The dynamic effects of
    health on the labor force transitions of older workers, Labour Economics, 6, 179-202.
Cole, J. A., H. Norman, et al. (2006). “Drug Copayment and Adherence in Chronic Heart Failure:
    Effect on Cost and Outcomes.” Pharmacotherapy: The Journal of Human Pharmacology and
    Drug Therapy, 26(8): 1157-1164.
Deb, P., P. K. Trivedi, et al. (2009). Dynamic Cost-Offsets of Prescription Drug Expenditures:
    Panel Data Analysis Using a Copula-Based Hurdle Model. NBER Working Paper Series,
    NBER.
Duggan, M. and F. S. Morton (2010). “The Effect of Medicare Part D on Pharmaceutical Prices
    and Utilization.” The American Economic Review, 100(1): 590-607.
Goldman, D. and T. J. Philipson (2007). “Integrated Insurance Design in the Presence of Multiple
    Medical Technologies.” The American Economic Review, 97(2): 427-432.
Goldman, D. P., G. F. Joyce, et al. (2004). “Pharmacy benefits and the use of drugs by the
    chronically ill.” Journal of the American Medical Association, 291(19): 2344-2350.
Hausman, J. A., B. H. Hall, and Z. Griliches (1984), “Econometric Models For Count Data with
    an Application to the Patents R and D Relationship,” Econometrica, 52, 909-938.
Health and Retirement Study (HRS), AHEAD 1995 public use dataset. Produced and distributed
    by the University of Michigan with funding from the National Institute on Aging (grant number
    NIA U01AG009740). Ann Arbor, MI, (1995).
Health and Retirement Study (HRS), HRS 1996-2010 public use dataset. Produced and distributed
    by the University of Michigan with funding from the National Institute on Aging (grant number
    NIA U01AG009740). Ann Arbor, MI, (1996-2010).
Heisler, M., K. M. Langa, et al. (2004). “The Health Effects of Restricting Prescription Medication
    Use Because of Cost.” Medical Care, 42(7): 626-634.
Hsu, J., M. Price, et al. (2006). “Unintended Consequences of Caps on Medicare Drug Benefits.”
    The New England Journal of Medicine, 354(22): 2349-2359.

                                                30
You can also read