The state level cost efficiencies are estimated using the stochastic frontier function (SFF) method. The SFF method assumes that a decision making unit, combines inputs (that cost a certain amount) to produce the services. Because we are concerned with cost efficiencies of LTSS provided by state Medicaid agencies, we may assume that the Medicaid agency for each state is the decision making unit for that state’s LTSS program, and has an organized system for providing those services for its beneficiaries. The SFF method follows that of Greene and Segal ([2004]) which also estimates cost efficiency using cross-section-time-series data. However, our method takes further advantage of the panel structure of the state level LTSS data and adopts the random effect models that incorporate time varying nature of the inefficiency (Greene [2005]). The estimation was carried out in two steps. We used the SFF method to estimate the cost efficiencies of states’ in LTSS delivery in the first step. In the second step, we regressed the logarithm of these scores obtained from the first step on the penetration of HCBS, and HCBS waivers following the approach developed by Greene ([2004]). The use of the second step addresses cross-state heterogeneity and effectively overcomes the concerns raised in early research using the SFF analysis in the health care sector (Skinner [1994]; Dor [1994]; Newhouse [1994]). More recently, McGlynn ([2008]) again advocated the SFF analysis in the health care sector. The detailed description of the model and its estimation are discussed below.

### Model specification and estimation

Cost efficiency (*CE*), in the econometric literature is defined as the ratio of the observed number of beneficiaries served for a given level of inputs and, consequently, for a given cost and the maximum number that could have been served at that cost. Figure 1 illustrates this concept. The solid curve represents the maximum number (*q*) of LTSS clients that can be served by a state for a given level of inputs (*x*). This is the state’s production frontier. The possible input–output combinations in real life however, are always below the frontier and represent varying levels of efficiency.

In Figure 1 the point A (*x*
_{
1
}, *q*
_{
1
}) represents the observed output *q*
_{
1
} for a given input vector (*x*
_{
1
}) and the point B (*x*
_{
1
}, *q*
_{
2
}) represents the efficient level of output serving a larger number of clients *q*
_{
2
} but using the same level of input x_{1}. A state’s CE is defined as the ratio *q*
_{
1
}/*q*
_{
2
} ([Fare et al. 1994]).

In general, the production function of LTSS services in a state is represented by the equation:

q=F\left(f\left(x,\beta \right)*\mathit{CE}\right),

(1)

where *q* is the number of LTSS clients served by the state Medicaid agency, *x* is the vector of inputs, and *β* is a vector of parameters to be estimated. The function *f*(*x*, *β*) represents the optimal number of LTSS users who can be served with *x* resources, while *CE* representing a state’s cost efficiency is such that 0 ≤ *CE* ≤ 1. Using the theory of duality, the production frontier is transformed to a cost frontier

C={F}^{-1}\left(q\right)*c\left(w\right)*C{E}^{-1}

(2)

here *w* is the vector of input prices and *c*(*w*) is the unit cost function.

### Cost function specification

We assume that each state produces two types of long-term care services, institutional care and non-institutional care (HCBS). Each state also faces a fixed set of input prices, such as wages and costs of facility usage that determine the cost of providing LTSS to the Medicaid enrollees.

Applying the two-output cost function, proposed by Burgess ([1974]) and later used by Truett and Truett ([2003]), the cost function that we estimate for our 50-states’ LTSS production is:

logc={\alpha}_{0}+{\alpha}_{{q}_{1}}log{q}_{1}+{\alpha}_{{q}_{2}}log{q}_{2}+\u03f5

(3)

In the above specification, *q*
_{
1
} and *q*
_{
2
} represent the numbers of clients in institutional care and HCBS, respectively and *ϵ* = *v* + *u* is the error term consisting of a random component (*v*), and an independently distributed cost efficiency term (*u* = | log(*CE*)|) (Aigner et al. [1977]). This specification is the most appropriate because there are no restrictions on the parameters (Christensen and Greene [1976]). Thus, an efficient cost frontier is given by the following

logc={\alpha}_{0}+{\alpha}_{{q}_{1}}log{q}_{1}+{\alpha}_{{q}_{2}}log{q}_{2}+v

(4)

According to this specification cost is both increasing in outputs and conforming to usual economies and diseconomies of scale reflecting long run average U-shaped cost curves. States are also assumed to be on their long-run cost curves.

In the above formulation, states that achieve this cost-to-output relationship are deemed to be cost efficient, i.e. *log*(*CE*) = 0. In other words, state Medicaid agencies that operate on this cost-frontier are successful in their attempts to maximize output given the cost, but are still subject to stochastic shocks not in their control (Kumbkakar and Lovell [2000]). State Medicaid agencies that are further away from the cost frontier are less efficient in cost minimization (or service maximization) as compared to those that operate relatively closer to the frontier. For inefficient states the absolute value of log (*CE*) can range from 0 to infinity. Thus, states can be ranked by their cost efficiency score of their Medicaid LTSS programs and this variability can provide an insight into factors that may be associated with a state’s level of efficiency in LTSS delivery.

### Estimation of cost efficiency

In the first step estimation, we use the SFF analysis. There is another method used for cost efficiency estimation, namely the Data Envelopment Analysis (DEA) approach. The DEA approach is often criticized (Dor [1994]) because it ignores the statistical distribution of the cost uncertainties and is based solely on the specific data sample at hand. For these reasons, *we chose not to use the DEA approach*, *using instead the SFF method*. We examined three different distributional assumptions of the efficiency term in SFF, namely truncated normal, half-normal, and exponential. However, the parameter estimations of the SFF were similar across different distributional assumptions. We present the results based on truncated normal efficiency. Using panel data for all 50 states and District of Columbia for 9 years, we allowed efficiency to be time-varying and used the random effects approach to estimate it. Specification of the time-varying efficiency was based on the Battese and Coelli ([1988, 1992]) parameterization. This model is more general and inclusive than the models in which efficiencies are forced to be fixed as time changes. The proposed model, however, does not automatically imply that efficiency will improve. It is quite possible that efficiency may decrease, depending upon parameter estimates. In addition to the variables specified in equation (4), we also added eight year dummies to separate out confounding by year.

### Regression specification for estimating the effect of HCBS on cost efficiency

Following Greene ([2004, 2005]) and Greene and Segal ([2004]), we gauge the independent impact of the state practices on its CE for LTSS delivery in the second step estimation. The use of a two-step approach over the single step approach for cost estimation is still debated in the literature. Unfortunately, a single-step approach, although allows estimation of the cost function, it does not allow an analysis of the effect of important policy tools, such as managed care or the effect of waivers versus state plans that are contributing to the inefficiency in service delivery – an important focus of the present study.

In the second estimation step therefore, we regressed the logarithm of the efficiency scores (*CE*), on a set of state LTSS characteristics controlling for socioeconomics. The LTSS characteristics included the percentage of HCBS participants among LTSS users in a state, percentage of waiver participants among HCBS users in a state, the percentage of ICFMR residents among the institutionalized population in the state, presence of managed LTSS and CON program in the state. State population, income, and unemployment were used to control for differences in state socioeconomic conditions.

### Estimation of the effect of HCBS on cost efficiency

In the first step estimation, the estimated random effects model with year dummies produced year specific estimates of CE scores for each state and year. These scores constituted the dependent variable in our second step analysis. Because we have multiple years of data for each state, we examined fixed effects and random effects models to account for state specific unobserved effects in the second step. However, the Hausman ([1978]) test rejected the random effect specification and we chose the fixed effects model. A major strength of the fixed effects approach is that it captures and eliminates all time-constant heterogeneity among states that remain unobserved in the model, which makes the analysis less likely to be biased. The availability of managed care and the implementation of CON, however varied by year for each state, and these were introduced as time varying covariates in the model.

Our primary variables of interest are the size of the HCBS population in a state and its distribution into waiver plans and state-only plans. Recognizing that state economic environment could influence state expenditure in LTSS and therefore, its efficiency score, we controlled for confounders such as, unemployment rate and per capita income. We included the total population in a state as an indicator of the level of service needs. In addition, we introduced a variable capturing the relative size of ICFMR residents to elderly and disabled nursing home residents as a broad measure of service need and the age distribution among the LTSS users. The developmentally disabled/mentally retarded population represents a unique group of LTSS beneficiaries with somewhat different service needs. We controlled for the presence of CON and managed LTSS in a state to capture state level variation in the policy environment likely to influence the size of HCBS and CE of the state. The availability of managed care, and the implementation of CON, however, varied by year for each state. These were introduced as time varying covariates in the model. We report robust standard errors and associated P-values using STATA 10 (StataCorp [2007]).

### Data sources

Data for this study come from several sources. To estimate CE of states’ LTSS programs, we use data on Medicaid LTSS expenditure and beneficiaries from 49 states and the District of Columbia for years 1999–2007. We excluded Arizona from our analysis because LTSS beneficiaries in Arizona receive services from managed care exclusively. Because it operates on a capitated system, data on expenditure by type of service are not available. Vermont also transitioned to an 1115 global waiver in 2005, and we excluded 2006 & 2007 data for Vermont. We chose our study period to reflect the a time of rapid growth in which HCBS coinciding with a range of policy initiatives directed towards re-balancing of Medicaid LTSS away from institutional care. These are also the most recent years with information on number of participants in non-institutional services.

We obtained expenditure data from the quarterly CMS-64 expense report compiled annually by Thomson Reuters available from http://medicaid.gov/Medicaid-CHIP-Program-Information/By-Topics/Data-and-Systems/MBES/CMS-64-Quarterly-Expense-Report.html. These data cover expenditure in both institutional services, such as nursing homes and intermediate care facilities for people with mental retardation (ICFMR), and non-institutional services provided through waivers and state plans. Participant data for institutional care services were downloaded from the CMS MSIS system (http://www.cms.gov/Research-Statistics-Data-and-Systems/Computer-Data-and-Systems/MedicaidDataSourcesGenInfo/MSIS-Mart-Home.html), and non-institutional services are compiled from Kaiser Family Foundation and University of California at San Francisco’s report of CMS form 372 for each State Medicaid director available at http://www.kff.org/medicaid/upload/7720-04.pdf.

Information on the presence of managed LTSS program in a state was culled from the CMS MSIS system. Data on the presence of PACE, and CON in any given year in a state are obtained from agencies that direct and oversee the respective programs. For example, information on the presence of PACE program was gathered from the website for the National Pace Association at http://www.npaonline.org./website/download.asp?id=1741. This information was combined with information from MSIS to determine if a state operated any managed LTSS program. Data on presence of CON was obtained from the website for the National Conference of State Legislatures available at http://www.ncsl.org/default.aspx?tabid=14373. Information on total state population, state per-capita income, and state unemployment rate were drawn from the Census Bureau. These two variables were used primarily as controls to reflect the general economic condition of the states that can potentially affect the cost efficiency in LTSS delivery. All expenditure data are converted to year 2005 dollar equivalent to account for inflation using the CPI as a measure of inflation. We did not use the health inflation measure because it seemed unlikely that personal care services would experience the high inflation rate of the health care sector, where cost escalation has been attributed primarily to expensive new medical technology (Cutler [2001]).