The final step is to take the predicted expected utility from the hospital network of each plan, for each type of individual, and include it as an input to the plan demand equation. I begin with a simple logit model, ignoring the distribution of individual characteristics within markets, before moving on to a fuller model where variation in individual attributes is taken into account.
5.3.1. The Benchmark Model: Logit Formulation
The logit framework assumes common coefficients for all individuals. It has the advantage of generating an equation that is easy to estimate, making it a useful benchmark model. However, estimating such a model requires a plan-level, rather than plan–individual-level variable representing expected utility from the hospital network. I therefore define a representative agent in each market as an individual living in the most populated ZCTA and having the weighted average probability of diagnoses and of hospital admission of people resident in that ZCTA. I define the plan-level expected utility as that of the representative agent in the relevant market, that is:
where is the weighted average probability of diagnoses of individuals in the most populated ZCTA, and is the vector of other characteristics of an individual in that ZCTA (income and location, the only individual characteristics other than diagnosis that are not set to zero, are both defined at the ZCTA level). I control for the unobserved η first by assuming η = 0 and then by using the instrumental variables methodology already described. The utility of consumer i from choosing plan j in market m is therefore given by
where premium is included in the observed plan characteristics z, and I assume that the ωijm is distributed iid according to a Type 1 extreme value distribution.41 Normalizing the utility of all consumers from the outside good (good 0) to be zero, I obtain the standard equation for estimation:
where s0m is the share of the outside good in market m.
The logit model can be estimated using a simple two-stage least squares methodology (instrumenting for premiums since these are likely to be correlated with the unobserved quality variable ξ). The basic plan characteristics included in z are: premium per member per month,42 number of physicians per 1000 population in the market and age of the plan (coded into four dummies: less than 3 years, 3–5 years, 6–9 years and over 10 years of age). The HEDIS measures used are the breast cancer and cervical cancer screening rates; the rate of check-ups after live deliveries; the proportion of diabetic patients with annual eye exams; the proportion of adolescents receiving final immunizations before their 13th birthday; the proportion of smokers advised by their physician to quit; and the proportion of patients seen on an outpatient basis within 30 days of discharge from a mental illness admission. The CAHPS measures are ‘getting needed care’ and ‘getting care quickly’. Each is an aggregation of responses to several CAHPS questions. The CAHPS measures of plan performance are highly correlated with one another; so are the HEDIS measures. Missing variable dummies are included in all specifications. Insurer fixed effects are included for insurers that are active in at least 10 of the major markets defined by AIS,43 and market dummies are also included in some specifications.
The model is completed by defining the outside good. The simplest definition to implement would be a composite of non-managed care private coverage and uninsured (I exclude Medicare by considering only the non-elderly population, and exclude Medicaid by assumption; see Appendix A for details). However, indemnity coverage and no coverage are at opposite ends of the spectrum in terms of price and many aspects of quality so this outside good would be non-homogeneous. Instead I define the outside good as ‘choosing to be uninsured’ and create a separate choice in each market, defined as ‘choosing indemnity or PPO insurance’, and assumed to be homogeneous within each market.44 None of the data sources provides information on non-managed care coverage, so assumptions must be made to complete the dataset. Indemnity plans are assumed to be over 10 years old; to have premiums equal to the highest managed care premium in the relevant market; and to offer a physician network size equal to the largest offered by a managed care plan in the market. Indemnity plan performance ratings (both HEDIS and CAHPS) are assumed to equal the average of managed care plans in the market.45 Average quality for uninsured consumers is not identified in the plan choice model (unless I make more assumptions or normalize one of the ‘inside’ goods) so I normalize it to zero.
The instruments used for the premium variable, in addition to the usual set of plan characteristics (the z′s), are the average hourly hospital wage and the average weekly nurse wage across the markets in which each health plan is observed to be active. Hospital wage data come from the Centers for Medicaid and Medicare Services (CMS) 1999; nurse wage data are from the Bureau of Labor Statistics 1999. The assumption required for these to be valid instruments is that health plan costs are correlated with premiums but not with unobserved health plan quality. The choice of instruments is a difficult issue. I use average wages across the plan's markets, rather than same-market wages, to generate variation across plans within each market. However, this requires an additional assumption that plans set their premiums partly centrally, in response to cost changes across all markets served, which may not be realistic.46,47
The last issue is the need to adjust the estimated standard errors for the three-step estimation process being used here.48 I need to take into account the fact that the expected utility variable is constructed from estimated parameters. This has no impact on the consistency of the results but, since the estimator is not adaptive, will affect the standard error estimates.
To incorporate the effect of these estimated parameters into the estimates of the variance–covariance matrix, I take advantage of the GMM structure of the estimation procedure. Incorporating estimated parameters into a GMM estimator is fairly straightforward. An extension of Pakes (1997) shows that if there exist first-stage parameter estimates β such that
and second-stage parameter estimates α such that
then, under standard regularity conditions, the third-stage parameter estimates θn are distributed asymptotically normal as
where Γ is the derivative of the moment condition with respect to the parameters, A is the weight matrix and V is given by
where gjm are the third-stage moments, M1 is a matrix of derivatives of g(z, x, ν, α0, β0, θ0) with respect to the elements of β0, M2 is a matrix of derivatives of g(z, x, ν, α0, β0, θ0) with respect to the elements of α0, and vc1 and vc2 are the variance–covariance matrices from the first two stages. The covariance terms can be reduced to an expression depending solely on the expectations of products of M1, M2, L1, L2, and the individual moment conditions. Further details of the methodology used are given in Appendix C.
5.3.2. Full Demand Specification
Since the focus of the analysis is on the weight consumers attach to the expected utility variable in the plan demand equation, the obvious extension to the logit model is to allow a richer specification of this expected utility term. The full demand specification includes the full variable EUijm rather than the summary measure used in the logit formulation and therefore accounts for the impact of the distribution of individual locations, income and demographics within each market on plan market shares. This has the additional advantage of avoiding the logit model's well-known unattractive implication: the imposition of the independence of irrelevant alternatives (IIA) substitution pattern for an individual patient's choice of plan. The IIA assumption implies that cross-price effects are a function solely of plan shares and are independent of plans' relative positions in the characteristic space. This is clearly inaccurate to the extent that consumers substitute more readily between plans that are ‘closer’ in terms of characteristics (for example, a consumer switching from a high-premium, high-choice plan is more likely to choose another high-premium plan with similar qualities than a low-premium competitor that offers restricted choice of providers).
The utility of consumer i choosing plan j in market m in the full demand specification is given by
where premjm is plan j's premium in market m, yi is the median family income of individual-type i (defined by ZCTA), and the other variables are as specified above. This is similar to the model introduced in BLP (1995) in which random coefficients, which are functions of demographic variables taken from market-level census data, are interacted with product characteristics. The difference is that, in this equation, both consumer-specific terms (EUijm and yi) are observed, either at the ZCTA level (in the case of income) or at the ZCTA–age–gender level (for the expected utility variable). Therefore simulation methods are not needed to evaluate the estimation algorithm.49 The share equation reduces to
where ni is the number of individuals in consumer-type i, nm is the number in the market, and sijm(ϑ, γ), the share of type—i individuals choosing plan j in market m, is defined by
The presence of the unobserved quality measure ξ implies that MLE cannot be used to estimate the model (as it was for the plan demand equation).50 Instead, the contraction mapping introduced in BLP (1995) is used to transform equation (22) into a linear equation for ξ and the coefficients estimated using a GMM methodology, again as in BLP (1995).51 The variables included in z are the same as those used in the logit specification. I instrument for premium for the same reasons as in the logit framework; I add two instruments to the hospital and nurse wage variables used before. These are the average expected utility in the most populated ZCTA in the market (the EUrepjm variable defined for the logit model) and the average income across the other markets in which the plan is observed to be active. It is clear that EUrepjm should be correlated with premiums and the set-up of the model already implies that it is uncorrelated with unobserved quality ξ as required for a valid instrument. Some additional assumptions are needed to include the average income across the rest of the plan's markets as an instrument. First, the average income in each market must affect aspects of the plan's unobserved quality, such as promotional activity, in that market but not in others, and this activity must affect plan costs. Second, plans must set premiums in market j taking account of costs in other markets. Put simply, the assumption is that plans determine variables such as promotional activity locally but set premiums (at least partly) centrally.52
Standard error adjustments are again needed to account for the variance introduced in the first two steps of estimation; the methodology used is exactly analogous to that described for the logit model.