How different are hospitals’ responses to a financial reform? The impact on efficiency of activity-based financing

Abstract

For policy-makers the heterogeneity of hospital response to reforms is of crucial concern. Even though a reform may entail a positive effect on average efficiency, policy-makers will consider the reform as less attractive if the variation in hospital efficiency increases. The reason is that increased variance of efficiency across hospitals is likely to increase the impact of geography on access to hospital services. This paper examines the heterogeneity with respect to the impact of a financial reform—Activity Based Financing (ABF)—on hospital efficiency in Norway. From a theoretical model we find an ambiguous effect of hospital heterogeneity on the effect of ABF on efficiency. The data set is from a contiguous 10-year panel of 47 hospitals covering both pre-ABF years and years after its imposition. Substantial heterogeneity in the responses, as measured by both estimated and predicted coefficients, is found. We did not find any significant correlation between pre-ABF measures of efficiency and the effect of ABF on efficiency. We did however find a strongly significant correlation between the effect of ABF and post-ABF efficiency. Thus, the analysis confirms the impression that, whereas pre-ABF efficiency did not play any role in how hospitals responded to ABF, those responding generally ended up as better-performing hospitals. Hence, for the type of reform studied in this article we find that policy-makers do not need to worry about the impact of location on patients’ access to hospital services.

Appendix

1.1 A A theoretical framework

In this appendix, we outline a theoretical framework which may motivate the reduced form econometric equation, Eq. 1, on which estimation and testing in Section 3 are based. The model draws on the framework in Biørn et al. [20], reformulated to account for hospital-specific heterogeneity in its coefficient structure. The hospital’s objective function contains three types of arguments: the utility from treating patients, the utility of profit and the disutility of effort. Hence, the function is of the same type as in models suggested by Chalkley and Malcomson [28]. The manager of hospital i is assumed to choose levels of effort, e _i , and the number of employees, v _i , in order to maximize

$$ U_{i}=u_{i}[f_{i}(v_{i},e_{i})]+h_{i}[A_{i}+pf_{i}(v_{i},e_{i}) -wv_{i}-K_{i}]-\gamma _{i}(e_{i}) $$

(2)

where u _i (·), f _i (·), h _i (·), and γ _i (·) are functions which jointly determine the form of the objective function. The function f _i (v _i ,e _i ) expresses how the number of treated patients, n _i , in hospital i depends on the number of employees (only one type of employee is, for simplicity, assumed) and the level of effort in hospital i. Heterogeneity in the production function may be related to, for example, heterogeneity in the quality of buildings and other physical assets. The function u _i (·) expresses the utility of treating patients, included to take intrinsic motivation into account. The function h _i (·) expresses the utility from profit, where A _i is a fixed income component, p is a fee received per treatment, w is gross expenditure per employee, and K _i is a fixed cost. Profit is included since, other things equal, a surplus adds to a manager’s prestige while a deficit reduces it. Finally, the function γ _i (·) captures the manager’s disutility of exerting effort. A _i , K _i , w and p are all considered to be exogenous. Equation 2 reflects that hospital heterogeneity is involved not only in the production structure, but also in the disutility of effort, and in the potential trade-off between treatment of patients, profit and effort.

Proceeding with all of the types of heterogeneity in Eq. 2 makes the comparative statics of the model intractable. Hence, for simplicity, we concentrate on heterogeneity in disutility of effort, and drop the hospital subscript on the functions $u_{i}({\displaystyle\cdot })$, $f_{i}({\displaystyle\cdot })$ and $h_{i}({\displaystyle\cdot })$ in the further derivations. For notational convenience we also omit the subscript on the variables. A parameter $\theta \in \lbrack 0,1]$ takes care of heterogeneity regarding the disutility of effort; the higher is θ, the stronger is, cet. par., the manager’s disutility. The objective function then becomes

$$ U=u[f(v,e)]+h[A+pf(v,e)-wv-K]-\theta \gamma (e), $$

(3)

where the functions u(·), f(·), h(·), and γ(·) are assumed to be the same for all hospitals. The interpretation of the parameter θ can be indicated as follows: Say, for instance, that changes are being considered for the system for physicians on call in order to increase the number of surgeons available for elective surgery. The manager’s disutility related to this effort depends on whether or not bottlenecks occur. If, for instance, operating rooms are in short supply, the disutility of reorganizing the surgical activities increases because more effort must be put into the management of operating room scheduling. Then θ is presumed to be large. Hospital organization also varies regarding mutual trust and cooperation between management and employees. If suggestions for organizational changes in general are met with strong opposition, θ is also presumed to be large.

We assume that f(·) has the following properties: f _k (v,e) > 0, f _kk (v,e) < 0 (k = 1,2), and f ₁₂(v,e) = f ₂₁(v,e) > 0, subscript k denoting the partial derivative with respect to the kth argument. We also make standard assumptions regarding u(·), h(·) , and γ(·): u(·) and h(·) have positive first order derivatives and negative second order derivatives (denoted by ^′ and ^″, respectively), while γ(·) has both positive first and second order derivative.

Maximizing Eq. 3 with respect to v and e gives, from the first-order conditions for an interior solution:

$$\begin{array}{l} u^{\,\prime}f_1(v,e) + h^{\,\prime}[pf_1(v,e)-w] = 0, \\[5pt] u^{\,\prime}f _2(v,e) + h^{\,\prime}[pf_2(v,e)]-\theta\gamma^{\,\prime}(e) = 0. \end{array} $$

(4)

The first of these equations implies that a necessary condition for obtaining an interior solution is [pf ₁(v,e) − w] < 0, which means that the fee per treatment only covers a proportion of the cost of a marginal employee. This is in accordance with the kind of financing system we study. The second-order conditions are: $U_{vv}^{}<0,\ U_{ee}^{}<0,\ D=U_{vv}^{}U_{ee}^{}\!- \!U_{ve}^2>0$, where $U_{vv}^{}$ and $U_{ee}^{}$ are the second-order derivatives of Eq. 3 with respect to v and e, respectively, and $U_{ve}^{}$ is the cross derivative. Equations 4 determine the optimal v and e as functions of the exogenous variables, i.e., reduced form equations, of the form

$$ \begin{array}{l} v = g_v(p,w,A,\theta,\ldots), \\[5pt] e = g_e(p,w,A,\theta,\ldots).\end{array} $$

(5)

In particular, we find a negative effect of θ on e, while its effect on v is indeterminate.

We model the effect of introducing ABF as a change where an increase in p occurs simultaneously with a decrease in A such that the previously optimal number of patients, $n_{{}}^{0}$, is still feasible. By differentiating the first-order conditions (4) with respect to p and A we find the effect on effort of a change to ABF to be:

$$\begin{array}{lll} \Delta \!&=&\!\frac{\partial e}{\partial p}-n_{{}}^{0}\frac{\partial e}{\partial A} \!=\! \frac{h^{\,\prime }(\cdot )}{D}\big\{\theta \gamma ^{\prime }(e)[f_{1}(\cdot )f_{12}(\cdot )-f_{2}(\cdot )f_{11}(\cdot )]\\ &&{\kern83pt}+\,h^{\prime \prime }(\cdot )[pf_{1}(\cdot )-w]wf_{2}(\cdot )\big\}>0. \end{array}$$

where the term preceding the curly bracket and the term within the curly bracket are both positive. Hence, the introduction of ABF initiates an increase in effort. The intuition is that effort is now more rewarding in terms of treatments and profit, since an increase in effort results in increased income because of the increase in the number of treatments.

We are, in particular, interested in finding a relation between the initial level of effort that stems from heterogeneity in θ and the effect of ABF on effort, i.e., the sign of $\partial \Delta /\partial \theta$. The model provides us with an ambiguous sign of this derivative. On the one hand, a hospital with a small θ has a small marginal disutility of increasing e. This pulls in the direction of a larger increase in e compared with a hospital with a large θ. On the other hand, a hospital with a small θ chooses a high e initially, which pulls in the direction of a relatively high marginal disutility of increasing e further. Hence, we cannot from standard economic theory predict whether hospitals with the highest level of initial effort or the hospitals with a catch-up potential are likely to show the highest effect on effort from the introduction of ABF. This ambiguity, which also applies to other types of heterogeneity considered in Eq. 3, is a primary motivation for the econometric analysis. In the estimations we assume that hospital efficiency is positively related to the level of effort exerted by hospital management and staff. Given the environment, a hospital that exerts effort in planning for a high level of capacity utilization and a low level of slack is then assumed to achieve a higher level of efficiency than a hospital that exerts no effort for this purpose.

1.2 B Estimation and coefficient prediction

In this appendix we describe the estimation procedures for the random coefficient model and the procedure for predicting the unit (hospital) specific coefficients from the estimation results, programmed for the present study in the Gauss software code.

We consider a linear, panel data regression model for N units (hospitals) observed in T periods, with G equations. Equation g has K _g regressors (including a one associated with the intercept) and a distinct coefficient vector, so that the total number of coefficients is $K=\sum_{g=1}^GK_g^{}$. Let, for Equation g (g = 1,...,G), unit i (i = 1,...,N), period t (t = 1,...,T), the (T ×1) vector of observations of the regressand be $\mbox{$\boldsymbol y$}_{gi}$, the $(T \times K_g^{})$ regressor matrix be $\mbox{$\boldsymbol X$} _{gi}$ (including a vector of ones associated with the intercept), and $\mbox{$\boldsymbol u$}_{gi}$ be a zero mean (T ×1) vector of disturbances. We allow for unit-specific heterogeneity to be represented, for equation g, unit i, by the random coefficient vector ${\beta}_{gi}^{}= {\beta}_g^{} + \mbox{$\boldsymbol \delta$}_{gi}^{}$ where β _g is a fixed constant vector and $\mbox{$\boldsymbol \delta$}_{gi}^{}$ its random shift variable with zero mean. We let ${ \mbox{$\boldsymbol 0$}}_{m,n}$ be the (m ×n) zero matrix and $\mbox{$\boldsymbol I$}_m^{}$ the m-dimensional identity matrix and assume that $\mbox{$\boldsymbol X$}_{gi}$, $\mbox{$\boldsymbol u$}_{gi}$, and $\mbox{$\boldsymbol \delta$}_{gi}$ are independent and that

$${\mathsf{E}}\big[\mbox{$\boldsymbol \delta$}_{gi}^{}\mbox{$\boldsymbol \delta$} _{hi}^{\,\prime}\big] = \mbox{$\boldsymbol \Sigma$}_{gh}^{\,\delta}, \ \ \ \ { \mathsf{E}}\big[\mbox{$\boldsymbol u$}_{gi}^{}\mbox{$\boldsymbol u$} _{hi}^{\,\prime}\big] = \sigma_{gh}^{\,u}\mbox{$\boldsymbol I$}_T^{}, \ \ \ \ g,h=1,\ldots,G. $$

(6)

Let

$$\begin{array}{lll} \mbox{$\boldsymbol \Sigma$}^u &=& \left[ \begin{array}{ccc} \sigma_{11}^u & \cdots & \sigma_{1G}^u \\[3pt] \vdots & & \vdots \\[3pt] \sigma_{G1}^u & \cdots & \sigma_{GG}^u \end{array} \right], \ \ \mbox{$\boldsymbol \Sigma$}_{}^{\delta} = \left[ \begin{array}{ccc} \mbox{$\boldsymbol \Sigma$}_{11}^{\delta} & \cdots & \mbox{$\boldsymbol \Sigma$} _{1G}^{\delta} \\[3pt] \vdots & & \vdots \\[3pt] \mbox{$\boldsymbol \Sigma$}_{G1}^{\delta} & \cdots & \mbox{$\boldsymbol \Sigma$} _{GG}^{\delta} \end{array} \right], \\[3pt] \mbox{$\boldsymbol X$}_i &=& \left[ \begin{array}{ccc} \mbox{$\boldsymbol X$}_{1i}^{} & \cdots & {\mbox{$\boldsymbol 0$}} \\[3pt] \vdots & \ddots & \vdots \\[3pt] {\mbox{$\boldsymbol 0$}} & \cdots & \mbox{$\boldsymbol X$}_{Gi}^{} \end{array} \right], \end{array}$$

$\mbox{$\boldsymbol y$}_i{\kern-1.5pt} ={\kern-1.5pt}(\mbox{$\boldsymbol y$}_{1i}^{\,\prime}, \ldots, \mbox{$\boldsymbol y$}_{Gi}^{\prime})^{\prime}$, ${\beta}{\kern-1.5pt}={\kern-1.5pt} ({\beta}_1^{\prime},\ldots,{\beta}_G^{\prime})^{\prime}$, $\mbox{$\boldsymbol \delta$}_i^{}{\kern-1.5pt}={\kern-1.5pt} ( \mbox{$\boldsymbol \delta$}_{1i}^{\prime},\ldots,\mbox{$\boldsymbol \delta$}_{Gi}^{\prime})^{ \prime}$, $\mbox{$\boldsymbol u$}_i^{} =(\mbox{$\boldsymbol u$} _{1i}^{\prime},\ldots,\mbox{$\boldsymbol u$}_{Gi}^{\prime})^{\prime}$, $\mbox{$\boldsymbol \eta$}_i^{}\!=\!(\mbox{$\boldsymbol \eta$}_{1i}^{\prime}, \ldots,\mbox{$\boldsymbol \eta$}_{Gi}^{\prime})^{\prime}$.

Since Equation g for unit i is

$$ \mbox{$\boldsymbol y$}_{gi}^{} = \mbox{$\boldsymbol X$}_{gi}^{} {\beta}_{gi}^{} + \mbox{$\boldsymbol u$}_{gi}^{} = \mbox{$\boldsymbol X$} _{gi}^{}{\beta}_g^{} + \mbox{$\boldsymbol \eta$}_{gi}^{}, \ \ \mbox{$\boldsymbol \eta$}_{gi}^{} = \mbox{$\boldsymbol X$}_{gi}^{} \mbox{$\boldsymbol \delta$}_{gi}^{} + \mbox{$\boldsymbol u$}_{gi}^{}, $$

(7)

we can write the model compactly as

$$ \mbox{$\boldsymbol y$}_i^{} = \mbox{$\boldsymbol X$}_i^{} {\beta} + \mbox{$\boldsymbol \eta$}_i^{},\;\;\;\;\;\;\;\;\;\; \mbox{$\boldsymbol \eta$}_i^{} = \mbox{$\boldsymbol X$}_i^{}\mbox{$\boldsymbol \delta$}_i^{} + \mbox{$\boldsymbol u$}_i^{}, $$

(8)

$$ \begin{array}{l}{\mathsf{V}}(\mbox{$\boldsymbol \delta$}_i^{})=\mbox{$\boldsymbol \Sigma$} _{}^{\delta}, \ \ \ {\mathsf{V}}(\mbox{$\boldsymbol u$}_i^{})= \mbox{$\boldsymbol I$}_T^{} \otimes \mbox{$\boldsymbol \Sigma$}_{}^u,\\[6pt] {\mathsf{V}}( \mbox{$\boldsymbol \eta$}_i^{}) = \mbox{$\boldsymbol \Omega$}_i^{} = \mbox{$\boldsymbol X$}_i^{}\mbox{$\boldsymbol \Sigma$}_{}^{\delta} \mbox{$\boldsymbol X$}_i^{\prime} + \mbox{$\boldsymbol I$}_T^{} \otimes \mbox{$\boldsymbol \Sigma$}_{}^u.\end{array} $$

(9)

The vector of OLS estimators for unit i is [this presumes that T > K _g for all g]

$$ \begin{array}{lll}\widehat{{\beta}}_i^{} &=& \left[\! \begin{array}{c} \widehat{{\beta}}_{1i}^{} \\ \vdots \\ \widehat{{\beta}}_{Gi}^{} \end{array} \! \right] = \big[\mbox{$\boldsymbol X$}_i^{\prime}\mbox{$\boldsymbol X$}_i^{}\big]^{-1} \mbox{$\boldsymbol X$}_i^{\prime}\mbox{$\boldsymbol y$}_i^{} = \!\left[ \begin{array}{c} \big(\mbox{$\boldsymbol X$}_{1i}^{\prime}\mbox{$\boldsymbol X$}_{1i}^{}\big)^{-1} \mbox{$\boldsymbol X$}_{1i}^{\prime}\mbox{$\boldsymbol y$}_{1i}^{}\\ \vdots \\ \!\big(\mbox{$\boldsymbol X$}_{Gi}^{\prime}\mbox{$\boldsymbol X$}_{Gi}^{}\big)^{-1} \mbox{$\boldsymbol X$}_{Gi}^{\prime}\mbox{$\boldsymbol y$}_{Gi}^{}\! \end{array} \right]{\kern-1.5pt} , \\ i&=&1,\ldots,N. \end{array} $$

(10)

A first-step estimator of the common expectation of the unit specific coefficient vectors, β, based on observations of all units is the unweighted mean

$$ \widehat{{\beta}}_{}^{} = \frac{1}{N}\sum\limits_{i=1}^N \widehat{{\beta}}_i^{} = \frac{1}{N}\sum\limits_{i=1}^N \big[ \mbox{$\boldsymbol X$}_i^{\prime}\mbox{$\boldsymbol X$}_i^{}]^{-1} \big[ \mbox{$\boldsymbol X$}_i^{\prime}\mbox{$\boldsymbol y$}_i^{}\big]. $$

(11)

We construct

$$ \widehat{\mbox{$\boldsymbol u$}}_i =\mbox{$\boldsymbol y$}_i - \mbox{$\boldsymbol X$}_i\widehat{{\beta}}_i^{}, \ \ \ \ \ \widehat{ \mbox{$\boldsymbol U$}}_i^{\prime} = \big[\widehat{\mbox{$\boldsymbol u$}} _{1i}^{\prime},\ldots,\widehat{\mbox{$\boldsymbol u$}}_{Gi}^{\prime}\big]^{\prime}, $$

where element (g,t) in $\widehat{\mbox{$\boldsymbol U$}}_i$ is the t’th OLS residual of unit i in the g’th equation. We estimate $\mbox{$\boldsymbol \Sigma$}_{}^u$ and $\mbox{$\boldsymbol \Sigma$}_{}^{\delta}$ by

$$ \widehat{\mbox{$\boldsymbol \Sigma$}}_{}^u = \displaystyle\frac{1}{NT} \sum\limits_{i=1}^N \widehat{\mbox{$\boldsymbol U$}}_i^{}\widehat{\mbox{$\boldsymbol U$} }_i^{\prime}, \ \ \ \ \ \widehat{ \mbox{$\boldsymbol \Sigma$}}_{}^{\delta} = \displaystyle\frac{1}{N}\sum\limits_{i=1}^N ( \widehat{{\beta}}_i^{}- \widehat{{\beta}}) (\widehat{ {\beta}}_i^{}- \widehat{{\beta}})^{\,\prime}. $$

(12)

These estimators are consistent if both T and N go to infinity and are always positive definite; $\widehat{\mbox{$\boldsymbol \Sigma$}}_{}^{\delta}$, however, is biased in finite samples. Inserting $\widehat{\mbox{$\boldsymbol \Sigma$}}_{}^u$ and $\widehat{ \mbox{$\boldsymbol \Sigma$}}_{}^{\delta}$ into Eq. 9, we get the following estimator of $\mbox{$\boldsymbol \Omega$}_i$ :

$$ \widehat{\mbox{$\boldsymbol \Omega$}}_i^{} = \mbox{$\boldsymbol X$}_i^{}\widehat{ \mbox{$\boldsymbol \Sigma$}}_{}^{\delta}\mbox{$\boldsymbol X$}_i^{\prime} + \mbox{$\boldsymbol I$}_T^{} \otimes \widehat{\mbox{$\boldsymbol \Sigma$}} _{}^{\,u}, \;\;\;\;\;\;\;\;\;\; i=1,\ldots,N. $$

(13)

The GLS estimator of β _i and its covariance matrix are

$$ \widetilde{{\beta}}_i^{} = \big[\mbox{$\boldsymbol X$}_i^{\prime} \mbox{$\boldsymbol \Omega$}_i^{-1}\mbox{$\boldsymbol X$}_i^{}\big]^{-1} \big[\mbox{$\boldsymbol X$}_i^{\prime}\mbox{$\boldsymbol \Omega$}_i^{-1} \mbox{$\boldsymbol y$}_i^{}\big], \ \ \ {\mathsf{V}}(\widetilde{ {\beta}}_i^{}) =\big[\mbox{$\boldsymbol X$}_i^{\prime} \mbox{$\boldsymbol \Omega$} _i^{-1}\mbox{$\boldsymbol X$}_i^{}\big]^{-1}. $$

(14)

A corresponding estimator of β and its covariance matrix, based all observations, are

$$ \begin{array}{lll}\widetilde{{\beta}}&\!=\!&\left[\!\textstyle\sum_{i=1}^N\! \mathsf{V}(\widetilde{ {\beta}}_i^{})^{-1}\!\right]^{-1}\! \left[\!\textstyle \sum_{i=1}^N\!{\mathsf{V}}(\widetilde{{\beta}}_i^{})^{-1}\!\widetilde{{\beta}}_i^{}\!\right]\\ &\!=\!& \left[\!\textstyle\sum_{i=1}^N\!\mbox{$\boldsymbol X$}_i^{\prime} \mbox{$\boldsymbol \Omega$}_i^{-1}\mbox{$\boldsymbol X$}_i^{}\!\right]^{-1}\! \left[\!\textstyle \sum_{i=1}^N\!\mbox{$\boldsymbol X$}_i^{\prime}\mbox{$\boldsymbol \Omega$} _i^{-1} \mbox{$\boldsymbol y$}_i^{}\!\right], \end{array} $$

(15)

$$ \begin{array}{lll} &&{\kern-17pt}{\mathsf{V}}(\widetilde{{\beta}})\!=\!\left[\!\textstyle \sum_{i=1}^N\!{\mathsf{V}}(\widetilde{{\beta}}_i^{})^{-1}\! \right]^{-1}\!=\!\left[\!\textstyle\sum_{i=1}^N\!\mbox{$\boldsymbol X$} _i^{\prime} \mbox{$\boldsymbol \Omega$}_i^{-1}\mbox{$\boldsymbol X$}_i^{}\! \right]^{-1}. \end{array} $$

(16)

The FGLS estimators of β _i (conditional on unit i) and β (unconditional mean) and their estimated covariance matrices are obtained by inserting Eq. 13 into Eqs. 14–16.

We construct the GLS residual vector corresponding to $\mbox{$\boldsymbol u$}_i$ from

$$\widetilde{\mbox{$\boldsymbol u$}}_i =\mbox{$\boldsymbol y$}_i - \mbox{$\boldsymbol X$}_i\widetilde{{\beta}}_i^{}, \ \ \ \ \widetilde{ \mbox{$\boldsymbol U$}}_i = \big[\widetilde{\mbox{$\boldsymbol u$}} _{1i}^{\prime},\cdots,\widetilde{\mbox{$\boldsymbol u$}}_{Gi}^{\prime}\big]^{ \prime}. $$

The second step estimator of $\mbox{$\boldsymbol \Sigma$}_{}^u$ and $\mbox{$\boldsymbol \Sigma$}_{}^{\delta}$ and the recomputed overall estimator of $\mbox{$\boldsymbol \Omega$}_i$ are

$$ \widetilde{\mbox{$\boldsymbol \Sigma$}}_{}^u \!=\! \displaystyle\frac{1}{NT} \displaystyle\sum_{i=1}^N\widetilde{\mbox{$\boldsymbol U$}}_i^{}\widetilde{ \mbox{$\boldsymbol U$}}_i^{\prime}, \ \ \widetilde{\mbox{$\boldsymbol \Sigma$}} _{}^{\delta} \!=\! \displaystyle\frac{1}{N}\displaystyle\sum_{i=1}^N\! \big(\widetilde{ {\beta}}_i^{}{\kern-.7pt}-{\kern-.7pt}\widetilde{{\beta}}\,\big) \big(\widetilde{{\beta}}_i^{}{\kern-.7pt}-{\kern-.7pt}\widetilde{{\beta}} \,\big)^{\prime}, $$

(17)

$$\widetilde{\mbox{$\boldsymbol \Omega$}}_i^{} \!=\! \mbox{$\boldsymbol X$}_i^{} \widetilde{\mbox{$\boldsymbol \Sigma$}}_{}^{\,\delta}\mbox{$\boldsymbol X$} _i^{\prime} + \mbox{$\boldsymbol I$}_T^{} \otimes \widetilde{\mbox{$\boldsymbol \Sigma$}}_{}^u, \;\;\;\;\;\;\;\;\;\; i=1,\ldots,N. $$

(18)

Revised FGLS estimators of β _i and β and their estimated covariance matrices are obtained by inserting Eq. 18 into Eqs. 14–16.

The random coefficient vector of unit i, β _i , is predicted by means of

$$ \begin{array}{lll} {\beta}_{i}^{\ast }&=&\widetilde{{\beta}} \!+\!\mbox{$\boldsymbol \Sigma$}^{\delta }\mbox{$\boldsymbol X$}_{i}^{\prime } \mbox{$\boldsymbol \Omega$}_{i}^{-1}\big(\mbox{$\boldsymbol y$}_{i}^{{}}\!-\! \mbox{$\boldsymbol X$}_{i}^{{}}\widetilde{{\beta}}\,\big)\\ &=& \widetilde{{\beta}}\!+\!\mbox{$\boldsymbol \Sigma$}^{\delta } \mbox{$\boldsymbol X$}_{i}^{\prime }\big(\mbox{$\boldsymbol X$}_{i}^{{}} \mbox{$\boldsymbol \Sigma$}_{{}}^{\delta }\mbox{$\boldsymbol X$}_{i}^{\prime }\!+\!\mbox{$\boldsymbol I$}_{T}\!\otimes \!\mbox{$\boldsymbol \Sigma$}_{{}}^{u}\big)^{-1}\big(\mbox{$\boldsymbol y$}_{i}^{{}}\!-\!\mbox{$\boldsymbol X$}_{i}^{{}}\widetilde{ {\beta}}\,\big), \end{array} $$

(19)

cf. Lee and Griffiths, Section 4 [29] and Hsiao p. 147 [30]. It has the interpretation as the minimum variance linear unbiased predictor (MVLUP) of the stochastic variable β _i . Combining Eq. 19 with Eq. 14 this predictor can be rewritten as

$$ \begin{array}{lll}{\beta}_{i}^{\ast }&=&\mbox{$\boldsymbol A$}_{i}\widetilde{{\beta}}_{i}+(\mbox{$\boldsymbol I$}_{K}\!-\! \mbox{$\boldsymbol A$}_{i})\widetilde{{\beta}},\ \\ \mbox{$\boldsymbol A$} _{i}&=&\mbox{$\boldsymbol \Sigma$}_{{}}^{\delta }{\mathsf{V}}(\widetilde{ {\beta}}_{i})^{-1}=\mbox{$\boldsymbol \Sigma$}_{{}}^{\delta }\mbox{$\boldsymbol X$}_{i}^{\prime }\big(\mbox{$\boldsymbol X$}_{i}^{{}}\mbox{$\boldsymbol \Sigma$} _{{}}^{\delta }\mbox{$\boldsymbol X$}_{i}^{\prime }+\mbox{$\boldsymbol I$} _{T}\otimes \mbox{$\boldsymbol \Sigma$}_{{}}^{u}\big)^{-1}\mbox{$\boldsymbol X$} _{i}^{{}}. \end{array} $$

(20)

Equation 20 shows that the predictor ${\beta} _{i}^{\ast }$ can be interpreted as a matrix weighted mean of the unit specific GLS estimator $\widetilde{{\beta}}_{i}$ and the overall GLS estimator $\widetilde{{\beta}}$. The predictor is ‘closer to’ the unit specific GLS estimator, i.e., $\mbox{$\boldsymbol A$}_{i}$ is ‘large’ in relation to $\mbox{$\boldsymbol I$}_{K}$, (i) the ‘larger’ is the dispersion of the random coefficient vector, as measured by $\mbox{$\boldsymbol \Sigma$}^{\delta }$, and (ii) the ‘more certain’ is the unit specific GLS estimator, as measured by the inverse of its covariance matrix ${\mathsf{V}}( \widetilde{{\beta}}_{i})^{-1}=\mbox{$\boldsymbol X$} _{i}^{\prime }\mbox{$\boldsymbol \Omega$}_{i}^{-1}\mbox{$\boldsymbol X$}_{i}^{{}}$. It is closer to the estimator of the common expected coefficient when $\mbox{$\boldsymbol \Sigma$}_{{}}^{\delta }{\mathsf{V}}( \widetilde{{\beta}}_{i})^{-1}$ is ‘small’ relative to $\mbox{$\boldsymbol I$}_{K}$. Intuitively, this is quite reasonable. We obtain the numerical values of the predicted coefficients reported in the text by inserting Eq. 13, or Eq. 18, into Eq. 20.

1.3 C Supplementary results

The strength of association between efficiency in any 2 years can be measured by rank correlation coefficients (RCCs), i.e., coefficients of correlation of the ranking numbers across hospitals [see Zar, p. 578 [31] and Kraemer [32]]. Table 6 reports RCCs for all pairs of years. The RRCs of neighbouring years are in general high, but tend to decrease with increasing time distance. The year 1997, when the ABF reform became effective for the first hospitals, gives a characteristic example: The RCCs for this years against 1998, 1999, 2000, and 2001 are, respectively, (0.81, 0.72, 0.57, 0.33) for efficiency measure CE, and (0.86, 0.58, 0.42, 0.29) for TE.

Table 6 Rank correlation coefficients of efficiency across hospitals

Table 7 Hospital-specific coefficient of ABF dummy

Table 8 puts the quartiles of the distribution of the ABF coefficients in focus, by specifying the time path of the efficiency indicators of the hospitals ranked, according to the ABF coefficients, at the lower and upper quartiles, respectively (ranking numbers 36 and 12). An improvement in efficiency from the pre-ABF to the ABF years is most clearly visible in the TE indexes. The upper quartile hospital ABF coefficient estimate (Table 8, Part I) had an increase in its TE measure from 81.7 in 1996 to 89.4 in 1997, showing some fluctuations in the next following years. This pattern is also seen in the ranking numbers. Considering the upper-quartile hospital according to the predicted ABF coefficient (Table 8, Part II) we find an increase in its TE measure from 79.8 to 81.9 and an increase also in the following years. For the ranking numbers, however, there is first a decline, from 23 in 1996 to to 28 in 1997, and then a substantial improvement in 1998–2001, with ranking numbers 17, 8, 9 and 3. There are signs that also the lower-quartile hospital had some initial deterioration of its technical efficiency and improvement thereafter: the ranking numbers in the years 1995–2001 are 3, 3, 9, 7, 2, 2, and 2.

Table 8 Efficiency of hospital with ABF coefficient at the lower (L) and at the upper (U) quartile