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Federal state differentials in the efficiency of health production in Germany: an artifact of spatial dependence?

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Abstract

Due to regional competition and patient migration, the efficiency of healthcare provision at the regional level is subject to spatial dependence. We address this issue by applying a spatial autoregressive model to longitudinal data from Germany at the district (‘Kreis’) level. The empirical model is specified to explain efficiency scores, which we derive through non-parametric order-m efficiency analysis of regional health production. The focus is on the role of health policy of federal states (‘Bundesländer’) for district efficiency. Regression results reveal significant spatial spillover effects. Notably, accounting for spatial dependence does not decrease but increases the estimated effect of federal states on district efficiency. It appears that genuinely more efficient states are less affected by positive efficiency spillovers, so that taking into account spatial dependence clarifies the importance of health policy at the state level.

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Notes

  1. See Hollingsworth [30, 31] and Hollingsworth et al. [32] for comprehensive surveys. Pilyavsky and Staat [47] is most relevant for the present analysis as it also employs the order-m efficiency approach; see Sect. 4.1. Key references include Staat [54], Herr [28], and Herr et al. [29] as they address the German health care system.

  2. To some degree, Schwierz and Wübker [50] allow for inter-regional correlation by carrying out a ‘multi-level analysis’. But in fact they account for correlation at the state level rather than spatial dependence.

  3. \( (1.63 - 0.5)(0.57 - 0.5)^{ - 1} \approx 16 \). The absolute values have no economic meaning as the value 0.5 is arbitrarily assigned to the parameter μ.

  4. The observation period is restricted to 3 years for several reasons. First, there are missing values in the data, e.g. the ‘INKAR’ data base did not release an issue every year. Second, a major district reform in East Germany in 2007 and 2008 impedes the construction of a consistent district-level panel for a longer period of time.

  5. Allowing for super efficient units has great appeal with our data, as they are presumably insufficient for comprehensively describing the physical process of health production. Hence, super-efficiency may capture unobserved yet relevant information (cf. [20]).

  6. For order-m efficiency, the distinction between input- and output-oriented efficiency is even more essential than for DEA. Input- and output-orientations do not only differ with respect to the direction in which the distance from the production frontier is measured, but also with respect to the frontiers themselves.

  7. m = 100 is a large value compared to other applications (e.g. [10]; [47]). But as the present analysis considers one output only, the share of observations satisfying v is  ≥ v 0s for s = 1,…,S is typically larger than for multi-output applications. Hence, when a large number of potential benchmark observations is available, a relatively large sample of actual observations can be drawn.

  8. Daraio and Simar [20] suggest that the substantially smaller value of B = 200 is sufficient. But with our data, increasing the number of iterations to B ≫ 200 markedly changes at least some estimated scores.

  9. This is not uncommon in applied health economics; see Hall and Jones [26] for a recent example.

  10. While DEA would necessarily require such a transformation (cf. [49]), order-m analysis allows the use of the original variable—with no effect on estimated efficiencies—if the condition v is  ≤ v 0s is used in (1) instead of v is  ≥ v 0s .

  11. A similar pattern is also found in regions other than Franconia, yet, due to the layout of districts, it is less obvious in our data.

  12. Maximum likelihood estimation represents an alternative approach. Arraiz et al. [4] argue, however, that ML performs poorly even for small deviations from distributional assumptions and involves computational difficulties. We therefore prefer the Kelejian and Prucha [35] estimator.

  13. All moment restrictions receive equal weight; hence a GMM technically coincides with non-linear least squares.

  14. Computation of HAC standard errors is based on: (1) a Parzen kernel (2) raw travel time as distance measure (3) travel time of 45 min as bandwidth. Point estimates obtained from initial 2SLS are close to the preferred three stage results. HAC standard errors estimated for 2SLS are somewhat larger than their conventional counterparts, computed for the three stage procedure. Nevertheless, in terms of significance, the majority of the qualitative findings still hold for the more robust model variant.

  15. \( \widetilde{x}_{it} \equiv x_{it} - \widehat{\theta }_{i} \overline{x}_{i} \) with \( \widehat{\theta }_{i} = 1 - \widehat{\sigma }_{\varsigma } (T_{i} \widehat{\sigma }_{\mu }^{2} + \widehat{\sigma }_{\varsigma }^{2} )^{ - 0.5} \) holds for a quasi-within-transformed variable \( \tilde{x}_{it} \), where T i denotes the number of observations on district i and \( \bar{x}_{i} \) denotes the district mean of x it (see e.g. [23]). For the fixed effects model, \( \theta_{j} \) takes on the value 1.

  16. Instead, we employ a random effects estimator at the first stage. The moment restrictions (cf. [33]) exploited at the second stage are derived under general fixed effects conditions, and are also valid for the special case of random effects.

  17. City-states are not taken into account. Berlin cannot be classified as either east or west.

  18. Results for tests on general cross-section dependence (cf. [7])—not necessarily spatial dependence—strongly argue in favor of the presence of cross-section dependence in the data.

  19. \( \widehat{\sigma }_{\mu }^{2} \left( {\widehat{\sigma }_{\mu }^{2} + \widehat{\sigma }_{\varsigma }^{2} } \right)^{ - 1} \) = 0.86.

  20. Fixed effects estimation does not allow for the inclusion of time-invariant explanatory variables. Hence, we cannot address state effects directly. But regressing estimates for μ i on the time-invariant regressors yields highly significant state effects.

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Acknowledgments

The authors are grateful to Rüdiger Budde for generating the regional distance matrix used in the empirical analysis, to Peter Grösche for helpful comments, Simon Decker and Adam Pilny for research assistance, and to Miriam Krieger for editorial assistance. Data provision by the Federal Association of Statutory Health Insurance Physicians (Kassenärztliche Bundesvereinigung) is gratefully acknowledged.

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Authors and Affiliations

  1. University of Basel and CINCH Centre of Health Economics Research, Peter Merian-Weg 6, 4002, Basel, Switzerland

    Stefan Felder

  2. Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI) and CINCH Centre of Health Economics Research, Hohenzollernstraße 1-3, 45128, Essen, Germany

    Harald Tauchmann

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Correspondence to Harald Tauchmann.

Appendices

Appendix 1: A modified Simar and Wilson procedure

Following the estimation of \( \hat{\beta } \), \( \hat{\lambda } \), \( \hat{\rho } \), and \( \hat{\sigma } \) obtained by the Kelejian and Prucha [35] approach, we apply and attune the four-step procedure by Simar and Wilson [51] to the spatial autoregressive model:

  1. 1.

    A vector \( \tilde{\varepsilon }^{d} \) of independent errors is drawn from the \( N\left( {0,\,\hat{\sigma }^{2} } \right) \) distribution and an artificial, spatially correlated error vector \( \tilde{\xi }^{d} \) is computed as \( \left( {I - \hat{\rho }M} \right)^{ - 1} \tilde{\varepsilon }^{d} \).

  2. 2.

    From this, a vector of artificial bootstrap efficiency scores is derived as \( \tilde{y}^{d} = \left( {I - \hat{\lambda }W} \right)^{ - 1} \left( {X\hat{\beta } + \tilde{\xi }^{d} } \right) \).

  3. 3.

    \( \tilde{y}^{d} \) is used as dependent variable to obtain bootstrap estimates \( \hat{\beta }^{d} \) and \( \hat{\lambda }^{d} \), once again following the estimation procedure of Kelejian and Prucha [35].

  1. 4.

    Steps 1–3 are repeated D times, with d = 1,…,D, and standard errors for \( \hat{\beta } \) and \( \hat{\lambda } \) are computed from the bootstrap distribution of \( \hat{\beta }^{d} \) and \( \hat{\lambda }^{d} \).

In contrast to the original Simar and Wilson [51] procedure, neither drawing from the truncated normal distribution nor truncated regression is required, as order-m efficiencies—unlike DEA efficiency scores—are not bounded from above. In our application we choose D = 250. While the described procedure yields standard errors for \( \hat{\beta } \) and \( \hat{\lambda } \) that are robust to efficiency measurement-induced error correlation, this does not apply to \( \hat{\rho } \) and \( \hat{\sigma } \), as these estimates enter the distribution from which the bootstrap error vector \( \tilde{\xi }^{d} \) is drawn. Moreover, the estimate for ρ will eventually capture efficiency measurement-induced error correlation along with spatial error correlation.

Appendix 2: Results for alternative model specifications

See the Tables 5, 6, 7, 8.

Table 5 Descriptive statistics for alternative efficiency measures (pooled)
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Table 6 Regression results for alternative efficiency measures (pooled regression)
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Table 7 Results for the model using contiguity spatial weighting matrix
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Table 8 Initial 2SLS estimates (pooled model)
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Felder, S., Tauchmann, H. Federal state differentials in the efficiency of health production in Germany: an artifact of spatial dependence?. Eur J Health Econ 14, 21–39 (2013). https://doi.org/10.1007/s10198-011-0345-8

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