## Abstract

The introduction of prospective hospital reimbursement based on diagnosis related groups (DRG) in 2004 has been a conspicuous attempt to increase hospital efficiency in the German health sector. As a consequence of the reform a rise of competition for (low cost) patients could be expected. In this paper the competition between hospitals, quantified as spatial spillover estimates of hospital efficiency, is analyzed for periods before and after the reform. We implement a two-stage efficiency model that allows for spatial interdependence among hospitals. Hospital efficiency is determined by means of non-parametric and parametric econometric frontier models. We diagnose a significant increase of negative spatial spillovers characterizing hospital performance in Germany, and thus, confirm the expected rise of competition.

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## Notes

As it turns out, empirical results on the effect of prospective payment on spatial spillovers are qualitatively similar for alternative threshold values applied for outlier detection.

Regarding the applied two-stage approach some remarks are in order. Firstly, two-step estimation of the effects of explanatory variables on \(\theta^{S}_{i}\) might obtain biased results in both steps [27, 49]. In the first step, the estimation of the stochastic production function 1 ignoring the explanatory variables

**z**_{ i }yield biased efficiency scores. This leads to biased results in the second step. Wang and Schmidt [49] show in a simulation experiment that the biases persist asymptotically. Unfortunately, to our knowledge, a one-step treatment of SFA models accounting for spatial error and lag dependence has not been proposed yet. Since the focus of this study is rather on the comparison of estimated spatial spillovers over time we consider the neglect of the bias problem as tenable. Secondly, Simar and Wilson [42] mention that in finite samples the estimated DEA efficiency scores are biased and serially correlated in a complicated fashion. The convergence rate of \(\theta^{D}_{i}\) depends on the number of inputs and outputs and is typically lower than the parametric convergence rate. Maximum Likelihood estimates of regressions involving \(\theta^{D}_{i}\) are consistent, but inference based on the inverse of the negative Hessian of the log-likelihood is generally invalid. To analyze the robustness of inferential results, we apply a bootstrap procedure suggested by Simar and Wilson [42]. However, the difference between bootstrap based and asymptotic results is negligible and, therefore, we only document the latter.As a robustness check we have applied two further input specifications. Firstly, the number of employees is replaced by wage expenditures. Secondly, in order to minimize measurement errors in the labor variables the number of full time equivalent employees is used instead of the number of employees. However, for these variables data are only available for physicians and non-physicians. In summary, the diagnostic results are qualitatively very similar across all considered input measures.

For the SARAR model, we only document results for specifications with the same imposed pattern for the spatial lag and error dependence, since log-likelihood statistics of alternative specifications (

*W*_{ d }and*M*_{ n }, and*W*_{ n }and*M*_{ d }) do not offer any improvements.Augurzky et al. [2] mention the importance of the geographic area where the market share is built up. In a rural area a higher market share can be the result of being the only provider of inpatient treatments invoking an inefficient production of medical care, due to the lack of competitors. At the opposite, a higher market share in an urban area might be the result of efficient performance under strengthened competition. We incorporate an interaction of

*ms*and an agglomeration dummy variable. However, there is no considerable difference between the market share impact in rural and urban areas.As a robustness check we have estimated a more restricted Cobb–Douglas production function, i.e.

*δ*_{ kl }= 0 ∀*k*,*l*in Eq. 1. It obtains positive effects of the inputs on output for almost all years. Due to space considerations results are not shown here, but available from the authors upon request. A comparison of the corresponding log-likelihood statistics is in favor of the more general translog model. Diagnostic results of the hypothesis about the effect of prospective payment on spatial spillovers are qualitatively similar for the translog and Cobb-Douglas production function.We consider Spearman rank correlation rather than linear correlation to account for a potentially non-linear relationship between SFA and DEA efficiency scores.

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## Acknowledgements

We thank three anonymous referees, Uwe Jensen, Andrew Street, the participants of the XVth. Spring meeting of young economists 2010 in Luxembourg, Jahrestagung des Vereins für Socialpolitik 2010 in Kiel and 15. Nachwuchsworkshop der DStatG 2009 in Merseburg for helpful comments and discussions on earlier versions of this manuscript. We also thank Alexander Vogel and Hendrik Tietje of the Forschungsdatenzentrum der Statistischen Landesämter—Standort Kiel/Hamburg for their cooperation.

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## Appendices

### A Efficiency measurement

### 1.1 A.1 DEA efficiency scores

The efficiency score, \(\theta^{D}_{i}\), is obtained under the assumption of variable returns to scale [4] by solving the following linear program

where *q*
_{
ri
}, \(x^N_{ji}\) and \(x^D_{ki}\) denote output, non-discretionary and discretionary input variables of hospital *i*. The numbers of outputs, non- and discretionary inputs, and reference hospitals are *s*, *m*
^{N}, *m*
^{D}, and *N*, respectively.

### 1.2 A.2 SFA efficiency scores

Given Maximum Likelihood estimates of the model parameters in Eq. 1 the technical efficiency of the *i*-th hospital is obtained as [26]

where \(\hat{\nu}_i\!=\!\ln q_i - \hat{\alpha}_0 - \sum\limits_{k} \hat{\alpha}_k \ln x_{ik} - \sum\limits_{k} \sum\limits_{k\geq l} \hat{\delta}_{kl} \ln x_{ik} \ln x_{il}\), and Φ and *ϕ* are the Gaussian cumulative and probability density function, respectively [29].

### B Maximum Likelihood estimation

Model 2 can be written as

where ** B** =

*I*_{ N }−

*ρ*

**and**

*W***=**

*A*

*I*_{ N }−

*λ*

**. Assuming a multivariate normal distribution of the error terms, the log likelihood function is given by**

*W*where *ϵ* = ** B**
\(\left(\boldsymbol{A} y - \boldsymbol{Z}\beta\right)\) and

*σ*

^{2}=

*ϵ*′

*ϵ*/

*N*. The ML estimator is

where \(\widehat{\boldsymbol{B}}= \boldsymbol{I}_N-\hat{\rho}_{ML} \boldsymbol{M}\) and \(\widehat{\boldsymbol{A}}=\boldsymbol{I}_N-\hat{\lambda}_{ML} \boldsymbol{W}\).

### C Construction of case mix weights

The more time the treatments of cases belonging to the *j*-th clinical department take relative to all other treatments, the higher the weight, *π*
_{
j
}, of the respective cases. Let *c*
_{
ij
} be the number of cases in the *j*-th clinical department of the *i*-th hospital. Then, the weighted cases of hospital *i* are calculated as

where *π*
_{
j
} = *los*
_{
j
}/*los*
_{
G
}, \(los_j=(\sum_{i=1}^N days_{ij}/c_{ij})/N\) is the mean length of stay for the cases belonging to the *j*-th clinical department over all hospitals and \(los_G=(\sum_{j=1}^J los_j) /J\) is the mean length of stay over all clinical departments and all hospitals.

### D Outlier detection

Table 5 shows the share of hospitals in the population and in the sample of the first and second stage over different subgroups characterized by the number of beds and ownership form. In the sample of both stages hospitals with less than 100 beds and profit oriented hospitals are under-represented.

### E First stage results

This appendix provides further details on the results of the parameter estimates of the SFA model and descriptive statistics of SFA and DEA efficiency scores.

### 1.1 E.1 SFA parameter estimates

The results of the SFA model are shown in Table 6. The parameter estimates are characterized by substantial heterogeneity over time. The estimated variance of the noise component, \(\sigma_v^2\), decreases remarkably after 1996 and 2001 suggesting less measurement errors and a higher quality of the data. Similarly, the estimated variance of the inefficiency term, \(\sigma_u^2\), shrinks after 2001. Moreover, the ratio of both variances, \(\sigma_u^2/\sigma_v^2\), indicates a lower proportion of signal to noise after 2001.

To facilitate the interpretation of the estimated translog input coefficients output elasticities with respect to inputs are also reported. The estimated output elasticities are mostly significantly positive for the most years and, similar to other estimates, vary substantially over time. After 2001 the amount of material expenses and the number of non-medical employees appear to have a (theoretically implausible) negative effect on the number of weighted treated cases. However, these estimates are mostly insignificant. The exogenously fixed input variable *beds* obtains a negative estimate, indicating a positive effect of the number of beds on efficiency. The magnitude of the estimated impact is decreasing after 2001.^{Footnote 6}

The variation in the estimates over time might be explained by two reasons. Firstly, the definition of the costs has changed in 1996 and 2002. Secondly, the application of minimally invasive procedures has been expanded over the last years leading to shorter rehabilitation time and decreased hospital costs [50]. Both instances might have an effect on the stochastic noise in the hospital data and, thus, influence the parameter estimates.

### 1.2 E.2 DEA and SFA efficiency scores

Descriptive statistics of estimated DEA and SFA efficiency scores are shown in Table 7. For all subgroups mean SFA efficiency scores are markedly higher than their DEA counterparts. This could be seen as a reflection of the fact that in the DEA model all deviations from the frontier are assigned as inefficiency, while it is separated from stochastic noise in the SFA model. The difference between both efficiency measures varies over beds and ownership form. For instance, profit oriented hospitals are characterized by the highest DEA but lowest SFA efficiency scores. In total, 3% of the hospitals are DEA-efficient, i.e. \(\theta^D_i=1\). The share of these hospitals varies over the groups. Almost 11% of small (<100 beds) and privately owned hospitals are clarified as efficient by means of DEA, while in the other groups the share varies between 0.87% (100–299 beds) to 3.97% (>500 beds). Similar to the SFA parameter estimates the Spearman rank correlation between the alternative efficiency measures vary over time.^{Footnote 7} In the years 1995 and 1996, the correlation between both efficiency scores is negative. This might be explained by the poor quality of the data as suggested by the relative large variance of the noise component \(\hat{\sigma}^2_{v}\) (Table 6). Measurement errors are in general more detrimental to DEA, since it exploits the noise as inefficiency (e.g. [5, 48]). In the SFA model an assumption about the distribution of the inefficiency term is required to separate inefficiency from noise. However, the decomposition of the error term could fail under large measurement errors [48]. Therefore, in the presence of large amounts of noise both methods may perform poorly. This might explain the remarkable differences between the efficiency scores. After 1996 the variance of the noise component drops to one third of its previous magnitude and the correlation between DEA and SFA efficiency scores reaches almost 78% in 1997. It decreases over time to 40% in 2006. The decline in the correlation is in line with the higher proportion of the noise component after 2001 as indicated by the decreased ratio of estimated variances \(\hat{\sigma}^2_u/\hat{\sigma}^2_v\). Hospitals with more than 300 beds are characterized by the strongest similarity between both efficiency measures. In summary, the statistics document the heterogeneity between parametric and non-parametric efficiency measures as also documented in other comparative studies of hospital performance (e.g. [10, 16, 23, 24]).

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Herwartz, H., Strumann, C. On the effect of prospective payment on local hospital competition in Germany.
*Health Care Manag Sci* **15**, 48–62 (2012). https://doi.org/10.1007/s10729-011-9180-9

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DOI: https://doi.org/10.1007/s10729-011-9180-9

### Keywords

- Hospital efficiency
- Stochastic frontier analysis
- Data envelopment analysis
- Spatial analysis
- Diagnosis related groups