Abstract
This paper considers the provision of some important municipal services and applies the non-parametric double bootstrap Simar and Wilson (J Econ 136(1):31–64, 2007) model based on a truncated-regression to estimate the effect of a group of relevant factors, which include the political sign of the governing party and the type of management, on robust DEA (Data Envelopment Analysis) estimates. Previous conditions, like separability, must hold for meaningful first- stage efficiency estimates and second-stage regression. After some confusion in the literature, Simar and Wilson (J Prod Anal 36(2):205–218, 2011b) clarify that their work of 2007 actually defines a statistical model where truncated (but not censored, i.e., Tobit, not Ordinary Least Square) regression yields a consistent estimation of model features. They demonstrate that conventional, likelihood-based approaches to inference are invalid, and they develop a bootstrap approach that yields valid inference in second stage regressions when these are appropriate. The results reveal a significant relation between efficiency and all the variables analysed and that municipalities governed by progressive parties are more efficient.
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Notes
This means that the rate of convergence decreases when the dimension of the attainable set increases. The rate of convergence of the FDH estimator is \(n^{{{1 \mathord{\left/ {\vphantom {1 {p + q}}} \right. \kern-0pt} {p + q}}}}\)whereas for the DEA with the additional assumption of convexity, the achieved rate is \(n^{{{2 \mathord{\left/ {\vphantom {2 {p + q}}} \right. \kern-0pt} {p + q}} + 1}}\). See Simar and Zelenyuk (2011).
See Bãdin et al. (2010b).
The probabilistic framework will be of great help later on this paper to analyze the Simar and Wilson (2007) separability assumption.
The asymptotics of the conditional order-m estimator was analyzed in Cazals et al. (2002) and recently in Jeong et al. (2010). As the conditional efficiency approach relies on the estimation of nonparametric kernel functions to select the appropriate reference partners, it relies on the choice of bandwidth parameters: although the estimates avoid the separability condition, their bandwidths rely on it.
In their simulated examples, they illustrate how the standard DEA is very sensitive to outliers. Moreover, once the size of the noise increases, the DEA estimator behaves badly.
See Sect. 1 in Wilson (1993, pp. 320–321). “S” is a set of n firms, S = (1,2,…,n); L ⊂ S contains i elements, i < n. \(R_{\hbox{min} }^{(i)}\)denote the observed minimum value of \(R_{L}^{(i)}\) for all \(\left( \begin{gathered} n \hfill \\ i \hfill \\ \end{gathered} \right)\) possible subsets L of size i. The numerator is the Andrews and Pregibon (1978) statistic that Wilson (1993) extends to the case for more than one. The denominator expresses the minimum value for a certain number of subsets of size i observations. As Wilson (1993) indicates for large data sets, Andrews and Pregibon (1978) suggest a graphical representation of the log ratios. Examination of the separation between the smallest ratios indicates possible outliers.
There are 470 articles from 2010 to May 2012, of which 62 correspond to 2012.
The DEA estimators to be computed in our case will have variable returns to scale and an input orientation that seems to be the most appropriate given that there is a demand minimization policy. The BCC model appeared first in Banker et al. (1984). Scales of returns to scale were also tested but rejected in favour of the VRS. For more details on the returns to scale test, refer to Simar and Wilson (2002). As regards traditional efficiency estimators we refer to section 2 in Wheelock and Wilson (2008). Since the command DEA of FEAR 1.15 computes estimates by Shephard (1970) distance function, efficiency has been measured in terms of Shephard’s input distance function, δi.
Daraio et al. (2010, p. 4): “In the empirical literature, researchers have typically assumed ψ(z i ,β) = z i ·β, where β is a vector of parameters. In addition to Assumptions 2.1–2.3, Simar and Wilson ( 2007 ) assume the error, ε, is distributed (truncated) normal in order to reflect the empirical literature. Alternatively, ψ(z i ) and the distribution of ε can be assumed to be nonparametric; see Park et al. ( 2008 ) for details”.
For the steps and details refer to Simar and Wilson (2007). To this purpose, FEAR 1.15 (March 2010) includes DEA, boot.sw98, rnorm.trunc and treg among its routines.
The statistic \(\hat{T}_{5n}\)results in significant p value of 0.0316, leading to rejection of the null hypothesis of independence.
To understand the importance of the separability condition see Simar and Wilson (2011b).
Only observations of the conditional DEA technology set are used for the construction of the DEA frontier for DMUi that lie within the chosen bandwidth, h, around zi. Efficiency scores for each observation can then be derived from this technology set. For a generalized approach of conditional efficiency measurement see Bãdin et al. (2010a). For the DEA conditional estimator, provided the elements of Z are continuous, the bandwidth parameter h can be optimized using the least-squares cross-validation (LSCV) technique discussed by Bãdin et al. (2010b). The appendix provides a Matlab routine that computes the bandwidth based on the LSCV criterion for the output oriented version. Following Daraio et al. (2010, p. 9): “if Z contains qualitative variables, then the sample observations must first be divided into groups defined by the qualitative variables”. Asymptotic results for DEA conditional estimators are given by Jeong et al. (2010).
Efficiency is measured in terms of Shephard’s input distance function. Measure is hence one or larger.
See Simar and Wilson (2000, p. 790). The authors advise that bias-correction should only be used when the ratio is well above unity, something which we will consider.
For example, the municipality Vera, which is efficient when considering the estimation based on the initial distance function, has a corrected coefficient of 1.128, indicating that to obtain the same level of output it could reduce its input by about 10%. Specifically, the confidence interval at 95% for this unit indicates that it could reduce its inputs by between 3 and 27%.
We follow Simar and Wilson (2007, p. 44): the choice of the number of replications L1 in Algorithm #2 determines the number of bootstrap replications used to compute the bias-corrected estimates, \(\hat{\hat{\delta }}_{i} .\) They found that 100 replications are sufficient for this purpose. For L2 they use 2000 replications. The values of the variables are scaled appropriately; signs of estimated coefficients can be interpreted in a Farrell–Debreu direction.
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Benito, B., Solana, J. & Moreno, MR. Explaining efficiency in municipal services providers. J Prod Anal 42, 225–239 (2014). https://doi.org/10.1007/s11123-013-0358-7
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DOI: https://doi.org/10.1007/s11123-013-0358-7