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Dynamic directional nonparametric profit efficiency analysis for a single decision-making unit: an aggregation approach

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Abstract

We propose a simple and intuitive nonparametric technique to assess the profit performances of a single decision-making unit over time. The particularity of our approach lies in recognizing that technological change may be present in the profit evaluation exercise. We partition the periods of time into several time intervals, in such a way that the technology is fixed within intervals but may differ between intervals. Attractively, our approach defines a new Luenberger-type indicator for dynamic profit performance evaluation when a single decision-making unit is of interest, and provides a coherent and systematic way to compare the profit performance changes between the periods of time and the time intervals. To define the interval-level concepts, we rely on a flexible weighting linear aggregation scheme. We also show how the new indicator can be decomposed into several dimensions. We illustrate the usefulness of our methodology with the case of the Chinese low-end hotel industry in 2005–2015. Our results highlight a performance regression, which is mainly due to the technical components of the indicator decomposition.

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Notes

  1. Note that solutions have been proposed when the prices are not observed, for example, reconstructing the prices using the input–output data (Färe and Zelenyuk 2003; Zelenyuk 2006) and relying on shadow prices (Cherchye et al. 2016; Walheer 2018b).

  2. Introducing the presence of heterogeneity in nonparametric efficiency analysis is not new (Battese et al. 2004; O’Donnell et al. 2008; Walheer 2018a). Previous works have considered the presence of heterogeneity between DMUs, while we present a methodology to capture heterogeneity in terms of technology between time intervals for a single DMU. The suggested methodology thus shares the willingness of incorporating technology heterogeneity in performance evaluation methods with these existing techniques. At this point, we highlight that it is important to justify the number of time intervals chosen. The common practice, previous works, important events, descriptive statistics, or statistical methods (e.g. cluster analysis) may help at this stage. See Sect. 3.1 for an illustration. Window analysis, which suggests a window width of three or four time periods because it tends to yield the best balance of informativeness and stability, may be used as an inspiration for selecting the number of time periods. Finally, one should keep in mind that differences between time intervals are not systematically due to technology changes. The method suggested in this paper can be used to test the correctness of such claims.

  3. Considering a netput representation instead of the more standard input–output representation allows us to considerably reduce our notation. Note that if we denote the inputs by \({\mathbf {x}}_t^n\) and the outputs by \({\mathbf {y}}_t^n\), and their respective price by \({\mathbf {p}}_{x,t}^n\) and \({\mathbf {p}}_{y,t}^n\), the netputs and their price are defined by: \({\mathbf {z}}_t^n = \left[ \begin{array}{c} {\mathbf {y}}_t^n \\ -{\mathbf {x}}_t^n \end{array} \right] \) and \({\mathbf {p}}^n_t = \left[ \begin{array}{c} {\mathbf {p}}_{y,t}^n \\ {\mathbf {p}}_{x,t}^n \end{array} \right] \).

  4. Note that very weak conditions are needed for profit efficiency evaluation (see, for example, Cherchye et al. 2016 for more discussion).

  5. Document ‘Guiding Opinions on Promoting Development of all-for-one tourism’ by the General Office of the State Council, March 2018.

  6. Let us illustrate the concept of netput using our application. Let the number of employees, the number of rooms, and total fixed assets be denoted by \(x_{1}\), \(x_{2}\), and \(x_{3}\), respectively; the total revenue by y. The netput vector is thus giving by \({\mathbf {z}} = \left[ \begin{array}{c} y \\ -x_{1}\\ -x_{2}\\ -x_{3} \end{array} \right] \).

  7. Formally, we obtain that \({\mathbf {g}}_{{\mathbf {z}}_t^{n}}={\mathbf {z}}_t^{n}\) for period t in interval n. Clearly, other exogenous and endogenous ways are possible to define the directional vectors (Hampf and Kruger 2014; Atkinson and Tsionas 2016; and Färe et al. 2017).

  8. Formally, the weights are defined, in our case, as follows: \(\omega ^n_t({\mathbf {z}}^n,{\mathbf {p}}^n,{\mathbf {g}}_{{\mathbf {z}}^n})=\frac{{\mathbf {p}}^{{n^\prime }}_t {\mathbf {z}}_t^n}{\sum _{t \in n} {\mathbf {p}}^{{n^\prime }}_t{\mathbf {z}}_t^n}.\)

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  1. HEC Liège, Université de Liège, Liège, Belgium

    Barnabé Walheer

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We are grateful to the Editor-in-Chief Rainer Kolisch, the Associated Editor, and the two referees for their insightful comments, which greatly improved this paper. We also thank Linjia Zhang for providing us the data.

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Walheer, B. Dynamic directional nonparametric profit efficiency analysis for a single decision-making unit: an aggregation approach. OR Spectrum 41, 1123–1149 (2019). https://doi.org/10.1007/s00291-019-00564-x

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