SpDEA
1 “Addressing spatial dependence when estimating technical efficiency: A spatialized data envelopment analysis of regional productive performance in the European Union”, Growth and Change, Volume 55, Issue 1, March 2024 (https://doi.org/10.1111/grow.12711)
1.1 Spatialized data envelopment analysis (SpDEA)
1.1.1 The standard (output-oriented) DEA method
The paper addresses the issue of the relative technical efficiency by means of the DEA approach that does not require the specification of the functional form of the production function being estimated. In order to measure the efficiency of regions, we define a set of p inputs \(\mathbf{x} = \left( x_{1},x_{2},\ldots,x_{p} \right) \in \mathbb{R}_{+}^{p}\) that are used to produce a vector of r outputs \(\mathbf{q} = \left( q_{1},q_{2},\ldots,q_{r} \right) \in \mathbb{R}_{+}^{r}\). Then, the technology set of all feasible input-output combinations \((\mathbf{x},\mathbf{q})\) can be defined as:
\[\psi = \left\{ (\mathbf{x},\mathbf{q}) \in \mathbb{R}_{+}^{p + r}|\ \mathbf{x}\ \text{can\ produce}\ \mathbf{q} \right\}\]
The (unconditional) output-oriented Farrell-Debreu technical efficiency DEA score, \({\widehat{\lambda}}_{DEA}\), of a DMU can be obtained by solving the following linear programming problem:
\[{\widehat{\lambda}}_{DEA}\left( \mathbf{x},\mathbf{q} \right) = sup\left\{ \lambda > 0\ |\ \left( \mathbf{x},\lambda\mathbf{q} \right) \in \widehat{\psi} \right\}\]
where
\(\widehat{\psi} = \left\{ (\mathbf{x},\mathbf{q}) \in \mathbb{R}_{+}^{p + r}\ |\ \mathbf{q} \leq \sum_{i = 1}^{n}{\gamma_{i}\mathbf{q}_{i}\ ,}\ \mathbf{x} \geq \sum_{i = 1}^{n}{\gamma_{i}\mathbf{x}_{i}}\ ,\sum_{i = 1}^{n}\gamma_{i} = 1\ ,\ \gamma_{i} \geq 0 \right\}\) is the attainable set estimated from an observed random sample of DMUs \(\left\{ (\mathbf{x}_{i},\mathbf{q}_{i})|\ i = 1,2,\ldots,n \right\}\), and \(\lambda\) is the efficiency parameter to be evaluated for the productive unit operating at level \((\mathbf{x},\mathbf{q})\). Using this definition, \({\widehat{\lambda}}_{DEA}(\mathbf{x},\mathbf{q}) = 1\) denotes an efficient production unit, while \({1/{\widehat{\lambda}}_{DEA}\left( \mathbf{x},\mathbf{q} \right)}\ < 1\) implies that the corresponding DMU is inefficient.
1.1.2 The influence of contextual variables and the conditional DEA estimator
Using the conditional frontier method of Daraio and Simar (2005, 2007), the conditional DEA (cDEA) output measure of technical efficiency can be obtained as:
\[{\widehat{\lambda}}_{cDEA}\left( \mathbf{x},\mathbf{q}|\mathbf{z} \right) = sup\left\{ \lambda > 0\ |\ {\widehat{S}}_{Q|X,Z}(\lambda\mathbf{q}|\mathbf{X} \leq \mathbf{x},\mathbf{Z} = \mathbf{z}) > 0 \right\}\]
where by construction \({{\widehat{\lambda}}_{DEA}\left( \mathbf{x},\mathbf{q} \right)}^{- 1} \leq {{\widehat{\lambda}}_{cDEA}\left( \mathbf{x},\mathbf{q}|\mathbf{z} \right)}^{- 1} \leq 1\) since for all \(\mathbf{z}\), \(\psi^{z} = \left\{ (\mathbf{x},\mathbf{q})|\mathbf{Z} = \mathbf{z},\ \mathbf{x}\ \text{can\ produce}\ \mathbf{q} \right\} \subseteq \psi\).
The estimated survival function \({\widehat{S}}\) is more difficult to evaluate than in the unconditional case (the support set \(\widehat{\psi}\) in the DEA method), because it requires the use of smoothing techniques for the external variables in\(\ \mathbf{z}\) (see Bădin et al. 2010):
\[{\widehat{S}}_{Q|X,Z}\left( \mathbf{q} \middle| \mathbf{X} \leq \mathbf{x},\mathbf{Z} = \mathbf{z} \right) = \frac{\sum_{i = 1}^{n}{\mathbb{I}\left( \mathbf{x}_{i} \leq \mathbf{x},\mathbf{q}_{i} \geq \mathbf{q} \right)K_{h}(\frac{\mathbf{z}_{i} - \mathbf{z}}{h})}}{\sum_{i = 1}^{n}{\mathbb{I}\left( \mathbf{x}_{i} \leq \mathbf{x} \right)K_{h}(\frac{\mathbf{z}_{i} - \mathbf{z}}{h})}}\]
where \(\mathbb{I}( \bullet )\) is the indicator function. Therefore, this approach relies on the selection of a product kernel function \({\ K}_{h}( \bullet )\) and an optimal bandwidth parameter \(h>0\) selected using any choice method. In the paper the data-driven selection approach developed by Bădin et al. (2010) is adopted.
1.1.3 Spatialized frontier models and the SpDEA approach
In the empirical application, we aim to construct a spatial frontier model for the EU28 regions. To achieve this, we start by formulating a regional production function \(Y_{it} = F(K_{it},L_{it})\), where \(Y_{it}\) represents the aggregate output in region i at year t, \(K_{it}\) denotes the respective stock of physical capital, and \(L_{it}\) is the employed labor force. As is usual in the productivity growth literature at the macro level, the constant returns-to-scale (CRS) assumption is used. Hence, in the production frontier, the output per worker ratio -labor productivity- (\(q_{it} = \frac{Y_{it}}{L_{it}}\)) is a function only of the capital per worker input -capital deepening- (\(x_{it} = \frac{K_{it}}{L_{it}}\)):
\[q_{it} = f(x_{it})\]
To introduce the spatial dimension in the DEA model, accounting for the spatial dependence observed in the regional data, a spatial lag of the output per worker ratio (a weighted average of the level of productivity in neighboring regions), \(z_{1i} = \sum_{j = 1}^{n}{w_{ij}q_{j}} = \mathbf{w}_{i}^{'}\mathbf{q}\), and a spatial lag of the capital-labor ratio (a weighted average of capital intensity in nearby regions), \(z_{2i} = \sum_{j = 1}^{n}{w_{ij}x_{j}} = \mathbf{w}_{i}^{'}\mathbf{x}\), are used as components of external conditioning factors of the vector \(\mathbf{z}\) in the DEA efficiency approach. The spatial weights, \(w_{ij}\), capture the spatial interaction between regions i and j (these elements are known a priori and satisfy the conditions \(w_{ij} \geq 0\), \(w_{ii} = 0\), and \(\mathbf{w}_{i}^{'}\mathbf{1 =}\sum_{j = 1}^{n}{w_{ij} = 1}\)), and the resulting n by n matrix W with elements \(w_{ij}\) describes the connectivity of the n regions.
1.2 Empirical application
1.2.1 Data
In this application we focus on analysing the year 2000 performance of 263 NUTS-2 regions in the 28 countries of the European Union (before 2020), excluding the overseas territories of Finland, France, Portugal and Spain.
1.2.2 Results
First, aspatial DEA scores are estimated and then, the SpDEA technical efficiency of an observed region \(\left( x_{it} = \frac{K_{it}}{L_{it}},q_{it} = \frac{Y_{it}}{L_{it}} \right)\) facing external conditions \(\left( z_{1i} = \mathbf{w}_{i}^{'}\mathbf{q},{z_{2i} = \mathbf{w}}_{i}^{'}\mathbf{x} \right)\) is calculated by:
\[{\widehat{\lambda}}_{SpDEA}\left( x_{i},q_{i}|z_{1i},z_{2i} \right) = sup\left\{ \lambda > 0\ |\ {\widehat{S}}_{Q|X,Z_{1}Z_{2}}(\lambda q_{i}|X \leq x_{i},Z_{1} = \mathbf{w}_{i}^{'}\mathbf{q},\ Z_{2} = \mathbf{w}_{i}^{'}\mathbf{x}) > 0 \right\}\]
The table below presents the descriptive statistics on DEA and SpDEA efficiency scores:
| DEA | SpDEA | |
| Mean | 0 .424 | 0.586 |
| Std. Dev. | 0 .150 | 0.223 |
| Min. | 0 .123 | 0.123 |
| Median | 0 .431 | 0.627 |
| Max. | 1 | 1 |
Pearson’s correlation (p-value) |
0 .823 (0. 000) |
|
Spearman’s rank correlation (p-value) |
0 .835 (0. 000) |