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Heterogeneous spillovers among Spanish provinces: a generalized spatial stochastic frontier model

Abstract

This paper introduces new spatial stochastic frontier models to examine Spanish provinces’ efficiency and its evolution over the period 2000–2013. We use a heteroscedastic version of the spatial stochastic frontier models introduced by Glass et al. (J Econ 190(2):289–300, 2016) that, in addition, allows us to identify the determinants of the spatial dependence among provinces. We contribute to the heterogeneous spatial models that have been introduced in recent years, such as Aquaro et al. (Working Paper No. 15-17. USC Dornsife Institute for New Economic Thinking, 2015) and Malikov and Sun (J Econ 199(1):13–34, 2017), allowing measures of spatial dependence specific to each observation. This feature of the model lets us rank all Spanish provinces in accordance with their degree of spatial dependence, information that will aid policymakers to better allocate public resources between provinces. The period examined is of special interest given that it coincides with a break in the economic growth tendency, which leads to a deterioration in Spain´s economic situation.

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Notes

  1. 1.

    This literature has been enriched with the articles introducing Bayesian and Markov Chain Monte Carlo (MCMC) mixture estimation methods (Cornwall 2016; Cornwall and Parent 2017; LeSage and Chih (forthcoming)).

  2. 2.

    See also the semiparametric models formerly introduced by Su (2012), Zhang (2013) and Sun et al. (2014).

  3. 3.

    Traditional growth-accounting exercises decompose economic growth into contributions due to factor accumulation and technological progress. Under this approach technical change and total factor productivity (TFP) growth are often used as synonymous because it is often assumed that all regions operate efficiently. This precludes the existence of catching-up effects among regions/countries. Färe et al. (1994) and Kumar and Russell (2002) among other studies used a production frontier approach to capture catching-up effects. In this sense, several measures of TFP growth are often decomposed into three basic sources: technical change, scale effects, and technical efficiency change. Applications using Spanish data are, for instance, Maudos et al. (2000) and Badunenko and Romero-Ávila (2014). In addition, the latter considers sectorial interactions, showing that the aggregated productivity changes are driven by intra-sectorial productivity dynamics.

  4. 4.

    There are other functional forms that could have been used to impose the lower and upper bounds for the spatial coefficients. For instance, we could have used the simple cumulative density function of a standard normal variable (probit model) which is quite popular in other research fields where the variables to be predicted are probabilities, or limited dependent variables are somehow involved in the model. Like the logistic function, it lies between zero and one. As the density function associated with the logit function is very close to a standard normal distribution, we have found in other papers (see e.g. Orea et al. 2015) that the results of the probit specification are in practice very similar to those obtained with the logistic specification.

  5. 5.

    There is not empirical evidence on this issue yet. Just anecdotal evidence, such as that provided by Sun (2016) who have used nonparametric techniques to estimate a functional-coefficient spatial autoregressive model. The spatial function estimated in this paper (see his Fig. 2) is non-positive everywhere and gradually vanishes as the contextual factor increases. Like in a logistic specification, this implies a positive but decreasing marginal effect of the contextual factor on the spatial function.

  6. 6.

    Other papers construct the weights so that they reflect the commercial relationships among regions (e.g. Álvarez et al. 2003; and Cohen and Morrison 2004). The idea behind this approach can be incorporated into our specification by adding, for instance, the (average) freight traffic with neighbouring regions divided by the freight traffic within the region as a new determinant of the spatial spillovers between regions.

  7. 7.

    In turn, the λ(ni)(Wy)it term can be viewed as a weighted average of two spatial lags of the dependent variable. Indeed, if the autoregressive parameter were a linear function, the spatial lag term λ(ni)(Wy)it can be rewritten as:

    \(\lambda \left( {n_i} \right)\left( {Wy} \right)_{it} = \lambda \left[ {\tau _0{\mathrm{\Sigma }}W_iy_{it} + \tau _1{\mathrm{\Sigma }}\left( {n_iW_i} \right)y_{it}} \right]\)

    where λ = λ01, and the relative magnitude τj = λj/λ of each individual coefficient in λ(ni) allows us to tune the relative importance of each definition of the spatial weight matrix.

  8. 8.

    In addition, this empirical strategy helps obtaining parameter estimates because we have found convergence problems when we allowed for input-specific multipliers (i.e. mkit ≠ mhit).

  9. 9.

    The defining feature of models with the scaling property is that provinces differ in their mean efficiencies, but not in the shape of the distribution of inefficiency. More details on this specification that satisfies the so-called scaling property can be found in Wang and Schmidt (2002); Álvarez et al. (2006); and Parmeter and Kumbhakar, (2014).

  10. 10.

    The multiple-step pseudo maximum likelihood (PML) estimation procedure used in Glass et al. (2016) cannot be used in our application as our inefficiency term depends on a set of covariates. Multi-step estimation has some practical appeals as its facilitates model convergence and it is consistent, but it is statistically less efficient and the error components must be homoscedastic.

  11. 11.

    See also Anselin (1988) and Elhorst (2009).

  12. 12.

    See also Chamberlain (1980). An alternative is to estimate the models with a set of provincial dummies in the same fashion as the True Fixed Effect frontier model (TFE) introduced by Greene (2005). This empirical strategy yields however convergence problems in our iterative procedures to maximize the likelihood function.

  13. 13.

    Note that the globally normalized binary matrices is just a scaled version of the simple (i.e. non-normalized) binary matrix. So, both models only differ in the intercept of the degree of spatial dependence.

  14. 14.

    For a recent review on the current status of this topic, see Groot et al. (2016).

  15. 15.

    Similar figures have been found using the pre- and post-crisis periods.

  16. 16.

    We have examined this issue using the following decomposition of the spatial-based technological differences: \(S_{it}^ \ast = \lambda _{it}^ \ast \cdot \left( {Wy} \right)_{it} + \overline \lambda \cdot \left( {Wy} \right)_{it}^ \ast\) where Sit = λit(Wy)it, \(S_{it}^ \ast = S_{it} - \overline \lambda \cdot \overline {Wy}\), \(\lambda _{it}^ \ast = \lambda _{it} - \overline \lambda\), and \(\left( {Wy} \right)_{it}^ \ast = \left( {Wy} \right)_{it} - \overline {Wy}\). Thus, while the first term captures deviations due to differences among provinces in the autoregressive parameter (given their neighbors’ size), the second term captures deviations in neighbors’ size.

  17. 17.

    The efficiency scores REit are calculated as the aggregation of a direct efficiency score and an indirect efficiency measure that captures imported efficiency spillovers. We do not provide an alternative relative total efficiency measure provided by Glass et al. (2016) that includes efficiency spillovers from each province to the rest of provinces because the results using simple and generalized models were quite similar. This similarity is due to the fact that the average autoregressive parameters of neighbouring provinces is akin in both models.

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Acknowledgements

This research was partially funded by the Government of the Principality of Asturias and the European Regional Development Fund (ERDF). The authors also thank the support of the Oviedo Efficiency Group, and. the anonymous referees for their helpful comments and suggestions.

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Correspondence to Luis Orea.

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Appendix

Appendix

Histograms of the estimated global and local spatial spillovers (GSARF and GSDF models)

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Gude, A., Álvarez, I. & Orea, L. Heterogeneous spillovers among Spanish provinces: a generalized spatial stochastic frontier model. J Prod Anal 50, 155–173 (2018). https://doi.org/10.1007/s11123-018-0540-z

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Keywords

  • Heterogeneous spatial spillovers
  • Spatial stochastic frontier models
  • Spanish provinces