Abstract
This paper proposes a new two-step stochastic frontier approach to estimate technical efficiency (TE) scores for firms in different groups adopting distinct technologies. Analogous to Battese et al. (J Prod Anal 21:91–103, 2004), the metafrontier production function allows for calculating comparable TE measures, which can be decomposed into group specific TE measures and technology gap ratios. The proposed approach differs from Battese et al. (J Prod Anal 21:91–103, 2004) and O’Donnell et al. (Empir Econ 34:231–255, 2008) mainly in the second step, where a stochastic frontier analysis model is formulated and applied to obtain the estimates of the metafrontier, instead of relying on programming techniques. The so-derived estimators have the desirable statistical properties and enable the statistical inferences to be drawn. While the within-group variation in firms’ technical efficiencies is frequently assumed to be associated with firm-specific exogenous variables, the between-group variation in technology gaps can be specified as a function of some exogenous variables to take account of group-specific environmental differences. Two empirical applications are illustrated and the results appear to support the use of our model.
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Notes
See Huang and Liu (1994), Battese and Coelli (1995), Wang (2002), Huang (2005), and Lai and Huang (2010) for the specification, interpretation, and testing of various efficiency models related to the truncated-normal specification \(N^{ + } \left( {\mu^{j} (Z_{jit} ), \, \sigma_{u}^{j2} (Z_{jit} )} \right)\).
In the case that the group-specific frontiers are known, Schmidt (1976) has shown that the optimization programming problems in (12) and (13) correspond to the maximum likelihood estimates if the \(U_{jit}^{M}\) follows an exponential distribution and a half-normal distribution, respectively. Even in this case, however, the statistical properties of the maximum likelihood estimators cannot be obtained since the regularity condition for the maximum likelihood method is violated.
The randomness of the technology gap component \(U_{jit}^{M} \ge 0\) is justified on the assumption of the existence of a population distribution of an array of (possibly continuous) group frontiers \(f_{t}^{j} \left( {X_{jit} } \right)\). Given \(X_{jit}\), the metafrontier is defined as the upper boundary of the support of the distribution of \(f_{t}^{j} \left( {X_{jit} } \right)\), i.e., \(f_{t}^{M} \left( {X_{jit} } \right) = \sup \left( {f_{t}^{j} \left( {X_{jit} } \right) \, \begin{array}{*{20}c} | \\ | \\ \end{array} \, G\left( {f_{t}^{j} \left( {X_{jit} } \right)} \right) < 1} \right)\) where \(G\left( {f_{t}^{j} \left( {X_{jit} } \right)} \right)\) is the distribution function of \(f_{t}^{j} \left( {X_{jit} } \right)\). Thus, in a random sample of J groups, the metafrontier \(f_{t}^{M} \left( {X_{jit} } \right)\) is the Jth order statistic. The non-negative technology gap component \(U_{jit}^{M} = \ln f_{t}^{M} \left( {X_{jit} } \right) - \ln f_{t}^{j} \left( {X_{jit} } \right) \ge 0\) is random and is assumed to be distributed as a truncated-normal. This definition of the metafrontier differs from the standard metafrontier literature where the group frontiers \(f_{t}^{j} \left( {X_{jit} } \right)\), j = 1, 2,…, J are nonstochastic with \(f_{t}^{M} \left( {X_{jit} } \right) = \hbox{max} \left( {f_{t}^{j} \left( {X_{jit} } \right) \, \begin{array}{*{20}c} | \\ | \\ \end{array} \, X_{jit} ,\quad j = 1,2, \ldots ,J} \right)\). We gratefully appreciate a referee for the constructive critiques on the stochastic nature of the randomness of the technology gap that has clarified and greatly improved the argument and exposition from the early version.
We thank an anonymous referee for making this observation.
Let \(\ln \left( \theta \right)\)be the log-likelihood function of the parameter \(\theta\). The standard ML estimator has the inverse of the Fisher information matrix \(I\left( \theta \right) = - E\left( {\frac{{\partial^{2} \ln \left( \theta \right)}}{{\partial \theta \, \partial \theta^{T} }}} \right)\)as the covariance matrix of the estimator \(\hat{\theta }\). However, the QML's covariance matrix has the so-called sandwich form: \(I^{ - 1} \left( \theta \right)\left[ {S\left( \theta \right)S^{T} \left( \theta \right)} \right]I^{ - 1} \left( \theta \right)\)where \(S\left( \theta \right) = E\left( {\frac{\partial \ln \left( \theta \right)}{\partial \theta }} \right)\) is the score function. Johnston and DiNardo (1992), pages 428-430, has a brief discussion of the quasi-maximum likelihood estimation of misspecified models and the derivation of the covariance matrix.
For example, the conditional expectations of (7) is \(E\left( {\ln Y_{jit} |X_{jit} } \right) = \ln f_{t}^{j} \left( {X_{jit} } \right) - E\left( {U_{jit} } \right)\), while from (18) is \(E\left( {\ln Y_{jit} |X_{jit} } \right) = \ln f_{t}^{M} \left( {X_{jit} } \right) - E\left( {U_{jit}^{*} } \right)\). These dual conditional expectations are inconsistent unless \(\ln f_{t}^{j} \left( {X_{jit} } \right) = \ln f_{t}^{M} \left( {X_{jit} } \right) - E\left( {U_{jit} } \right) - E\left( {U_{jit}^{*} } \right)\), which is unlikely from the DGP of Battese and Rao (2002) specification. On the other hand, the current proposed SMF model has a unique DGP derived either from (7) or from \(\ln Y_{jit} = \ln f_{t}^{M} \left( {X_{jit} } \right) + V_{jit} - U_{jit} - U_{jit}^{M}\) after substituting (11) into (7). This latter expression has the conditional expectation \(E\left( {\ln Y_{jit} |X_{jit} } \right) = \ln f_{t}^{M} \left( {X_{jit} } \right) - E\left( {U_{jit} } \right) - E\left( {U_{jit}^{M} } \right)\), which is identical to the one obtained from (7) because of (11), \(\ln f_{t}^{j} \left( {X_{jit} } \right) = \ln f_{t}^{M} \left( {X_{jit} } \right) - E\left( {U_{jit}^{M} } \right)\).
To see the decomposition from the proposed two-step regressions in (7) and (15), we observe that, from (6), \(MTE_{jit} \equiv \frac{{Y_{jit} e^{{ - V_{jit} }} }}{{f_{t}^{M} }} = \frac{{f_{t}^{j} }}{{f_{t}^{M} }} \times e^{{ - U_{jit} }}\). Since, from (14), \(\hat{f}_{t}^{j} = f_{t}^{j} \times e^{{V_{jit}^{M} }}\), we have \(MTE_{jit} = \frac{{\hat{f}_{t}^{j} }}{{f_{t}^{M} e^{{V_{jit}^{M} }} }} \times e^{{ - U_{jit} }}\). Thus, from (15), we have the exact decomposition: \(MTE_{jit} = e^{{ - U_{jit}^{M} }} \times e^{{ - U_{jit} }} = TGR_{jit} \times TE_{jit}\).
We are grateful to Professor C.J. O’Donnell for providing the raw data. For a more detailed description of the input/output variables in our empirical application, please see O’Donnell et al. (2008).
Note that the QML sandwich estimated standard errors are calculated and presented in Tables 1 (for FAO data) and 5 (for hotel data). The unadjusted estimated standard errors without taking into account heteroscedasticity are not shown to save space. For FAO data, most of the unadjusted standard errors are slightly greater than those of QML sandwich estimates and the number of significant parameter estimates, at least at the 10 % level, are the same. However, for hotel data, the reverse is true, i.e., most of the unadjusted standard errors are somewhat less than those of the QML sandwich estimates. Consequently, there are fewer coefficients reaching statistical significance in Table 5.
In standard stochastic frontier analysis, it is the variance ratio \({{\gamma^{M} = \sigma_{u}^{M2} } \mathord{\left/ {\vphantom {{\gamma^{M} = \sigma_{u}^{M2} } {\left( {\sigma_{v}^{M2} + \sigma_{u}^{M2} } \right)}}} \right. \kern-0pt} {\left( {\sigma_{v}^{M2} + \sigma_{u}^{M2} } \right)}}\) that tests the hypothesis of \(U_{jit}^{M} = 0\), i.e., the average verse frontier model. On the other hand, the complement of the variance ratio, \({{\bar{\gamma }^{M} = 1 - \gamma^{M} = \sigma_{v}^{M2} } \mathord{\left/ {\vphantom {{\bar{\gamma }^{M} = 1 - \gamma^{M} = \sigma_{v}^{M2} } {\left( {\sigma_{v}^{M2} + \sigma_{u}^{M2} } \right)}}} \right. \kern-0pt} {\left( {\sigma_{v}^{M2} + \sigma_{u}^{M2} } \right)}}\), allows us to test the hypothesis of \(V_{jit}^{M} = 0\), i.e., which is the deterministic verse the stochastic frontier model.
An environmental variable that has a positive (negative) coefficient implies that the variable exerts a negative (positive) impact on technical efficiency.
Because the results from the LP model are almost identical to those from the QP model, we report only the results from the QP model to save space.
We perform a t test for the hypothesis that the average value of the TGR of the chain-operated hotels is the same as that of the independently operated hotels. The p value of the test statistic is equal to 0.0564, which is significant at the 10 % level. Empirically this might indicate that the average TGR of the former type of hotels is greater than that of the latter type of hotels.
By construction, the LP and QP estimates always result in a few TGR scores to be 1 as shown in the Table 6. However, as argued, the LP and QP estimates are likely on average to be smaller than the MTE estimates.
We do not show the estimation results from the models without considering industry-wide environment variables to save space. The results are available upon request from the authors.
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Huang, C.J., Huang, TH. & Liu, NH. A new approach to estimating the metafrontier production function based on a stochastic frontier framework. J Prod Anal 42, 241–254 (2014). https://doi.org/10.1007/s11123-014-0402-2
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DOI: https://doi.org/10.1007/s11123-014-0402-2