Abstract
We show how a wide range of stochastic frontier models can be estimated relatively easily using variational Bayes. We derive approximate posterior distributions and point estimates for parameters and inefficiency effects for (a) time invariant models with several alternative inefficiency distributions, (b) models with time varying effects, (c) models incorporating environmental effects, and (d) models with more flexible forms for the regression function and error terms. Despite the abundance of stochastic frontier models, there have been few attempts to test the various models against each other, probably due to the difficulty of performing such tests. One advantage of the variational Bayes approximation is that it facilitates the computation of marginal likelihoods that can be used to compare models. We apply this idea to test stochastic frontier models with different inefficiency distributions. Estimation and testing is illustrated using three examples.
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Notes
Given a sample of MCMC draws θ1,θ2,…,θK, it is also possible to estimate an upper bound for the log of the marginal likelihood using \(\ln \overline {ML} = \frac{1}{K}\sum\limits_{k = 1}^k {\ln } \left( {L(y\vert {\theta ^k})p({\theta ^k})/q({\theta ^k})} \right)\). This may not be a tight bound, however and is calculated by simulation which may be prone to simulation errors (Ji et al. 2010).
See Fruhwirth-Schnatter (2006) for a review of alternative methods.
The same general conclusions can be drawn from the other data sets. Comparable tables and figures are available from the authors on request.
The DIC (deviance information criterion) is a goodness-of-fit with-penalty measure similar in nature to the Akaike and Bayesian information criteria. Because it uses averages rather than maxima as estimates, it is readily computed from MCMC output.
It is possible to consider other distributions such as truncated normal. We consider a lognormal distribution because it is more convenient for obtaining conditional distributions for the parameters μ that appear in the inefficiency distribution.
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Appendices
Appendix A: Models with other inefficiency distributions
Half Normal Inefficiency
Optimal Densities
Coordinate ascent algorithm
Iterate the following quantities until the change in \({\mathrm{ln}}\underline {ML}\) is negligible
where αi = μi/υ and m(·) = ?(·)/Φ(·).
Lower bound for marginal likelihood
Truncated-Normal Inefficiency
Optimal densities
Coordinate ascent algorithm
Iterate the following quantities until the change in \(\ln \underline {ML}\) is negligible.
where \(q^ \ast \left( \mu \right) = q\left( \mu \right)/C\), αi = μi/υ and m(·) = ?(·)/Φ(·).
Lower bound for marginal likelihood
Lognormal inefficiency
Model \(y_{it} = {\mathbf{x}}_{it}{\mathbf{\beta }} \mp u_i{\mathbf{ + }}v_{it}\)
Priors \({\mathbf{\beta }} \sim N\left( {\underline {\mathbf{\beta }} ,\underline {\mathbf{V}} _{\mathbf{\beta }}} \right)\quad \sigma ^{ - 2} \sim G\left( {\underline A _\sigma ,\underline B _\sigma } \right)\quad \mu \sim N\left( {\underline \mu ,\underline V {\kern 1pt} _\mu } \right)\quad \lambda ^{ - 2} \sim G\left( {\underline A _\lambda ,\underline B _\lambda } \right)\)
Optimal densities
Coordinate ascent algorithm
Iterate the following quantities until the change in \(\ln \underline {ML}\) is negligible.
Lower bound for marginal likelihood
Appendix B: Model with zero-one environmental variables
Here we return to model Eq. (2.1), \(y_{it} = {\mathbf{x}}_{it}{\mathbf{\beta }} \mp u_i{\mathbf{ + }}v_{it}\) as suggested by Koop et al. (1997), we can derive the following optimal densities:
Interestingly, all these densities are standard forms; we therefore can easily set-up a coordinate ascent algorithm by iterating over
where \(\alpha _i = \overline \mu _i{\mathrm{/}}\upsilon\). The marginal likelihood lower bound used to assess convergence is
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Hajargasht, G., Griffiths, W.E. Estimation and testing of stochastic frontier models using variational Bayes. J Prod Anal 50, 1–24 (2018). https://doi.org/10.1007/s11123-018-0531-0
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DOI: https://doi.org/10.1007/s11123-018-0531-0
Keywords
- Technical efficiency
- Marginal likelihood
- Time-varying panel
- Environmental effects
- Mixture
- Semiparametric model