Abstract
A Bayesian estimator is proposed for a stochastic frontier model with errors in variables. The model assumes a truncated-normal distribution for the inefficiency and accommodates exogenous determinants of inefficiency. An empirical example of Tobin’s Q investment model is provided, in which the Q variable is known to suffer from measurement error. Results show that correcting for measurement error in the Q variable has an important effect on the estimation results.
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Notes
Except for u i and the related terms, the distribution assumptions adopted in the model are similar to those discussed in Chapter 9 of Carroll et al. (2006).
We tested the specification with an intercept in (48) thus allowing for systematic measurement errors. The estimated coefficient is small and insignificant and therefore the intercept was dropped subsequently. The independence assumption between the Q and the cash flow and asset variables may be a problem. Given that our estimation results, as shown in Tables 1 and 2, are consistent with findings in the literature, we suspect that the violation of the independence assumption may not of great concern. Further studies are needed to clarify the issue.
One may have concern about the feasibility of obtaining reliable estimates of the variances of three error components. To investigate, we calculate the correlation coefficients from the posterior distributions. The results are: corr\((\sigma_v^2,\sigma_\epsilon^2)=-0.635,\) corr(σ 2 v , σ 2 u ) = − 0.051, and corr\((\sigma_u^2, \sigma_\epsilon^2)=-0.052, \)where corr\((\cdot)\) is the correlation coefficient. In addition, we also have corr(β0, δ0) = 0.005. It appears that most of the correlations are very minor. Although the correlation between σ 2 v and \(\sigma_\epsilon^2\) is moderately high, results in Tables 1 and 2 indicate that the Bayesian estimates are consistent with the investment theory and with those from another bias-corrected GMM estimator.
It may seem somewhat surprising that the cash flow variable is not an important determinant of financing constraints. We found that in the 2000–2006 sample period the mean cash flow to asset ratio was 0.249, which is significantly larger than the mean ratio of 0.127 in the 1988–1996 sample period studied by Wang (2003). Therefore, one possible explanation is that the increased availability of cash flow, though itself was insufficient to finance the entire investment, was enough so that cash flow is rendered useless in discriminating between firms with different degrees of financing constraints.
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Acknowledgments
We thank **n-Yu Ho and Luke Lin for excellent research assistance and two anonymous referees for very helpful comments. Financial support from National Science Council is gratefully acknowledged (NSC 95-2415-H-001-003).
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Chang, SK., Chen, YY. & Wang, HJ. A Bayesian estimator for stochastic frontier models with errors in variables. J Prod Anal 38, 1–9 (2012). https://doi.org/10.1007/s11123-011-0242-2
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DOI: https://doi.org/10.1007/s11123-011-0242-2