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Estimation of cost inefficiency in panel data models with firm specific and sub-company specific effects

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Abstract

This paper proposes a dual-level inefficiency model for analysing datasets with a sub-company structure, which permits firm inefficiency to be decomposed into two parts: a component that varies across different sub-companies within a firm (internal inefficiency); and a persistent component that applies across all sub-companies in the same firm (external inefficiency). We adapt the models developed by Kumbhakar and Hjalmarsson (J Appl Econom 10:33–47, 1995) and Kumbhakar and Heshmati (Am J Agric Econ 77:660–674, 1995), making the same distinction between persistent and residual inefficiency, but in our case across sub-companies comprising a firm, rather than over time. The proposed model is important in a regulatory context, where datasets with a sub-company structure are commonplace, and regulators are interested in identifying and eliminating both persistent and sub-company varying inefficiency. Further, as regulators often have to work with small cross-sections, the utilisation of sub-company data can be seen as an additional means of expanding cross-sectional datasets for efficiency estimation. Using an international dataset of rail infrastructure managers we demonstrate the possibility of separating firm inefficiency into its persistent and sub-company varying components. The empirical illustration highlights the danger that failure to allow for the dual-level nature of inefficiency may cause overall firm inefficiency to be underestimated.

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Notes

  1. Even under incentive-based (RPI-X) regulation, regulators are interested in the sources of inefficiency in order to assess the deliverability of savings (without compromising safety and quality), and to monitor progress. Understanding the split between internal and external inefficiency thus provides important information for the regulator.

  2. At least in terms of efficiency levels. Some regulators have compared trends in efficiency/productivity between different industries however.

  3. As widely noted in the literature, the model can easily be translated into a production function by reversing the sign on \( {\text{u}}_{\text{its}} \).

  4. We use the terms persistent and residual inefficiency as in Kumbhakar and Hjalmarsson (1995).

  5. Since the sub-company varying component is an absolute measure of inefficiency, the efficiency scores for each sub-company unit are measured relative to a theoretical frontier and for a given sample it will not necessarily be the case that one sub-company within each firm will be on the frontier.

  6. Note that by effects by firm per time period we do not mean that this has two way effects in firm and time. Instead we mean there is one set of effects, with one effect for each year and firm. This is very general. We could replace this with an assumption that the persistent inefficiency of sub-companies in a firm is also time invariant, in which case \( \alpha_{\text{it}} = \alpha_{\text{i}} = \alpha + \mu_{\text{i}} \). This is the assumption we use in our empirical example. A further assumption could be that \( \alpha_{\text{it}} = \alpha_{{{\text{i}}1}} + \alpha_{{{\text{i}}2}} {\text{t}} + \alpha_{{{\text{i}}3}} {\text{t}}^{2} \), that is that the persistent inefficiency follows a Cornwell et al. (1990) type variation over time.

  7. Conditional on the (consistent) estimates in the first stage.

  8. Note that we reverse the sign on \( {{\varpi_{\text{its}} \lambda } \mathord{\left/ {\vphantom {{\varpi_{\text{its}} \lambda } \sigma }} \right. \kern-\nulldelimiterspace} \sigma } \) vis-à-vis Kumbhakar and Heshmati (1995) since we are estimating a cost frontier.

  9. Provided in the GLS case the regressors and \( \mu_{\text{it}} \) are uncorrelated as discussed earlier.

  10. See Smith et al. (2008) and ORR (2008) for details of the work undertaken. Note that the railway companies considered are slightly different in the analysis for this paper than in the Periodic Review analysis.

  11. Note ProElect is not normalised to the sample mean.

  12. We note that while the terminology “pooled model” accurately describes the pooled nature of the data over sub-companies, it should be noted that time invariance is assumed. As such the model is actually an analogue to the time invariant model first proposed by Pitt and Lee (1981).

  13. See Caves et al. (1981, 1984) for use of the terms returns to scale (RTS) and returns to density (RTD) in empirical applications.

  14. In interpreting these results it should be noted that the final two studies in Table 3 utilise firm-level data, whilst the other studies utilise sub-company data of varying levels of disaggregation.

  15. Note that we find the degree of RTS to increase with track length, which if interpreted literally and simply extrapolated, would imply a single region within each company. Of course we are more confident in the findings of our model at the sample mean than at the extremes of the sample (or even out of sample).

  16. Owing to lack of data, this is an estimate based solely on Network Rail data.

  17. As noted in Sects. 3 and 5.1, given the unbalanced nature of the observations over time and the generally small number of time periods for most IMs, both the firm-specific and sub-company inefficiency components are time invariant in our model.

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Acknowledgments

This work was funded partly by the British Office of Rail Regulation and partly by a part-time PhD scholarship provided by the UK Engineering and Physical Sciences Research Council. We also gratefully acknowledge the contributions of the individual infrastructure managers who provided data and commented on this work, as well as comments on the analysis and assistance with data collection from the British Office of Rail Regulation. Finally, we acknowledge the comments of two anonymous referees. All remaining errors are the responsibility of the authors.

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  1. Institute for Transport Studies and Leeds University Business School, University of Leeds, Leeds, LS2 9JT, UK

    Andrew S. J. Smith

  2. Institute for Transport Studies, University of Leeds, Leeds, UK

    Phill Wheat

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Correspondence to Andrew S. J. Smith.

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Smith, A.S.J., Wheat, P. Estimation of cost inefficiency in panel data models with firm specific and sub-company specific effects. J Prod Anal 37, 27–40 (2012). https://doi.org/10.1007/s11123-011-0220-8

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