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Estimating SUR system with random coefficients: the unbalanced panel data case


A system of regression equations for analyzing panel data with random heterogeneity in intercepts and coefficients, and unbalanced panel data is considered. A maximum likelihood (ML) procedure for joint estimation of all parameters is described. Since its implementation for numerical computation is complicated, simplified procedures are presented. The simplifications essentially concern the estimation of the covariance matrices of the random coefficients. The application and ‘anatomy’ of the proposed algorithm for modified ML estimation are illustrated by using panel data for output, inputs and costs for 111 manufacturing firms observed up to 22 years.

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  1. 1.

    Subscripts denoting the calendar period may be attached. This may be convenient for data documentation and in formulating dynamic models, but will not be necessary for the static model considered here. For example, in a dataset with \(P\!=\!20\), from the years 1981–2000, some units in the \(p\!=\!18\) group may be observed in the years 1981–1998, some in 1982–1999, some in 1981–1990 and 1992–1999, etc.

  2. 2.

    For notational simplicity, the conditioning on \(({\varvec{X}}_{g(ip)},{\varvec{X}}_{h(ip)})\) is suppressed.

  3. 3.

    We here neglect possible cross-equational coefficient restrictions.

  4. 4.

    This estimator is positive definite and consistent if both \(p\) and \(N_p\) go to infinity. It is not, however, unbiased in finite samples. Modified estimators for similar balanced situations are considered in (Hsiao 2003, pp. 146–147).

  5. 5.

    Note that while \(\widehat{{\varvec{\Omega }}}_{(ip)}^{}\) is constructed from observations from all units, \(\widehat{{\varvec{\Sigma }}}_{}^{\,u}\) and \(\widehat{{\varvec{\Sigma }}}_{}^{\,\delta }\) are constructed from observations from units observed at least \(q\) times.

  6. 6.

    The data are from virtually the same source as that of (Biørn et al. 2002).

  7. 7.

    Technical change is, for simplicity, disregarded. The underlying total cost also includes energy cost, which is not modelled in the example. Neither capital cost nor capital input is represented in the model. Hence, we have no ‘full’ cost function represented.


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An earlier version of the paper was presented at the Sixteenth International Conference on Panel Data, Amsterdam, July 2010. I am grateful to Xuehui Han for excellent programming assistance and Terje Skjerpen and a referee for comments.

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Correspondence to Erik Biørn.



See Tables 7, 8, 9, and 10.

Table 7 Block \(p=22\) specific OLS estimates
Table 8 Distribution of firm-specific OLS estimates
Table 9 FGLS Estimates. Block specific results, \({\varvec{\beta }}_{(p)}^*\), obtained from Eqs. (33)–(34)
Table 10 Alternative FGLS estimates: Single equation, random intercept only

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Biørn, E. Estimating SUR system with random coefficients: the unbalanced panel data case. Empir Econ 47, 451–468 (2014).

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  • Panel data
  • Unbalanced data
  • Random coefficients
  • Heterogeneity
  • Regression systems
  • Iterated maximum likelihood

JEL Classification

  • C33
  • C51
  • C63
  • D24