Panel Data, Factor Models, and the Solow Residual



In this paper we discuss the Solow residual (Solow, Rev. Econ. Stat. 39:312–320, 1957) and how it has been interpreted and measured in the neoclassical production literature and in the complementary literature on productive efficiency. We point out why panel data are needed to measure productive efficiency and innovation and thus link the two strands of literatures. We provide a discussion on the various estimators used in the two literatures, focusing on one class of estimators in particular, the factor model. We evaluate in finite samples the performance of a particular factor model, the model of Kneip, Sickles, and Song (A New Panel Data Treatment for Heterogeneity in Time Trends, Econometric Theory, 2011), in identifying productive efficiencies. We also point out that the measurement of the two main sources of productivity growth, technical change and technical efficiency change, may be not be feasible in many empirical settings and that alternative survey based approaches offer advantages that have yet to be exploited in the productivity accounting literature.


Technical Efficiency Productivity Growth Total Factor Productivity Technical Change Stochastic Frontier 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This paper is based in part on keynote lectures given by Sickles at the Preconference Workshop of the 2008 Asia-Pacific Productivity Conference, July 17–19, Department of Economics, National Taiwan University, Taipei, Taiwan, 2008; Anadolu University International Conference in Economics: Developments in Economic Theory, Modeling and Policy, Eskişehir, Turkey, June 17–19, 2009; and the 15th Conference on Panel Data, Bonn, July 3–5, 2009. The authors would like to thank Paul Wilson, Co-Editor of this festschrift honoring Leopold Simar, for his insightful comments and editorial oversight on our paper. The usual caveat applies.


  1. Adams, R.M., Berger, A.N. & Sickles, R.C. (1999). Semiparametric approaches to stochastic panel frontiers with applications in the banking industry. Journal of Business and Economic Statistics, 17, 349–358.Google Scholar
  2. Adams, R.M., & Sickles, R.C. (2007). Semi-parametric efficient distribution free estimation of panel models. Communication in Statistics: Theory and Methods, 36, 2425–2442.MathSciNetMATHCrossRefGoogle Scholar
  3. Afriat, S. (1972). Efficiency estimation of a production function. International Economic Review, 13, 568–598.MathSciNetMATHCrossRefGoogle Scholar
  4. Ahn, S.C., Lee, Y., & Schmidt, P.J. (2005). Panel data models with multiple time-varying individual effects: application to a stochastic frontier production model. mimeo, Michigan State University.Google Scholar
  5. Aigner, D.J., Lovell, C.A.K., & Schmidt, P. (1977) Formulation and estimation of stochastic frontier models. Journal of Econometrics, 6, 21–37.MathSciNetMATHCrossRefGoogle Scholar
  6. Arrow K.J. (1962). The economic implications of learning by doing. Review of Economic Studies, 29, 155–173.CrossRefGoogle Scholar
  7. Battese, G.E., & Cora, G.S. (1977). Estimation of a production frontier model: with application to the pastoral zone of eastern Australia. Australian Journal of Agricultural Economics, 21, 169–179.CrossRefGoogle Scholar
  8. Bai, J. (2003). Inferential theory for factor models of large dimensions. Econometrica, 71, 135–171.MathSciNetMATHCrossRefGoogle Scholar
  9. Bai, J. (2005). Panel data models with interactive fixed effects. April 2005, mimeo, Department of Economics, New York University.Google Scholar
  10. Bai, J., & Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70, 191–221.MathSciNetMATHCrossRefGoogle Scholar
  11. Bai, J., & Ng, S. (2007). Determining the number of primitive shocks in factor models. Journal of Business and Economic Statistics, 25, 52–60.MathSciNetCrossRefGoogle Scholar
  12. Bai, J., Kao, C., & Ng, S. (2007). Panel cointegration with global stochastic trends. Center for Policy Research Working Papers 90, Center for Policy Research, Maxwell School, Syracuse University.Google Scholar
  13. Balk, B. (2009). Price and quantity index numbers: models for measuring aggregate change and difference. New York: Cambridge University Press.Google Scholar
  14. Baltagi, B., Egger, P., & Pfaffermayr, M. (2003). A generalized design for bilateral trade flow models. Economics Letters, 80, 391–397.MathSciNetMATHCrossRefGoogle Scholar
  15. Baltagi, B. (2005). Econometric Analysis of Panel Data, 3rd edition, New Jersey: Wiley.Google Scholar
  16. Battese, G.E. & Coelli, T.J. (1992). Frontier production functions, technical efficiency and panel data: with application to paddy farmers in India. Journal of Productivity Analysis, 3, 153–169.CrossRefGoogle Scholar
  17. Berger, A.N. (1993). “Distribution-Free” estimates of efficiency in U.S. banking industry and tests of the standard distributional assumption. Journal of Productivity Analysis, 4, 261–292.Google Scholar
  18. Bernanke, B.S., & Boivin, J. (2003). Monetary policy in a data-rich environment. Journal of Monetary Economics, 50, 525–546.CrossRefGoogle Scholar
  19. Blazek, D., & Sickles, R.C. (2010). The impact of knowledge accumulation and geographical spillovers on productivity and efficiency: the case of U.S. shipbuilding during WWII. In Hall, S.G., Klein, L.R., Tavlas, G.S. & Zellner, A. (eds.), Economic Modelling, 27, 1484–1497.Google Scholar
  20. Breitung, J., & Eickmeier, S. (2005). Dynamic factor models. Discussion Paper Series 1: Economic Studies, No 38/2005. Frankfurt: Deutsche Bundesbank.Google Scholar
  21. Breitung, J., & Kretschmer, U. (2005). Identification and estimation of dynamic factors from large macroeconomic panels. Mimeo: Universitat Bonn.Google Scholar
  22. Brumback, B.A., & Rice, J.A. (1998). Smoothing spline models for the analysis of nested and crossed samples of curves (with discussion). Journal of the American Statistical Association, 93, 961–94.MathSciNetMATHCrossRefGoogle Scholar
  23. de Boor, C. (1978). A Practical Guide to Splines. New York: Springer.MATHCrossRefGoogle Scholar
  24. Caves, D., Christensen, L.R, & Diewert, W.E. (1982). Multilateral comparisons of output, input, and productivity using superlative index numbers. Economic Journal, 92, 73–86.CrossRefGoogle Scholar
  25. Carriero, A., Kapetanios, G., & Marcellino, M. (2008). Forecasting large datasets with reduced rank multivariate models. Working Papers 617, Queen Mary, University of London, School of Economics and Finance.Google Scholar
  26. Chang, Y. (2004). Bootstrap unit root tests in panels with cross-sectional dependency. Journal of Econometrics, 120, 263–293.MathSciNetCrossRefGoogle Scholar
  27. Chamberlain, G., & Rothschild, M. (1983). Arbitrage, factor structure and mean-variance analysis in large asset markets. Econometrica, 51, 1305–1324.MathSciNetMATHCrossRefGoogle Scholar
  28. Charnes, A., Cooper, W.W., & Rhodes, E.L. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444.MathSciNetMATHCrossRefGoogle Scholar
  29. Chun, A. (2009). Forecasting interest rates and inflation: blue chip clairvoyants or econometrics? EFA 2009 Bergen Meetings Paper, Bergen, Norway.Google Scholar
  30. Cornwell, C., Schmidt, P., & Sickles, R.C. (1990). Production frontiers with cross-sectional and time-series variation in efficiency levels. Journal of Econometrics, 46, 185–200.CrossRefGoogle Scholar
  31. Debreu, G. (1951). The coefficient of resource utilization. Econometrica, 19, 273–292.MATHCrossRefGoogle Scholar
  32. Diebold, F.X., & Li, C. (2006). Forecasting the term structure of government bond yields. Journal of Econometrics, 130, 337–364.MathSciNetCrossRefGoogle Scholar
  33. Doz, C., Giannone, D., & Reichlin, L. (2006). A quasi maximum likelihood approach for large approximate dynamic factor models. ECB Working Paper 674.Google Scholar
  34. Engle, R., Granger, C., Rice, J., & Weiss, A. (1986). Nonparametric estimates of the relation between weather and electricity sales. Journal of American Statistical Association, 81, 310–320.CrossRefGoogle Scholar
  35. Eubank, R.L. (1988). Nonparametric regression and spline smoothing. New York: Marcel Dekker.MATHGoogle Scholar
  36. Färe, R., Grosskopf, S., Lindgren, B., & Roos, P. (1992). Productivity changes in Swedish pharamacies 1980–1989: a non-parametric Malmquist approach. Journal of Productivity Anlaysis, 3, 85–101.CrossRefGoogle Scholar
  37. Färe R., Grosskopf, S., Norris, M., & Zhang, Z. (1994). Productivity growth, technical progress and efficiency change in industrialized countries. American Economic Review, 84, 66–83.Google Scholar
  38. Farrell, M. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A, 120, 253–282.CrossRefGoogle Scholar
  39. Ferré, L. (1995). Improvement of some multivariate estimates by reduction of dimensionality. Journal of Multivariate Analysis, 54, 147–162.MathSciNetMATHCrossRefGoogle Scholar
  40. Fisher, I. (1927). The Making of Index Numbers. Boston: Houghton-Mifflin.Google Scholar
  41. Forni, M., & Lippi, M. (1997). Aggregation and the microfoundations of dynamic macroeconomics. Oxford: Oxford University Press.MATHGoogle Scholar
  42. Forni, M., & Reichlin, L. (1998). Let’s get real: a factor analytic approach to disaggregated business cycle dynamics. Review of Economic Studies, 65, 653–473.CrossRefGoogle Scholar
  43. Forni, M., Hallin, M., Lippi, M., & Reichlin, L. (2000). The generalized dynamic factor model: identification and estimation. Review of Economics and Statistics, 82, 540–554.CrossRefGoogle Scholar
  44. Førsund, F., & Hjalmarsson, L.(2008). Dynamic Analysis of Structural Change and Productivity Measurement. Unpublished Working Paper, Mimeo.Google Scholar
  45. Fried, H.O., Lovell, C.A.K., & Schmidt, S.S. (2008). The measurement of productive efficiency and productivity growth. Oxford University Press, Oxford.CrossRefGoogle Scholar
  46. Getachew, L., & Sickles, R.C. (2007). Allocative distortions and technical efficiency change in Egypt’s private sector manufacturing industries: 1987–1996. Journal of the Applied Econometrics, 22, 703–728.MathSciNetCrossRefGoogle Scholar
  47. Greene, W. (2004). Fixed and random effects in stochastic frontier models. Journal of Productivity Analysis, 23, 7–32.MathSciNetCrossRefGoogle Scholar
  48. Greene, W. (2005). Reconsidering heterogeneity in panel data estimators of the stochastic frontier model. Journal of Econometrics, 126, 269–303.MathSciNetCrossRefGoogle Scholar
  49. Greene, W. (2008). In Fried, H., Lovell, C.A.K., & Schmidt, S. (eds.) The Measurement of Productive Efficiency and Productivity Change, Chap. 2. Oxford: Oxford University Press.Google Scholar
  50. Härdle, W., Liang, H., & Gao, J. (2000). Partially linear models. Heidelberg: Physica-Verlag.MATHCrossRefGoogle Scholar
  51. Jeon, B.M., & Sickles, R.C. (2004). The Role of environmental factors in growth accounting: a nonparametric analysis. Journal of the Applied Economics, 19, 567–591.CrossRefGoogle Scholar
  52. Jorgenson, D.W., & Griliches, Z. (1972). Issues in growth accounting: a reply to Edward F. Denison. Survey of Current Business, 55 (part 2), 65–94.Google Scholar
  53. Kao, C., & Chiang, M.H. (2000). On the estimation and inference of a cointegrated regression in panel data. Advances in Econometrics, 15, 179–222.MathSciNetCrossRefGoogle Scholar
  54. Kapetanios, G., & Marcellino, M., (2009). A parametric estimation method for dynamic factor models of large dimensions, Journal of Time Series Analysis, 30, 208–238.MathSciNetCrossRefGoogle Scholar
  55. Klee, E.C., & Natalucci, F.M. (2005). Profits and balance sheet developments at U.S. commercial banks in 2004. Federal Reserve Bulletin, Spring.Google Scholar
  56. Kendrick, J. (1961). Productivity trends in the United States. Princeton: Princeton University Press for the National Bureau of Economic Research.Google Scholar
  57. Koenig, E. (2000). Productivity growth. Federal Reserve Bank of Dallas, Expand Your Insight, March 1,
  58. Koop, G.M., & Poirier, D. (2004). Bayesian variants of some classical semiparametric regression techniques. Journal of Econometrics, 123(2), 259–282.MathSciNetMATHCrossRefGoogle Scholar
  59. Kneip, A. (1994). Nonparametric estimation of common regressors for similar curve data. Annals of Statistics, 22, 1386–1427.MathSciNetMATHCrossRefGoogle Scholar
  60. Kneip, A., & Utikal, K.J. (2001). Inference for density families using functional principal component analysis. Journal of American Statistical Association, 96, 519–532.MathSciNetMATHCrossRefGoogle Scholar
  61. Kneip, A., Sickles, R.C., & Song, W. (2011). A new panel data treatment for heterogeneity in time trends. Econometric Theory, to appear.Google Scholar
  62. Kumbhakar, S.C. (1990). Production Frontiers, panel data and time-varying technical inefficiency. Journal of Econometrics, 46, 201–211.CrossRefGoogle Scholar
  63. Kumbhakar, S., & Lovell, C.A.K. (2000). Stochastic Frontier Analysis. Cambridge: Cambridge University Press.MATHCrossRefGoogle Scholar
  64. Lovell, C.A.K., Richardson, S., Travers, P., & Wood, L.L. (1994). Resources and functionings: a new view of inequality in Australia. In Eichorn, W. (Ed.) Models and Measurement of Welfare and Inequality, pp. 787–807. Berlin, Heidelberg, New York: Springer.CrossRefGoogle Scholar
  65. Lucas, R.E. (1988). On the Mechanics of Economic Development. Journal of Monetary Economics, 22, 3–42.CrossRefGoogle Scholar
  66. Maddala, G.S., & Kim, I.M. (1998). Unit Roots, cointegration and structural change. Cambridge: Cambridge University Press.Google Scholar
  67. Mark, N.C., & Sul, D. (2003). Cointegration vector estimation by panel dlos and long-run money demand. Oxford Bulletin of Economics and Statistics, 65, 655–680.CrossRefGoogle Scholar
  68. Marcellino, M., & Schumacher, C. (2007). Factor-midas for now- and forecasting with ragged-edge data: a model comparison for German gdp. Discussion Paper Series 1: Economic Studies,34, Deutsche Bundesbank, Research Centre.Google Scholar
  69. Meeusen, W., & van den Broeck, J. (1977). Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review, 18, 435–444.MATHCrossRefGoogle Scholar
  70. Nelson, C.R., & Plosser, C.I. (1982). Trends and random walks in macroeconomics time series: some evidence and implications. Journal of Monetary Economics, 10, 139–162.CrossRefGoogle Scholar
  71. Orea, C., & Kumbhakar, S. (2004). Efficiency measurement using a latent class stochastic frontier model. Empirical Economics, 29, 169–184.CrossRefGoogle Scholar
  72. Park, B.U., & Simar, L. (1994). Efficient semiparametric estimation in stochastic frontier models. Journal of the American Statistical Association, 89, 929–936.MathSciNetMATHCrossRefGoogle Scholar
  73. Park, B.U., Sickles, R.C., & Simar, L. (1998). Stochastic frontiers: a semiparametric approach. Journal of Econometrics, 84, 273–301.MathSciNetMATHCrossRefGoogle Scholar
  74. Park, B.U., Sickles, R.C., & Simar, L. (2003). Semiparametric efficient estimation of AR(1) panel data models. Journal of Econometrics, 117, 279–309.MathSciNetMATHCrossRefGoogle Scholar
  75. Park, B.U., Sickles, R.C., & Simar, L. (2007). Semiparametric efficient estimation of dynamic panel data models. Journal of Econometrics, 136, 281–301.MathSciNetCrossRefGoogle Scholar
  76. Pesaran, M.H. (2006). Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica, 74, 967–1012.MathSciNetMATHCrossRefGoogle Scholar
  77. Pitt, M., & Lee, L.-F. (1981). The measurement and sources of technical inefficiency in the Indonesian weaving industry. Journal of Development Economics, 9, 43–64.CrossRefGoogle Scholar
  78. Ramsay, J., & Silverman, B. (1997). Functional data analysis. Heidelberg: Springer.MATHGoogle Scholar
  79. Rao, C.R. (1958). Some statistical methods for the comparison of growth curves. Biometrics, 14, 1–17.MATHCrossRefGoogle Scholar
  80. Reikard, G. (2005). Endogenous technical advance and the stochastic trend in output: A neoclassical approach. Research Policy, 34, 1476–1490.CrossRefGoogle Scholar
  81. Romer, P.M. (1986). Increasing returns and long-run growth. Journal of Political Economy, 94, 1002–1037.CrossRefGoogle Scholar
  82. Schmidt, P., & Sickles, R.C. (1984). Production frontiers and panel data. Journal of Business and Economic Statistics, 2, 367–374.Google Scholar
  83. Sickles, R.C., & Streitwieser, M. (1992). Technical inefficiency and productive decline in the U.S. interstate natural gas pipeline industry under the U.S. interstate natural gas policy act. Journal of Productivity Analysis (Lewin, A., & Lovell, C.A.K. Eds.), 3, 115–130. Reprinted in International Applications for Productivity and Efficiency Analysis, (Thomas R. Gulledge, Jr., & Knox Lovell, C.A. Eds.). Boston: Kluwer.Google Scholar
  84. Sickles, R.C., & Streitwieser, M. (1998). The structure of technology, substitution and productivity in the interstate natural gas transmission industry under the natural gas policy act of l978. Journal of Applied Econometrics, 13, 377–395.CrossRefGoogle Scholar
  85. Sickles, R.C. (2005). Panel estimators and the identification of firm-specific efficiency levels in parametric, semiparametric and nonparametric settings. Journal of Econometrics, 50, 126, 305–334.Google Scholar
  86. Sickles, R.C., & Tsionas, E.G. (2008). A panel data model with nonparametric time effects. Mimeo: Rice University.Google Scholar
  87. Simar, L., & Wilson, P.W. (2010). Inference from cross-sectional stochastic frontier models. Econometric Reviews, 29, 62–98.MathSciNetMATHCrossRefGoogle Scholar
  88. Shephard, R.W. (1970). Theory of cost and production functions. Princeton: Princeton University Press.MATHGoogle Scholar
  89. Solow, Robert M. (1957). Technical change and the aggregate production function. Review of Economics and Statistics, 39, 312–320.CrossRefGoogle Scholar
  90. Speckman, P. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B, 50, 413–436.MathSciNetMATHGoogle Scholar
  91. Stock, J.H., & Watson, M.W. (2002). Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association, 97, 1167–1179.MathSciNetMATHCrossRefGoogle Scholar
  92. Stock, J.H., & Watson, M.W. (2005). Implications of dynamic factor models for VAR analysis. Mimeo: Princeton University.CrossRefGoogle Scholar
  93. Tsionas, E.G., & Greene, W. (2003). A panel data model with nonparametric time effects, Athens University of Business and Economics.Google Scholar
  94. Utreras, F. (1983). Natural spline functions, their associated eigenvalue problem. Numerical Mathematics, 42, 107–117.MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of BonnBonnGermany
  2. 2.Department of Economics - MS 22Rice UniversityHoustonUSA

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