American Journal of Applied Mathematics and Statistics
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American Journal of Applied Mathematics and Statistics. 2016, 4(2), 46-58
DOI: 10.12691/ajams-4-2-4
Open AccessArticle

Generalized Random Coefficient Estimators of Panel Data Models: Asymptotic and Small Sample Properties

Mohamed Reda Abonazel1,

1Department of Applied Statistics and Econometrics, Institute of Statistical Studies and Research, Cairo University, Egypt

Pub. Date: May 23, 2016

Cite this paper:
Mohamed Reda Abonazel. Generalized Random Coefficient Estimators of Panel Data Models: Asymptotic and Small Sample Properties. American Journal of Applied Mathematics and Statistics. 2016; 4(2):46-58. doi: 10.12691/ajams-4-2-4


This paper provides a generalized model for the random-coefficients panel data model where the errors are cross-sectional heteroskedastic and contemporaneously correlated as well as with the first-order autocorrelation of the time series errors. Of course, the conventional estimators, which used in standard random-coefficients panel data model, are not suitable for the generalized model. Therefore, the suitable estimator for this model and other alternative estimators have been provided and examined in this paper. Moreover, the efficiency comparisons for these estimators have been carried out in small samples and also we examine the asymptotic distributions of them. The Monte Carlo simulation study indicates that the new estimators are more reliable (more efficient) than the conventional estimators in small samples.

classical pooling estimation contemporaneous covariance first-order autocorrelation heteroscedasticity mean group estimation; monte carlo simulation random coefficient regression

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[1]  Abonazel, M. R. (2009). Some Properties of Random Coefficients Regression Estimators. MSc thesis. Institute of Statistical Studies and Research. Cairo University.
[2]  Abonazel, M. R. (2014). Some estimation methods for dynamic panel data models. PhD thesis. Institute of Statistical Studies and Research. Cairo University.
[3]  Alcacer, J., Chung, W., Hawk, A., Pacheco-de-Almeida, G. (2013). Applying random coefficient models to strategy research: testing for firm heterogeneity, predicting firm-specific coefficients, and estimating Strategy Trade-Offs. Working Paper, No. 14-022. Harvard Business School Strategy Unit.
[4]  Anh, V. V., Chelliah, T. (1999). Estimated generalized least squares for random coefficient regression models. Scandinavian journal of statistics 26(1):31-46.‏
[5]  Baltagi, B. H. (2011). Econometrics. 5th ed. Berlin: Springer-Verlag Berlin Heidelberg.
[6]  Baltagi, B. H. (2013). Econometric Analysis of Panel Data. 5th ed. Chichester: John Wiley and Sons.
[7]  Beck, N., Katz, J. N. (2007). Random coefficient models for time-series–cross-section data: Monte Carlo experiments. Political Analysis 15(2):182-195.‏
[8]  Beran, R., Millar, P. W. (1994). Minimum distance estimation in random coefficient regression models. The Annals of Statistics 22(4):1976-1992.‏
[9]  Bodhlyera, O., Zewotir, T., Ramroop, S. (2014). Random coefficient model for changes in viscosity in dissolving pulp. Wood Research 59(4):571-582.‏
[10]  Boness, A. J., Frankfurter, G. M. (1977). Evidence of Non-Homogeneity of capital costs within “risk-classes”. The Journal of Finance 32(3):775-787.
[11]  Boot, J. C., Frankfurter, G. M. (1972). The dynamics of corporate debt management, decision rules, and some empirical evidence. Journal of Financial and Quantitative Analysis 7(04):1957-1965.
[12]  Chelliah, N. (1998). A new covariance estimator in random coefficient regression model. The Indian Journal of Statistics, Series B 60(3):433-436.‏
[13]  Cheng, J., Yue, R. X., Liu, X. (2013). Optimal Designs for Random Coefficient Regression Models with Heteroscedastic Errors. Communications in Statistics-Theory and Methods 42(15):2798-2809.
[14]  Cooley, T. F., Prescott, E. C. (1973). Systematic (non-random) variation models: varying parameter regression: a theory and some applications. Annals of Economic and Social Measurement 2(4): 463-473.‏
[15]  Dielman, T. E. (1983). Pooled cross-sectional and time series data: a survey of current statistical methodology. The American Statistician 37(2):111-122.
[16]  Dielman, T. E. (1989). Pooled Cross-Sectional and Time Series Data Analysis. New York: Marcel Dekker.
[17]  Dielman, T. E. (1992a). Misspecification in random coefficient regression models: a Monte Carlo simulation. Statistical Papers 33(1):241-260.‏
[18]  Dielman, T. E. (1992b). Small sample properties of random coefficient regression estimators: a Monte Carlo simulation. Communications in Statistics-Simulation and Computation 21(1):103-132.‏
[19]  Dwivedi, T.D., Srivastava, V.K. (1978). Optimality of least squares in the seemingly unrelated regression equation model. Journal of Econometrics 7:391-395.
[20]  Dziechciarz, J. (1989). Changing and random coefficient models. A survey. In: Hackl, P., ed. Statistical Analysis and Forecasting of Economic Structural Change. Berlin: Springer Berlin Heidelberg.‏
[21]  Elhorst, J. P. (2014). Spatial Econometrics: From Cross-Sectional Data to Spatial Panels. Heidelberg, New York, Dordrecht, London: springer.‏
[22]  Elster, C., Wübbeler, G. (2016). Bayesian inference using a noninformative prior for linear Gaussian random coefficient regression with inhomogeneous within-class variances. Computational Statistics (in press). DOI: 10.1007/s00180-015-0641-3.
[23]  Feige, E. L., Swamy, P. A. V. B. (1974). A random coefficient model of the demand for liquid assets. Journal of Money, Credit and Banking, 6(2):241-252.
[24]  Fu, K. A., Fu, X. (2015). Asymptotics for the random coefficient first-order autoregressive model with possibly heavy-tailed innovations. Journal of Computational and Applied Mathematics 285:116-124.
[25]  Horváth, L., Trapani, L. (2016). Statistical inference in a random coefficient panel model. Journal of Econometrics 193(1):54-75.
[26]  Hsiao, C. (2014). Analysis of Panel Data. 3rd ed. Cambridge: Cambridge University Press.
[27]  Hsiao, C., Pesaran, M. H. (2008). Random coefficient models. In: Matyas, L., Sevestre, P., eds. The Econometrics of Panel Data. Vol. 46. Berlin: Springer Berlin Heidelberg.
[28]  Judge, G. G., Griffiths, W. E., Hill, R. C., Lütkepohl, H., Lee, T. C. (1985). The Theory and Practice of Econometrics, 2nd ed. New York: Wiley.
[29]  Livingston, M., Erickson, K., Mishra, A. (2010). Standard and Bayesian random coefficient model estimation of US Corn–Soybean farmer risk attitudes. In Ball, V. E., Fanfani, R., Gutierrez, L., eds. The Economic Impact of Public Support to Agriculture. Springer New York.
[30]  Mousa, A., Youssef, A. H., Abonazel, M. R. (2011). A Monte Carlo study for Swamy’s estimate of random coefficient panel data model. Working paper, No. 49768. University Library of Munich, Germany.
[31]  Murtazashvili, I., Wooldridge, J. M. (2008). Fixed effects instrumental variables estimation in correlated random coefficient panel data models. Journal of Econometrics 142:539-552.
[32]  Parks, R. W. (1967). Efficient Estimation of a System of regression equations when disturbances are both serially and contemporaneously correlated. Journal of the American Statistical Association 62:500-509.
[33]  Pesaran, M.H., Smith, R. (1995). Estimation of long-run relationships from dynamic heterogeneous panels. Journal of Econometrics 68:79-114.
[34]  Poi, B. P. (2003). From the help desk: Swamy’s random-coefficients model. The Stata Journal 3(3):302-308.‏
[35]  Rao, C. R. (1973). Linear Statistical Inference and Its Applications. 2nd ed. New York: John Wiley & Sons.
[36]  Rao, C. R., Mitra, S. (1971). Generalized Inverse of Matrices and Its Applications. John Wiley and Sons Ltd.
[37]  Rao, U. G. (1982). A note on the unbiasedness of Swamy's estimator for the random coefficient regression model. Journal of econometrics 18(3):395-401.‏
[38]  Rausser, G.C., Mundlak, Y., Johnson, S.R. (1982). Structural change, updating, and forecasting. In: Rausser, G.C., ed. New Directions in Econometric Modeling and Forecasting US Agriculture. Amsterdam: North-Holland.
[39]  Sant, D. (1977). Generalized least squares applied to time-varying parameter models. Annals of Economic and Social Measurement 6(3):301-314.
[40]  Srivastava, V. K., Giles, D. E. A. (1987). Seemingly Unrelated Regression Equations Models: Estimation and Inference. New York: Marcel Dekker.
[41]  Swamy, P. A. V. B. (1970). Efficient inference in a random coefficient regression model. Econometrica 38:311-323.
[42]  Swamy, P. A. V. B. (1971). Statistical Inference in Random Coefficient Regression Models. New York: Springer-Verlag.
[43]  Swamy, P. A. V. B. (1973). Criteria, constraints, and multicollinearity in random coefficient regression model. Annals of Economic and Social Measurement 2(4):429-450.
[44]  Swamy, P. A. V. B. (1974). Linear models with random coefficients. In: Zarembka, P., ed. Frontiers in Econometrics. New York: Academic Press.
[45]  Swamy, P. A. V. B., Mehta, J. S., Tavlas, G. S., Hall, S. G. (2015). Two applications of the random coefficient procedure: Correcting for misspecifications in a small area level model and resolving Simpson's paradox. Economic Modelling 45:93-98.
[46]  Westerlund, J., Narayan, P. (2015). A random coefficient approach to the predictability of stock returns in panels. Journal of Financial Econometrics 13(3):605-664.
[47]  Youssef, A. H., Abonazel, M. R. (2009). A comparative study for estimation parameters in panel data model. Working paper, No. 49713. University Library of Munich, Germany.
[48]  Youssef, A. H., Abonazel, M. R. (2015). Alternative GMM estimators for first-order autoregressive panel model: an improving efficiency approach. Communications in Statistics-Simulation and Computation (in press).
[49]  Youssef, A. H., El-sheikh, A. A., Abonazel, M. R. (2014a). Improving the efficiency of GMM estimators for dynamic panel models. Far East Journal of Theoretical Statistics 47:171-189.
[50]  Youssef, A. H., El-sheikh, A. A., Abonazel, M. R. (2014b). New GMM estimators for dynamic panel data models. International Journal of Innovative Research in Science, Engineering and Technology 3:16414-16425.
[51]  Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests of aggregation bias. Journal of the American Statistical Association 57:348-368.