Likelihood approach to dynamic panel models with interactive effects

Bai, Jushan (2013): Likelihood approach to dynamic panel models with interactive effects.

Abstract

This paper considers dynamic panel models with a factor error structure that is correlated with the regressors. Both short panels (small T) and long panels (large T) are considered. With a small T, consistent estimation requires either a suitable formulation of the reduced form or an appropriate conditional equation for the first observation. Also needed is a suitable control for the correlation between the effects and the regressors. Under the factor error structure, the panel system implies parameter constraints between the mean vector and the covariance matrix. We explore the constraints through a quasi-FIML approach.

The factor process is treated as parameters and it can have arbitrary dynamics under both fixed and large T. The large T setting involves incidental parameters because the number of parameters (including the time effects, the factor process, the heteroskedasticity parameters) increases with T. Even though an increasing number of parameters are estimated, we show that there is no incidental parameters bias to affect the limiting distributions; the estimator is centered at zero even scaled by the fast convergence rate of root-NT. We also show that the quasi-FIML approach is efficient under both fixed and large T, despite non-normality, heteroskedasticity, and incidental parameters. Finally we develop a feasible and fast algorithm for computing the quasi-FIML estimators under interactive effects.

• Item Type: MPRA Paper
• Language: English
• Keywords: factor structure, interactive effects, incidental parameters, predetermined regressors, heterogeneity and endogeneity, quasi-FIML, efficiency
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C31 - Cross-Sectional Models ; Spatial Models ; Treatment Effect Models ; Quantile Regressions ; Social Interaction Models

References

Ahn, S.G., Y.H. Lee and P. Schmidt (2001): ``GMM Estimation of Linear Panel Data Models with Time-varying Individual Effects," Journal of Econometrics, 102, 219-255.

Ahn, S.G., Y.H. Lee and P. Schmidt (2013): ``Panel Data Models with Multiple Time-varying Effects," Journal of Econometrics, 174, 1-14.

Alvarez, J. and M. Arellano (2003): The Time Series and Cross-Section Asymptotics of Dynamic Panel Data Estimators. Econometrica 71, 1121-1159.

Alvarez, J. and M. Arellano (2004): Robust likelihood estimation of dynamic panel data models. Unpublished manuscript, CEMFI.

Amemiya, T. (1985): Advanced Econometrics, Harvard University Press, Cambridge, MA.

Amemiya, Y., W.A. Fuller, and S.G. Pantula (1987), The Asymptotic Distributions of Some Estimators for a Factor Analysis Model, Journal of Multivariate Analysis, 22, 51-64.

Anderson, T.W. and Y. Amemiya (1988). The asymptotic normal distribution of estimators in factor analysis under general conditions, Annals of Statistics, 16 759-771

Anderson, T.W., and C. Hsiao (1981): ``Estimation of dynamic models with error components," Journal of American Statistical Association, 76, 598-606.

Anderson, T.W., and C. Hsiao (1982): ``Formulation and estimation of dynamic Models with Error Components," Journal of Econometrics, 76, 598-606.

Anderson, T.W. and H. Rubin (1956): ``Statistical Inference in Factor Analysis," in J. Neyman, ed., Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Vol 5, 111-150.

Arellano, M. (2003): Panel Data Econometrics, Oxford University Press.

Bai, J. (2009). Panel data models with interactive fixed effects, Econometrica, 77 1229-1279.

Bai, J. (2013). Fixed effects dynamic panel data models, a factor analytical method, Econometrica, 81, 285-314.

Bai, J. and K.P. Li (2012). Statistical analysis of factor models of high dimension. Annals of Statistics, 40, 436-465.

Baltagi, B.H. (2005): Econometric Analysis of Panel Data, John Wiley: Chichester.

Bhargava, A. and J.D. Sargan (1983): ``Estimating Dynamic Random Effects Models from Panel Data Covering Short Time Periods," Econometrica, 51, 1635-1659.

Blundell R. and S. Bond (1998). Initial conditions and moment restrictions in dynamic panel data models, Journal of Econometrics, 87 115-143.

Blundell R. and R.J. Smith (1991). Initial conditions and efficient estimation in dynamic panel data models. Annales d'Economie et de Statistique 20/21, 109-123.

Chamberlain, G. (1982): ``Multivariate regression models for panel data," Journal of Econometrics, 18, 5-46.

Chamberlain, G. (1984): ``Panel Data," in Handbook of Econometrics, Vol. 2, ed. by Z. Griliches and M. Intriligator. Amsterdam: North-Holland

Chamberlain, G. and M.J. Moreira (2009), Decision Theory Applied To A Linear Panel Data Model, Econometrica 77, 107-133.

Dahm, P.F. and W.A. Fuller (1986): ``Generalized Least Squares Estimation of the Functional Multivariate Linear Errors-in-variables Model," Journal of Multivariate Analysis 19, 132-141.

Dempster, A.P. N.M. Laird, and D.B. Rubin (1977), Maximum likelihood from incomplete data via the EM algorithm. Journal of The Royal Statistical Society B, 39, 1-38.

Doz, C., Giannone, D. and L Reichlin (2008). A quasi maximum likelihood approach for large approximate dynamic factor models. ECARES and CEPR.

Engle, R., D.F. Hendry, and J.F. Richard (1983): ``Exogeneity," Econometrica, 51, 277-304.

Hahn, J., and G. Kuersteiner (2002): ``Asymptotically Unbiased Inference for a Dynamic Panel Model with Fixed Effects when Both n and T Are Large," Econometrica, 70, 1639-1657.

Holtz-Eakin, D., W. Newey, and H. Rosen (1988): ``Estimating Vector Autoregressions with Panel Data", Econometrica, 56, 1371-1395.

Hsiao, C. (2003): Analysis of Panel Data. Cambridge University Press, New York.

Iwakura, H. and R. Okui (2012). Asymptotic Efficiency in Dynamic Panel Data Models with Factor Structure, unpublished manuscript, Institute of Economic Research, Kyoto University.

Jungbacker, B. and S.J. Koopman (2008). Likelihood-based analysis for dynamic factor models, memio.

Kiviet, J. (1995): ``On Bias, Inconsistency, and Efficiency of Various Estimators in Dynamic Panel Data Models", Journal of Econometrics, 68, 53-78.

Kruiniger, H. (2008). Maximum likelihood estimation and inference methods for the covariance stationary panel AR(1)/unit root model. Journal of Econometrics, 447-464.

Lancaster, T. (2000): ``The incidental parameter problem since 1948," Journal of Econometrics, 95 391-413.

Lancaster, T. (2002): ``Orthogonal parameters and panel data," Review of Economics Studies. 69 647-666.

Lawley, D.N. and A.E. Maxwell (1971): Factor Analysis as a Statistical Method, London: Butterworth.

Liu, C. and D.B. Rubin (1994). The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika, 81, 633-648.

Magnus, J.R. and H. Neudecker (1999): Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley: New York.

McLachlan, G.J, and T. Krishnan (1996): The EM Algorithm and Extensions, Wiley, New York.

Meng, X.L and D.B. Rubin (1993). Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika, 80(2), 267-278.

Moffitt, R., and P. Gottschalk (2002): ``Trends in the Transitory Variance of Earnings in the United States," The Economic Journal, 112, C68-C73.

Moon H.R. and M. Weidner (2010a). ``Dynamic Linear Panel Regression Models with Interactive Fixed Effects", unpublished manuscript, USC.

Moon, H.R. and M. Weidner (2010b). ``Linear regression for panel with unknown number of factors as interactive fixed effects," unpublished manuscript, USC.

Moreira, M.J. (2009), A Maximum Likelihood Method for the Incidental Parameter Problem, Annals of Statistics, 37, 3660-3696.

Mundlak, Y. (1978): ``On the pooling of time series and cross section data," Econometrica, 46, 69-85.

Newey, W. and D. McFadden (1994): ``Large Sample Estimation and Hypothesis Testing," in Engle, R.F. and D. McFadden (eds.) Handbook of Econometrics, North Holland.

Neyman, J., and E. L. Scott (1948): ``Consistent Estimates Based on Partially Consistent Observations," Econometrica, 16, 1-32.

Nickell, S. (1981): ``Biases in Dynamic Models with Fixed Effects," Econometrica, 49, 1417-1426.

Pesaran, M. H. (2006): ``Estimation and Inference in Large Heterogeneous panels with a Multifactor Error Structure," Econometrica, 74, 967-1012.

Proietti, T. (2008). Estimation of common factors under cross-sectional and temporal aggregation constraints: nowcasting monthly GDP and its main components. MPRA Paper 6860, University Library of Munich, Germany.

Quad, Q. and T. Sargent (1993). A Dynamic Index Model for Large Cross Sections. CEP Discussion Paper No. 0132.

Rao, C.R. and S.K. Mitra (1971). Generalized Inverse of Matrices and its Applications, Wiley: New York.

Rubin, D.B. and D.T. Thayer (1982). EM algorithm for ML factor analysis. Psychometrika, 47 69-76.

Shapiro, A. (1986). Asymptotic theory of overparameterized structural models. Journal of the American Statistical Association 81, 142-149.

Sims, C.A. (2000). Using a likelihood perspective to sharpen econometric discourse: Three examples. Journal of Econometrics, 95 443-462.

Stock, J.~H. and M.~W. Watson (2011): Dynamic Factor Models, The Oxford Handbook of Economic Forecasting, Edited by M.P. Clements and D.F. Hendry, Oxford University Press.

van der Vaart, A.W. and J.A. Wellner (1996): Weak Convergence and Empirical Processes. Springer, New York.

Watson, M.W. and R.F. Engle (1983): ``Alternative algorithms for the estimation of the dynamic factor, MIMIC, and varying coefficient regression models" Journal of Econometrics, Vol. 23, pp. 385-400.

Westerlund, J. and J.P. Urbain (2013). On the estimation and inference in factor-augmented panel regressions with correlated loadings. Economics Letters, 119(3), 247-250.