A further look at Modified ML estimation of the panel AR(1) model with fixed effects and arbitrary initial conditions.

Kruiniger, Hugo (2018): A further look at Modified ML estimation of the panel AR(1) model with fixed effects and arbitrary initial conditions.

Abstract

In this paper we consider two kinds of generalizations of Lancaster's (Review of Economic Studies, 2002) Modified ML estimator (MMLE) for the panel AR(1) model with fixed effects and arbitrary initial conditions and possibly covariates when the time dimension, T, is fixed. When the autoregressive parameter ρ=1, the limiting modified profile log-likelihood function for this model has a stationary point of inflection and ρ is first-order underidentified but second-order identified. We show that the generalized MMLEs exist w.p.a.1 and are uniquely defined w.p.1. and consistent for any value of |ρ|≤1. When ρ=1, the rate of convergence of the MMLEs is N^{1/4}, where N is the cross-sectional dimension of the panel. We then develop an asymptotic theory for GMM estimators when one of the parameters is only second-order identified and use this to derive the limiting distributions of the MMLEs. They are generally asymmetric when ρ=1. We also show that Quasi LM tests that are based on the modified profile log-likelihood and use its expected rather than observed Hessian, with an additional modification for ρ=1, and confidence regions that are based on inverting these tests have correct asymptotic size in a uniform sense when |ρ|≤1. Finally, we investigate the finite sample properties of the MMLEs and the QLM test in a Monte Carlo study.

• Item Type: MPRA Paper
• Language: English
• Keywords: dynamic panel data, expected Hessian, fixed effects, Generalized Method of Moments (GMM), inflection point, Modified Maximum Likelihood, Quasi LM test, second-order identification, singular information matrix, weak moment conditions.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C23 - Panel Data Models ; Spatio-temporal Models

References

Ahn, S.C., and G.M. Thomas, 2004, Likelihood-based inference for dynamic panel data models, unpublished manuscript, Arizona State University, revised in 2006.

Alvarez, J., and M. Arellano, 2004, Robust likelihood estimation of dynamic panel data models, unpublished manuscript, University of Alicante and CEMFI, Madrid.

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

Anderson, T.W., and C. Hsiao, 1982, Formulation and estimation of dynamic models using panel data, Journal of econometrics 18, 47-82.

Arellano, M., and S. Bond, 1991, Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations, Review of Economic Studies 58, 277--297.

Arellano, M., and S. Bonhomme, 2009, Robust priors in nonlinear panel data models, Econometrica 77, 489-536.

Arellano, M., and O. Bover, 1995, Another look at the instrumental variable estimation of error-components models, Journal of econometrics 68, 29-51.

Bester, C.A., and C. Hansen, 2007, Frequentist bias reduction via Bayesian priors, unpublished manuscript, Graduate School of Business, University of Chicago.

Bottai, M., 2003, Confidence regions when the Fisher information is zero, Biometrika 90, 73-84.

Bun, M.J.G., and M.A. Carree, 2005, Bias-Corrected Estimation in Dynamic Panel Data Models, Journal of business & economic statistics 23, 200-210; Correction: 363.

Chamberlain, G., 1980, Analysis of covariance with qualitative data, Review of economic studies XLVII, 225-238.

Dhaene, G., and K. Jochmans, 2015, Likelihood inference in an autoregression with fixed effects, Econometric Theory 32, 1178-1215.

Dovonon, P., and A. Hall, 2018, The asymptotic properties of GMM and Indirect Inference under second-order identification, Journal of econometrics 205, 76-111.

Dovonon, P., and E. Renault, 2013, Testing for conditionally heteroskedastic factors, Econometrica 81, 2561-2586.

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.

Hallin, M., and C. Ley, 2014, Skew-symmetric distributions and Fisher information: The double sin of the skew-normal, Bernoulli 20, 1432-1453.

Han, C., P.C.B. Phillips, and D. Sul, 2014, X-differencing and dynamic model estimation, Econometric Theory 30, 201-251.

Harris, R.D.F., and E. Tzavalis, 1999, Inference for unit roots in dynamic panels where the time dimension is fixed, Journal of econometrics 91, 201-226.

Hsiao, C., M.H. Pesaran, and A.K. Tahmiscioglu, 2002, Maximum likelihood estimation of fixed effects dynamic panel models covering short time periods, Journal of econometrics 109, 107-150.

Juodis, A., 2013, A note on bias-corrected estimation in dynamic panel data models, Economics Letters 118, 435-438.

Kruiniger, H., 2001, On the estimation of panel regression models with fixed effects, appeared in 2002 as working paper no. 450, Queen Mary, University of London, revised in 2003.

Kruiniger, H., 2008, Maximum likelihood estimation and inference methods for the covariance stationary panel AR(1)/unit root model, Journal of Econometrics 144, 447-464.

Kruiniger, H., 2009, GMM estimation and inference in dynamic panel data models with persistent data, Econometric theory 25, 1348-1391.

Kruiniger, H., 2013, Quasi ML estimation of the panel AR(1) model with arbitrary initial conditions, Journal of Econometrics 173, 175-188.

Kruiniger, H., 2015, Uniform Quasi ML based inference for the panel AR(1) model with arbitrary initial conditions, mimeo.

Lancaster, T., 2002, Orthogonal parameters and panel data, Review of economic studies LXIX, 647-667.

Lee, L.F., and A. Chesher, 1986, Specification testing when score test statistics are identically zero, Journal of econometrics 31, 121-149.

Madsen, E., 2009, GMM-based inference in the AR(1) panel data model for parameter values where local identification fails, unpublished manuscript, Centre for Applied Microeconometrics (CAM), University of Copenhagen.

Moreira, M.J., 2009, A maximum likelihood method for the incidental parameter problem, Annals of Statistics 37, 3660--3696.

Newey, W., and D. McFadden, 1994, Large sample estimation and hypothesis testing, Ch. 36 in Handbook of econometrics, Vol. 4, R.F. Engle and D. McFadden (eds.), North-Holland, Elsevier, Amsterdam.

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.

Rotnitzky, A., D.R. Cox, M. Bottai, and J. Robins, 2000, Likelihood-based inference with singular information matrix, Bernoulli 6, 243-284.

Sargan, J. D., 1983. Identification and lack of identification, Econometrica, 51, 1605-1633.

Van der Vaart, A.W., 2000, Asymptotic statistics, Cambridge University Press.