- Volume 26 Issue 4
DOI QR Code
Restricted maximum likelihood estimation of a censored random effects panel regression model
- Lee, Minah (Data Analysis Team, Samsung SDS) ;
- Lee, Seung-Chun (Department of Applied Statistics, Hanshin University)
- Received : 2019.01.02
- Accepted : 2019.02.18
- Published : 2019.07.31
Panel data sets have been developed in various areas, and many recent studies have analyzed panel, or longitudinal data sets. Maximum likelihood (ML) may be the most common statistical method for analyzing panel data models; however, the inference based on the ML estimate will have an inflated Type I error because the ML method tends to give a downwardly biased estimate of variance components when the sample size is small. The under estimation could be severe when data is incomplete. This paper proposes the restricted maximum likelihood (REML) method for a random effects panel data model with a censored dependent variable. Note that the likelihood function of the model is complex in that it includes a multidimensional integral. Many authors proposed to use integral approximation methods for the computation of likelihood function; however, it is well known that integral approximation methods are inadequate for high dimensional integrals in practice. This paper introduces to use the moments of truncated multivariate normal random vector for the calculation of multidimensional integral. In addition, a proper asymptotic standard error of REML estimate is given.
- Amemiya T (1984). Tobit models: a survey, Journal of Econometrics, 24, 3-61. https://doi.org/10.1016/0304-4076(84)90074-5
- Arellano M and Bond S (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations, The Review of Economic Studies, 58, 227-297.
- Arismendi JC (2013). Multivariate truncated moments, Journal of Multivariate Analysis, 117, 41-75. https://doi.org/10.1016/j.jmva.2013.01.007
- Croissant Y and Millo G (2008). Panel data econometrics in R: The plm package, Journal of Statistical Software, 27, 1-43.
- Drum ML and McCullagh P (1993). REML estimation with exact covariance in the logistic mixed model, Biometrics, 49, 677-689. https://doi.org/10.2307/2532189
- Duan JC and Fulop A (2011). A stable estimator of the information matrix under EM for dependent data, Statistics and Computing, 21, 83-91. https://doi.org/10.1007/s11222-009-9149-4
- Eddelbuettel D, Francois R, Allaire J, Ushey K, Kou Q, Russell N, Bates D, and Chambers J (2018). Seamless R and C++ Integration. Available from: http://www.rcpp.org, http://dirk.eddelbuettel.com/code/rcpp.html
- Eddelbuettel D and Sanderson C (2014). RcppArmadillo: accelerating R with high-performance C++ linear algebra, Computational Statistics and Data Analysis, 71, 1054-1063. https://doi.org/10.1016/j.csda.2013.02.005
- Efron B and Hinkley DV (1978). The observed versus expected information, Biometrika, 65, 457-487. https://doi.org/10.1093/biomet/65.3.457
- Green W (2004). Fixed effects and bias due to the incidental parameters problem in the Tobit model, Econometric Review, 23, 125-147. https://doi.org/10.1081/ETC-120039606
- Henningsen A (2017). censReg: Censored Regression (Tobit) Models. R package version 0.5. Avail-able from: http://CRAN.R-Project.org/package=censReg
- Hughes JP (1999). Mixed effects models with censored data with application to HIV RNA levels, Biometrics, 55, 625-629. https://doi.org/10.1111/j.0006-341X.1999.00625.x
- Kan R and Robotti C (2017). On moments of folded and truncated multivariate normal distributions, Unpublished manuscript. Available from: https://sites.google.com/site/cesarerobotti/krJCGS.pdf
- Kleiber C and Zeileis A (2009). AER: Applied Econometrics with R, R package version 1.1. Available from: http://CRAN.R-project.org/package=AER
- Lancaster T (2000). The incidental parameter problem since 1948, Journal of Econometrics, 95, 391-413. https://doi.org/10.1016/S0304-4076(99)00044-5
- Lee L (2017) Nondetects And Data Analysis for environmental data. Available from: http://cran.r-project.org/ pack-age=NADA
- Lee SC (2016). A Bayesian inference for fixed effects panel probit model, Communications for sta-tistical Applications and Methods, 23, 179-187. https://doi.org/10.5351/CSAM.2016.23.2.179
- Lesaffre E and Spiessens B (2001). On the effect of the number of quadrature points in a logistic random effects model: an example, Journal of Royal Statistical Society, Applied Statistics, Series C, 50, 325-335. https://doi.org/10.1111/1467-9876.00237
- Lee Y and Nelder JA (2001). Hierarchical generalized linear models: a synthesis of generalized linear models, random-effect model and structure dispersion, Biometrika, 88, 987-1006. https://doi.org/10.1093/biomet/88.4.987
- Louis TA (1982). Finding the observed information matrix when using the EM algorithm, Journal of the Royal Statistical Society, Series B, 62, 257-270.
- Maddala GS (1983). Limited-Dependent and Qualitative Variables in Econometrics, Cambridge University Press, New York.
- Meilijson I (1989). A fast improvement to the EM algorithm on its own terms, Journal of the Royal Statistical Society, Series B, 51, 127-138.
- Meng XL and Rubin DB (1991). Using EM to obtain asymptotic variance-covariance matrices: the SEM algorithm, Journal of the American Statistical Association, 86, 899-909. https://doi.org/10.1080/01621459.1991.10475130
- McCulloch CE (1994). Maximum likelihood variance components estimation for binary data, Journal of the American Statistical Association, 89, 330-335. https://doi.org/10.1080/01621459.1994.10476474
- McCulloch CE (1996). Fixed and random effects and best prediction. In Proceedings of the Kansas State Conference on Applied Statistics in Agriculture.
- Noh M and Lee Y (2007). REML estimation for binary data in GLMMs, Journal of Multivariate Analysis, 98, 896-915. https://doi.org/10.1016/j.jmva.2006.11.009
- Patterson H and Thomson R (1971). Recovery of inter-block information when block sizes are unequal, Biometrika, 31, 100-109.
- R Core Team (2017). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Available from: https://www.R-project.org/
- SAS (2011). SAS/ETS 9.3 User's Guide.
- Searle SR, Casella G, and McCulloch CE (2006). Variance Components, John Wiley & Sons, New York.
- Stata (2017). Finite Mixture Models Reference Manual, Stata press.
- Tobin J (1958). Estimation of relationships for limited dependent variables, Econometrica, 26, 24-36. https://doi.org/10.2307/1907382
- Zhang H, Lu N, Feng C, Thurston SW, Xia Y, Zhu L, and Tu XM (2011). On fitting generalized linear mixed-effects models for binary responses using different statistical packages, Statistics in Medicine, 30, 2562-2572. https://doi.org/10.1002/sim.4265