Modified Two-stage Least Squares Methods for Estimating Parameters in Nonlinear Regression Models with Correlated Errors

Wannapa   Pukdee

Authors

Wannapa Pukdee Department of General Science, Faculty of Sciences and Engineering, Kasetsart University Chalermphrakiat Sakon Nakhon Province Campus

Keywords:

nonlinear regression , correlated errors , least squares estimation , the two-stage least squares method

Abstract

The two-stage least squares method is used to fit nonlinear regression models with correlated errors based on a stationary autoregressive process of order one. This method tends to underestimate the standard error of parameter estimates. Therefore, this paper presents modified two-stage least squares methods by using residuals from the one-way ANOVA model and estimating the correlation coefficient from the conditional least squares procedure to construct a weight matrix. These methods are used to estimate all parameters of nonlinear regression models. A simulation study covers a wide range of correlation levels on the models. The study result shows that the modified two-stage least squares methods can improve the efficient statistical inference since they produce unbiased estimators of parameters and reduce bias in estimating standard errors for parameter estimates.

References

Asikgil, B., & Erar, A. (2009). Modified two-stage least squares method. In Proceedings of the XIII International Conference on Applied Stochastic Models and Data Analysis. (pp.124-128). Lithuania: Vilnius.

Bates, D.M, & Watts, D.G. (1988). Nonlinear regression analysis and its applications. New York: John Wiley & Sons.

Bender, R., & Heinemann, L. (1995). Fitting nonlinear regression models with correlated errors to individual pharmacodynamics data using SAS software. Journal of Pharmacokinetics and Biopharmaceutics, 23(1), 87-100.

Bender, R. (1995). Calculating summary measures of unimodal response curve by means of nonlinear regression models. Journal of Theoretical Medicine, 2(1), 73-80.

Crawley, M.J. (2013). The R book. West Sussex: John Wiley & Sons.

Gallant, A.R., & Goebel, J.J. (1976). Nonlinear regression with autoregressive errors. Journal of the American Statistical Association, 71(356), 961-967.

Izumo, M., Johnson, C.H., & Yamazaki, S. (2003). Circadian gene expression in mammalian fibroblasts revealed by real-time luminescence reporting: Temperature compensation and damping. The National Academy of Sciences of the USA, 100(26), 16089-16094.

Maier, B., Wendt, S., Vanselow, J.T., Wallach, T., Reischl, S., Oehmke, S., Schlosser, A., & Kramer,A. (2009). A large-scale functional RNAi screen reveals a role for CK2 in the mammalian circadian clock. Genes & Development, 23(1), 708-718.

Park, R.E., & Mitchell, B.M. (1980). Estimating the autocorrelated error model with trended data. Journal of Econometrics, 13(2), 185-201.

Pukdee, W., Polsen, O., & Baksh, M.F. (2018) Improved methods for the analysis of circadian rhythms in correlated gene expression data. Songklanakarin Journal of Science and Technology, 40(3), 692-700.

Pukdee, W., Polsen, O., & Baksh, M.F. (2020). A Modified Two-Stage Method for Parameter Estimation in Sinusoidal Models of Correlated Gene Expression Profiles. Thailand Statistician, 18(1), 77-89.

R Core Team. (2013). A language and environment for statistical computing. R Foundation for Statistical Computing. Retrieved March 3, 2015, from: http://www.R-project.org/

Ritz, C., & Streibig, J.C. (2008). Nonlinear regression with R. New York: Springer.

Seber, G.A.F., & Wild, C.J. (2003). Nonlinear regression. New York: Wiley.