Business - Beta Regression Practical Issues in Estimation 1...

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1/09/2005 Beta Regression: Practical Issues in Estimation 1 Beta Regression: Practical Issues in Estimation Michael Smithson and Jay Verkuilen, The Australian National University and University of Illinois at Urbana-Champaign General Procedures This document supplements the paper by Smithson and Verkuilen (2005) on beta regression, and focuses on maximum likelihood estimation procedures in several statistical packages. Maximizing the likelihood function can be achieved using a Newton-Raphson or a quasi-Newton method. Ferrari and Cribari-Neto (2004) use Fisher scoring. Buckley (2002) has used MCMC estimation in winBUGS, which provides a Bayesian posterior density. Buckley (2002) also provided Stata code and Paolino (2001) provided Gauss code, both of which compute maximum likelihood estimates. We have estimated beta regressions using R, SPlus, SAS, SPSS, Mathematica, and winBUGs. Syntax and/or script files for all of these packages are freely available on this site. Different packages require more or less information from the user. SAS, for instance, only requires the likelihood and analytically computes the derivatives for Newton methods, whereas Mathematica’s Newton-Raphson routine requires the user to supply expressions for the derivatives. The major difference between Newton-Raphson and quasi-Newton is in the number of function evaluations per iteration (more for Newton-Raphson) and the number of iterations necessary (more for quasi-Newton). The domain of convergence for different algorithms, and hence importance of good starting values, will differ across algorithms as well. The trust region Newton method implemented in SAS seems to be particularly stable on hard problems and we recommend its use when convergence might be a problem. In general, speed of execution for ML estimation is proportional to N + p 2 , where N is the sample size and p is the number of parameters. We have never observed a well-specified model given
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1/09/2005 Beta Regression: Practical Issues in Estimation 2 good starting values taking longer than a few seconds to converge, even on a fairly modest laptop. If a Newton or quasi-Newton method is used, asymptotic standard errors usually are estimated from the inverse of the final Hessian matrix. Bayesian estimation gives posterior densities from which the Bayesian analogs of frequentist stability measures can be taken, e.g., the 2.5% and 97.5% quantiles of the posterior density as analogous quantities to a 95% confidence interval. Though it is generally recommended in the literature that the Newton estimate of the Hessian be used to provide asymptotic standard errors, we have tried both methods on several data sets and it has never seemed to make an appreciable difference. Well-chosen starting values are needed to ensure convergence when more than a few
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Business - Beta Regression Practical Issues in Estimation 1...

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