dougherty3e_ch10 - Dougherty: Introduction to Econometrics...

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Dougherty: Introduction to Econometrics 3e Study Guide Chapter 10 Binary choice and limited dependent variable models, and maximum likelihood estimation Overview The first part of this chapter describes the linear probability model, logit analysis, and probit analysis, three techniques for fitting regression models where the dependent variable is a qualitative characteristic. Next it discusses tobit analysis, a censored regression model fitted using a combination of linear regression analysis and probit analysis. This leads to sample selection models and heckman analysis. The second part of the chapter introduces maximum likelihood estimation, the method used to fit all of these models except the linear probability model. Further material Limiting distributions and the properties of maximum likelihood estimators Provided that weak regularity conditions involving the differentiability of the likelihood function are satisfied, maximum likelihood (ML) estimators have the following attractive properties in large samples: (1) They are consistent. (2) They are asymptotically normally distributed. (3) They are asymptotically efficient. The meaning of the first property is familiar. It implies that the probability density function of the estimator collapses to a spike at the true value. This being the case, what can the other assertions mean? If the distribution becomes degenerate as the sample size becomes very large, how can it be described as having a normal distribution? And how can it be described as being efficient, when its variance, and the variance of any other consistent estimator, tend to zero? To discuss the last two properties, we consider what is known as the limiting distribution of an estimator. This is the distribution of the estimator when the divergence between it and its population mean is multiplied by n . If we do this, the distribution of a typical estimator remains nondegenerate as n becomes large, and this enables us to say meaningful things about its shape and to make comparisons with the distributions of other estimators (also multiplied by n ). To put this mathematically, suppose that there is one parameter of interest, θ , and that is its ML estimator. Then (2) says that ˆ ( ) ( ) 2 , 0 ~ ˆ σϑ N n for some variance . (3) says that , given any other consistent estimator 2 σ ~ , ( ) θθ ~ n cannot have smaller variance. © Christopher Dougherty, 2007 The material in this book has been adapted and developed from material originally produced for the degrees and diplomas by distance learning offered by the University of London External System ( )
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Dougherty: Introduction to Econometrics 3e Study Guide Test procedures for maximum likelihood estimation This section on ML tests contains material that is a little advanced for an introductory econometrics course. It is provided because likelihood ratio tests are encountered in the sections on binary choice models and because a brief introduction may be of help to those who proceed to a more advanced course.
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dougherty3e_ch10 - Dougherty: Introduction to Econometrics...

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