Maximum Likelihood Estimation with Stata, 4th Edition

by William Gould, Jeffrey Pitblado, and Brian Poi

 Maximum Likelihood Estimation with Stata, 4th Edition, is the essential reference and guide for researchers in all disciplines who wish to write maximum likelihood (ML) estimators in Stata. Beyond providing comprehensive coverage of Stata’s ml command for writing ML estimators, the book presents an overview of the underpinnings of maximum likelihood and how to think about ML estimation.

The book shows you how to take full advantage of the ml command’s noteworthy features:

    * linear constraints
    * four optimization algorithms (Newton–Raphson, DFP, BFGS, and BHHH)
    * observed information matrix (OIM) variance estimator
    * outer product of gradients (OPG) variance estimator
    * Huber/White/sandwich robust variance estimator
    * cluster-robust variance estimator
    * complete and automatic support for survey data analysis
    * direct support of evaluator functions written in Mata

When appropriate options are used, many of these features are provided automatically by ml and require no special programming or intervention by the researcher writing the estimator.

The fourth edition has been updated to include new features introduced in Stata 11. Such features include new methods for handling scores, more consistent arguments for likelihood-evaluator programs, a general quadratic form evaluator for problems like nonlinear least squares and generalized methods of moments (GMM), and support for likelihood evaluators written in Mata (Stata’s matrix programming language). The authors illustrate how to write your estimation command so that it fully supports factor variable notation and the svy prefix for estimation with survey data. They have also restructured the chapters that introduce ml in a way that gets you started quicker. This edition is essential for anyone using Stata 11.

In the final chapter, the authors illustrate the major steps required to get from log-likelihood function to fully operational estimation command. This is done using several different models: logit and probit, linear regression, Weibull regression, the Cox proportional hazards model, random-effects regression, and seemingly unrelated regression.

The authors provide extensive advice for developing your own estimation commands. With a little care and the help of this book, users will be able to write their own estimation commands—commands that look and behave just like the official estimation commands in Stata.

Whether you want to fit a special ML estimator for your own research or wish to write a general-purpose ML estimator for others to use, you need this book.

Table of contents

List of Tables

List of Figures

Preface to the fourth edition

Notation and typography

Versions of Stata

1 Theory and Practice

1.1 The likelihood maximization problem

1.2 Likelihood theory

1.2.1 All results are asymptotic

1.2.2 Variance estimates and hypothesis tests

1.2.3 Likelihood-ratio tests and Wald tests

1.2.4 The outer product of gradients variance estimator

1.2.5 Robust variance estimates

1.3 The maximization problem

1.3.1 Numerical root finding

      Newton's method

      The Newton-Raphson algorithm

1.3.2 Quasi-Newton methods

      The BHHH algorithm

      The DFP and BFGS algorithms

1.3.3 Numerical maximization

1.3.4 Numerical derivatives

1.3.5 Numerical second derivatives

1.4 Monitoring convergence

2 Introduction to of ml

2.1 The probit model

2.2 Normal linear regression 

2.3 Robust standard errors 

2.4 Weighted estimation

2.5 Other features of method-gf0 evaluators

2.6 Limitations 

3 Overview of ml

3.1 The terminology of ml 

3.2 Equations in ml 

3.3 Likelihood-evaluator methods 

3.4 Tools for the ml programmer 

3.5 Common ml options

                    3.5.1 Subsamples
                    3.5.2 Weights
                    3.5.3 OPG estimates of variance
                    3.5.4 Robust estimates of variance
                    3.5.5 Survey data
                    3.5.6 Constraints
                    3.5.7 Choosing among the optimization algorithms

3.6 Maximizing your own likelihood functions 

4 Methods lf

5 Methods lf0, lf1, and lf2

5.1 Comparing these methods
5.2 Outline of evaluators of methods lf0, lf1, and lf2
          5.2.1 The todo argument
          5.2.2 The b argument
                   Using mleval to obtain values from each equation
          5.2.3 The lnfj argument
          5.2.4 Arguments for scores
          5.2.5 The H argument
                   Using mlmatsum to define H
          5.2.6 Aside: Stata’s scalars
5.3 Summary of methods lf0, lf1, and lf2
          5.3.1 Method lf0
          5.3.2 Method lf1
          5.3.3 Method lf2
5.4 Examples
          5.4.1 The probit model
          5.4.2 Normal linear regression
          5.4.3 The Weibull model 

6 Methods d0, d1, and d2

6.1 Comparing these methods
6.2 Outline of method d0, d1, and d2 evaluators
          6.2.1 The todo argument
          6.2.2 The b argument
          6.2.3 The lnf argument
                  Using lnf to indicate that the likelihood cannot be calculated
                  Using mlsum to define lnf
          6.2.4 The g argument
                  Using mlvecsum to define g
          6.2.5 The H argument
6.3 Summary of methods d0, d1, and d2
          6.3.1 Method d0
          6.3.2 Method d1
          6.3.3 Method d2
6.4 Panel-data likelihoods
          6.4.1 Calculating lnf
          6.4.2 Calculating g
          6.4.3 Calculating H
                  Using mlmatbysum to help define H
6.5 Other models that do not meet the linear-form restrictions

7 Debugging likelihood evaluators

7.1 ml check
7.2 Using the debug methods
          7.2.1 First derivatives
          7.2.2 Second derivatives
7.3 ml trace

8 Setting initial values

8.1 ml search
8.2 ml plot
8.3 ml init

9 Interactive maximization

9.1 The iteration log
9.2 Pressing the Break key
9.3 Maximizing difficult likelihood functions

10 Final results

10.1 Graphing convergence
10.2 Redisplaying output

11Mata-based likelihood evaluators

11.1 Introductory examples
           11.1.1 The probit model
           11.1.2 The Weibull model
11.2 Evaluator function prototypes
           Method-lf evaluators
           lf-family evaluators
           d-family evaluators
11.3 Utilities
           Dependent variables
           Obtaining model parameters
           Summing individual or group-level log likelihoods
           Calculating the gradient vector
           Calculating the Hessian
11.4 Random-effects linear regression
           11.4.1 Calculating lnf
           11.4.2 Calculating g
           11.4.3 Calculating H
           11.4.4 Results at last 

12 Writing do-files to maximize likelihoods

12.1 The structure of a do-file
12.2 Putting the do-file into production

13 Writing ado-files to maximize likelihoods

13.1 Writing estimation commands
13.2 The standard estimation-command outline
13.3 Outline for estimation commands using ml
13.4 Using ml in noninteractive mode
13.5 Advice
           13.5.1 Syntax
           13.5.2 Estimation subsample
           13.5.3 Parsing with help from mlopts
           13.5.4 Weights
           13.5.5 Constant-only model
           13.5.6 Initial values
           13.5.7 Saving results in e()
           13.5.8 Displaying ancillary parameters
           13.5.9 Exponentiated coefficients
           13.5.10 Offsetting linear equations
           13.5.11 Program properties

14 Writing ado-files for survey data analysis

14.1 Program properties
14.2 Writing your own predict command

References

Author Index

Subject Index

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