Introduction to Modern Bayesian Econometrics

by Lancaster, Tony

• ISBN13: 9781405117203
• ISBN10: 1405117206
• Edition: 1st
• Format: Paperback
• Copyright: 2004-06-18
• Publisher: Wiley-Blackwell

Summary

In this new and expanding area, Tony Lancaster's text is the first comprehensive introduction to the Bayesian way of doing applied economics. Uses clear explanations and practical illustrations and problems to present innovative, computer-intensive ways for applied economists to use the Bayesian method; Emphasizes computation and the study of probability distributions by computer sampling; Covers all the standard econometric models, including linear and non-linear regression using cross-sectional, time series, and panel data; Details causal inference and inference about structural econometric models; Includes numerical and graphical examples in each chapter, demonstrating their solutions using the S programming language and Bugs software Supported by online supplements, including Data Sets and Solutions to Problems, at www.blackwellpublishing.com/lancaster

Author Biography

Tony Lancaster is Herbert H. Goldberger Professor of Economics and Professor of Community Health at Brown University. He is the author of The Econometric Analysis of Transition Data (1990), an Econometric Society Monograph.

Table of Contents

Preface	xi
Introduction	xiii
1 The Bayesian Algorithm	1	(69)
1.1 Econometric Analysis	1	(1)
1.2 Statistical Analysis	2	(1)
1.3 Bayes' Theorem	3	(7)
1.3.1 Parameters and data	8	(1)
1.3.2 The Bayesian algorithm	9	(1)
1.4 The Components of Bayes' Theorem	10	(48)
1.4.1 The likelihood ρ(y|theta)	10	(19)
1.4.2 The prior ρ(theta)	29	(12)
1.4.3 The posterior ρ(theta|y)	41	(15)
1.4.4 Decisions	56	(2)
1.5 Conclusion and Summary	58	(1)
1.6 Exercises and Complements	58	(6)
1.7 Appendix to Chapter 1: Some Probability Distributions	64	(3)
1.8 Bibliographic Notes	67	(3)
2 Prediction and Model Criticism	70	(42)
2.1 Methods of Model Checking	71	(1)
2.2 Informal Model Checks	72	(5)
2.2.1 Residual QQ plots	73	(4)
2.3 Uncheckable Beliefs?	77	(2)
2.4 Formal Model Checks	79	(9)
2.4.1 Predictive distributions	79	(1)
2.4.2 The prior predictive distribution	79	(3)
2.4.3 Using the prior predictive distribution to check your model	82	(2)
2.4.4 Improper prior predictive distributions	84	(1)
2.4.5 Prediction from training samples	84	(4)
2.5 Posterior Prediction	88	(9)
2.5.1 Posterior model checking	90	(1)
2.5.2 Sampling the predictive distribution	91	(6)
2.6 Posterior Odds and Model Choice	97	(5)
2.6.1 Two approximations to Bayes factors	100	(1)
2.6.2 Model averaging	101	(1)
2.7 Enlarging the Model	102	(4)
2.8 Summary	106	(1)
2.9 Exercises	107	(4)
2.10 Bibliographic Notes	111	(1)
3 Linear Regression Models	112	(71)
3.1 Introduction	112	(1)
3.2 Economists and Regression Models	112	(2)
3.2.1 Mean independence	114	(1)
3.3 Linear Regression Models	114	(27)
3.3.1 Independent, normal, homoscedastic errors	116	(4)
3.3.2 Vague prior beliefs about β andτ	120	(3)
3.3.3 The two marginals under a vague prior	123	(2)
3.3.4 Highest posterior density intervals and regions	125	(5)
3.3.5 The least squares line	130	(3)
3.3.6 Informative prior beliefs	133	(1)
3.3.7 Sampling the posterior density of β	134	(1)
3.3.8 An approximate joint posterior distribution	135	(6)
3.4 A Multinomial Approach to Linear Regression	141	(6)
3.4.1 Comments on the multinomial approach	146	(1)
3.5 Checking the Normal Linear Model	147	(5)
3.6 Extending the Normal Linear Model	152	(19)
3.6.1 Criticizing the gasoline model	152	(6)
3.6.2 Generalizing the error distribution	158	(10)
3.6.3 Model choice	168	(3)
3.7 Conclusion and Summary of the Argument	171	(1)
3.8 Appendix to Chapter 3	172	(8)
3.8.1 Analytical results in the normal linear model	172	(3)
3.8.2 Simulating dirichlet variates	175	(2)
3.8.3 Some probability distributions	177	(3)
3.9 Exercises and Complements	180	(1)
3.10 Bibliographic Notes	181	(2)
4 Bayesian Calculations	183	(44)
4.1 Normal Approximations	186	(2)
4.2 Exact Sampling in One Step	188	(4)
4.2.1 Rejection sampling	188	(2)
4.2.2 Inverting the distribution function	190	(2)
4.3 Markov Chain Monte Carlo	192	(15)
4.3.1 Markov chains and transition kernels	192	(2)
4.3.2 The state distribution, ρt(χ)	194	(1)
4.3.3 Stationary distributions	195	(1)
4.3.4 Finding the stationary distribution given a kernel	195	(4)
4.3.5 Finite discrete chains	199	(2)
4.3.6 More general chains	201	(2)
4.3.7 Convergence	203	(1)
4.3.8 Ergodicity	204	(1)
4.3.9 Speed	205	(1)
4.3.10 Finding kernels with a given stationary distribution	206	(1)
4.4 Two General Methods of Constructing Kernels	207	(17)
4.4.1 The Gibbs Sampler	207	(5)
4.4.2 The Metropolis method	212	(2)
4.4.3 Metropolis-Hastings	214	(1)
4.4.4 Practical convergence	215	(2)
4.4.5 Using samples from the posterior	217	(4)
4.4.6 Calculating the prior predictive density	221	(1)
4.4.7 Implementing markov chain monte carlo	222	(2)
4.5 Conclusion	224	(1)
4.6 Exercises and Complements	224	(2)
4.7 Bibliographic Notes	226	(1)
5 Non-linear Regression Models	227	(38)
5.1 Estimation of Production Functions	227	(4)
5.1.1 Criticisms of this model	230	(1)
5.2 Binary Choice	231	(5)
5.2.1 Probit likelihoods	232	(2)
5.2.2 Criticisms of the probit model	234	(1)
5.2.3 Other models for binary choice	234	(2)
5.3 Ordered Multinomial Choice	236	(3)
5.3.1 Data augmentation	238	(1)
5.3.2 Parameters of interest	238	(1)
5.4 Multinomial Choice	239	(1)
5.5 Tobit Models	240	(6)
5.5.1 Censored linear models	240	(1)
5.5.2 Censoring and truncation	240	(4)
5.5.3 Selection models	244	(2)
5.6 Count Data	246	(5)
5.6.1 Unmeasured heterogeneity in non-linear regression	247	(3)
5.6.2 Time series of counts	250	(1)
5.7 Duration Data	251	(5)
5.7.1 Exponential durations	253	(1)
5.7.2 Weibull durations	254	(1)
5.7.3 Piecewise constant hazards	255	(1)
5.7.4 Heterogeneous duration models	255	(1)
5.8 Concluding Remarks	256	(1)
5.9 Exercises	257	(1)
5.10 Appendix to Chapter 5: Some Further Distributions	258	(5)
5.10.1 The lognormal family	258	(1)
5.10.2 Truncated normal distributions	258	(2)
5.10.3 The poisson family	260	(1)
5.10.4 The heterogeneous poisson or negative binomial family	260	(1)
5.10.5 The weibull family	261	(2)
5.11 Bibliographic Notes	263	(2)
6 Randomized, Controlled, and Observational Data	265	(12)
6.1 Introduction	265	(1)
6.2 Designed Experiments	266	(6)
6.2.1 Randomization	266	(1)
6.2.2 Controlled experimentation	267	(1)
6.2.3 Randomization and control in economics	268	(1)
6.2.4 Exogeneity and endogeneity in economics	269	(3)
6.3 Simpson's Paradox	272	(3)
6.4 Conclusions	275	(1)
6.5 Appendix to Chapter 6: Koopmans' Views on Exogeneity	275	(1)
6.6 Bibliographic Notes	275	(2)
7 Models for Panel Data	277	(34)
7.1 Panel Data	277	(1)
7.2 How Do Panels Help?	278	(3)
7.3 Linear Models on Panel Data	281	(15)
7.3.1 Likelihood	282	(2)
7.3.2 A uniform prior on the individual effects	284	(7)
7.3.3 Exact sampling	291	(1)
7.3.4 A hierarchical prior	291	(2)
7.3.5 BUGS program	293	(1)
7.3.6 Shrinkage	294	(1)
7.3.7 A richer prior for the individual effects	295	(1)
7.4 Panel Counts	296	(4)
7.4.1 A uniform prior on the individual effects	297	(2)
7.4.2 A gamma prior for the individual effects	299	(1)
7.4.3 Calculation in the panel count model	300	(1)
7.5 Panel Duration Data	300	(2)
7.6 Panel Binary Data	302	(6)
7.6.1 Parameters of interest	303	(1)
7.6.2 Choices of prior	303	(2)
7.6.3 Orthogonal reparametrizations	305	(1)
7.6.4 Implementation of the model	306	(2)
7.7 Concluding Remarks	308	(1)
7.8 Exercises	308	(1)
7.9 Bibliographic Notes	309	(2)
8 Instrumental Variables	311	(31)
8.1 Introduction	311	(1)
8.2 Randomizers and Instruments	311	(1)
8.3 Models and Instrumental Variables	312	(3)
8.4 The Structure of a Recursive Equations Model	315	(2)
8.4.1 Identification	316	(1)
8.5 Inference in a Recursive System	317	(3)
8.5.1 Likelihood surfaces with weak instruments	318	(2)
8.6 A Numerical Study of Inference with Instrumental Variables	320	(5)
8.6.1 Generating data for a simulation study	321	(1)
8.6.2 A BUGS model statement	321	(1)
8.6.3 Simulation results	322	(3)
8.7 An Application of IV Methods to Wages and Education	325	(10)
8.7.1 Is education endogenous?	332	(3)
8.8 Simultaneous Equations	335	(5)
8.8.1 Likelihood identification	338	(1)
8.8.2 Inference in simultaneous equations models	339	(1)
8.9 Concluding Remarks About Instrumental Variables	340	(1)
8.10 Bibliographic Notes	341	(1)
9 Some Time Series Models	342	(17)
9.1 First Order Autoregression	342	(12)
9.1.1 Likelihoods and priors	346	(3)
9.1.2 BUGS implementation	349	(1)
9.1.3 Some calculations	350	(4)
9.2 Stochastic Volatility	354	(2)
9.3 Extensions	356	(1)
9.4 Exercises	356	(1)
9.5 Bibliographic Notes	357	(2)
Appendix 1: A Conversion Manual	359	(6)
A1.1 The Frequentist Approach	359	(2)
A1.2 The Bayesian Contrast	361	(4)
Appendix 2: Programming	365	(10)
A2.1 S	365	(2)
A2.2 WinBUGS	367	(8)
A2.2.1 Formulating the model and inputting data from within BUGS	368	(2)
A2.2.2 Running the sampler	370	(2)
A2.2.3 Running BUGS from R	372	(1)
A2.2.4 Special likelihoods and the ones trick	373	(1)
A2.2.5 Computing references	373	(2)
Appendix 3: BUGS Code	375	(8)
A3.1 Normal Linear Model	375	(1)
A3.2 Heteroscedastic Regression	375	(1)
A3.3 Regression with Autocorrelated Errors	376	(1)
A3.4 CES Production Function	376	(1)
A3.5 Probit Model	377	(1)
A3.6 Tobit Model	377	(1)
A3.7 Truncated Normal	378	(1)
A3.8 Ordered Probit	378	(1)
A3.9 Poisson Regression	379	(1)
A3.10 Heterogeneous Poisson Regression	379	(1)
A3.11 Right Censored Weibull Data	379	(1)
A3.12 A Censored Heterogeneous Weibull Model	380	(1)
A3.13 An Over-identified Recursive Equations Model	380	(1)
A3.14 A Just Identified Simultaneous Equations Model	381	(1)
A3.15 A Panel Data Linear Model	381	(1)
A3.16 A Second Order Autoregression	382	(1)
A3.17 Stochastic Volatility	382	(1)
References	383	(8)
Index	391