Microeconometrics Using Stata, Second Edition, Volume II: Nonlinear Models and Casual Inference Methods

Contents
Tables
Figures
16 Nonlinear optimization methods
	16.1 Introduction
	16.2 Newton–Raphson method
		16.2.1 NR method
		16.2.2 NR method for Poisson
		16.2.3 Poisson NR example using Mata
	16.3 Gradient methods
		16.3.1 Maximization options
		16.3.2 Gradient methods
		16.3.3 Messages during iterations
		16.3.4 Stopping criteria
		16.3.5 Nonconvergence and possible lack of identification
		16.3.6 Multiple maximums
		16.3.7 Numerical derivatives
		16.3.8 Constraints on parameters
	16.4 Overview of ml, moptimize(), and optimize()
		16.4.1 Evaluator types
		16.4.2 Optimization techniques
	16.5 The ml command: lf method
		16.5.1 The ml commands
		16.5.2 The lf method
		16.5.3 Poisson example: Single-index model
		16.5.4 Negative binomial example: Two-index model
		16.5.5 NLS example: Nonlikelihood model
	16.6 Checking the program
		16.6.1 Program debugging using ml check and ml trace
		16.6.2 Getting the program to run
		16.6.3 Checking the data
		16.6.4 Multicollinearity and near collinearity
		16.6.5 Checking parameter estimation
		16.6.6 Checking standard error estimation
	16.7 The ml command: lf0–lf2, d0–d2, and gf0 methods
		16.7.1 ml evaluator functions
		16.7.2 ml methods lf0, lf1 and lf2
		16.7.3 ml methods d0, d1, and d2
		16.7.4 ml method gf0
	16.8 Nonlinear instrumental-variables (GMM) example
		16.8.1 Nonlinear IV example
		16.8.2 GMM using the Mata optimize() function
	16.9 Additional resources
	16.10 Exercises
17 Binary outcome models
	17.1 Introduction
	17.2 Some parametric models
		17.2.1 Basic model
		17.2.2 Logit, probit, linear probability, and complementary log–log models
		17.2.3 Latent-variable interpretation and identification
	17.3 Estimation
		17.3.1 ML estimation
		17.3.2 The logit and probit commands
		17.3.3 Robust estimate of the variance–covariance matrix of the estimator
		17.3.4 OLS estimation of linear probability model
		17.3.5 Estimation with proportions or fractional response data
	17.4 Example
		17.4.1 Data description
		17.4.2 Logit regression
		17.4.3 Coefficient interpretation
		17.4.4 Comparison of binary models and parameter estimates
		17.4.5 Wald tests
		17.4.6 Likelihood-ratio tests
		17.4.7 Model comparison
		17.4.8 Generalized linear model estimation
	17.5 Goodness of fit and prediction
		17.5.1 Pseudo-R measures
		17.5.2 Comparing predicted frequencies with sample frequencies
		17.5.3 Comparing predicted outcomes with actual outcomes
		17.5.4 The predict command for fitted probabilities
		17.5.5 Receiver operator characteristics curve
		17.5.6 The margins command for fitted probabilities
		17.5.7 The prvalue command for fitted probabilities
	17.6 Marginal effects
		17.6.1 AME
		17.6.2 MEM
		17.6.3 MER
		17.6.4 The prchange command for MEs
	17.7 Clustered data
	17.8 Additional models
		17.8.1 Heteroskedastic probit model
		17.8.2 Generalized logit
		17.8.3 Alternative-specific random parameters logit or mixed logit
		17.8.4 Nonparametric logit estimation
	17.9 Endogenous regressors
		17.9.1 Example
		17.9.2 Structural model
		17.9.3 Structural-model estimators
		17.9.4 Linear two-stage least-squares approach
		17.9.5 Nonlinear IV approach
	17.10 Grouped and fractional data
		17.10.1 Grouped dataset
		17.10.2 Comparison of various grouped estimators
	17.11 Additional resources
	17.12 Exercises
18 Multinomial models
	18.1 Introduction
	18.2 Multinomial models overview
		18.2.1 Probabilities and MEs
		18.2.2 Maximum likelihood estimation
		18.2.3 Robust standard errors
		18.2.4 Case-specific and alternative-specific regressors
		18.2.5 Additive random-utility model
		18.2.6 Stata multinomial model commands
	18.3 Multinomial example: Choice of fishing mode
		18.3.1 Data description
		18.3.2 Case-specific regressors
		18.3.3 Alternative-specific regressors
	18.4 Multinomial logit model
		18.4.1 The mlogit command
		18.4.2 Application of the mlogit command
		18.4.3 Coefficient interpretation
		18.4.4 Predicted probabilities
		18.4.5 MEs
	18.5 Alternative-specific conditional logit model
		18.5.1 Creating long-form data from wide-form data
		18.5.2 The cmset command
		18.5.3 The cmclogit command
		18.5.4 Application of the cmclogit command
		18.5.5 Relationship to MNL model
		18.5.6 Coefficient interpretation
		18.5.7 Predicted probabilities
		18.5.8 MEs
		18.5.9 The clogit command
		18.5.10 Rank-ordered logit
	18.6 Nested logit model
		18.6.1 Relaxing the independence of irrelevant alternatives assumption
		18.6.2 NL model
		18.6.3 The nlogit command
		18.6.4 Model estimates
		18.6.5 Predicted probabilities
		18.6.6 MEs
		18.6.7 Comparison of various MNL models
	18.7 Multinomial probit model
		18.7.1 MNP
		18.7.2 The mprobit command
		18.7.3 Maximum simulated likelihood
		18.7.4 The cmmprobit command
		18.7.5 Application of the cmmprobit command
		18.7.6 Predicted probabilities and MEs
		18.7.7 Rank-ordered probit
	18.8 Alternative-specific random-parameters logit
		18.8.1 Alternative-specific RPL
		18.8.2 The cmmixlogit command
		18.8.3 Application of the cmmixlogit command
		18.8.4 Berry–Levinson–Pakes market demand model
	18.9 Ordered outcome models
		18.9.1 Data summary
		18.9.2 Ordered outcomes
		18.9.3 Application of the ologit command
		18.9.4 Predicted probabilities
		18.9.5 MEs
		18.9.6 Other ordered models
	18.10 Clustered data
	18.11 Multivariate outcomes
		18.11.1 Bivariate probit
		18.11.2 Nonlinear SUR
	18.12 Additional resources
	18.13 Exercises
19 Tobit and selection models
	19.1 Introduction
	19.2 Tobit model
		19.2.1 Regression with censored data
		19.2.2 Tobit model setup
		19.2.3 Tobit estimation
		19.2.4 Robust standard errors
		19.2.5 The tobit command
		19.2.6 Unknown censoring point
	19.3 Tobit model example
		19.3.1 Data summary
		19.3.2 Tobit analysis
		19.3.3 Prediction after tobit
		19.3.4 MEs
		19.3.5 Truncated regression
		19.3.6 Additional commands for censored and truncated regression
		19.3.7 Clustered data
	19.4 Tobit for lognormal data
		19.4.1 Data example
		19.4.2 Setting the censoring point for data in logs
		19.4.3 Results
		19.4.4 Two-limit tobit
		19.4.5 Model diagnostics
		19.4.6 Tests of normality and homoskedasticity
		19.4.7 Next step?
	19.5 Two-part model in logs
		19.5.1 Model structure
		19.5.2 Part 1 specification
		19.5.3 Part 2 of the two-part model
	19.6 Selection models
		19.6.1 Model structure and assumptions
		19.6.2 ML estimation of the sample-selection model
		19.6.3 Estimation without exclusion restrictions
		19.6.4 Two-step estimation
		19.6.5 Estimation with exclusion restrictions
	19.7 Nonnormal models of selection
		19.7.1 Copula models
		19.7.2 Copula application
	19.8 Prediction from models with outcome in logs
		19.8.1 Predictions from tobit in logs
		19.8.2 Predictions from two-part model in logs
		19.8.3 Predictions from selection model
	19.9 Endogenous regressors
		19.9.1 Tobit model with endogenous regressors
		19.9.2 Richer models with endogenous regressors
		19.9.3 Endogenous-switching regression model
	19.10 Missing data
		19.10.1 Missing-data mechanisms
		19.10.2 Complete-case analysis
		19.10.3 Inverse probability weighting
		19.10.4 Endogenous sample selection
		19.10.5 Imputation
	19.11 Panel attrition
		19.11.1 Attrition empirical example
		19.11.2 Unbalanced and balanced panel analysis
		19.11.3 Inverse-probability weighting
		19.11.4 Selection-based panel attrition model
		19.11.5 Sample refreshment
	19.12 Additional resources
	19.13 Exercises
20 Count-data models
	20.1 Introduction
	20.2 Modeling strategies for count data
		20.2.1 Generated Poisson data
		20.2.2 Overdispersion and NB data
		20.2.3 Conditional mean modeling strategies
		20.2.4 Poisson regression for continuous nonnegative dependent variable
		20.2.5 Fully parametric modeling strategies
	20.3 Poisson and negative binomial models
		20.3.1 Data summary
		20.3.2 Poisson model
		20.3.3 NB2 model
		20.3.4 Fitted probabilities for Poisson and NB2 models
		20.3.5 Generalized NB model
		20.3.6 Nonlinear least-squares estimation
		20.3.7 Censored and truncated count regression
	20.4 Hurdle model
		20.4.1 Hurdle model
		20.4.2 Variants of the hurdle model
		20.4.3 Application of the hurdle model
	20.5 Finite-mixture models
		20.5.1 FMM specification
		20.5.2 Simulated finite-mixture sample with comparisons
		20.5.3 ML estimation of the FMM
		20.5.4 Application: Poisson FMM
		20.5.5 Application: NB2 FMM
		20.5.6 Latent class posterior probabilities
		20.5.7 Testing the equality of coefficients of mixture components
		20.5.8 Mixtures with varying mixing probability
		20.5.9 Mixtures with different sets of regressors in each component
		20.5.10 Model selection
		20.5.11 A cautionary note
	20.6 Zero-inflated models
		20.6.1 Data summary
		20.6.2 Models for zero-inflated data
		20.6.3 Results for the NB2 model
		20.6.4 Results for the ZINB model
		20.6.5 Point-mass version of zip and zinb commands
		20.6.6 Model comparison
	20.7 Endogenous regressors
		20.7.1 ML estimator of structural model
		20.7.2 Control function estimator of structural equation
		20.7.3 Nonlinear IV (GMM) estimators
		20.7.4 Weak instruments
	20.8 Clustered data
	20.9 Quantile regression for count data
		20.9.1 Quantile count regression
		20.9.2 The qcount command
		20.9.3 Doctor-visits data
		20.9.4 Results for NB
		20.9.5 Results for QCR
	20.10 Additional resources
	20.11 Exercises
21 Survival analysis for duration data
	21.1 Introduction
	21.2 Data and data summary
		21.2.1 Summary statistics
		21.2.2 The stset command
		21.2.3 Survival data organization
	21.3 Survivor and hazard functions
		21.3.1 Densities for complete and incomplete spells
		21.3.2 Survivor function
		21.3.3 Hazard rate and cumulative hazard function
	21.4 Semiparametric regression model
		21.4.1 The stcox command
		21.4.2 Cox proportional hazards model
		21.4.3 Cox PH estimates
		21.4.4 Prediction and marginal effects
		21.4.5 The stcurve command
		21.4.6 Diagnostics for the PH model
		21.4.7 Competing risks models
	21.5 Fully parametric regression models
		21.5.1 The streg command
		21.5.2 Parametric Weibull model
		21.5.3 Prediction and MEs
		21.5.4 Proportional hazards models
		21.5.5 Accelerated failure-time models
		21.5.6 Comparison of hazard functions
		21.5.7 Comparison of mean duration and associated AMEs
		21.5.8 Models with frailty
	21.6 Multiple-records data
		21.6.1 Expanding the dataset
	21.7 Discrete-time hazards logit model
		21.7.1 Relationship to Cox PH model
	21.8 Time-varying regressors
	21.9 Clustered data
	21.10 Additional resources
	21.11 Exercises
22 Nonlinear panel models
	22.1 Introduction
	22.2 Nonlinear panel-data overview
		22.2.1 FE models
		22.2.2 RE models
		22.2.3 Pooled models or PA models
		22.2.4 CRE models
		22.2.5 Prediction and MEs
		22.2.6 Stata nonlinear panel commands
		22.2.7 Cluster–robust inference
	22.3 Nonlinear panel-data example
		22.3.1 Data description and summary statistics
		22.3.2 Panel-data organization
		22.3.3 Within and between variation
		22.3.4 FE or RE model for these data?
	22.4 Binary outcome and ordered outcome models
		22.4.1 Panel summary of the dependent variable
		22.4.2 Pooled logit estimator
		22.4.3 The xtlogit command
		22.4.4 The xtgee command
		22.4.5 PA logit estimator
		22.4.6 RE logit estimator
		22.4.7 FE logit estimator
		22.4.8 Bias-corrected logit dummy-variable FE estimator
		22.4.9 CRE logit estimator
		22.4.10 Comparison of panel logit parameter estimates
		22.4.11 Panel logit prediction and MEs
		22.4.12 Mixed-effects logit estimator
		22.4.13 Dynamic panel logit models
		22.4.14 Panel probit models
		22.4.15 Panel ordered logit and probit models
	22.5 Tobit and interval-data models
		22.5.1 Panel summary of the dependent variable
		22.5.2 RE tobit model
		22.5.3 The xttobit command
		22.5.4 The xtheckman command
		22.5.5 The xtintreg and xtfrontier commands
	22.6 Count-data models
		22.6.1 The xtpoisson command
		22.6.2 Panel summary of the dependent variable
		22.6.3 Pooled Poisson estimator
		22.6.4 PA Poisson estimator
		22.6.5 RE Poisson estimators
		22.6.6 FE Poisson estimator
		22.6.7 Panel Poisson estimators comparison
		22.6.8 Panel probit prediction and MEs
		22.6.9 Negative binomial estimators
	22.7 Panel quantile regression
		22.7.1 Pooled QR estimator
		22.7.2 Panel QR with fixed effects
	22.8 Endogenous regressors in nonlinear panel models
	22.9 Additional resources
	22.10 Exercises
23 Parametric models for heterogeneity and endogeneity
	23.1 Introduction
	23.2 Finite mixtures and unobserved heterogeneity
		23.2.1 Finite mixtures model
		23.2.2 The fmm prefix
	23.3 Empirical examples of FMMs
		23.3.1 Example 1: Gamma regression for expenditures
		23.3.2 Example 2: Logit and probit regression
		23.3.3 Example 3: Multinomial logit regression
		23.3.4 Example 4: Tobit regression
		23.3.5 Example 5: Poisson regression
		23.3.6 Example 6: Mixture distribution with point mass
		23.3.7 Example 7: Mixture application using survival data
		23.3.8 Heterogeneity and quantile regression
	23.4 Nonlinear mixed-effects models
		23.4.1 Mixed-effects GLM and the meglm command
		23.4.2 Poisson and negative binomial mixed-effects example
		23.4.3 Prediction and MEs
	23.5 Linear structural equation models
		23.5.1 SEMs for linear regression
		23.5.2 SEM linear regression example
		23.5.3 SEM model specification
		23.5.4 The sem command
		23.5.5 Some sem command examples
		23.5.6 sem path builder
		23.5.7 Measurement-error example
		23.5.8 Measurement-error model estimates using a latent variable
		23.5.9 SEM endogenous regressors example
		23.5.10 SEM estimation in general
	23.6 Generalized structural equation models
		23.6.1 GSEM estimation
		23.6.2 The gsem command
		23.6.3 Some GSEM examples
		23.6.4 Poisson with latent normal variable for overdispersion
		23.6.5 Poisson model with endogeneity
		23.6.6 Prediction and MEs
		23.6.7 GSEM for FMM with multiple outcomes
	23.7 ERM commands for endogeneity and selection
		23.7.1 Overview of ERM commands
		23.7.2 The eprobit command
		23.7.3 The eprobit command application
	23.8 Additional resources
	23.9 Exercises
24 Randomized control trials and exogenous treatment effects
	24.1 Introduction
	24.2 Potential outcomes
	24.3 Randomized control trials
		24.3.1 Simple RCT setting of difference in means
		24.3.2 Optimal sample size and power analysis
		24.3.3 Sample size and power calculations in Stata
		24.3.4 Some examples
		24.3.5 Stratified randomization and clustering
		24.3.6 Limitations of size and power calculations
		24.3.7 Covariate balance
	24.4 Regression in an RCT
		24.4.1 Adding covariates
		24.4.2 Covariates interacted with treatment status
		24.4.3 Covariate balance
		24.4.4 Simulation-based example of regression for an RCT
		24.4.5 Marginal effects and nonlinearity
		24.4.6 Cluster correction
		24.4.7 Design-based inference
		24.4.8 RCT as a “gold standard” of treatment evaluation
	24.5 Treatment evaluation with exogenous treatment
	24.6 Treatment evaluation methods and estimators
		24.6.1 Regression adjustment
		24.6.2 Inverse-probability weighting
		24.6.3 Propensity-score overlap
		24.6.4 Regressor balance
		24.6.5 Doubly robust augmented IPW and IPW regression adjustment
		24.6.6 Matching methods
		24.6.7 Propensity-score matching
		24.6.8 Blocking and stratification
		24.6.9 Nearest neighbors matching
		24.6.10 Machine-learning methods
		24.6.11 Assessing unconfoundedness
		24.6.12 Robust confidence intervals for ATE
	24.7 Stata commands for treatment evaluation
		24.7.1 The teffects commands and telasso command
		24.7.2 ERM commands when treatment is exogenous
	24.8 Oregon Health Insurance Experiment example
		24.8.1 The OHIE
		24.8.2 Data summary
		24.8.3 Initial regression analysis
	24.9 Treatment-effect estimates using the OHIE data
		24.9.1 RA estimates
		24.9.2 IPW estimates
		24.9.3 Propensity-score overlap
		24.9.4 Regressor balance
		24.9.5 AIPW and IPW-RA estimates
		24.9.6 PSM estimates
		24.9.7 Blocking and stratified estimate
		24.9.8 NNM estimates
	24.10 Multilevel treatment effects
		24.10.1 Multivariate TEs methods
		24.10.2 Multivalued TEs application
		24.10.3 RA estimates
		24.10.4 AIPW estimates
	24.11 Conditional quantile TEs
	24.12 Additional resources
	24.13 Exercises
25 Endogenous treatment effects
	25.1 Introduction
	25.2 Parametric methods for endogenous treatment
		25.2.1 Canonical model
		25.2.2 Assumptions
		25.2.3 Estimation methods
		25.2.4 Computing TEs
	25.3 ERM commands for endogenous treatment
		25.3.1 The ERM commands
		25.3.2 Interpretation of ET effects
		25.3.3 Endogenous multivalued TEs application
	25.4 ET commands for binary endogenous treatment
		25.4.1 The etregress command
		25.4.2 Endogenous binary treatment application
		25.4.3 The etpoisson command
		25.4.4 Application to count data with ET
		25.4.5 Manual computation of TEs
		25.4.6 The eteffects command
	25.5 The LATE estimator for heterogeneous effects
		25.5.1 LATE with binary treatment and binary instrument
		25.5.2 Assumptions for LATE
		25.5.3 Further discussion of LATE
		25.5.4 Application of LATE
		25.5.5 Marginal TEs
	25.6 Difference-in-differences and synthetic control
		25.6.1 DID
		25.6.2 Synthetic control
	25.7 Regression discontinuity design
		25.7.1 Sharp regression discontinuity design
		25.7.2 SRD numerical and graphical illustration
		25.7.3 rdrobust package and application
		25.7.4 The rdplot command
		25.7.5 The rdrobust command
		25.7.6 The rdbwselect command
		25.7.7 Fuzzy regression discontinuity design
		25.7.8 Further discussion
	25.8 Conditional quantile regression with endogenous regressors
		25.8.1 Local conditional quantile TE with endogenous binary regressor
		25.8.2 The ivqte command
		25.8.3 Alternative estimators of conditional QTE with endogeneity
		25.8.4 Rank invariance and rank similarity
	25.9 Unconditional quantiles
		25.9.1 Unconditional QTE using inverse-propensity score weighting
		25.9.2 Unconditional QTE using recentered influence functions
		25.9.3 Counterfactual distributions
		25.9.4 Unconditional QTE with endogenous discrete binary regressor
	25.10 Additional resources
	25.11 Exercises
26 Spatial regression
	26.1 Introduction
	26.2 Overview of spatial regression models
	26.3 Geospatial data
		26.3.1 Spatial dataset
		26.3.2 Geospatial shapefile
		26.3.3 Heat maps
		26.3.4 Geospatial distance
	26.4 The spatial weighting matrix
		26.4.1 Creating a spatial weighting matrix
		26.4.2 Creating spatial lag variables
	26.5 OLS regression and test for spatial correlation
	26.6 Spatial dependence in the error
		26.6.1 Spatial heteroskedastic- and autocorrelation-consistent standard errors
		26.6.2 FGLS estimation
	26.7 Spatial autocorrelation regression models
		26.7.1 The spregress command
		26.7.2 SAR(1) in mean model
		26.7.3 Prediction and marginal effects
		26.7.4 SAR(1) in error model
		26.7.5 SARAR(1,1) model
		26.7.6 SARAR(p,q) model
	26.8 Spatial instrumental variables
	26.9 Spatial panel-data models
	26.10 Additional resources
	26.11 Exercises
27 Semiparametric regression
	27.1 Introduction
	27.2 Kernel regression
		27.2.1 Kernel-weighted local constant regression
		27.2.2 Kernel-weighted local linear regression
		27.2.3 Kernel functions
		27.2.4 Bandwidth choice
		27.2.5 The npregress kernel command
		27.2.6 Kernel-weighted local linear m-estimation
	27.3 Series regression
		27.3.1 Series regression model
		27.3.2 The npregress series command
	27.4 Nonparametric single regressor example
		27.4.1 Basic kernel-weighted local linear regression
		27.4.2 Observations not identified
		27.4.3 Trimmed means
		27.4.4 Conditional means and partial effects for different regressor values
		27.4.5 Plots for different regressor values
		27.4.6 Comparison with the lpoly command
		27.4.7 Basic series regression
	27.5 Nonparametric multiple regressor example
	27.6 Partial linear model
	27.7 Single-index model
	27.8 Generalized additive models
	27.9 Additional resources
	27.10 Exercises
28 Machine learning for prediction and inference
	28.1 Introduction
	28.2 Measuring the predictive ability of a model
		28.2.1 Generated data example
		28.2.2 Mean squared error
		28.2.3 Information criteria and related penalty measures
		28.2.4 The splitsample command
		28.2.5 Single-split cross-validation
		28.2.6 K-fold cross-validation
		28.2.7 Leave-one-out cross-validation
		28.2.8 Best subsets selection and stepwise selection
	28.3 Shrinkage estimators
		28.3.1 Ridge regression
		28.3.2 Lasso
		28.3.3 Elastic net
		28.3.4 Finite sample distribution of lasso-related estimators
	28.4 Prediction using lasso, ridge, and elasticnet
		28.4.1 The lasso command
		28.4.2 Lasso linear regression example
		28.4.3 Lasso postestimation commands example
		28.4.4 Adaptive lasso
		28.4.5 elasticnet command and ridge regression
		28.4.6 In-sample comparison of shrinkage estimators
		28.4.7 Shrinkage for logit, probit, and Poisson models
	28.5 Dimension reduction
		28.5.1 Principal components
	28.6 Machine learning methods for prediction
		28.6.1 Supervised learning for continuous outcome
		28.6.2 Neural networks
		28.6.3 Regression trees
		28.6.4 Bagging
		28.6.5 Random forests
		28.6.6 Boosting
		28.6.7 Supervised learning for categorical outcome (classification)
		28.6.8 Unsupervised learning (cluster analysis)
	28.7 Prediction application
		28.7.1 Training and holdout samples
		28.7.2 Various predictors
		28.7.3 Comparison of predictors
	28.8 Machine learning for inference in partial linear model
		28.8.1 Partial effects in the partial linear model
		28.8.2 Partial linear model application
		28.8.3 Partialing-out estimator
		28.8.4 The poregress command and related commands
		28.8.5 Plugin penalty parameter
		28.8.6 Partialing-out application
		28.8.7 Clustered errors application
		28.8.8 Orthogonalization
		28.8.9 Cross-fit partialing-out estimator
		28.8.10 Double-selection estimator
	28.9 Machine learning for inference in other models
		28.9.1 Estimators for exponential conditional mean models
		28.9.2 Estimators for the logit model
		28.9.3 Partialing-out for IV estimation
		28.9.4 Further discussion
	28.10 Additional resources
	28.11 Exercises
29 Bayesian methods: Basics
	29.1 Introduction
	29.2 Bayesian introductory example
		29.2.1 The bayes prefix
		29.2.2 Bayesian estimates using default priors
	29.3 Bayesian methods overview
		29.3.1 Posterior distribution
		29.3.2 MCMC methods
		29.3.3 Metropolis–Hastings algorithm
		29.3.4 Sampling efficiency
		29.3.5 Gibbs sampling algorithm
		29.3.6 Bayesian and classical approaches compared
		29.3.7 The bayesmh command
	29.4 An i.i.d. example
		29.4.1 MLE
		29.4.2 Bayesian analysis
		29.4.3 Bayesian inference
		29.4.4 MCMC diagnostics
		29.4.5 MCMC diagnostics using different chains
		29.4.6 Sensitivity analysis
		29.4.7 Further analysis of the draws
		29.4.8 Analytical results
	29.5 Linear regression
		29.5.1 Normal regression with variance known
		29.5.2 Prior distributions for and
		29.5.3 Posterior densities for normal regression with variance unknown
	29.6 A linear regression example
		29.6.1 MLE
		29.6.2 Specifying the prior
		29.6.3 Bayesian analysis
		29.6.4 A supposed uninformative prior can be informative
		29.6.5 A prior can lead to an unidentified model being identified
	29.7 Modifying the MH algorithm
		29.7.1 Blocking parameters
		29.7.2 Gibbs sampler within MH
	29.8 RE model
		29.8.1 RE MLE
		29.8.2 Bayesian RE models using hierarchical priors
	29.9 Bayesian model selection
		29.9.1 Bayes factors
		29.9.2 Posterior odds ratio
	29.10 Bayesian prediction
		29.10.1 Posterior predictive distribution
		29.10.2 The bayespredict command
		29.10.3 Bayesian prediction example
	29.11 Probit example
		29.11.1 MLE
		29.11.2 Bayesian analysis
		29.11.3 MEs
	29.12 Additional resources
	29.13 Exercises
30 Bayesian methods: Markov chain Monte Carlo algorithms
	30.1 Introduction
	30.2 User-provided log likelihood
		30.2.1 Data and probit maximum-likelihood estimator
		30.2.2 The bayesmh command
		30.2.3 The bayesmh command with user-provided evaluator
	30.3 MH algorithm in Mata
		30.3.1 The log posterior
		30.3.2 Random-walk Metropolis algorithm
		30.3.3 Numerical example in Mata
	30.4 Data augmentation and the Gibbs sampler in Mata
		30.4.1 Data augmentation
		30.4.2 Gibbs sampler data-augmentation algorithm for probit
		30.4.3 Numerical example
		30.4.4 Further examples
	30.5 Multiple imputation
		30.5.1 Missingness mechanisms
		30.5.2 Estimation and inference for multiple imputation
		30.5.3 Regression-based imputation
		30.5.4 Imputation by data augmentation
		30.5.5 The mi import, mi impute, and mi estimate commands
	30.6 Multiple-imputation example
		30.6.1 Describing missing-value patterns
		30.6.2 Data imputation
		30.6.3 Estimation using imputed data
	30.7 Additional resources
	30.8 Exercises
Glossary of abbreviations
Glossary of abbreviations
References
References
Author index
Author index
Subject index
Subject index