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Econometrics
Comment from the Stata technical groupFranco Peracchi’s Econometrics offers concise advanced theoretical discussions of a wide range of econometric topics. This book provides a good theorem–proof treatment of many topics in econometrics. Although there are no empirical examples, this book reminds the reader of just how much econometrics can be learned by concentrating on the assumptions and derivations. The technical details that make up the heart of each chapter are framed between enlightening introductions and bibliographic surveys. Aside from covering the standard topics that surround the linear statistical model, the text also treats nonparametric methods, M-estimation, and adaptive and robust regression estimators. The author includes several important topics that are frequently overlooked in advanced theoretical econometrics textbooks; for instance, Peracchi discusses stratified sampling, projection pursuit regression estimation, and the generalized linear model. Table of contentsPreface
Notation
Mathematical symbols
Statistical symbols Abbreviations 1. Regression Models
1.1 Parametric statistical models
1.1.1 Examples of parametric models
1.2 Conditional parametric models1.1.2 Location and scale models 1.1.3 The quantile function 1.1.4 Hazard rate models 1.1.5 Identifiability 1.1.6 Regular parametric models 1.1.7 Exponential families
1.2.1 Conditioning
1.3 Nonparametric and semiparametric problems1.2.2 Exogeneity 1.2.3 Conditional location and scale models 1.2.4 Conditional parametrizations
1.3.1 Unconditional prediction problems
1.4 Statistical models as approximations1.3.2 Conditional prediction problems 1.3.3 Models of the conditional mean 1.3.4 Models of the conditional variance 1.3.5 Models of conditional quantiles
1.4.1 Parametric models as approximations
1.4.2 Linear prediction 1.4.3 Relation between the BLP and the CMF 2. Sampling
2.1 Sampling from a finite population
2.1.1 SRS with replacement
2.2 Sampling from an infinite population2.1.2 SRS without replacement 2.1.3 Unequal probability sampling 2.1.4 Stratified sampling 2.1.5 Optimal sample allocation
2.2.1 Properties of the sample mean and variance
2.3 Selective sampling2.2.2 The empirical distribution function
2.3.1 Censored and truncated sampling
2.4 Unobserved heterogeneity and mixture models2.3.2 The Kaplan–Meier estimator 2.3.3 Bounds on the regression function under censoring 2.3.4 Response-based sampling 2.3.5 Flow and stock sampling 2.5 Measurement errors
2.5.1 The classical measurement error model
2.5.2 Other measurement error models 2.5.3 Contaminated sampling 3. Time Series
3.1 Univariate time series
3.1.1 Stochastic processes
3.2 ARMA processes3.1.2 Stationarity 3.1.3 Autocovariances and autocorrelations 3.1.4 Stability and Invertibility 3.1.5 Prediction 3.1.6 The lag operator 3.1.7 Generating functions 3.1.8 Linear processes 3.1.9 The spectral density
3.2.1 Stationarity, stability and invertibility
3.3 Multivariate time series3.2.2 Prediction 3.2.3 ARMA-ARCH processes
3.3.1 Stationarity and autocovariances
3.4 Models for nonstationary time series3.3.2 Multivariate ARMA processes 3.3.3 Impulse response analysis 3.3.4 Granger noncausality 3.3.5 Exogeneity
3.4.1 Deterministic component models
3.4.2 Integrated processes 3.4.3 Cointegration 4. Point Estimation
4.1 The analogy principle
4.2 Estimating moments and quantiles
4.2.1 Sample moments
4.3 Estimating conditional moments and quantiles4.2.2 Estimating the moments of a stationary time series 4.2.3 The method of moments 4.2.4 Sample quantiles
4.3.1 Nonparametric methods
4.4 Maximum likelihood estimation4.3.2 Parametric methods 4.3.3 Ordinary least squares
4.4.1 ML estimates
4.5 Minimum variance unbiasedness4.4.2 Likelihood equations and local ML estimates
4.5.1 Sufficiency
4.6 Asymptotic properties of ML estimators4.5.2 The Cramér–Rao bound
4.6.1 Consistency
4.7 Bayes methods4.6.2 Asymptotic normality
4.7.1 Bayes theorem
4.8 Statistical decision problems4.7.2 Conjugate priors 4.7.3 Diffuse priors 4.7.4 The binomial model 4.7.5 Bayes point estimates 4.7.6 The Gaussian location model
4.8.1 Statistical games
4.8.2 The minimax and Bayes principles 4.8.3 The complete class theorem 5. Statistical Accuracy and Hypothesis Testing
5.1 Assessing variability and bias
5.1.1 Assessing variability
5.2 Confidence sets5.1.2 The nonparametric bootstrap 5.1.3 The parametric bootstrap 5.1.4 Jackknife estimates of variance 5.1.5 Assessing bias
5.2.1 Classical confidence sets
5.3 Hypothesis testing5.2.2 Bootstrap confidence intervals 5.2.3 Bayesian confidence sets
5.3.1 Statistical tests
5.4 Likelihood-based tests5.3.2 Duality between confidence sets and critical regions 5.3.3 Optimality in testing 5.3.4 The Neyman–Pearson theorem 5.3.5 Composite alternatives
5.4.1 The likelihood ratio principle
5.5 Bayesian hypothesis testing5.4.2 The Wald principle 5.4.3 The score principle 5.4.4 Partitioned parameters 5.4.5 Sampling distribution 5.4.6 Bootstrap hypothesis testing
5.5.1 Simple hypotheses
5.5.2 Composite hypotheses 5.5.3 Multiple hypotheses 6. The Classical Linear Model: Estimation
6.1 Elements of a linear regression model
6.1.1 Types of covariate
6.2 Ordinary least squares6.1.2 Equivariance properties
6.2.1 The OLS estimate
6.3 Sampling properties of the OLS estimator6.2.2 Geometric interpretation 6.2.3 Fitted values and residuals 6.2.4 Goodness-of-fit and equivariance properties 6.2.5 Multivariate OLS 6.2.6 Partitioned regression 6.2.7 Adding and excluding observations
6.3.1 The classical linear model
6.4 The Gaussian linear model6.3.2 Stochastic regressors 6.3.3 Estimates of precision
6.4.1 Sampling distribution of the OLS estimator
6.5 Constrained OLS6.4.2 The Gaussian ML estimator 6.4.3 Minimum variance unbiased estimation 6.4.4 Confidence sets 6.6 Bayesian estimators
6.6.1 Uninformative priors
6.7 Shrinkage estimators6.6.2 Conjugate priors
6.7.1 Stein estimators
6.7.2 James–Stein estimators 7. Violations of the Ideal Conditions for OLS
7.1 Collinearity
7.1.1 Quasi-collinearity
7.2 Misspecification of the regression function7.1.2 Ridge-regression estimators 7.3 Misspecification of the variance function
7.3.1 Pure heteroskedasticity
7.4 Generalized least squares7.3.2 Autocorrelation 7.3.3 Seemingly unrelated regression equations
7.4.1 Aitken theorem
7.5 Estimating the precision of OLS7.4.2 Feasible GLS estimators
7.5.1 Heteroskedasticity
7.6 Non-normality7.5.2 Autocorrelation
7.6.1 Efficiency of OLS
7.6.2 Transformations of the response variable 8. Diagnostics Based on the OLS Estimates
8.1 OLS residuals
8.2 Transformations of the OLS residuals
8.2.1 Predicted residuals
8.3 Influence and leverage8.2.2 Studentized residuals 8.2.3 LUS and BLUS residuals 8.2.4 Recursive residuals 9. The Classical Linear Model: Hypothesis Testing
9.1 The classical t- and F-tests
9.1.1 The t-test
9.2 Specification tests9.1.2 The F-test 9.1.3 Relations with likelihood-based tests 9.1.4 Classical pre-test estimation
9.2.1 Reset
9.3 Classical model selection criteria9.2.2 The difference test 9.2.3 Relations with the classical tests 9.2.4 The Durbin–Watson test
9.3.1 R2 and adjusted R2
9.4 Bayesian hypothesis testing9.3.2 The Cp procedure 9.3.3 Cross-validation 9.3.4 Information criteria 10. Asymptotic Properties of Least Squares Methods
10.1 Consistency of the OLS estimator
10.1.1 The main result
10.2 Asymptotic normality of the OLS estimator10.1.2 Some primitive assumptions 10.1.3 Violations of the fundamental assumption 10.1.4 Trends in the covariates
10.2.1 The main result
10.3 Asymptotic properties of the GLS estimator10.2.2 Some primitive assumptions 10.2.3 Estimates of statistical precision 10.4 Asymptotic properties of t- and F-tests
10.4.1 Consistency
10.5 Likelihood-based tests10.4.2 Local alternatives 10.4.3 Nonscalar variance 10.4.4 Tests based on the OLS Criterion 10.4.5 Nonlinear constraints
10.5.1 Likelihood ratio tests
10.6 Testing non-nested hypotheses10.5.2 Wald tests 10.5.3 Score tests
10.6.1 The comprehensive approach
10.7 The information matrix test10.6.2 The Cox test 10.6.3 Encompassing 11. The Instrumental Variables Method
11.1 Instrumental variables estimation
11.1.1 The simple IV estimator
11.2 Asymptotic properties of IV estimators11.1.2 The class of IV estimators
11.2.1 Consistency
11.3 Hypothesis testing11.2.2 Asymptotic normality 11.2.3 Asymptotic efficiency 11.2.4 Estimates of precision 11.2.5 The generalized 2SLS estimator
11.3.1 Wald tests
11.4 The generalized method of moments11.3.2 Tests of overidentifying restrictions 11.3.3 Difference tests 12. Linear Models for Panel Data
12.1 The basic model
12.2 The fixed effects model and the within group estimator 12.3 The random effects model
12.3.1 The GLS and the between group estimators
12.4 Unbalanced panels12.3.2 Testing for individual effects 12.3.3 Specification tests 12.5 Minimum distance estimation 12.6 IV estimation
12.6.1 Asymptotically efficient IV estimation
12.7 Time series of repeated cross-sections12.6.2 Choice of instruments 12.6.3 Autoregressive models for panel data 13. Linear Simultaneous Equation Models
13.1 The statistical problem
13.2 Identifiability
13.2.1 Conditions for identifiability: First method
13.3 Single equation estimation13.2.2 Separable constraints 13.2.3 Normalization and exclusion constraints 13.2.4 Conditions for identifiability: Second method
13.3.1 OLS estimation
13.4 System estimation13.3.2 IV estimation
13.4.1 IV estimation
13.4.2 Minimum distance estimation 14. Nonparametric Methods
14.1 Density estimation
14.1.1 Empirical densities
14.2 Statistical properties of the kernel method14.1.2 The kernel method
14.2.1 Local properties
14.3 Other methods for density estimation14.2.2 Global properties 14.2.3 Optimal choice of bandwidth and kernel function
14.3.1 The nearest neighbor method
14.4 Regression smoothers14.3.2 The maximum penalized likelihood method
14.4.1 Regression splines
14.5 Statistical properties of linear smoothers14.4.2 The kernel method 14.4.3 The nearest neighbor method 14.4.4 Smoothing splines and penalized LS
14.5.1 Measures of accuracy
14.6 Other methods for high-dimensional data14.5.2 Equivalent kernels and equivalent degrees of freedom 14.5.3 Asymptotic properties of kernel regression estimators 14.5.4 Tests of parametric models
14.6.1 The curse of dimensionality problem
14.6.2 Projection pursuit density estimation 14.6.3 Projection pursuit regression estimation 14.6.4 Additive regression 15. M-Estimators
15.1 The class of M-estimators
15.2 Consistency
15.2.1 Sufficient conditions for consistency
15.3 Asymptotic normality15.2.2 Conditions for uniform convergence in probability
15.3.1 The standard case
15.4 Robustness15.3.2 One-step M-estimators 15.3.3 Asymptotic normality under nonstandard conditions
15.4.1 Qualitative and quantitative robustness
15.4.2 The influence function 15.4.3 B-robustness 15.4.4 The influence function of M-estimators 15.4.5 Asymptotic efficiency 15.4.6 Optimal B-robust estimators 15.4.7 The breakdown point 16. Adaptive and Robust Regression Estimators
16.1 Adaptive estimators
16.1.1 Partially adaptive estimators
16.2 Quantile regression16.1.2 Fully adaptive estimators
16.2.1 Definitions and algebraic properties
16.3 Robust estimators16.2.2 Computational aspects 16.2.3 Robustness properties 16.2.4 Asymptotic distribution 16.2.5 Some drawbacks of ALAD estimators
16.3.1 Huber estimator of regression
16.3.2 Optimal bounded influence estimators 16.3.3 Computational aspects 16.3.4 High-breakdown estimators 17. Models for Discrete Responses
17.1 Models for binary responses
17.1.1 Binomial regression models
17.2 Models for multinomial responses17.1.2 ML estimation 17.1.3 Asymptotic properties of ML estimators 17.1.4 The logit link 17.1.5 Discriminant analysis and logit 17.1.6 Other links 17.1.7 Grouped data 17.1.8 Unobserved heterogeneity 17.1.9 Semiparametric estimation
17.2.1 The multinomial log-likelihood
17.3 Panel data models17.2.2 Ordered categorical responses 17.2.3 Multinomial logit 17.2.4 Nested logit 17.2.5 Multinomial choice models 17.2.6 The method of simulated moments
17.3.1 The fixed effects approach
17.4 Generalized linear models17.3.2 The random effects approach
17.4.1 One-parameter exponential families
17.5 Estimation of a GLM17.4.2 Elements of a GLM
17.5.1 ML estimation
17.6 Poisson regression17.5.2 Residuals and diagnostics 17.5.3 Least squares estimation 17.5.4 Pseudo ML methods 18. Models for Truncated and Censored Data
18.1 Distribution and moments of truncated and censored data
18.1.1 Truncated data
18.2 The Roy model18.1.2 Censored data 18.3 Least squares estimation
18.3.1 OLS estimation
18.4 ML estimation18.3.2 Nonlinear least squares 18.3.3 Two-step estimation
18.4.1 Gaussian ML estimators
18.5 Semiparametric estimation18.4.2 An alternative parametrization 18.4.3 Nongaussian ML estimators
18.5.1 Semiparametric least squares
18.6 Bivariate models18.5.2 Quantile restrictions 18.5.3 Symmetry restrictions
18.6.1 Type-2 tobit models
18.6.2 Type-3 tobit models References
Appendix A Review of Linear Algebra
A.1 Vector spaces and linear transformations
A.2 Matrices and matrix operations A.3 Determinant and inverse of a matrix A.4 Rank and trace of a matrix A.5 Eigenvalues and eigenvectors A.6 Quadratic forms and definite matrices A.7 Partitioned matrices A.8 Kronecker product and VEC operator A.9 Differentiation of matrices and vectors A.10 Generalized inverses A.11 Projection theorems Appendix B Methods of Numerical Maximization
B.1 Algorithms
B.2 Direct search methods B.3 Newton-type methods
B.3.1 Newton–Raphson
B.4 The EM algorithmB.3.2 Method of Scoring B.3.3 BHHH B.3.4 Gauss–Newton B.5 Numerical derivatives Appendix C Review of Probability
C.1 Conditional distributions
C.2 Conditional expectations C.3 Stochastic inequalities C.4 Moment generating and characteristic functions C.5 The Gamma and Beta distributions C.6 Distributions related to the exponential C.7 Distributions related to the Gaussian C.8 Quadratic forms in Gaussian random variables C.9 Moments of the truncated Gaussian distribution Appendix D Elements of Asymptotic Theory
D.1 Convergence of sequences of random variables
D.2 Weak convergence D.3 Orders in probability D.4 Laws of large numbers D.5 Central limit theorems D.6 Convergence of sequences of transformed random variables D.7 The asymptotic delta method D.8 The functional central limit theorem Index
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