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Contents of volume IV
CONTENTS OF VOLUME
IV
Introduction to the Series
v
Contents of the Handbook
vii
Preface to the Handbook PART
9 -
Xlll
ECONOMETRIC THEORY
Chapter 36
Large Sample Estimation and Hypothesis Testing WHITNEY K. NEWEY and DANIEL McFADDEN
Abstract 1. Introduction 2. Consistency 2.1. The basic consistency theorem 2.2. Identification 2.3. Uniform convergence and continuity 2.4. Consistency of maximum likelihood 2.5. Consistency of GMM 2.6. Consistency without compactness 2.7. Stochastic equicontinuity and uniform convergence 2.8. Least absolute deviations examples
3. Asymptotic normality 3.1. The basic results 3.2. Asymptotic normality for MLE 3.3. Asymptotic normality for GMM 3.4. One-step theorems 3.5. Technicalities
4.
Consistent asymptotic variance estimation 4.1. The basic results 4.2. Variance estimation for MLE 4.3. Asymptotic variance estimation for GMM
5. Asymptotic efficiency 5.1. Efficiency of maximum likelihood estimation 5.2. Optimal minimum distance estimation 5.3. A general efficiency framework
2113 2113 2120 2121 2124 2129 2131 2132 2133 2136 2138 2141 2143 2146 2148 2150 2152 2153 2155 2157 2160 2162 2162 2164 2165
xvi
Contents of Volume I V
6.
7.
8.
9.
5.4.
Solving for the smallest asymptotic variance
5.5.
Feasible efficient estimation
5.6.
Technicalities
Two-step estimators 6.1.
Two-step estimators as joint G M M estimators
6.2.
The effect of first-step estimation on second-step standard errors
6.3.
Consistent asymptotic variance estimation for two-step estimators
Asymptotic normality with nonsmooth objective functions 7.1. 7.2.
The basic results Stochastic equicontinuity for Lipschitz moment functions
7.3.
Asymptotic variance estimation
7.4.
Technicalities
Semiparametric two-step estimators 8.1.
Asymptotic normality and consistent variance estimation
8.2. 8.3.
V-estimators First-step kernel estimation
8.4.
Technicalities
Hypothesis testing with GMM estimators 9.1.
The null hypothesis and the constrained G M M estimator
9.2.
The test statistics
9.3.
One-step versions of the trinity
9.4. 9.5. 9.6.
Special cases Tests for overidentifying restrictions Specification tests in linear models
9.7.
Specification testing in multinomial .models
9.8.
Technicalities
References
2168 2171 2173 2175 2176 2179 2182 2184 2185 2188 2189 2191 2194 2196 2200 2203 2214 2215 2217 2220 2226 2228 2231 2234 2236 2239 2241
Chapter 3 7
Empirical Process Methods in Econometrics D O N A L D W.K. ANDREWS
Abstract 1. Introduction 2. Weak convergence and stochastic equicontinuity 3. Applications 3.1. Review of applications 3.2. Parametric M-estimators based on non-differentiable criterion functions 3.3. Tests when a nuisance parameter is present only under the alternative 3.4. Semiparametric estimation
4.
Stochastic equicontinuity via symmetrization 4.1.
Primitive conditions for stochastic equicontinuity
4.2.
Examples
5. Stochastic equicontinuity via bracketing 6. Conclusion
2248 2248 2249 2253 2253 2255 2259 2263 2267 2267 2273 2276 2283
Contents of Volume IV
Appendix References
xvii 2284 2292
Chapter 38
Applied N o n p a r a m e t r i c M e t h o d s WOLFGANG H,~RDLE and OLIVER LINTON Abstract 1. N o n p a r a m e t r i c e s t i m a t i o n in e c o n o m e t r i c s 2. D e n s i t y e s t i m a t i o n 2.1. Kernels as windows 2.2. Kernels and ill-posed problems 2.3. Properties of kernels 2.4. Properties of the kernel density estimator 2.5. Estimation of multivariate densities, their derivatives and bias reduction 2.6. Fast implementation of density estimation 3. Regression e s t i m a t i o n 3.1. Kernelestimators 3.2. k-Nearest neighbor estimators 3.3. Local polynomial estimators 3.4. Spline estimators 3.5, Series estimators 3.6. Kernels, k-NN, splines, and series 3.7. Confidence intervals 3.8. Regression derivatives and quantiles 4. O p t i m a l i t y a n d b a n d w i d t h choice 4.1. Optimality 4.2. Choice of smoothing parameter 5. A p p l i c a t i o n to time series 5.1. Autoregression 5.2. Correlated errors 6. Applications to s e m i p a r a m e t r i c e s t i m a t i o n 6.1. The partially linear model 6.2. Heteroskedastic nonlinear regression 6.3. Single index models 7. C o n c l u s i o n s References
2297 2297 2300 2300 2301 2302 2303 2304 2306 2308 2308 2310 2311 2312 2313 2314 2315 2318 2319 2319 232l 2325 2326 2327 2328 2329 2330 2331 2334 2334
Chapter 39
M e t h o d o l o g y a n d T h e o r y for the B o o t s t r a p PETER HALL Abstract 1. I n t r o d u c t i o n 2. A formal definition of the b o o t s t r a p principle 3. I t e r a t i n g the principle
2342 2342 2345 2352
Contents of Volume I V
xviii
Asymptotic theory 4.1. Summary 4.2. Edgeworth and Cornish-Fisher expansions 4.3. Edgeworth and Cornish-Fisher expansions of bootstrap distributions 4.4. Different versions of bootstrap confidenceintervals 4.5. Order of correctness of bootstrap approximations to critical points 4.6. Coverage error of confidenceintervals 4.7. Simple linear regression References .
2357 2357 2358 2362 2364 2367 2369 2375 2379
Chapter 40
Classical Estimation Methods for LDV Models Using Simulation VASSILIS A. HAJIVASSILIOU and P A U L A. R U U D
1. 2.
3.
Introduction Limited dependent variable models 2.1.
The latent normal regression model
2.2.
Censoring
2.3.
Truncation
2.4.
Mixtures
2.5.
Time series models
2.6.
Score functions
2.7.
The computational intractability of L D V models
Simulation methods 3.1.
4.
Overview
3.2.
Censored simulation
3.3.
Truncated simulation
Simulation and estimation of LDV models 4.1.
Overview
4.2.
Simulation of the log-likelihood function
4.3.
Simulation of moment functions
4.4.
Simulation of the score function
4.5.
Bias corrections
5. Conclusion 6. Acknowledgements References
2384 2386 2386 2386 2392 2394 2395 2397 2399 2400 2400 2402 2406 2408 2408 2412 2421 2428 2435 2437 2438 2438
Chapter 41
Estimation of Semiparametric Models JAMES L. P O W E L L
Abstract 1. Introduction 1.1. Overview 1.2. Definition of"semiparametric"
2444 2444 2444 2449
Contents of Volume I V 1.3. Stochastic restrictions and structural models 1.4. Objectives and techniques of asymptotic theory
2.
Stochastic restrictions 2.1. Conditional mean restriction 2.2. Conditional quantile restrictions 2.3. Conditional symmetry restrictions 2.4, Independence restrictions 2.5. Exclusion and index restrictions
3. Structural models 3.1. Discrete response models 3.2. Transformation models 3.3. Censored and truncated regression models 3.4. Selection models 3.5. Nonlinear panel data models
4. Summary and conclusions References
xix
2452 2460 2465 2466 2469 2474 2476 2482 2487 2487 2492 2500 2506 2511 2513 2514
Chapter 42
Restrictions of Economic Theory in Nonparametric Methods ROSA L. MATZKIN
Abstract 1. Introduction 2. Identification of nonparametric models using economic restrictions 2.1. Definition ofnonparametric identification 2.2. Identification of limited dependent variable models 2.3. Identification of functions generating regression functions 2.4. Identification of simultaneous equations models
3. Nonparametric estimation using economic restrictions 3.1. Estimators that depend on the shape of the estimated function 3.2. Estimation using seminonparametric methods 3.3. Estimation using weighted average methods
4.
Nonparametric tests using economic restrictions 4.1. Nonstatistical tests 4.2. Statistical tests
5. Conclusions References
2524 2524 2528 2528 2530 2535 2536 2537 2538 2544 2546 2548 2548 2551 2554 2554
Chapter 43
Analog Estimation of Econometric Models CHARLES F. MANSKI
Abstract 1. Introduction 2. Preliminaries
2560 2560 2561
xx
Contents of Volume I V
2.1. The analogy principle 2.2. Moment problems 2.3. Econometric models 3. M e t h o d - o f - m o m e n t s e s t i m a t i o n of separable models 3.L Mean independence 3.2, Median independence 3.3, Conditional symmetry 3.4, Variance independence 3.5, Statistical independence 3.6. A historical note 4. M e t h o d - o f - m o m e n t s e s t i m a t i o n of response models 4,1. Likelihood models 4.2. Invertible models 4.3. Mean independent linear models 4.4. Quantile independent monotone models 5. E s t i m a t i o n of general separable a n d response models 5.1. Closest-empirical-distributionestimation of separable models 5.2. Minimum-distance estimation of response models 6. C o n c l u s i o n References
2561 2563 2565 2566 2567 2568 2569 2570 2570 2571 2571 2572 2574 2574 2575 2577 2577 2580 2581 2581
Chapter 44
Testing N o n - N e s t e d Hypotheses c. GOURIEROUX and A. MONFORT 1. 2.
3,
4.
Introduction N o n - n e s t e d hypotheses 2.1. Definitions 2.2. Pseudo-true values 2.3. Semi-parametric hypotheses 2.4. Examples 2.5. Symmetryof the problem Testing procedures 3.1. Maximum likelihood estimator under misspecification 3.2. The extended Wald test 3.3. The extended score test 3.4. The Cox procedure 3.5. Application to the choice of regressors in linear models 3.6. Applications to qualitative models Artificial nesting models 4.1. Examples 4.2. Localexpansions of artificial nesting models 4.3. A score test based on a modified Atkinson's compound model 4.4. The partially modified Atkinson's compound model
2585 2587 2587 2589 2590 2591 2596 2597 2597 2598 2600 2602 2605 2608 2610 2610 2614 2618 2621
xxi
Contents of Volume I V
5.
Comparison of testing procedures 5.1. Asymptotic equivalence of test statistics 5.2. Asymptotic comparisons of power functions 5.3. Exactfinite sample results 5.4. Monte Carlo studies 6. Encompassing 6.1. The encompassing principle 6.2. The encompassing tests References
P A R T 10
THEORY AND METHODS FOR DEPENDENT
2621 2622 2622 2624 2625 2626 2626 2628 2633
PROCESSES
Chapter 45
Estimation and Inference for Dependent Processes JEFFREY M. WOOLDRIDGE Abstract Part I. I n t r o d u c t i o n and overview 1. 2. 3.
Introduction Examples of stochastic processes Types of estimation techniques
Part II. 4.
5. 6.
7.
The essentially stationary, weakly dependent case
Asymptotic properties of M-estimators 4.1. Introduction 4.2. Consistency 4.3. Asymptotic normality 4.4. Adjustment for nuisance parameters 4.5. Estimating the asymptotic variance 4.6. Hypothesis testing M a x i m u m likelihood estimation Q u a s i - m a x i m u m likelihood estimation ( Q M L E ) 6.1. Conditional mean estimation 6.2. QMLE for the mean and variance Generalized m e t h o d of m o m e n t s estimation 7.1. Introduction 7.2. Consistency 7.3. Asymptotic normality 7.4. Estimating the asymptotic variance 7.5. Asymptotic efficiency 7.6. Testing
2641 2641 2641 2643 2648 2649 2649 2649 2651 2654 2658 2659 2665 2670 2680 2681 2688 2691 2691 2694 2695 2697 2699 2700
xxii
Contents of Volume I V
P a r t III. 8.
9.
T h e g l o b a l l y n o n s t a t i o n a r y , w e a k l y d e p e n d e n t case
G e n e r a l results 8.1. Introduction 8.2. Asymptotic normality of an abstract estimator A s y m p t o t i c n o r m a l i t y of M - e s t i m a t o r s 9.1. Asymptotic normality 9.2. Estimating the asymptotic variance
P a r t IV.
The n o n e r g o d i c case
10.
G e n e r a l results 10.1. Introduction 10.2. Abstract limiting distribution result 11. S o m e results for linear m o d e l s 12. A p p l i c a t i o n s to n o n l i n e a r m o d e l s Appendix References
2701 2701 2701 2702 2706 2706 2710 2710 2710 2710 2711 2713 2723 2725 2733
Chapter 46
U n i t Roots, S t r u c t u r a l Breaks a n d T r e n d s JAMES H. STOCK Abstract 1. I n t r o d u c t i o n 2. M o d e l s a n d p r e l i m i n a r y a s y m p t o t i c t h e o r y 2.1. Basic concepts and notation 2.2: The functional central limit theorem and related tools 2.3. Examples and preliminary results 2.4. Generalizations and additional references 3. U n i t autoregressive r o o t s 3.1. Point estimation 3.2. Hypothesis tests 3.3. Interval estimation 4. U n i t m o v i n g average r o o t s 4.1. Point estimation 4.2. Hypothesis tests 5. S t r u c t u r a l b r e a k s a n d b r o k e n t r e n d s 5.1. Breaks in coefficients in time series regression 5.2. Trend breaks and tests for autoregressive unit roots 6. Tests of the I(1) a n d I(0) hypotheses: links a n d p r a c t i c a l limitations 6.1. Parallels between the I(0) and I(1) testing problems 6.2. Decision-theoretic classification schemes 6.3. Practical and theoretical limitations in the ability to distinguish I(0) and I(1) processes References
2740 2740 2744 2745 2748 2751 2756 2757 2758 2763 2785 2788 2790 2792 2805 2807 2817 2821 2821 2822 2825 2831
Contents of Volume I V
xxiii
Chapter 47
Vector Autoregressions a n d C o i n t e g r a t i o n MARK W. WATSON Abstract 1. I n t r o d u c t i o n 2. Inference in VARs with integrated regressors 2.1. Introductory comments 2.2. An example 2.3. A useful lemma 2.4. Continuing with the example 2.5. A general framework 2.6. Applications 2.7. Implications for econometric practice 3. C o i n t e g r a t e d systems 3.1. Introductory comments 3.2. Representations for the I(1) cointegrated model 3.3. Testing for cointegration in I(1) systems 3.4. Estimating cointegrating vectors 3.5. The role of constants and trends 4. Structural vector autoregressions 4.1. Introductory comments 4.2. The structural moving average model, impulse response functions and variance decompositions 4.3. The structural VAR representation 4.4. Identification of the structural VAR 4.5. Estimating structural VAR models References
2844 2844 2848 2848 2848 2850 2852 2854 2860 2866 2870 2870 2870 2876 2887 2894 2898 2898 2899 2900 2902 2906 2910
Chapter 48
Aspects of M o d e l l i n g N o n l i n e a r Time Series TIMO TERASVIRTA,DAG TJ(2)STHEIM and CLIVE W.J. GRANGER Abstract 1. I n t r o d u c t i o n 2. Types of n o n l i n e a r models 2.1. Modelsfrom economic theory 2.2. Modelsfrom time series theory 2.3. Flexible statistical parametric models 2.4. State-dependent, time-varying parameter and long-memory models 2.5. Nonparametric models 3. Testing linearity 3.1. Testsagainst a specific alternative 3.2. Testswithout a specific alternative 3.3. Constancy of conditional variance
2919 2919 2921 2921 2922 2923 2924 2925 2926 2927 2930 2933
Contents of Volume I V
xxiv
4. 5.
Specification of nonlinear models Estimation in nonlinear time series 5.1. 5.2.
Estimation of parameters in parametric models Estimation ofnonparametric functions
5.3.
Estimation of restricted nonparametric and semiparametric models
6. Evaluation of estimated models 7. Example 8. Conclusions References
2934 2937 2937 2938 2942 2945 2946 2952 2953
Chapter 49
ARCH Models TIM BOLLERSLEV, ROBERT F. E N G L E and DANIEL B. N E L S O N
Abstract 1. Introduction 1.1. Definitions 1.2. Empirical regularities of asset returns 1.3. Univariate parametric models 1.4. ARCH in mean models 1.5. Nonparametric and semiparametric methods
2.
3.
4.
5.
Inference procedures 2.1.
Testing for ARCH
2.2. 2.3.
Maximum likelihood methods Quasi-maximum likelihood methods
2.4.
Specification checks
Stationary and ergodic properties 3.1.
Strict stationarity
3.2.
Persistence
Continuous time methods 4.1.
ARCH models as approximations to diffusions
4.2. 4.3.
Diffusions as approximations to ARCH models ARCH models as filters and forecasters
Aggregation and forecasting 5.1.
Temporal aggregation
5.2. Forecast error distributions
6.
Multivariate specifications 6.1.
Vector ARCH and diagonal ARCH
6.2. Factor ARCH 6.3. Constant conditional correlations 6.4. 6.5.
7. 8.
Bivariate EGARCH Stationarity and co-persistence
Model selection Alternative measures for volatility
2961 2961 2961 2963 2967 2972 2972 2974 2974 2977 2983 2984 2989 2989 2990 2992 2994 2996 2997 2999 2999 3001 3002 3003 3005 3007 3008 3009 3010 3012
Contents of Volume I V
9.
Empirical examples 9.1. U.S. Dollar/Deutschmark exchange rates 9.2.
U.S. stock prices
10. Conclusion References
xxv
3014 3014 3017 3030 3031
Chapter 50
State-Space Models JAMES D. H A M I L T O N
Abstract 1. The state-space representation of a linear dynamic system 2. The. Kalman filter
3.
4.
5.
2.1.
Overview of the Kalman filter
2.2. 2.3.
Derivation of the Kalman filter Forecasting with the Kalman filter
2.4. 2.5.
Smoothed inference Interpretation of the Kalman filter with non-normal disturbances
2.6. 2.7.
Time-varying coefficient models Other extensions
Statistical inference about unknown parameters using the Kalman filter 3.1.
Maximum likelihood estimation
3.2. 3.3. 3.4.
Identification Asymptotic properties of maximum likelihood estimates Confidence intervals for smoothed estimates and forecasts
3.5.
Empirical application - an analysis of the real interest rate
Discrete-valued state variables 4.1.
Linear state-space representation of the Markov-switching model
4.2.
Optimal filter when the state variable follows a Markov chain
4.3.
Extensions
4.4. 4.5. 4.6.
Forecasting Smoothed probabilities Maximum likelihood estimation
4.7.
Asymptotic properties of maximum likelihood estimates
4.8.
Empirical application - another look at the real interest rate
Non-normal and nonlinear state-space models
3041 3041 3046 3047 3048 3051 3051 3052 3053 3054 3055 3055 3057 3058 3060 3060 3062 3063 3064 3067 3068 3069 3070 3071 3071 3073
5.1. Kitagawa's grid approximation for nonlinear, non-normal state-space 5.2.
models Extended Kalman filter
5.3.
Other approaches to nonlinear state-space models
References
3073 3076 3077 3077
xxvi
Contents of Volume I V
Chapter 51
Structural E s t i m a t i o n of M a r k o v Decision Processes JOHN RUST 1. 2.
3082 3088
Introduction Solving M D P ' s via d y n a m i c p r o g r a m m i n g : A brief review 2.1. Finite-horizon dynamic programming and the optimality of Markovian decision rules 2.2. Infinite-horizon dynamic programming and Bellman's equation 2.3. Bellman'sequation, contraction mappings and optimality 2.4. A geometric series representation for MDP's 2.5. Overview of solution methods 3. E c o n o m e t r i c m e t h o d s for discrete decision processes 3.1. Alternative models of the "error term" 3.2. Maximum likelihood estimation of DDP's 3.3. Alternative estimation methods: Finite-horizon DDP problems 3.4. Alternative estimation methods: Infinite-horizon DDP's 3.5. The identification problem 4. Empirical applications 4.l. Optimal replacement of bus engines 4.2. Optimal retirement from a firm References
3089 3091 3091 3094 3095 3099 3100 3101 3118 3123 3125 3130 3130 3134 3139
List of T h e o r e m s
3145
Index
3147
IV
Introduction to the Series
v
Contents of the Handbook
vii
Preface to the Handbook PART
9 -
Xlll
ECONOMETRIC THEORY
Chapter 36
Large Sample Estimation and Hypothesis Testing WHITNEY K. NEWEY and DANIEL McFADDEN
Abstract 1. Introduction 2. Consistency 2.1. The basic consistency theorem 2.2. Identification 2.3. Uniform convergence and continuity 2.4. Consistency of maximum likelihood 2.5. Consistency of GMM 2.6. Consistency without compactness 2.7. Stochastic equicontinuity and uniform convergence 2.8. Least absolute deviations examples
3. Asymptotic normality 3.1. The basic results 3.2. Asymptotic normality for MLE 3.3. Asymptotic normality for GMM 3.4. One-step theorems 3.5. Technicalities
4.
Consistent asymptotic variance estimation 4.1. The basic results 4.2. Variance estimation for MLE 4.3. Asymptotic variance estimation for GMM
5. Asymptotic efficiency 5.1. Efficiency of maximum likelihood estimation 5.2. Optimal minimum distance estimation 5.3. A general efficiency framework
2113 2113 2120 2121 2124 2129 2131 2132 2133 2136 2138 2141 2143 2146 2148 2150 2152 2153 2155 2157 2160 2162 2162 2164 2165
xvi
Contents of Volume I V
6.
7.
8.
9.
5.4.
Solving for the smallest asymptotic variance
5.5.
Feasible efficient estimation
5.6.
Technicalities
Two-step estimators 6.1.
Two-step estimators as joint G M M estimators
6.2.
The effect of first-step estimation on second-step standard errors
6.3.
Consistent asymptotic variance estimation for two-step estimators
Asymptotic normality with nonsmooth objective functions 7.1. 7.2.
The basic results Stochastic equicontinuity for Lipschitz moment functions
7.3.
Asymptotic variance estimation
7.4.
Technicalities
Semiparametric two-step estimators 8.1.
Asymptotic normality and consistent variance estimation
8.2. 8.3.
V-estimators First-step kernel estimation
8.4.
Technicalities
Hypothesis testing with GMM estimators 9.1.
The null hypothesis and the constrained G M M estimator
9.2.
The test statistics
9.3.
One-step versions of the trinity
9.4. 9.5. 9.6.
Special cases Tests for overidentifying restrictions Specification tests in linear models
9.7.
Specification testing in multinomial .models
9.8.
Technicalities
References
2168 2171 2173 2175 2176 2179 2182 2184 2185 2188 2189 2191 2194 2196 2200 2203 2214 2215 2217 2220 2226 2228 2231 2234 2236 2239 2241
Chapter 3 7
Empirical Process Methods in Econometrics D O N A L D W.K. ANDREWS
Abstract 1. Introduction 2. Weak convergence and stochastic equicontinuity 3. Applications 3.1. Review of applications 3.2. Parametric M-estimators based on non-differentiable criterion functions 3.3. Tests when a nuisance parameter is present only under the alternative 3.4. Semiparametric estimation
4.
Stochastic equicontinuity via symmetrization 4.1.
Primitive conditions for stochastic equicontinuity
4.2.
Examples
5. Stochastic equicontinuity via bracketing 6. Conclusion
2248 2248 2249 2253 2253 2255 2259 2263 2267 2267 2273 2276 2283
Contents of Volume IV
Appendix References
xvii 2284 2292
Chapter 38
Applied N o n p a r a m e t r i c M e t h o d s WOLFGANG H,~RDLE and OLIVER LINTON Abstract 1. N o n p a r a m e t r i c e s t i m a t i o n in e c o n o m e t r i c s 2. D e n s i t y e s t i m a t i o n 2.1. Kernels as windows 2.2. Kernels and ill-posed problems 2.3. Properties of kernels 2.4. Properties of the kernel density estimator 2.5. Estimation of multivariate densities, their derivatives and bias reduction 2.6. Fast implementation of density estimation 3. Regression e s t i m a t i o n 3.1. Kernelestimators 3.2. k-Nearest neighbor estimators 3.3. Local polynomial estimators 3.4. Spline estimators 3.5, Series estimators 3.6. Kernels, k-NN, splines, and series 3.7. Confidence intervals 3.8. Regression derivatives and quantiles 4. O p t i m a l i t y a n d b a n d w i d t h choice 4.1. Optimality 4.2. Choice of smoothing parameter 5. A p p l i c a t i o n to time series 5.1. Autoregression 5.2. Correlated errors 6. Applications to s e m i p a r a m e t r i c e s t i m a t i o n 6.1. The partially linear model 6.2. Heteroskedastic nonlinear regression 6.3. Single index models 7. C o n c l u s i o n s References
2297 2297 2300 2300 2301 2302 2303 2304 2306 2308 2308 2310 2311 2312 2313 2314 2315 2318 2319 2319 232l 2325 2326 2327 2328 2329 2330 2331 2334 2334
Chapter 39
M e t h o d o l o g y a n d T h e o r y for the B o o t s t r a p PETER HALL Abstract 1. I n t r o d u c t i o n 2. A formal definition of the b o o t s t r a p principle 3. I t e r a t i n g the principle
2342 2342 2345 2352
Contents of Volume I V
xviii
Asymptotic theory 4.1. Summary 4.2. Edgeworth and Cornish-Fisher expansions 4.3. Edgeworth and Cornish-Fisher expansions of bootstrap distributions 4.4. Different versions of bootstrap confidenceintervals 4.5. Order of correctness of bootstrap approximations to critical points 4.6. Coverage error of confidenceintervals 4.7. Simple linear regression References .
2357 2357 2358 2362 2364 2367 2369 2375 2379
Chapter 40
Classical Estimation Methods for LDV Models Using Simulation VASSILIS A. HAJIVASSILIOU and P A U L A. R U U D
1. 2.
3.
Introduction Limited dependent variable models 2.1.
The latent normal regression model
2.2.
Censoring
2.3.
Truncation
2.4.
Mixtures
2.5.
Time series models
2.6.
Score functions
2.7.
The computational intractability of L D V models
Simulation methods 3.1.
4.
Overview
3.2.
Censored simulation
3.3.
Truncated simulation
Simulation and estimation of LDV models 4.1.
Overview
4.2.
Simulation of the log-likelihood function
4.3.
Simulation of moment functions
4.4.
Simulation of the score function
4.5.
Bias corrections
5. Conclusion 6. Acknowledgements References
2384 2386 2386 2386 2392 2394 2395 2397 2399 2400 2400 2402 2406 2408 2408 2412 2421 2428 2435 2437 2438 2438
Chapter 41
Estimation of Semiparametric Models JAMES L. P O W E L L
Abstract 1. Introduction 1.1. Overview 1.2. Definition of"semiparametric"
2444 2444 2444 2449
Contents of Volume I V 1.3. Stochastic restrictions and structural models 1.4. Objectives and techniques of asymptotic theory
2.
Stochastic restrictions 2.1. Conditional mean restriction 2.2. Conditional quantile restrictions 2.3. Conditional symmetry restrictions 2.4, Independence restrictions 2.5. Exclusion and index restrictions
3. Structural models 3.1. Discrete response models 3.2. Transformation models 3.3. Censored and truncated regression models 3.4. Selection models 3.5. Nonlinear panel data models
4. Summary and conclusions References
xix
2452 2460 2465 2466 2469 2474 2476 2482 2487 2487 2492 2500 2506 2511 2513 2514
Chapter 42
Restrictions of Economic Theory in Nonparametric Methods ROSA L. MATZKIN
Abstract 1. Introduction 2. Identification of nonparametric models using economic restrictions 2.1. Definition ofnonparametric identification 2.2. Identification of limited dependent variable models 2.3. Identification of functions generating regression functions 2.4. Identification of simultaneous equations models
3. Nonparametric estimation using economic restrictions 3.1. Estimators that depend on the shape of the estimated function 3.2. Estimation using seminonparametric methods 3.3. Estimation using weighted average methods
4.
Nonparametric tests using economic restrictions 4.1. Nonstatistical tests 4.2. Statistical tests
5. Conclusions References
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Chapter 43
Analog Estimation of Econometric Models CHARLES F. MANSKI
Abstract 1. Introduction 2. Preliminaries
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Contents of Volume I V
2.1. The analogy principle 2.2. Moment problems 2.3. Econometric models 3. M e t h o d - o f - m o m e n t s e s t i m a t i o n of separable models 3.L Mean independence 3.2, Median independence 3.3, Conditional symmetry 3.4, Variance independence 3.5, Statistical independence 3.6. A historical note 4. M e t h o d - o f - m o m e n t s e s t i m a t i o n of response models 4,1. Likelihood models 4.2. Invertible models 4.3. Mean independent linear models 4.4. Quantile independent monotone models 5. E s t i m a t i o n of general separable a n d response models 5.1. Closest-empirical-distributionestimation of separable models 5.2. Minimum-distance estimation of response models 6. C o n c l u s i o n References
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Chapter 44
Testing N o n - N e s t e d Hypotheses c. GOURIEROUX and A. MONFORT 1. 2.
3,
4.
Introduction N o n - n e s t e d hypotheses 2.1. Definitions 2.2. Pseudo-true values 2.3. Semi-parametric hypotheses 2.4. Examples 2.5. Symmetryof the problem Testing procedures 3.1. Maximum likelihood estimator under misspecification 3.2. The extended Wald test 3.3. The extended score test 3.4. The Cox procedure 3.5. Application to the choice of regressors in linear models 3.6. Applications to qualitative models Artificial nesting models 4.1. Examples 4.2. Localexpansions of artificial nesting models 4.3. A score test based on a modified Atkinson's compound model 4.4. The partially modified Atkinson's compound model
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5.
Comparison of testing procedures 5.1. Asymptotic equivalence of test statistics 5.2. Asymptotic comparisons of power functions 5.3. Exactfinite sample results 5.4. Monte Carlo studies 6. Encompassing 6.1. The encompassing principle 6.2. The encompassing tests References
P A R T 10
THEORY AND METHODS FOR DEPENDENT
2621 2622 2622 2624 2625 2626 2626 2628 2633
PROCESSES
Chapter 45
Estimation and Inference for Dependent Processes JEFFREY M. WOOLDRIDGE Abstract Part I. I n t r o d u c t i o n and overview 1. 2. 3.
Introduction Examples of stochastic processes Types of estimation techniques
Part II. 4.
5. 6.
7.
The essentially stationary, weakly dependent case
Asymptotic properties of M-estimators 4.1. Introduction 4.2. Consistency 4.3. Asymptotic normality 4.4. Adjustment for nuisance parameters 4.5. Estimating the asymptotic variance 4.6. Hypothesis testing M a x i m u m likelihood estimation Q u a s i - m a x i m u m likelihood estimation ( Q M L E ) 6.1. Conditional mean estimation 6.2. QMLE for the mean and variance Generalized m e t h o d of m o m e n t s estimation 7.1. Introduction 7.2. Consistency 7.3. Asymptotic normality 7.4. Estimating the asymptotic variance 7.5. Asymptotic efficiency 7.6. Testing
2641 2641 2641 2643 2648 2649 2649 2649 2651 2654 2658 2659 2665 2670 2680 2681 2688 2691 2691 2694 2695 2697 2699 2700
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Contents of Volume I V
P a r t III. 8.
9.
T h e g l o b a l l y n o n s t a t i o n a r y , w e a k l y d e p e n d e n t case
G e n e r a l results 8.1. Introduction 8.2. Asymptotic normality of an abstract estimator A s y m p t o t i c n o r m a l i t y of M - e s t i m a t o r s 9.1. Asymptotic normality 9.2. Estimating the asymptotic variance
P a r t IV.
The n o n e r g o d i c case
10.
G e n e r a l results 10.1. Introduction 10.2. Abstract limiting distribution result 11. S o m e results for linear m o d e l s 12. A p p l i c a t i o n s to n o n l i n e a r m o d e l s Appendix References
2701 2701 2701 2702 2706 2706 2710 2710 2710 2710 2711 2713 2723 2725 2733
Chapter 46
U n i t Roots, S t r u c t u r a l Breaks a n d T r e n d s JAMES H. STOCK Abstract 1. I n t r o d u c t i o n 2. M o d e l s a n d p r e l i m i n a r y a s y m p t o t i c t h e o r y 2.1. Basic concepts and notation 2.2: The functional central limit theorem and related tools 2.3. Examples and preliminary results 2.4. Generalizations and additional references 3. U n i t autoregressive r o o t s 3.1. Point estimation 3.2. Hypothesis tests 3.3. Interval estimation 4. U n i t m o v i n g average r o o t s 4.1. Point estimation 4.2. Hypothesis tests 5. S t r u c t u r a l b r e a k s a n d b r o k e n t r e n d s 5.1. Breaks in coefficients in time series regression 5.2. Trend breaks and tests for autoregressive unit roots 6. Tests of the I(1) a n d I(0) hypotheses: links a n d p r a c t i c a l limitations 6.1. Parallels between the I(0) and I(1) testing problems 6.2. Decision-theoretic classification schemes 6.3. Practical and theoretical limitations in the ability to distinguish I(0) and I(1) processes References
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Chapter 47
Vector Autoregressions a n d C o i n t e g r a t i o n MARK W. WATSON Abstract 1. I n t r o d u c t i o n 2. Inference in VARs with integrated regressors 2.1. Introductory comments 2.2. An example 2.3. A useful lemma 2.4. Continuing with the example 2.5. A general framework 2.6. Applications 2.7. Implications for econometric practice 3. C o i n t e g r a t e d systems 3.1. Introductory comments 3.2. Representations for the I(1) cointegrated model 3.3. Testing for cointegration in I(1) systems 3.4. Estimating cointegrating vectors 3.5. The role of constants and trends 4. Structural vector autoregressions 4.1. Introductory comments 4.2. The structural moving average model, impulse response functions and variance decompositions 4.3. The structural VAR representation 4.4. Identification of the structural VAR 4.5. Estimating structural VAR models References
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Chapter 48
Aspects of M o d e l l i n g N o n l i n e a r Time Series TIMO TERASVIRTA,DAG TJ(2)STHEIM and CLIVE W.J. GRANGER Abstract 1. I n t r o d u c t i o n 2. Types of n o n l i n e a r models 2.1. Modelsfrom economic theory 2.2. Modelsfrom time series theory 2.3. Flexible statistical parametric models 2.4. State-dependent, time-varying parameter and long-memory models 2.5. Nonparametric models 3. Testing linearity 3.1. Testsagainst a specific alternative 3.2. Testswithout a specific alternative 3.3. Constancy of conditional variance
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4. 5.
Specification of nonlinear models Estimation in nonlinear time series 5.1. 5.2.
Estimation of parameters in parametric models Estimation ofnonparametric functions
5.3.
Estimation of restricted nonparametric and semiparametric models
6. Evaluation of estimated models 7. Example 8. Conclusions References
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Chapter 49
ARCH Models TIM BOLLERSLEV, ROBERT F. E N G L E and DANIEL B. N E L S O N
Abstract 1. Introduction 1.1. Definitions 1.2. Empirical regularities of asset returns 1.3. Univariate parametric models 1.4. ARCH in mean models 1.5. Nonparametric and semiparametric methods
2.
3.
4.
5.
Inference procedures 2.1.
Testing for ARCH
2.2. 2.3.
Maximum likelihood methods Quasi-maximum likelihood methods
2.4.
Specification checks
Stationary and ergodic properties 3.1.
Strict stationarity
3.2.
Persistence
Continuous time methods 4.1.
ARCH models as approximations to diffusions
4.2. 4.3.
Diffusions as approximations to ARCH models ARCH models as filters and forecasters
Aggregation and forecasting 5.1.
Temporal aggregation
5.2. Forecast error distributions
6.
Multivariate specifications 6.1.
Vector ARCH and diagonal ARCH
6.2. Factor ARCH 6.3. Constant conditional correlations 6.4. 6.5.
7. 8.
Bivariate EGARCH Stationarity and co-persistence
Model selection Alternative measures for volatility
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Contents of Volume I V
9.
Empirical examples 9.1. U.S. Dollar/Deutschmark exchange rates 9.2.
U.S. stock prices
10. Conclusion References
xxv
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Chapter 50
State-Space Models JAMES D. H A M I L T O N
Abstract 1. The state-space representation of a linear dynamic system 2. The. Kalman filter
3.
4.
5.
2.1.
Overview of the Kalman filter
2.2. 2.3.
Derivation of the Kalman filter Forecasting with the Kalman filter
2.4. 2.5.
Smoothed inference Interpretation of the Kalman filter with non-normal disturbances
2.6. 2.7.
Time-varying coefficient models Other extensions
Statistical inference about unknown parameters using the Kalman filter 3.1.
Maximum likelihood estimation
3.2. 3.3. 3.4.
Identification Asymptotic properties of maximum likelihood estimates Confidence intervals for smoothed estimates and forecasts
3.5.
Empirical application - an analysis of the real interest rate
Discrete-valued state variables 4.1.
Linear state-space representation of the Markov-switching model
4.2.
Optimal filter when the state variable follows a Markov chain
4.3.
Extensions
4.4. 4.5. 4.6.
Forecasting Smoothed probabilities Maximum likelihood estimation
4.7.
Asymptotic properties of maximum likelihood estimates
4.8.
Empirical application - another look at the real interest rate
Non-normal and nonlinear state-space models
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5.1. Kitagawa's grid approximation for nonlinear, non-normal state-space 5.2.
models Extended Kalman filter
5.3.
Other approaches to nonlinear state-space models
References
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Chapter 51
Structural E s t i m a t i o n of M a r k o v Decision Processes JOHN RUST 1. 2.
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Introduction Solving M D P ' s via d y n a m i c p r o g r a m m i n g : A brief review 2.1. Finite-horizon dynamic programming and the optimality of Markovian decision rules 2.2. Infinite-horizon dynamic programming and Bellman's equation 2.3. Bellman'sequation, contraction mappings and optimality 2.4. A geometric series representation for MDP's 2.5. Overview of solution methods 3. E c o n o m e t r i c m e t h o d s for discrete decision processes 3.1. Alternative models of the "error term" 3.2. Maximum likelihood estimation of DDP's 3.3. Alternative estimation methods: Finite-horizon DDP problems 3.4. Alternative estimation methods: Infinite-horizon DDP's 3.5. The identification problem 4. Empirical applications 4.l. Optimal replacement of bus engines 4.2. Optimal retirement from a firm References
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List of T h e o r e m s
3145
Index
3147