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Econometric analysis

Author: Greene, William H. Publisher: Macmillan, 1993.Edition: 2nd ed.Language: EnglishDescription: 791 p. ; 26 cm.ISBN: 0023463910Type of document: BookNote: Doriot: for 2013-2014 coursesBibliography/Index: Includes bibliographical references and index
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Doriot: for 2013-2014 courses

Includes bibliographical references and index

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Econometric Analysis Contents 1 INTRODUCTION 1.1. Econometrics 1 1.2. Econometric Modeling 1 1.3. Theoretical and Applied Econometrics 1.4. Plan of the Book 3 1 3 2 MATRIX ALGEBRA 2.1. Introduction 5 2.2. Some Terminology 5 2.3. Algebraic Manipulation of Matrices 6 2.3.1. Equality of Matrices 7 2.3.2. Transposition 7 2.3.3. Matrix Addition 7 2.3.4. Matrix Multiplication 8 2.3.5. Sums of Values 10 2.3.6. A Useful Idempotent Matrix 11 2.4. Geometry of Matrices 13 2.4.1. Vector Spaces 13 2.4.2. Linear Combinations of Vectors and Basis Vectors 2.4.3. Linear Dependence 16 2.4.4. Subspaces 16 2.4.5. Rank of a Matrix 17 2.4.6. Determinant of a Matrix 19 2.4.7. A Least Squares Problem 21 2.5. Solution of a System of Equations 23 2.5.1. Systems of Linear Equations 23 2.5.2. Inverse Matrices 24 2.5.3. Nonhomogeneous Systems of Equations 26 2.6. Partitioned Matrices 26 2.6.1. Addition and Multiplication of Partitioned Matrices 2.6.2. Determinants of Partitioned Matrices 27 2.6.3. Inverses of Partitioned Matrices 27 2.6.4. Deviations from Means 28 2.6.5. Kronecker Products 28 5 14 26 xi xii Contents 2.7. Characteristic Roots and Vectors 29 2.7.1. The Characteristic Equation 29 2.7.2. Characteristic Vectors 30 2.7.3. General Results for Characteristic Roots and Vectors 2.7.4. Diagonalization of a Matrix 31 2.7.5. Rank of a Matrix 32 2.7.6. Condition Number of a Matrix 32 2.7.7. Trace of a Matrix 33 2.7.8. Determinant of a Matrix 34 2.7.9. Spectral Decomposition of a Matrix 34 2.7.10. Powers of a Matrix 34 2.7.11. Idempotent Matrices 36 2.7.12. Factoring a Matrix 36 2.7.13. The Generalized Inverse of a Matrix 37 2.8. Quadratic Forms and Definite Matrices 38 2.8.1. Nonnegative Definite Matrices 38 2.8.2. Idempotent Quadratic Forms 39 2.8.3. Ranking Matrices 39 2.9. Calculus and Matrix Algebra 40 2.9.1. Differentiation and the Taylor Series 41 2.9.2. Optimization 44 2.9.3. Constrained Optimization 46 2.9.4. Transformations 48 Exercises 49 31 3 PROBABILITY AND DISTRIBUTION THEORY 53 3.1. Introduction 53 3.2. Random Variables 53 3.2.1. Probability Distributions 53 3.2.2. Cumulative Distribution Function 54 3.3. Expectations of a Random Variable 55 3.4. Some Specific Probability Distributions 57 3.4.1. The Normal Distribution 58 3.4.2. The Chi-Squared, t, and F Distributions 58 3.4.3. Distributions with Large Degrees of Freedom 59 3.4.4. Size Distributions--The Lognormal Distribution 60 3.4.5. The Gamma Distribution 61 3.4.6. The Beta Distribution 61 3.5. The Distribution of a Function of a Random Variable 61 3.6. Joint Distributions 63 3.6.1. Marginal Distributions 64 3.6.2. Expectations in a Joint Distribution 64 3.6.3. Covariance and Correlation 65 3.6.4. Distribution of Functions of Bivariate Random Variables 66 3.7. Conditioning in a Bivariate Distribution 67 3.7.1. Regression--The Conditional Mean 67 3.7.2. Conditional Variance 68 Contents xiii 3.7.3. Relationships Among Marginal and Conditional Moments 69 3.7.4. The Analysis of Variance 71 3.8. The Bivariate Normal Distribution 72 3.9. Multivariate Distributions 73 3.9.1. Moments 73 3.9.2. Sets of Linear Functions 74 3.9.3. Nonlinear Functions 75 3.10. The Multivariate Normal Distribution 75 3.10.1. Marginal and Conditional Distributions 76 3.10.2. Linear Functions of a Normal Vector 77 3.10.3. Quadratic Forms in a Standard Normal Vector 77 3.10.3a. Idempotent Quadratic Forms 77 3.10.3b. Independence of Idempotent Quadratic Forms 79 3.10.4. The F Distribution 79 3.10.5. A Full Rank Quadratic Form 79 3.10.6. Independence of a Linear and a Quadratic Form 80 Exercises 81 Appendix: Integration--The Gamma Function 85 4 STATISTICAL INFERENCE 87 4.1. Introduction 87 4.2. Samples and Sampling Distributions 87 4.2.1. Random Sampling 87 4.2.2. Descriptive Statistics 88 4.2.3. Sampling Distributions 90 4.3. Point Estimation of Parameters 91 4.3.1. Estimation in a Finite Sample 93 4.3.2. Efficient Unbiased Estimation 95 4.4. Large-Sample Distribution Theory 99 4.4.1. Convergence in Probability 99 4.4.2. Convergence in Distribution--Limiting Distributions 102 4.4.3. Asymptotic Distributions 106 4.4.4. Asymptotic Distribution of a Nonlinear Function 108 4.4.5. Asymptotic Expectations 109 4.5. Efficient Estimation--Maximum Likelihood 110 4.5.1. Properties of Maximum Likelihood Estimators 114 4.5.2. Estimating the Variance of the Maximum Likelihood Estimator 115 4.6. Consistent Estimation--The Method of Moments 117 4.6.1. Random Sampling and Estimating the Parameters of Distributions 117 4.6.2. Computing the Variance of a Method of Moments Estimator 121 4.7. Interval Estimation 123 4.8. Hypothesis Testing 125 4.8.1. Testing Procedures 126 4.8.2. Tests Based on Confidence Intervals 128 xiv Contents 4.8.3. 4.8.4. Exercises Three Asymptotically Equivalent Test Procedures 4.8.3a. The Likelihood Ratio Test 130 4.8.3b. The Wald Test 131 4.8.3c. The Lagrange Multiplier Test 133 An Example of the Test Procedures 134 4.8.4a. Confidence Interval Test 135 4.8.4b. Likelihood Ratio Test 135 4.8.4c. Wald Test 135 4.8.4d. Lagrange Multiplier Test 136 136 129 5 THE CLASSICAL LINEAR REGRESSION MODEL 140 5.1. Introduction 140 5.2. Specifying the Regression Model--An Example 140 5.3. The Assumptions of the Linear Regression Model 143 5.3.1. Functional Form and Nonlinear Models 144 5.3.2. The Regressor 146 5.3.3. The Disturbance 146 5.4. Least Squares 148 5.4.1. The Least Squares Coefficients 148 5.4.2. Evaluating the Fit of the Regression 150 5.5. Statistical Properties of the Least Squares Estimator 155 5.6. Statistical Inference 159 5.6.1. Estimating the Sampling Distribution 159 5.6.2. Testing a Hypothesis About ß 160 5.6.3. Tests Based on the Fit of the Regression 162 5.7. Prediction 164 Exercises 166 6 MULTIPLE REGRESSION 170 6.1. Introduction 170 6.2. Assumptions of the Linear Model 170 6.3. Least Squares Regression 172 6.3.1. The Least Squares Coefficient Vector 173 6.3.2. Some Examples 174 6.3.3. Algebraic Aspects of the Solution 177 6.3.4. Partitioned Regression and Partial Regression 179 6.3.5. Partial Regression and Partial Correlation Coefficients 180 6.3.6. Deviations from Means--Regression on a Constant 181 6.4. Statistical Properties of the Least Squares Estimator 182 6.4.1. Nonstochastic Regressors 182 6.4.2. Stochastic Regressors 183 6.5. Statistical Inference 184 6.5.1. Testing a Hypothesis About a Coefficient 184 6.5.2. Testing a Linear Restriction 187 6.5.3. Test Statistics with Stochastic X and Normal 190 Contents xv 6.6. Goodness of Fit and the Analysis of Variance 191 6.7. Testing the Significance of the Regression 194 6.8. Prediction 195 6.8.1. A Convenient Method of Computing the Forecasts 6.8.2. Measuring the Accuracy of Forecasts 197 Exercises 198 196 7 HYPOTHESIS TESTS WITH THE MULTIPLE REGRESSION MODEL Introduction 203 7.2. Testing Restrictions 203 7.2.1. Two Approaches to Testing Hypotheses 203 7.2.2. Testing a Set of Linear Restrictions 204 7.2.3. The Restricted Least Squares Estimator 205 7.2.4. Testing the Restrictions 206 7.2.5. Examples and Some General Procedures 206 7.3. Tests of Structural Change 211 7.3.1. Different Parameter Vectors 211 7.3.2. Different Constant Terms 212 7.3.3. Change in a Subset of Coefficients 213 7.3.4. Insufficient Observations 214 7.4. Tests of Structural Change with Unequal Variances 7.5. Alternative Tests of Model Stability 216 7.6. Testing Nonlinear Restrictions 218 7.7. Choosing Between Nonnested Models 222 Exercises 225 7.1. 203 215 8 FUNCTIONAL FORM, NONLINEARITY, AND SPECIFICATION 8.1. Introduction 229 8.2. Dummy Variables 229 8.2.1. Comparing Two Means 229 8.2.2. Binary Variables in Regression 231 8.2.3. Several Categories 232 8.2.4. Several Groupings 233 8.2.5. Threshold Effects 234 8.2.6. Interactions and Spline Regression 235 8.3. Nonlinearity in the Variables 238 8.3.1. Functional Forms 238 8.3.2. Identifying Nonlinearity 240 8.3.3. Intrinsic Linearity and Identification 242 8.4. Specification Analysis 244 8.4.1. Selection of Variables 244 8.4.2. Omission of Relevant Variables 245 8.4.3. Inclusion of Irrelevant Variables 248 229 xvi Contents 8.5. Biased Estimators and Pretest Estimators 248 8.5.1. The Mean-Squared-Error Test 249 8.5.2. Pretest Estimators 251 8.5.3. Inequality Restrictions 253 8.6. Bayesian Estimation 255 8.6.1. Bayesian Analysis of the Classical Regression Model 8.6.2. Estimation with an Informative Prior Density 258 Exercises 260 255 9 DATA PROBLEMS 9.1. Introduction 266 9.2. Multicollinearity 266 9.2.1. Perfect Collinearity 267 9.2.2. Near Multicollinearity 267 9.2.3. The Symptoms of Multicollinearity 267 9.2.4. Suggested Remedies for the Multicollinearity Problem 9.3. Missing Observations 273 9.4. Grouped Data 277 9.5. Measurement Error and Proxy Variables 279 9.5.1. One Badly Measured Variable 280 9.5.2. Multiple Regression with Measurement Error 283 9.5.3. The Method of Instrumental Variables 284 9.5.4. Proxy Variables 286 9.5.5. A Specification Test for Measurement Error 287 9.6. Regression Diagnostics and Influential Data Points 287 Exercises 289 266 270 10 LARGE-SAMPLE RESULTS FOR THE CLASSICAL REGRESSION MODEL 292 10.1. Introduction 292 10.2. The Finite-Sample Properties of Least Squares 292 10.3. Asymptotic Distribution Theory for the Classical Regression Model 293 10.3.1. Consistency of the Least Squares Coefficient Vector 293 10.3.2. Asymptotic Normality of the Least Squares Estimator 295 10.3.3. Asymptotic Distribution of a Function of b--The Delta Method 297 10.3.4. Asymptotic Behavior of the Standard Test Statistics 299 10.3.5. Consistency of s2 and the Estimator of Asy.Var[ b] 301 10.4. Stochastic Regressors and Lagged Dependent Variables 302 10.5. Normally Distributed Disturbances 305 10.5.1. Asymptotic Efficiency--Maximum Likelihood Estimation 305 10.5.2. Cases in Which Least Squares Is Inefficient 307 Contents xvii 10.5.3. Alternative Estimation Criteria 308 10.5.4. Detecting Departures from Normality Exercises 311 309 11 NONLINEAR REGRESSION MODELS 314 11.1. Introduction 314 11.2. Nonlinear Regression Models 314 11.2.1. The Linearized Regression 315 11.2.2. Nonlinear Least Squares Estimation 316 11.2.3. A Specification Test for Nonlinear Regressions: Testing for Linear Versus Log-Linear Specification 321 11.3. Parametric Transformations of the Dependent Variable 324 11.4. The Box--Cox Transformation 329 11.4.1. Transforming the Independent Variables 329 11.4.2. Transforming the Model 331 11.4.3. A Test for (Log-) Linearity 334 11.5. Hypothesis Testing and Parametric Restrictions 335 11.5.1. An Asymptotically Valid F Test 336 11.5.2. Wald Test 336 11.5.3. Likelihood Ratio Test 336 11.5.4. Lagrange Multiplier Test 337 Exercises 341 12 AN INTRODUCTION TO NONLINEAR OPTIMIZATION 12.1. Introduction 343 12.2. Optimization Problems 343 12.3. Grid Search 344 12.4. General Characteristics of Algorithms 344 12.5. Gradient Methods 346 12.5.1. Steepest Ascent 346 12.5.2. Newton's Method 347 12.5.3. Maximum Likelihood Estimation 347 12.5.4. Alternatives to Newton's Method 348 12.5.5. Quasi-Newton Methods--Davidon­Fletcher­Powell 12.6. Some Practical Considerations 350 12.7. Examples 352 12.8. The Concentrated Log Likelihood 355 Exercises 356 343 350 13 NONSPHERICAL DISTURBANCES 13.1. Introduction 358 13.2. Consequences for Ordinary Least Squares 13.2.1. Finite-Sample Properties 359 13.2.2. Asymptotic Properties of Least Squares 359 360 358 xviii Contents 13.3. Efficient Estimation 361 13.3.1. Generalized Least Squares 361 13.3.2. Maximum Likelihood Estimation 364 13.4. Estimation When fl is Unknown 365 13.4.1. Feasible Generalized Least Squares 365 13.4.2. Maximum Likelihood Estimation 366 13.5. The Generalized Method of Moments (GMM) Estimator 370 13.5.1. Method of Moments Estimators 370 13.5.2. Generalizing the Method of Moments 372 13.5.3. Testing the Validity of the Moment Restrictions 374 13.5.4. GMM Estimation of Econometric Models 375 13.5.5. Testing Restrictions 379 Exercises 381 14 HETEROSCEDASTICITY 384 14.1. Introduction 384 14.2. Ordinary Least Squares Estimation 386 14.2.1. Inefficiency of Least Squares 387 14.2.2. The Estimated Covariance Matrix of b 388 14.2.3. Estimating the Appropriate Covariance Matrix for Ordinary Least Squares 391 14.3. Testing for Heteroscedasticity 392 14.3.1. White's General Test 392 14.3.2. The Goldfeld­Quandt Test 393 14.3.3. The Breusch­Pagan/Godfrey Test 394 14.3.4. Testing for Groupwise Heteroscedasticity 395 14.3.5. Tests Based on Regressions--Glesjer's Tests 396 14.4. Generalized Least Squares When ^ Is Known 397 14.5. Estimation When Contains Unknown Parameters 399 14.5.1. Two-Step Estimation 399 14.5.2. Maximum Likelihood Estimation 402 14.6. General Conclusions 407 Exercises 408 15 AUTOCORRELATED DISTURBANCES 411 15.1. Introduction 411 15.2. The Analysis of Time-Series Data 413 15.3. Disturbance Processes 415 15.3.1. Characteristics of Disturbance Processes 415 15.3.2. AR(1) Disturbances 416 15.4. Least Squares Estimation 418 15.4.1. OLS Estimation with Lagged Dependent Variables 419 15.4.2. Efficiency of Least Squares 420 15.4.3. Estimating the Variance of the Least Squares Estimator 422 15.5. Testing for Autocorrelation 423 15.5.1. The Durbin­Watson Test 423 Contents xix 15.5.2. Other Testing Procedures 426 15.5.3. Testing in the Presence of Lagged Dependent Variables 428 15.6. Efficient Estimation When Is Known 428 15.6.1. Generalized Least Squares 428 15.6.2. Maximum Likelihood Estimation 430 15.7. Estimation When Is Unknown 431 15.7.1. AR(1) Disturbances 431 15.7.2. AR(2) Disturbances 435 15.7.3. Estimation with a Lagged Dependent Variable 435 15.8. Forecasting in the Presence of Autocorrelation 437 15.9. Autoregressive Conditional Heteroscedasticity 438 Exercises 442 16 MODELS THAT USE BOTH CROSSSECTION AND TIME-SERIES DATA 16.1. Introduction 444 16.2. Time-Series­Cross-Section Data 444 16.3. Models of Several Time Series 447 16.3.1. Cross-Sectional Heteroscedasticity 448 16.3.2. Cross-Sectional Correlation 452 16.3.3. Autocorrelation 455 16.3.4. A Random Coefficients Model 459 16.3.5. Summary 463 16.4. Longitudinal Data 464 16.4.1. A Basic Model of Heterogeneity 465 16.4.2. Fixed Effects 466 16.4.3. Random Effects 469 16.4.3a. Generalized Least Squares 470 16.4.3b. Feasible Generalized Least Squares When Is Unknown 474 16.4.4. Fixed or Random Effects? 479 Exercises 480 444 17 SYSTEMS OF REGRESSION EQUATIONS 17.1. Introduction 486 17.2. The Seemingly Unrelated Regressions Model 487 17.2.1. Generalized Least Squares 488 17.2.2. Feasible Generalized Least Squares 489 17.2.3. Maximum Likelihood Estimation 493 17.2.3a. Iterated FGLS 493 17.2.3b. Direct Maximum Likelihood Estimation 17.2.4. Autocorrelation 497 17.3. Systems of Demand Equations--Singular Systems 17.4. Flexible Functional Forms--Translog Cost Function Exercises 508 486 493 499 503 xx Contents 18 REGRESSIONS WITH LAGGED VARIABLES 511 18.1. Introduction 511 18.2. Distributed Lag Models 511 18.2.1. Lagged Effects in a Regression Model 512 18.2.2. The Lag and Difference Operators 514 18.3. Unrestricted Finite Distributed Lag Models 515 18.4. Polynomial Distributed Lag Models 519 18.4.1. Estimation by Restricted Least Squares 523 18.4.2. Determining the Degree of the Polynomial 523 18.4.3. Determining the Lag Length 524 18.5. The Geometric Lag Model 525 18.5.1. Economic Models with Geometric Lags 526 18.5.2. Stochastic Specifications in the Geometric Lag Models 528 18.5.3. Estimating the Moving Average Form of the Geometric Lag Model 529 18.5.3a. Uncorrelated Disturbances 529 18.5.3b. Autocorrelated Disturbances 531 18.5.3c. Estimating the MA Form of the Partial Adjustment Model 534 18.5.4. Estimating the Autoregressive Form of the Geometric Lag Model 534 18.5.4a. Uncorrelated Disturbances 534 18.5.4b. Autocorrelated Disturbances--Instrumental Variables 535 18.5.4c. Autoregressive Disturbances--Hatanaka's Estimator and the Maximum Likelihood Estimator 535 18. 5.4d. MA Disturbances--The Adaptive Expectations Model 536 18.5.5. Disturbance Specifications in the Geometric Lag Model 537 18.6. Dynamic Regression Models 538 18.6.1. Nonlinear Least Squares Estimation of ARIMA and ARMAX Models 539 18.6.2. Computation of the Lag Weights in the ARMAX Model 542 18.6.3. Stability of a Dynamic Equation 543 18.6.4. Forecasting 544 Appendix: Nonlinear Least Squares Estimation 545 Exercises 547 19 TIME-SERIES MODELS 19.1. Introduction 549 19.2. Stationary Stochastic Processes 549 19.2.1. Autoregressive--Moving Average Processes 19.2.2. Vector Autoregressions 552 549 550 Contents xxi 19.2.3. Stationarity and Invertibility 554 19.2.4. Autocorrelations of a Stationary Process 557 19.3. Integrated Processes and Differencing 559 19.4. Random Walks, Trends, and Spurious Regressions 560 19.5. Unit Roots in Economic Data 563 19.6. Cointegration and Error Correction 566 19.7. Generalized Autoregressive Conditional Heteroscedasticity 568 19.7.1. Maximum Likelihood Estimation of the GARCH Model 570 19.7.2. Pseudo-Maximum Likelihood Estimation 574 19.7.3. Testing for GARCH Effects 575 19.7.4. An Example 576 Exercises 577 20 SIMULTANEOUS EQUATIONS MODELS 578 20.1. Introduction 578 20.2. Fundamental Issues in Simultaneous Equations Models 578 20.2.1. Illustrative Systems of Equations 579 20.2.2. A General Notation for Simultaneous Equations Models 582 20.3. The Problem of Identification 585 20.3.1. The Rank and Order Conditions for Identification 589 20.3.2. Identification Through Nonsample Information 595 20.4. Methods of Estimation 598 20.4.1. Ordinary Least Squares and Triangular Systems 599 20.4.2. Indirect Least Squares 601 20.4.3. Estimation by Instrumental Variables 601 20.4.4. Single-Equation Instrumental Variable Methods 602 20.4.4a. Estimating an Exactly Identified Equation 602 20.4.4b. Two-Stage Least Squares 603 20.4.4c. Limited Information Maximum Likelihood 605 20.4.4d. Two-Stage Least Squares with Autocorrelation 608 20.4.4e. Two-Stage Least Squares in Models That Are Nonlinear in Variables 609 20.4.5. System Methods of Estimation 610 20.4.5a. Three-Stage Least Squares 611 20.4.5b. Full-Information Maximum Likelihood 612 20.4.6. Comparison of Methods 615 20.5. Specification Tests 616 20.6. Properties of Dynamic Models 619 20.6.1. Dynamic Models and Their Multipliers 619 20.6.2. Stability 622 20.6.3. Adjustment to Equilibrium 624 Exercises 626 Appendix: Yearly Data on the U.S. Economy 628 xxii Contents 21 MODELS WITH DISCRETE DEPENDENT VARIABLES 635 21.1. Introduction 635 21.2. Discrete Choice Models 635 21.3. Models for Binary Choice 636 21.3.1. The Regression Approach 636 21.3.2. Index Function and Random Utility Models 642 21.4. Estimation and Inference in Binary Choice Models 643 21.4.1. Specification Tests in Binary Choice Models 648 21.4.1a. Testing for Omitted Variables 649 21.4.1b. Testing for Heteroscedasticity 649 21.4.2. Measuring Goodness of Fit 651 21.4.3. Analysis of Proportions Data 653 21.5. Recent Developments 655 21.5.1. Fixed and Random Effects Models for Panel Data 655 21.5.2. Semiparametric Estimation 657 21.5.3. Nonparametric Estimation--The Maximum Score Estimator 658 21.6. Bivariate and Multivariate Probit Models 660 21.6.1. Maximum Likelihood Estimation 660 21.6.2. Extensions 663 21.6.2a. A Multivariate Probit Model 663 21.6.2b. A Model with Censoring 664 21.7. Models for Multiple Choices 664 21.7.1. Unordered Multiple Choices 664 21.7.1a. The Multinomial Logit Model 666 21.7. lb. The Conditional Logit Model 668 21.7.1c. The Independence of Irrelevant Alternatives 670 21.7.2. Ordered Data 672 21.8. A Poisson Model for Count Data 676 Exercises 679 22 LIMITED DEPENDENT VARIABLE AND DURATION MODELS 682 22.1. Introduction 682 22.2. Truncation 682 22.2.1. Truncated Distributions 683 22.2.2. Moments of Truncated Distributions 684 22.2.3. The Truncated Regression Model 687 22.2.3a. Least Squares Estimation 689 22.2.3b. Maximum Likelihood Estimation 22.3. Censored Data 691 22.3.1. The Censored Normal Distribution 691 22.3.2. The Censored Regression Model--Tobit Analysis 22.3.3. Estimation 696 689 694 Contents xxiii 22.3.4. Some Issues in Specification 698 22.3.4a. Heteroscedasticity 698 22.3.4b. Misspecification of Prob[y* < 0] 700 22.3.4c. Nonnormality 701 22.3.4d. Conditional Moment Tests--An Introduction 701 22.4. Selection--Incidental Truncation 706 22.4.1. Incidental Truncation in a Bivariate Distribution 707 22.4.2. Regression in a Model of Selection 708 22.4.3. Estimation 711 22.4.4. Treatment Effects 713 22.4.5. The Normality Assumption 714 22.5. Models for Duration Data 715 22.5.1. Duration Data 715 22.5.2. A Regression-Like Approach--Parametric Models of Duration 716 22.5.2a. Theoretical Background 716 22.5.2b. Models of the Hazard Rate 717 22.5.2c. Maximum Likelihood Estimation 719 22.5.2d. Exogenous Variables 721 22.5.2e. Specification Analysis 722 22.5.2f. Heterogeneity 724 22.5.3. Other Approaches 725 Exercises 726 Appendix Tables 728 1. Cumulative Normal Distribution 729 2. Ordinates of the Standard Normal Density 730 3. Percentiles of the Student's t Distribution 731 4. Percentiles of the Chi-Squared Distribution 732 5. 95th Percentiles of the F Distribution 734 6. 99th Percentiles of the F Distribution 736 7. Durbin­Watson Test 738 8. Five Percent Significance Point of d4,L and d4,U for Regressions Without Quarterly Dummy Variables ( k = k' + 1) 744 References 745 Author Index 777 Subject Index 783

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