Principles of Econometrics, 5th Ed.

Cover......Page 1
Title Page......Page 3
Copyright......Page 4
Preface......Page 7
Contents......Page 14
List of Examples......Page 23
1.1 Why Study Econometrics?......Page 29
1.2 What Is Econometrics About?......Page 30
1.2.1 Some Examples......Page 31
1.3 The Econometric Model......Page 32
1.4 How Are Data Generated?......Page 33
1.4.2 Quasi-Experimental Data......Page 34
1.5.1 Time-Series Data......Page 35
1.5.2 Cross-Section Data......Page 36
1.6 The Research Process......Page 37
1.7.2 A Format for Writing a Research Report......Page 39
1.8.1 Links to Economic Data on the Internet......Page 41
1.8.3 Obtaining the Data......Page 42
Probability Primer......Page 43
P.1 Random Variables......Page 44
P.2 Probability Distributions......Page 45
P.3.1 Marginal Distributions......Page 48
P.3.3 Statistical Independence......Page 49
P.4 A Digression: Summation Notation......Page 50
P.5 Properties of Probability Distributions......Page 51
P.5.1 Expected Value of a Random Variable......Page 52
P.5.3 Rules for Expected Values......Page 53
P.5.4 Variance of a Random Variable......Page 54
P.5.6 Covariance Between Two Random Variables......Page 55
P.6 Conditioning......Page 57
P.6.1 Conditional Expectation......Page 58
P.6.2 Conditional Variance......Page 59
P.6.3 Iterated Expectations......Page 60
P.6.4 Variance Decomposition......Page 61
P.7 The Normal Distribution......Page 62
P.7.1 The Bivariate Normal Distribution......Page 65
P.8 Exercises......Page 67
Chapter 2: The Simple Linear Regression Model......Page 74
2.1 An Economic Model......Page 75
2.2 An Econometric Model......Page 77
2.2.1 Data Generating Process......Page 79
2.2.2 The Random Error and Strict Exogeneity......Page 80
2.2.3 The Regression Function......Page 81
2.2.4 Random Error Variation......Page 82
2.2.7 Generalizing the Exogeneity Assumption......Page 84
2.2.8 Error Correlation......Page 85
2.2.9 Summarizing the Assumptions......Page 86
2.3 Estimating the Regression Parameters......Page 87
2.3.1 The Least Squares Principle......Page 89
2.3.2 Other Economic Models......Page 93
2.4 Assessing the Least Squares Estimators......Page 94
2.4.1 The Estimator b2......Page 95
2.4.2 The Expected Values of b1 and b2......Page 96
2.4.4 The Variances and Covariance of b1 and b2......Page 97
2.5 The Gauss-Markov Theorem......Page 100
2.6 The Probability Distributions of the Least Squares Estimators......Page 101
2.7.1 Estimating the Variances and Covariance of the Least Squares Estimators......Page 102
2.7.2 Interpreting the Standard Errors......Page 104
2.8.2 Using a Quadratic Model......Page 105
2.8.3 A Log-Linear Function......Page 107
2.8.4 Using a Log-Linear Model......Page 108
2.9 Regression with Indicator Variables......Page 110
2.10.1 Random and Independent x......Page 112
2.10.2 Random and Strictly Exogenous x......Page 114
2.10.3 Random Sampling......Page 115
2.11.1 Problems......Page 117
2.11.2 Computer Exercises......Page 121
Appendix 2A Derivation of the Least Squares Estimates......Page 126
Appendix 2B Deviation from the Mean Form of b2......Page 127
Appendix 2E Deriving the Conditional Variance of b2......Page 128
Appendix 2F Proof of the Gauss-Markov Theorem......Page 130
2G.2 The Random and Independent x Case......Page 131
2G.3 The Random and Strictly Exogenous x Case......Page 133
2H.1 The Regression Function......Page 134
2H.3 Theoretically True Values......Page 135
2H.4 Creating a Sample of Data......Page 136
2H.6 Monte Carlo Results......Page 137
2H.7 Random-x Monte Carlo Results......Page 138
Chapter 3: Interval Estimation and Hypothesis Testing......Page 140
3.1.1 The t-Distribution......Page 141
3.1.2 Obtaining Interval Estimates......Page 143
3.1.3 The Sampling Context......Page 144
3.2.2 The Alternative Hypothesis......Page 146
3.2.4 The Rejection Region......Page 147
3.3.1 One-Tail Tests with Alternative "Greater Than" (>)......Page 148
3.3.2 One-Tail Tests with Alternative "Less Than" (<)......Page 149
3.3.3 Two-Tail Tests with Alternative "Not Equal To" (≠)......Page 150
3.4 Examples of Hypothesis Tests......Page 151
3.5 The p-Value......Page 154
3.6 Linear Combinations of Parameters......Page 157
3.6.1 Testing a Linear Combination of Parameters......Page 159
3.7.1 Problems......Page 161
3.7.2 Computer Exercises......Page 167
Appendix 3A Derivation of the t-Distribution......Page 172
Appendix 3B Distribution of the t-Statistic under H1......Page 173
Appendix 3C Monte Carlo Simulation......Page 175
3C.1 Sampling Properties of Interval Estimators......Page 176
3C.3 Choosing the Number of Monte Carlo Samples......Page 177
3C.4 Random-x Monte Carlo Results......Page 178
Chapter 4: Prediction, Goodness-of-Fit, and Modeling Issues......Page 180
4.1 Least Squares Prediction......Page 181
4.2 Measuring Goodness-of-Fit......Page 184
4.2.2 Correlation Analysis and R2......Page 186
4.3.1 The Effects of Scaling the Data......Page 188
4.3.2 Choosing a Functional Form......Page 189
4.3.3 A Linear-Log Food Expenditure Model......Page 191
4.3.4 Using Diagnostic Residual Plots......Page 193
4.3.5 Are the Regression Errors Normally Distributed?......Page 195
4.3.6 Identifying Influential Observations......Page 197
4.4.1 Quadratic and Cubic Equations......Page 199
4.5 Log-Linear Models......Page 201
4.5.1 Prediction in the Log-Linear Model......Page 203
4.5.2 A Generalized R2 Measure......Page 204
4.6 Log-Log Models......Page 205
4.7.1 Problems......Page 207
4.7.2 Computer Exercises......Page 213
Appendix 4A Development of a Prediction Interval......Page 220
Appendix 4C Mean Squared Error: Estimation and Prediction......Page 221
Chapter 5: The Multiple Regression Model......Page 224
5.1.1 The Economic Model......Page 225
5.1.2 The Econometric Model......Page 226
5.1.3 The General Model......Page 230
5.1.4 Assumptions of the Multiple Regression Model......Page 231
5.2.1 Least Squares Estimation Procedure......Page 233
5.2.2 Estimating the Error Variance σ2......Page 235
5.2.3 Measuring Goodness-of-Fit......Page 236
5.2.4 Frisch-Waugh-Lovell (FWL) Theorem......Page 237
5.3 Finite Sample Properties of the Least Squares Estimator......Page 239
5.3.1 The Variances and Covariances of the Least Squares Estimators......Page 240
5.3.2 The Distribution of the Least Squares Estimators......Page 242
5.4.1 Interval Estimation for a Single Coefficient......Page 244
5.4.2 Interval Estimation for a Linear Combination of Coefficients......Page 245
5.5 Hypothesis Testing......Page 246
5.5.1 Testing the Significance of a Single Coefficient......Page 247
5.5.2 One-Tail Hypothesis Testing for a Single Coefficient......Page 248
5.5.3 Hypothesis Testing for a Linear Combination of Coefficients......Page 249
5.6 Nonlinear Relationships......Page 250
5.7.1 Consistency......Page 255
5.7.2 Asymptotic Normality......Page 257
5.7.3 Relaxing Assumptions......Page 258
5.7.4 Inference for a Nonlinear Function of Coefficients......Page 260
5.8.1 Problems......Page 262
5.8.2 Computer Exercises......Page 268
Appendix 5A Derivation of Least Squares Estimators......Page 275
5B.1 Nonlinear Function of a Single Parameter......Page 276
5B.2 Nonlinear Function of Two Parameters......Page 277
5C.1 Least Squares Estimation with Chi-Square Errors......Page 278
5C.2 Monte Carlo Simulation of the Delta Method......Page 280
Appendix 5D Bootstrapping......Page 282
5D.1 Resampling......Page 283
5D.3 Bootstrap Standard Error......Page 284
5D.4 Bootstrap Percentile Interval Estimate......Page 285
5D.5 Asymptotic Refinement......Page 286
Chapter 6: Further Inference in the Multiple Regression Model......Page 288
6.1 Testing Joint Hypotheses: The F-test......Page 289
6.1.1 Testing the Significance of the Model......Page 292
6.1.2 The Relationship Between t- and F-Tests......Page 293
6.1.3 More General F-Tests......Page 295
6.1.4 Using Computer Software......Page 296
6.1.5 Large Sample Tests......Page 297
6.2 The Use of Nonsample Information......Page 299
6.3.1 Causality versus Prediction......Page 301
6.3.2 Omitted Variables......Page 303
6.3.3 Irrelevant Variables......Page 305
6.3.4 Control Variables......Page 306
6.3.5 Choosing a Model......Page 308
6.3.6 RESET......Page 309
6.4 Prediction......Page 310
6.4.1 Predictive Model Selection Criteria......Page 313
6.5 Poor Data, Collinearity, and Insignificance......Page 316
6.5.1 The Consequences of Collinearity......Page 317
6.5.2 Identifying and Mitigating Collinearity......Page 318
6.5.3 Investigating Influential Observations......Page 321
6.6 Nonlinear Least Squares......Page 322
6.7.1 Problems......Page 325
6.7.2 Computer Exercises......Page 331
Appendix 6A The Statistical Power of F-Tests......Page 339
Appendix 6B Further Results from the FWL Theorem......Page 343
Chapter 7: Using Indicator Variables......Page 345
7.1.1 Intercept Indicator Variables......Page 346
7.1.2 Slope-Indicator Variables......Page 348
7.2.1 Interactions Between Qualitative Factors......Page 351
7.2.2 Qualitative Factors with Several Categories......Page 352
7.2.3 Testing the Equivalence of Two Regressions......Page 354
7.2.4 Controlling for Time......Page 356
7.3 Log-Linear Models......Page 357
7.3.2 An Exact Calculation......Page 358
7.4 The Linear Probability Model......Page 359
7.5 Treatment Effects......Page 360
7.5.2 Analysis of the Difference Estimator......Page 362
7.5.3 The Differences-in-Differences Estimator......Page 366
7.6.1 The Nature of Causal Effects......Page 370
7.6.2 Treatment Effect Models......Page 371
7.6.3 Decomposing the Treatment Effect......Page 372
7.6.4 Introducing Control Variables......Page 373
7.6.6 Regression Discontinuity Designs......Page 375
7.7.1 Problems......Page 379
7.7.2 Computer Exercises......Page 386
Appendix 7B Derivation of the Differences-in-Differences Estimator......Page 394
Appendix 7C The Overlap Assumption: Details......Page 395
Chapter 8: Heteroskedasticity......Page 396
8.1 The Nature of Heteroskedasticity......Page 397
8.2 Heteroskedasticity in the Multiple Regression Model......Page 398
8.2.1 The Heteroskedastic Regression Model......Page 399
8.2.2 Heteroskedasticity Consequences for the OLS Estimator......Page 401
8.3 Heteroskedasticity Robust Variance Estimator......Page 402
8.4.1 Transforming the Model: Proportional Heteroskedasticity......Page 403
8.4.2 Weighted Least Squares: Proportional Heteroskedasticity......Page 405
8.5 Generalized Least Squares: Unknown Form of Variance......Page 407
8.5.1 Estimating the Multiplicative Model......Page 409
8.6 Detecting Heteroskedasticity......Page 411
8.6.2 The Goldfeld-Quandt Test......Page 412
8.6.3 A General Test for Conditional Heteroskedasticity......Page 413
8.6.4 The White Test......Page 415
8.6.5 Model Specification and Heteroskedasticity......Page 416
8.7 Heteroskedasticity in the Linear Probability Model......Page 418
8.8.1 Problems......Page 419
8.8.2 Computer Exercises......Page 429
Appendix 8A Properties of the Least Squares Estimator......Page 435
Appendix 8B Lagrange Multiplier Tests for Heteroskedasticity......Page 436
Appendix 8C Properties of the Least Squares Residuals......Page 438
Appendix 8D Alternative Robust Sandwich Estimators......Page 439
Appendix 8E Monte Carlo Evidence: OLS, GLS, and FGLS......Page 442
Chapter 9: Regression with Time-Series Data: Stationary Variables......Page 445
9.1 Introduction......Page 446
9.1.1 Modeling Dynamic Relationships......Page 448
9.1.2 Autocorrelations......Page 452
9.2 Stationarity and Weak Dependence......Page 455
9.3 Forecasting......Page 458
9.3.1 Forecast Intervals and Standard Errors......Page 461
9.3.2 Assumptions for Forecasting......Page 463
9.3.3 Selecting Lag Lengths......Page 464
9.3.4 Testing for Granger Causality......Page 465
9.4 Testing for Serially Correlated Errors......Page 466
9.4.1 Checking the Correlogram of the Least Squares Residuals......Page 467
9.4.2 Lagrange Multiplier Test......Page 468
9.5 Time-Series Regressions for Policy Analysis......Page 471
9.5.1 Finite Distributed Lags......Page 473
9.5.2 HAC Standard Errors......Page 476
9.5.3 Estimation with AR(1) Errors......Page 480
9.5.4 Infinite Distributed Lags......Page 484
9.6.1 Problems......Page 491
9.6.2 Computer Exercises......Page 496
Appendix 9A The Durbin-Watson Test......Page 504
9A.1 The Durbin-Watson Bounds Test......Page 506
Appendix 9B Properties of an AR(1) Error......Page 507
Chapter 10: Endogenous Regressors and Moment-Based Estimation......Page 509
10.1 Least Squares Estimation with Endogenous Regressors......Page 510
10.1.1 Large Sample Properties of the OLS Estimator......Page 511
10.1.2 Why Least Squares Estimation Fails......Page 512
10.1.3 Proving the Inconsistency of OLS......Page 514
10.2.1 Measurement Error......Page 515
10.2.2 Simultaneous Equations Bias......Page 516
10.2.4 Omitted Variables......Page 517
10.3.1 Method of Moments Estimation of a Population Mean and Variance......Page 518
10.3.2 Method of Moments Estimation in the Simple Regression Model......Page 519
10.3.3 Instrumental Variables Estimation in the Simple Regression Model......Page 520
10.3.4 The Importance of Using Strong Instruments......Page 521
10.3.5 Proving the Consistency of the IV Estimator......Page 522
10.3.6 IV Estimation Using Two-Stage Least Squares (2SLS)......Page 523
10.3.7 Using Surplus Moment Conditions......Page 524
10.3.8 Instrumental Variables Estimation in the Multiple Regression Model......Page 526
10.3.9 Assessing Instrument Strength Using the First-Stage Model......Page 528
10.3.10 Instrumental Variables Estimation in a General Model......Page 530
10.3.11 Additional Issues When Using IV Estimation......Page 532
10.4.1 The Hausman Test for Endogeneity......Page 533
10.4.2 The Logic of the Hausman Test......Page 535
10.4.3 Testing Instrument Validity......Page 536
10.5.1 Problems......Page 538
10.5.2 Computer Exercises......Page 544
Appendix 10A Testing for Weak Instruments......Page 548
10A.1 A Test for Weak Identification......Page 549
Appendix 10B Monte Carlo Simulation......Page 553
10B.1 Illustrations Using Simulated Data......Page 554
10B.2 The Sampling Properties of IV/2SLS......Page 556
Chapter 11: Simultaneous Equations Models......Page 559
11.1 A Supply and Demand Model......Page 560
11.2 The Reduced-Form Equations......Page 562
11.3.1 Proving the Failure of OLS......Page 563
11.4 The Identification Problem......Page 564
11.5 Two-Stage Least Squares Estimation......Page 566
11.5.1 The General Two-Stage Least Squares Estimation Procedure......Page 567
11.5.2 The Properties of the Two-Stage Least Squares Estimator......Page 568
11.6.1 Problems......Page 573
11.6.2 Computer Exercises......Page 579
11A.1 The k-Class of Estimators......Page 585
11A.2 The LIML Estimator......Page 586
11A.3 Monte Carlo Simulation Results......Page 590
Chapter 12: Regression with Time-Series Data: Nonstationary Variables......Page 591
12.1 Stationary and Nonstationary Variables......Page 592
12.1.1 Trend Stationary Variables......Page 595
12.1.2 The First-Order Autoregressive Model......Page 598
12.1.3 Random Walk Models......Page 600
12.2 Consequences of Stochastic Trends......Page 602
12.3.1 Unit Roots......Page 604
12.3.3 Dickey-Fuller Test with Intercept and No Trend......Page 605
12.3.4 Dickey-Fuller Test with Intercept and Trend......Page 607
12.3.5 Dickey-Fuller Test with No Intercept and No Trend......Page 608
12.3.6 Order of Integration......Page 609
12.4 Cointegration......Page 610
12.4.1 The Error Correction Model......Page 612
12.5 Regression When There Is No Cointegration......Page 613
12.6 Summary......Page 615
12.7.1 Problems......Page 616
12.7.2 Computer Exercises......Page 620
Chapter 13: Vector Error Correction and Vector Autoregressive Models......Page 625
13.1 VEC and VAR Models......Page 626
13.2 Estimating a Vector Error Correction Model......Page 628
13.3 Estimating a VAR Model......Page 629
13.4.1 Impulse Response Functions......Page 631
13.4.2 Forecast Error Variance Decompositions......Page 633
13.5.1 Problems......Page 635
13.5.2 Computer Exercises......Page 636
Appendix 13A The Identification Problem3......Page 640
Chapter 14: Time-Varying Volatility and ARCH Models......Page 642
14.1 The ARCH Model......Page 643
14.2 Time-Varying Volatility......Page 644
14.3 Testing, Estimating, and Forecasting......Page 648
14.4.1 The GARCH Model-Generalized ARCH......Page 650
14.4.2 Allowing for an Asymmetric Effect......Page 651
14.4.3 GARCH-in-Mean and Time-Varying Risk Premium......Page 652
14.4.4 Other Developments......Page 653
14.5.1 Problems......Page 654
14.5.2 Computer Exercises......Page 655
Chapter 15: Panel Data Models......Page 662
15.1 The Panel Data Regression Function......Page 664
15.1.1 Further Discussion of Unobserved Heterogeneity......Page 666
15.1.3 Using OLS to Estimate the Panel Data Regression......Page 667
15.2.1 The Difference Estimator: T = 2......Page 668
15.2.2 The Within Estimator: T = 2......Page 670
15.2.3 The Within Estimator: T > 2......Page 671
15.2.4 The Least Squares Dummy Variable Model......Page 672
15.3 Panel Data Regression Error Assumptions......Page 674
15.3.1 OLS Estimation with Cluster-Robust Standard Errors......Page 676
15.3.2 Fixed Effects Estimation with Cluster-Robust Standard Errors......Page 678
15.4 The Random Effects Estimator......Page 679
15.4.1 Testing for Random Effects......Page 681
15.4.2 A Hausman Test for Endogeneity in the Random Effects Model......Page 682
15.4.3 A Regression-Based Hausman Test......Page 684
15.4.4 The Hausman-Taylor Estimator......Page 686
15.4.5 Summarizing Panel Data Assumptions......Page 688
15.4.6 Summarizing and Extending Panel Data Model Estimation......Page 689
15.5.1 Problems......Page 691
15.5.2 Computer Exercises......Page 698
Appendix 15A Cluster-Robust Standard Errors: Some Details......Page 705
Appendix 15B Estimation of Error Components......Page 707
Chapter 16: Qualitative and Limited Dependent Variable Models......Page 709
16.1 Introducing Models with Binary Dependent Variables......Page 710
16.1.1 The Linear Probability Model......Page 711
16.2 Modeling Binary Choices......Page 713
16.2.1 The Probit Model for Binary Choice......Page 714
16.2.2 Interpreting the Probit Model......Page 715
16.2.3 Maximum Likelihood Estimation of the Probit Model......Page 718
16.2.4 The Logit Model for Binary Choices......Page 721
16.2.5 Wald Hypothesis Tests......Page 723
16.2.6 Likelihood Ratio Hypothesis Tests......Page 724
16.2.8 Binary Choice Models with a Continuous Endogenous Variable......Page 726
16.2.9 Binary Choice Models with a Binary Endogenous Variable......Page 727
16.2.10 Binary Endogenous Explanatory Variables......Page 728
16.2.11 Binary Choice Models and Panel Data......Page 729
16.3 Multinomial Logit......Page 730
16.3.2 Maximum Likelihood Estimation......Page 731
16.3.3 Multinomial Logit Postestimation Analysis......Page 732
16.4.1 Conditional Logit Choice Probabilities......Page 735
16.4.2 Conditional Logit Postestimation Analysis......Page 736
16.5 Ordered Choice Models......Page 737
16.5.1 Ordinal Probit Choice Probabilities......Page 738
16.5.2 Ordered Probit Estimation and Interpretation......Page 739
16.6.1 Maximum Likelihood Estimation of the Poisson Regression Model......Page 741
16.6.2 Interpreting the Poisson Regression Model......Page 742
16.7.1 Maximum Likelihood Estimation of the Simple Linear Regression Model......Page 745
16.7.3 Censored Samples and Regression......Page 746
16.7.4 Tobit Model Interpretation......Page 748
16.7.5 Sample Selection......Page 751
16.8.1 Problems......Page 753
16.8.2 Computer Exercises......Page 761
16A.1 Standard Error of Marginal Effect at a Given Point......Page 767
16A.2 Standard Error of Average Marginal Effect......Page 768
16B.1 Binary Choice Model......Page 769
16B.2 Probit or Logit?......Page 770
16C.1 Tobit (Tobit Type I)......Page 771
16C.2 Heckit (Tobit Type II)......Page 772
Appendix 16D A Tobit Monte Carlo Experiment......Page 773
Appendix A: Mathematical Tools......Page 776
A.1.3 Scientific Notation......Page 777
A.1.4 Logarithms and the Number e......Page 778
A.1.6 Logarithms and Percentages......Page 779
A.2 Linear Relationships......Page 780
A.3 Nonlinear Relationships......Page 781
A.3.1 Rules for Derivatives......Page 782
A.3.3 Second Derivatives......Page 785
A.3.4 Maxima and Minima......Page 786
A.3.5 Partial Derivatives......Page 787
A.3.6 Maxima and Minima of Bivariate Functions......Page 788
A.4.1 Computing the Area Under a Curve......Page 790
A.5 Exercises......Page 792
Appendix B: Probability Concepts......Page 796
B.1.1 Expected Value of a Discrete Random Variable......Page 797
B.1.2 Variance of a Discrete Random Variable......Page 798
B.1.3 Joint, Marginal, and Conditional Distributions......Page 799
B.1.4 Expectations Involving Several Random Variables......Page 800
B.1.5 Covariance and Correlation......Page 801
B.1.8 Variance Decomposition......Page 802
B.1.9 Covariance Decomposition......Page 805
B.2 Working with Continuous Random Variables......Page 806
B.2.1 Probability Calculations......Page 807
B.2.2 Properties of Continuous Random Variables......Page 808
B.2.3 Joint, Marginal, and Conditional Probability Distributions......Page 809
B.2.4 Using Iterated Expectations with Continuous Random Variables......Page 813
B.2.5 Distributions of Functions of Random Variables......Page 815
B.3 Some Important Probability Distributions......Page 817
B.3.2 The Binomial Distribution......Page 818
B.3.3 The Poisson Distribution......Page 819
B.3.4 The Uniform Distribution......Page 820
B.3.5 The Normal Distribution......Page 821
B.3.6 The Chi-Square Distribution......Page 822
B.3.7 The t-Distribution......Page 824
B.3.8 The F-Distribution......Page 825
B.3.9 The Log-Normal Distribution......Page 827
B.4 Random Numbers......Page 828
B.4.1 Uniform Random Numbers......Page 833
B.5 Exercises......Page 834
Appendix C: Review of Statistical Inference......Page 840
C.1 A Sample of Data......Page 841
C.2 An Econometric Model......Page 842
C.3 Estimating the Mean of a Population......Page 843
C.3.1 The Expected Value of Y......Page 844
C.3.3 The Sampling Distribution of Y......Page 845
C.3.4 The Central Limit Theorem......Page 846
C.4 Estimating the Population Variance and Other Moments......Page 848
C.4.2 Estimating Higher Moments......Page 849
C.5.1 Interval Estimation: σ2 Known......Page 850
C.5.2 Interval Estimation: σ2 Unknown......Page 853
C.6.1 Components of Hypothesis Tests......Page 854
C.6.2 One-Tail Tests with Alternative "Greater Than" (>)......Page 856
C.6.4 Two-Tail Tests with Alternative "Not Equal To" (≠)......Page 857
C.6.5 The p-Value......Page 859
C.6.6 A Comment on Stating Null and Alternative Hypotheses......Page 860
C.6.8 A Relationship Between Hypothesis Testing and Confidence Intervals......Page 861
C.7.2 Testing the Equality of Two Population Means......Page 862
C.7.3 Testing the Ratio of Two Population Variances......Page 863
C.7.4 Testing the Normality of a Population......Page 864
C.8 Introduction to Maximum Likelihood Estimation......Page 865
C.8.1 Inference with Maximum Likelihood Estimators......Page 868
C.8.2 The Variance of the Maximum Likelihood Estimator......Page 869
C.8.3 The Distribution of the Sample Proportion......Page 870
C.8.4 Asymptotic Test Procedures......Page 871
C.9.1 Derivation of Least Squares Estimator......Page 876
C.9.2 Best Linear Unbiased Estimation......Page 877
C.10 Kernel Density Estimator......Page 879
C.11.1 Problems......Page 882
C.11.2 Computer Exercises......Page 885
TableD.1 Cumulative Probabilities for the Standard Normal Distribution ��(z) = P(Z ≤ z)......Page 890
TableD.2 Percentiles of the t-distribution......Page 891
TableD.3 Percentiles of the Chi-square Distribution......Page 892
TableD.4 95th Percentile for the F-distribution......Page 893
TableD.5 99th Percentile for the F-distribution......Page 894
TableD.6 Standard Normal pdf Values ��(z)......Page 895
Index......Page 897
EULA......Page 907