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Principles of Econometrics, 3rd Edition

by Hill, R. Carter, Louisiana State University; Griffiths, William E., University of Melbourne, Australia; Lim, Guay C., University of Melbourne, Australia

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Copyright, iv
Brief Contents, vii
Preface, ix
Acknowledgments, xiii
Contents, xv
Chapter 1. An Introduction to Econometrics, 1
1.1. Why Study Econometrics?, 1
1.2. What is Econometrics About?, 2
1.2.1. Some Examples, 3
1.3. The Econometric Model, 4
1.4. How Do We Obtain Data?, 5
1.4.1. Experimental Data, 5
1.4.2. Nonexperimental Data, 5
1.5. Statistical Inference, 6
1.6. A Research Format, 7
Chapter 2. The Simple Linear Regression Model, 8
Learning Objectives, 8
Keywords, 9
2.1. An Economic Model, 9
2.2. An Econometric Model, 12
2.2.1. Introducing the Error Term, 15
2.3. Estimating the Regression Parameters, 18
2.3.1. The Least Squares Principle, 20
2.3.2. Estimates for the Food Expenditure Function, 22
2.3.3. Interpreting the Estimates, 22
2.3.3a. Elasticities, 23
2.3.3b. Prediction, 24
2.3.3c. Computer Output, 24
2.3.4. Other Economic Models, 24
2.4. Assessing the Least Squares Estimators, 26
2.4.1. The Estimator b₂, 27
2.4.2. The Expected Values of b₁ and b₂, 27
2.4.3. Repeated Sampling, 28
2.4.4. The Variances and Covariance of b₁ and b₂, 29
2.5. The Gauss-Markov Theorem, 31
2.6. The Probability Distributions of the Least Squares Estimators, 32
2.7. Estimating the Variance of the Error Term, 33
2.7.1. Estimating the Variances and Covariances of the Least Squares Estimators, 34
2.7.2. Calculations for the Food Expenditure Data, 35
2.8. Exercises, 36
2.8.1. Problems, 36
2.8.2. Computer Exercises, 39
Appendix 2A Derivation of the Least Squares Estimates, 42
Appendix 2B Deviation from the Mean Form of b₂, 43
Appendix 2C b₂ is a Linear Estimator, 44
Appendix 2D Derivation of Theoretical Expression for b₂, 44
Appendix 2E Deriving the Variance of b₂, 45
Appendix 2F Proof of the Gauss-Markov Theorem, 46
Chapter 3. Interval Estimation and Hypothesis Testing, 48
Learning Objectives, 48
Keywords, 48
3.1. Interval Estimation, 49
3.1.1. The t-distribution, 49
3.1.2. Obtaining Interval Estimates, 51
3.1.3. An Illustration, 52
3.1.4. The Repeated Sampling Context, 53
3.2. Hypothesis Tests, 54
3.2.1. The Null Hypothesis, 55
3.2.2. The Alternative Hypothesis, 55
3.2.3. The Test Statistic, 55
3.2.4. The Rejection Region, 55
3.2.5. A Conclusion, 56
3.3. Rejection Regions for Specific Alternatives, 56
3.3.1. One-Tail Tests with Alternative “Greater Than” (>), 56
3.3.2. One-Tail Tests with Alternative “Less Than” (<), 57
3.3.3. Two-Tail Tests with Alternative “Not Equal To” (≠), 58
3.4. Examples of Hypothesis Tests, 59
3.4.1. Right-Tail Tests, 59
3.4.1a. One-Tail Test of Significance, 59
3.4.1b. One-Tail Test of an Economic Hypothesis, 60
3.4.2. Left-Tail Tests, 61
3.4.3. Two-Tail Tests, 62
3.4.3a. Two-Tail Test of an Economic Hypothesis, 62
3.4.3b. Two-Tail Test of Significance, 63
3.5. The p-value, 64
3.5.1. p-value for a Right-Tail Test, 65
3.5.2. p-value for a Left-Tail Test, 66
3.5.3. p-value for a Two-Tail Test, 66
3.5.4. p-value for a Two-Tail Test of Significance, 67
3.6. Exercises, 68
3.6.1. Problems, 68
3.6.2. Computer Exercises, 69
Appendix 3A Derivation of the t-Distribution, 72
Appendix 3B Distribution of the t-Statistic Under H₁, 73
Chapter 4. Prediction, Goodness-of-Fit, and Modeling Issues, 75
Learning Objectives, 75
Keywords, 76
4.1. Least Squares Prediction, 76
4.1.1. Prediction in the Food Expenditure Model, 79
4.2. Measuring Goodness-of-Fit, 80
4.2.1. Correlation Analysis, 81
4.2.2. Correlation Analysis and R², 82
4.2.3. The Food Expenditure Example, 82
4.2.4. Reporting the Results, 83
4.3. Modeling Issues, 84
4.3.1. The Effects of Scaling the Data, 84
4.3.2. Choosing a Functional Form, 86
4.3.3. The Food Expenditure Model, 87
4.3.4. Are the Regression Errors Normally Distributed?, 89
4.3.5. Another Empirical Example, 90
4.4. Log-Linear Models, 93
4.4.1. A Growth Model, 94
4.4.2. A Wage Equation, 94
4.4.3. Prediction in the Log-Linear Model, 95
4.4.4. A Generalized R² Measure, 96
4.4.5. Prediction Intervals in the Log-Linear Model, 96
4.5. Exercises, 97
4.5.1. Problems, 97
4.5.2. Computer Exercises, 98
Appendix 4A Development of a Prediction Interval, 101
Appendix 4B The Sum of Squares Decomposition, 103
Appendix 4C The Log-Normal Distribution, 103
Chapter 5. The Multiple Regression Model, 105
Learning Objectives, 105
Keywords, 105
5.1. Introduction, 106
5.1.1. The Economic Model, 106
5.1.2. The Econometric Model, 108
5.1.2a. The General Model, 109
5.1.2b. The Assumptions of the Model, 110
5.2. Estimating the Parameters of the Multiple Regression Model, 111
5.2.1. Least Squares Estimation Procedure, 111
5.2.2. Least Squares Estimates Using Hamburger Chain Data, 112
5.2.3. Estimation of the Error Variance σ², 114
5.3. Sampling Properties of the Least Squares Estimator, 115
5.3.1. The Variances and Covariances of the Least Squares Estimators, 115
5.3.2. The Properties of the Least Squares Estimators Assuming Normally Distributed Errors, 117
5.4. Interval Estimation, 118
5.5. Hypothesis Testing for a Single Coefficient, 120
5.5.1. Testing the Significance of a Single Coefficient, 120
5.5.2. One-Tail Hypothesis Testing for a Single Coefficient, 122
5.5.2a. Testing For Elastic Demand, 122
5.5.2b. Testing Advertising Effectiveness, 123
5.6. Measuring Goodness-of-Fit, 124
5.6.1. Reporting the Regression Results, 126
5.7. Exercises, 127
5.7.1. Problems, 127
5.7.2. Computer Exercises, 129
Appendix 5A Derivation of Least Squares Estimators, 133
Chapter 6. Further Inference in the Multiple Regression Model, 134
Learning Objectives, 134
Keywords, 135
6.1. The F-Test, 135
6.1.1. The Relationship Between t- and F-Tests, 138
6.2. Testing the Significance of a Model, 138
6.3. An Extended Model, 140
6.4. Testing Some Economic Hypotheses, 142
6.4.1. The Significance of Advertising, 142
6.4.2. The Optimal Level of Advertising, 142
6.4.2a. A One-Tail Test with More than One Parameter, 144
6.4.3. Using Computer Software, 145
6.5. The Use of Nonsample Information, 146
6.6. Model Specification, 148
6.6.1. Omitted Variables, 149
6.6.2. Irrelevant Variables, 150
6.6.3. Choosing the Model, 151
6.6.3a. The RESET Test, 151
6.7. Poor Data, Collinearity, and Insignificance, 153
6.7.1. The Consequences of Collinearity, 153
6.7.2. An Example, 154
6.7.3. Identifying and Mitigating Collinearity, 155
6.8. Prediction, 156
6.9. Exercises, 157
6.9.1. Problems, 157
6.9.2. Computer Exercises, 160
Appendix 6A Chi-Square and F-Tests: More Details, 163
Appendix 6B Omitted-Variable Bias: A Proof, 165
Chapter 7. Nonlinear Relationships, 166
Learning Objectives, 166
Keywords, 166
7.1. Polynomials, 167
7.1.1. Cost and Product Curves, 167
7.1.2. A Wage Equation, 169
7.2. Dummy Variables, 170
7.2.1. Intercept Dummy Variables, 171
7.2.1a. Choosing the Reference Group, 172
7.2.2. Slope Dummy Variables, 172
7.2.3. An Example: The University Effect on House Prices, 174
7.3. Applying Dummy Variables, 175
7.3.1. Interactions between Qualitative Factors, 175
7.3.2. Qualitative Factors with Several Categories, 177
7.3.3. Testing the Equivalence of Two Regressions, 179
7.3.4. Controlling for Time, 181
7.3.4a. Seasonal Dummies, 181
7.3.4b. Annual Dummies, 182
7.3.4c. Regime Effects, 182
7.4. Interactions Between Continuous Variables, 182
7.5. Log-Linear Models, 184
7.5.1. Dummy Variables, 185
7.5.1a. A Rough Calculation, 185
7.5.1b. An Exact Calculation, 185
7.5.2. Interaction and Quadratic Terms, 186
7.6. Exercises, 186
7.6.1. Problems, 186
7.6.2. Computer Exercises, 190
Appendix 7A Details of Log-Linear Model Interpretation, 195
Chapter 8. Heteroskedasticity, 197
Learning Objectives, 197
Keywords, 197
8.1. The Nature of Heteroskedasticity, 197
8.2. Using the Least Squares Estimator, 201
8.3. The Generalized Least Squares Estimator, 202
8.3.1. Transforming The Model, 203
8.3.2. Estimating the Variance Function, 205
8.3.3. A Heteroskedastic Partition, 208
8.4. Detecting Heteroskedasticity, 211
8.4.1. Residual Plots, 211
8.4.2. The Goldfeld-Quandt Test, 211
8.4.3. Testing the Variance Function, 212
8.4.3a. The White Test, 215
8.4.3b. Testing the Food Expenditure Example, 215
8.5. Exercises, 216
8.5.1. Problems, 216
8.5.2. Computer Exercises, 219
Appendix 8A Properties of the Least Squares Estimator, 222
Appendix 8B Variance Function Tests for Heteroskedasticity, 224
Chapter 9. Dynamic Models, Autocorrelation and Forecasting, 226
Learning Objectives, 226
Keywords, 226
9.1. Introduction, 227
9.2. Lags in the Error Term: Autocorrelation, 230
9.2.1. Area Response Model for Sugar Cane, 230
9.2.2. First Order Autoregressive Errors, 231
9.3. Estimating an AR(1) Error Model, 235
9.3.1. Least Squares Estimation, 235
9.3.2. Nonlinear Least Squares Estimation, 236
9.3.2a. Generalized Least Squares Estimation, 237
9.3.3. Estimating a More General Model, 237
9.4. Testing for Autocorrelation, 239
9.4.1. Residual Correlogram, 239
9.4.2. A Lagrange Multiplier Test, 242
9.4.3. Recapping and Looking Forward, 243
9.5. An Introduction to Forecasting: Autoregressive Models, 244
9.6. Finite Distributed Lags, 248
9.7. Autoregressive Distributed Lag Models, 250
9.8. Exercises, 253
9.8.1. Problems, 253
9.8.2. Computer Exercises, 255
Appendix 9A Generalized Least Squares Estimation, 259
Appendix 9B The Durbin-Watson Test, 261
9B.1. The Durbin-Watson Bounds Test, 263
Appendix 9C Deriving ARDL Lag Weights, 264
9C.1. The Geometric Lag, 264
9C.2. Lag Weights for More General ARDL Models, 265
Appendix 9D Forecasting:Exponential Smoothing, 266
Chapter 10. Random Regressors and Moment Based Estimation, 268
Learning Objectives, 268
Keywords, 269
10.1. Linear Regression with Random x’s, 270
10.1.1. The Small Sample Properties of the Least Squares Estimator, 270
10.1.2. Asymptotic Properties of the Least Squares Estimator: x Not Random, 271
10.1.3. Asymptotic Properties of the Least Squares Estimator: x Random, 272
10.1.4. Why Least Squares Fails, 273
10.2. Cases in Which x and e are Correlated, 274
10.2.1. Measurement Error, 274
10.2.2. Omitted Variables, 275
10.2.3. Simultaneous Equations Bias, 276
10.2.4. Lagged Dependent Variable Models with Serial Correlation, 276
10.3. Estimators Based on the Method of Moments, 276
10.3.1. Method of Moments Estimation of a Population Mean and Variance, 277
10.3.2. Method of Moments Estimation in the Simple Linear Regression Model, 278
10.3.3. Instrumental Variables Estimation in the Simple Linear Regression Model, 278
10.3.3a. The Importance of Using Strong Instruments, 279
10.3.3b. An Illustration Using Simulated Data, 280
10.3.3c. An Illustration Using a Wage Equation, 281
10.3.4. Instrumental Variables Estimation with Surplus Instruments, 282
10.3.4a. An Illustration Using Simulated Data, 284
10.3.4b. An Illustration Using a Wage Equation, 284
10.3.5. Instrumental Variables Estimation in a General Model, 285
10.3.5a. Hypothesis Testing with Instrumental Variables Estimates, 286
10.3.5b. Goodness-of-Fit with Instrumental Variables Estimates, 286
10.4. Specification Tests, 286
10.4.1. The Hausman Test for Endogeneity, 287
10.4.2. Testing for Weak Instruments, 288
10.4.3. Testing Instrument Validity, 289
10.4.4. Numerical Examples Using Simulated Data, 290
10.4.4a. The Hausman Test, 290
10.4.4b. Test for Weak Instruments, 290
10.4.4c. Testing Surplus Moment Conditions, 291
10.4.5. Specification Tests for the Wage Equation, 291
10.5. Exercises, 292
10.5.1. Problems, 292
10.5.2. Computer Exercises, 293
Appendix 10A Conditional and Iterated Expectations, 297
10A.1. Conditional Expectations, 297
10A.2. Iterated Expectations, 298
10A.3. Regression Model Applications, 298
Appendix 10B The Inconsistency of Least Squares, 299
Appendix 10C The Consistency of the IV Estimator, 300
Appendix 10D The Logic of the Hausman Test, 301
Chapter 11. Simultaneous Equations Models, 303
Learning Objectives, 303
Keywords, 303
11.1. A Supply and Demand Model, 304
11.2. The Reduced Form Equations, 306
11.3. The Failure of Least Squares, 307
11.4. The Identification Problem, 307
11.5. Two-Stage Least Squares Estimation, 309
11.5.1. The General Two-Stage Least Squares Estimation Procedure, 310
11.5.2. The Properties of the Two-Stage Least Squares Estimator, 311
11.6. An Example of Two-Stage Least Squares Estimation, 311
11.6.1. Identification, 312
11.6.2. The Reduced Form Equations, 312
11.6.3. The Structural Equations, 313
11.7. Supply and Demand at the Fulton Fish Market, 314
11.7.1. Identification, 315
11.7.2. The Reduced Form Equations, 315
11.7.3. Two-Stage Least Squares Estimation of Fish Demand, 317
11.8. Exercises, 318
11.8.1. Problems, 318
11.8.2. Computer Exercises, 319
Appendix 11A An Algebraic Explanation of the Failure of Least Squares, 323
Chapter 12. Nonstationary Time–Series Data and Cointegration, 325
Learning Objectives, 325
Keywords, 325
12.1. Stationary and Nonstationary Variables, 326
12.1.1. The First-Order Autoregressive Model, 328
12.1.2. Random Walk Models, 331
12.2. Spurious Regressions, 333
12.3. Unit Root Tests for Stationarity, 335
12.3.1. Dickey-Fuller Test 1 (No Constant and No Trend), 335
12.3.2. Dickey-Fuller Test 2 (With Constant But No Trend), 335
12.3.3. Dickey-Fuller Test 3 (With Constant and With Trend), 336
12.3.4. The Dickey-Fuller Testing Procedure, 336
12.3.5. The Dickey-Fuller Tests: An Example, 337
12.3.6. Order of Integration, 338
12.4. Cointegration, 339
12.4.1. An Example of a Cointegration Test, 340
12.5. Regression When There is No Cointegration, 340
12.5.1. First Difference Stationary, 341
12.5.2. Trend Stationary, 342
12.6. Exercises, 342
12.6.1. Problems, 342
12.6.2. Computer Exercises, 344
Chapter 13. VEC and VAR Models: An Introduction to Macroeconometrics, 346
Learning Objectives, 346
Keywords, 346
13.1. VEC and VAR Models, 347
13.2. Estimating a Vector Error Correction Model, 349
13.2.1. Example, 349
13.3. Estimating a VAR Model, 351
13.4. Impulse Responses and Variance Decompositions, 352
13.4.1. Impulse Response Functions, 352
13.4.1a. The Univariate Case, 352
13.4.1b. The Bivariate Case, 353
13.4.2. Forecast Error Variance Decompositions, 355
13.4.2a. Univariate Analysis, 355
13.4.2b. Bivariate Analysis, 356
13.4.2c. The General Case, 357
13.5. Exercises, 357
13.5.1. Problems, 357
13.5.2. Computer Exercises, 358
Appendix 13A The Identification Problem, 361
Chapter 14. Time-Varying Volatility and ARCH Models: An Introduction to Financial Econometrics, 363
Learning Objectives, 363
Keywords, 363
14.1. The ARCH Model, 364
14.1.1. Conditional and Unconditional Forecasts, 365
14.2. Time-Varying Volatility, 365
14.3. Testing, Estimating and Forecasting, 369
14.3.1. Testing for ARCH Effects, 369
14.3.2. Estimating ARCH Models, 369
14.3.3. Forecasting Volatility, 370
14.4. Extensions, 371
14.4.1. The GARCH Model—Generalized ARCH, 371
14.4.2. Allowing for an Asymmetric Effect, 372
14.4.3. GARCH-in-Mean and Time-Varying Risk Premium, 374
14.5. Exercises, 375
14.5.1. Problems, 375
14.5.2. Computer Exercises, 376
Chapter 15. Panel Data Models, 382
Learning Objectives, 382
Keywords, 382
15.1. Grunfeld’s Investment Data, 384
15.2. Sets of Regression Equations, 385
15.3. Seemingly Unrelated Regressions, 387
15.3.1. Separate or Joint Estimation?, 389
15.3.2. Testing Cross-Equation Hypotheses, 390
15.4. The Fixed Effects Model, 391
15.4.1. A Dummy Variable Model, 391
15.4.2. The Fixed Effects Estimator, 393
15.4.3. Fixed Effects Estimation Using a Microeconomic Panel, 396
15.5. The Random Effects Model, 398
15.5.1. Error Term Assumptions, 399
15.5.2. Testing for Random Effects, 400
15.5.3. Estimation of the Random Effects Model, 401
15.5.4. An Example Using the NLS Data, 402
15.5.5. Comparing Fixed and Random Effects Estimators, 403
15.5.5a. Endogeneity in the Random Effects Model, 403
15.5.5b. The Fixed Effects Estimator in a Random Effects Model, 404
15.5.5c. A Hausman Test, 404
15.6. Exercises, 406
15.6.1. Problems, 406
15.6.2. Computer Exercises, 408
Appendix 15A Estimation of Error Components, 415
Chapter 16. Qualitative and Limited Dependent Variable Models, 417
Learning Objectives, 417
Keywords, 417
16.1. Models with Binary Dependent Variables, 418
16.1.1. The Linear Probability Model, 419
16.1.2. The Probit Model, 421
16.1.3. Interpretation of the Probit Model, 422
16.1.4. Maximum Likelihood Estimation of the Probit Model, 423
16.1.5. An Example, 424
16.2. The Logit Model for Binary Choice, 425
16.3. Multinomial Logit, 426
16.3.1. Multinomial Logit Choice Probabilities, 427
16.3.2. Maximum Likelihood Estimation, 427
16.3.3. Post-Estimation Analysis, 428
16.3.4. An Example, 429
16.4. Conditional Logit, 431
16.4.1. Conditional Logit Choice Probabilities, 431
16.4.2. Post-Estimation Analysis, 432
16.4.3. An Example, 433
16.5. Ordered Choice Models, 433
16.5.1. Ordinal Probit Choice Probabilities, 434
16.5.2. Estimation and Interpretation, 435
16.5.3. An Example, 437
16.6. Models for Count Data, 437
16.6.1. Maximum Likelihood Estimation, 438
16.6.2. Interpretation in the Poisson Regression Model, 439
16.6.3. An Example, 440
16.7. Limited Dependent Variables, 441
16.7.1. Censored Data, 441
16.7.2. A Monte Carlo Experiment, 442
16.7.3. Maximum Likelihood Estimation, 444
16.7.4. Tobit Model Interpretation, 445
16.7.5. An Example, 446
16.7.6. Sample Selection, 447
16.7.6a. The Econometric Model, 448
16.7.6b. Heckit Example: Wages of Married Women, 449
16.8. Exercises, 450
Chapter 17. Writing an Empirical Research Report, and Sources of Economic Data, 457
17.1. Selecting a Topic for an Economics Project, 457
17.1.1. Choosing a Topic, 457
17.1.2. Writing an Abstract, 458
17.2. A Format for Writing a Research Report, 458
17.3. Sources of Economic Data, 460
17.3.1. Links to Economic Data on the Internet, 460
17.3.2. Traditional Sources of Economic Data, 461
17.3.3. Interpreting Economic Data, 461
17.4. Exercises, 462
Appendix A. Review of Math Essentials, 463
Learning Objectives, 463
Keywords, 463
A.1. Summation, 464
A.2. Some Basics, 465
A.2.1. Numbers, 465
A.2.2. Exponents, 466
A.2.3. Scientific Notation, 466
A.2.4. Logarithms and the Number e, 466
A.3. Linear Relationships, 468
A.3.1. Elasticity, 469
A.4. Nonlinear Relationships, 470
A.4.1. Quadratic Function, 471
A.4.2. Cubic Function, 471
A.4.3. Reciprocal Function, 472
A.4.4. Log-Log Function, 473
A.4.5. Log-Linear Function, 473
A.4.6. Approximating Logarithms, 473
A.4.7. Approximating Logarithms in the Log-Linear Model, 474
A.4.8. Linear-Log Function, 475
A.5. Exercises, 476
Appendix B. Review of Probability Concepts, 478
Learning Objectives, 478
Keywords, 479
B.1. Random Variables, 479
B.2. Probability Distributions, 480
B.3. Joint, Marginal, and Conditional Probability Distributions, 483
B.3.1. Marginal Distributions, 484
B.3.2. Conditional Probability, 484
B.3.3. A Simple Experiment, 486
B.4. Properties of Probability Distributions, 487
B.4.1. Mean, Median, and Mode, 487
B.4.2. Expected Values of Functions of a Random Variable, 488
B.4.3. Expected Values of Several Random Variables, 490
B.4.4. The Simple Experiment Again, 492
B.5. Some Important Probability Distributions, 493
B.5.1. The Normal Distribution, 493
B.5.2. The Chi-Square Distribution, 495
B.5.3. The t-Distribution, 495
B.5.4. The F-Distribution, 496
B.6. Exercises, 497
Appendix C. Review of Statistical Inference, 501
Learning Objectives, 501
Keywords, 502
C.1. A Sample of Data, 502
C.2. An Econometric Model, 504
C.3. Estimating the Mean of a Population, 504
C.3.1. The Expected Value of Y, 506
C.3.2. The Variance of Y, 506
C.3.3. The Sampling Distribution of Y, 507
C.3.4. The Central Limit Theorem, 508
C.3.5. Best Linear Unbiased Estimation, 509
C.4. Estimating the Population Variance and Other Moments, 509
C.4.1. Estimating the Population Variance, 510
C.4.2. Estimating Higher Moments, 511
C.4.3. The Hip Data, 511
C.4.4. Using the Estimates, 512
C.5. Interval Estimation, 512
C.5.1. Interval Estimation: σ² Known, 513
C.5.2. A Simulation, 514
C.5.3. Interval Estimation: σ² Unknown, 515
C.5.4. A Simulation (Continued), 516
C.5.5. Interval Estimation Using the Hip Data, 517
C.6. Hypothesis Tests About a Population Mean, 517
C.6.1. Components of Hypothesis Tests, 517
C.6.1a. The Null Hypothesis, 518
C.6.1b. The Alternative Hypothesis, 518
C.6.1c. The Test Statistic, 518
C.6.1d. The Rejection Region, 519
C.6.1e. A Conclusion, 519
C.6.2. One-Tail Tests with Alternative “Greater Than” (>), 519
C.6.3. One-Tail Tests with Alternative “Less Than” (<), 519
C.6.4. Two-Tail Tests with Alternative “Not Equal To” (≠), 520
C.6.5. Example of a One-Tail Test Using the Hip Data, 520
C.6.6. Example of a Two-Tail Test Using Hip Data, 521
C.6.7. The p-Value, 522
C.6.8. A Comment on Stating Null and Alternative Hypotheses, 523
C.6.9. Type I and Type II Errors, 524
C.6.10. A Relationship Between Hypothesis Testing and Confidence Intervals, 525
C.7. Some Other Useful Tests, 525
C.7.1. Testing the Population Variance, 525
C.7.2. Testing the Equality of Two Population Means, 526
C.7.3. Testing the Ratio of Two Population Variances, 527
C.7.4. Testing the Normality of a Population, 527
C.8. Introduction to Maximum Likelihood Estimation, 528
C.8.1. Inference with Maximum Likelihood Estimators, 532
C.8.2. The Variance of the Maximum Likelihood Estimator, 533
C.8.3. The Distribution of the Sample Proportion, 534
C.8.4. Asymptotic Test Procedures, 536
C.8.4a. The Likelihood Ratio (LR) Test, 536
C.8.4b. The Wald Test, 538
C.8.4c. The Lagrange Multiplier (LM) Test, 539
C.9. Algebraic Supplements (Optional), 541
C.9.1. Derivation of Least Squares Estimator, 541
C.9.2. Best Linear Unbiased Estimation, 543
C.10. Exercises, 544
Appendix D. Answers to Selected Exercises, 548
Appendix E. Tables, 572
Table 1 Cumulative Probabilities for the Standard Normal Distribution, 572
Table 2 Percentiles for the t-Distribution, 573
Table 3 Percentiles for the Chi-square Distribution, 574
Table 4 95th Percentile for the F-Distribution, 575
Table 5 99th Percentile for the F-Distribution, 576
Index, 577

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