Multicollinearity 269

7.1 Introduction 269

7.2 Some Illustrative Examples 270

7.3 Some Measures of Multicollinearity 274

7.4 Problems with Measuring Multicollinearity 276

7.5 Solutions to the Multicollinearity Problem: Ridge Regression 280

7.6 Principal Component Regression 284

7.7 Dropping Variables 289

7.8 Miscellaneous Other Solutions 292 Summary 294

Exercises 295

Appendix to Chapter 7 296

Dummy Variables and Truncated Variables 305

8.1 Introduction 306

8.2 Dummy Variables for Changes in the Intercept Term 306

8.3 Dummy Variables for Changes in Slope Coefficients 312

8.4 Dummy Variables for Cross-Equation Constraints 315

8.5 Dummy Variables for Testing Stability of Regression Coefficients 318

8.6 Dummy Variables Under Heteroskedasticity and Autocorrelation 320

8.7 Dummy Dependent Variables 322

6.8 A General Test for Higher-Order Serial Correlation: The LM Test 250

6.9 Strategies When the DW Test Statistic Is Significant 252

%.10 Trends and Random Walks 258

%.]1 ARCH Models and Serial Correlation 264

Summary 265

Exercises 267

8.8 The Linear Probability Model and the Linear Discriminant Function 323

8.9 The Probit and Logit Models 327

8.10 Illustrative Example 335

8.11 Truncated Variables: The Tobit Model 338 Summary 344

Exercises 345

Simultaneous Equations Models 355

9.1 Introduction 356

9.2 Endogenous and Exogenous Variables 357

9.3 The Identification Problem: Identification Through Reduced Form 358

9.4 Necessary and Sufficient Conditions for Identification 363

9.5 Methods of Estimation: The Instrumental Variable Method 366

9.6 Methods of Estimation: The Two-Stage Least Squares Method 373

9.7 The Question of Normalization 377

*9.8 The Limited-Information Maximum Likelihood

Method 381 *9.9 On the Use of OLS in the Estimation of Simultaneous-Equations Models 383 *9.10 Exogeneity and Causality 389 Summary 395 Exercises 397 Appendix to Chapter 9 400

Models of Expectations 405

10.1 Introduction 405

10.2 Naive Models of Expectations 406

10.3 The Adaptive Expectations Model 408

10.4 Estimation with the Adaptive Expectations Model 410

Errors in Variables 447

11.1 Introduction 447

11.2 The Classical Solution for a Single-Equation Model with One Explanatory Variable 448

11.3 The Single-Equation Model with Two Explanatory Variables 451

11.4 Reverse Regression 459

11.5 Instrumental Variable Methods 461

11.6 Proxy Variables 464

11.7 Some Other Problems 468 Summary 470

Exercises 472

Diagnostic Checldng, Model Selection, and Specification Testing 475

12.1 Introduction 476

12.2 Diagnostic Tests Based on Least Squares Residuals 477

12.3 Problems with Least Squares Residuals 479

12.4 Some Other Types of Residuals 481

10.5 Two Illustrative Examples 413

10.6 Expectational Variables and Adjustment Lags 415

10.7 Partial Adjustment with Adaptive Expectations 421

10.8 Alternative Distributed Lag Models: Polynomial Lags 423

10.9 Rational Lags 429

10.10 Rational Expectations 431

10.11 Tests for Rationality 434

10.12 Estimation of a Demand and Supply Model Under Rational Expectations 436

10.13 The Serial Correlation Problem in Rational Expectations Models 443

Summary 444 Exercises 445