BASIC ECONOMETRICS FOURTH EDITION Damodar N. Gujarati United States Military Academy, West Point Me Graw Boston Burr Ridge, IL . Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto CONTENTS PREFACE 1.1 1.2 1.3 Introduction 1 WHAT IS ECONOMETRICS? WHY A SEPARATE DISCIPLINE? METHODOLOGY OF ECONOMETRICS 1 2 3 1. Statement of Theory or Hypothesis 2. Specification of the Mathematical Model of Consumption 3. Specification of the Econometric Model of Consumption 4 4 5 4. Obtaining Data 5. Estimation of the Econometric Model 6. Hypothesis Testing ' 7. Forecasting or Prediction 8. Use of the Model for Control or Policy Purposes Choosing among Competing Models 6 7 8 8 9 10 TYPES OF ECONOMETRICS MATHEMATICAL AND STATISTICAL PREREQUISITES THE ROLE OF THE COMPUTER SUGGESTIONS FOR FURTHER READING 12 12 13 13 I SINGLE-EQUATION REGRESSION MODELS 15 1 The Nature of Regression Analysis 17 HISTORICAL ORIGIN OF THE TERM REGRESSION THE MODERN INTERPRETATION OF REGRESSION Examples STATISTICAL VERSUS DETERMINISTIC RELATIONSHIPS 17 18 18 22 1.4 1.5 1.6 1.7 PART XXV 1.1 1.2 1.3 VII Viii CONTENTS 1.4 1.5 1.6 1.7 \ 1.8 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 REGRESSION VERSUS CAUSATION REGRESSION VERSUS CORRELATION TERMINOLOGY AND NOTATION THE NATURE AND SOURCES OF DATA FOR ECONOMIC ANALYSIS Types of Data The Sources of Data The Accuracy of Data A Note on the Measurement Scales of Variables SUMMARY AND CONCLUSIONS 22 23 24 25 25 29 29 30 31 EXERCISES 32 Two-Variable Regression Analysis: Some Basic Ideas 37 A HYPOTHETICAL EXAMPLE THE CONCEPT OF POPULATION REGRESSION FUNCTION (PRF) THE MEANING OF THE TERM LINEAR Linearity in the Variables Linearity in the Parameters STOCHASTIC SPECIFICATION OF PRF THE SIGNIFICANCE OF THE STOCHASTIC DISTURBANCE TERM THE SAMPLE REGRESSION FUNCTION (SRF) AN ILLUSTRATIVE EXAMPLE SUMMARY AND CONCLUSIONS 45 47 51 52 EXERCISES 52 Two-Variable Regression Model: The Problem of Estimation 58 THE METHOD OF ORDINARY LEAST SQUARES THE CLASSICAL LINEAR REGRESSION MODEL: THE ASSUMPTIONS UNDERLYING THE METHOD OF LEAST SQUARES A Word about These Assumptions PRECISION OR STANDARD ERRORS OF LEAST-SQUARES ESTIMATES PROPERTIES OF LEAST-SQUARES ESTIMATORS: THE GAUSS-MARKOV THEOREM THE COEFFICIENT OF DETERMINATION r2: A MEASURE OF "GOODNESS OF FIT" A NUMERICAL EXAMPLE ILLUSTRATIVE EXAMPLES A NOTE ON MONTE CARLO EXPERIMENTS 37 41 42 42 42 43 58 65 75 76 79 81 87 90 91 CONTENTS 3.9 SUMMARY AND CONCLUSIONS EXERCISES 3A.1 3A.2 3A.3 3A.4 3A.5 3A.6 3A.7 4 4.1 4.2 4.3 4.4 4.5 4A.1 4A.2 APPENDIX 3A DERIVATION OF LEAST-SQUARES ESTIMATES LINEARITY AND UNBIASEDNESS PROPERTIES OF LEAST-SQUARES ESTIMATORS VARIANCES AND STANDARD ERRORS OF LEAST-SQUARES ESTIMATORS COVARIANCE BETWEEN /§i AND /§2 THE LEAST-SQUARES ESTIMATOR OF a2 MINIMUM-VARIANCE PROPERTY OF LEAST-SQUARES ESTIMATORS CONSISTENCY OF LEAST-SQUARES ESTIMATORS 5.1 5.2 5.3 5.4 5.5 5.6 - Classical Normal Linear Regression Model (CNLRM) THE PROBABILITY DISTRIBUTION OF DISTURBANCES U; THE NORMALITY ASSUMPTION FOR u. Why the Normality Assumption? PROPERTIES OF OLS ESTIMATORS UNDER THE NORMALITY ASSUMPTION THE METHOD OF MAXIMUM LIKELIHOOD (ML) SUMMARY AND CONCLUSIONS APPENDIX 4A MAXIMUM LIKELIHOOD ESTIMATION OF TWO-VARIABLE REGRESSION MODEL MAXIMUM LIKELIHOOD ESTIMATION OF FOOD EXPENDITURE IN INDIA APPENDIX 4A EXERCISES 5 c Two-Variable Regression: Interval Estimation and Hypothesis Testing STATISTICAL PREREQUISITES INTERVAL ESTIMATION: SOME BASIC IDEAS CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS # AND ft. Confidence Interval for p2 Confidence Interval for /?i Confidence Interval for /^ and ft Simultaneously CONFIDENCE INTERVAL FOR a2 HYPOTHESIS TESTING: GENERAL COMMENTS HYPOTHESIS TESTING: THE CONFIDENCE-INTERVAL APPROACH Two-Sided or Two-Tail Test One-Sided or One-Tail Test ix X CONTENTS 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5A.1 5A.2 5A.3 5A.4 6 6.1 6.2 6.3 6.4 6.5 6.6 HYPOTHESIS TESTING: THE TEST-OF-SIGNIFICANCE APPROACH Testing the Significance of Regression Coefficients: The t Test Testing the Significance of a2: The x 2 Test HYPOTHESIS TESTING: SOME PRACTICAL ASPECTS The Meaning of "Accepting" or "Rejecting" a Hypothesis The "Zero" Null Hypothesis and the "2-f" Rule of Thumb Forming the Null and Alternative Hypotheses Choosing a, the Level of Significance The Exact Level of Significance: The p Value Statistical Significance versus Practical Significance The Choice between Confidence-Interval and Test-of-Significance Approaches to Hypothesis Testing REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE APPLICATION OF REGRESSION ANALYSIS: THE PROBLEM OF PREDICTION Mean Prediction Individual Prediction REPORTING THE RESULTS OF REGRESSION ANALYSIS EVALUATING THE RESULTS OF REGRESSION ANALYSIS Normality Tests Other Tests of Model Adequacy 129 129 133 134 134 134 135 136 137 138 139 140 142 142 144 145 146 147 149 SUMMARY AND CONCLUSIONS 150 EXERCISES 151 APPENDIX 5A PROBABILITY DISTRIBUTIONS RELATED TO THE NORMAL DISTRIBUTION ' DERIVATION OF EQUATION (5.3.2) DERIVATION OF EQUATION (5.9.1) DERIVATIONS OF EQUATIONS (5.10.2) AND (5.10.6) Variance of Mean Prediction Variance of Individual Prediction 159 Extensions of the Two-Variable Linear Regression Model 164 REGRESSION THROUGH THE ORIGIN r2 for Regression-through-Origin Model SCALING AND UNITS OF MEASUREMENT A Word about Interpretation REGRESSION ON STANDARDIZED VARIABLES FUNCTIONAL FORMS OF REGRESSION MODELS HOW TO MEASURE ELASTICITY: THE LOG-LINEAR MODEL SEMILOG MODELS: LOG-LIN AND LIN-LOG MODELS How to Measure the Growth Rate: The Log-Lin Model The Lin-Log Model 164 167 169 173 173 175 175 178 178 181 159 161 162 162 162 163 CONTENTS Xi 6.7 6.8 *6.9 RECIPROCAL MODELS Log Hyperbola or Logarithmic Reciprocal Model 183 189 CHOICE OF FUNCTIONAL FORM 190 A NOTE ON THE NATURE OF THE STOCHASTIC ERROR TERM: ADDITIVE VERSUS MULTIPLICATIVE STOCHASTIC ERROR TERM 191 6.10 SUMMARY AND CONCLUSIONS 192 EXERCISES 194 198 6A.1 APPENDIX 6A DERIVATION OF LEAST-SQUARES ESTIMATORS FOR REGRESSION THROUGH T H E ORIGIN PROOF THAT A STANDARDIZED VARIABLE HAS ZERO MEAN AND UNIT VARIANCE 198 200 Multiple Regression Analysis: The Problem of Estimation 202 6A.2 7 7.1 THE THREE-VARIABLE MODEL: NOTATION AND ASSUMPTIONS 7.2 7.3 7.4 INTERPRETATION OF MULTIPLE REGRESSION EQUATION THE MEANING OF PARTIAL REGRESSION COEFFICIENTS OLS AND ML ESTIMATION OF THE PARTIAL REGRESSION COEFFICIENTS OLS Estimators Variances and Standard Errors of OLS Estimators Properties of OLS Estimators Maximum Likelihood Estimators 7.5 7.6 7.7 7.8 7.9 7.10 *7.11 202 205 205 207 207 208 210 211 THE MULTIPLE COEFFICIENT OF DETERMINATION R2 AND THE MULTIPLE COEFFICIENT OF CORRELATION R EXAMPLE 7 . 1 : CHILD MORTALITY IN RELATION TO PER CAPITA GNP AND FEMALE LITERACY RATE Regression on Standardized Variables 213 215 SIMPLE REGRESSION IN THE CONTEXT OF MULTIPLE REGRESSION: INTRODUCTION TO SPECIFICATION BIAS R2 AND THE ADJUSTED R2 Comparing Two Rz Values Allocating R2 among Regressors The "Game" of Maximizing R2 215 217 219 222 222 212 EXAMPLE 7.3: THE C O B B - D O U G L A S PRODUCTION FUNCTION: MORE ON FUNCTIONAL FORM 223 POLYNOMIAL REGRESSION MODELS Empirical Results 226 229 PARTIAL CORRELATION COEFFICIENTS Explanation of Simple and Partial Correlation Coefficients Interpretation of Simple and Partial Correlation Coefficients 230 230 231 Xii CONTENTS 7.12 7A.1 7A.2 7A.3 7A.4 7A.5 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 *8.10 SUMMARY AND CONCLUSIONS 232 EXERCISES 233 APPENDIX 7A DERIVATION OF OLS ESTIMATORS GIVEN IN EQUATIONS (7.4.3) TO (7.4.5) EQUALITY BETWEEN THE COEFFICIENTS OF PGNP IN (7.3.5) AND (7.6.2) DERIVATION OF EQUATION (7.4.19) MAXIMUM LIKELIHOOD ESTIMATION OF THE MULTIPLE REGRESSION MODEL SAS OUTPUT OF THE COBB-DOUGLAS PRODUCTION FUNCTION (7.9.4) 243 Multiple Regression Analysis: The Problem of Inference 248 THE NORMALITY ASSUMPTION ONCE AGAIN EXAMPLE 8.1: CHILD MORTALITY EXAMPLE REVISITED HYPOTHESIS TESTING IN MULTIPLE REGRESSION: GENERAL COMMENTS HYPOTHESIS TESTING ABOUT INDIVIDUAL REGRESSION COEFFICIENTS TESTING THE OVERALL SIGNIFICANCE OF THE SAMPLE REGRESSION The Analysis of Variance Approach to Testing the Overall Significance of an Observed Multiple Regression: The FTest Testing the Overall Significance of a Multiple Regression: The FTest An Important Relationship between R2 and F Testing the Overall Significance of a Multiple Regression in Terms of Rz The "Incremental" or "Marginal" Contribution of an Explanatory Variable TESTING THE EQUALITY OF TWO REGRESSION COEFFICIENTS RESTRICTED LEAST SQUARES: TESTING LINEAR EQUALITY RESTRICTIONS The f-Test Approach The F-Test Approach: Restricted Least Squares General FTesting TESTING FOR STRUCTURAL OR PARAMETER STABILITY OF REGRESSION MODELS: THE CHOW TEST PREDICTION WITH MULTIPLE REGRESSION THE TROIKA OF HYPOTHESIS TESTS: THE LIKELIHOOD RATIO (LR), WALD (W), AND LAGRANGE MULTIPLIER (LM) TESTS 248 249 243 244 245 246 247 250 250 253 254 257 258 259 260 264 266 267 267 271 273 279 280 CONTENTS 8.11 Xiii TESTING THE FUNCTIONAL FORM OF REGRESSION: CHOOSING BETWEEN LINEAR AND LOG-LINEAR REGRESSION MODELS SUMMARY AND CONCLUSIONS 280 282 EXERCISES 283 APPENDIX 8A: LIKELIHOOD RATIO (LR) TEST 294 Dummy Variable Regression Models 297 297 298 301 304 9.11 THE NATURE OF DUMMY VARIABLES ANOVA MODELS Caution in the Use of Dummy Variables ANOVA MODELS WITH TWO QUALITATIVE VARIABLES REGRESSION WITH A MIXTURE OF QUANTITATIVE AND QUALITATIVE REGRESSORS: THE ANCOVA MODELS THE DUMMY VARIABLE ALTERNATIVE TO THE CHOW TEST INTERACTION EFFECTS USING DUMMY VARIABLES THE USE OF DUMMY VARIABLES IN SEASONAL ANALYSIS PIECEWISE LINEAR REGRESSION . PANEL DATA REGRESSION MODELS SOME TECHNICAL ASPECTS OF THE DUMMY VARIABLE TECHNIQUE The Interpretation of Dummy Variables in Semilogarithmic Regressions Dummy Variables and Heteroscedasticity Dummy Variables and Autocorrelation What Happens if the Dependent Variable Is a Dummy Variable? TOPICS FOR FURTHER STUDY 9.12 SUMMARY AND CONCLUSIONS 323 8.12 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 PART II 10 10.1 10.2 304 306 310 .312 317 320 320 320 321 322 322 322 EXERCISES 324 APPENDIX 9A: SEMILOGARITHMIC REGRESSION WITH DUMMY REGRESSOR 333 RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL 335 Multicollinearity: What Happens if the Regressors Are Correlated? 341 THE NATURE OF MULTICOLLINEARITY ESTIMATION IN THE PRESENCE OF PERFECT MULTICOLLINEARITY 342 345 Xiv CONTENTS 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 ESTIMATION IN THE PRESENCE OF "HIGH" BUT "IMPERFECT" MULTICOLLINEARITY MULTICOLLINEARITY: MUCH ADO ABOUT NOTHING? THEORETICAL CONSEQUENCES OF MULTICOLLINEARITY PRACTICAL CONSEQUENCES OF MULTICOLLINEARITY Large Variances and Covariances of OLS Estimators Wider Confidence Intervals "Insignificant" t Ratios A High R2 but Few Significant f Ratios Sensitivity of OLS Estimators and Their Standard Errors to Small Changes in Data Consequences of Micronumerosity AN ILLUSTRATIVE EXAMPLE: CONSUMPTION EXPENDITURE IN RELATION TO INCOME AND WEALTH DETECTION OF MULTICOLLINEARITY REMEDIAL MEASURES Do Nothing Rule-of-Thumb Procedures IS MULTICOLLINEARITY NECESSARILY BAD? MAYBE NOT IF THE OBJECTIVE IS PREDICTION ONLY AN EXTENDED EXAMPLE: THE LONGLEY DATA SUMMARY AND CONCLUSIONS 347 348 350 350 353 354 354 354 356 356 359 363 363 364 369 370 374 EXERCISES 375 Heteroscedasticity: What Happens if the Error Variance Is Nonconstant? 387 THE NATURE OF HETEROSCEDASTICITY OLS ESTIMATION IN THE PRESENCE OF HETEROSCEDASTICITY THE METHOD OF GENERALIZED LEAST SQUARES (GLS) Difference between OLS and GLS CONSEQUENCES OF USING OLS IN THE PRESENCE OF HETEROSCEDASTICITY OLS Estimation Allowing for Heteroscedasticity OLS Estimation Disregarding Heteroscedasticity A Technical Note DETECTION OF HETEROSCEDASTICITY Informal Methods Formal Methods REMEDIAL MEASURES When af Is Known: The Method of Weighted Least Squares When af Is Not Known CONCLUDING EXAMPLES A CAUTION ABOUT OVERREACTING TO HETEROSCEDASTICITY 387 393 394 397 398 398 398 400 400 401 403 415 415 417 422 426 CONTENTS 11.9 11A.1 11A.2 11A.3 11 A.4 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 XV SUMMARY AND CONCLUSIONS 427 EXERCISES 428 APPENDIX 11A PROOF OF EQUATION (11.2.2) THE METHOD OF WEIGHTED LEAST SQUARES PROOF THAT £ ( a 2 ) ^ a2 IN THE PRESENCE OF HETEROSCEDASTICITY WHITE'S ROBUST STANDARD ERRORS 437 437 437 438 439 Autocorrelation: What Happens if the Error Terms Are Correlated 441 THE NATURE OF THE PROBLEM OLS ESTIMATION IN THE PRESENCE OF AUTOCORRELATION THE BLUE ESTIMATOR IN THE PRESENCE OF AUTOCORRELATION CONSEQUENCES OF USING OLS IN THE PRESENCE OF AUTOCORRELATION OLS Estimation Allowing for Autocorrelation OLS Estimation Disregarding Autocorrelation RELATIONSHIP BETWEEN WAGES AND PRODUCTIVITY IN THE BUSINESS SECTOR OF THE UNITED STATES, 1959-1998 DETECTING AUTOCORRELATION I. Graphical Method ' II. The Runs Test III. Durbin-Watson dTest IV. A General Test of Autocorrelation: The Breusch-Godfrey (BG) Test V. Why So Many Tests of Autocorrelation? WHAT TO DO WHEN YOU FIND AUTOCORRELATION: REMEDIAL MEASURES MODEL MIS-SPECIFICATION VERSUS PURE AUTOCORRELATION CORRECTING FOR (PURE) AUTOCORRELATION: THE METHOD OF GENERALIZED LEAST SQUARES (GLS) When p Is Known When p Is Not Known THE NEWEY-WEST METHOD OF CORRECTING THE OLS STANDARD ERRORS OLS VERSUS FGLS AND HAC FORECASTING WITH AUTOCORRELATED ERROR TERMS ADDITIONAL ASPECTS OF AUTOCORRELATION Dummy Variables and Autocorrelation ARCH and GARCH Models Coexistence of Autocorrelation and Heteroscedasticity 442 449 453 454 454 455 460 462 462 465 467 472 474 475 475 477 477 478 484 485 485 487 487 488 488 XVi CONTENTS 12.14 12A.1 12A.2 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 SUMMARY AND CONCLUSIONS 488 EXERCISES 490 APPENDIX 12A PROOF THATTHE ERROR TERM v,\N (12.1.11) IS AUTOCORRELATED PROOF OF EQUATIONS (12.2.3), (12.3.4), AND (12.3.5) 504 Econometric Modeling: Model Specification and Diagnostic Testing MODEL SELECTION CRITERIA TYPES OF SPECIFICATION ERRORS CONSEQUENCES OF MODEL SPECIFICATION ERRORS Underfitting a Model (Omitting a Relevant Variable) Inclusion of an Irrelevant Variable (Overfitting a Model) TESTS OF SPECIFICATION ERRORS Detecting the Presence of Unnecessary Variables (Overfitting a Model) Tests for Omitted Variables and Incorrect Functional Form ERRORS OF MEASUREMENT Errors of Measurement in the Dependent Variable Y Errors of Measurement in the Explanatory Variable X INCORRECT SPECIFICATION OF THE STOCHASTIC ERROR TERM NESTED VERSUS NON-NESTED MODELS TESTS OF NON-NESTED HYPOTHESES The Discrimination Approach The Discerning Approach MODEL SELECTION CRITERIA The R2 Criterion Adjusted R2 Akaike Information Criterion (AIC) Schwarz Information Criterion (SIC) Mallows's Cp Criterion A Word of Caution about Model Selection Criteria Forecast Chi-Square (x2) ADDITIONAL TOPICS IN ECONOMETRIC MODELING Outliers, Leverage, and Influence Recursive Least Squares Chow's Prediction Failure Test 13.11 13.12 13.13 504 504 506 507 508 510 510 513 514 515 517 524 524 526 529 529 530 530 531 536 536 537 537 537 538 538 539 540 540 542 543 A CONCLUDING EXAMPLE: A MODEL OF HOURLY WAGE DETERMINATION A WORD TO THE PRACTITIONER SUMMARY AND CONCLUSIONS 544 546 547 EXERCISES 548 CONTENTS 13A.1 13A.2 13A.3 13A.4 PART III 14 14.1 14.2 14.3 14.4 14.5 14.6 14A.1 14A.2 14A.3 15 15.1 15.2 15.3 15.4 15.5 15.6 XVii APPENDIX 13A THE PROOF THAT E(612) = /32 + fybsz [EQUATION (13.3.3)] THE CONSEQUENCES OF INCLUDING AN IRRELEVANT VARIABLE: THE UNBIASEDNESS PROPERTY THE PROOF OF EQUATION (13.5.10) THE PROOF OF EQUATION (13.6.2) 556 556 TOPICS IN ECONOMETRICS 561 Nonlinear Regression Models 563 INTRINSICALLY LINEAR AND INTRINSICALLY NONLINEAR REGRESSION MODELS ESTIMATION OF LINEAR AND NONLINEAR REGRESSION MODELS ESTIMATING NONLINEAR REGRESSION MODELS: THE TRIAL-AND-ERROR METHOD APPROACHES TO ESTIMATING NONLINEAR REGRESSION MODELS Direct Search or Trial-and-Error or Derivative-Free Method Direct Optimization Iterative Linearization Method ILLUSTRATIVE EXAMPLES SUMMARY AND CONCLUSIONS : 557 558 559 563 565 566 ' 568 568 569 569 570 573 EXERCISES 573 APPENDIX 14A DERIVATION OF EQUATIONS (14.2.4) AND (14.2.5) THE LINEARIZATION METHOD LINEAR APPROXIMATION OF THE EXPONENTIAL FUNCTION GIVEN IN (14.2.2) 575 575 576 Qualitative Response Regression Models 580 THE NATURE OF QUALITATIVE RESPONSE MODELS THE LINEAR PROBABILITY MODEL (LPM) Non-Normality of the Disturbances u-, Heteroscedastic Variances of the Disturbances Nonfulfillment of 0 < E{ Y, \ X) < 1 Questionable Value of R2 as a Measure of Goodness of Fit APPLICATIONS OF LPM ALTERNATIVES TO LPM THE LOGIT MODEL ESTIMATION OF THE LOGIT MODEL Data at the Individual Level Grouped or Replicated Data 580 582 584 584 58 586 589 593 595 597 597 598 577 XViii CONTENTS 15.7 15.8 15.9 15.10 15.11 15.12 15.13 15.14 15A.1 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 THE GROUPED LOGIT (GLOGIT) MODEL: A NUMERICAL EXAMPLE Interpretation of the Estimated Logit Model THE LOGIT MODEL FOR UNGROUPED OR INDIVIDUAL DATA THE PROBIT MODEL Probit Estimation with Grouped Data: gprobit The Probit Model for Ungrouped or Individual Data The Marginal Effect of a Unit Change in the Value of a Regressor in the Various Regression Models LOGIT AND PROBIT MODELS THE TOBIT MODEL Illustration of the Tobit Model: Ray Fair's Model of Extramarital Affairs MODELING COUNT DATA: THE POISSON REGRESSION MODEL FURTHER TOPICS IN QUALITATIVE RESPONSE REGRESSION MODELS Ordinal Logit and Probit Models Multinomial Logit and Probit Models Duration Models 600 600 604 608 610 612 613 614 616 618 620 623 623 623 623 SUMMARY AND CONCLUSIONS 624 EXERCISES APPENDIX 15A MAXIMUM LIKELIHOOD ESTIMATION OF THE LOGIT AND PROBIT MODELS FOR INDIVIDUAL (UNGROUPED) DATA 625 633 633 Panel Data Regression Models 636 WHY PANEL DATA? PANEL DATA: AN ILLUSTRATIVE EXAMPLE ESTIMATION OF PANEL DATA REGRESSION MODELS: THE FIXED EFFECTS APPROACH 1. All Coefficients Constant across Time and Individuals 2. Slope Coefficients Constant but the Intercept Varies across Individuals: The Fixed Effects or Least-Squares Dummy Variable (LSDV) Regression Model 3. Slope Coefficients Constant but the Intercept Varies over Individuals As Well As Time 4. All Coefficients Vary across Individuals ESTIMATION OF PANEL DATA REGRESSION MODELS: THE RANDOM EFFECTS APPROACH FIXED EFFECTS (LSDV) VERSUS RANDOM EFFECTS MODEL PANEL DATA REGRESSIONS: SOME CONCLUDING COMMENTS SUMMARY AND CONCLUSIONS 637 638 651 652 EXERCISES 652 640 641 642 644 644 647 650 CONTENTS 17 17.1 17.2 17.3 17.4 17.5 17.6 *17.7 17.8 17.9 17.10 17.11 17.12 17.13 17.14 17.15 17A.1 PART IV 18 18.1 18.2 18.3 18.4 18.5 Dynamic Econometric Models: Autoregressive and Distributed-Lag Models XiX 656 THE ROLE OF "TIME," OR "LAG," IN ECONOMICS THE REASONS FOR LAGS ESTIMATION OF DISTRIBUTED-LAG MODELS Ad Hoc Estimation of Distributed-Lag Models THE KOYCK APPROACH TO DISTRIBUTED-LAG MODELS The Median Lag The Mean Lag RATIONALIZATION OF THE KOYCK MODEL: THE ADAPTIVE EXPECTATIONS MODEL ANOTHER RATIONALIZATION OF THE KOYCK MODEL: THE STOCK ADJUSTMENT, OR PARTIAL ADJUSTMENT, MODEL COMBINATION OF ADAPTIVE EXPECTATIONS AND PARTIAL ADJUSTMENT MODELS ESTIMATION OF AUTOREGRESSIVE MODELS THE METHOD OF INSTRUMENTAL VARIABLES (IV) DETECTING AUTOCORRELATION IN AUTOREGRESSIVE MODELS: DURBIN/? TEST A NUMERICAL EXAMPLE: THE DEMAND FOR MONEY IN CANADA, 1979-I TO 1988-IV ILLUSTRATIVE EXAMPLES ' THE ALMON APPROACH TO DISTRIBUTED-LAG MODELS: THE ALMON OR POLYNOMIAL DISTRIBUTED LAG (PDL) CAUSALITY IN ECONOMICS: THE GRANGER CAUSALITY TEST The Granger Test A Note on Causality and Exogeneity 657 662 663 663 665 668 668 696 696 701 SUMMARY AND CONCLUSIONS 702 EXERCISES 703 APPENDIX 17A THE SARGAN TEST FOR THE VALIDITY OF INSTRUMENTS 713 713 SIMULTANEOUS-EQUATION MODELS 715 Simultaneous-Equation Models 717 THE NATURE OF SIMULTANEOUS-EQUATION MODELS EXAMPLES OF SIMULTANEOUS-EQUATION MODELS THE SIMULTANEOUS-EQUATION BIAS: INCONSISTENCY OF OLS ESTIMATORS THE SIMULTANEOUS-EQUATION BIAS: A NUMERICAL EXAMPLE SUMMARY AND CONCLUSIONS 717 718 EXERCISES 730 670 673 675 676 678 679 681 684 687 724 727 729 XX CONTENTS 19 19.1 19.2 19.3 19.4 "19.5 19.6 20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20A.1 20A.2 21 21.1 21.2 21.3 21.4 21.5 21.6 21.7 The Identification Problem 735 NOTATIONS AND DEFINITIONS THE IDENTIFICATION PROBLEM Underidentification Just, or Exact, Identification Overidentification RULES FOR IDENTIFICATION The Order Condition of Identifiability The Rank Condition of Identifiability A TEST OF SIMULTANEITY Hausman Specification Test TESTS FOR EXOGENEITY SUMMARY AND CONCLUSIONS 735 739 739 742 746 747 748 750 753 754 756 757 EXERCISES 758 Simultaneous-Equation Methods 762 APPROACHES TO ESTIMATION RECURSIVE MODELS AND ORDINARY LEAST SQUARES ESTIMATION OF A JUST IDENTIFIED EQUATION: THE METHOD OF INDIRECT LEAST SQUARES (ILS) An Illustrative Example Properties of ILS Estimators ESTIMATION OF AN OVERIDENTIFIED EQUATION: THE METHOD OF TWO-STAGE LEAST SQUARES (2SLS) 2SLS: A NUMERICAL EXAMPLE ILLUSTRATIVE EXAMPLES SUMMARY AND CONCLUSIONS EXERCISES 762 764 767 767 770 770 775 778 784 785 APPENDIX 20A BIAS IN THE INDIRECT LEAST-SQUARES ESTIMATORS ESTIMATION OF STANDARD ERRORS OF 2SLS ESTIMATORS 789 789 791 Time Series Econometrics: Some Basic Concepts 792 A LOOKAT SELECTED U.S. ECONOMIC TIME SERIES KEY CONCEPTS STOCHASTIC PROCESSES Stationary Stochastic Processes Nonstationary Stochastic Processes UNIT ROOT STOCHASTIC PROCESS TREND STATIONARY (TS) AND DIFFERENCE STATIONARY (DS) STOCHASTIC PROCESSES INTEGRATED STOCHASTIC PROCESSES Properties of Integrated Series THE PHENOMENON OF SPURIOUS REGRESSION 793 796 796 797 798 802 802 804 805 806 CONTENTS 21.8 TESTS OF STATIONARITY 1. Graphical Analysis 2. Autocorrelation Function (ACF) and Correlogram Statistical Significance of Autocorrelation Coefficients 21.9 THE UNIT ROOT TEST The Augmented Dickey-Fuller (ADF) Test Testing the Significance of More Than One Coefficient: The FTest The Phillips-Perron (PP) Unit Root Tests A Critique of the Unit Root Tests 21.10 TRANSFORMING NONSTATIONARYTIME SERIES Difference-Stationary Processes 21.11 COINTEGRATION: REGRESSION OF A UNIT ROOT TIME Trend-Stationary Process SERIES ON ANOTHER UNIT ROOT TIME SERIES Testing for Cointegration Cointegration and Error Correction Mechanism (ECM) 21.12 21.13 SOME ECONOMIC APPLICATIONS SUMMARY AND CONCLUSIONS EXERCISES 22 Time Series Econometrics: Forecasting 22.1 APPROACHES TO ECONOMIC FORECASTING Exponential Smoothing Methods Single-Equation Regression Models Simultaneous-Equation Regression Models ARIMA Models VAR Models 22.2 AR, MA, AND ARIMA MODELING OF TIME SERIES DATA An Autoregressive (AR) Process A Moving Average (MA) Process An Autoregressive and Moving Average (ARMA) Process An Autoregressive Integrated Moving Average (ARIMA) Process 22.3 22.4 22.5 22.6 22.7 THE BOX-JENKINS (BJ) METHODOLOGY IDENTIFICATION ESTIMATION OF THE ARIMA MODEL DIAGNOSTIC CHECKING FORECASTING 22.8 22.9 FURTHER ASPECTS OF THE BJ METHODOLOGY VECTOR AUTOREGRESSION (VAR) Estimation or VAR Forecasting with VAR VAR and Causality Some Problems with VAR Modeling An Application of VAR: A VAR Model of the Texas Economy XXi XXii CONTENTS 22.10 MEASURING VOLATILITY IN FINANCIAL TIME SERIES: THE ARCH AND GARCH MODELS What To Do if ARCH Is Present A Word on the Durbin-Watson d and the ARCH Effect A Note on the GARCH Model CONCLUDING EXAMPLES SUMMARY AND CONCLUSIONS 861 861 862 864 EXERCISES 865 A Review of Some Statistical Concepts 869 A.1 A.2 A.3 SUMMATION AND PRODUCT OPERATORS SAMPLE SPACE, SAMPLE POINTS, AND EVENTS PROBABILITY AND RANDOM VARIABLES Probability Random Variables 869 870 870 870 871 A.4 PROBABILITY DENSITY FUNCTION (PDF) Probability Density Function of a Discrete Random Variable Probability Density Function of a Continuous Random Variable Joint Probability Density Functions Marginal Probability Density Function Statistical Independence 872 872 873 874 874 876 A.5 CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS Expected Value Properties of Expected Values Variance 878 878 879 880 22.11 22.12 Appendix A Properties of Variance Covariance Properties of Covariance Correlation Coefficient Conditional Expectation and Conditional Variance Properties of Conditional Expectation and Conditional Variance Higher Moments of Probability Distributions A.6 A.7 856 861 881 881 882 883 884 885 886 SOME IMPORTANT THEORETICAL PROBABILITY DISTRIBUTIONS Normal Distribution The x 2 (Chi-Square) Distribution Student's t Distribution The F Distribution The Bernoulli Binomial Distribution Binomial Distribution The Poisson Distribution 887 887 890 892 893 894 894 895 STATISTICAL INFERENCE: ESTIMATION 895 Point Estimation 896 Interval Estimation 896 Methods of Estimation 898 CONTENTS XXiii Small-Sample Properties Large-Sample Properties A.8 Appendix B B.1 STATISTICAL INFERENCE: HYPOTHESIS TESTING The Confidence Interval Approach The Test of Significance Approach 899 902 905 906 910 REFERENCES 912 Rudiments of Matrix Algebra 913 DEFINITIONS Matrix Column Vector Row Vector Transposition 913 913 914 914 914 914 915 915 915 915 915 915 916 916 916 916 916 916 917 917 918 919 919 920 920 921 922 923 923 923 925 Submatrix B.2 TYPES OF MATRICES Square Matrix Diagonal Matrix Scalar Matrix Identity, or Unit, Matrix Symmetric Matrix Null Matrix Null Vector B.3 MATRIX OPERATIONS Equal Matrices Matrix Addition Matrix Subtraction Scalar Multiplication Matrix Multiplication Properties of Matrix Multiplication Matrix Transposition Matrix Inversion B.4 B.5 B.6 Appendix C C.1 C.2 DETERMINANTS Evaluation of a Determinant Properties of Determinants Rank of a Matrix Minor Cofactor FINDING THE INVERSE OF A SQUARE MATRIX MATRIX DIFFERENTIATION REFERENCES 925 The Matrix Approach to Linear Regression Model 926 THE /c-VARIABLE LINEAR REGRESSION MODEL ASSUMPTIONS OF THE CLASSICAL LINEAR REGRESSION MODEL IN MATRIX NOTATION 926 928 XXiV CONTENTS C.3 C.11 C.12 OLS ESTIMATION An Illustration Variance-Covariance Matrix of p - - Properties of OLS Vector p~ THE COEFFICIENT OF DETERMINATION, R2 IN MATRIX NOTATION THE CORRELATION MATRIX HYPOTHESIS TESTING ABOUT INDIVIDUAL REGRESSION COEFFICIENTS IN MATRIX NOTATION TESTING THE OVERALL SIGNIFICANCE OF REGRESSION: ANALYSIS OF VARIANCE IN MATRIX NOTATION TESTING LINEAR RESTRICTIONS: GENERAL FTESTING USING MATRIX NOTATION PREDICTION USING MULTIPLE REGRESSION: MATRIX FORMULATION Mean Prediction Variance of Mean Prediction Individual Prediction Variance of Individual Prediction SUMMARY OF THE MATRIX APPROACH: AN ILLUSTRATIVE EXAMPLE GENERALIZED LEAST SQUARES (GLS) SUMMARY AND CONCLUSIONS EXERCISES 949 CA.1 CA.2 CA.3 CA.4 APPENDIX CA DERIVATIVE OF k NORMAL OR SIMULTANEOUS EQUATIONS MATRIX DERIVATION OF NORMAL EQUATIONS VARIANCE-COVARIANCE MATRIX OF 0 BLUE PROPERTY OF OLS ESTIMATORS 955 955 956 956 957 Appendix D Statistical Tables 959 Appendix E Economic Data on the World Wide Web 976 SELECTED BIBLIOGRAPHY 979 C.4 C.5 C.6 C.7 C.8 C.9 C.10 931 933 934 936 936 937 938 939 940 940 941 941 942 942 942 947 948