{"product_id":"principles-of-econometrics","title":"Principles of Econometrics","description":"\u003cp\u003ePreface v\u003c\/p\u003e \u003cp\u003eList of Examples xxi\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 \u003c\/b\u003e\u003cb\u003eAn Introduction to Econometrics \u003c\/b\u003e\u003cb\u003e1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Why Study Econometrics? 1\u003c\/p\u003e \u003cp\u003e1.2 What Is Econometrics About? 2\u003c\/p\u003e \u003cp\u003e1.2.1 Some Examples 3\u003c\/p\u003e \u003cp\u003e1.3 The Econometric Model 4\u003c\/p\u003e \u003cp\u003e1.3.1 Causality and Prediction 5\u003c\/p\u003e \u003cp\u003e1.4 How Are Data Generated? 5\u003c\/p\u003e \u003cp\u003e1.4.1 Experimental Data 6\u003c\/p\u003e \u003cp\u003e1.4.2 Quasi-Experimental Data 6\u003c\/p\u003e \u003cp\u003e1.4.3 Nonexperimental Data 7\u003c\/p\u003e \u003cp\u003e1.5 Economic Data Types 7\u003c\/p\u003e \u003cp\u003e1.5.1 Time-Series Data 7\u003c\/p\u003e \u003cp\u003e1.5.2 Cross-Section Data 8\u003c\/p\u003e \u003cp\u003e1.5.3 Panel or Longitudinal Data 9\u003c\/p\u003e \u003cp\u003e1.6 The Research Process 9\u003c\/p\u003e \u003cp\u003e1.7 Writing an Empirical Research Paper 11\u003c\/p\u003e \u003cp\u003e1.7.1 Writing a Research Proposal 11\u003c\/p\u003e \u003cp\u003e1.7.2 A Format for Writing a Research Report 11\u003c\/p\u003e \u003cp\u003e1.8 Sources of Economic Data 13\u003c\/p\u003e \u003cp\u003e1.8.1 Links to Economic Data on the Internet 13\u003c\/p\u003e \u003cp\u003e1.8.2 Interpreting Economic Data 14\u003c\/p\u003e \u003cp\u003e1.8.3 Obtaining the Data 14\u003c\/p\u003e \u003cp\u003eProbability Primer 15\u003c\/p\u003e \u003cp\u003eP.1 Random Variables 16\u003c\/p\u003e \u003cp\u003eP.2 Probability Distributions 17\u003c\/p\u003e \u003cp\u003eP.3 Joint, Marginal, and Conditional Probabilities 20\u003c\/p\u003e \u003cp\u003eP.3.1 Marginal Distributions 20\u003c\/p\u003e \u003cp\u003eP.3.2 Conditional Probability 21\u003c\/p\u003e \u003cp\u003eP.3.3 Statistical Independence 21\u003c\/p\u003e \u003cp\u003eP.4 A Digression: Summation Notation 22\u003c\/p\u003e \u003cp\u003eP.5 Properties of Probability Distributions 23\u003c\/p\u003e \u003cp\u003eP.5.1 Expected Value of a Random Variable 24\u003c\/p\u003e \u003cp\u003eP.5.2 Conditional Expectation 25\u003c\/p\u003e \u003cp\u003eP.5.3 Rules for Expected Values 25\u003c\/p\u003e \u003cp\u003eP.5.4 Variance of a Random Variable 26\u003c\/p\u003e \u003cp\u003eP.5.5 Expected Values of Several Random Variables 27\u003c\/p\u003e \u003cp\u003eP.5.6 Covariance Between Two Random Variables 27\u003c\/p\u003e \u003cp\u003eP.6 Conditioning 29\u003c\/p\u003e \u003cp\u003eP.6.1 Conditional Expectation 30\u003c\/p\u003e \u003cp\u003eP.6.2 Conditional Variance 31\u003c\/p\u003e \u003cp\u003eP.6.3 Iterated Expectations 32\u003c\/p\u003e \u003cp\u003eP.6.4 Variance Decomposition 33\u003c\/p\u003e \u003cp\u003eP.6.5 Covariance Decomposition 34\u003c\/p\u003e \u003cp\u003eP.7 The Normal Distribution 34\u003c\/p\u003e \u003cp\u003eP.7.1 The Bivariate Normal Distribution 37\u003c\/p\u003e \u003cp\u003eP.8 Exercises 39\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 \u003c\/b\u003e\u003cb\u003eThe Simple Linear Regression Model \u003c\/b\u003e\u003cb\u003e46\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 An Economic Model 47\u003c\/p\u003e \u003cp\u003e2.2 An Econometric Model 49\u003c\/p\u003e \u003cp\u003e2.2.1 Data Generating Process 51\u003c\/p\u003e \u003cp\u003e2.2.2 The Random Error and Strict Exogeneity 52\u003c\/p\u003e \u003cp\u003e2.2.3 The Regression Function 53\u003c\/p\u003e \u003cp\u003e2.2.4 Random Error Variation 54\u003c\/p\u003e \u003cp\u003e2.2.5 Variation in x 56\u003c\/p\u003e \u003cp\u003e2.2.6 Error Normality 56\u003c\/p\u003e \u003cp\u003e2.2.7 Generalizing the Exogeneity Assumption 56\u003c\/p\u003e \u003cp\u003e2.2.8 Error Correlation 57\u003c\/p\u003e \u003cp\u003e2.2.9 Summarizing the Assumptions 58\u003c\/p\u003e \u003cp\u003e2.3 Estimating the Regression Parameters 59\u003c\/p\u003e \u003cp\u003e2.3.1 The Least Squares Principle 61\u003c\/p\u003e \u003cp\u003e2.3.2 Other Economic Models 65\u003c\/p\u003e \u003cp\u003e2.4 Assessing the Least Squares Estimators 66\u003c\/p\u003e \u003cp\u003e2.4.1 The Estimator b\u003csub\u003e2\u003c\/sub\u003e 67\u003c\/p\u003e \u003cp\u003e2.4.2 The Expected Values of b1 and b2 68\u003c\/p\u003e \u003cp\u003e2.4.3 Sampling Variation 69\u003c\/p\u003e \u003cp\u003e2.4.4 The Variances and Covariance of b1 and b2 69\u003c\/p\u003e \u003cp\u003e2.5 The Gauss–Markov Theorem 72\u003c\/p\u003e \u003cp\u003e2.6 The Probability Distributions of the Least Squares Estimators 73\u003c\/p\u003e \u003cp\u003e2.7 Estimating the Variance of the Error Term 74\u003c\/p\u003e \u003cp\u003e2.7.1 Estimating the Variances and Covariance of the Least Squares Estimators 74\u003c\/p\u003e \u003cp\u003e2.7.2 Interpreting the Standard Errors 76\u003c\/p\u003e \u003cp\u003e2.8 Estimating Nonlinear Relationships 77\u003c\/p\u003e \u003cp\u003e2.8.1 Quadratic Functions 77\u003c\/p\u003e \u003cp\u003e2.8.2 Using a Quadratic Model 77\u003c\/p\u003e \u003cp\u003e2.8.3 A Log-Linear Function 79\u003c\/p\u003e \u003cp\u003e2.8.4 Using a Log-Linear Model 80\u003c\/p\u003e \u003cp\u003e2.8.5 Choosing a Functional Form 82\u003c\/p\u003e \u003cp\u003e2.9 Regression with Indicator Variables 82\u003c\/p\u003e \u003cp\u003e2.10 The Independent Variable 84\u003c\/p\u003e \u003cp\u003e2.10.1 Random and Independent x 84\u003c\/p\u003e \u003cp\u003e2.10.2 Random and Strictly Exogenous x 86\u003c\/p\u003e \u003cp\u003e2.10.3 Random Sampling 87\u003c\/p\u003e \u003cp\u003e2.11 Exercises 89\u003c\/p\u003e \u003cp\u003e2.11.1 Problems 89\u003c\/p\u003e \u003cp\u003e2.11.2 Computer Exercises 93\u003c\/p\u003e \u003cp\u003eAppendix 2A Derivation of the Least Squares Estimates 98\u003c\/p\u003e \u003cp\u003eAppendix 2B Deviation from the Mean Form of b\u003csub\u003e2\u003c\/sub\u003e 99\u003c\/p\u003e \u003cp\u003eAppendix 2C b\u003csub\u003e2\u003c\/sub\u003e Is a Linear Estimator 100\u003c\/p\u003e \u003cp\u003eAppendix 2D Derivation of Theoretical Expression for b\u003csub\u003e2\u003c\/sub\u003e 100\u003c\/p\u003e \u003cp\u003eAppendix 2E Deriving the Conditional Variance of b\u003csub\u003e2\u003c\/sub\u003e 100\u003c\/p\u003e \u003cp\u003eAppendix 2F Proof of the Gauss–Markov Theorem 102\u003c\/p\u003e \u003cp\u003eAppendix 2G Proofs of Results Introduced in Section 2.10 103\u003c\/p\u003e \u003cp\u003e2G.1 The Implications of Strict Exogeneity 103\u003c\/p\u003e \u003cp\u003e2G.2 The Random and Independent x Case 103\u003c\/p\u003e \u003cp\u003e2G.3 The Random and Strictly Exogenous x Case 105\u003c\/p\u003e \u003cp\u003e2G.4 Random Sampling 106\u003c\/p\u003e \u003cp\u003eAppendix 2H Monte Carlo Simulation 106\u003c\/p\u003e \u003cp\u003e2H.1 The Regression Function 106\u003c\/p\u003e \u003cp\u003e2H.2 The Random Error 107\u003c\/p\u003e \u003cp\u003e2H.3 Theoretically True Values 107\u003c\/p\u003e \u003cp\u003e2H.4 Creating a Sample of Data 108\u003c\/p\u003e \u003cp\u003e2H.5 Monte Carlo Objectives 109\u003c\/p\u003e \u003cp\u003e2H.6 Monte Carlo Results 109\u003c\/p\u003e \u003cp\u003e2H.7 Random-x Monte Carlo Results 110\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 \u003c\/b\u003e\u003cb\u003eInterval Estimation and Hypothesis Testing \u003c\/b\u003e\u003cb\u003e112\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Interval Estimation 113\u003c\/p\u003e \u003cp\u003e3.1.1 The t-Distribution 113\u003c\/p\u003e \u003cp\u003e3.1.2 Obtaining Interval Estimates 115\u003c\/p\u003e \u003cp\u003e3.1.3 The Sampling Context 116\u003c\/p\u003e \u003cp\u003e3.2 Hypothesis Tests 118\u003c\/p\u003e \u003cp\u003e3.2.1 The Null Hypothesis 118\u003c\/p\u003e \u003cp\u003e3.2.2 The Alternative Hypothesis 118\u003c\/p\u003e \u003cp\u003e3.2.3 The Test Statistic 119\u003c\/p\u003e \u003cp\u003e3.2.4 The Rejection Region 119\u003c\/p\u003e \u003cp\u003e3.2.5 A Conclusion 120\u003c\/p\u003e \u003cp\u003e3.3 Rejection Regions for Specific Alternatives 120\u003c\/p\u003e \u003cp\u003e3.3.1 One-Tail Tests with Alternative ‘‘Greater Than’’ (\u0026gt;) 120\u003c\/p\u003e \u003cp\u003e3.3.2 One-Tail Tests with Alternative ‘‘Less Than’’ (\u0026lt;) 121\u003c\/p\u003e \u003cp\u003e3.3.3 Two-Tail Tests with Alternative ‘‘Not Equal To’’ (≠) 122\u003c\/p\u003e \u003cp\u003e3.4 Examples of Hypothesis Tests 123\u003c\/p\u003e \u003cp\u003e3.5 The p-Value 126\u003c\/p\u003e \u003cp\u003e3.6 Linear Combinations of Parameters 129\u003c\/p\u003e \u003cp\u003e3.6.1 Testing a Linear Combination of Parameters 131\u003c\/p\u003e \u003cp\u003e3.7 Exercises 133\u003c\/p\u003e \u003cp\u003e3.7.1 Problems 133\u003c\/p\u003e \u003cp\u003e3.7.2 Computer Exercises 139\u003c\/p\u003e \u003cp\u003eAppendix 3A Derivation of the t-Distribution 144\u003c\/p\u003e \u003cp\u003eAppendix 3B Distribution of the t-Statistic under H\u003csub\u003e1\u003c\/sub\u003e 145\u003c\/p\u003e \u003cp\u003eAppendix 3C Monte Carlo Simulation 147\u003c\/p\u003e \u003cp\u003e3C.1 Sampling Properties of Interval Estimators 148\u003c\/p\u003e \u003cp\u003e3C.2 Sampling Properties of Hypothesis Tests 149\u003c\/p\u003e \u003cp\u003e3C.3 Choosing the Number of Monte Carlo Samples 149\u003c\/p\u003e \u003cp\u003e3C.4 Random-x Monte Carlo Results 150\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 \u003c\/b\u003e\u003cb\u003ePrediction, Goodness-of-Fit, and Modeling Issues \u003c\/b\u003e\u003cb\u003e152\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Least Squares Prediction 153\u003c\/p\u003e \u003cp\u003e4.2 Measuring Goodness-of-Fit 156\u003c\/p\u003e \u003cp\u003e4.2.1 Correlation Analysis 158\u003c\/p\u003e \u003cp\u003e4.2.2 Correlation Analysis and R2 158\u003c\/p\u003e \u003cp\u003e4.3 Modeling Issues 160\u003c\/p\u003e \u003cp\u003e4.3.1 The Effects of Scaling the Data 160\u003c\/p\u003e \u003cp\u003e4.3.2 Choosing a Functional Form 161\u003c\/p\u003e \u003cp\u003e4.3.3 A Linear-Log Food Expenditure Model 163\u003c\/p\u003e \u003cp\u003e4.3.4 Using Diagnostic Residual Plots 165\u003c\/p\u003e \u003cp\u003e4.3.5 Are the Regression Errors Normally Distributed? 167\u003c\/p\u003e \u003cp\u003e4.3.6 Identifying Influential Observations 169\u003c\/p\u003e \u003cp\u003e4.4 Polynomial Models 171\u003c\/p\u003e \u003cp\u003e4.4.1 Quadratic and Cubic Equations 171\u003c\/p\u003e \u003cp\u003e4.5 Log-Linear Models 173\u003c\/p\u003e \u003cp\u003e4.5.1 Prediction in the Log-Linear Model 175\u003c\/p\u003e \u003cp\u003e4.5.2 A Generalized R2 Measure 176\u003c\/p\u003e \u003cp\u003e4.5.3 Prediction Intervals in the Log-Linear Model 177\u003c\/p\u003e \u003cp\u003e4.6 Log-Log Models 177\u003c\/p\u003e \u003cp\u003e4.7 Exercises 179\u003c\/p\u003e \u003cp\u003e4.7.1 Problems 179\u003c\/p\u003e \u003cp\u003e4.7.2 Computer Exercises 185\u003c\/p\u003e \u003cp\u003eAppendix 4A Development of a Prediction Interval 192\u003c\/p\u003e \u003cp\u003eAppendix 4B The Sum of Squares Decomposition 193\u003c\/p\u003e \u003cp\u003eAppendix 4C Mean Squared Error: Estimation and Prediction 193\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 \u003c\/b\u003e\u003cb\u003eThe Multiple Regression Model \u003c\/b\u003e\u003cb\u003e196\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Introduction 197\u003c\/p\u003e \u003cp\u003e5.1.1 The Economic Model 197\u003c\/p\u003e \u003cp\u003e5.1.2 The Econometric Model 198\u003c\/p\u003e \u003cp\u003e5.1.3 The General Model 202\u003c\/p\u003e \u003cp\u003e5.1.4 Assumptions of the Multiple Regression Model 203\u003c\/p\u003e \u003cp\u003e5.2 Estimating the Parameters of the Multiple Regression Model 205\u003c\/p\u003e \u003cp\u003e5.2.1 Least Squares Estimation Procedure 205\u003c\/p\u003e \u003cp\u003e5.2.2 Estimating the Error Variance σ2 207\u003c\/p\u003e \u003cp\u003e5.2.3 Measuring Goodness-of-Fit 208\u003c\/p\u003e \u003cp\u003e5.2.4 Frisch–Waugh–Lovell (FWL) Theorem 209\u003c\/p\u003e \u003cp\u003e5.3 Finite Sample Properties of the Least Squares Estimator 211\u003c\/p\u003e \u003cp\u003e5.3.1 The Variances and Covariances of the Least Squares Estimators 212\u003c\/p\u003e \u003cp\u003e5.3.2 The Distribution of the Least Squares Estimators 214\u003c\/p\u003e \u003cp\u003e5.4 Interval Estimation 216\u003c\/p\u003e \u003cp\u003e5.4.1 Interval Estimation for a Single Coefficient 216\u003c\/p\u003e \u003cp\u003e5.4.2 Interval Estimation for a Linear Combination of Coefficients 217\u003c\/p\u003e \u003cp\u003e5.5 Hypothesis Testing 218\u003c\/p\u003e \u003cp\u003e5.5.1 Testing the Significance of a Single Coefficient 219\u003c\/p\u003e \u003cp\u003e5.5.2 One-Tail Hypothesis Testing for a Single Coefficient 220\u003c\/p\u003e \u003cp\u003e5.5.3 Hypothesis Testing for a Linear Combination of Coefficients 221\u003c\/p\u003e \u003cp\u003e5.6 Nonlinear Relationships 222\u003c\/p\u003e \u003cp\u003e5.7 Large Sample Properties of the Least Squares Estimator 227\u003c\/p\u003e \u003cp\u003e5.7.1 Consistency 227\u003c\/p\u003e \u003cp\u003e5.7.2 Asymptotic Normality 229\u003c\/p\u003e \u003cp\u003e5.7.3 Relaxing Assumptions 230\u003c\/p\u003e \u003cp\u003e5.7.4 Inference for a Nonlinear Function of Coefficients 232\u003c\/p\u003e \u003cp\u003e5.8 Exercises 234\u003c\/p\u003e \u003cp\u003e5.8.1 Problems 234\u003c\/p\u003e \u003cp\u003e5.8.2 Computer Exercises 240\u003c\/p\u003e \u003cp\u003eAppendix 5A Derivation of Least Squares Estimators 247\u003c\/p\u003e \u003cp\u003eAppendix 5B The Delta Method 248\u003c\/p\u003e \u003cp\u003e5B.1 Nonlinear Function of a Single Parameter 248\u003c\/p\u003e \u003cp\u003e5B.2 Nonlinear Function of Two Parameters 249\u003c\/p\u003e \u003cp\u003eAppendix 5C Monte Carlo Simulation 250\u003c\/p\u003e \u003cp\u003e5C.1 Least Squares Estimation with Chi-Square Errors 250\u003c\/p\u003e \u003cp\u003e5C.2 Monte Carlo Simulation of the Delta Method 252\u003c\/p\u003e \u003cp\u003eAppendix 5D Bootstrapping 254\u003c\/p\u003e \u003cp\u003e5D.1 Resampling 255\u003c\/p\u003e \u003cp\u003e5D.2 Bootstrap Bias Estimate 256\u003c\/p\u003e \u003cp\u003e5D.3 Bootstrap Standard Error 256\u003c\/p\u003e \u003cp\u003e5D.4 Bootstrap Percentile Interval Estimate 257\u003c\/p\u003e \u003cp\u003e5D.5 Asymptotic Refinement 258\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 \u003c\/b\u003e\u003cb\u003eFurther Inference in the Multiple Regression Model \u003c\/b\u003e\u003cb\u003e260\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Testing Joint Hypotheses: The F-test 261\u003c\/p\u003e \u003cp\u003e6.1.1 Testing the Significance of the Model 264\u003c\/p\u003e \u003cp\u003e6.1.2 The Relationship Between t- and F-Tests 265\u003c\/p\u003e \u003cp\u003e6.1.3 More General F-Tests 267\u003c\/p\u003e \u003cp\u003e6.1.4 Using Computer Software 268\u003c\/p\u003e \u003cp\u003e6.1.5 Large Sample Tests 269\u003c\/p\u003e \u003cp\u003e6.2 The Use of Nonsample Information 271\u003c\/p\u003e \u003cp\u003e6.3 Model Specification 273\u003c\/p\u003e \u003cp\u003e6.3.1 Causality versus Prediction 273\u003c\/p\u003e \u003cp\u003e6.3.2 Omitted Variables 275\u003c\/p\u003e \u003cp\u003e6.3.3 Irrelevant Variables 277\u003c\/p\u003e \u003cp\u003e6.3.4 Control Variables 278\u003c\/p\u003e \u003cp\u003e6.3.5 Choosing a Model 280\u003c\/p\u003e \u003cp\u003e6.3.6 RESET 281\u003c\/p\u003e \u003cp\u003e6.4 Prediction 282\u003c\/p\u003e \u003cp\u003e6.4.1 Predictive Model Selection Criteria 285\u003c\/p\u003e \u003cp\u003e6.5 Poor Data, Collinearity, and Insignificance 288\u003c\/p\u003e \u003cp\u003e6.5.1 The Consequences of Collinearity 289\u003c\/p\u003e \u003cp\u003e6.5.2 Identifying and Mitigating Collinearity 290\u003c\/p\u003e \u003cp\u003e6.5.3 Investigating Influential Observations 293\u003c\/p\u003e \u003cp\u003e6.6 Nonlinear Least Squares 294\u003c\/p\u003e \u003cp\u003e6.7 Exercises 297\u003c\/p\u003e \u003cp\u003e6.7.1 Problems 297\u003c\/p\u003e \u003cp\u003e6.7.2 Computer Exercises 303\u003c\/p\u003e \u003cp\u003eAppendix 6A The Statistical Power of F-Tests 311\u003c\/p\u003e \u003cp\u003eAppendix 6B Further Results from the FWL Theorem 315\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 \u003c\/b\u003e\u003cb\u003eUsing Indicator Variables \u003c\/b\u003e\u003cb\u003e317\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Indicator Variables 318\u003c\/p\u003e \u003cp\u003e7.1.1 Intercept Indicator Variables 318\u003c\/p\u003e \u003cp\u003e7.1.2 Slope-Indicator Variables 320\u003c\/p\u003e \u003cp\u003e7.2 Applying Indicator Variables 323\u003c\/p\u003e \u003cp\u003e7.2.1 Interactions Between Qualitative Factors 323\u003c\/p\u003e \u003cp\u003e7.2.2 Qualitative Factors with Several Categories 324\u003c\/p\u003e \u003cp\u003e7.2.3 Testing the Equivalence of Two Regressions 326\u003c\/p\u003e \u003cp\u003e7.2.4 Controlling for Time 328\u003c\/p\u003e \u003cp\u003e7.3 Log-Linear Models 329\u003c\/p\u003e \u003cp\u003e7.3.1 A Rough Calculation 330\u003c\/p\u003e \u003cp\u003e7.3.2 An Exact Calculation 330\u003c\/p\u003e \u003cp\u003e7.4 The Linear Probability Model 331\u003c\/p\u003e \u003cp\u003e7.5 Treatment Effects 332\u003c\/p\u003e \u003cp\u003e7.5.1 The Difference Estimator 334\u003c\/p\u003e \u003cp\u003e7.5.2 Analysis of the Difference Estimator 334\u003c\/p\u003e \u003cp\u003e7.5.3 The Differences-in-Differences Estimator 338\u003c\/p\u003e \u003cp\u003e7.6 Treatment Effects and Causal Modeling 342\u003c\/p\u003e \u003cp\u003e7.6.1 The Nature of Causal Effects 342\u003c\/p\u003e \u003cp\u003e7.6.2 Treatment Effect Models 343\u003c\/p\u003e \u003cp\u003e7.6.3 Decomposing the Treatment Effect 344\u003c\/p\u003e \u003cp\u003e7.6.4 Introducing Control Variables 345\u003c\/p\u003e \u003cp\u003e7.6.5 The Overlap Assumption 347\u003c\/p\u003e \u003cp\u003e7.6.6 Regression Discontinuity Designs 347\u003c\/p\u003e \u003cp\u003e7.7 Exercises 351\u003c\/p\u003e \u003cp\u003e7.7.1 Problems 351\u003c\/p\u003e \u003cp\u003e7.7.2 Computer Exercises 358\u003c\/p\u003e \u003cp\u003eAppendix 7A Details of Log-Linear Model Interpretation 366\u003c\/p\u003e \u003cp\u003eAppendix 7B Derivation of the Differences-in-Differences Estimator 366\u003c\/p\u003e \u003cp\u003eAppendix 7C The Overlap Assumption: Details 367\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 \u003c\/b\u003e\u003cb\u003eHeteroskedasticity \u003c\/b\u003e\u003cb\u003e368\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 The Nature of Heteroskedasticity 369\u003c\/p\u003e \u003cp\u003e8.2 Heteroskedasticity in the Multiple Regression Model 370\u003c\/p\u003e \u003cp\u003e8.2.1 The Heteroskedastic Regression Model 371\u003c\/p\u003e \u003cp\u003e8.2.2 Heteroskedasticity Consequences for the OLS Estimator 373\u003c\/p\u003e \u003cp\u003e8.3 Heteroskedasticity Robust Variance Estimator 374\u003c\/p\u003e \u003cp\u003e8.4 Generalized Least Squares: Known Form of Variance 375\u003c\/p\u003e \u003cp\u003e8.4.1 Transforming the Model: Proportional Heteroskedasticity 375\u003c\/p\u003e \u003cp\u003e8.4.2 Weighted Least Squares: Proportional Heteroskedasticity 377\u003c\/p\u003e \u003cp\u003e8.5 Generalized Least Squares: Unknown Form of Variance 379\u003c\/p\u003e \u003cp\u003e8.5.1 Estimating the Multiplicative Model 381\u003c\/p\u003e \u003cp\u003e8.6 Detecting Heteroskedasticity 383\u003c\/p\u003e \u003cp\u003e8.6.1 Residual Plots 384\u003c\/p\u003e \u003cp\u003e8.6.2 The Goldfeld–Quandt Test 384\u003c\/p\u003e \u003cp\u003e8.6.3 A General Test for Conditional Heteroskedasticity 385\u003c\/p\u003e \u003cp\u003e8.6.4 The White Test 387\u003c\/p\u003e \u003cp\u003e8.6.5 Model Specification and Heteroskedasticity 388\u003c\/p\u003e \u003cp\u003e8.7 Heteroskedasticity in the Linear Probability Model 390\u003c\/p\u003e \u003cp\u003e8.8 Exercises 391\u003c\/p\u003e \u003cp\u003e8.8.1 Problems 391\u003c\/p\u003e \u003cp\u003e8.8.2 Computer Exercises 401\u003c\/p\u003e \u003cp\u003eAppendix 8A Properties of the Least Squares Estimator 407\u003c\/p\u003e \u003cp\u003eAppendix 8B Lagrange Multiplier Tests for Heteroskedasticity 408\u003c\/p\u003e \u003cp\u003eAppendix 8C Properties of the Least Squares Residuals 410\u003c\/p\u003e \u003cp\u003e8C.1 Details of Multiplicative Heteroskedasticity Model 411\u003c\/p\u003e \u003cp\u003eAppendix 8D Alternative Robust Sandwich Estimators 411\u003c\/p\u003e \u003cp\u003eAppendix 8E Monte Carlo Evidence: OLS, GLS, and FGLS 414\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 \u003c\/b\u003e\u003cb\u003eRegression with Time-Series Data: Stationary Variables \u003c\/b\u003e\u003cb\u003e417\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Introduction 418\u003c\/p\u003e \u003cp\u003e9.1.1 Modeling Dynamic Relationships 420\u003c\/p\u003e \u003cp\u003e9.1.2 Autocorrelations 424\u003c\/p\u003e \u003cp\u003e9.2 Stationarity and Weak Dependence 427\u003c\/p\u003e \u003cp\u003e9.3 Forecasting 430\u003c\/p\u003e \u003cp\u003e9.3.1 Forecast Intervals and Standard Errors 433\u003c\/p\u003e \u003cp\u003e9.3.2 Assumptions for Forecasting 435\u003c\/p\u003e \u003cp\u003e9.3.3 Selecting Lag Lengths 436\u003c\/p\u003e \u003cp\u003e9.3.4 Testing for Granger Causality 437\u003c\/p\u003e \u003cp\u003e9.4 Testing for Serially Correlated Errors 438\u003c\/p\u003e \u003cp\u003e9.4.1 Checking the Correlogram of the Least Squares Residuals 439\u003c\/p\u003e \u003cp\u003e9.4.2 Lagrange Multiplier Test 440\u003c\/p\u003e \u003cp\u003e9.4.3 Durbin–Watson Test 443\u003c\/p\u003e \u003cp\u003e9.5 Time-Series Regressions for Policy Analysis 443\u003c\/p\u003e \u003cp\u003e9.5.1 Finite Distributed Lags 445\u003c\/p\u003e \u003cp\u003e9.5.2 HAC Standard Errors 448\u003c\/p\u003e \u003cp\u003e9.5.3 Estimation with AR(1) Errors 452\u003c\/p\u003e \u003cp\u003e9.5.4 Infinite Distributed Lags 456\u003c\/p\u003e \u003cp\u003e9.6 Exercises 463\u003c\/p\u003e \u003cp\u003e9.6.1 Problems 463\u003c\/p\u003e \u003cp\u003e9.6.2 Computer Exercises 468\u003c\/p\u003e \u003cp\u003eAppendix 9A The Durbin–Watson Test 476\u003c\/p\u003e \u003cp\u003e9A.1 The Durbin–Watson Bounds Test 478\u003c\/p\u003e \u003cp\u003eAppendix 9B Properties of an AR(1) Error 479\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 \u003c\/b\u003e\u003cb\u003eEndogenous Regressors and Moment-Based Estimation \u003c\/b\u003e\u003cb\u003e481\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Least Squares Estimation with Endogenous Regressors 482\u003c\/p\u003e \u003cp\u003e10.1.1 Large Sample Properties of the OLS Estimator 483\u003c\/p\u003e \u003cp\u003e10.1.2 Why Least Squares Estimation Fails 484\u003c\/p\u003e \u003cp\u003e10.1.3 Proving the Inconsistency of OLS 486\u003c\/p\u003e \u003cp\u003e10.2 Cases inWhich x and e are Contemporaneously Correlated 487\u003c\/p\u003e \u003cp\u003e10.2.1 Measurement Error 487\u003c\/p\u003e \u003cp\u003e10.2.2 Simultaneous Equations Bias 488\u003c\/p\u003e \u003cp\u003e10.2.3 Lagged-Dependent Variable Models with Serial Correlation 489\u003c\/p\u003e \u003cp\u003e10.2.4 Omitted Variables 489\u003c\/p\u003e \u003cp\u003e10.3 Estimators Based on the Method of Moments 490\u003c\/p\u003e \u003cp\u003e10.3.1 Method of Moments Estimation of a Population Mean and Variance 490\u003c\/p\u003e \u003cp\u003e10.3.2 Method of Moments Estimation in the Simple Regression Model 491\u003c\/p\u003e \u003cp\u003e10.3.3 Instrumental Variables Estimation in the Simple Regression Model 492\u003c\/p\u003e \u003cp\u003e10.3.4 The Importance of Using Strong Instruments 493\u003c\/p\u003e \u003cp\u003e10.3.5 Proving the Consistency of the IV Estimator 494\u003c\/p\u003e \u003cp\u003e10.3.6 IV Estimation Using Two-Stage Least Squares (2SLS) 495\u003c\/p\u003e \u003cp\u003e10.3.7 Using Surplus Moment Conditions 496\u003c\/p\u003e \u003cp\u003e10.3.8 Instrumental Variables Estimation in the Multiple Regression Model 498\u003c\/p\u003e \u003cp\u003e10.3.9 Assessing Instrument Strength Using the First-Stage Model 500\u003c\/p\u003e \u003cp\u003e10.3.10 Instrumental Variables Estimation in a General Model 502\u003c\/p\u003e \u003cp\u003e10.3.11 Additional Issues When Using IV Estimation 504\u003c\/p\u003e \u003cp\u003e10.4 Specification Tests 505\u003c\/p\u003e \u003cp\u003e10.4.1 The Hausman Test for Endogeneity 505\u003c\/p\u003e \u003cp\u003e10.4.2 The Logic of the Hausman Test 507\u003c\/p\u003e \u003cp\u003e10.4.3 Testing Instrument Validity 508\u003c\/p\u003e \u003cp\u003e10.5 Exercises 510\u003c\/p\u003e \u003cp\u003e10.5.1 Problems 510\u003c\/p\u003e \u003cp\u003e10.5.2 Computer Exercises 516\u003c\/p\u003e \u003cp\u003eAppendix 10A Testing for Weak Instruments 520\u003c\/p\u003e \u003cp\u003e10A.1 A Test for Weak Identification 521\u003c\/p\u003e \u003cp\u003e10A.2 Testing for Weak Identification: Conclusions 525\u003c\/p\u003e \u003cp\u003eAppendix 10B Monte Carlo Simulation 525\u003c\/p\u003e \u003cp\u003e10B.1 Illustrations Using Simulated Data 526\u003c\/p\u003e \u003cp\u003e10B.2 The Sampling Properties of IV\/2SLS 528\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 \u003c\/b\u003e\u003cb\u003eSimultaneous Equations Models \u003c\/b\u003e\u003cb\u003e531\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 A Supply and Demand Model 532\u003c\/p\u003e \u003cp\u003e11.2 The Reduced-Form Equations 534\u003c\/p\u003e \u003cp\u003e11.3 The Failure of Least Squares Estimation 535\u003c\/p\u003e \u003cp\u003e11.3.1 Proving the Failure of OLS 535\u003c\/p\u003e \u003cp\u003e11.4 The Identification Problem 536\u003c\/p\u003e \u003cp\u003e11.5 Two-Stage Least Squares Estimation 538\u003c\/p\u003e \u003cp\u003e11.5.1 The General Two-Stage Least Squares Estimation Procedure 539\u003c\/p\u003e \u003cp\u003e11.5.2 The Properties of the Two-Stage Least Squares Estimator 540\u003c\/p\u003e \u003cp\u003e11.6 Exercises 545\u003c\/p\u003e \u003cp\u003e11.6.1 Problems 545\u003c\/p\u003e \u003cp\u003e11.6.2 Computer Exercises 551\u003c\/p\u003e \u003cp\u003eAppendix 11A 2SLS Alternatives 557\u003c\/p\u003e \u003cp\u003e11A.1 The k-Class of Estimators 557\u003c\/p\u003e \u003cp\u003e11A.2 The LIML Estimator 558\u003c\/p\u003e \u003cp\u003e11A.3 Monte Carlo Simulation Results 562\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 \u003c\/b\u003e\u003cb\u003eRegression with Time-Series Data: Nonstationary Variables \u003c\/b\u003e\u003cb\u003e563\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Stationary and Nonstationary Variables 564\u003c\/p\u003e \u003cp\u003e12.1.1 Trend Stationary Variables 567\u003c\/p\u003e \u003cp\u003e12.1.2 The First-Order Autoregressive Model 570\u003c\/p\u003e \u003cp\u003e12.1.3 Random Walk Models 572\u003c\/p\u003e \u003cp\u003e12.2 Consequences of Stochastic Trends 574\u003c\/p\u003e \u003cp\u003e12.3 Unit Root Tests for Stationarity 576\u003c\/p\u003e \u003cp\u003e12.3.1 Unit Roots 576\u003c\/p\u003e \u003cp\u003e12.3.2 Dickey–Fuller Tests 577\u003c\/p\u003e \u003cp\u003e12.3.3 Dickey–Fuller Test with Intercept and No Trend 577\u003c\/p\u003e \u003cp\u003e12.3.4 Dickey–Fuller Test with Intercept and Trend 579\u003c\/p\u003e \u003cp\u003e12.3.5 Dickey–Fuller Test with No Intercept and No Trend 580\u003c\/p\u003e \u003cp\u003e12.3.6 Order of Integration 581\u003c\/p\u003e \u003cp\u003e12.3.7 Other Unit Root Tests 582\u003c\/p\u003e \u003cp\u003e12.4 Cointegration 582\u003c\/p\u003e \u003cp\u003e12.4.1 The Error Correction Model 584\u003c\/p\u003e \u003cp\u003e12.5 Regression When There Is No Cointegration 585\u003c\/p\u003e \u003cp\u003e12.6 Summary 587\u003c\/p\u003e \u003cp\u003e12.7 Exercises 588\u003c\/p\u003e \u003cp\u003e12.7.1 Problems 588\u003c\/p\u003e \u003cp\u003e12.7.2 Computer Exercises 592\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 \u003c\/b\u003e\u003cb\u003eVector Error Correction and Vector Autoregressive Models \u003c\/b\u003e\u003cb\u003e597\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 VEC and VAR Models 598\u003c\/p\u003e \u003cp\u003e13.2 Estimating a Vector Error Correction Model 600\u003c\/p\u003e \u003cp\u003e13.3 Estimating a VAR Model 601\u003c\/p\u003e \u003cp\u003e13.4 Impulse Responses and Variance Decompositions 603\u003c\/p\u003e \u003cp\u003e13.4.1 Impulse Response Functions 603\u003c\/p\u003e \u003cp\u003e13.4.2 Forecast Error Variance Decompositions 605\u003c\/p\u003e \u003cp\u003e13.5 Exercises 607\u003c\/p\u003e \u003cp\u003e13.5.1 Problems 607\u003c\/p\u003e \u003cp\u003e13.5.2 Computer Exercises 608\u003c\/p\u003e \u003cp\u003eAppendix 13A The Identification Problem 612\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 \u003c\/b\u003e\u003cb\u003eTime-Varying Volatility and ARCH Models \u003c\/b\u003e\u003cb\u003e614\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 The ARCH Model 615\u003c\/p\u003e \u003cp\u003e14.2 Time-Varying Volatility 616\u003c\/p\u003e \u003cp\u003e14.3 Testing, Estimating, and Forecasting 620\u003c\/p\u003e \u003cp\u003e14.4 Extensions 622\u003c\/p\u003e \u003cp\u003e14.4.1 The GARCH Model—Generalized ARCH 622\u003c\/p\u003e \u003cp\u003e14.4.2 Allowing for an Asymmetric Effect 623\u003c\/p\u003e \u003cp\u003e14.4.3 GARCH-in-Mean and Time-Varying Risk Premium 624\u003c\/p\u003e \u003cp\u003e14.4.4 Other Developments 625\u003c\/p\u003e \u003cp\u003e14.5 Exercises 626\u003c\/p\u003e \u003cp\u003e14.5.1 Problems 626\u003c\/p\u003e \u003cp\u003e14.5.2 Computer Exercises 627\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 \u003c\/b\u003e\u003cb\u003ePanel Data Models \u003c\/b\u003e\u003cb\u003e634\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 The Panel Data Regression Function 636\u003c\/p\u003e \u003cp\u003e15.1.1 Further Discussion of Unobserved Heterogeneity 638\u003c\/p\u003e \u003cp\u003e15.1.2 The Panel Data Regression Exogeneity Assumption 639\u003c\/p\u003e \u003cp\u003e15.1.3 Using OLS to Estimate the Panel Data Regression 639\u003c\/p\u003e \u003cp\u003e15.2 The Fixed Effects Estimator 640\u003c\/p\u003e \u003cp\u003e15.2.1 The Difference Estimator: T = 2 640\u003c\/p\u003e \u003cp\u003e15.2.2 The Within Estimator: T = 2 642\u003c\/p\u003e \u003cp\u003e15.2.3 The Within Estimator: T \u0026gt; 2 643\u003c\/p\u003e \u003cp\u003e15.2.4 The Least Squares Dummy Variable Model 644\u003c\/p\u003e \u003cp\u003e15.3 Panel Data Regression Error Assumptions 646\u003c\/p\u003e \u003cp\u003e15.3.1 OLS Estimation with Cluster-Robust Standard Errors 648\u003c\/p\u003e \u003cp\u003e15.3.2 Fixed Effects Estimation with Cluster-Robust Standard Errors 650\u003c\/p\u003e \u003cp\u003e15.4 The Random Effects Estimator 651\u003c\/p\u003e \u003cp\u003e15.4.1 Testing for Random Effects 653\u003c\/p\u003e \u003cp\u003e15.4.2 A Hausman Test for Endogeneity in the Random Effects Model 654\u003c\/p\u003e \u003cp\u003e15.4.3 A Regression-Based Hausman Test 656\u003c\/p\u003e \u003cp\u003e15.4.4 The Hausman–Taylor Estimator 658\u003c\/p\u003e \u003cp\u003e15.4.5 Summarizing Panel Data Assumptions 660\u003c\/p\u003e \u003cp\u003e15.4.6 Summarizing and Extending Panel Data Model Estimation 661\u003c\/p\u003e \u003cp\u003e15.5 Exercises 663\u003c\/p\u003e \u003cp\u003e15.5.1 Problems 663\u003c\/p\u003e \u003cp\u003e15.5.2 Computer Exercises 670\u003c\/p\u003e \u003cp\u003eAppendix 15A Cluster-Robust Standard Errors: Some Details 677\u003c\/p\u003e \u003cp\u003eAppendix 15B Estimation of Error Components 679\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 \u003c\/b\u003e\u003cb\u003eQualitative and Limited Dependent Variable Models \u003c\/b\u003e\u003cb\u003e681\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Introducing Models with Binary Dependent Variables 682\u003c\/p\u003e \u003cp\u003e16.1.1 The Linear Probability Model 683\u003c\/p\u003e \u003cp\u003e16.2 Modeling Binary Choices 685\u003c\/p\u003e \u003cp\u003e16.2.1 The Probit Model for Binary Choice 686\u003c\/p\u003e \u003cp\u003e16.2.2 Interpreting the Probit Model 687\u003c\/p\u003e \u003cp\u003e16.2.3 Maximum Likelihood Estimation of the Probit Model 690\u003c\/p\u003e \u003cp\u003e16.2.4 The Logit Model for Binary Choices 693\u003c\/p\u003e \u003cp\u003e16.2.5 Wald Hypothesis Tests 695\u003c\/p\u003e \u003cp\u003e16.2.6 Likelihood Ratio Hypothesis Tests 696\u003c\/p\u003e \u003cp\u003e16.2.7 Robust Inference in Probit and Logit Models 698\u003c\/p\u003e \u003cp\u003e16.2.8 Binary Choice Models with a Continuous Endogenous Variable 698\u003c\/p\u003e \u003cp\u003e16.2.9 Binary Choice Models with a Binary Endogenous Variable 699\u003c\/p\u003e \u003cp\u003e16.2.10 Binary Endogenous Explanatory Variables 700\u003c\/p\u003e \u003cp\u003e16.2.11 Binary Choice Models and Panel Data 701\u003c\/p\u003e \u003cp\u003e16.3 Multinomial Logit 702\u003c\/p\u003e \u003cp\u003e16.3.1 Multinomial Logit Choice Probabilities 703\u003c\/p\u003e \u003cp\u003e16.3.2 Maximum Likelihood Estimation 703\u003c\/p\u003e \u003cp\u003e16.3.3 Multinomial Logit Postestimation Analysis 704\u003c\/p\u003e \u003cp\u003e16.4 Conditional Logit 707\u003c\/p\u003e \u003cp\u003e16.4.1 Conditional Logit Choice Probabilities 707\u003c\/p\u003e \u003cp\u003e16.4.2 Conditional Logit Postestimation Analysis 708\u003c\/p\u003e \u003cp\u003e16.5 Ordered Choice Models 709\u003c\/p\u003e \u003cp\u003e16.5.1 Ordinal Probit Choice Probabilities 710\u003c\/p\u003e \u003cp\u003e16.5.2 Ordered Probit Estimation and Interpretation 711\u003c\/p\u003e \u003cp\u003e16.6 Models for Count Data 713\u003c\/p\u003e \u003cp\u003e16.6.1 Maximum Likelihood Estimation of the Poisson Regression Model 713\u003c\/p\u003e \u003cp\u003e16.6.2 Interpreting the Poisson Regression Model 714\u003c\/p\u003e \u003cp\u003e16.7 Limited Dependent Variables 717\u003c\/p\u003e \u003cp\u003e16.7.1 Maximum Likelihood Estimation of the Simple Linear Regression Model 717\u003c\/p\u003e \u003cp\u003e16.7.2 Truncated Regression 718\u003c\/p\u003e \u003cp\u003e16.7.3 Censored Samples and Regression 718\u003c\/p\u003e \u003cp\u003e16.7.4 Tobit Model Interpretation 720\u003c\/p\u003e \u003cp\u003e16.7.5 Sample Selection 723\u003c\/p\u003e \u003cp\u003e16.8 Exercises 725\u003c\/p\u003e \u003cp\u003e16.8.1 Problems 725\u003c\/p\u003e \u003cp\u003e16.8.2 Computer Exercises 733\u003c\/p\u003e \u003cp\u003eAppendix 16A Probit Marginal Effects: Details 739\u003c\/p\u003e \u003cp\u003e16A.1 Standard Error of Marginal Effect at a Given Point 739\u003c\/p\u003e \u003cp\u003e16A.2 Standard Error of Average Marginal Effect 740\u003c\/p\u003e \u003cp\u003eAppendix 16B Random Utility Models 741\u003c\/p\u003e \u003cp\u003e16B.1 Binary Choice Model 741\u003c\/p\u003e \u003cp\u003e16B.2 Probit or Logit? 742\u003c\/p\u003e \u003cp\u003eAppendix 16C Using Latent Variables 743\u003c\/p\u003e \u003cp\u003e16C.1 Tobit (Tobit Type I) 743\u003c\/p\u003e \u003cp\u003e16C.2 Heckit (Tobit Type II) 744\u003c\/p\u003e \u003cp\u003eAppendix 16D A Tobit Monte Carlo Experiment 745\u003c\/p\u003e \u003cp\u003e\u003cb\u003eA \u003c\/b\u003e\u003cb\u003eMathematical Tools \u003c\/b\u003e\u003cb\u003e748\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1 Some Basics 749\u003c\/p\u003e \u003cp\u003eA.1.1 Numbers 749\u003c\/p\u003e \u003cp\u003eA.1.2 Exponents 749\u003c\/p\u003e \u003cp\u003eA.1.3 Scientific Notation 749\u003c\/p\u003e \u003cp\u003eA.1.4 Logarithms and the Number e 750\u003c\/p\u003e \u003cp\u003eA.1.5 Decimals and Percentages 751\u003c\/p\u003e \u003cp\u003eA.1.6 Logarithms and Percentages 751\u003c\/p\u003e \u003cp\u003eA.2 Linear Relationships 752\u003c\/p\u003e \u003cp\u003eA.2.1 Slopes and Derivatives 753\u003c\/p\u003e \u003cp\u003eA.2.2 Elasticity 753\u003c\/p\u003e \u003cp\u003eA.3 Nonlinear Relationships 753\u003c\/p\u003e \u003cp\u003eA.3.1 Rules for Derivatives 754\u003c\/p\u003e \u003cp\u003eA.3.2 Elasticity of a Nonlinear Relationship 757\u003c\/p\u003e \u003cp\u003eA.3.3 Second Derivatives 757\u003c\/p\u003e \u003cp\u003eA.3.4 Maxima and Minima 758\u003c\/p\u003e \u003cp\u003eA.3.5 Partial Derivatives 759\u003c\/p\u003e \u003cp\u003eA.3.6 Maxima and Minima of Bivariate Functions 760\u003c\/p\u003e \u003cp\u003eA.4 Integrals 762\u003c\/p\u003e \u003cp\u003eA.4.1 Computing the Area Under a Curve 762\u003c\/p\u003e \u003cp\u003eA.5 Exercises 764\u003c\/p\u003e \u003cp\u003e\u003cb\u003eB \u003c\/b\u003e\u003cb\u003eProbability Concepts \u003c\/b\u003e\u003cb\u003e768\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB.1 Discrete Random Variables 769\u003c\/p\u003e \u003cp\u003eB.1.1 Expected Value of a Discrete Random Variable 769\u003c\/p\u003e \u003cp\u003eB.1.2 Variance of a Discrete Random Variable 770\u003c\/p\u003e \u003cp\u003eB.1.3 Joint, Marginal, and Conditional Distributions 771\u003c\/p\u003e \u003cp\u003eB.1.4 Expectations Involving Several Random Variables 772\u003c\/p\u003e \u003cp\u003eB.1.5 Covariance and Correlation 773\u003c\/p\u003e \u003cp\u003eB.1.6 Conditional Expectations 774\u003c\/p\u003e \u003cp\u003eB.1.7 Iterated Expectations 774\u003c\/p\u003e \u003cp\u003eB.1.8 Variance Decomposition 774\u003c\/p\u003e \u003cp\u003eB.1.9 Covariance Decomposition 777\u003c\/p\u003e \u003cp\u003eB.2 Working with Continuous Random Variables 778\u003c\/p\u003e \u003cp\u003eB.2.1 Probability Calculations 779\u003c\/p\u003e \u003cp\u003eB.2.2 Properties of Continuous Random Variables 780\u003c\/p\u003e \u003cp\u003eB.2.3 Joint, Marginal, and Conditional Probability Distributions 781\u003c\/p\u003e \u003cp\u003eB.2.4 Using Iterated Expectations with Continuous Random Variables 785\u003c\/p\u003e \u003cp\u003eB.2.5 Distributions of Functions of Random Variables 787\u003c\/p\u003e \u003cp\u003eB.2.6 Truncated Random Variables 789\u003c\/p\u003e \u003cp\u003eB.3 Some Important Probability Distributions 789\u003c\/p\u003e \u003cp\u003eB.3.1 The Bernoulli Distribution 790\u003c\/p\u003e \u003cp\u003eB.3.2 The Binomial Distribution 790\u003c\/p\u003e \u003cp\u003eB.3.3 The Poisson Distribution 791\u003c\/p\u003e \u003cp\u003eB.3.4 The Uniform Distribution 792\u003c\/p\u003e \u003cp\u003eB.3.5 The Normal Distribution 793\u003c\/p\u003e \u003cp\u003eB.3.6 The Chi-Square Distribution 794\u003c\/p\u003e \u003cp\u003eB.3.7 The t-Distribution 796\u003c\/p\u003e \u003cp\u003eB.3.8 The F-Distribution 797\u003c\/p\u003e \u003cp\u003eB.3.9 The Log-Normal Distribution 799\u003c\/p\u003e \u003cp\u003eB.4 Random Numbers 800\u003c\/p\u003e \u003cp\u003eB.4.1 Uniform Random Numbers 805\u003c\/p\u003e \u003cp\u003eB.5 Exercises 806\u003c\/p\u003e \u003cp\u003e\u003cb\u003eC \u003c\/b\u003e\u003cb\u003eReview of Statistical Inference \u003c\/b\u003e\u003cb\u003e812\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eC.1 A Sample of Data 813\u003c\/p\u003e \u003cp\u003eC.2 An Econometric Model 814\u003c\/p\u003e \u003cp\u003eC.3 Estimating the Mean of a Population 815\u003c\/p\u003e \u003cp\u003eC.3.1 The Expected Value of Y 816\u003c\/p\u003e \u003cp\u003eC.3.2 The Variance of Y 817\u003c\/p\u003e \u003cp\u003eC.3.3 The Sampling Distribution of Y 817\u003c\/p\u003e \u003cp\u003eC.3.4 The Central Limit Theorem 818\u003c\/p\u003e \u003cp\u003eC.3.5 Best Linear Unbiased Estimation 820\u003c\/p\u003e \u003cp\u003eC.4 Estimating the Population Variance and Other Moments 820\u003c\/p\u003e \u003cp\u003eC.4.1 Estimating the Population Variance 821\u003c\/p\u003e \u003cp\u003eC.4.2 Estimating Higher Moments 821\u003c\/p\u003e \u003cp\u003eC.5 Interval Estimation 822\u003c\/p\u003e \u003cp\u003eC.5.1 Interval Estimation: σ2 Known 822\u003c\/p\u003e \u003cp\u003eC.5.2 Interval Estimation: σ2 Unknown 825\u003c\/p\u003e \u003cp\u003eC.6 Hypothesis Tests About a Population Mean 826\u003c\/p\u003e \u003cp\u003eC.6.1 Components of Hypothesis Tests 826\u003c\/p\u003e \u003cp\u003eC.6.2 One-Tail Tests with Alternative ‘‘Greater Than’’ (\u0026gt;) 828\u003c\/p\u003e \u003cp\u003eC.6.3 One-Tail Tests with Alternative ‘‘Less Than’’ (\u0026lt;) 829\u003c\/p\u003e \u003cp\u003eC.6.4 Two-Tail Tests with Alternative ‘‘Not Equal To’’ (≠) 829\u003c\/p\u003e \u003cp\u003eC.6.5 The p-Value 831\u003c\/p\u003e \u003cp\u003eC.6.6 A Comment on Stating Null and Alternative Hypotheses 832\u003c\/p\u003e \u003cp\u003eC.6.7 Type I and Type II Errors 833\u003c\/p\u003e \u003cp\u003eC.6.8 A Relationship Between Hypothesis Testing and Confidence Intervals 833\u003c\/p\u003e \u003cp\u003eC.7 Some Other Useful Tests 834\u003c\/p\u003e \u003cp\u003eC.7.1 Testing the Population Variance 834\u003c\/p\u003e \u003cp\u003eC.7.2 Testing the Equality of Two Population Means 834\u003c\/p\u003e \u003cp\u003eC.7.3 Testing the Ratio of Two Population Variances 835\u003c\/p\u003e \u003cp\u003eC.7.4 Testing the Normality of a Population 836\u003c\/p\u003e \u003cp\u003eC.8 Introduction to Maximum Likelihood Estimation 837\u003c\/p\u003e \u003cp\u003eC.8.1 Inference with Maximum Likelihood Estimators 840\u003c\/p\u003e \u003cp\u003eC.8.2 The Variance of the Maximum Likelihood Estimator 841\u003c\/p\u003e \u003cp\u003eC.8.3 The Distribution of the Sample Proportion 842\u003c\/p\u003e \u003cp\u003eC.8.4 Asymptotic Test Procedures 843\u003c\/p\u003e \u003cp\u003eC.9 Algebraic Supplements 848\u003c\/p\u003e \u003cp\u003eC.9.1 Derivation of Least Squares Estimator 848\u003c\/p\u003e \u003cp\u003eC.9.2 Best Linear Unbiased Estimation 849\u003c\/p\u003e \u003cp\u003eC.10 Kernel Density Estimator 851\u003c\/p\u003e \u003cp\u003eC.11 Exercises 854\u003c\/p\u003e \u003cp\u003eC.11.1 Problems 854\u003c\/p\u003e \u003cp\u003eC.11.2 Computer Exercises 857\u003c\/p\u003e \u003cp\u003e\u003cb\u003eD \u003c\/b\u003e\u003cb\u003eStatistical Tables \u003c\/b\u003e\u003cb\u003e862\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eTableD.1 Cumulative Probabilities for the Standard Normal Distribution 𝚽(z) = P(Z ≤ z) 862\u003c\/p\u003e \u003cp\u003eTableD.2 Percentiles of the t-distribution 863\u003c\/p\u003e \u003cp\u003eTableD.3 Percentiles of the Chi-square Distribution 864\u003c\/p\u003e \u003cp\u003eTableD.4 95th Percentile for the F-distribution 865\u003c\/p\u003e \u003cp\u003eTableD.5 99th Percentile for the F-distribution 866\u003c\/p\u003e \u003cp\u003eTableD.6 Standard Normal pdf Values 𝛟(z) 867\u003c\/p\u003e \u003cp\u003eIndex 869\u003c\/p\u003e","brand":"Guay C. 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