Principles of Econometrics

Häftad, Engelska, 2024

Av R. Carter Hill, William E. Griffiths, Guay C. Lim, R. Carter (Louisiana State University) Hill, William E. (University of Melbourne) Griffiths, Australia) Lim, Guay C. (University of Melbourne

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Principles of Econometrics, Fifth Edition, is an introductory book for undergraduate students in economics and finance, as well as first-year graduate students in a variety of fields that include economics, finance, accounting, marketing, public policy, sociology, law, and political science. Students will gain a working knowledge of basic econometrics so they can apply modeling, estimation, inference, and forecasting techniques when working with real-world economic problems. Readers will also gain an understanding of econometrics that allows them to critically evaluate the results of others’ economic research and modeling, and that will serve as a foundation for further study of the field. This new edition of the highly-regarded econometrics text includes major revisions that both reorganize the content and present students with plentiful opportunities to practice what they have read in the form of chapter-end exercises.

Produktinformation

Utgivningsdatum2024-09-19
Mått203 x 252 x 33 mm
Vikt1 565 g
FormatHäftad
SpråkEngelska
Antal sidor912
Upplaga5
FörlagJohn Wiley & Sons Inc
ISBN9781118452271

Tillhör följande kategorier

Mikroekonomi inom Ekonomi och ledarskap

Preface vList of Examples xxi1 An Introduction to Econometrics 11.1 Why Study Econometrics? 11.2 What Is Econometrics About? 21.2.1 Some Examples 31.3 The Econometric Model 41.3.1 Causality and Prediction 51.4 How Are Data Generated? 51.4.1 Experimental Data 61.4.2 Quasi-Experimental Data 61.4.3 Nonexperimental Data 71.5 Economic Data Types 71.5.1 Time-Series Data 71.5.2 Cross-Section Data 81.5.3 Panel or Longitudinal Data 91.6 The Research Process 91.7 Writing an Empirical Research Paper 111.7.1 Writing a Research Proposal 111.7.2 A Format for Writing a Research Report 111.8 Sources of Economic Data 131.8.1 Links to Economic Data on the Internet 131.8.2 Interpreting Economic Data 141.8.3 Obtaining the Data 14Probability Primer 15P.1 Random Variables 16P.2 Probability Distributions 17P.3 Joint, Marginal, and Conditional Probabilities 20P.3.1 Marginal Distributions 20P.3.2 Conditional Probability 21P.3.3 Statistical Independence 21P.4 A Digression: Summation Notation 22P.5 Properties of Probability Distributions 23P.5.1 Expected Value of a Random Variable 24P.5.2 Conditional Expectation 25P.5.3 Rules for Expected Values 25P.5.4 Variance of a Random Variable 26P.5.5 Expected Values of Several Random Variables 27P.5.6 Covariance Between Two Random Variables 27P.6 Conditioning 29P.6.1 Conditional Expectation 30P.6.2 Conditional Variance 31P.6.3 Iterated Expectations 32P.6.4 Variance Decomposition 33P.6.5 Covariance Decomposition 34P.7 The Normal Distribution 34P.7.1 The Bivariate Normal Distribution 37P.8 Exercises 392 The Simple Linear Regression Model 462.1 An Economic Model 472.2 An Econometric Model 492.2.1 Data Generating Process 512.2.2 The Random Error and Strict Exogeneity 522.2.3 The Regression Function 532.2.4 Random Error Variation 542.2.5 Variation in x 562.2.6 Error Normality 562.2.7 Generalizing the Exogeneity Assumption 562.2.8 Error Correlation 572.2.9 Summarizing the Assumptions 582.3 Estimating the Regression Parameters 592.3.1 The Least Squares Principle 612.3.2 Other Economic Models 652.4 Assessing the Least Squares Estimators 662.4.1 The Estimator b2 672.4.2 The Expected Values of b1 and b2 682.4.3 Sampling Variation 692.4.4 The Variances and Covariance of b1 and b2 692.5 The Gauss–Markov Theorem 722.6 The Probability Distributions of the Least Squares Estimators 732.7 Estimating the Variance of the Error Term 742.7.1 Estimating the Variances and Covariance of the Least Squares Estimators 742.7.2 Interpreting the Standard Errors 762.8 Estimating Nonlinear Relationships 772.8.1 Quadratic Functions 772.8.2 Using a Quadratic Model 772.8.3 A Log-Linear Function 792.8.4 Using a Log-Linear Model 802.8.5 Choosing a Functional Form 822.9 Regression with Indicator Variables 822.10 The Independent Variable 842.10.1 Random and Independent x 842.10.2 Random and Strictly Exogenous x 862.10.3 Random Sampling 872.11 Exercises 892.11.1 Problems 892.11.2 Computer Exercises 93Appendix 2A Derivation of the Least Squares Estimates 98Appendix 2B Deviation from the Mean Form of b2 99Appendix 2C b2 Is a Linear Estimator 100Appendix 2D Derivation of Theoretical Expression for b2 100Appendix 2E Deriving the Conditional Variance of b2 100Appendix 2F Proof of the Gauss–Markov Theorem 102Appendix 2G Proofs of Results Introduced in Section 2.10 1032G.1 The Implications of Strict Exogeneity 1032G.2 The Random and Independent x Case 1032G.3 The Random and Strictly Exogenous x Case 1052G.4 Random Sampling 106Appendix 2H Monte Carlo Simulation 1062H.1 The Regression Function 1062H.2 The Random Error 1072H.3 Theoretically True Values 1072H.4 Creating a Sample of Data 1082H.5 Monte Carlo Objectives 1092H.6 Monte Carlo Results 1092H.7 Random-x Monte Carlo Results 1103 Interval Estimation and Hypothesis Testing 1123.1 Interval Estimation 1133.1.1 The t-Distribution 1133.1.2 Obtaining Interval Estimates 1153.1.3 The Sampling Context 1163.2 Hypothesis Tests 1183.2.1 The Null Hypothesis 1183.2.2 The Alternative Hypothesis 1183.2.3 The Test Statistic 1193.2.4 The Rejection Region 1193.2.5 A Conclusion 1203.3 Rejection Regions for Specific Alternatives 1203.3.1 One-Tail Tests with Alternative ‘‘Greater Than’’ (>) 1203.3.2 One-Tail Tests with Alternative ‘‘Less Than’’ (<) 1213.3.3 Two-Tail Tests with Alternative ‘‘Not Equal To’’ (≠) 1223.4 Examples of Hypothesis Tests 1233.5 The p-Value 1263.6 Linear Combinations of Parameters 1293.6.1 Testing a Linear Combination of Parameters 1313.7 Exercises 1333.7.1 Problems 1333.7.2 Computer Exercises 139Appendix 3A Derivation of the t-Distribution 144Appendix 3B Distribution of the t-Statistic under H1 145Appendix 3C Monte Carlo Simulation 1473C.1 Sampling Properties of Interval Estimators 1483C.2 Sampling Properties of Hypothesis Tests 1493C.3 Choosing the Number of Monte Carlo Samples 1493C.4 Random-x Monte Carlo Results 1504 Prediction, Goodness-of-Fit, and Modeling Issues 1524.1 Least Squares Prediction 1534.2 Measuring Goodness-of-Fit 1564.2.1 Correlation Analysis 1584.2.2 Correlation Analysis and R2 1584.3 Modeling Issues 1604.3.1 The Effects of Scaling the Data 1604.3.2 Choosing a Functional Form 1614.3.3 A Linear-Log Food Expenditure Model 1634.3.4 Using Diagnostic Residual Plots 1654.3.5 Are the Regression Errors Normally Distributed? 1674.3.6 Identifying Influential Observations 1694.4 Polynomial Models 1714.4.1 Quadratic and Cubic Equations 1714.5 Log-Linear Models 1734.5.1 Prediction in the Log-Linear Model 1754.5.2 A Generalized R2 Measure 1764.5.3 Prediction Intervals in the Log-Linear Model 1774.6 Log-Log Models 1774.7 Exercises 1794.7.1 Problems 1794.7.2 Computer Exercises 185Appendix 4A Development of a Prediction Interval 192Appendix 4B The Sum of Squares Decomposition 193Appendix 4C Mean Squared Error: Estimation and Prediction 1935 The Multiple Regression Model 1965.1 Introduction 1975.1.1 The Economic Model 1975.1.2 The Econometric Model 1985.1.3 The General Model 2025.1.4 Assumptions of the Multiple Regression Model 2035.2 Estimating the Parameters of the Multiple Regression Model 2055.2.1 Least Squares Estimation Procedure 2055.2.2 Estimating the Error Variance σ2 2075.2.3 Measuring Goodness-of-Fit 2085.2.4 Frisch–Waugh–Lovell (FWL) Theorem 2095.3 Finite Sample Properties of the Least Squares Estimator 2115.3.1 The Variances and Covariances of the Least Squares Estimators 2125.3.2 The Distribution of the Least Squares Estimators 2145.4 Interval Estimation 2165.4.1 Interval Estimation for a Single Coefficient 2165.4.2 Interval Estimation for a Linear Combination of Coefficients 2175.5 Hypothesis Testing 2185.5.1 Testing the Significance of a Single Coefficient 2195.5.2 One-Tail Hypothesis Testing for a Single Coefficient 2205.5.3 Hypothesis Testing for a Linear Combination of Coefficients 2215.6 Nonlinear Relationships 2225.7 Large Sample Properties of the Least Squares Estimator 2275.7.1 Consistency 2275.7.2 Asymptotic Normality 2295.7.3 Relaxing Assumptions 2305.7.4 Inference for a Nonlinear Function of Coefficients 2325.8 Exercises 2345.8.1 Problems 2345.8.2 Computer Exercises 240Appendix 5A Derivation of Least Squares Estimators 247Appendix 5B The Delta Method 2485B.1 Nonlinear Function of a Single Parameter 2485B.2 Nonlinear Function of Two Parameters 249Appendix 5C Monte Carlo Simulation 2505C.1 Least Squares Estimation with Chi-Square Errors 2505C.2 Monte Carlo Simulation of the Delta Method 252Appendix 5D Bootstrapping 2545D.1 Resampling 2555D.2 Bootstrap Bias Estimate 2565D.3 Bootstrap Standard Error 2565D.4 Bootstrap Percentile Interval Estimate 2575D.5 Asymptotic Refinement 2586 Further Inference in the Multiple Regression Model 2606.1 Testing Joint Hypotheses: The F-test 2616.1.1 Testing the Significance of the Model 2646.1.2 The Relationship Between t- and F-Tests 2656.1.3 More General F-Tests 2676.1.4 Using Computer Software 2686.1.5 Large Sample Tests 2696.2 The Use of Nonsample Information 2716.3 Model Specification 2736.3.1 Causality versus Prediction 2736.3.2 Omitted Variables 2756.3.3 Irrelevant Variables 2776.3.4 Control Variables 2786.3.5 Choosing a Model 2806.3.6 RESET 2816.4 Prediction 2826.4.1 Predictive Model Selection Criteria 2856.5 Poor Data, Collinearity, and Insignificance 2886.5.1 The Consequences of Collinearity 2896.5.2 Identifying and Mitigating Collinearity 2906.5.3 Investigating Influential Observations 2936.6 Nonlinear Least Squares 2946.7 Exercises 2976.7.1 Problems 2976.7.2 Computer Exercises 303Appendix 6A The Statistical Power of F-Tests 311Appendix 6B Further Results from the FWL Theorem 3157 Using Indicator Variables 3177.1 Indicator Variables 3187.1.1 Intercept Indicator Variables 3187.1.2 Slope-Indicator Variables 3207.2 Applying Indicator Variables 3237.2.1 Interactions Between Qualitative Factors 3237.2.2 Qualitative Factors with Several Categories 3247.2.3 Testing the Equivalence of Two Regressions 3267.2.4 Controlling for Time 3287.3 Log-Linear Models 3297.3.1 A Rough Calculation 3307.3.2 An Exact Calculation 3307.4 The Linear Probability Model 3317.5 Treatment Effects 3327.5.1 The Difference Estimator 3347.5.2 Analysis of the Difference Estimator 3347.5.3 The Differences-in-Differences Estimator 3387.6 Treatment Effects and Causal Modeling 3427.6.1 The Nature of Causal Effects 3427.6.2 Treatment Effect Models 3437.6.3 Decomposing the Treatment Effect 3447.6.4 Introducing Control Variables 3457.6.5 The Overlap Assumption 3477.6.6 Regression Discontinuity Designs 3477.7 Exercises 3517.7.1 Problems 3517.7.2 Computer Exercises 358Appendix 7A Details of Log-Linear Model Interpretation 366Appendix 7B Derivation of the Differences-in-Differences Estimator 366Appendix 7C The Overlap Assumption: Details 3678 Heteroskedasticity 3688.1 The Nature of Heteroskedasticity 3698.2 Heteroskedasticity in the Multiple Regression Model 3708.2.1 The Heteroskedastic Regression Model 3718.2.2 Heteroskedasticity Consequences for the OLS Estimator 3738.3 Heteroskedasticity Robust Variance Estimator 3748.4 Generalized Least Squares: Known Form of Variance 3758.4.1 Transforming the Model: Proportional Heteroskedasticity 3758.4.2 Weighted Least Squares: Proportional Heteroskedasticity 3778.5 Generalized Least Squares: Unknown Form of Variance 3798.5.1 Estimating the Multiplicative Model 3818.6 Detecting Heteroskedasticity 3838.6.1 Residual Plots 3848.6.2 The Goldfeld–Quandt Test 3848.6.3 A General Test for Conditional Heteroskedasticity 3858.6.4 The White Test 3878.6.5 Model Specification and Heteroskedasticity 3888.7 Heteroskedasticity in the Linear Probability Model 3908.8 Exercises 3918.8.1 Problems 3918.8.2 Computer Exercises 401Appendix 8A Properties of the Least Squares Estimator 407Appendix 8B Lagrange Multiplier Tests for Heteroskedasticity 408Appendix 8C Properties of the Least Squares Residuals 4108C.1 Details of Multiplicative Heteroskedasticity Model 411Appendix 8D Alternative Robust Sandwich Estimators 411Appendix 8E Monte Carlo Evidence: OLS, GLS, and FGLS 4149 Regression with Time-Series Data: Stationary Variables 4179.1 Introduction 4189.1.1 Modeling Dynamic Relationships 4209.1.2 Autocorrelations 4249.2 Stationarity and Weak Dependence 4279.3 Forecasting 4309.3.1 Forecast Intervals and Standard Errors 4339.3.2 Assumptions for Forecasting 4359.3.3 Selecting Lag Lengths 4369.3.4 Testing for Granger Causality 4379.4 Testing for Serially Correlated Errors 4389.4.1 Checking the Correlogram of the Least Squares Residuals 4399.4.2 Lagrange Multiplier Test 4409.4.3 Durbin–Watson Test 4439.5 Time-Series Regressions for Policy Analysis 4439.5.1 Finite Distributed Lags 4459.5.2 HAC Standard Errors 4489.5.3 Estimation with AR(1) Errors 4529.5.4 Infinite Distributed Lags 4569.6 Exercises 4639.6.1 Problems 4639.6.2 Computer Exercises 468Appendix 9A The Durbin–Watson Test 4769A.1 The Durbin–Watson Bounds Test 478Appendix 9B Properties of an AR(1) Error 47910 Endogenous Regressors and Moment-Based Estimation 48110.1 Least Squares Estimation with Endogenous Regressors 48210.1.1 Large Sample Properties of the OLS Estimator 48310.1.2 Why Least Squares Estimation Fails 48410.1.3 Proving the Inconsistency of OLS 48610.2 Cases inWhich x and e are Contemporaneously Correlated 48710.2.1 Measurement Error 48710.2.2 Simultaneous Equations Bias 48810.2.3 Lagged-Dependent Variable Models with Serial Correlation 48910.2.4 Omitted Variables 48910.3 Estimators Based on the Method of Moments 49010.3.1 Method of Moments Estimation of a Population Mean and Variance 49010.3.2 Method of Moments Estimation in the Simple Regression Model 49110.3.3 Instrumental Variables Estimation in the Simple Regression Model 49210.3.4 The Importance of Using Strong Instruments 49310.3.5 Proving the Consistency of the IV Estimator 49410.3.6 IV Estimation Using Two-Stage Least Squares (2SLS) 49510.3.7 Using Surplus Moment Conditions 49610.3.8 Instrumental Variables Estimation in the Multiple Regression Model 49810.3.9 Assessing Instrument Strength Using the First-Stage Model 50010.3.10 Instrumental Variables Estimation in a General Model 50210.3.11 Additional Issues When Using IV Estimation 50410.4 Specification Tests 50510.4.1 The Hausman Test for Endogeneity 50510.4.2 The Logic of the Hausman Test 50710.4.3 Testing Instrument Validity 50810.5 Exercises 51010.5.1 Problems 51010.5.2 Computer Exercises 516Appendix 10A Testing for Weak Instruments 52010A.1 A Test for Weak Identification 52110A.2 Testing for Weak Identification: Conclusions 525Appendix 10B Monte Carlo Simulation 52510B.1 Illustrations Using Simulated Data 52610B.2 The Sampling Properties of IV/2SLS 52811 Simultaneous Equations Models 53111.1 A Supply and Demand Model 53211.2 The Reduced-Form Equations 53411.3 The Failure of Least Squares Estimation 53511.3.1 Proving the Failure of OLS 53511.4 The Identification Problem 53611.5 Two-Stage Least Squares Estimation 53811.5.1 The General Two-Stage Least Squares Estimation Procedure 53911.5.2 The Properties of the Two-Stage Least Squares Estimator 54011.6 Exercises 54511.6.1 Problems 54511.6.2 Computer Exercises 551Appendix 11A 2SLS Alternatives 55711A.1 The k-Class of Estimators 55711A.2 The LIML Estimator 55811A.3 Monte Carlo Simulation Results 56212 Regression with Time-Series Data: Nonstationary Variables 56312.1 Stationary and Nonstationary Variables 56412.1.1 Trend Stationary Variables 56712.1.2 The First-Order Autoregressive Model 57012.1.3 Random Walk Models 57212.2 Consequences of Stochastic Trends 57412.3 Unit Root Tests for Stationarity 57612.3.1 Unit Roots 57612.3.2 Dickey–Fuller Tests 57712.3.3 Dickey–Fuller Test with Intercept and No Trend 57712.3.4 Dickey–Fuller Test with Intercept and Trend 57912.3.5 Dickey–Fuller Test with No Intercept and No Trend 58012.3.6 Order of Integration 58112.3.7 Other Unit Root Tests 58212.4 Cointegration 58212.4.1 The Error Correction Model 58412.5 Regression When There Is No Cointegration 58512.6 Summary 58712.7 Exercises 58812.7.1 Problems 58812.7.2 Computer Exercises 59213 Vector Error Correction and Vector Autoregressive Models 59713.1 VEC and VAR Models 59813.2 Estimating a Vector Error Correction Model 60013.3 Estimating a VAR Model 60113.4 Impulse Responses and Variance Decompositions 60313.4.1 Impulse Response Functions 60313.4.2 Forecast Error Variance Decompositions 60513.5 Exercises 60713.5.1 Problems 60713.5.2 Computer Exercises 608Appendix 13A The Identification Problem 61214 Time-Varying Volatility and ARCH Models 61414.1 The ARCH Model 61514.2 Time-Varying Volatility 61614.3 Testing, Estimating, and Forecasting 62014.4 Extensions 62214.4.1 The GARCH Model—Generalized ARCH 62214.4.2 Allowing for an Asymmetric Effect 62314.4.3 GARCH-in-Mean and Time-Varying Risk Premium 62414.4.4 Other Developments 62514.5 Exercises 62614.5.1 Problems 62614.5.2 Computer Exercises 62715 Panel Data Models 63415.1 The Panel Data Regression Function 63615.1.1 Further Discussion of Unobserved Heterogeneity 63815.1.2 The Panel Data Regression Exogeneity Assumption 63915.1.3 Using OLS to Estimate the Panel Data Regression 63915.2 The Fixed Effects Estimator 64015.2.1 The Difference Estimator: T = 2 64015.2.2 The Within Estimator: T = 2 64215.2.3 The Within Estimator: T > 2 64315.2.4 The Least Squares Dummy Variable Model 64415.3 Panel Data Regression Error Assumptions 64615.3.1 OLS Estimation with Cluster-Robust Standard Errors 64815.3.2 Fixed Effects Estimation with Cluster-Robust Standard Errors 65015.4 The Random Effects Estimator 65115.4.1 Testing for Random Effects 65315.4.2 A Hausman Test for Endogeneity in the Random Effects Model 65415.4.3 A Regression-Based Hausman Test 65615.4.4 The Hausman–Taylor Estimator 65815.4.5 Summarizing Panel Data Assumptions 66015.4.6 Summarizing and Extending Panel Data Model Estimation 66115.5 Exercises 66315.5.1 Problems 66315.5.2 Computer Exercises 670Appendix 15A Cluster-Robust Standard Errors: Some Details 677Appendix 15B Estimation of Error Components 67916 Qualitative and Limited Dependent Variable Models 68116.1 Introducing Models with Binary Dependent Variables 68216.1.1 The Linear Probability Model 68316.2 Modeling Binary Choices 68516.2.1 The Probit Model for Binary Choice 68616.2.2 Interpreting the Probit Model 68716.2.3 Maximum Likelihood Estimation of the Probit Model 69016.2.4 The Logit Model for Binary Choices 69316.2.5 Wald Hypothesis Tests 69516.2.6 Likelihood Ratio Hypothesis Tests 69616.2.7 Robust Inference in Probit and Logit Models 69816.2.8 Binary Choice Models with a Continuous Endogenous Variable 69816.2.9 Binary Choice Models with a Binary Endogenous Variable 69916.2.10 Binary Endogenous Explanatory Variables 70016.2.11 Binary Choice Models and Panel Data 70116.3 Multinomial Logit 70216.3.1 Multinomial Logit Choice Probabilities 70316.3.2 Maximum Likelihood Estimation 70316.3.3 Multinomial Logit Postestimation Analysis 70416.4 Conditional Logit 70716.4.1 Conditional Logit Choice Probabilities 70716.4.2 Conditional Logit Postestimation Analysis 70816.5 Ordered Choice Models 70916.5.1 Ordinal Probit Choice Probabilities 71016.5.2 Ordered Probit Estimation and Interpretation 71116.6 Models for Count Data 71316.6.1 Maximum Likelihood Estimation of the Poisson Regression Model 71316.6.2 Interpreting the Poisson Regression Model 71416.7 Limited Dependent Variables 71716.7.1 Maximum Likelihood Estimation of the Simple Linear Regression Model 71716.7.2 Truncated Regression 71816.7.3 Censored Samples and Regression 71816.7.4 Tobit Model Interpretation 72016.7.5 Sample Selection 72316.8 Exercises 72516.8.1 Problems 72516.8.2 Computer Exercises 733Appendix 16A Probit Marginal Effects: Details 73916A.1 Standard Error of Marginal Effect at a Given Point 73916A.2 Standard Error of Average Marginal Effect 740Appendix 16B Random Utility Models 74116B.1 Binary Choice Model 74116B.2 Probit or Logit? 742Appendix 16C Using Latent Variables 74316C.1 Tobit (Tobit Type I) 74316C.2 Heckit (Tobit Type II) 744Appendix 16D A Tobit Monte Carlo Experiment 745A Mathematical Tools 748A.1 Some Basics 749A.1.1 Numbers 749A.1.2 Exponents 749A.1.3 Scientific Notation 749A.1.4 Logarithms and the Number e 750A.1.5 Decimals and Percentages 751A.1.6 Logarithms and Percentages 751A.2 Linear Relationships 752A.2.1 Slopes and Derivatives 753A.2.2 Elasticity 753A.3 Nonlinear Relationships 753A.3.1 Rules for Derivatives 754A.3.2 Elasticity of a Nonlinear Relationship 757A.3.3 Second Derivatives 757A.3.4 Maxima and Minima 758A.3.5 Partial Derivatives 759A.3.6 Maxima and Minima of Bivariate Functions 760A.4 Integrals 762A.4.1 Computing the Area Under a Curve 762A.5 Exercises 764B Probability Concepts 768B.1 Discrete Random Variables 769B.1.1 Expected Value of a Discrete Random Variable 769B.1.2 Variance of a Discrete Random Variable 770B.1.3 Joint, Marginal, and Conditional Distributions 771B.1.4 Expectations Involving Several Random Variables 772B.1.5 Covariance and Correlation 773B.1.6 Conditional Expectations 774B.1.7 Iterated Expectations 774B.1.8 Variance Decomposition 774B.1.9 Covariance Decomposition 777B.2 Working with Continuous Random Variables 778B.2.1 Probability Calculations 779B.2.2 Properties of Continuous Random Variables 780B.2.3 Joint, Marginal, and Conditional Probability Distributions 781B.2.4 Using Iterated Expectations with Continuous Random Variables 785B.2.5 Distributions of Functions of Random Variables 787B.2.6 Truncated Random Variables 789B.3 Some Important Probability Distributions 789B.3.1 The Bernoulli Distribution 790B.3.2 The Binomial Distribution 790B.3.3 The Poisson Distribution 791B.3.4 The Uniform Distribution 792B.3.5 The Normal Distribution 793B.3.6 The Chi-Square Distribution 794B.3.7 The t-Distribution 796B.3.8 The F-Distribution 797B.3.9 The Log-Normal Distribution 799B.4 Random Numbers 800B.4.1 Uniform Random Numbers 805B.5 Exercises 806C Review of Statistical Inference 812C.1 A Sample of Data 813C.2 An Econometric Model 814C.3 Estimating the Mean of a Population 815C.3.1 The Expected Value of Y 816C.3.2 The Variance of Y 817C.3.3 The Sampling Distribution of Y 817C.3.4 The Central Limit Theorem 818C.3.5 Best Linear Unbiased Estimation 820C.4 Estimating the Population Variance and Other Moments 820C.4.1 Estimating the Population Variance 821C.4.2 Estimating Higher Moments 821C.5 Interval Estimation 822C.5.1 Interval Estimation: σ2 Known 822C.5.2 Interval Estimation: σ2 Unknown 825C.6 Hypothesis Tests About a Population Mean 826C.6.1 Components of Hypothesis Tests 826C.6.2 One-Tail Tests with Alternative ‘‘Greater Than’’ (>) 828C.6.3 One-Tail Tests with Alternative ‘‘Less Than’’ (