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2.7.3. Violation of SLR.4 . . . . .  89  7. Multiple Regression Analysis with Qual-

                   2.7.4. Violation of SLR.5 . . . . .  89  itative Regressors                   135

                                                            7.1. Linear Regression with Dummy

           3. Multiple Regression Analysis: Estima-             Variables as Regressors . . . . . . .  135

              tion                                   91     7.2. Logical Variables . . . . . . . . . .  137

              3.1. Multiple Regression in Practice . .  91  7.3. Factor variables . . . . . . . . . . .  138

              3.2. OLS in Matrix Form . . . . . . . .  95   7.4. Breaking a Numeric Variable Into

                                                                Categories . . . . . . . . . . . . . .  139

              3.3. Ceteris Paribus Interpretation and

                   Omitted Variable Bias . . . . . . .  97  7.5. Interactions and Differences in Re-

                                                                gression Functions Across Groups  141

              3.4. Standard Errors, Multicollinearity,

                   and VIF . . . . . . . . . . . . . . . .  99

                                                         8. Heteroscedasticity                   143

                                                            8.1. Heteroscedasticity-Robust Inference 143

           4. Multiple Regression Analysis: Inference103

                                                            8.2. Heteroscedasticity Tests . . . . . .  147

              4.1. The t Test . . . . . . . . . . . . . . .  103

                                                            8.3. Weighted Least Squares . . . . . .  150

                   4.1.1. General Setup . . . . . . . .  103

                   4.1.2. Standard case . . . . . . . .  104  9. More on Specification and Data Issues155

                   4.1.3. Other hypotheses . . . . . .  106  9.1. Functional Form Misspecification .  155

              4.2. Confidence Intervals . . . . . . . .  108  9.2. Measurement Error . . . . . . . . .  157

              4.3. Linear Restrictions: F-Tests . . . .  109  9.3. Missing Data and Nonrandom

              4.4. Reporting Regression Results . . .  113      Samples . . . . . . . . . . . . . . .  160

                                                            9.4. Outlying Observations . . . . . . .  163

           5. Multiple Regression Analysis:     OLS         9.5. Least Absolute Deviations (LAD)

              Asymptotics                          115          Estimation . . . . . . . . . . . . . .  164

              5.1. Simulation Exercises . . . . . . . .  115

                   5.1.1. Normally Distributed Error

                         Terms . . . . . . . . . . . . .  115  II. Regression Analysis with Time

                   5.1.2. Non-Normal Error Terms .  116      Series Data                        165

                   5.1.3. (Not) Conditioning on the

                         Regressors . . . . . . . . . .  119  10.Basic Regression Analysis with Time Se-

              5.2. LM Test . . . . . . . . . . . . . . . .  121  ries Data                       167

                                                            10.1. Static Time Series Models . . . . .  167

                                                            10.2. Time Series Data Types in R . . . .  168

           6. Multiple Regression Analysis: Further Is-

              sues                                 123          10.2.1. Equispaced Time Series in R 168

                                                                10.2.2. Irregular Time Series in R .  170

              6.1. Model Formulae . . . . . . . . . . .  123

                                                            10.3. Other Time Series Models . . . . .  173

                   6.1.1. Data Scaling:  Arithmetic

                                                                10.3.1. The dynlm Package . . . .  173

                         Operations Within a Formula 123

                                                                10.3.2. Finite  Distributed  Lag

                   6.1.2. Standardization: Beta Coef-

                                                                       Models . . . . . . . . . . . .  173

                         ficients . . . . . . . . . . . .  125

                                                                10.3.3. Trends . . . . . . . . . . . .  176

                   6.1.3. Logarithms . . . . . . . . .  126

                                                                10.3.4. Seasonality . . . . . . . . .  177

                   6.1.4. Quadratics and Polynomials 126

                   6.1.5. Interaction Terms . . . . . .  128  11.Further Issues In Using OLS with Time Se-

              6.2. Prediction . . . . . . . . . . . . . .  130  ries Data                        179

                   6.2.1. Confidence  Intervals  for         11.1. Asymptotics with Time Series . . .  179

                         Predictions . . . . . . . . . .  130  11.2. The Nature of Highly Persistent

                   6.2.2. Prediction Intervals . . . . .  132   Time Series . . . . . . . . . . . . . .  182

                   6.2.3. Effect Plots for Nonlinear        11.3. Differences of Highly Persistent

                         Specifications . . . . . . . .  133     Time Series . . . . . . . . . . . . . .  185
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