Contents 13.4 Quasi-Experiments 490 Examples 490 The Differences-in-Differences Estimator 492 Instrumental Variables Estimators 494 Regression Discontinuity Estimators 495 13.5 Potential Problems with Quasi-Experiments 496 Threats to Internal Validity 496 Threats to External Validity 498 13.6 Experimental and Quasi-Experimental Estimates in Heterogeneous Populations 498 OLS with Heterogeneous Causal Effects499 IV Regression with Heterogeneous Causal Effects 500 13.7 Conclusion 503 APPENDIX 13.1 The Project STAR Data Set 510 APPENDIX 13.2 IV Estimation When the Causal Effect Varies Across Individuals 511 PENDIX 13.3 The Potential O tcomes Framework for Analyzing Data from Experiments 512 CHAPTER 14 Prediction with Many Regressors and Big Data 514 14.1 What Is"Big Data"?515 14.2 The Many-Predictor Problem and OLS 516 The Mean Squared Prediction Error 518 The First Least Squares Assumption for Prediction 519 The Predictive Regression Model with Standardized Regressors 519 The MSPE of OLS and the Principle of Shrinkage 521 Estimation of the MSPE 522 14.3 Ridge Regression 524 Shrinkage via Penalization and Ridge Regression 524 Estimation of the Ridge Shrinkage Parameter by Cross Validation 525 Application to School Test Scores 526 14.4 The Lasso 527 Shrinkage Using the Lasso 528 Application to School Test Scores 531 14.5 Principal Components 532 Principals Components with Two Variables 532 Principal Components with k Variables 534 Application to School Test Scores 536 14.6 Predicting School Test Scores with Many Predictors 537
13.4 Quasi-Experiments 490 Examples 490 The Differences-in-Differences Estimator 492 Instrumental Variables Estimators 494 Regression Discontinuity Estimators 495 13.5 Potential Problems with Quasi-Experiments 496 Threats to Internal Validity 496 Threats to External Validity 498 13.6 Experimental and Quasi-Experimental Estimates in Heterogeneous Populations 498 OLS with Heterogeneous Causal Effects 499 IV Regression with Heterogeneous Causal Effects 500 13.7 Conclusion 503 APPENDIX 13.1 The Project STAR Data Set 510 APPENDIX 13.2 IV Estimation When the Causal Effect Varies Across Individuals 511 APPENDIX 13.3 The Potential Outcomes Framework for Analyzing Data from Experiments 512 CHAPTER 14 Prediction with Many Regressors and Big Data 514 14.1 What Is “Big Data”? 515 14.2 The Many-Predictor Problem and OLS 516 The Mean Squared Prediction Error 518 The First Least Squares Assumption for Prediction 519 The Predictive Regression Model with Standardized Regressors 519 The MSPE of OLS and the Principle of Shrinkage 521 Estimation of the MSPE 522 14.3 Ridge Regression 524 Shrinkage via Penalization and Ridge Regression 524 Estimation of the Ridge Shrinkage Parameter by Cross Validation 525 Application to School Test Scores 526 14.4 The Lasso 527 Shrinkage Using the Lasso 528 Application to School Test Scores 531 14.5 Principal Components 532 Principals Components with Two Variables 532 Principal Components with k Variables 534 Application to School Test Scores 536 14.6 Predicting School Test Scores with Many Predictors 537 Contents 15 A01_STOC4455_04_GE_FM.indd 15 06/12/18 10:52 AM
Contents 14.7 Conclusion 542 APPENDIX 14.1 The California School Test Score Data Set 551 APPENDIX 14.2 Derivation of Equation(14.4)for k =1 551 APPENDIX 14.3 The Ridge Regression Estimator When k =1 551 APPENDIX 14.4 The Lasso Estimator When k 1 552 negciaereg8tosanpePmedtionsintheSandrdzad Regression Model PART FOUR Regression Analysis of Economic Time Series Data CHAPTER 15 Introduction to Time Series Regression and Forecasting 554 15.1 Introduction to Time Series Data and Serial Correlation 555 Real GDP in the United States 555 Lags,First Differences,Logarithms,and Growth Rates 555 relation 558 Other Examples of Economic Time Series 560 15.2 Stationarity and the Mean Squared Forecast Error 561 Stationarity 561 Forecasts and Forecast frrors 562 The Mean Squared Forecast Error 563 15.3 Autoregressions 565 The First-Order Autoregressive Model 565 The pth-Order Autoregressive Model 567 15.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model 568 Using the Term Spread 569 The Autoregressive Distributed Lag Model 570 The Least Squares Assumptions for Forecasting with Multiple Predictors 571 15.5 Estimation of the MSFE and Forecast Intervals 573 15.6 Estimating the Lag Length Using Information Criteria 578 Determining the Order of an Autoregression 578 Lag Length Selection in Time Series Regression with Multiple Predictors 581 15.7 Nonstationarity I:Trends 582 What Is a Trend?582 Problems Caused by Stochastic Trends 584 Detecting Stochastic Trends:Testing for a Unit AR Root 586 Avoiding the Problems Caused by Stochastic Trends 588
14.7 Conclusion 542 APPENDIX 14.1 The California School Test Score Data Set 551 APPENDIX 14.2 Derivation of Equation (14.4) for k = 1 551 APPENDIX 14.3 The Ridge Regression Estimator When k = 1 551 APPENDIX 14.4 The Lasso Estimator When k = 1 552 APPENDIX 14.5 Computing Out-of-Sample Predictions in the Standardized Regression Model 552 PART FOUR Regression Analysis of Economic Time Series Data CHAPTER 15 Introduction to Time Series Regression and Forecasting 554 15.1 Introduction to Time Series Data and Serial Correlation 555 Real GDP in the United States 555 Lags, First Differences, Logarithms, and Growth Rates 555 Autocorrelation 558 Other Examples of Economic Time Series 560 15.2 Stationarity and the Mean Squared Forecast Error 561 Stationarity 561 Forecasts and Forecast Errors 562 The Mean Squared Forecast Error 563 15.3 Autoregressions 565 The First-Order Autoregressive Model 565 The pth-Order Autoregressive Model 567 15.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model 568 Forecasting GDP Growth Using the Term Spread 569 The Autoregressive Distributed Lag Model 570 The Least Squares Assumptions for Forecasting with Multiple Predictors 571 15.5 Estimation of the MSFE and Forecast Intervals 573 Estimation of the MSFE 573 Forecast Uncertainty and Forecast Intervals 576 15.6 Estimating the Lag Length Using Information Criteria 578 Determining the Order of an Autoregression 578 Lag Length Selection in Time Series Regression with Multiple Predictors 581 15.7 Nonstationarity I: Trends 582 What Is a Trend? 582 Problems Caused by Stochastic Trends 584 Detecting Stochastic Trends: Testing for a Unit AR Root 586 Avoiding the Problems Caused by Stochastic Trends 588 16 Contents A01_STOC4455_04_GE_FM.indd 16 06/12/18 10:52 AM
Contents 15.8 Nonstationarity ll:Breaks 589 What Is a Break?589 Testing for Breaks 589 Detecting Breaks Using Pseudo Out-of-Sample Forecasts 594 Avoiding the Problems Caused by Breaks 595 15.9 Conclusion 596 APPENDIX 15.1 Time Series Data Used in Chapter 15 604 APPENDIX 15.2 Stationarity in the AR(1)Model 605 APPENDIX 15.3 Lag Operator Notation 606 APPENDIX 15.4 ARMA Models 607 APPENDIX 15.5 Consistency of the BIC Lag Length Estimator 607 CHAPTER 16 Estimation of Dynamic Causal Effects 609 16.1 An Initial Taste of the Orange Juice Data 610 16.2 Dynamic Causal Effects 612 Causal Effects and Time Series Data 612 Two Types of Exogeneity 615 16.3 Estimation of Dynamic Causal Effects with Exogenous Regressors 617 The Distributed Lag Model Assumptions 617 Autocorrelated u Standard Errors,and Inference 618 Dynamic Multipliers and Cumulative Dynamic Multipliers 618 16.4 Heteroskedasticity-and Autocorrelation-Consistent Standard Errors 620 imator with Autocorrelated Errors 620 HAC Standard Errors 621 16.5 Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors 624 The Distributed Lag Model with AR(1)Errors 625 e ADL Model 627 GLS Estimation 628 16.6 Orange Juice Prices and Cold Weather 630 16.7 Is Exogeneity Plausible?Some Examples 637 US.Income and Australian Exports637 Oil Prices and Inflation 637 Monetary Policy and Inflation 638 The Growth Rate of GDP and the Term Spread 638 16.8 Conclusion 639 APPENDIX 16.1 The Orange Juice Data Set 646 APPENDIX 16.2 The ADL Model and Generalized Least Squares in Lag Operator Notation 647
15.8 Nonstationarity II: Breaks 589 What Is a Break? 589 Testing for Breaks 589 Detecting Breaks Using Pseudo Out-of-Sample Forecasts 594 Avoiding the Problems Caused by Breaks 595 15.9 Conclusion 596 APPENDIX 15.1 Time Series Data Used in Chapter 15 604 APPENDIX 15.2 Stationarity in the AR(1) Model 605 APPENDIX 15.3 Lag Operator Notation 606 APPENDIX 15.4 ARMA Models 607 APPENDIX 15.5 Consistency of the BIC Lag Length Estimator 607 CHAPTER 16 Estimation of Dynamic Causal Effects 609 16.1 An Initial Taste of the Orange Juice Data 610 16.2 Dynamic Causal Effects 612 Causal Effects and Time Series Data 612 Two Types of Exogeneity 615 16.3 Estimation of Dynamic Causal Effects with Exogenous Regressors 617 The Distributed Lag Model Assumptions 617 Autocorrelated ut, Standard Errors, and Inference 618 Dynamic Multipliers and Cumulative Dynamic Multipliers 618 16.4 Heteroskedasticity- and Autocorrelation-Consistent Standard Errors 620 Distribution of the OLS Estimator with Autocorrelated Errors 620 HAC Standard Errors 621 16.5 Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors 624 The Distributed Lag Model with AR(1) Errors 625 OLS Estimation of the ADL Model 627 GLS Estimation 628 16.6 Orange Juice Prices and Cold Weather 630 16.7 Is Exogeneity Plausible? Some Examples 637 U.S. Income and Australian Exports 637 Oil Prices and Inflation 637 Monetary Policy and Inflation 638 The Growth Rate of GDP and the Term Spread 638 16.8 Conclusion 639 APPENDIX 16.1 The Orange Juice Data Set 646 APPENDIX 16.2 The ADL Model and Generalized Least Squares in Lag Operator Notation 647 Contents 17 A01_STOC4455_04_GE_FM.indd 17 06/12/18 10:52 AM
Contents CHAPTER 17 Additional Topics in Time Series Regression 649 17.1 Vector Autoregressions 649 The VAR Model 650 A VAR Model of the Growth Rate of GDP and the Term Spread 653 17.2 Multi-period Forecasts 654 Iterated Multi-period Forecasts 654 17.3 Orders of Integration and the Nonnormality of Unit Root Test Statistics 658 Other Models of Trends and Orders of Integ ation 659 Why Do Unit Root Tests Hat ve Nonnormal Distributions?61 17.4 Cointegration 663 Cointegration and Error Correction 663 How Can You Tell Whether Two Variables Are Cointegrated?664 Estimation of Cointegrating Coefficients 665 Extension to Multiple Cointegrated Variables 666 17.5 Volatility Clustering and Autoregressive Conditional Heteroskedasticity 667 Volatility Clustering 667 Realized Volatility 668 oskedasticity 669 17.6 Forecasting with Many Predictors Using Dynamic Factor Models and Principal Components The Dynamic Factor Model 672 The DEM:Estimation and Forecasting 673 Application to U.S.Macroeconomic Data 676 17.7 Conclusion 682 APPENDIX 17.1 The Quarterly U.S.Macro Data Set 686 PART FIVE Regression Analysis of Economic Time Series Data CHAPTER 18 The Theory of Linear Regression with One Regressor 687 18.1 The Extended Least Squares Assumptions and the OLS Estimator 688 The Extended Least Squares Assumptions68 The OLS Estimator 689 18.2 Fundamentals of Asymptotic Distribution Theory 690 nd Convergence in Distribution 692
CHAPTER 17 Additional Topics in Time Series Regression 649 17.1 Vector Autoregressions 649 The VAR Model 650 A VAR Model of the Growth Rate of GDP and the Term Spread 653 17.2 Multi-period Forecasts 654 Iterated Multi-period Forecasts 654 Direct Multi-period Forecasts 656 Which Method Should You Use? 658 17.3 Orders of Integration and the Nonnormality of Unit Root Test Statistics 658 Other Models of Trends and Orders of Integration 659 Why Do Unit Root Tests Have Nonnormal Distributions? 661 17.4 Cointegration 663 Cointegration and Error Correction 663 How Can You Tell Whether Two Variables Are Cointegrated? 664 Estimation of Cointegrating Coefficients 665 Extension to Multiple Cointegrated Variables 666 17.5 Volatility Clustering and Autoregressive Conditional Heteroskedasticity 667 Volatility Clustering 667 Realized Volatility 668 Autoregressive Conditional Heteroskedasticity 669 Application to Stock Price Volatility 670 17.6 Forecasting with Many Predictors Using Dynamic Factor Models and Principal Components 671 The Dynamic Factor Model 672 The DFM: Estimation and Forecasting 673 Application to U.S. Macroeconomic Data 676 17.7 Conclusion 682 APPENDIX 17.1 The Quarterly U.S. Macro Data Set 686 PART FIVE Regression Analysis of Economic Time Series Data CHAPTER 18 The Theory of Linear Regression with One Regressor 687 18.1 The Extended Least Squares Assumptions and the OLS Estimator 688 The Extended Least Squares Assumptions 688 The OLS Estimator 689 18.2 Fundamentals of Asymptotic Distribution Theory 690 Convergence in Probability and the Law of Large Numbers 690 The Central Limit Theorem and Convergence in Distribution 692 18 Contents A01_STOC4455_04_GE_FM.indd 18 06/12/18 10:52 AM
Contents Slutsky's Theorem and the Continuous Mapping Theorem 693 Application to the t-Statistic Based on the Sample Mean 694 18.3 Asymptotic Distribution of the OLS Estimator and t-Statistic 695 Consistency and Asymptotic Normality of the OLS Estimators 695 Consistency of Heteroskedasticity-Robust Standard Errors 695 Asymptotic Normality of the Heteroskedasticity-Robust t-Statistic 696 18.4 Exact Sampling Distributions When the Errors Are Normally Distributed 697 Distribution of B with Normal Errors 697 Distribution of the Homoskedasticity-Only t-Statistic 698 18.5 Weighted Least Squares 699 700 70 Heteroskedasticity-Robust Standard Errors or WLS?703 APPENDIX 18.1 The Normal and Related Distributions and Moments of Continuous Random Variables 709 APPENDIX 182 Two Inequalities 711 CHAPTER 19 The Theory of Multiple Regression 713 19.1 The Linear Multiple Regression Model and OLS Estimator in Matrix Form 714 The Multiple Regression Model in Matrix Notation 714 The Extended Least Squares Assumptions 715 The OLS Estimator 716 19.2 Asymptotic Distribution of the OLS Estimator and t-Statistic 717 718 Asy Central Limit Theorem rmality of718 asticity-Robust Standard Errors 719 Confidence Intervals for Predicted Effects 720 Asymptotic Distribution of the t-Statistic 720 19.3 Tests of Joint Hypotheses 721 Joint Hypotheses in Matrix Notation 721 19.4 Distribution of Regression Statistics with Normal Errors 722 Matrix Representations of OLS Regression Statistics 723 Distribution of B with Independent Normal Errors 724 Distribution of s是724 Homoskedasticity-Only Standard Errors 724 Distribution of the t-Statistic 725 Distribution of the F-Statistic 725
Slutsky’s Theorem and the Continuous Mapping Theorem 693 Application to the t-Statistic Based on the Sample Mean 694 18.3 Asymptotic Distribution of the OLS Estimator and t-Statistic 695 Consistency and Asymptotic Normality of the OLS Estimators 695 Consistency of Heteroskedasticity-Robust Standard Errors 695 Asymptotic Normality of the Heteroskedasticity-Robust t-Statistic 696 18.4 Exact Sampling Distributions When the Errors Are Normally Distributed 697 Distribution of b n 1 with Normal Errors 697 Distribution of the Homoskedasticity-Only t-Statistic 698 18.5 Weighted Least Squares 699 WLS with Known Heteroskedasticity 700 WLS with Heteroskedasticity of Known Functional Form 701 Heteroskedasticity-Robust Standard Errors or WLS? 703 APPENDIX 18.1 The Normal and Related Distributions and Moments of Continuous Random Variables 709 APPENDIX 18.2 Two Inequalities 711 CHAPTER 19 The Theory of Multiple Regression 713 19.1 The Linear Multiple Regression Model and OLS Estimator in Matrix Form 714 The Multiple Regression Model in Matrix Notation 714 The Extended Least Squares Assumptions 715 The OLS Estimator 716 19.2 Asymptotic Distribution of the OLS Estimator and t-Statistic 717 The Multivariate Central Limit Theorem 718 Asymptotic Normality of b n 718 Heteroskedasticity-Robust Standard Errors 719 Confidence Intervals for Predicted Effects 720 Asymptotic Distribution of the t-Statistic 720 19.3 Tests of Joint Hypotheses 721 Joint Hypotheses in Matrix Notation 721 Asymptotic Distribution of the F-Statistic 721 Confidence Sets for Multiple Coefficients 722 19.4 Distribution of Regression Statistics with Normal Errors 722 Matrix Representations of OLS Regression Statistics 723 Distribution of b n with Independent Normal Errors 724 Distribution of su 2 724 Homoskedasticity-Only Standard Errors 724 Distribution of the t-Statistic 725 Distribution of the F-Statistic 725 N Contents 19 A01_STOC4455_04_GE_FM.indd 19 06/12/18 10:52 AM