ECONOMICS 7200 MODERN TIME SERIES ANALYSIS Econometric Theory and Applications

Transcription

1 ECONOMICS 7200 MODERN TIME SERIES ANALYSIS Econometric Theory and Applications Yongmiao Hong Department of Economics & Department of Statistical Sciences Cornell University Spring 2019 Time and uncertainty are two most important factors when economic agent make decisions. Time series econometrics is important and useful empirical studies in economics and nance because it provides statistical methods and tools to investigate dynamic relations of economic and nancial time series. Economic theories usually do not indicate a linear dynamic relationship among economic time series, and in many cases, they suggest a complicated nonlinear relationship. Rational expectations, nancial derivatives pricing, asymmetric business cycles, nonlinear pricing schemes, asymmetric costs of adjustments, and volatility clustering are well-known examples of nonlinear phenomena in economics and nance. Time series econometrics, however, has mainly focused on linear time series analysis of economic and nancial time series data. Linear time series econometrics has achieved a mature stage in statistical theory and methods, and has been widely used in economics and nance. However, linear time series models cannot capture the stylized facts of nonlinear phenomena, such as asymmetry, time irreversibility, amplitude-dependent adjustment, regime-shifts, volatility clustering, and jumps or outliers. Over the past four decades, nonlinear time series analysis has been advancing rather rapidly, thanks to the demands for capturing nonlinear dynamics, the availability of large time series data, the progress of computer technology, and the application of nonparametric analysis in time series. 1

2 Empirical experience indicates that nonlinear time series analysis, from a practical point of view, and in many cases, can lead to improved methods of model tting and forecasting. Nevertheless, unlike linear time series econometrics, nonlinear time series econometrics has not achieved a mature stage in theory and methods, and there has been no uni ed account of nonlinear time series econometrics. This has apparently hindered the application of nonlinear time series analytic tools in economics and nance. This book is an attempt to provide a systematic and uni ed treatment of linear and nonlinear time series econometrics, in both theory and applications. It is hoped that this book will stimulate more interests in developing and applying nonlinear time series econometric methods and tools in economics and nance. Course prerequisites: Advanced calculus and a rst year graduate course on probability and statistics. Some background in linear time series analysis and/or stochastic process is useful but not essential. The course is self-contained. Course Requirements: Attending lectures and Submitting a term paper that can be a survey of some topic or a research work (either econometric theoretical or empirical) related to course materials. Textbook: Lecture Notes on Modern Time Series Analysis: Theory and Applications, Yongmiao Hong Schedule and Classroom: TBA Instructor Contact Information: Yongmiao Hong, 424 Uris Hall, yh20@cornell.edu & yhong.cornell@gmail.com. O ce Hours: 1:00-3:00pm, Thursday, or by appointment. 2

3 Preface Chapter 1: Introduction 1. Objectives of This Book Outline of Contents 2. Economic System and Data Generating Process 3. Stylized Facts of Macroeconomic and Financial Time Series 4. General Approach to Economic and Financial Analysis 5. Time Series Econometrics: Advantages and Limitations 6. What Can We Learn from This Book? Some Motivating Examples 7. Summary Chapter 2: Basic Concepts in Time Series Analysis 1. Stochastic Time Series Processes 2. Economic System and Data Generating Process 2.1 Two Fundamental Axioms in Time Series Econometrics 2.2 Conditional Probability Distribution and Economic Dynamics 3. General Modelling Strategy in Time Series Analysis 4. Basic Building Blocks in Time Series Analysis 4.1 White Noise (W.N.) 4.2 Martingale Di erence Sequence (M.D.S.) 4.3 Independence and Identical Distribution (I.I.D.) 5. Stationarity 5.1 Weak Stationarity 5.2 Strict Stationarity 5.3 N-th Order Stationarity 5.4 Nonstationary Time Series Di erence Stationary Time Series Trend Stationary Time Series Locally Stationary Time Series 6. Ergodicity and Memory 6.1 Ergodicity 6.2 Mixing 6.3 Long Memory Processes 3

4 7. Gaussian and Non-Gaussian Processes 8. Markov and Non-Markov Processes 9. Linear and Nonlinear Time Series Processes 9.1 Linear Time Series with I.I.D. Innovations 9.2 Linear Time Series with M.D.S. Innovations 9.3 Linear Time Series with White Noise Innovations 9.4 Nonlinear Time Series 10. Reversibility 11. Invertability 12. Summary Chapter 3: Linear Time Series Analysis 1. Dynamics and Serial Dependence 2. Measures of Serial Correlation 2.1 Autocorrelation Function 2.2 Spectral Density 3. Interpretation of Spectrum 4. Spectral Density of ARMA Processes 5. Spectral Applications in Econometrics 6. Linearity of (j) and h(!) 7. Time Domain Analysis and Frequency Domain Analysis 8. Summary Chapter 4: Nonlinear Measures of Serial Dependence 1. Motivation 2. Stylized Features of Nonlinear Time Series 3. Third Order Cumulants 4. Bispectrum 5. Higher Order Cumulants and Polyspectra 6. Density-Based Measures for Serial Dependence 7. CDF-Based Measures for Serial Dependence 8. Inferences on Patterns of Serial Dependence 8.1 Autoregression Function 8.2 Granger and Terasvirta s (1993) Characterization 4

5 8.3 Campbell, Lo and MacKinlay s (1997) Characterization 9. Summary Chapter 5: Generalized Spectral Analysis 1. Moment Generating Function 2. Characteristic Function 3. Generalized Spectrum 4. Inferences on Patterns of Serial Dependence 4.1 Inferences on Serial Dependence in Mean 4.2 Inferences on Serial Dependence in Variance 5. Summary and Directions for Further Research Chapter 6: Nonparametric Methods in Time Series 1. Motivation 2. Kernel Density Method 2.1 Kernel Density Estimation Kernel Function Consistency of a Kernel Density Estimator Optimal Choice of a Bandwidth 2.2 Kernel Estimation of a Multivariate Density 3. Nonparametric Regression Estimation 3.1 Kernel Regression Estimation 3.2 Local Polynomial Estimation 4. Nonparametric Kernel Method in Frequency Domain 4.1 Periodogram and Motivation 4.2 Kernel Spectral Estimation 4.3 Consistency of Kernel Spectral Estimators 5. Summary Chapter 7: Inferences on Conditional Mean Dynamics and Tests of the Martingale Hypothesis 1. Why is Conditional Mean Important? A Statistical Perspective 2. Why is Conditional Mean Important? An Economic Perspective 2.1 E cient Market Hypothesis 5

6 2.2 Rational Expectations General Framework: The optimizing Approach Hall s Martingale Theory of Consumption Martingale Theory of Stock Prices Dynamic Capital Asset Pricing 2.3 Derivative Pricing 3. Inference on the Martingale Hypothesis 3.1 Hypotheses of Interest 3.2 Existing Conventional Methods Box-Pierce Portmanteau Test Spectral Distribution Test Variance Ratio Test Spectral Density Test 3.3 New Approaches Indicator Function Tests Generalized Spectral Derivative Tests 4. Empirical Application: Do Foreign Exchange Rates Follow a Martingale? 5. Summary Chapter 8: Modelling Conditional Mean Dynamics 1. Model Speci cation for Conditional Mean Dynamics 2. Linear Time Series Models 2.1 Exponential Smoothing 2.2 ARMA Models 3. Nonlinear Time Series Models 3.1 Nonlinear Phenomena in Economics and Finance 3.2 Nonlinear Autoregressive Models Threshold Autoregressive Model Smooth Transition Autoregressive Model Markov Chain Regime Switching Autoregressive Model Amplitude-Dependent Exponential Autoregressive Model Random Coe cient Autoregressive Model A. Jump Model B. Stochastic Unit Root Model 6

7 3.2.6 Bilinear Autoregressive Model Nonlinear Moving Average Model Priestley s State Space Model Additive Autoregressive Model Functional Coe cient Autoregressive Model AR Model with Periodic Coe cients Locally Stationary ARMA Model 4. Estimation of Conditional Mean Models 4.1 Conditional Least Squares Method 4.2 Quasi-Maximum Likelihood Method 4.3 Generalized Method of Moments (GMM) Estimation 5. Diagnostic Checking for Conditional Mean Models 5.1 Linearity Testing Bispectral Tests Hamilton s Random Field Test Keenan s Test Tsay s Test White s Neural Network Test Generalized Spectral Derivative Test 5.2 Speci cation Testing for Nonlinear Time Series Models 6. Empirical Application: Predictability and Nonlinearity in Mean for U.S. Stock Markets 7. Summary Chapter 9: Modeling Conditional Variance Dynamics 1. Introduction 1.1 Stylized Facts 1.2 Generalized Modeling Strategy 2. Strong Form Volatility Modeling 2.1 Linear ARCH Models ARCH(q) GARCH(p; q) IGARCH RiskMetrics 7

8 2.1.5 Long Memory Volatility Model 2.2 Nonlinear Volatility Models EGARCH(p; q) Threshold GARCH(p; q) Markov Regime-Switching GARCH Model 2.3 GARCH-in-Mean Model 2.4 Estimation of Strong Form Volatility Models Quasi-Maximum Likelihood Estimation Consequence of Misspeci cation in the Innovation Distribution and Dynamics 3. Stochastic Volatility Models 3.1 Motivation 3.2 Generalized Modeling Strategy 3.3 Special Models SV(1) Model Long Memory SV Model 3.4 Estimation of Stochastic Volatility Models 4. Weak Form Volatility Modeling 4.1 Non-i.i.d. Innovation 4.2 Estimation of Weak Form Volatility Models 4.3 Consequence of Misspeci cation in the Innovation Dynamics and Distribution 5. Diagnostic Tests 5.1 Testing for ARCH E ects Engle s LM Test McLeod and Li s Portmanteau Test One-sided ARCH Test 5.2 Speci cation Testing for Volatility Models Invalidity of Box-Pierce Portmanteau Test Mak and Li s Test Generalized Spectral Derivative Test A. Test for Volatility Models with i.i.d. Innovation B. Test for Volatility Models with non-i.i.d. Innovation 5.3 Testing for Adequacy of a Strong Form Volatility Model for Full Dynamics Generalized Spectral Density Test 6. Empirical Applications 8

9 7. Summary Chapter 10: Modeling Conditional Probability Distributions 1. Introduction 1.1 Why are Conditional Density Models Important? A Statistical Perspective 1.2 Why are Conditional Density Models Important? A Decision-Theoretic Perspective 2. Discrete-Time Conditional Density Models 2.1 Strong Form Volatility Models 2.2 Markov-Chain Regime Switching Model 2.3 Hansen s Conditional Autoregressive Density Model 2.4 Hermite Polynomial Model 3. Maximum Likelihood Estimation of Discrete-Time Conditional Density Models 4. Continuous-time Di usion Models 4.1 One-factor Di usion Models 4.2 Time-Dependent Di usion Models 4.3 Jump Di usion Models 4.4 Time-Dependent Jump Di usion Models 5. Estimation of Continuous-time Models 5.1 Quasi-Maximum Likelihood Estimation 5.2 Simulation-Based Estimation 5.3 E cient Method of Moments 5.4 Approximated Maximum Likelihood Estimation 5.5 Characteristic Function Approach 5.6 MCMC Method 6. Speci cation Tests for Conditional Density Models 6.1 Univariate Conditional Density Models Marginal Density-Based Test Transition Density-Based Test 6.2 Multivariate Conditional Density Models E cient Method of Moment Test Generalized Spectral Test 7. Empirical Applications 7.1 Evaluation of Discrete-time Spot Interest Rate Models 9

10 7.2 Evaluation of Continuous-time A ne Term Structure Models 7.3 Evaluation of Autoregressive Duration Models for Foreign Exchange Rate Changes 8. Summary Chapter 11: Conclusion 1. What we have learnt? 2. Directions for Future Development in Nonlinear Time Series 3. Conclusion 10