FINANCIAL ECONOMETRICS i

Transcription

1 Handbook of FINANCIAL ECONOMETRICS i Tools and Techniques VOLUME Edited by YACINEAIT-SAHALIA Bendheim Center for Finance Princeton University Princeton, NJ LARS PETER HANSEN Department of Economics The University of Chicago Chicago, IL ELSEVIER Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo North-Holland is an imprint of Elsevier

2 List of Contributors xxv Volume 1: Tools and Techniques 1 Operator Methods for Continuous-Time Markov Processes 1 Yacine ATt-Sahalia, Lars Peter Hansen, and Jose A. Scheinkman 1. Introduction 2 2. Alternative Ways to Model a Continuous-Time Markov Process Transition Functions Semigroup of Conditional Expectations Infinitesimal Generators Quadratic Forms Stochastic Differential Equations Extensions Time Deformation Semigroup Pricing Parametrizations of the Stationary Distribution: Calibrating the Long Run Wong's Polynomial Models ' Stationary Distributions Fitting the Stationary Distribution Nonparametric Methods for Inferring Drift or Diffusion Coefficients Transition Dynamics and Spectral Decomposition Quadratic Forms and Implied Generators Implied Generator Syrrimetrization Principal Components Existence Spectral Decomposition Dependence Applications ZipfsLaw Stationarity and Volatility Approximating Variance Processes Imitating Long Memory Processes 33 viil

3 viii Contents 5. Hermite and Related Expansions of a Transition Density Exponential Expansion Hermite Expansion of the Transition Function Change of Variable and Rescaling Coefficients of the Expansion Local Expansions of the Log-Transition Function Expansion in A Leading Term Next Two Terms! Remaining Terms - - ^ Expansions in Powers of x xo Observable Implications and Tests Local Characterization Total Positivity and Testing for Jumps Principal Component Approach Testing the Specification of Transitions Testing Markovianity Testing Symmetry Random Time Changes The Properties of Parameter Estimators Maximum Likelihood Estimation Estimating the Diffusion Coefficient in the Presence of Jumps Maximum Likelihood Estimation with Random Sampling Times Conclusions 61 Acknowledgments 62 References 62 2 Parametric and Nonparametric Volatility Measurement 67 Torben G. Andersen, Tim Bollerslev, and Francis X. Diebold 1. Introduction Volatility Definitions Continuous-Time No-Arbitrage Price Processes Notional, Expected, and Instantaneous Volatility Volatility Modeling and Measurement Parametric Methods ' Continuous-Time Models Continuous Sample Path Diffusions Jump Diffusions and Levy-Driven Processes Discrete-Time Models ARCH Models Stochastic Volatility Models 99

4 Contents ix 4. Nonparametric Methods ARCH Filters and Smoothers Realized Volatility Directions for Future Research 124 Acknowledgments 124 References Nonstationary Continuous-Time Processes 139 Federico M. Bandi and Peter C. B. Phillips i 1. Introduction Intuition and Conditions Scalar Diffusion Processes Generalized Density Estimation for SDPs NW Kernel Estimation of the Infinitesimal Moments of an SDP The Choice of Bandwidth Extensions in Kernel Estimation for SDPs Double-Smoothing Local Linear and Polynomial Estimation Finite Sample Refinements Using Nonparametric Information to Estimate and Test Parametric Models for SDPs ' Time-lnhomogeneous SDPs An Empirical Application: Stochastic Volatility Scalar Jump-Diffusion Processes Generalized Density Estimation for SJDPs NW Kernel Estimation of the Infinitesimal Moments of an SJDP An Empirical Application: Stochastic Volatility ' Multivariate Diffusion Processes Generalized Density Estimation for MDPs NW Kernel Estimation of the Infinitesimal Moments of an MDP Concluding Remarks. 194 Acknowledgments 196 References Estimating Functions for Discretely Sampled Diffusion-Type Models 203 Bo Martin Bibby, Martin Jacobsen, and Michael Sorensen 1. Introduction Estimating Functions Martingale Estimating Functions Estimating Functions for Diffusion-Type Processes Limit Results for Diffusion Processes Maximum Likelihood Estimation 215

5 Contents 3.3. Martingale Estimating Functions for Diffusion Models Constructing Estimating Functions by Simulation Explicit Estimating Functions A Diffusion with Jumps Non-Markovian Models Optimal Estimating Functions for Diffusion Models Optimal Linear Combinations of Relationships between Consecutive Observations Approximately Optimal Estimating Functions Simple Diffusion Models for Interest Rates'" Small A-optimality 254 Acknowledgments 262 References Portfolio Choice Problems 269 Michael W. Brandt 1. Introduction Theoretical Problem Markowitz Paradigm Intertemporal Expected Utility Maximization Discrete Time Formulation Continuous-Time Formulation When is it Optimal to Invest Myopically? Constant Investment Opportuntities Stochastic but Unhedgable Investment Opportunities Logarithmic Utility Modeling Issues and Extensions Preferences Intermediate Consumption Complete Markets Infinite or Random Horizon Frictions and Background Risks Traditional Econometric Approaches Plug-In Estimation Theory Finite Sample Properties Decision Theory Parameter Uncertainty Incorporating Economic Views and Models Model Uncertainty 319

6 Contents xi 4. Alternative Econometric Approach Parametric Portfolio Weights Conditional Portfolio Choice by Augmenting the Asset Space Large-Scale Portfolio Choice with Parametric Weights Nonparametric Portfolio Weights 327 Acknowledgments 329 References Heterogeneity and Portfolio Choice: Theory and Evidence 337 Stephanie Curcuru, John Heaton, Deborah Lucas, and Damien Moore 1. Introduction Summary Statistics on Stock Market Participation and Portfolio Choice Theories of Portfolio Choice Basic Analytic Framework Time Variation in Returns Uninsurable Background Risk Trading Frictions Life-Cycle Effects Nonparticipation Generalized Preferences Quantitative Analyses The Consumption of Stockholders and Nonstockholders Calibrated Models with Background Risk Labor Income Business Income Housing Restricted Pension Investments Explaining Nonparticipation Exploiting the Covariance of Background and Market Risks Empirical Evidence and Issues An Illustrative Example Aggregate Income Statistics Evidence on Background Risk Labor Income Business Ownership Employer Stock ' Pension Investments Housing Conclusions 374 Acknowledgments 376 References 376

7 xii Contents 7 Analysis of High-Frequency Data 383 Jeffrey R. Russell and Robert F. Engle 1. Introduction ' Data Characteristics Irregular Temporal Spacing Discreteness Diurnal Patterns Temporal Dependence Types of Economic Data Economic Questions Econometric Framework Examples of Point Processes The ACD Model Thinning Point Processes Modeling in Tick Time-the Marks VAR Models for Prices and Trades in Tick Time Volatility Models in TickTime Models for Discrete Prices Calendar Time Conversion Bivariate Relationships Conclusion 421 Appendix A: EACD(3,3) Parameter Estimates Using EVIEWS GARCH Module 423 Appendix B: VAR Parameter Estimates 423 References Simulated Score Methods and Indirect Inference for Continuous-time Models 427 A. Ronald Gallant and George Tauchen 1. Introduction and Overview Estimation and Model Evaluation Overview Simulated Score Estimation Indirect Inference Estimation Details Projection: General Guidelines on the Score Generator An Initial Look at Efficiency Misspecification Nonnested Models Dynamic Stability A General Purpose Score Generator Efficiency Comparisons SNP: A General Purpose Score Generator 448

8 Contents xiii 5. Reprojection: Analysis of Postestimation Simulations Simple Illustration of Volatility Extraction General Theory of Reprojection Applications Multifactor Stochastic Volatility Models for Stock Returns Jump Diffusions Alternative Models Volatility Index Models Term Structure of Interest Rates j Affine Term Structure Models ~ ~" Regime-Switching Affine Term Structure Models Nonaffine Models Exchange Rates General Equilibrium Models Additional Applications Software and Practical Issues Code Troubleshooting, Numerical Stability, and Convergence Problems Start Value Problems and Scaling Enforcing Dynamic Stability Bulletproofing the Data Generating Process The Chernozukov-Hong Method Conclusion 472 References The Econometrics of Option Pricing 479 Rene Garcia, Eric Ghysels, and Eric Renault 1. Introduction and Overview Pricing Kernels, Risk-Neutral Probabilities, and Option Pricing Equivalent Martingale Measure and Volatility Smile How to Graph the Smile? Stochastic Discount Factors and Pricing Kernels Black-Scholes-lmplied Volatility as a Calibrated Parameter Black-Scholes-lmplied Volatility as an Expected Average Volatility Generalized Black-Scholes Option Pricing Formula Modeling Asset Price Dynamics via Diffusions for the Purpose of Option Pricing The Affine Jump-Diffusion Class of Models Models with a Single Volatility Factor Multiple Volatility Factors Other Continuous-Time Processes Nonaffine Index Models 501

9 xiv Contents Levy Processes and Time Deformation Long-Memory in Continuous Time Pricing Options Based on Objective Parameters Implied Risk-Neutral Probabilities Econometric Model of Option Pricing Errors Maximum Likelihood-Based Inference Implied-StateGMM Joint Estimation of Risk-Neutral and Objective Distributions Nonparametric Approaches Semiparametric Approaches to Derivative Pricing Canonical Valuation and Implied Binomial Trees Canonical Valuation Implied Binomial Trees A SDF Alternative to Implied Binomial Trees Comparing the Unconditional and Conditional Methodologies for Extracting Risk-Neutral Distributions Extended Method of Moments Other SNP Estimators An Economic Application of Nonparametric Methods: Extraction of Preferences Conclusion 542 Acknowledgments 544 References Value at Risk 553 Christian Gourieroux and Joann Jasiak 1. Introduction Value at Risk Definition Examples The Gaussian VaR Comparison of Tails Term Structure of the VaR _ Conditional and Marginal VaR Sensitivity of the VaR Estimation of the Marginal VaR Historical Simulation Parametric Methods Semiparametric Analysis Estimation from Sample Quantiles The Use of the Hill Estimator The i.i.d. Assumption 572

10 Contents xv 4. Estimation of the Conditional VaR Conditionally Heteroskedastic Autoregressive Models Estimation Given the Information on Portfolio Value Estimation Given the Full Information Nonparametric Methods Miscellaneous Switching Regimes Conditional Autoregressive VaR Dynamic Quantile Models! Local maximum likelihood ' "~ VaR for Portfolios with Derivatives Parametric Monte Carlo Method Taylor Expansion of Nonlinear Portfolios The Delta Method The Delta-Gamma Method Linearization of Nonlinear Portfolios The Normality Assumption in the Case of Option Prices Credit Risk Spread of Interest Rates Assessment of Default Rates Recovering Default Rates from Market Prices of Bonds Recovering Default Rates from Equity Prices Recovering Default Rates from Individual Credit Histories The Credit Migration Approach The Model Statistical Inference when the State is Observable Unobservable States VaR for Credit Portfolio The Future Portfolio Value The Credit Migration Approach, The Cohort Approach Default Correlation Future Directions for Research and Development Coherent Risk Measures, Infrequent Extreme Risks and Utility Functions The Dynamics of Infrequent Extreme Risks Large Portfolio Extreme Value Theory Concluding Remarks 609 Acknowledgments ' 609 References 609

11 xvi Contents 11 Measuring and Modeling Variation in the Risk-Return Trade-off 617 Martin Lettau and Sydney C. Ludvigson 1. Introduction The Conditional Mean of Stock Returns Origins of Predictability Evidence Linking the Macroeconomy to Conditional Mean Excess Returns Consumption, Aggregate Wealth, and Expected Stock Market Returns Popular Predictor Variables for Excess Stock Returns The Forecastability of Stock Market Retums:-Empirical Evidence Statistical Issues with Forecasting Returns Problems with Overlapping Data Problems with Persistent, Predetermined Regressors Problems with Interpreting Long-Horizon Forecasts Conceptual Issues with Forecasting Returns Expected Returns versus Average Realized Returns Cointegration and Return Forecasting Use of Macroeconomic Data in Empirical Asset Pricing When Is "Look-Ahead Bias" a Concern? Structural Change In-Sample versus Out-of-Sample Prediction The Conditional Volatility of Stock Returns and Its Relation to the Conditional Mean Updated Evidence on Risk and Return Econometric Framework Forecasts of Volatility Empirical Results on Risk and Return The Conditional Sharpe Ratio Conclusion 681 Appendix: Data Description 682 Acknowledgments 684 References Affine Term Structure Models 691 Monika Piazzesi 1. Introduction Overview Why Care About Bond Yields? Why Care About Cross-Equation Restrictions? Basics Bond Pricing in Continuous Time Local Expectations Hypothesis 698

12 Contents xvii 2.3. Partial Differential Equation for Bond Prices with LEH Without LEH Affine Models Affine Short Rate Affine Diffusions Mean Variance Affine Bond Pricing with LEH Without LEH ) Jumps V - - ' Calendar Time Does Not Matter Calendar Time Matters Risk Adjustment with Jumps ' Negative Short Rates and Jump Intensities Expected Returns 717 4^ Affine General Equilibrium Models Some Famous Affine Models Labels Based on Moments of the Short Rate Labels Based on Fundamentals Estimation Methods for Affine Models Stochastic singularity Likelihood-Based Methods Closed Form Densities Quasi-Maximum Likelihood Fourier Inversion of the Characteristic Function Solving the PDE for the Density Simulated Maximum Likelihood Hermite Expansions Matching Moments Identification Empirical Evidence on Affine Models Data Issues ShortYields Long Yields of Zero-Coupon Bonds Long Yields of Treasuries Long Yields for Swaps, Other Data Level, Slope, and Curvature Cross-Sectional Performance Unconditional First Moments (Positive Slope) Conditional First Moments (Persistence, Predictability, Nonlinearity) Persistence 742

13 xviii Contents Predictability of Excess Returns Affine Diffusions Under Both Measures Risk-Neutral Affine Diffusions with Nonlinear Data-Generating Process More on Nonlinearities Unconditional Second Moments (Vol Curve) Conditional Second Moments (Stochastic Vol) Higher Order Moments (Jumps and Regimes) Seasonalities (Settlement Wednesdays and Macro Releases) Fitting Errors at the Short End ' Joint System with Other Macroeconomic Variables Monetary Policy Inflation Other Macroeconomic Variables 757 Acknowledgments 758 References 758 Index 767