Financial Econometrics Notes. Kevin Sheppard University of Oxford

Transcription

1 Financial Econometrics Notes Kevin Sheppard University of Oxford Monday 15 th January, 2018

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3 This version: 22:52, Monday 15 th January, Kevin Sheppard

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5 Contents 1 Probability, Random Variables and Expectations Axiomatic Probability Univariate Random Variables Multivariate Random Variables Expectations and Moments Estimation, Inference, and Hypothesis Testing Estimation Convergence and Limits for Random Variables Properties of Estimators Distribution Theory Hypothesis Testing The Bootstrap and Monte Carlo Inference on Financial Data Analysis of Cross-Sectional Data Model Description Functional Form Estimation Assessing Fit Assumptions Small Sample Properties of OLS estimators Maximum Likelihood Small Sample Hypothesis Testing Large Sample Assumption Large Sample Properties Large Sample Hypothesis Testing Violations of the Large Sample Assumptions Model Selection and Specification Checking

6 iv CONTENTS 3.14 Projection A Selected Proofs Analysis of a Single Time Series Stochastic Processes Stationarity, Ergodicity, and the Information Set ARMA Models Difference Equations Data and Initial Estimates Autocorrelations and Partial Autocorrelations Estimation Inference Forecasting Nonstationary Time Series Nonlinear Models for Time-Series Analysis Filters A Computing Autocovariance and Autocorrelations Analysis of Multiple Time Series Vector Autoregressions Companion Form Empirical Examples VAR forecasting Estimation and Identification Granger causality Impulse Response Function Cointegration Cross-sectional Regression with Time-series Data A Cointegration in a trivariate VAR Generalized Method Of Moments (GMM) Classical Method of Moments Examples General Specification Estimation Asymptotic Properties Covariance Estimation Special Cases of GMM Diagnostics

7 CONTENTS v 6.9 Parameter Inference Two-Stage Estimation Weak Identification Considerations for using GMM Univariate Volatility Modeling Why does volatility change? ARCH Models Forecasting Volatility Realized Variance Implied Volatility and VIX A Kurtosis of an ARCH(1) B Kurtosis of a GARCH(1,1) Value-at-Risk, Expected Shortfall and Density Forecasting Defining Risk Value-at-Risk (VaR) Expected Shortfall Density Forecasting Coherent Risk Measures Multivariate Volatility, Dependence and Copulas Introduction Preliminaries Simple Models of Multivariate Volatility Multivariate ARCH Models Realized Covariance Measuring Dependence Copulas A Bootstrap Standard Errors

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9 List of Figures 1.1 Set Operations Bernoulli Random Variables Normal pdf and cdf Poisson and χ 2 distributions Bernoulli Random Variables Joint and Conditional Distributions Joint distribution of the FTSE 100 and S&P Simulation and Numerical Integration Modes Convergence in Distribution Consistency and Central Limits Central Limit Approximations Data Generating Process and Asymptotic Covariance of Estimators Power Standard Normal cdf and Empirical cdf CRSP Value Weighted Market (VWM) Excess Returns Rejection regions of a t Bivariate F distributions Rejection region of a F 5,30 distribution Location of the three test statistic statistics Effect of correlation on the variance of ˆβ IV Gains of using GLS Neglected Nonlinearity and Residual Plots Rolling Parameter Estimates in the 4-Factor Model Recursive Parameter Estimates in the 4-Factor Model Influential Observations Correct and Incorrect use of Robust Estimators Weights of an S&P 500 Tracking Portfolio Dynamics of linear difference equations Stationarity of an AR(2)

10 viii LIST OF FIGURES 4.3 VWM and Default Spread ACF and PACF for ARMA Processes ACF and PACF for ARMA Processes Autocorrelations and Partial Autocorrelations for the VWM and the Default Spread M1, M1 growth, and the ACF and PACF of M1 growth Time Trend Models of GDP Unit Root Analysis of ln C P I and the Default Spread Ideal Filters Actual Filters Cyclical Component of U.S. Real GDP Markov Switching Processes Self Exciting Threshold Autoregression Processes Exercise Plots for question 2(b) Comparing forecasts from a VAR(1) and an AR(1) ACF and CCF Impulse Response Functions Cointegration Detrended CAY Residuals Impulse Response of Level-Slope-Curvature Step GMM Objective Function Surface Returns of the S&P 500 and WTI Squared returns of the S&P 500 and WTI Absolute returns of the S&P 500 and WTI News impact curves Various estimated densities for the S&P Effect of distribution on volatility estimates ACF and PACF of S&P 500 squared returns ACF and PACF of WTI squared returns Realized Variance and sampling frequency RV AC 1 and sampling frequency Volatility Signature Plot for SPDR RV Black-Scholes Implied Volatility VIX and alternative measures of volatility Graphical representation of Value-at-Risk Estimated % VaR for the S&P S&P 500 Returns and a Parametric Density Empirical and Smoothed empirical CDF

11 LIST OF FIGURES ix 8.5 Naïve and Correct Density Forecasts Fan plot QQ plot Kolmogorov-Smirnov plot Returns, Historical Simulation VaR and Normal GARCH VaR Lag weights in RiskMetrics methodologies Rolling Window Correlation Measures Observable and Principal Component Correlation Measures Volatility from Multivariate Models Small Cap - Large Cap Correlation Small Cap - Long Government Bond Correlation Large Cap - Bond Correlation Symmetric and Asymmetric Dependence Rolling Dependence Measures Exceedance Correlation Copula Distributions and Densities Copula Densities with Standard Normal Margins S&P FTSE 100 Diagnostics S&P 500 and FTSE 100 Exceedance Correlations

12 x LIST OF FIGURES

13 List of Tables 1.1 Monte Carlo and Numerical Integration Parameter Values of Mixed Normals Outcome matrix for a hypothesis test Inference on the Market Premium Inference on the Market Premium Comparing the Variance of the NASDAQ and S&P Comparing the Variance of the NASDAQ and S&P Wald, LR and LM Tests Fama-French Data Description Descriptive Statistics of the Fama-French Data Set Regression Coefficient on the Fama-French Data Set Centered and Uncentered R 2 and R Centered and Uncentered R 2 and R2 with Regressor Changes t -stats for the Big-High Portfolio Likelihood Ratio Tests on the Big-High Portfolio Comparison of Small- and Large- Sample t -Statistics Comparison of Small- and Large- Sample Wald, LR and LM Statistic OLS and GLS Parameter Estimates and t -stats Estimates from Time-Series Models ACF and PACF for ARMA processes Seasonal Model Estimates Unit Root Analysis of ln C P I Parameter estimates from Campbell s VAR AIC and SBIC in Campbell s VAR Granger Causality Johansen Methodology Unit Root Tests Parameter Estimates from a Consumption-Based Asset Pricing Model

14 xii LIST OF TABLES 6.2 Stochastic Volatility Model Parameter Estimates Effect of Covariance Estimator on GMM Estimates Stochastic Volatility Model Monte Carlo Tests of a Linear Factor Model Fama-MacBeth Inference Summary statistics for the S&P 500 and WTI Parameter estimates from ARCH-family models Bollerslev-Wooldridge Covariance estimates GARCH-in-mean estimates Model selection for the S&P Model selection for WTI Estimated model parameters and quantiles Unconditional VaR of the S&P Principal Component Analysis of the S&P Correlation Measures for the S&P CCC GARCH Correlation Multivariate GARCH Model Estimates Refresh-time sampling Dependence Measures for Weekly FTSE and S&P 500 Returns Copula Tail Dependence Unconditional Copula Estimates Conditional Copula Estimates