Financial Econometrics Notes Kevin Sheppard University of Oxford Monday 15 th January, 2018
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This version: 22:52, Monday 15 th January, 2018 2018 Kevin Sheppard
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Contents 1 Probability, Random Variables and Expectations 1 1.1 Axiomatic Probability................................... 1 1.2 Univariate Random Variables.............................. 8 1.3 Multivariate Random Variables.............................. 22 1.4 Expectations and Moments............................... 35 2 Estimation, Inference, and Hypothesis Testing 61 2.1 Estimation........................................ 61 2.2 Convergence and Limits for Random Variables..................... 72 2.3 Properties of Estimators................................. 76 2.4 Distribution Theory.................................... 83 2.5 Hypothesis Testing.................................... 100 2.6 The Bootstrap and Monte Carlo............................. 114 2.7 Inference on Financial Data............................... 118 3 Analysis of Cross-Sectional Data 137 3.1 Model Description.................................... 137 3.2 Functional Form..................................... 141 3.3 Estimation........................................ 144 3.4 Assessing Fit....................................... 147 3.5 Assumptions....................................... 151 3.6 Small Sample Properties of OLS estimators....................... 153 3.7 Maximum Likelihood................................... 155 3.8 Small Sample Hypothesis Testing............................ 157 3.9 Large Sample Assumption................................ 173 3.10 Large Sample Properties................................ 175 3.11 Large Sample Hypothesis Testing............................ 177 3.12 Violations of the Large Sample Assumptions...................... 182 3.13 Model Selection and Specification Checking....................... 196
iv CONTENTS 3.14 Projection........................................ 213 3.A Selected Proofs..................................... 216 4 Analysis of a Single Time Series 227 4.1 Stochastic Processes.................................. 227 4.2 Stationarity, Ergodicity, and the Information Set..................... 228 4.3 ARMA Models...................................... 230 4.4 Difference Equations................................... 239 4.5 Data and Initial Estimates................................ 247 4.6 Autocorrelations and Partial Autocorrelations...................... 247 4.7 Estimation........................................ 260 4.8 Inference......................................... 263 4.9 Forecasting....................................... 264 4.10 Nonstationary Time Series................................ 270 4.11 Nonlinear Models for Time-Series Analysis....................... 282 4.12 Filters.......................................... 282 4.A Computing Autocovariance and Autocorrelations.................... 297 5 Analysis of Multiple Time Series 323 5.1 Vector Autoregressions................................. 323 5.2 Companion Form.................................... 329 5.3 Empirical Examples................................... 330 5.4 VAR forecasting..................................... 333 5.5 Estimation and Identification............................... 334 5.6 Granger causality.................................... 338 5.7 Impulse Response Function............................... 340 5.8 Cointegration....................................... 345 5.9 Cross-sectional Regression with Time-series Data................... 361 5.A Cointegration in a trivariate VAR............................. 367 6 Generalized Method Of Moments (GMM) 379 6.1 Classical Method of Moments.............................. 379 6.2 Examples........................................ 380 6.3 General Specification.................................. 384 6.4 Estimation........................................ 388 6.5 Asymptotic Properties.................................. 392 6.6 Covariance Estimation.................................. 397 6.7 Special Cases of GMM.................................. 401 6.8 Diagnostics....................................... 406
CONTENTS v 6.9 Parameter Inference................................... 408 6.10 Two-Stage Estimation.................................. 411 6.11 Weak Identification.................................... 414 6.12 Considerations for using GMM.............................. 415 7 Univariate Volatility Modeling 417 7.1 Why does volatility change?............................... 417 7.2 ARCH Models...................................... 419 7.3 Forecasting Volatility................................... 451 7.4 Realized Variance.................................... 455 7.5 Implied Volatility and VIX................................. 464 7.A Kurtosis of an ARCH(1)................................. 471 7.B Kurtosis of a GARCH(1,1)................................ 472 8 Value-at-Risk, Expected Shortfall and Density Forecasting 483 8.1 Defining Risk....................................... 483 8.2 Value-at-Risk (VaR)................................... 484 8.3 Expected Shortfall.................................... 500 8.4 Density Forecasting................................... 502 8.5 Coherent Risk Measures................................. 512 9 Multivariate Volatility, Dependence and Copulas 521 9.1 Introduction....................................... 521 9.2 Preliminaries....................................... 522 9.3 Simple Models of Multivariate Volatility.......................... 525 9.4 Multivariate ARCH Models................................ 534 9.5 Realized Covariance................................... 546 9.6 Measuring Dependence................................. 553 9.7 Copulas......................................... 561 9.A Bootstrap Standard Errors................................ 575
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List of Figures 1.1 Set Operations...................................... 3 1.2 Bernoulli Random Variables............................... 9 1.3 Normal pdf and cdf.................................... 13 1.4 Poisson and χ 2 distributions............................... 15 1.5 Bernoulli Random Variables............................... 18 1.6 Joint and Conditional Distributions............................ 29 1.7 Joint distribution of the FTSE 100 and S&P 500..................... 33 1.8 Simulation and Numerical Integration.......................... 37 1.9 Modes.......................................... 43 2.1 Convergence in Distribution............................... 73 2.2 Consistency and Central Limits............................. 80 2.3 Central Limit Approximations............................... 81 2.4 Data Generating Process and Asymptotic Covariance of Estimators........... 96 2.5 Power.......................................... 105 2.6 Standard Normal cdf and Empirical cdf......................... 115 2.7 CRSP Value Weighted Market (VWM) Excess Returns................. 122 3.1 Rejection regions of a t 10................................ 161 3.2 Bivariate F distributions................................. 163 3.3 Rejection region of a F 5,30 distribution.......................... 165 3.4 Location of the three test statistic statistics........................ 174 3.5 Effect of correlation on the variance of ˆβ IV....................... 190 3.6 Gains of using GLS................................... 195 3.7 Neglected Nonlinearity and Residual Plots........................ 202 3.8 Rolling Parameter Estimates in the 4-Factor Model................... 205 3.9 Recursive Parameter Estimates in the 4-Factor Model.................. 206 3.10 Influential Observations................................. 208 3.11 Correct and Incorrect use of Robust Estimators.................... 212 3.12 Weights of an S&P 500 Tracking Portfolio........................ 215 4.1 Dynamics of linear difference equations......................... 245 4.2 Stationarity of an AR(2)................................. 248
viii LIST OF FIGURES 4.3 VWM and Default Spread................................ 249 4.4 ACF and PACF for ARMA Processes.......................... 253 4.5 ACF and PACF for ARMA Processes.......................... 254 4.6 Autocorrelations and Partial Autocorrelations for the VWM and the Default Spread... 258 4.7 M1, M1 growth, and the ACF and PACF of M1 growth.................. 272 4.8 Time Trend Models of GDP............................... 274 4.9 Unit Root Analysis of ln C P I and the Default Spread.................. 281 4.10 Ideal Filters....................................... 284 4.11 Actual Filters....................................... 288 4.12 Cyclical Component of U.S. Real GDP......................... 291 4.13 Markov Switching Processes............................... 295 4.14 Self Exciting Threshold Autoregression Processes.................... 297 4.15 Exercise 4.9....................................... 317 4.16 Plots for question 2(b)................................... 322 5.1 Comparing forecasts from a VAR(1) and an AR(1).................... 334 5.2 ACF and CCF...................................... 336 5.3 Impulse Response Functions.............................. 343 5.4 Cointegration....................................... 347 5.5 Detrended CAY Residuals................................ 360 5.6 Impulse Response of Level-Slope-Curvature...................... 374 6.1 2-Step GMM Objective Function Surface........................ 393 7.1 Returns of the S&P 500 and WTI............................ 424 7.2 Squared returns of the S&P 500 and WTI........................ 426 7.3 Absolute returns of the S&P 500 and WTI........................ 433 7.4 News impact curves................................... 436 7.5 Various estimated densities for the S&P 500....................... 446 7.6 Effect of distribution on volatility estimates........................ 447 7.7 ACF and PACF of S&P 500 squared returns....................... 449 7.8 ACF and PACF of WTI squared returns......................... 450 7.9 Realized Variance and sampling frequency....................... 460 7.10 RV AC 1 and sampling frequency............................. 461 7.11 Volatility Signature Plot for SPDR RV.......................... 463 7.12 Black-Scholes Implied Volatility............................. 466 7.13 VIX and alternative measures of volatility........................ 470 8.1 Graphical representation of Value-at-Risk........................ 485 8.2 Estimated % VaR for the S&P 500............................ 492 8.3 S&P 500 Returns and a Parametric Density....................... 496 8.4 Empirical and Smoothed empirical CDF......................... 505
LIST OF FIGURES ix 8.5 Naïve and Correct Density Forecasts.......................... 506 8.6 Fan plot......................................... 507 8.7 QQ plot......................................... 508 8.8 Kolmogorov-Smirnov plot................................ 510 8.9 Returns, Historical Simulation VaR and Normal GARCH VaR............... 517 9.1 Lag weights in RiskMetrics methodologies........................ 527 9.2 Rolling Window Correlation Measures.......................... 534 9.3 Observable and Principal Component Correlation Measures.............. 535 9.4 Volatility from Multivariate Models............................ 546 9.5 Small Cap - Large Cap Correlation........................... 547 9.6 Small Cap - Long Government Bond Correlation..................... 548 9.7 Large Cap - Bond Correlation.............................. 549 9.8 Symmetric and Asymmetric Dependence........................ 555 9.9 Rolling Dependence Measures............................. 557 9.10 Exceedance Correlation................................. 558 9.11 Copula Distributions and Densities............................ 569 9.12 Copula Densities with Standard Normal Margins.................... 570 9.13 S&P 500 - FTSE 100 Diagnostics............................ 574 9.14 S&P 500 and FTSE 100 Exceedance Correlations.................... 579
x LIST OF FIGURES
List of Tables 1.1 Monte Carlo and Numerical Integration......................... 60 2.1 Parameter Values of Mixed Normals........................... 95 2.2 Outcome matrix for a hypothesis test.......................... 103 2.3 Inference on the Market Premium............................ 121 2.4 Inference on the Market Premium............................ 121 2.5 Comparing the Variance of the NASDAQ and S&P 100................. 123 2.6 Comparing the Variance of the NASDAQ and S&P 100................. 125 2.7 Wald, LR and LM Tests................................. 131 3.1 Fama-French Data Description.............................. 140 3.2 Descriptive Statistics of the Fama-French Data Set................... 141 3.3 Regression Coefficient on the Fama-French Data Set.................. 147 3.4 Centered and Uncentered R 2 and R2.......................... 151 3.5 Centered and Uncentered R 2 and R2 with Regressor Changes............. 152 3.6 t -stats for the Big-High Portfolio............................. 167 3.7 Likelihood Ratio Tests on the Big-High Portfolio..................... 170 3.8 Comparison of Small- and Large- Sample t -Statistics.................. 183 3.9 Comparison of Small- and Large- Sample Wald, LR and LM Statistic.......... 184 3.10 OLS and GLS Parameter Estimates and t -stats..................... 196 4.1 Estimates from Time-Series Models........................... 248 4.2 ACF and PACF for ARMA processes........................... 252 4.3 Seasonal Model Estimates................................ 273 4.4 Unit Root Analysis of ln C P I.............................. 281 5.1 Parameter estimates from Campbell s VAR....................... 332 5.2 AIC and SBIC in Campbell s VAR............................ 338 5.3 Granger Causality.................................... 340 5.4 Johansen Methodology................................. 357 5.5 Unit Root Tests...................................... 359 6.1 Parameter Estimates from a Consumption-Based Asset Pricing Model......... 392
xii LIST OF TABLES 6.2 Stochastic Volatility Model Parameter Estimates..................... 394 6.3 Effect of Covariance Estimator on GMM Estimates................... 401 6.4 Stochastic Volatility Model Monte Carlo......................... 402 6.5 Tests of a Linear Factor Model.............................. 408 6.6 Fama-MacBeth Inference................................ 413 7.1 Summary statistics for the S&P 500 and WTI...................... 431 7.2 Parameter estimates from ARCH-family models..................... 432 7.3 Bollerslev-Wooldridge Covariance estimates....................... 441 7.4 GARCH-in-mean estimates............................... 444 7.5 Model selection for the S&P 500............................. 448 7.6 Model selection for WTI................................. 451 8.1 Estimated model parameters and quantiles....................... 493 8.2 Unconditional VaR of the S&P 500............................ 496 9.1 Principal Component Analysis of the S&P 500...................... 530 9.2 Correlation Measures for the S&P 500.......................... 534 9.3 CCC GARCH Correlation................................ 544 9.4 Multivariate GARCH Model Estimates.......................... 545 9.5 Refresh-time sampling.................................. 552 9.6 Dependence Measures for Weekly FTSE and S&P 500 Returns............. 557 9.7 Copula Tail Dependence................................. 568 9.8 Unconditional Copula Estimates............................. 573 9.9 Conditional Copula Estimates.............................. 573