Modeling Financial Time Series with S-PLUS®

Modeling Financial Time Series with S-PLUS®

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English

Description

The field of financial econometrics has exploded over the last decade This book represents an integration of theory, methods, and examples using the S-PLUS statistical modeling language and the S+FinMetrics module to facilitate the practice of financial econometrics. This is the first book to show the power of S-PLUS for the analysis of time series data. It is written for researchers and practitioners in the finance industry, academic researchers in economics and finance, and advanced MBA and graduate students in economics and finance. Readers are assumed to have a basic knowledge of S-PLUS and a solid grounding in basic statistics and time series concepts.



This Second Edition is updated to cover S+FinMetrics 2.0 and includes new chapters on copulas, nonlinear regime switching models, continuous-time financial models, generalized method of moments, semi-nonparametric conditional density models, and the efficient method of moments.



Eric Zivot is an associate professor and Gary Waterman Distinguished Scholar in the Economics Department, and adjunct associate professor of finance in the Business School at the University of Washington. He regularly teaches courses on econometric theory, financial econometrics and time series econometrics, and is the recipient of the Henry T. Buechel Award for Outstanding Teaching. He is an associate editor of Studies in Nonlinear Dynamics and Econometrics. He has published papers in the leading econometrics journals, including Econometrica, Econometric Theory, the Journal of Business and Economic Statistics, Journal of Econometrics, and the Review of Economics and Statistics.



Jiahui Wang is an employee of Ronin Capital LLC. He received a Ph.D. in Economics from the University of Washington in 1997. He has published in leading econometrics journals such as Econometrica and Journal of Business and Economic Statistics, and is the Principal Investigator of National Science Foundation SBIR grants. In 2002 Dr. Wang was selected as one of the "2000 Outstanding Scholars of the 21st Century" by International Biographical Centre.

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Published 10 October 2007
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EAN13 9780387323480
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Contents
Preface
1
2
SandSPLUS 1.1 Introduction . . . . . . . . . . . . . . . . . 1.2SObjects . . . . . . . . . . . . . . . . . . 1.2.1 Assignment . . . . . . . . . . . . . 1.2.2 Class . . . . . . . . . . . . . . . . . 1.2.3 Method . . . . . . . . . . . . . . . 1.3 Modeling Functions inS+FinMetrics. . 1.3.1 Formula Specification . . . . . . . 1.3.2 Method . . . . . . . . . . . . . . . 1.4SPLUSResources . . . . . . . . . . . . . . 1.4.1 Books . . . . . . . . . . . . . . . . 1.4.2 Internet . . . . . . . . . . . . . . . 1.5 References . . . . . . . . . . . . . . . . . .
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Time Series Specification, Manipulation, and Visualization inSPLUS 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Specification of “timeSeries” Objects inSPLUS. . . 2.2.1 Basic Manipulations . . . . . . . . . . . . . . . . . . 2.2.2SPLUStimeDate” Objects . . . . . . . . . . . . . . 2.2.3 Creating Common “timeDate” Sequences . . . . . . 2.2.4 Miscellaneous Time and Date Functions . . . . . . .
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2.2.5 Creating timeSeries. . . . . . . . . . .” Objects 2.2.6 Aggregating and Disaggregating Time Series . . . . 2.2.7 Merging Time Series . . . . . . . . . . . . . . . . . . 2.2.8 Dealing with Missing Values Using the S+FinMetricsFunctioninterpNA. . . . . . . . . . Time Series Manipulation inSPLUS. . . . . . . . . . . . . 2.3.1 Creating Lags and Dierences . . . . . . . . . . . . . 2.3.2 Return Definitions . . . . . . . . . . . . . . . . . . . 2.3.3 Computing Asset Returns Using the S+FinMetricsFunctiongetReturns. . . . . . . . . Visualizing Time Series inSPLUS. . . . . . . . . . . . . . 2.4.1 Plotting timeSeries” Using theSPLUS GenericplotFunction . . . . . . . . . . . . . . . . . 2.4.2 Plotting timeSeries” Using theS+FinMetrics Trellis Plotting Functions . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time Series Concepts 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Univariate Time Series . . . . . . . . . . . . . . . . . . . 3.2.1 Stationary and Ergodic Time Series . . . . . . . 3.2.2 Linear Processes and ARMA Models . . . . . . . 3.2.3 Autoregressive Models . . . . . . . . . . . . . . . 3.2.4 Moving Average Models . . . . . . . . . . . . . . 3.2.5 ARMA(p,q) Models . . . . . . . . . . . . . . . . 3.2.6 Estimation of ARMA Models and Forecasting . . 3.2.7 Martingales and Martingale Dierence Sequences 3.2.8 Longrun Variance . . . . . . . . . . . . . . . . . 3.2.9 Variance Ratios . . . . . . . . . . . . . . . . . . . 3.3 Univariate Nonstationary Time Series . . . . . . . . . . 3.4 Long Memory Time Series . . . . . . . . . . . . . . . . . 3.5 Multivariate Time Series . . . . . . . . . . . . . . . . . . 3.5.1 Stationary and Ergodic Multivariate Time Series 3.5.2 Multivariate Wold Representation . . . . . . . . 3.5.3 Long Run Variance . . . . . . . . . . . . . . . . . 3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Unit Root Tests 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Testing for Nonstationarity and Stationarity . . . . . . . . . 4.3 Autoregressive Unit Root Tests . . . . . . . . . . . . . . . . 4.3.1 Simulating the DF and Normalized Bias Distributions . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Trend Cases . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 DickeyFuller Unit Root Tests . . . . . . . . . . . . .
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4.3.4 PhillipsPerron Unit Root Tests . . . . . . . . . . . . Stationarity Tests . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Simulating the KPSS Distributions . . . . . . . . . . 4.4.2 Testing for Stationarity Using theS+FinMetrics FunctionstationaryTest. . . . . . . . . . . . . . . Some Problems with Unit Root Tests . . . . . . . . . . . . . E. . . . . . . . . . . . . . . . . . .cient Unit Root Tests 4.6.1 Point Optimal Tests . . . . . . . . . . . . . . . . . . 4.6.2 DFGLS Tests . . . . . . . . . . . . . . . . . . . . . 4.6.3 Modified E. . . . . . . . . . . . . .cient PP Tests 2 4.6.4 Estimating. . . . . . . . . . . . . . . . . . . . . . 4.6.5 Choosing Lag Lengths to Achieve Good Size and Power . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Modeling Extreme Values 141 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2 Modeling Maxima and Worst Cases . . . . . . . . . . . . . . 142 5.2.1 The FisherTippet Theorem and the Generalized Extreme Value Distribution . . . . . . . . . . . . . . 143 5.2.2 Estimation of the GEV Distribution . . . . . . . . . 147 5.2.3 Return Level . . . . . . . . . . . . . . . . . . . . . . 153 5.3 Modeling Extremes Over High Thresholds . . . . . . . . . . 157 5.3.1 The Limiting Distribution of Extremes Over High Thresholds and the Generalized Pareto Distribution . . . . . . . . . . . . . . . . . . . . . . . 159 5.3.2 Estimating the GPD by Maximum Likelihood . . . . 164 5.3.3 Estimating the Tails of the Loss Distribution . . . . 165 5.3.4 Risk Measures . . . . . . . . . . . . . . . . . . . . . 171 5.4 Hill’s Nonparametric Estimator of Tail Index . . . . . . . . 174 5.4.1 Hill Tail and Quantile Estimation . . . . . . . . . . . 175 5.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Time Series Regression Modeling 181 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.2 Time Series Regression Model . . . . . . . . . . . . . . . . . 182 6.2.1 Least Squares Estimation . . . . . . . . . . . . . . . 183 6.2.2 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . 183 6.2.3 Hypothesis Testing . . . . . . . . . . . . . . . . . . . 184 6.2.4 Residual Diagnostics . . . . . . . . . . . . . . . . . . 185 6.3 Time Series Regression Using theS+FinMetrics FunctionOLS185. . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Dynamic Regression . . . . . . . . . . . . . . . . . . . . . . 201 6.4.1 Distributed Lags and Polynomial Distributed Lags . 205 6.4.2 Polynomial Distributed Lag Models . . . . . . . . . 207
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Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation . . . . . . . . . . . . . . . . . 6.5.1 The EickerWhite Heteroskedasticity Consistent (HC) Covariance Matrix Estimate . . . . . . . . . . 6.5.2 Testing for Heteroskedasticity . . . . . . . . . . . . . 6.5.3 The NeweyWest Heteroskedasticity and Autocorrelation Consistent (HAC) Covariance Matrix Estimate . . . . . . . . . . . . . . . . . . . . Recursive Least Squares Estimation . . . . . . . . . . . . . 6.6.1 CUSUM and CUSUMSQ Tests for Parameter Stability . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Computing Recursive Least Squares Estimates Using theS+FinMetricsFunctionRLS. . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Univariate GARCH Modeling 223 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.2 The Basic ARCH Model . . . . . . . . . . . . . . . . . . . . 224 7.2.1 Testing for ARCH E. . . . . . . . . . . . . . 228ects . 7.3 The GARCH Model and Its Properties . . . . . . . . . . . . 229 7.3.1 ARMA Representation of GARCH Model . . . . . . 230 7.3.2 GARCH Model and Stylized Facts . . . . . . . . . . 230 7.4 GARCH Modeling UsingS+FinMetrics. . . . . . . . . . . 232 7.4.1 GARCH Model Estimation . . . . . . . . . . . . . . 232 7.4.2 GARCH Model Diagnostics . . . . . . . . . . . . . . 235 7.5 GARCH Model Extensions . . . . . . . . . . . . . . . . . . 240 7.5.1 Asymmetric Leverage Eects and News Impact . . . 241 7.5.2 Two Components Model . . . . . . . . . . . . . . . . 247 7.5.3 GARCHintheMean Model . . . . . . . . . . . . . . 250 7.5.4 ARMA Terms and Exogenous Variables in Conditional Mean Equation . . . . . . . . . . . . . . 252 7.5.5 Exogenous Explanatory Variables in the Conditional Variance Equation . . . . . . . . . . . . 254 7.5.6 NonGaussian Error Distributions . . . . . . . . . . 257 7.6 GARCH Model Selection and Comparison . . . . . . . . . . 260 7.6.1 Constrained GARCH Estimation . . . . . . . . . . . 261 7.7 GARCH Model Prediction . . . . . . . . . . . . . . . . . . . 262 7.8 GARCH Model Simulation . . . . . . . . . . . . . . . . . . 265 7.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Long Memory Time Series Modeling 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Long Memory Time Series . . . . . . . . . . . . . . . . . . . 8.3 Statistical Tests for Long Memory . . . . . . . . . . . . . .
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8.3.1 R/S Statistic . . . . . . . . . . . . . . . . . . . . . . 8.3.2 GPH Test . . . . . . . . . . . . . . . . . . . . . . . . Estimation of Long Memory Parameter . . . . . . . . . . . 8.4.1 R/S Analysis . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Periodogram Method . . . . . . . . . . . . . . . . . . 8.4.3 Whittle’s Method . . . . . . . . . . . . . . . . . . . . Estimation of FARIMA and SEMIFAR Models . . . . . . . 8.5.1 Fractional ARIMA Models . . . . . . . . . . . . . . 8.5.2 SEMIFAR Model . . . . . . . . . . . . . . . . . . . . Long Memory GARCH Models . . . . . . . . . . . . . . . . 8.6.1 FIGARCH and FIEGARCH Models . . . . . . . . . 8.6.2 Estimation of Long Memory GARCH Models . . . . 8.6.3 Custom Estimation of Long Memory GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . Prediction from Long Memory Models . . . . . . . . . . . . 8.7.1 Prediction from FARIMA/SEMIFAR Models . . . . 8.7.2 Prediction from FIGARCH/FIEGARCH Models . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rolling Analysis of Time Series 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Rolling Descriptive Statistics . . . . . . . . . . . . . . . . . 9.2.1 Univariate Statistics . . . . . . . . . . . . . . . . . . 9.2.2 Bivariate Statistics . . . . . . . . . . . . . . . . . . . 9.2.3 Exponentially Weighted Moving Averages . . . . . . 9.2.4 Moving Average Methods for Irregularly Spaced High Frequency Data . . . . . . . . . . . . . . . . . 9.2.5 Rolling Analysis of Miscellaneous Functions . . . . . 9.3 Technical Analysis Indicators . . . . . . . . . . . . . . . . . 9.3.1 Price Indicators . . . . . . . . . . . . . . . . . . . . . 9.3.2 Momentum Indicators and Oscillators . . . . . . . . 9.3.3 Volatility Indicators . . . . . . . . . . . . . . . . . . 9.3.4 Volume Indicators . . . . . . . . . . . . . . . . . . . 9.4 Rolling Regression . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Estimating Rolling Regressions Using the S+FinMetricsFunctionrollOLS. . . . . . . . . . . 9.4.2 Rolling Predictions and Backtesting . . . . . . . . . 9.5 Rolling Analysis of General Models Using theS+FinMetrics Functionroll. . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10 Systems of Regression Equations 361 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 10.2 Systems of Regression Equations . . . . . . . . . . . . . . . 362 10.3 Linear Seemingly Unrelated Regressions . . . . . . . . . . . 364
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10.3.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Analysis of SUR Models with theS+FinMetrics FunctionSUR. . . . . . . . . . . . . . . . . . . . . . 10.4 Nonlinear Seemingly Unrelated Regression Models . . . . . 10.4.1 Analysis of Nonlinear SUR Models with the S+FinMetricsFunctionNLSUR. . . . . . . . . . . . 10.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11 Vector Autoregressive Models for Multivariate Time Series 385 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 11.2 The Stationary Vector Autoregression Model . . . . . . . . 386 11.2.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . 388 11.2.2 Inference on Coecients . . . . . . . . . . . . . . . . 390 11.2.3 Lag Length Selection . . . . . . . . . . . . . . . . . . 390 11.2.4 Estimating VAR Models Using theS+FinMetrics FunctionVAR390. . . . . . . . . . . . . . . . . . . . . . 11.3 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 11.3.1 Traditional Forecasting Algorithm . . . . . . . . . . 398 11.3.2 SimulationBased Forecasting . . . . . . . . . . . . . 402 11.4 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . 406 11.4.1 Granger Causality . . . . . . . . . . . . . . . . . . . 407 11.4.2 Impulse Response Functions . . . . . . . . . . . . . . 409 11.4.3 Forecast Error Variance Decompositions . . . . . . . 414 11.5 An Extended Example . . . . . . . . . . . . . . . . . . . . . 416 11.6 Bayesian Vector Autoregression . . . . . . . . . . . . . . . . 424 11.6.1 An Example of a Bayesian VAR Model . . . . . . . 424 11.6.2 Conditional Forecasts . . . . . . . . . . . . . . . . . 427 11.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
12 Cointegration 431 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 12.2 Spurious Regression and Cointegration . . . . . . . . . . . . 432 12.2.1 Spurious Regression . . . . . . . . . . . . . . . . . . 432 12.2.2 Cointegration . . . . . . . . . . . . . . . . . . . . . . 435 12.2.3 Cointegration and Common Trends . . . . . . . . . . 437 12.2.4 Simulating Cointegrated Systems . . . . . . . . . . . 437 12.2.5 Cointegration and Error Correction Models . . . . . 441 12.3 ResidualBased Tests for Cointegration . . . . . . . . . . . . 444 12.3.1 Testing for Cointegration When the Cointegrating Vector Is Prespecified . . . . . . . . . . . . . . . . . 444 12.3.2 Testing for Cointegration When the Cointegrating Vector Is Estimated . . . . . . . . . . . . . . . . . . 447 12.4 RegressionBased Estimates of Cointegrating Vectors and Error Correction Models . . . . . . . . . . . . . . . . . . . . 450
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12.4.1 Least Square Estimator . . . . . . . . . . . . . . . . 12.4.2 Stock and Watson’s E.cient Lead/Lag Estimator 12.4.3 Estimating Error Correction Models by Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 VAR Models and Cointegration . . . . . . . . . . . . . . . . 12.5.1 The Cointegrated VAR . . . . . . . . . . . . . . . . 12.5.2 Johansen’s Methodology for Modeling Cointegration . . . . . . . . . . . . . . . . . . . . . . 12.5.3 Specification of Deterministic Terms . . . . . . . . . 12.5.4 Likelihood Ratio Tests for the Number of Cointegrating Vectors . . . . . . . . . . . . . . . . . 12.5.5 Testing Hypothesis on Cointegrating Vectors Using theS+FinMetricsFunctioncoint. . . . . . 12.5.6 Maximum Likelihood Estimation of the Cointegrated VECM . . . . . . . . . . . . . . . . . . 12.5.7 Maximum Likelihood Estimation of the Cointegrated VECM Using theS+FinMetrics FunctionVECM. . . . . . . . . . . . . . . . . . . . . 12.5.8 Forecasting from the VECM . . . . . . . . . . . . . 12.6 Appendix: Maximum Likelihood Estimation of a Cointegrated VECM . . . . . . . . . . . . . . . . . . . . . . 12.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 Multivariate GARCH Modeling 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 13.2 Exponentially Weighted Covariance Estimate . . . . 13.3 Diagonal VEC Model . . . . . . . . . . . . . . . . . . 13.4 Multivariate GARCH Modeling inS+FinMetrics. . 13.4.1 Multivariate GARCH Model Estimation . . . 13.4.2 Multivariate GARCH Model Diagnostics . . . 13.5 Multivariate GARCH Model Extensions . . . . . . . 13.5.1 MatrixDiagonal Models . . . . . . . . . . . . 13.5.2 BEKK Models . . . . . . . . . . . . . . . . . 13.5.3 Univariate GARCHbased Models . . . . . . 13.5.4 ARMA Terms and Exogenous Variables . . . 13.5.5 Multivariate Conditional tDistribution . . . 13.6 Multivariate GARCH Prediction . . . . . . . . . . . 13.7 Custom Estimation of GARCH Models . . . . . . . . 13.7.1 GARCH Model Objects . . . . . . . . . . . . 13.7.2 Revision of GARCH Model Estimation . . . . 13.8 Multivariate GARCH Model Simulation . . . . . . . 13.9 References . . . . . . . . . . . . . . . . . . . . . . . .
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14 State Space Models 519 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
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14.2 State Space Representation . . . . . . . . . . . . . . . . . . 14.2.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . 14.2.2 State Space Representation in S+FinMetrics/SsfPack. . . . . . . . . . . . . . . . 14.2.3 Missing Values . . . . . . . . . . . . . . . . . . . . . 14.2.4S+FinMetrics/SsfPackFunctions for Specifying the State Space Form for Some Common Time Series Models . . . . . . . . . . . . . . . . . . . . . . 14.2.5 Simulating Observations from the State Space Model . . . . . . . . . . . . . . . . . . . . . . 14.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Kalman Smoother . . . . . . . . . . . . . . . . . . . 14.3.3 Smoothed State and Response Estimates . . . . . . 14.3.4 Smoothed Disturbance Estimates . . . . . . . . . . . 14.3.5 Forecasting . . . . . . . . . . . . . . . . . . . . . . . 14.3.6S+FinMetrics/SsfPackImplementation of State Space Modeling Algorithms . . . . . . . . . . . . . . 14.4 Estimation of State Space Models . . . . . . . . . . . . . . . 14.4.1 Prediction Error Decomposition of LogLikelihood . . . . . . . . . . . . . . . . . . . . . 14.4.2 Fitting State Space Models Using the S+FinMetrics/SsfPackFunctionSsfFit. . . . . . 14.4.3 QuasiMaximum Likelihood Estimation . . . . . . . 14.5 Simulation Smoothing . . . . . . . . . . . . . . . . . . . . . 14.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15 Factor Models for Asset Returns 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . 15.2 Factor Model Specification . . . . . . . . . . . . . 15.3 Macroeconomic Factor Models for Returns . . . . 15.3.1 Sharpe’s Single Index Model . . . . . . . 15.3.2 The General Multifactor Model . . . . . . 15.4 Fundamental Factor Model . . . . . . . . . . . . 15.4.1 BARRAtype Single Factor Model . . . . 15.4.2 BARRAtype Industry Factor Model . . . 15.5 Statistical Factor Models for Returns . . . . . . . 15.5.1 Factor Analysis . . . . . . . . . . . . . . . 15.5.2 Principal Components . . . . . . . . . . . 15.5.3 Asymptotic Principal Components . . . . 15.5.4 Determining the Number of Factors . . . 15.6 References . . . . . . . . . . . . . . . . . . . . . .
16 Term Structure of Interest Rates 16.1 Introduction . . . . . . . . . . . . . . . .
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16.2 Discount, Spot and Forward Rates . . . . . . . . . . . . . . 16.2.1 Definitions and Rate Conversion . . . . . . . . . . . 16.2.2 Rate Conversion inS+FinMetrics. . . . . . . . . . 16.3 Quadratic and Cubic Spline Interpolation . . . . . . . . . . 16.4 Smoothing Spline Interpolation . . . . . . . . . . . . . . . . 16.5 NelsonSiegel Function . . . . . . . . . . . . . . . . . . . . . 16.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17 Robust Change Detection 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 17.2 REGARIMA Models . . . . . . . . . . . . . . . . . . . 17.3 Robust Fitting of REGARIMA Models . . . . . . . . . 17.4 Prediction Using REGARIMA Models . . . . . . . . . 17.5 Controlling Robust Fitting of REGARIMA Models . . 17.5.1 Adding Seasonal E. . . . . . . . . . . .ects . 17.5.2 Controlling Outlier Detection . . . . . . . . . . 17.5.3 Iterating the Procedure . . . . . . . . . . . . . 17.6 Algorithms of Filtered. . . . . . . . .Estimation . 17.6.1 Classical Maximum Likelihood Estimates . . . 17.6.2 Filtered. . . . . . . . . . . . . .Estimates . 17.7 References . . . . . . . . . . . . . . . . . . . . . . . . .
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18 Nonlinear Time Series Models 653 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 18.2 BDS Test for Nonlinearity . . . . . . . . . . . . . . . . . . . 654 18.2.1 BDS Test Statistic . . . . . . . . . . . . . . . . . . . 655 18.2.2 Size of BDS Test . . . . . . . . . . . . . . . . . . . . 655 18.2.3 BDS Test as a Nonlinearity Test and a Misspecification Test . . . . . . . . . . . . . . . . . . 657 18.3 Threshold Autoregressive Models . . . . . . . . . . . . . . . 662 18.3.1 TAR and SETAR Models . . . . . . . . . . . . . . . 663 18.3.2 Tsay’s Approach . . . . . . . . . . . . . . . . . . . . 664 18.3.3 Hansen’s Approach . . . . . . . . . . . . . . . . . . . 671 18.4 Smooth Transition Autoregressive Models . . . . . . . . . . 678 18.4.1 Logistic and Exponential STAR Models . . . . . . . 678 18.4.2 Test for STAR Nonlinearity . . . . . . . . . . . . . . 680 18.4.3 Estimation of STAR Models . . . . . . . . . . . . . . 683 18.5 Markov Switching State Space Models . . . . . . . . . . . . 687 18.5.1 Discrete State Markov Process . . . . . . . . . . . . 688 18.5.2 Markov Switching AR Process . . . . . . . . . . . . 690 18.5.3 Markov Switching State Space Models . . . . . . . . 691 18.6 An Extended Example: Markov Switching Coincident Index 701 18.6.1 State Space Representation of Markov Switching Co incident Index Model . . . . . . . . . . . . . . . . . . 702
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18.6.2 Approximate MLE of Markov Switching Coincident Index . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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19 Copulas 713 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 19.2 Motivating Example . . . . . . . . . . . . . . . . . . . . . . 714 19.3 Definitions and Basic Properties of Copulas . . . . . . . . . 722 19.3.1 Properties of Distributions . . . . . . . . . . . . . . 722 19.3.2 Copulas and Sklar’s Theorem . . . . . . . . . . . . . 724 19.3.3 Dependence Measures and Copulas . . . . . . . . . . 726 19.4 Parametric Copula Classes and Families . . . . . . . . . . . 729 19.4.1 Normal Copula . . . . . . . . . . . . . . . . . . . . . 729 19.4.2 Normal Mixture Copula . . . . . . . . . . . . . . . . 730 19.4.3 Extreme Value Copula Class . . . . . . . . . . . . . 730 19.4.4 Archimedean Copulas . . . . . . . . . . . . . . . . . 732 19.4.5 Archimax Copulas . . . . . . . . . . . . . . . . . . . 735 19.4.6 Representation of Copulas inS+FinMetrics735. . . . . 19.4.7 Creating Arbitrary Bivariate Distributions . . . . . . 743 19.4.8 Simulating from Arbitrary Bivariate Distributions . 745 19.5 Fitting Copulas to Data . . . . . . . . . . . . . . . . . . . . 747 19.5.1 Empirical Copula . . . . . . . . . . . . . . . . . . . . 747 19.5.2 Maximum Likelihood Estimation . . . . . . . . . . . 750 19.5.3 Fitting Copulas Using theS+FinMetrics/EVANESCE Functionfit.copula. . . . . . . . . . . . . . . . . 751 19.6 Risk Management Using Copulas . . . . . . . . . . . . . . . 754 19.6.1 Computing Portfolio Risk Measures Using Copulas . 754 19.6.2 Computing VaR and ES by Simulation . . . . . . . . 755 19.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757
20 ContinuousTime Models for Financial Time Series 759 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 20.2 SDEs: Background . . . . . . . . . . . . . . . . . . . . . . . 760 20.3 Approximating Solutions to SDEs . . . . . . . . . . . . . . 761 20.4S+FinMetrics765. . . . . . . . . Functions for Solving SDEs . 20.4.1 ProblemSpecific Simulators . . . . . . . . . . . . . . 765 20.4.2 General Simulators . . . . . . . . . . . . . . . . . . . 771 20.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782
21 Generalized Method of Moments 785 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 21.2 Single Equation Linear GMM . . . . . . . . . . . . . . . . . 786 21.2.1 Definition of the GMM Estimator . . . . . . . . . . 787 21.2.2 Specification Tests in Overidentified Models . . . . . 791