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A Non-Gaussian Asymmetric Volatility Model

A Non-Gaussian Asymmetric Volatility Model. Geert Bekaert Columbia University and NBER Eric Engstrom Federal Reserve Board of Governors* * The views expressed herein do not necessarily reflect those of the Board of Governors of Federal Reserve System, or its staff. Overview.

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A Non-Gaussian Asymmetric Volatility Model

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  1. A Non-Gaussian Asymmetric Volatility Model Geert Bekaert Columbia University and NBER Eric Engstrom Federal Reserve Board of Governors* * The views expressed herein do not necessarily reflect those of the Board of Governors of Federal Reserve System, or its staff.

  2. Overview • We extend asymmetric volatility models in the GARCH class • accommodates time-varying skewness, kurtosis, and tail behavior • provides simple, closed-form expressions for higher order conditional moments • outperforms a wide set of extant models in an application to equity return data

  3. Motivation

  4. Standard GARCH • The Glosten, Jagannathan, and Runkle (1993) extension of GARCH (GJR-GARCH) has been found to fit stock return data quite well • Engle and Ng (1993)

  5. Our Extension • First, we define the “BEGE” distribution

  6. CenteredGamma Distributions

  7. Examples of the BEGE Density

  8. Examples of the BEGE Density

  9. Examples of the BEGE Density

  10. Examples of the BEGE Density

  11. Reasonable Acronym? Bad Environment Good Environment

  12. Narcissistic? Bekaert Engstrom Geert Eric

  13. Bee Gee Wannabes?

  14. Moments under BEGE • Simple, closed-form solutions

  15. Embed BEGE inGJR-GARCH • Shape parameters follow GJR GARCH-like process

  16. Application • Monthly (log) stock return data 1926-2010 • Estimate by maximum likelihood • Compare performance of a variety of models • Standard GARCH (Gaussian and Student t) • GJR-GARCH (Gaussian and Student t) • Regime switching models (2,3 states, with and without “jumps”) • BEGE GJR GARCH (including restricted versions)

  17. Comparing Models:Information Criteria • BEGE also dominates in a variety of other tests

  18. BEGE: Filtered Series

  19. BEGE: Impact Curves

  20. Out of Sample Test: VIX • The VIX index is the one-month ahead volatility of the stock market implied by equity option prices under the Q-measure.

  21. VIX Hypotheses • Assume that investors have CRRA utility with respect to stock market wealth

  22. VIX versus Vol

  23. VIX Test Results • Regression (1990-2012, monthly) • Orthogonality test

  24. Portfolio Application • An investor invests, period-by-period, in the risk free rate and the stock market. The portfolio return is

  25. Risk Management • GJR weights are more aggressive • GJR: “1 percent” VaR breached in 15 of 1050 periods • BEGE: 1 percent VaR breached in 10 of 1050 periods

  26. Macroeconomic Series Slowdown = four quarter MA < 1% (annual)

  27. Monetary Policy • Should policymakers care about upside versus downside risks to real growth or inflation? • standard “loss function” suggests maybe not • But • typically arises from a second order approximation to agents’ utility function. Why not third order? • is it plausible? • evidence of asymmetries in reaction functions (Dolado, Maria-Dolores, Naveira (2003))

  28. Conclusion • The BEGE distribution in a GARCH setting • Accommodates time-varying tail risk behavior in a tractable fashion • Fits historical return data better than some models • Helps explain observed option prices • Applications to macroeconomic time series analysis, term structure modeling, and monetary policy are planned.

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