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. Centered Gamma Distributions

7. Examples of the BEGE Density

11. Reasonable Acronym? B ad E nvironment G ood E nvironment

12. Narcissistic? B ekaert E ngstrom G eert E ric

13. Bee Gee Wannabes?

14. Moments under BEGE Simple, closed-form solutions

15. Embed BEGE in GJR-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) 0.00011.15 (0.0001)(0.03) Orthogonality test -0.590.15 (0.15)(0.04)

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.