A Non-Gaussian Asymmetric Volatility Model
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A Non-Gaussian Asymmetric Volatility Model
Geert BekaertColumbia University and NBER
Eric EngstromFederal 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.
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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
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Motivation
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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)
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Our Extension
• First, we define the “BEGE” distribution
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CenteredGamma Distributions
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Examples of the BEGE Density
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Examples of the BEGE Density
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Examples of the BEGE Density
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Examples of the BEGE Density
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Reasonable Acronym?
Bad
Environment
Good
Environment
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Narcissistic?
Bekaert
Engstrom
Geert
Eric
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Bee Gee Wannabes?
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Moments under BEGE
• Simple, closed-form solutions
2 2 21
3 3 31
4 4 4 2 21 1
1
2
6 3
t t p t n t
t t p t n t
t t p t n t t t
E u p n
E u p n
E u p n E u
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Embed BEGE inGJR-GARCH
• Shape parameters follow GJR GARCH-like process
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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)
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Comparing Models:Information Criteria
• BEGE also dominates in a variety of other tests
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BEGE: Filtered Series
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BEGE: Impact Curves
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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.
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VIX Hypotheses
• Assume that investors have CRRA utility with respect to stock market wealth
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VIX versus Vol
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VIX Test Results
• Regression (1990-2012, monthly)
• Orthogonality test
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Portfolio Application
• An investor invests, period-by-period, in the risk free rate and the stock market. The portfolio return is
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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
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Macroeconomic Series
Slowdown = four quarter MA < 1% (annual)
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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))
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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.