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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.
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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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Centered Gamma Distributions
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Examples of the BEGE Density
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Reasonable Acronym? B ad E nvironment G ood E nvironment
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Narcissistic? B ekaert E ngstrom G eert E ric
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Bee Gee Wannabes?
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Moments under BEGE Simple, closed-form solutions
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Embed BEGE in GJR-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) 0.00011.15 (0.0001)(0.03) Orthogonality test -0.590.15 (0.15)(0.04)
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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.
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