a non-gaussian asymmetric volatility model
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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.
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
Motivation
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)
Our Extension
• First, we define the “BEGE” distribution
CenteredGamma Distributions
Examples of the BEGE Density
Examples of the BEGE Density
Examples of the BEGE Density
Examples of the BEGE Density
Reasonable Acronym?
Bad
Environment
Good
Environment
Narcissistic?
Bekaert
Engstrom
Geert
Eric
Bee Gee Wannabes?
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
Embed BEGE inGJR-GARCH
• Shape parameters follow GJR GARCH-like process
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)
Comparing Models:Information Criteria
• BEGE also dominates in a variety of other tests
BEGE: Filtered Series
BEGE: Impact Curves
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.
VIX Hypotheses
• Assume that investors have CRRA utility with respect to stock market wealth
VIX versus Vol
VIX Test Results
• Regression (1990-2012, monthly)
• Orthogonality test
Portfolio Application
• An investor invests, period-by-period, in the risk free rate and the stock market. The portfolio return is
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
Macroeconomic Series
Slowdown = four quarter MA < 1% (annual)
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))
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.