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.