Glosten-Jagannathan-Runkle (1993) On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks

Summary

Estimates the intertemporal risk-return relation for monthly U.S. stock returns using a modified GARCH-in-mean model — the specification now known as GJR-GARCH. Three modifications to the standard GARCH-M matter: (1) seasonal (monthly) patterns in volatility, (2) an asymmetric term letting positive and negative return innovations have different impacts on conditional variance, and (3) the nominal interest rate as a predictor of conditional variance. With these, the authors find a negative relation between conditional expected return and conditional variance, and that positive return surprises lower future volatility while negative surprises raise it (the leverage/asymmetry effect). They also find conditional volatility is less persistent than earlier symmetric-GARCH estimates suggested.

Key Claims

• The GJR asymmetric term. Adding a term $\gamma\, I_{t-1}\varepsilon_{t-1}^2$ (with $I_{t-1}=1$ when $\varepsilon_{t-1}<0$) to the GARCH variance equation lets bad news ($\varepsilon_{t-1}<0$) raise conditional variance by more than good news of equal size — a parsimonious model of the leverage effect that became a standard GARCH variant.
• Negative risk-return relation. In the modified GARCH-M, the estimated coefficient on conditional variance in the mean is negative: higher conditional variance is associated with lower conditional expected excess return — contrary to the simple positive risk-return tradeoff.
• Interest rates carry volatility information. Including the nominal short rate as a conditioning variable for the variance improves the model and matters for the risk-return estimate.
• Volatility less persistent than thought. Once asymmetry and the interest-rate/seasonal terms are included, estimated volatility persistence falls relative to symmetric GARCH.
• Seasonality in volatility. Monthly seasonal dummies in the variance equation are needed to avoid biasing the other coefficients.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH
• Risk-Return Tradeoff

Entities Mentioned

• Lawrence R. Glosten
• Ravi Jagannathan
• David E. Runkle

Quotes

"We find support for a negative relation between conditional expected monthly return and conditional variance of monthly return, using a GARCH-M model modified by allowing (1) seasonal patterns in volatility, (2) positive and negative innovations to returns having different impacts on conditional volatility, and (3) nominal interest rates to predict conditional variance."

My Take

Two lasting legacies from one paper. First, the GJR term — a single sign-indicator interaction on the lagged squared innovation — is the most-used parametric model of asymmetric volatility after EGARCH, precisely because it is so cheap to bolt onto standard GARCH; the GARCH page's news-impact-curve discussion (Engle-Ng 1993 rank GJR the best parametric fit on TOPIX) is downstream of it. Second, the negative in-mean coefficient sharpened the risk-return sign puzzle that French-Schwert-Stambaugh (1987) had opened and that Brandt-Kang and Bollerslev-Zhou later resolved: GJR is the canonical "negative GARCH-M" result, and its negativity is exactly the contemporaneous-leverage contamination those structural papers diagnose. The methodological care — seasonal terms, interest-rate conditioning, asymmetry all at once — is why the negative finding was taken seriously rather than dismissed as a specification artefact.