Engle-Lilien-Robins (1987) Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model

Summary

This paper introduces the ARCH-in-Mean (ARCH-M) model, which extends Engle's (1982) ARCH by letting the conditional variance enter the conditional mean — so that a time-varying second moment drives a time-varying risk premium. Motivated by the risk-return trade-off in finance, the authors postulate that the expected excess holding yield on a long bond depends on its conditional variance, estimate the model on three interest-rate datasets, and find both the ARCH process and the time-varying risk premium highly significant. The framework offers an econometric explanation for the documented failures of the expectations hypothesis of the term structure: apparent predictability of excess returns reflects a time-varying risk premium, not irrationality.

Key Claims

• The ARCH-M model. The mean equation carries a function of the conditional variance, $y_t=\beta'x_t+\delta\,g(h_t)+\varepsilon_t$, with $\varepsilon_t\mid\mathcal F_{t-1}\sim\mathcal N(0,h_t)$ and $h_t$ an ARCH process; common choices are $g(h_t)=h_t$, $\sqrt{h_t}$, or $\ln h_t$. The coefficient $\delta$ is the price of risk: expected return rises with conditional risk.
• A time-varying risk premium. Because $h_t$ moves over time, so does the risk premium $\delta\,g(h_t)$ — the model measures risk and its movement and feeds it into price within one likelihood, which static risk measures cannot do.
• Term-structure application. Applied to excess long-bond holding yields, the estimated $\delta$ is positive and significant: periods of high conditional volatility carry higher expected excess returns.
• Rescues the expectations hypothesis. The regression evidence against the expectations hypothesis — excess returns being forecastable from current information — is reinterpreted as a time-varying premium rather than irrational expectations; earlier attempts to model irrational expectations had failed.
• Estimation and diagnostics. Joint maximum-likelihood estimation of the mean and variance parameters, with a battery of Lagrange-multiplier (LM) diagnostic tests showing robustness to alternative volatility measures, regime changes, and interest-rate formulations.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH
• Risk-Return Tradeoff

Entities Mentioned

• Robert Engle
• David M. Lilien
• Russell P. Robins

Quotes

"Engle's (1982) ARCH model is extended to allow the conditional variance to be a determinant of the mean and is called ARCH-M."

"Any increase in the expected rate of return of an asset as it becomes more risky will be identified as a risk premium."

My Take

ARCH-M is the model that connected Engle's volatility machinery to asset pricing: ARCH gave you the conditional variance, but ARCH-M made that variance pay by putting it in the mean, which is exactly the object finance cares about — the compensation for bearing time-varying risk. Its cleverest move is interpretive: the "failure" of the expectations hypothesis becomes evidence for a sensible risk premium once you allow the premium to move with volatility. For the wiki it is the mean-side counterpart to Bollerslev's variance-side extensions of GARCH and the empirical engine behind the risk-return tradeoff page. The enduring caveat is fragility: the sign and significance of $\delta$ are notoriously sensitive to the volatility functional form ($h_t$ vs. $\sqrt{h_t}$ vs. $\ln h_t$) and to the conditional-mean specification, which is why the later risk-return literature keeps finding conflicting signs.