Brandt-Kang (2004) On the Relationship between the Conditional Mean and Volatility of Stock Returns: A Latent VAR Approach

Summary

Models the conditional mean and volatility of stock returns as two jointly evolving latent state variables following a bivariate vector autoregression, VAR(1), in log scale, without relying on exogenous predictors. Estimated by simulated maximum likelihood (SML) with an importance-sampling correction. Key finding: the contemporaneous correlation between mean and volatility innovations is strongly negative ($\hat{\rho} \approx -0.56$, $t = -5.80$), while the lag effects (volatility predicting future mean, mean predicting future volatility) are weak and insignificant. Both moments are countercyclical and peak at recession troughs.

Key Claims

• The literature's contradictory findings on the risk-return tradeoff arise from conflating the contemporaneous correlation $\rho$ with the lag coefficient $a_{12}$ (lag volatility → mean); reduced-form GARCH-in-mean (GARCH-M) models cannot cleanly separate the two.
• The contemporaneous innovation correlation $\rho \approx -0.56$ is strongly negative; a positive mean innovation is simultaneously associated with a drop in volatility.
• The lag risk-return tradeoff ($a_{12}$) is economically small and statistically insignificant.
• The unconditional correlation between conditional mean and volatility is positive — both peak in recession troughs, explaining positive risk-return correlations found in cross-sectional or long-run studies.
• Adding exogenous predictors (short rate, term premium, default premium) reduces $|\rho|$ to $\approx -0.45$ but the negative contemporaneous correlation remains highly significant.

Concepts Introduced or Extended

• Risk-Return Tradeoff
• Stochastic Volatility

Entities Mentioned

• Michael W. Brandt
• Qiang Kang

Quotes

"We find a strong and robust negative contemporaneous correlation between innovations to the conditional mean and volatility... The negative unconditional correlation is consistent with the notion of habit formation." (p. 218, paraphrase)

My Take

The latent VAR design is elegant: by keeping both moments latent and estimating them jointly, it sidesteps the misspecification inherent in reduced-form approaches. The SML estimator is well-motivated and the simulation study (Table 1) is reassuring. The key empirical finding — strongly negative contemporaneous correlation, essentially no lag tradeoff — is striking and well-identified. The limitation is that the exponential functional form for the mean and volatility ($\bar{\mu}e^{m_{t-1}}$, $\bar{\sigma}e^{v_{t-1}}$) is a modeling choice, and alternative parameterizations could yield different dynamics. The paper also does not deliver a structural interpretation for the negative $\rho$ — it documents but does not explain the mechanism.