Bansal and Yaron (2004) demonstrate, by calibration, that the Consumption-based Capital Asset Pricing Model (CCAPM) can be rescued by assuming that consumption growth rate follows a stochastic volatility model. They show that the conditional equity premium is a linear function of conditional consumption and market return volatilities, which can be estimated handily by various Generalized Autoregressive Conditonal Heteroskedasticity (GARCH) and Stochastic Volatility (SV) models. We find that conditional consumption and market volatilities are capable of explaining cross-sectional return differences. The Exponential GARCH (EGARCH) volatility can explain up to 55% variation of return and the EGARCH model augmented with cay^ — a cointegrating factor of consumption, labor income and asset wealth growth — greatly enhances model performance. We proceed to test another hypothesis: if Bansal and Yaron estimator is an unbiased estimator of true conditional equity premium, then the instrumental variables for estimating conditional equity premium should no longer be significant. We demonstrate that once the theoretical conditional risk premium is added to the model, it renders all instrumental variables redundant. Also, the model prediction is consistent with observed declining equity premium.
Fung, K. W. T., Lau, C. K. M., & Chan, K. H. (2014). The Conditional Equity Premium, Cross-Sectional Returns and Stochastic Volatility. Economic Modelling, 38, 316-327. https://doi.org/10.1016/j.econmod.2014.01.009