This paper proposes an approach to estimating the relation between risk (conditional variance) and expected returns in the aggregate stock market that allows us to escape some of the limitations of existing empirical analyses. First, we focus on a nonparametric volatility measure that is void of any specific functional form assumptions about the stochastic process generating returns. Second, we offer a solution to the error-in-variables problem that arises because of the use of a proxy for the volatility in estimating the risk-return relation. Third, our estimation strategy involves the Generalized Method of Moments approach that overcomes the endogeneity problem in a least squares regression of an estimate of the conditional mean on the corresponding estimate of the conditional variance, that arises because both the above quantities are endogenously determined within a general equilibrium asset pricing model. Finally, we use our approach to assess the plausibility of the prominent Long Run Risks asset pricing models studied in the literature based on the restrictions that they imply on the time series properties of expected returns and conditional variances of market aggregates.
