A kev implication of Consumption CAPM models, therefore, is that equilibrium returns are determined by a single factor: the IMRS. If there are complete markets and common informatioasets across all consumers, observed asset returns are sufficient to identify the IMRS. When either of these assumptions is violated one can derive relationships, implied by the observed asset returns, that impose restrictions on the unconditional moments of the IMRS. Hansen and Jagannathan (1991) derive bounds on the mean and variance of the benchmark portfolio from observed data on asset returns.
An alternative is to take a more structural approach based on the specification of a utility function. The orthogonality conditions implied by equation (3) or (4) for different assets can be used to estimate preference parameters in (1) and, provided the model is overidentified, test the over-identifying assumptions. This was the approach followed, for instance, by Hansen and Singleton (1982, 1983) who estimated several versions of equation (4) using aggregate time series data. Source
Equation (3) and (4) involve non-linear relationships. As we are using a synthetic panel approach and we want to allow for measurement error, we prefer to deal with relationships that are linear in parameters. Therefore, we log-linearize the Euler equation (3) under the assumption of CRRA (or iso-elastic) preferences to obtain:
where £^+] is a term that includes expectational errors and changes in unobserved heterogeneity, у is the inverse of the coefficient of relative risk aversion and kj is a term including the log of the discount factor as well as conditional higher moments of the return on asset j and of consumption growth (such as variances and covariances). If one assumes log normality of consumption growth and asset returns, kj is given by the following expression:
where the subscripts t indicate that the variance and covariances are conditional on the information available at time t. If there are instruments that are uncorrelated with £Jt+l and with the innovations to kj, the parameters of equation (5) can be estimated using GMM techniques.