Approximation bias in linearized Euler equations

This paper concerns pitfalls associated with the use of approximations to dynamic Euler equations. Two applications of the approximations are notable. First, tests for precautionary saving motives typically involve regressing consumption growth on uncertainty in expected consumption growth. The parameter estimates are used to measure the strength of precautionary motives, which is also related to the coefficient of relative risk aversion. Another application estimates the sensitivity of consumption growth to the expected real interest rate, with the coefficient on the latter equal to the intertemporal elasticity of substitution in consumption, often the inverse of the coefficient of relative risk aversion. The two literatures yield very different estimates of how prudent or risk averse consumers are or, alternatively, how willing they are to substitute consumption over time. We investigate one possible reason for these apparently contradictory results: both methods of estimation rely on linear approximations of Euler equations. We demonstrate that biases associated with these approximations can be substantial, and that the direction of the biases is consistent with the divergent estimates found in the literature.