¹²By including the assets in the estimated equation, we follow the literature on precautionary saving and liquidity constraints. The alternative justification for including $a$ is as a proxy for expected interest rates $Rt+1$ ; calibrated general equilibrium models imply that the relationship between $at$ and $Et[Rt+1]$ is very close to linear. If such models are a good way of interpreting the data, the $a$ term should therefore capture the interest rate effects implied by the theory. However, empirical estimates of Euler equations using macro data generally produce insignificant (or even implausible) coefficients on expected interest rates (see, e.g., Hall (1988) and table 3 of Campbell and Mankiw (1991); and Vissing-Jørgensen (2002) for evidence in micro consumption data).