Cross-variable restrictions in Euler equations and risk premia

Hansen and Singleton (1982) argue in a seminal paper that it is su cient to test the ® rst-order Euler equations implied by an economic model without having to solve for all endogenous variables. Many interesting economic models cannot be solved analytically, so restricting oneself to the class of solvable models would unnecessarily limit the scope of economic analysis. This paper argues that, while the argument in Hansen and Singleton is appreciated, just focusing on the Euler equation can lead to misleading results. Two examples are used to make this point: preference speci® cations suggested by Bakshi and Chen (1996) and Epstein and Zin (1989) in otherwise standard models. Bakshi and Chen (BC) (1996) argue that augmenting standard preferences by allowing wealth to enter directly the utility function helps to explain risk premia in capital markets. The motivation for this generalization comes from the observation that utility might not only be generated by consumption but also by the social status implied by high wealth. The most straightforward way to include this notation into neoclassical economics is to modify the utility function. BC propose various functional forms of preferences which have consumption, wealth and a socialwealth index as arguments. They focus on the implications of these preferences on risk premia by studying the volatility bounds of the intertemporal marginal rate of substitution (IMRS) as suggested by Hansen and Jagannathan (1991). They conclude that their preference speci® cation enter the volatility bounds for more reasonable parameter values than the standard CRRA utility function. In particular, a relative risk aversion coe cient of about 10 is su cient compared to about 100 for CRRA preferences. This paper argues that this conclusion can be misleading because it does not take into account all restrictions among variables implied by the model. In particular, the authors treat consumption and wealth as independent variables when they compute the Hansen± Jagannathan bounds of the IMRS. However, one of the implications of the model is that the consumption± wealth ratio is constant and innovations in consumption and wealth are perfectly correlated. Once these restrictions are used to substitute out wealth from the IMRS, the same IMRS is obtained as in the standard CRRA speci® cation (apart from a different discount factor) in which only consumption enters the IMRS. This implies that the preferences studied in BC have the same implications for risk premia as the standard CRRA preferences. For example, a high risk aversion coef® cient is needed to enter the Hansen± Jagannathan bounds. Hence, just focusing on the Euler equation as suggested by Hansen and Singleton (1982) can severely overstate the volatility of the IMRS implied by the model. The same issue arises in the context of Epstein± Zin (1989) preferences. Here, the Euler equation included consumption growth as well as the return on the market portfolio. However, just as in the Bakshi± Chen model there is a tight connection between these variables. If returns are i.i.d. consumption growth is perfectly correlated with the market return. Hence one variable can be used to substitute out the other and one obtains the original CRRA Euler equation. This has been shown by Kocherlakota (1990),