Model uncertainty: the Economist and the Evil agent

Méphistophélès in Faust’s study
Eugène Delacroix

Introducing  “Doubts and Variability” by Rhys Bidder (Federal Reserve Bank of San Francisco) and Matthew E. Smith (Federal Reserve Board). Working Paper (First Draft Sept 4, 2010 – current version Aug 4, 2011).

One of the most enduring puzzles in the macrofinance literature is the equity premium puzzle. A manifestation of this puzzle is the difficulty of designing a model that simultaneously generates a substantial market price of risk and a low risk free rate, while also respecting stylized facts regarding consumption dynamics, as discussed by Hansen and Singleton (1982) and Mehra and Prescott (1985).

In this paper Bidder and Smith show that the interaction of stochastic volatility in consumption with a fear of model misspecification can bring new pieces to the puzzle.

The agent in their heteroskedastic (with time-varying variance) endowment economy does not fully trust the joint conditional distribution of the volatility process and consumption series. The agent acknowledges that the model is an approximation to the true data generating process and fears it is misspecified in some way. These fears are expressed through alternative models, or probability distributions, that are distorted versions of the distribution implied by the approximating model and which the agent thinks may be generating the consumption and consumption volatility series. The agent's preference for robustness can be expressed using a two player game between the agent and his alter-ego, or a metaphorical `evil agent', who attempts to minimize his utility by choosing a worst case, probability distribution. The strength of the agent's preference for robustness is captured by a parameter θ which determines how strongly the evil agent is penalized for choosing distorted distributions that are different from the approximating model, where the degree to which they differ is captured by a relative entropy function.

In their model, a changing level of risk in the economy, captured by stochastic volatility of log consumption growth, interacts with an agent's fear that his model is misspecified. This interaction reflects the fact that the level of volatility in the economy affects the agent's ability to detect differences between his approximating model and alternative models. Consequently, the level of volatility affects the set of plausible models the agent entertains as a way of expressing his doubts over his approximating model.

Playing against the “Evil agent” means that the “Economist” is going to play against the worst case distribution among all alternative models. In order to obtain these characterizations of the worst case distributions it is necessary for the authors to extend the methodology of robust control to handle general nonlinear and non-Gaussian settings. One key contribution is that they manage to compute an approximation to the worst case model's pdf (via an approximation based on a third order Taylor polynomial of the Bellman value function).