Risk sensitive linear approximations

We propose a linear approximation to the solution of DSGE models that is sensitive to the effects of risk. If variables remain close to the approximation point in expectation, a second-order Taylor expansion to the equilibrium conditions reduces to a fixed-point problem characterized by a system of linear equations that depends on the second-order moments of the variables. The latter can be solved recursively using standard linear rational expectation methods. The resulting approximation captures the effects of risk in models with constant volatility, stochastic volatility, and GARCH effects through the intercept and/or the slopes of the decision rules. Relative to alternative approximations, our method yields approximation errors that are up to two orders of magnitude smaller, all while preserving a linear structure in the state variables. Finally, we show how to accommodate asymmetric effects from non-normal shocks within our linear approximation using information from a third-order Taylor expansion.