Second-Order Approximation of Dynamic Models with Time-Varying Risk

This paper provides first and second-order approximation methods for the solution of non-linear dynamic stochastic models in which the exogenous state variables follow conditionally linear stochastic processes displaying time-varying risk. The first-order approximation is consistent with a conditionally linear model in which risk is still time-varying but has no distinct role – separated from the primitive stochastic disturbances – in influencing the endogenous variables. The second-order approximation of the solution, instead, is sufficient to get this role. Moreover, risk premia, evaluated using only a first-order approximation of the solution, will be also time varying.