Perturbations in DSGE Models: Odd Derivatives Theorem
This paper proves a generalization of previous results in the perturbation literature. Perturbation methods compute policy functions to DSGE models using a multivariate Taylor series with respect to the state variables x and a perturbation parameter σ. Schmitt-Groh´e and Uribe (2004) shows that Taylor coeﬃcients of order x0σ1 and x1σ1 are zero. Andreasen (2012) extends this to order x2σ1, and shows the x0σ3 coeﬃcient is zero if innovations are symmetric. We show that Taylor coeﬃcients of order xrσ1 are zero for all r. Most generally, if odd moments of the innovations are zero up to some moment ¯ s, then coeﬃcients of order xrσs are zero for all r and odd s ≤ ¯ s. (The intuition for this comes from classical portfolio theory.) Eliminating these coeﬃcients signiﬁcantly reduces what needs to be computed and thereby runtime, memory usage, and numerical errors.