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.

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