_specifications

cultivars.engine._internals._specifications = <module 'cultivars.engine._internals._specifications' from '/home/runner/work/cultivars/cultivars/src/cultivars/engine/_internals/_specifications.py'>

Perturbation solution of rational-expectations models, to second order.

A model is a system of equilibrium conditions E_t f(y', y, x', x) = 0 with x the predetermined states (including any exogenous driving processes) and y the controls, closed by x' = h(x, sigma) + sigma eta eps'. Schmitt-Grohe and Uribe (2004) showed that the Taylor coefficients of the policy functions g and h around the deterministic steady state solve a sequence of linear problems once the first-order coefficients are known: the first order is the Blanchard-Kahn / Klein generalized-Schur problem, the second order in x is one linear system in (g_xx, h_xx), and the risk correction is one linear system in (g_ss, h_ss) – with the cross terms g_xs, h_xs identically zero.

Derivatives of f are taken numerically (central differences), so the model is supplied as a plain callable and no symbolic layer is needed; the cost is a step-size floor on accuracy of order 1e-8 in the second derivatives, immaterial next to the perturbation error itself.

The second-order solution is carried as the pruned state space of Kim, Kim, Schaumburg, and Sims (2008) in the form of Andreasen, Fernandez-Villaverde, and Rubio-Ramirez (2018): a first-order state and a second-order state evolve side by side, the quadratic term is fed only by the first-order state, and the system cannot explode. That is what makes it a well-posed customer of the nonlinear filters.

References

Schmitt-Grohe, S., & Uribe, M. (2004). Solving dynamic general

equilibrium models using a second-order approximation to the policy function. Journal of Economic Dynamics and Control, 28(4), 755-775.

Klein, P. (2000). Using the generalized Schur form to solve a

multivariate linear rational expectations model. Journal of Economic Dynamics and Control, 24(10), 1405-1423.

Kim, J., Kim, S., Schaumburg, E., & Sims, C. A. (2008). Calculating

and using second-order accurate solutions of discrete time dynamic equilibrium models. Journal of Economic Dynamics and Control, 32(11), 3397-3414.

Andreasen, M. M., Fernandez-Villaverde, J., & Rubio-Ramirez, J. F.

(2018). The pruned state-space system for non-linear DSGE models: Theory and empirical applications. Review of Economic Studies, 85(1), 1-49.

Fernandez-Villaverde, J., & Rubio-Ramirez, J. F. (2007). Estimating

macroeconomic models: A likelihood approach. Review of Economic Studies, 74(4), 1059-1087.