Accounting for Risk in a Linearized Solution: How to Approximate the Risky Steady State and Perturb Around It

44 Pages Posted: 31 Jul 2020 Last revised: 11 May 2022

See all articles by Pierlauro Lopez

Pierlauro Lopez

Federal Reserve Banks - Federal Reserve Bank of Cleveland

David Lopez-Salido

Board of Governors of the Federal Reserve System

Francisco Vazquez-Grande

Board of Governors of the Federal Reserve System

Date Written: July 5, 2020

Abstract

We propose a novel approximation of the risky steady state and construct first-order perturbations around it for a general class of dynamic equilibrium models with time-varying and non-Gaussian risk. We offer analytical formulas and conditions for their local existence and uniqueness. We apply this approximation technique to models featuring Campbell-Cochrane habits, recursive preferences, and time-varying disaster risk, and show how the proposed approximation represents the implications of the model similarly to global solution methods. We show that our approximation of the risky steady state cannot be generically replicated by higher-order perturbations around the deterministic steady state, which cannot account well for the effects of risk in our applications even up to third order. Finally, we argue that our perturbation can be viewed as a generalized version of the heuristic loglinear-lognormal approximations commonly used in the macro-finance literature.

Keywords: Perturbation methods, Risky steady state, Macroeconomic uncertainty, Risk modeling, Intertemporal choice under uncertainty

JEL Classification: C63, G12, E32, E44

Suggested Citation

Lopez, Pierlauro and Lopez-Salido, David and Vazquez-Grande, Francisco, Accounting for Risk in a Linearized Solution: How to Approximate the Risky Steady State and Perturb Around It (July 5, 2020). Available at SSRN: https://ssrn.com/abstract=3643629 or http://dx.doi.org/10.2139/ssrn.3643629

Pierlauro Lopez (Contact Author)

Federal Reserve Banks - Federal Reserve Bank of Cleveland ( email )

East 6th & Superior
Cleveland, OH 44101-1387
United States

David Lopez-Salido

Board of Governors of the Federal Reserve System ( email )

20th Street and Constitution Avenue NW
Washington, DC 20551
United States

Francisco Vazquez-Grande

Board of Governors of the Federal Reserve System ( email )

20th Street and Constitution Avenue NW
Washington, DC 20551
United States
202-973-7488 (Phone)

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