Working Paper Series no. 702: Risk-Adjusted Linearizations of Dynamic Equilibrium Models

People make decisions under uncertainty and are sensitive to risk. Accordingly, financial prices and the macroeconomy reflect, and move with, the exposure of future cashflows to risky events and the compensation for risk commanded by investors. Examples of the ingredients economists use to capture these facts include risk appetite that changes with the state of the economy, and a varying probability of the realization of some disastrous event such as a financial crisis or war. The success of these ingredients relies on solution methods that can capture correctly their dynamic equilibrium implications.

Numerical methods that reveal the global solution of a model are used increasingly by economists. Nevertheless, these methods are typically computationally intensive––a cost particularly large in estimation––and offer limited analytic insight into the main economic channels that drive the solution. Analytic tractability is key to understand the mechanics of a model. For one thing, it helps identifying the role of different assumptions and parameters in driving particular results. Furthermore, it helps talking about existence, uniqueness, or multiplicity of equilibrium dynamics.

In this context, perturbation methods provide analytic insight into the local structure of a model around a specific point. But for these properties to be meaningful, the expansion point should be a point the model actually fluctuates around. In fact, the conventional expansion point for perturbation methods––the deterministic steady state––and local dynamics around it are often inaccurate approximations of a model’s implications when risk matters. Higher-order perturbations do not always help and, in any event, analytic insight is quickly lost as the order of approximation increases.

We propose a simple risk-adjusted linear approximation to solve a large class of dynamic models with time-varying and non-Gaussian risk. Our approach generalizes loglinear-lognormal approximations commonly used in the finance literature, and we show that it coincides with first-order perturbations around the risky steady state recently developed in the macro literature. Therefore, we provide a formal foundation for approximation methods that remained so far heuristic, and we unify coexisting theories of risk-adjusted linearizations. Two strands of literature that developed independently are actually one and the same.

We make two main contributions. First, we generalize affine approximations. We extend risk adjustments to non-Gaussian distributed shocks using relative entropy––a generalized notion of variation––and the cumulant generating function of shocks. And we discuss the main features of this approximation that determine its accuracy.

Second, we root in formal ground our unified theory of risk-adjusted linearizations based on the implicit function and Taylor theorems. Our approach provides explicit formulas for the approximation coefficients, clarifies when the risky steady state is defined uniquely, and characterizes local existence and uniqueness of the approximate solution. Therefore, we are the first to provide a complete description of first-order perturbations around the risky steady state.

The top figure illustrates the importance of risk corrections in a basic example. With habit formation risk aversion is time-varying, and low-order perturbations around the deterministic steady state are inappropriate. They fail to capture the key role of precautionary savings in providing greater incentives to save in a risk-free security, especially during recessions when people are unwilling to take on more risk and invest in risky projects. Conventional third-order perturbations recover the global structure of the equilibrium risk-free rate but remain inappropriate to characterize equilibrium wealth. In contrast, affine perturbations offer an accurate description of local equilibrium dynamics.

Thus, affine approximations seem to be an appropriate perturbation choice to gain analytic understanding of the macroeconomic forces that drive macroeconomic quantities, asset prices, and welfare costs of fluctuations. To popularize these methods we provide a user-friendly computer code, flexible enough for application to most DSGE models.