This paper shows how particle filtering facilitates likelihood-based inference in dynamic macroeconomic models. The economies can be non-linear and/or non-normal. We describe how to use the output from the particle filter to estimate the structural parameters of the model, those characterizing preferences and technology, and to compare different economies. Both tasks can be implemented from either a classical or a Bayesian perspective. We illustrate the technique by estimating a business cycle model with investment-specific technological change, preference shocks, and stochastic volatility.

We identify structural vector autoregressions using narrative sign restrictions. Narrative sign restrictions constrain the structural shocks and/or the historical decomposition around key historical events, ensuring that they agree with the established narrative account of these episodes. Using models of the oil market and monetary policy, we show that narrative sign restrictions tend to be highly informative. Even a single narrative sign restriction may dramatically sharpen and even change the inference of SVARs originally identified via traditional sign restrictions. Our approach combines the appeal of narrative methods with the popularized usage of traditional sign restrictions.

There has been a call for caution regarding the standard procedure for Bayesian inference in set‐identified structural vector autoregressions on the grounds that the common practice of using a uniform prior over the set of orthogonal matrices induces a non‐uniform prior for individual impulse responses or other quantities of interest. This paper challenges this call by formally showing that when the focus is on joint inference, the uniform prior over the set of orthogonal matrices is not only sufficient but also necessary for inference based on a uniform joint prior distribution over the identified set for the vector of impulse responses. In addition, we show how to conduct inference based on a uniform joint prior distribution for the vector of impulse responses.

We propose an approach for Bayesian inference in time-varying structural vector autoregressions (SVARs) identified with sign restrictions. The linchpin of our approach is a class of rotation-invariant time-varying SVARs in which the prior and posterior densities of any sequence of structural parameters belonging to the class are invariant to orthogonal transformations of the sequence. Our methodology is new to the literature. In contrast to existing algorithms for inference based on sign restrictions, our algorithm is the first to draw from a uniform distribution over the sequences of orthogonal matrices given the reduced-form parameters. We illustrate our procedure for inference by analyzing the role played by monetary policy during the latest inflation surge.

We present a novel algorithm called Dynamic Perturbation for solving large-scale macroeconomic models. Our approach involves computing first-order Taylor expansions of the policy functions along the entire equilibrium path. This method applies to a wide range of models and offers significantly higher accuracy than traditional perturbation approaches. Remarkably, even when utilising first-order approximations, our method can effectively handle models with strong nonlinearities and occasionally binding constraints, such as the zero lower bound.

Structural vector autoregressions (SVARs) are widely used for policy analysis and to provide stylized facts for dynamic stochastic general equilibrium (DSGE) models; yet no workable rank conditions to ascertain whether an SVAR is globally identified have been established. Moreover, when nonlinear identifying restrictions are used, no efficient algorithms exist for small-sample estimation and inference. This paper makes four contributions towards filling these important gaps in the literature. First, we establish general rank conditions for global identification of both identified and exactly identified models. These rank conditions are sufficient for general identification and are necessary and sufficient for exact identification. Second, we show that these conditions can be easily implemented and that they apply to a wide class of identifying restrictions, including linear and certain nonlinear restrictions. Third, we show that the rank condition for exactly identified models amounts to a straightforward counting exercise. Fourth, we develop efficient algorithms for small-sample estimation and inference, especially for SVARs with nonlinear restrictions.

We show how changes in the volatility of the real interest rate at which small open emerging economies borrow have an important effect on variables like output, consumption, investment, and hours. We start by documenting the strong evidence of time-varying volatility in the real interest rates faced by four emerging economies: Argentina, Brazil, Ecuador, and Venezuela. We estimate a stochastic volatility process for real interest rates. Then, we feed this process in a standard small open economy business cycle model. We find that an increase in real interest rate volatility triggers a fall in output, consumption, investment, hours, and debt.

This article studies the pruned state-space system for higher-order perturbation approximations to dynamic stochastic general equilibrium (DSGE) models. We show the stability of the pruned approximation up to third order and provide closed-form expressions for first and second unconditional moments and impulse response functions. Our results introduce generalized method of moments (GMM) estimation and impulse-response matching for DSGE models approximated up to third order and provide a foundation for indirect inference and simulated method of moments (SMM). As an application,we consider a New Keynesian model with Epstein–Zin preferences and two novel feedback effects from long-term bonds to the real economy, allowing us to match the level and variability of the 10-year term premium in the U.S. with a low relative risk aversion of 5.

This paper studies the econometrics of computed dynamic models. Since these models generally lack a closed-form solution, their policy functions are approximated by numerical methods. Hence, the researcher can only evaluate an approximated likelihood associated with the approximated policy function rather than the exact likelihood implied by the exact policy function. What are the consequences for inference of the use of approximated likelihoods? First, we find conditions under which, as the approximated policy function converges to the exact policy, the approximated likelihood also converges to the exact likelihood. Second, we show that second order approximation errors in the policy function, which almost always are ignored by researchers, have first order effects on the likelihood function. Third, we discuss convergence of Bayesian and classical estimates. Finally, we propose to use a likelihood ratio test as a diagnostic device for problems derived from the use of approximated likelihoods.

The dynamics of a linear (or linearized) dynamic stochastic economic model can be expressed in terms of matrices (A, B, C, D) that define a state space system for a vector of observables. An associated state space system (A, ^ B,C, ^D) determines a vector autoregression for those same observables. We present a simple condition for checking when these two state space systems match up and when they do not when there are equal numbers of economic and VAR shocks. We illustrate our condition with a permanent income example.