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This article reviews the literature on the econometric relationship between DSGE and VAR models from the point of view of estimation and model validation. The mapping between DSGE and VAR models is broken down into three stages: (1) from DSGE to state-space model; (2) from state-space model to VAR( ∞ ); (3) from VAR( ∞ ) to finite-order VAR. The focus is on discussing what can go wrong at each step of this mapping and on critically highlighting the hidden assumptions. I also point out some open research questions and interesting new research directions in the literature on the econometrics of DSGE models. These include, in no particular order: understanding the effects of log-linearization on estimation and identification; dealing with multiplicity of equilibria; estimating nonlinear DSGE models; incorporating into DSGE models information from atheoretical models and from survey data; adopting flexible modeling approaches that combine the theoretical rigor of DSGE models and the econometric model’s ability to fit the data.

A state space representation of a linearized DSGE model implies a VAR in terms of observable variables. The model is said be non-invertible if there exists no linear rotation of the VAR innovations which can recover the economic shocks. Non-invertibility arises when the observed variables fail to perfectly reveal the state variables of the model. The imperfect observation of the state drives a wedge between the VAR innovations and the deep shocks, potentially invalidating conclusions drawn from structural impulse response analysis in the VAR. The principal contribution of this chapter is to show that non-invertibility should not be thought of as an “either/or” proposition – even when a model has a non-invertibility, the wedge between VAR innovations and economic shocks may be small, and structural VARs may nonetheless perform reliably. As an increasingly popular example, so-called “news shocks” generate foresight about changes in future fundamentals – such as productivity, taxes, or government spending – and lead to an unassailable missing state variable problem and hence non-invertible VAR representations. Simulation evidence from a medium scale DSGE model augmented with news shocks about future productivity reveals that structural VAR methods often perform well in practice, in spite of a known non-invertibility. Impulse responses obtained from VARs closely correspond to the theoretical responses from the model, and the estimated VAR responses are successful in discriminating between alternative, nested specifications of the underlying DSGE model. Since the non-invertibility problem is, at its core, one of missing information, conditioning on more information, for example through factor augmented VARs, is shown to either ameliorate or eliminate invertibility problems altogether.

This article provides an overview of the panel vector autoregressive models (VAR) used in macroeconomics and finance to study the dynamic relationships between heterogeneous assets, households, firms, sectors, and countries. We discuss what their distinctive features are, what they are used for, and how they can be derived from economic theory. We also describe how they are estimated and how shock identification is performed. We compare panel VAR models to other approaches used in the literature to estimate dynamic models involving heterogeneous units. Finally, we show how structural time variation can be dealt with.