VAR Models in Macroeconomics – New Developments and Applications: Essays in Honor of Christopher A. Sims: Volume 32

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.

Recently, it has been suggested that macroeconomic forecasts from estimated dynamic stochastic general equilibrium (DSGE) models tend to be more accurate out-of-sample than random walk forecasts or Bayesian vector autoregression (VAR) forecasts. Del Negro and Schorfheide (2013) in particular suggest that the DSGE model forecast should become the benchmark for forecasting horse-races. We compare the real-time forecasting accuracy of the Smets and Wouters (2007) DSGE model with that of several reduced-form time series models. We first demonstrate that none of the forecasting models is efficient. Our second finding is that there is no single best forecasting method. For example, typically simple AR models are most accurate at short horizons and DSGE models are most accurate at long horizons when forecasting output growth, while for inflation forecasts the results are reversed. Moreover, the relative accuracy of all models tends to evolve over time. Third, we show that there is no support to the common practice of using large-scale Bayesian VAR models as the forecast benchmark when evaluating DSGE models. Indeed, low-dimensional unrestricted AR and VAR forecasts may forecast more accurately.

This article investigates the robustness of impulse response estimators to near unit roots and near cointegration in vector autoregressive (VAR) models. We compare estimators based on VAR specifications determined by pretests for unit roots and cointegration as well as unrestricted VAR specifications in levels. Our main finding is that the impulse response estimators obtained from the levels specification tend to be most robust when the magnitude of the roots is not known. The pretest specification works well only when the restrictions imposed by the model are satisfied. Its performance deteriorates even for small deviations from the exact unit root for one or more model variables. We illustrate the practical relevance of our results through simulation examples and an empirical application.

This article surveys recent developments in the evaluation of point and density forecasts in the context of forecasts made by vector autoregressions. Specific emphasis is placed on highlighting those parts of the existing literature that are applicable to direct multistep forecasts and those parts that are applicable to iterated multistep forecasts. This literature includes advancements in the evaluation of forecasts in population (based on true, unknown model coefficients) and the evaluation of forecasts in the finite sample (based on estimated model coefficients). The article then examines in Monte Carlo experiments the finite-sample properties of some tests of equal forecast accuracy, focusing on the comparison of VAR forecasts to AR forecasts. These experiments show the tests to behave as should be expected given the theory. For example, using critical values obtained by bootstrap methods, tests of equal accuracy in population have empirical size about equal to nominal size.

Identification of shocks of interest is a central problem in structural vector autoregressive (SVAR) modeling. Identification is often achieved by imposing restrictions on the impact or long-run effects of shocks or by considering sign restrictions for the impulse responses. In a number of articles changes in the volatility of the shocks have also been used for identification. The present study focuses on the latter device. Some possible setups for identification via heteroskedasticity are reviewed and their potential and limitations are discussed. Two detailed examples are considered to illustrate the approach.

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.

The development of models for variables sampled at different frequencies has attracted substantial interest in the recent literature. In this article, we discuss classical and Bayesian methods of estimating mixed-frequency VARs, and use them for forecasting and structural analysis. We also compare mixed-frequency VARs with other approaches to handling mixed-frequency data.

This survey focuses on two families of nonlinear vector time series models, the family of vector threshold regression (VTR) models and that of vector smooth transition regression (VSTR) models. These two model classes contain incomplete models in the sense that strongly exogeneous variables are allowed in the equations. The emphasis is on stationary models, but the considerations also include nonstationary VTR and VSTR models with cointegrated variables. Model specification, estimation and evaluation is considered, and the use of the models illustrated by macroeconomic examples from the literature.

For over three decades, vector autoregressions have played a central role in empirical macroeconomics. These models are general, can capture sophisticated dynamic behavior, and can be extended to include features such as structural instability, time-varying parameters, dynamic factors, threshold-crossing behavior, and discrete outcomes. Building upon growing evidence that the assumption of linearity may be undesirable in modeling certain macroeconomic relationships, this article seeks to add to recent advances in VAR modeling by proposing a nonparametric dynamic model for multivariate time series. In this model, the problems of modeling and estimation are approached from a hierarchical Bayesian perspective. The article considers the issues of identification, estimation, and model comparison, enabling nonparametric VAR (or NPVAR) models to be fit efficiently by Markov chain Monte Carlo (MCMC) algorithms and compared to parametric and semiparametric alternatives by marginal likelihoods and Bayes factors. Among other benefits, the methodology allows for a more careful study of structural instability while guarding against the possibility of unaccounted nonlinearity in otherwise stable economic relationships. Extensions of the proposed nonparametric model to settings with heteroskedasticity and other important modeling features are also considered. The techniques are employed to study the postwar U.S. economy, confirming the presence of distinct volatility regimes and supporting the contention that certain nonlinear relationships in the data can remain undetected by standard models.

This article proposes a new approach to detecting the presence of common cyclical features when several time series are sampled at different frequencies. We generalize the common-frequency approach introduced by Engle and Kozicki (1993) and Vahid and Engle (1993). We start with the mixed-frequency VAR representation investigated in Ghysels (2012) for stationary time series. For non-stationary time series in levels, we show that one has to account for the presence of two sets of long-run relationships. The first set is implied by identities stemming from the fact that the differences of the high-frequency I (1) regressors are stationary. The second set comes from possible additional long-run relationships between one of the high-frequency series and the low-frequency variables. Our transformed vector error-correction model (VECM) representations extend the results of Ghysels (2012) and are important for determining the correct set of variables to be used in a subsequent common cycle investigation. This fact has implications for the distribution of test statistics and for forecasting. Empirical analyses with quarterly real gross national product (GNP) and monthly industrial production indices for, respectively, the United States and Germany illustrate our new approach. We also conduct a Monte Carlo study which compares our proposed mixed-frequency models with models stemming from classical temporal aggregation methods.

In this article we propose a multivariate dynamic probit model. Our model can be viewed as a nonlinear VAR model for the latent variables associated with correlated binary time-series data. To estimate it, we implement an exact maximum likelihood approach, hence providing a solution to the problem generally encountered in the formulation of multivariate probit models. Our framework allows us to study the predictive relationships among the binary processes under analysis. Finally, an empirical study of three financial crises is conducted.