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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.

This chapter investigates the performance of different dimension reduction approaches for large vector autoregressions in multi-step ahead forecasts. The authors consider factor augmented VAR models using principal components and partial least squares, random subset regression, random projection, random compression, and estimation via LASSO and Bayesian VAR. The authors compare the accuracy of iterated and direct multi-step point and density forecasts. The comparison is based on macroeconomic and financial variables from the FRED-MD data base. Our findings suggest that random subspace methods and LASSO estimation deliver the most precise forecasts.

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 research applies vector autoregression to estimate a system composed of market share and relative advertising expenditures of the seven major competitors in the U. S. replacement passenger tire market between 1972 and 1983. The results of the study suggest that a company's market share in this market cannot be predicted from its relative advertising expenditures.

This paper addresses the issue of improving the forecasting performance of vector autoregressions (VARs) when the set of available predictors is inconveniently large to handle with methods and diagnostics used in traditional small-scale models. First, available information from a large dataset is summarized into a considerably smaller set of variables through factors estimated using standard principal components. However, even in the case of reducing the dimension of the data the true number of factors may still be large. For that reason I introduce in my analysis simple and efficient Bayesian model selection methods. Model estimation and selection of predictors is carried out automatically through a stochastic search variable selection (SSVS) algorithm which requires minimal input by the user. I apply these methods to forecast 8 main U.S. macroeconomic variables using 124 potential predictors. I find improved out-of-sample fit in high-dimensional specifications that would otherwise suffer from the proliferation of parameters.

The authors propose the information matrix test to assess the constancy of mean and variance parameters in vector autoregressions (VAR). They additively decompose it into several orthogonal components: conditional heteroskedasticity and asymmetry of the innovations, and their unconditional skewness and kurtosis. Their Monte Carlo simulations explore both its finite size properties and its power against i.i.d. coefficients, persistent but stationary ones, and regime switching. Their procedures detect variation in the autoregressive coefficients and residual covariance matrix of a VAR for the US GDP growth rate and the statistical discrepancy, but they fail to detect any covariation between those two sets of coefficients.

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.

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.

Vector autoregressions (VAR) combined with Minnesota-type priors are widely used for macroeconomic forecasting. The fact that strong but sensible priors can substantially improve forecast performance implies VAR forecasts are sensitive to prior hyperparameters. But the nature of this sensitivity is seldom investigated. We develop a general method based on Automatic Differentiation to systematically compute the sensitivities of forecasts – both points and intervals – with respect to any prior hyperparameters. In a forecasting exercise using US data, we find that forecasts are relatively sensitive to the strength of shrinkage for the VAR coefficients, but they are not much affected by the prior mean of the error covariance matrix or the strength of shrinkage for the intercepts.

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.