Combining survey long-run forecasts and nowcasts with BVAR forecasts using relative entropy

This paper constructs hybrid forecasts that combine forecasts from vector autoregressive (VAR) model(s) with both short- and long-term expectations from surveys. Specifically, we use the relative entropy to tilt one-step-ahead and long-horizon VAR forecasts to match the nowcasts and long-horizon forecasts from the Survey of Professional Forecasters. We consider a variety of VAR models, ranging from simple fixed-parameter to time-varying parameters. The results across models indicate meaningful gains in multi-horizon forecast accuracy relative to model forecasts that do not incorporate long-term survey conditions. Accuracy improvements are achieved for a range of variables, including those that are not tilted directly but are affected through spillover effects from tilted variables. The accuracy gains for hybrid inflation forecasts from simple VARs are substantial, statistically significant, and competitive to time-varying VARs, univariate benchmarks, and survey forecasts. We view our proposal as an indirect approach to accommodating structural change and moving end points.     

Structural information in recursive VAR orderings

This paper investigates structural models that will permit a Cholesky decomposition of the covariance matrix of VAR residuals to identify some structural impulse response functions. Cholesky decompositions are found to be useful identification tools for the set of partially recursive structural models. A partially recursive structure is defined as any block recursive system where the equations in one block can be recursively ordered and where the structural shocks are uncorrelated. Using this class of models, we derive necessary and sufficient conditions for the moving average representation from a Cholesky decomposition to identify structure. The paper concludes by discussing implications of these results for empirical research.     

Forecasting aggregates of independent Arima processes

In this paper I derive the properties of linear aggregates of independent ARIMA processes, including those with seasonality. I show that such aggregates are ARIMA processes, but that multiplicative seasonal structure on the moving average side will generally break down in the aggregation. The forecasting efficiency of a direct model of an aggregate is contrasted with that of an optimal predictor which uses the structural information. The former is shown to suffer from a kind of aggregation bias, a bias which can be expressed as a mean zero stochastic process.     

The puzzling effects of monetary policy in VARs: Invalid identification or missing information?

Standard vector autoregressions (VARs) often find puzzling effects of monetary policy shocks. Is this due to an invalid (recursive) identification scheme, or because the underlying small‐scale VAR neglects important information? I employ factor methods and external instruments to answer this question and provide evidence that the root cause is missing information. In particular, while a recursively identified dynamic factor model yields conventional monetary policy effects across the board, a small‐scale VAR identified via external instruments does not. Importantly, the discrepancy between both models largely disappears once the information set of the VAR is augmented via factors. This finding is comforting news for the recent monetary literature. Two leading empirical advances with different underlying assumptions—namely external instruments (applied to a factor‐augmented VAR) and dynamic factor models (identified recursively)—find very similar effects of monetary policy shocks, cross‐verifying each other.     

Structural VARs and noninvertible macroeconomic models

We resume the line of research pioneered by C. A. Sims and Zha (Macroeconomic Dynamics, 2006, 10, 231–272) and make two novel contributions. First, we provide a formal treatment of partial fundamentalness—that is, the idea that a structural vector autoregression (VAR) can recover, either exactly or with good approximation, a single shock or a subset of shocks, even when the underlying model is nonfundamental. In particular, we extend the measure of partial fundamentalness proposed by Sims and Zha to the finite‐order case and study the implications of partial fundamentalness for impulse‐response and variance‐decomposition analysis. Second, we present an application where we validate a theory of news shocks and find it to be in line with the empirical evidence.     

Inference on impulse response functions in structural VAR models

Skepticism toward traditional identifying assumptions based on exclusion restrictions has led to a surge in the use of structural VAR models in which structural shocks are identified by restricting the sign of the responses of selected macroeconomic aggregates to these shocks. Researchers commonly report the vector of pointwise posterior medians of the impulse responses as a measure of central tendency of the estimated response functions, along with pointwise 68% posterior error bands. It can be shown that this approach cannot be used to characterize the central tendency of the structural impulse response functions. We propose an alternative method of summarizing the evidence from sign-identified VAR models designed to enhance their practical usefulness. Our objective is to characterize the most likely admissible model(s) within the set of structural VAR models that satisfy the sign restrictions. We show how the set of most likely structural response functions can be computed from the posterior mode of the joint distribution of admissible models both in the fully identified and in the partially identified case, and we propose a highest-posterior density credible set that characterizes the joint uncertainty about this set. Our approach can also be used to resolve the long-standing problem of how to conduct joint inference on sets of structural impulse response functions in exactly identified VAR models. We illustrate the differences between our approach and the traditional approach for the analysis of the effects of monetary policy shocks and of the effects of oil demand and oil supply shocks.     

Studying co-movements in large multivariate data prior to multivariate modelling

For non-stationary vector autoregressive models (VAR hereafter, or VAR with moving average, VARMA hereafter), we show that the presence of common cyclical features or cointegration leads to a reduction of the order of the implied univariate autoregressive-integrated-moving average (ARIMA hereafter) models. This finding can explain why we identify parsimonious univariate ARIMA models in applied research although VAR models of typical order and dimension used in macroeconometrics imply non-parsimonious univariate ARIMA representations.     

Hours per capita and productivity: evidence from correlated unobserved components models

Recent studies debate the effect of a permanent productivity shock on hours per capita within a structural VAR context. This paper examines the issue using a correlated unobserved components (UC) framework. The estimates show that permanent shocks to productivity are negatively correlated with transitory shocks to hours. This result is robust for non‐stationary or levels stationary specifications of hours. Model comparisons indicate that the data do not favor imposing VAR‐type restrictions on the UC models. Copyright © 2008 John Wiley & Sons, Ltd.     

Tests of Predictive Ability for Vector Autoregressions Used for Conditional Forecasting

Many forecasts are conditional in nature. For example, a number of central banks routinely report forecasts conditional on particular paths of policy instruments. Even though conditional forecasting is common, there has been little work on methods for evaluating conditional forecasts. This paper provides analytical, Monte Carlo and empirical evidence on tests of predictive ability for conditional forecasts from estimated models. In the empirical analysis, we examine conditional forecasts obtained with a VAR in the variables included in the DSGE model of Smets and Wouters (American Economic Review 2007; 97 : 586–606). Throughout the analysis, we focus on tests of bias, efficiency and equal accuracy applied to conditional forecasts from VAR models. Copyright © 2016 John Wiley & Sons, Ltd.     

STRUCTURAL VECTOR AUTOREGRESSIONS: CHECKING IDENTIFYING LONG‐RUN RESTRICTIONS VIA HETEROSKEDASTICITY

Long‐run restrictions have been used extensively for identifying structural shocks in vector autoregressive (VAR) analysis. Such restrictions are typically just‐identifying but can be checked by utilizing changes in volatility. This paper reviews and contrasts the volatility models that have been used for this purpose. Three main approaches have been used, exogenously generated changes in the unconditional residual covariance matrix, changing volatility modelled by a Markov switching mechanism and multivariate generalized autoregressive conditional heteroskedasticity models. Using changes in volatility for checking long‐run identifying restrictions in structural VAR analysis is illustrated by reconsidering models for identifying fundamental components of stock prices.     

How do stock prices respond to fundamental shocks in the case of the United States? Evidence from NASDAQ and DJIA

In this paper, we use both the Dow Jones and NASDAQ indices to test the robustness of Binswanger's (2004c) finding that US stock market dynamics are governed mostly by nonfundamental shocks or speculative bubbles after the 1982 debt crisis. We estimate a total of 72 SVAR models and 36 SVECM models. We determine that the findings are robust indeed and that fundamental shocks have become less and less important over the years, irrespective of which US stock market index is considered.     

CO‐INTEGRATING VAR MODELS AND ECONOMIC POLICY

This paper surveys some relevant contributions to the economic literature on co‐integrating vector autoregressive (VAR) models [vector error correction mechanisms (VECMs)], emphasizing their usefulness for economic policy. It further discusses some theoretical aspects that are necessary for a complete understanding of their potential. The theoretical introduction of the co‐integrating VAR model is followed by an illustration of its applications to monetary policy, fiscal policy and exchanges rates as well as in establishing the effects of structural bilateral shocks between countries (the so‐called global VAR, or GVAR, models). Special attention is paid to the VECM capacities of being used in conjunction with dynamic stochastic general equilibrium models and of jointly specifying the short‐ and long‐run dynamics, thus representing the steady‐state of economic systems (by means of the co‐integration relations) and the short‐run dynamics around it.     

Forecasting with vector autoregressive (VAR) models subject to business cycle restrictions

In the last decade VAR models have become a widely-used tool for forecasting macroeconomic time series. To improve the out-of-sample forecasting accuracy of these models, Bayesian random-walk prior restrictions are often imposed on VAR model parameters. This paper focuses on whether placing an alternative type of restriction on the parameters of unrestricted VAR models improves the out-of-sample forecasting performance of these models. The type of restriction analyzed here is based on the business cycle characteristics of U.S. macroeconomic data, and in particular, requires that the dynamic behavior of the restricted VAR model mimic the business cycle characteristics of historical data. The question posed in this paper is: would a VAR model, estimated subject to the restriction that the cyclical characteristics of simulated data from the model “match up” with the business cycle characteristics of U.S. data, generate more accurate out-of-sample forecasts than unrestricted or Bayesian VAR models?     

Methods for computing marginal data densities from the Gibbs output

We introduce two estimators for estimating the Marginal Data Density (MDD) from the Gibbs output. Our methods are based on exploiting the analytical tractability condition, which requires that some parameter blocks can be analytically integrated out from the conditional posterior densities. This condition is satisfied by several widely used time series models. An empirical application to six-variate VAR models shows that the bias of a fully computational estimator is sufficiently large to distort the implied model rankings. One of the estimators is fast enough to make multiple computations of MDDs in densely parameterized models feasible.     

Bayesian Graphical Models for STructural Vector Autoregressive Processes

This paper proposes a Bayesian, graph‐based approach to identification in vector autoregressive (VAR) models. In our Bayesian graphical VAR (BGVAR) model, the contemporaneous and temporal causal structures of the structural VAR model are represented by two different graphs. We also provide an efficient Markov chain Monte Carlo algorithm to estimate jointly the two causal structures and the parameters of the reduced‐form VAR model. The BGVAR approach is shown to be quite effective in dealing with model identification and selection in multivariate time series of moderate dimension, as those considered in the economic literature. In the macroeconomic application the BGVAR identifies the relevant structural relationships among 20 US economic variables, thus providing a useful tool for policy analysis. The financial application contributes to the recent econometric literature on financial interconnectedness. The BGVAR approach provides evidence of a strong unidirectional linkage from financial to non‐financial super‐sectors during the 2007–2009 financial crisis and a strong bidirectional linkage between the two sectors during the 2010–2013 European sovereign debt crisis. Copyright © 2015 John Wiley & Sons, Ltd.     

The linkage between insurance activity and banking credit: Some evidence from dynamic analysis

This paper investigates the long-run and short-run linkages between insurance activity and banking credit for G-7 countries. To minimize the pretest bias and overcome the structural changes, we adopt the bootstrap Granger causality test applied to full sample and subsamples with a fixed window size. The Johansen cointegration test with GMM-IV estimator finds a long-run positive relation between the series. The full sample results of bootstrap Granger causality test show that there is predictive power from life insurance activity to banking credit only for France and Japan, while the short-run causal relationships between nonlife insurance activity and banking credit are country-specific. However, parameter stability test results suggest that the short-run results in full sample are unreliable. The results of rolling VAR models report that the causal linkages between them are time-varying across various subsamples. These findings offer some useful insights for achieving the co-evolution between insurance and banking credit markets.     

A Note on the Identification of Dynamic Economic Models with Generalized Shock Processes

Dynamic stochastic general equilibrium (DSGE) models with generalized shock processes, such as shock processes which follow a vector autoregression (VAR), have been an active area of research in recent years. Unfortunately, the structural parameters governing DSGE models are not identified when the driving process behind the model follows an unrestricted VAR. This finding implies that parameter estimates derived from recent attempts to estimate DSGE models with generalized driving processes should be treated with caution, and that there always exists a tradeoff between identification and the risk of model misspecification. However, these results also make it easier to address the issue of model misspecification by making it computationally easier to check the validity of cross‐equation restrictions.     

Applied cointegration analysis in the mirror of macroeconomic theory

Cointegration analyses of macroeconomic time series are often not based on fully specified theoretical models. We use a theoretical model to scrutinize common procedures in applied cointegration analysis. Monte Carlo experiments show that (1) some tests of the cointegration vectors do not work well on series generated by an equilibrium business cycle model; (2) cointegration restrictions add little to forecasting; (3) structural VAR models based on weak long-run restrictions seem to work well. The main disadvantages of cointegration analysis without strong links to economic theory are that it makes it hard to estimate and interpret the cointegration vectors.     

Steady‐state priors for vector autoregressions

Bayesian priors are often used to restrain the otherwise highly over‐parametrized vector autoregressive (VAR) models. The currently available Bayesian VAR methodology does not allow the user to specify prior beliefs about the unconditional mean, or steady state, of the system. This is unfortunate as the steady state is something that economists usually claim to know relatively well. This paper develops easily implemented methods for analyzing both stationary and cointegrated VARs, in reduced or structural form, with an informative prior on the steady state. We document that prior information on the steady state leads to substantial gains in forecasting accuracy on Swedish macro data. A second example illustrates the use of informative steady‐state priors in a cointegration model of the consumption‐wealth relationship in the USA. Copyright © 2009 John Wiley & Sons, Ltd.     

On forecasting cointegrated seasonal time series

We analyze periodic and seasonal cointegration models for bivariate quarterly observed time series in an empirical forecasting study. We include both single equation and multiple equation methods for those two classes of models. A VAR model in first differences, with and without cointegration restrictions, and a VAR model in annual differences are also included in the analysis, where they serve as benchmark models. Our empirical results indicate that the VAR model in first differences without cointegration is best if one-step ahead forecasts are considered. For longer forecast horizons however, the VAR model in annual differences is better. When comparing periodic versus seasonal cointegration models, we find that the seasonal cointegration models tend to yield better forecasts. Finally, there is no clear indication that multiple equations methods improve on single equation methods.