Finite Order VAR Research Articles

A Matter of Perspective: Mapping Linear Rational Expectations Models into Finite-Order VAR Form

This paper considers the characterization of the reduced-form solution of a large class of linear rational expectations models. I show that under certain conditions, if a solution exists and is unique, it can be cast in finite-order VAR form. I also investigate the conditions for the VAR form to be stationary with a well-defined residual variance-covariance matrix in equilibrium, for the shocks to be recoverable, and for local identification of the structural parameters for estimation from the sample likelihood. An application to the workhorse New Keynesian model with accompanying Matlab codes illustrates the practical use of the finite-order VAR representation. In particular, I argue that the identification of monetary policy shocks based on structural VARs can be more closely aligned with theory using the finite-order VAR form of the model solution characterized in this paper.

A Matter of Perspective: Mapping Linear Rational Expectations Models into Finite-Order VAR Form

This paper considers the characterization of the reduced-form solution of a large class of linear rational expectations models. I show that under certain conditions, if a solution exists and is unique, it can be cast in finite-order VAR form. I also investigate the conditions for the VAR form to be stationary with a well-defined residual variance-covariance matrix in equilibrium, for the shocks to be recoverable, and for local identification of the structural parameters for estimation from the sample likelihood. An application to the workhorse New Keynesian model with accompanying Matlab codes illustrates the practical use of the finite-order VAR representation. In particular, I argue that the identification of monetary policy shocks based on structural VARs can be more closely aligned with theory using the finite-order VAR form of the model solution characterized in this paper.

Implications of Partial Information for Econometric Modeling of Macroeconomic Systems

Representative models of the macroeconomy (RMs), such as DSGE models, frequently contain unobserved variables. A finite-order VAR representation in the observed variables may not exist, and therefore the impulse responses of the RMs and SVAR models may differ. We demonstrate this divergence often is: (i) not substantial; (ii) reflects the omission of stock variables from the VAR; and (iii) when the RM features I (1) variables can be ameliorated by estimating a latent-variable VECM. We show that DSGE models utilize identifying restrictions stemming from common factor dynamics reflecting statistical, not economic, assumptions. We analyze the use of measurement error, and demonstrate that it may result in unintended consequences, particularly in models featuring I (1) variables.

Finite-Order VAR Representation of Linear Rational Expectations Models: In a Nutshell

This note summarizes the salient derivations of Martinez-Garcia (2018). Specifically, it shows that the solution to a large class of linear rational expectations (LRE) models can be represented in finite-order VAR form. Martinez-Garcia (2018) proposes a unified approach that uses a companion quadratic matrix equation to decouple the backward- and forward-looking parts of the canonical form of the LRE model and a Sylvester equation to pin down the solution and simplify its characterization so it can be expressed as a finite-order VAR.

Finite-Order VAR Representation of Linear Rational Expectations Models: With Some Lessons for Monetary Policy

This paper considers the solution of a large class of linear rational expectations (LRE) models and its characterization via finite-order VARs. The solution of the canonical LRE model can be cast in state-space form and solved for by the method of undetermined coefficients. In this paper I propose an approach that simplifies the systematic characterization of the solution into finite-order VAR form and checks existence and uniqueness based on the solution of a companion Sylvester equation. Solving LRE models with a finite-order VAR representation via the Sylvester equation is straightforward to implement, efficient, and can be handled easily with standard matrix algebra. An application to the workhorse New Keynesian model with accompanying Matlab codes is provided to illustrate the implementation of the procedure in practice.

System Reduction and Finite-Order VAR Solution Methods for Linear Rational Expectations Models

This paper considers the solution of a large class of linear rational expectations (LRE) models and its characterization via finite-order VARs. The solution of the canonical LRE model can be cast in state-space form and solved for by the method of undetermined coefficients. In this paper I propose a novel approach that simplifies the characterization of the solution into finite-order VAR form and checks its existence and uniqueness based on the solution of a Sylvester equation. Solving LRE models with a finite-order VAR representation by this method is straightforward to implement, efficient, and can be handled easily with standard matrix algebra. An application to the workhorse New Keynesian model with accompanying Matlab codes is provided to illustrate the practical implementation of the procedure.

System Reduction and Finite-Order VAR Solution Methods for Linear Rational Expectations Models

System Reduction and Finite-Order VAR Solution Methods for Linear Rational Expectations Models

System Reduction and Finite-Order VAR Solution Methods for Linear Rational Expectations Models

This paper considers the solution of a large class of linear rational expectations (LRE) models and its characterization via finite-order VARs. The solution of the canonical LRE model can be cast in state-space form and solved for by the method of undetermined coefficients. In this paper I propose an approach that simplifies the systematic characterization of the solution into finite-order VAR form and checks existence and uniqueness based on the solution of a companion Sylvester equation. Solving LRE models with a finite-order VAR representation via the Sylvester equation is straightforward to implement, efficient, and can be handled easily with standard matrix algebra. An application to the workhorse New Keynesian model with accompanying Matlab codes is provided to illustrate the implementation of the procedure in practice.

Forecasting Local Inflation with Global Inflation: When Economic Theory Meets the Facts

This paper provides both theoretical insight as well as empirical evidence in support of the view that inflation is largely a global phenomenon. First, we show that inflation across countries incorporates a significant common factor captured by global inflation. Second, we show that in theory a role for global inflation in local inflation dynamics emerges over the business cycle even without common shocks, and under flexible exchange rates and complete international asset markets. Third, we identify a strong correction mechanism that brings local inflation rates back in line with global inflation which explains the relative success of inflation forecasting models based on global inflation (e.g., Ciccarelli and Mojon (2010)). Fourth, we argue that the workhorse New Open Economy Macro (NOEM) model of Martinez-Garcia and Wynne (2010) can be approximated by a finite-order VAR and estimated using Bayesian techniques to forecast domestic inflation incorporating all relevant linkages with the rest of the world. This NOEM-BVAR provides a tractable model of inflation determination that can be tested empirically in forecasting. Finally, we use pseudo-out-of-sample forecasts to assess the NOEM-BVAR at different horizons (1 to 8 quarters ahead) across 17 OECD countries using quarterly data over the period 1980Q1-2014Q4. In general, we find that the NOEM-BVAR model produces a lower root mean squared prediction error (RMSPE) than its competitors --- which include most conventional forecasting models based on domestic factors and also the recent models based on global inflation. In a number of cases, the gains in smaller RMSPEs are statistically significant. The NOEM-BVAR model is also accurate in predicting the direction of change for inflation, and often better than its competitors along this dimension too.

Forecasting Local Inflation with Global Inflation: When Economic Theory Meets the Facts

This paper provides both theoretical insight as well as empirical evidence in support of the view that inflation is largely a global phenomenon. First, we show that inflation across countries incorporates a significant common factor captured by global inflation. Second, we show that in theory a role for global inflation in local inflation dynamics emerges over the business cycle even without common shocks, and under flexible exchange rates and complete international asset markets. Third, we identify a strong correction mechanism that brings local inflation rates back in line with global inflation which explains the relative success of inflation forecasting models based on global inflation (e.g., Ciccarelli and Mojon (2010). Fourth, we argue that the workhorse New Open Economy Macro (NOEM) model of Martinez-Garcia and Wynne (2010) can be approximated by a finiteorder VAR and estimated using Bayesian techniques to forecast domestic inflation incorporating all relevant linkages with the rest of the world. This NOEM-BVAR provides a tractable model of inflation determination that can be tested empirically in forecasting. Finally, we use pseudo-out-of-sample forecasts to assess the NOEM-BVAR at different horizons (1 to 8 quarters ahead) across 17 OECD countries using quarterly data over the period 1980Q1-2014Q4. In general, we find that the NOEM-BVAR model produces a lower root mean squared prediction error (RMSPE) than its competitors—which include most conventional forecasting models based on domestic factors and also the recent models based on global inflation. In a number of cases, the gains in smaller RMSPEs are statistically significant. The NOEM-BVAR model is also accurate in predicting the direction of change for inflation, and often better than its competitors along this dimension too.

Forecasting Local Inflation with Global Inflation: When Economic Theory Meets the Facts

This paper provides both theoretical insight as well as empirical evidence in support of the view that inflation is largely a global phenomenon. First, we show that inflation across countries incorporates a significant common factor captured by global inflation. Second, we show that in theory a role for global inflation in local inflation dynamics emerges over the business cycle even without common shocks, and under flexible exchange rates and complete international asset markets. Third, we identify a strong \error correction mechanism that brings local inflation rates back in line with global inflation which explains the relative success of inflation forecasting models based on global inflation (e.g., Ciccarelli and Mojon (2010). Fourth, we argue that the workhorse New Open Economy Macro (NOEM) model of Martnez-Garca and Wynne (2010) can be approximated by a finiteorder VAR and estimated using Bayesian techniques to forecast domestic inflation incorporating all relevant linkages with the rest of the world. This NOEM-BVAR provides a tractable model of inflation determination that can be tested empirically in forecasting. Finally, we use pseudo-out-of-sample forecasts to assess the NOEM-BVAR at different horizons (1 to 8 quarters ahead) across 17 OECD countries using quarterly data over the period 1980Q1-2014Q4. In general, we find that the NOEM-BVAR model produces a lower root mean squared prediction error (RMSPE) than its competitors?which include most conventional forecasting models based on domestic factors and also the recent models based on global inflation. In a number of cases, the gains in smaller RMSPEs are statistically significant. The NOEM-BVAR model is also accurate in predicting the direction of change for inflation, and often better than its competitors along this dimension too.

An asymptotic invariance property of the common trends under linear transformations of the data

It is well known that if Xt is a nonstationary process and Yt is a linear function of Xt, then cointegration of Yt implies cointegration of Xt. We want to find an analogous result for common trends if Xt is generated by a finite order VAR with i.i.d. (0,Ωx) errors εxt. We first show that Yt has an infinite order VAR representation in terms of its white noise prediction errors, εyt, which are a linear process in εxt, the prediction error for Xt. We then apply this result to show that the limit of the common trends for Yt generated by εyt, are linear functions of the common trends for Xt, generated by εxt.We illustrate the findings with a small analysis of the term structure of interest rates.

Factor-Augmented VARMA Models With Macroeconomic Applications

We study the relationship between vector autoregressive moving-average (VARMA) and factor representations of a vector stochastic process. We observe that, in general, vector time series and factors cannot both follow finite-order VAR models. Instead, a VAR factor dynamics induces a VARMA process, while a VAR process entails VARMA factors. We propose to combine factor and VARMA modeling by using factor-augmented VARMA (FAVARMA) models. This approach is applied to forecasting key macroeconomic aggregates using large U.S. and Canadian monthly panels. The results show that FAVARMA models yield substantial improvements over standard factor models, including precise representations of the effect and transmission of monetary policy.

Factor-Augmented VARMA Models with Macroeconomic Applications

We study the relationship between VARMA and factor representations of a vector stochastic process. We observe that, in general, vector time series and factors cannot both follow finite-order VAR models. Instead, a VAR factor dynamics induces a VARMA process, while a VAR process entails VARMA factors. We propose to combine factor and VARMA modeling by using factor-augmented VARMA (FAVARMA) models. This approach is applied to forecasting key macroeconomic aggregates using large U.S. and Canadian monthly panels. The results show that FAVARMA models yield substantial improvements over standard factor models, including precise representations of the effect and transmission of monetary policy.

A check for finite order VAR representations of DSGE models

The present paper shows that a DSGE model can be represented by a finite order VAR if and only if the eigenvalues of the matrix defined in Fernández-Villaverde et al. (2007) are all equal to zero. Further it shows that this condition is equivalent to the unimodularity condition presented in Ravenna (2007).

An Invariance Property of the Common Trends Under Linear Transformations of the Data

It is well known that if X(t) is a nonstationary process and Y(t) is a linear function of X(t), then cointegration of Y(t) implies cointegration of X(t). We want to find an analogous result for common trends if X(t) is generated by a finite order VAR. We first show that Y(t) has an infinite order VAR representation in terms of its prediction errors, which are a linear process in the prediction error for X(t). We then apply this result to show that the limit of the common trends for Y(t) are linear functions of the common trends for X(t). We illustrate the findings with a small analysis of the term structure of interest rates.

Vector autoregressions and reduced form representations of DSGE models

The performance of dynamic stochastic general equilibrium models is often tested against estimated VARs. This requires that the data-generating process consistent with the DSGE theoretical model has a finite order VAR representation. This paper discusses the assumptions needed for a finite order VAR( p) representation of a DSGE model to exist. When a VAR( p) is only an approximation to the exact infinite order VAR, the truncated VAR( p) may return largely incorrect estimates of the impulse response function. The results do not hinge on small-sample bias or on incorrect identification assumptions. But the bias introduced by truncation can lead to bias in the identification of the structural shocks. Identification strategies that work in the exact VAR representation perform poorly in the truncated VAR.

Vector Autoregressions and Reduced Form Representations of DSGE Models

Dynamic Stochastic General Equilibrium models are often tested against empirical VARs or estimated by minimizing the distance between the model's and the VAR impulse response functions. These methodologies require that the data-generating process consistent with the DSGE theoretical model has a VAR representation. This paper discusses the assumptions needed for a finite-order VAR(p) representation of any subset of a DSGE model variables to exist. When a VAR(p) is only an approximation to the true VAR, the paper shows that the truncated VAR(p) may return largely incorrect estimates of the impulse response function. The results do not hinge on an incorrect identification strategy or on small sample bias. But the bias introduced by truncation can lead to bias in the identification of the structural shocks. Identification strategies that are equivalent in the true VAR representation perform differently in the approximating VAR.

Granger-causality in cointegrated VAR processes The case of the term structure

Wald tests for Granger-causality in bivariate cointegrated finite order VAR processes are considered. It is argued that in the bivariate case these tests maintain their usual asymptotic properties. The relationship between short- and long-term interest rates in the U.S. is investigated in the light of this result. The long-term rate is found to cause the short-term rate in contrast to what is assumed in some other studies.