Vector Autoregressions and Macroeconomic Modeling: An Error Taxonomy

In this article, we investigate the theoretical behavior of finite lag VAR(n)$\text{VAR}(n)$ models fitted to time series that in truth come from an infinite-order VAR(∞)$\text{VAR}(\infty)$ data-generating mechanism. We show that the overall error can be broken down into two basic components, an estimation error that stems from the difference between the parameter estimates and their population ensemble VAR(n)$\text{VAR}(n)$ counterparts, and an approximation error that stems from the difference between the VAR(n)$\text{VAR}(n)$ and the true VAR(∞)$\text{VAR}(\infty)$. The two sources of error are shown to be present in other performance indicators previously employed in the literature to characterize, so-called, truncation effects. Our theoretical analysis indicates that the magnitude of the estimation error exceeds that of the approximation error, but experimental results based upon a prototypical real business cycle model and a practical example indicate that the approximation error approaches its asymptotic position far more slowly than does the estimation error, their relative orders of magnitude notwithstanding. The experimental results suggest that with sample sizes and lag lengths like those commonly employed in practice VAR(n)$\text{VAR}(n)$ models are likely to exhibit serious errors of both types when attempting to replicate the dynamics of the true underlying process and that inferences based on VAR(n)$\text{VAR}(n)$ models can be very untrustworthy.