Title: Unraveling the Enigma of VARs: A Check for Economic Shocks Informative Analysis

The Mystery of Interpreting Vector Autoregressions

In recent years, a heated debate has arisen over whether unrestricted Vector Autoregressions (VARs) provide valuable insights into how economic models react to preference, technology, and information shocks. This blog post offers a simple solution for determining if a theoretical model can accurately infer economic shocks and their responses from a VAR.

Two Recursive Representations of Observable Data

1. Recursive Representation of Equilibrium: Economists often represent equilibria in state space form, where observable data yt is expressed as a function of states xt and shocks wt. However, the question remains: under what conditions do economic shocks in this system match up with those associated with a VAR?

2. Invertibility Issue: A potential problem for interpreting VARs is the invertibility issue. We present a straightforward check to identify its presence and provide an example using a permanent income model.

The Hidden Connection Between Economic Models and VARs

In general, economic shocks wt in the state-space system are not equal to the VAR errors yt+1 - E(yt+1|yt) due to a prediction error term C(xt −E(xt|yt)). Our aim is to find conditions that eliminate this discrepancy.

A Simple Check for Invertibility

Our solution involves investigating whether wt+1 = Ω(yt+1 −E(yt+1|yt)) holds, where Ω is a matrix of constants that could be uncovered through Structural VAR (SVAR) analysis. When this equation does hold, impulse responses from the SVAR match those from the economic model.

Portfolio Implications and Asset Analysis

What does this mean for investors? Understanding the invertibility issue can help portfolio managers assess the reliability of their models and make informed decisions regarding asset allocation. Specifically, assets with strong correlations to economic shocks (e.g., C, QUAL, BAC, MS) may provide valuable insights when evaluating the informative value of unrestricted VARs.

Actionable Insight: Assessing Your VAR Models

In light of this analysis, investors and economists should consider incorporating a check for invertibility into their VAR models to ensure that economic shocks can be accurately inferred from the associated impulse responses. By doing so, they can improve the reliability of their predictions and make more informed decisions about asset allocation.