A, B, C's (and D)'s for Understanding VARs

Summary

National Bureau of Economic Research (NBER) Technical Working Paper 308. Derives a complete set of formulas mapping the $(A,B,C,D)$ state-space representation of any linear/linearized Dynamic Stochastic General Equilibrium (DSGE) model to the Vector Autoregression (VAR) it implies via the Kalman filter innovations representation $(A,K,C,\Sigma)$ . Provides a single, easy-to-check necessary and sufficient condition for invertibility — the requirement that VAR innovations span the full space of economic shocks — in terms of the eigenvalues of $A - BD^{-1}C$ . Four worked examples illustrate when the condition holds, when it fails, and when a benign borderline case (unit eigenvalue) prevents an infinite-order VAR from existing while preserving invertibility.

Key Claims

Any linear economic model implies a VAR via its innovations representation; the VAR is the model's conditional likelihood function, so a researcher who trusts her model cannot ignore it.
The economic model Impulse Response Function (IRF) ( $d_j = CA^{j-1}B$ ) and the VAR IRF ( $c_j = CA^{j-1}KG$ ) coincide if and only if the model is invertible and $G = D$ is the correct identification matrix.
Invertibility theorem (square case): when $D^{-1}$ exists, the VAR innovations span the economic shocks iff all eigenvalues of $A - BD^{-1}C$ are strictly less than 1 in modulus. If so, $K = BD^{-1}$ , $\Sigma = 0$ , and $G\varepsilon_t = Dw_t$ .
A unit eigenvalue of $A - BD^{-1}C$ is benign: invertibility holds but the infinite-order VAR representation does not exist; finite-order VARs still approximate well.
Non-invertibility is not universal: it depends on the choice of observables, not just the model structure. Switching from $(y_m, l_m)$ to $(c_m, l_m)$ in the household production model flips invertibility.
The VAR implied by an economic model is generically infinite-order (a Vector AutoRegressive Moving Average (VARMA)), so truncated finite-order VARs are approximations.
The paper provides a four-level hierarchy of challenges in mapping VAR shocks to economic shocks.

Concepts Introduced or Extended

State-Space Representation — $(A,B,C,D)$ formulation; innovations representation; Wold and VAR derivations
Invertibility Problem — central new concept; eigenvalue check; taxonomy of failures
Impulse Response Function — DSGE vs. VAR IRF; conditions for equivalence

Entities Mentioned

Quotes

"By describing how VAR shocks recombine current and past realizations of the economic shocks hitting preferences, technologies, information sets, and measurements, formula (25) helps us to express and evaluate diverse grounds for skepticism about VARs."

"If you fully trust your model, [estimating deep parameters by ML] is incontrovertible. However, the enterprise of identifying shocks and responses to them by identifying SVARs aims to coax interesting patterns from the data that will prevail across a set of incompletely specified and not fully trusted models."

My Take

This paper's contribution is the eigenvalue check — a single computable scalar criterion replacing vague appeals to "invertibility." The four examples are well chosen to cover the spectrum from catastrophic failure (permanent income) to clean success (Fisher 2003). The main limitation is the square-case restriction ( $k = m$ ); when the number of shocks differs from the number of observables the problem is much harder and this paper is largely silent on it. The paper also does not address the practical question of what to do when the model fails the check (change observables, add measurement error, etc.) — it diagnoses but does not prescribe.