ABCs (and Ds) of Understanding VARs

This 2007 American Economic Review paper by Fernández-Villaverde, Rubio-Ramírez, Sargent, and Watson asks when the structural economic shocks in a DSGE model’s state-space representation can be recovered from the one-step-ahead forecast errors (“innovations”) of a VAR estimated on the model’s observables — the “invertibility” problem. Writing the model as a state equation x_{t+1} = Ax_t + Bw_{t+1} and observable equation y_{t+1} = Cx_t + Dw_{t+1}, with w_t an i.i.d. Gaussian vector of structural economic shocks, they show that in the square case (number of observables k equals number of shocks m, and D is nonsingular) the VAR innovations equal the structural shocks if and only if the eigenvalues of A − BD^{-1}C are strictly less than one in modulus — a “poor man’s invertibility condition” that can be checked directly from the model’s own matrices without deriving a full VARMA representation. When this eigenvalue condition fails, the VAR instead recovers the model’s “innovations representation,” a distinct state-space system built from the Kalman-filtered state estimate x̂_t = E(x_t|y^t) rather than the true state x_t; because the state cannot then be fully inferred from current and past observables (Σ = var(x_t|y^t) > 0), the variance of the VAR’s innovations strictly exceeds that of the true structural shocks (D̂D̂’ > DD’), and the VAR’s estimated impulse responses can differ sharply — even in sign — from the model’s true responses. The paper illustrates this failure analytically in a permanent-income consumption model (Sargent 1987, chap. XII) calibrated with gross interest rate R = 1.2 and income shock scale σ_w = 1: when only the consumption-income surplus y_t − c_t is observed, A − BD^{-1}C = R > 1, so the eigenvalue condition fails and Σ = σ²_w(1 − R^{-2}) > 0. The resulting VAR — an AR(1) for the surplus — has impulse responses that are “markedly different” from the true model’s: consumption responds with the opposite sign to a VAR shock than it does to the true structural shock, and the surplus response has a positive present value in the VAR representation versus a present value of exactly zero in the true model (which imposes budget balance). The authors note that observing additional variables (such as consumption, income, or the value of accumulated assets) can restore invertibility, and conclude that despite this problem VARs remain informative about the shapes of impulse responses that theories should be disciplined to match, even when they cannot recover every structural shock exactly. The analysis is purely theoretical and methodological — it presents no empirical VAR estimation or Monte Carlo evidence — and is restricted to the square case (k = m) with Gaussian shocks and the time-invariant (steady-state) limits of the Kalman filter.

Jesús Fernández-Villaverde | Macro Paper Warehouse

ABCs (and Ds) of Understanding VARs