Juan F. Rubio-Ramírez | Macro Paper Warehouse

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.

The systematic component of monetary policy in SVARs: An agnostic identification procedure

This 2019 Journal of Monetary Economics paper by Jonas Arias, Dario Caldara, and Juan Rubio-Ramírez proposes identifying monetary policy shocks in a structural VAR by placing sign and zero restrictions directly on the coefficients of the monetary policy reaction function itself — the “systematic component” of policy — rather than on the impulse responses the shock is supposed to produce. The approach is partial and set identified: only the monetary policy shock is pinned down, out of a six-variable monthly VAR (real GDP, GDP deflator, a commodity price index, total reserves, nonborrowed reserves, and the federal funds rate) estimated with 12 lags over January 1965-June 2007 using a Bayesian uniform-normal-inverse-Wishart prior. Two restrictions on the contemporaneous federal funds rate equation carry the identification: the funds rate reacts to output and prices but not contemporaneously to total or nonborrowed reserves (ruling reserves out of the systematic rule), and its reactions to output and prices are both restricted to be positive, consistent with Taylor-type rules; critically, neither restriction touches the contemporaneous response of output to the shock, the assumption Uhlig (2005) and Ramey (2016) identify as the reason most VAR evidence finds monetary policy expansionary. Under these restrictions the posterior median response to a contractionary shock is an immediate output decline that remains significant for about 18 months, a protracted fall in the price level, and an on-impact funds-rate increase of roughly 20 basis points, while commodity prices and reserves show little systematic movement; the posterior median contemporaneous coefficients imply the funds rate reacts nearly one-for-one to output (0.84) and more than one-for-one to prices (2.73), though the posterior intervals are wide (95% interval for the output coefficient: 0.04 to 5.25), reflecting the “double-edged sword” of set identification. The results are qualitatively robust to restricting the sample to the 1983-2007 Great Moderation period, where the estimated standard deviation of the policy shock falls from about 0.9 to about 0.3 and the output elasticity to the funds rate rises in magnitude to roughly -2, in line with Gertler and Karadi (2015). Applying the same systematic-component restrictions to evaluate Uhlig’s (2005) admissible set of structural parameters, the paper finds that Uhlig’s IRF-based sign restrictions alone imply, with posterior probability 1.00, a systematic reaction of the funds rate to reserves and, with probability 0.63, a negative reaction to output — both violations of the paper’s restrictions — so that combining Uhlig’s restrictions with the systematic-component restrictions collapses the admissible set toward the paper’s own contractionary and Taylor-rule-consistent conclusions. The authors caveat that the identified set remains wide because only one shock is identified out of many admissible structural VARs, that the pre-2007 sample excludes the zero-lower-bound and unconventional-policy period, and that a robustness exercise bounding the output and price coefficients to (0,4) is used to guard against implausible admissible models in the spirit of the critique in Kilian and Murphy (2012).

Uniform Priors for Impulse Responses

Structural vector autoregressions (SVARs) identified with sign restrictions are a widely used tool for estimating dynamic causal effects in macroeconomics. Critics—notably Baumeister and Hamilton (2015) and Watson (2020)—have called for caution because the standard practice of using a uniform prior over the set of orthogonal matrices (with respect to the Haar measure) induces non-uniform marginal prior distributions over the identified sets of individual impulse responses. This paper formally challenges that caution: through an if-and-only-if theorem the authors show that the uniform prior over orthogonal matrices is not only sufficient but also necessary to induce a uniform joint prior distribution over the identified set for the vector of impulse responses—a result that holds for any prior distribution over the reduced-form parameters. The paper additionally shows how to conduct posterior inference based on a uniform joint prior for the vector of impulse responses, which requires modifying the prior for the reduced-form parameters away from the standard Minnesota prior while retaining the uniform prior over orthogonal matrices. An application to Watson’s (2020) empirical example finds that joint credible sets under this new prior are similar to, but wider than, those obtained under the standard approach, and that imposing tighter identifying restrictions sharpens inference under both priors.