Vector autoregressions and reduced form representations of DSGE models

This 2007 Journal of Monetary Economics paper by Federico Ravenna asks a purely methodological question with direct consequences for structural VAR (SVAR) practice: under what conditions does a DSGE model’s equilibrium solution imply that its observable variables follow a finite-order vector autoregression (VAR), and when that condition fails, how badly does a VAR fit to DSGE-generated data misrepresent the model’s true dynamics? Substituting the DSGE state-transition equations into the observation equation shows that the observables generically follow a VARMA process rather than a finite VAR: Proposition 2.1 derives the general VARMA(n+qm, n+q(m-1)) order for a model with n observables, m state variables, and exogenous-shock lag order q (VARMA(n+m, n+m-1) in the standard AR(1)-shock case), while Corollary 2.2 shows a finite-order VAR exists only under a knife-edge cancellation condition on the model’s coefficient matrices (P, Q, R, S) — the determinant of [I − (R − SQ^{-1}P)L] must be of degree zero in L — that fails generically and depends on the detailed matrix structure rather than on model size alone; Proposition 2.3 extends both the VARMA-order formula and the finite-VAR condition to the case where the observed vector mixes a subset of the state variables with a subset of the endogenous variables, showing the same results hold regardless of exactly which variables are observed. When the finite-VAR condition fails, truncating the true infinite-order VAR at a finite lag p introduces two additive but distinct sources of bias: pure coefficient-truncation bias, whose severity is governed by the largest eigenvalue of the matrix (R - SQ^{-1}P) — the closer this eigenvalue is to 1, the more slowly the true VARMA’s moving-average coefficients decay and the worse a fixed-lag VAR approximates them — and a separate identification bias that arises because a structural identification scheme (Cholesky, sign restrictions, or long-run restrictions) applied to the truncated VAR’s own misspecified impulse responses recovers the wrong identifying matrix even in population. Ravenna illustrates both channels with a Monte Carlo built on a two-shock Hansen (1985) RBC model (technology-shock persistence rho=0.35, sigma_z=0.0148; baseline labor-supply-shock persistence rho_d=0.8, sigma_d=0.009) with output growth and hours as the two observables, whose implied second moments are checked against US data over 1955:1-2006:1 and a 1980:1-2006:1 subsample: a correctly-identified VAR(2) fit to simulated data understates the true impulse response of hours to a technology shock by more than 60% at the 10-quarter horizon and lets it decay to zero by about 25 quarters (versus a much more persistent true response), and layering a Blanchard-Quah (1989) long-run identification scheme on top of the same truncated VAR compounds this into a roughly 75%-too-large impact response of hours — even though the truncated VAR’s own estimated structural shocks remain remarkably accurate (correlation with the true shocks of 0.99 for technology and 0.98 for labor supply, with relative RMSEs of 3.99% and 16.04% respectively, computed from simulations of 1.5 million observations). A further experiment shows that raising the labor-supply shock’s persistence to rho_d=0.97 — without changing the true technology-shock impulse response at all — sharply improves the VAR(2) approximation, because it shrinks the dominant eigenvalue of (R - SQ^{-1}P) and speeds the decay of the VARMA moving-average coefficients. The paper is explicit that these are Monte Carlo and analytical results about population-level approximation error in one two-variable RBC calibration, not empirical findings about the actual economy, and that adding lags reduces truncation bias but does not eliminate identification bias, so whether the combined bias is quantitatively important for a given empirical VAR is model- and parameterization-specific.

Federico Ravenna | Macro Paper Warehouse

Vector autoregressions and reduced form representations of DSGE models