Vector autoregressions and reduced form representations of DSGE models

Ravenna asks under what conditions a DSGE model’s solution implies its observable variables follow a finite-order VAR, and, when it does not, how severely a VAR estimated at a finite lag length misrepresents the model’s true dynamics — both in its reduced-form coefficients and in the structural shocks/impulse responses extracted from it. This matters because most SVAR practice implicitly assumes the fitted VAR is (at least approximately) the true data-generating process; the paper asks how costly that assumption is when the underlying economy is actually a DSGE model (Section 1, pp. 2048-2050).

Substituting the state-variable law of motion x_t = Rx_{t-1} + Sz_t into the observation equation y_t = Px_{t-1} + Qz_t produces a Wold representation with a moving-average (MA) component that generically does not cancel, so the observables Y-tilde_t follow a VARMA(infinity) process, not a finite VAR. The DSGE equilibrium law of motion (Eqs. 1-3) is the source of this: y_t depends on lagged states x_{t-1}, and x_{t-1} in turn depends on the entire history of shocks, so eliminating the unobserved states from the system leaves behind an MA term (Section 2, pp. 2050-2052).

Proposition 2.1 shows that when the observed vector consists of the model’s endogenous variables yt (with the number of exogenous shocks set equal to the number of those variables), it has a VARMA(n+qm, n+q(m-1)) representation, where n is the number of endogenous state variables, m is the number of exogenous shocks, and q is the autoregressive order of the exogenous shock process; for the common case of AR(1) shocks (q=1) this simplifies to VARMA(n+m, n+m-1) (Section 2, p. 2051). The order therefore rises mechanically with the number of state variables and shocks carried by the underlying DSGE model — larger, more disaggregated models imply higher-order VARMA representations for their observables — but the paper states this as a general proposition and does not work through a specific calibrated numerical example, nor does it characterize the bound as loose; it moves directly from Proposition 2.1 to the finite-order VAR existence condition in Corollary 2.2.

Corollary 2.2 restates the finite-order VAR condition from Proposition 2.1 as a determinant condition on the model’s state-space matrices: a finite-order VAR representation exists if and only if the determinant of [I − (R − SQ^{-1}P)L] is of degree zero in L — i.e., the VARMA’s moving-average component must cancel entirely, a knife-edge cancellation that depends on the detailed matrix structure of (P, Q, R, S) rather than on model size alone (Section 2, p. 2052). The paper states more generally that a finite-order VAR consistent with a given DSGE model requires either (a) all the endogenous state variables xt to be included among the observables, or (a′) this same determinant condition to hold (Section 2, p. 2052). Proposition 2.3 extends the VARMA-order formula and finite-VAR condition of Proposition 2.1 to the case where the observed vector instead mixes a subset of the state variables xt with a subset of the endogenous variables yt, showing the same order formula and existence condition apply regardless of exactly which variables are dropped from or retained in the observed set (Section 2, p. 2052).

Ravenna distinguishes pure truncation bias — biased reduced-form VAR coefficients and impulse responses from cutting off the true infinite-order VAR at a finite lag p, present even with no structural identification imposed — from a separate identification bias, which arises because applying a structural identification scheme (Cholesky, sign restrictions, or long-run restrictions) to the truncated VAR’s own impulse responses recovers the wrong identifying matrix, since those truncated-VAR impulse responses are not equal to the DSGE model’s true impulse responses even asymptotically. Truncation bias is governed by the largest eigenvalue of the matrix (R - SQ^{-1}P): the closer this eigenvalue is to 1, the more slowly the VARMA’s MA coefficients decay to zero and the worse any fixed-lag VAR approximates them (Section 3, pp. 2053-2055). The two bias sources are additive and can reinforce or partly offset one another, and adding lags shrinks truncation bias but does not, by itself, remove identification bias (Section 4, pp. 2062-2063).

Using a two-shock Hansen (1985) RBC model (technology-shock persistence rho=0.35, sigma_z=0.0148; labor-supply persistence rho_d=0.8, sigma_d=0.009) with output growth and hours as the two observables, a VAR(2) fit to the simulated data — even when identified with the theoretically correct identification matrix — understates the true impulse response of hours to a technology shock by more than 60% at the 10-quarter horizon, and the VAR(2) response decays to zero by about 25 quarters where the true response is substantially more persistent (Section 3, pp. 2058-2059, Fig. 1). The output-growth response is described as less severely biased than the hours response in the same exercise (Section 3, p. 2059).

Applying a Blanchard-Quah (1989) long-run restriction — which identifies the technology shock by assuming the labor-supply shock has no long-run effect on the level of log output — to the truncated VAR(2) compounds the truncation bias into an identification bias: the resulting impact response of hours to the technology shock is about 75% larger than the true theoretical response, even though the long-run restriction is often thought of as agnostic about short-run dynamics (Section 3, pp. 2059-2060). Ravenna notes Chari, Kehoe and McGrattan (2005) obtain an analogous result in a similar model, and explicitly contrasts this with Erceg, Guerrieri and Gust (2005), who conclude pure truncation bias is negligible in population and mainly a small-sample problem — a genuine disagreement in the literature that the paper flags rather than resolves (Section 3, p. 2059).

Table 2 shows the VAR(2)-estimated structural shocks (orthogonalized with the theoretical identification matrix) track the true simulated shocks closely: a 0.99 correlation and 3.99% relative RMSE for the technology shock, and a 0.98 correlation and 16.04% relative RMSE for the labor-supply shock, computed from simulations of 1.5 million observations. The paper attributes this robustness to the fact that shock estimates are built from the contemporaneous data vector X_t rather than from lagged estimates, so truncation error does not compound over time the way it does for impulse responses, which depend recursively on their own lagged values (Section 3, p. 2060, Table 2).

Raising the labor-supply shock’s persistence from rho_d=0.8 to rho_d=0.97 leaves the true technology-shock impulse response unchanged but sharply improves the VAR(2) approximation to it, because higher persistence in this model changes the matrix structure of (R - SQ^{-1}P) in a way that shrinks its dominant eigenvalue, speeding the decay of the underlying VARMA’s moving-average coefficients (Section 3, pp. 2060-2062). Ravenna is explicit that whether truncation/identification bias is quantitatively important is model- and parameterization-specific, not a generic property of DSGE models, and that the paper offers no practical diagnostic test for detecting when a given empirical VAR suffers from this kind of bias (Section 4, pp. 2062-2063). The whole exercise is a Monte Carlo/analytical illustration using a specific two-variable RBC calibration — the paper reports no empirical estimation on actual macroeconomic data.