VARMA representation of DSGE models
📄 Summarized from the full manuscript · Human-reviewed for faithfulness before publication
In brief
How compactly can a fully specified macroeconomic model be rewritten as a statistical time-series equation, and can its deep parameters be recovered from that form? This 2016 note shows an earlier general answer is far larger than necessary: for a widely used medium-sized model it implies nineteen lags, where the paper's own condition shows three suffice. It also shows the parameters recoverable from the full model's likelihood are exactly those recoverable from the compact form — 36 of 41 in that model, leaving 125 testable restrictions. Why it matters: a compact equivalent form makes such models cheaper to estimate and easier to test.
What this paper finds — and why it matters
This 2016 Economics Letters paper by Stephen D. Morris asks how concise the VARMA (vector autoregressive moving-average) representation of a DSGE model can be made, and whether the model’s structural parameters can be locally identified from that representation. Starting from the general “ABCD” state-space form of a DSGE model (Fernandez-Villaverde et al. 2007) – states X_t (m x 1), observables Y_t (n x 1), transition matrix A, and shock-loading matrices B, C, D – Morris shows that a prior general result (Ravenna 2007) implying every such model has a VARMA(n+m, n+m-1) representation is far larger than necessary in practice: for the Smets-Wouters (2007) model, with n=7 observables and m=12 states, that bound is a “prohibitively large” VARMA(19,18). Under Assumption 1 – observables no more numerous than states (n <= m) and an invertible shock-loading matrix D, i.e., the model is not stochastically singular – Morris derives a condition (Proposition 1) on the rank of an observability-type matrix Psi(kappa) that pins down a much lower-order VARMA(kappa+2, kappa+1) representation, where kappa is the minimum number of lags of the observables needed before the unobserved states become indirectly recoverable through them; corollaries collapse this further to VARMA(2,1), VARMA(1,1), or even VAR(1) representation under additional structure (n=m and C_X invertible; then A_Y=0 and C_Y=0; then an invertible map from X_t to Y_t). Applied to the Smets-Wouters (2007) model, checking that Psi(1) has full column rank 7 at reasonable parameter values shows the model actually has a VARMA(3,2) representation – far more concise than the VARMA(19,18) implied by Ravenna’s bound. Morris then shows that the largest subset of DSGE structural parameters that is locally identifiable from the entire model likelihood is also locally identifiable from the identifiable VARMA parameter subset alone – the nonzero AR coefficients phi together with vech of the innovation covariance matrix Omega – meaning no information outside this reduced parameter set aids local identification of the structural parameters; for Smets-Wouters, the paper reports the Jacobian of this map is 161x36 with full column rank 36 “for a range of reasonable parameterizations,” implying 36 of the model’s 41 structural parameters are locally identified this way and yielding 125 over-identifying restrictions useful for GMM-based testing. The paper is pure theory: it presents no data and no empirical estimation, its main results are stated as holding under Assumption 1 and as local (not global) identification results valid at generic rather than all parameter points, and it does not claim its VARMA(kappa+2, kappa+1) representation is the uniquely minimal one in every case.
Summary of a classic paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.
Questions & answers
Q1. What problem is this paper trying to solve, and why does it matter for applied DSGE work?
Morris asks how small the VARMA order of a DSGE model’s implied time-series representation can be made, because the previously known general bound was too large to be practically useful. Ravenna (2007) had shown that any DSGE model in the “ABCD” state-space form has a VARMA(n+m, n+m-1) representation, where n is the number of observables and m the number of states; for the Smets-Wouters (2007) model (n=7, m=12) this is VARMA(19,18), which the paper calls a “prohibitively large order for commonly utilized models with many states” (Section 2, p. 31). A more concise representation matters because the VARMA order determines how many autoregressive and moving-average lags a researcher must carry when working with the model’s reduced-form implications – for estimation, for computing the identifiable parameter subset, or for testing the model against unrestricted time-series behavior.
Q2. What is the ABCD framework this result is built on?
Morris adopts the ABCD state-space representation of a DSGE model from Fernandez-Villaverde et al. (2007): states X_t (m x 1) and observables Y_t (n x 1) evolve according to [X_t; Y_t] = A(theta)[X_{t-1}; Y_{t-1}] + B(theta)eps_t and Y_t = C(theta)[X_{t-1}; Y_{t-1}] + D(theta)eps_t, where eps_t is a k x 1 vector of structural shocks and theta the k x 1 vector of structural parameters (Section 2, p. 30). The matrix D maps structural shocks directly into observables; when D is invertible the model is not “stochastically singular” (there are no more shocks than observables), which is one of the two conditions (Assumption 1) needed for the paper’s main proposition.
Q3. What is the paper’s central theoretical result for shrinking the VARMA order?
Proposition 1 shows that under Assumption 1 (n <= m and D invertible), the ABCD model has a VARMA(kappa+2, kappa+1) representation whenever an observability-type matrix Psi(kappa) – built from C, powers of A, and the state-transition blocks A_X and C_X – exists and has full column rank n (Section 3, p. 31). The index kappa is the minimum number of lags of the observables Y_t needed before the unobservable states X_t become indirectly recoverable (“observable”) from that lagged history; the smaller the necessary kappa, the more concise the VARMA representation. Three corollaries give further reductions under additional structure: VARMA(2,1) if n=m and C_X is invertible (Corollary 1); VARMA(1,1) if additionally A_Y=0 and C_Y=0 (Corollary 2); and a plain VAR(1) if additionally there is an invertible matrix H with Y_t = HX_t (Corollary 3) (Section 3, p. 32).
Q4. What does the Smets-Wouters application show, concretely?
For the Smets-Wouters (2007) model (n=7 observables, m=12 states, D invertible), checking kappa=1 shows the 12x7 matrix Psi(1) is full column rank 7 “at reasonable parameterizations,” which means the model has a VARMA(3,2) representation rather than the VARMA(19,18) implied by Ravenna’s general bound (Section 3, p. 32). The paper reports that the third AR coefficient matrix has “three columns of exclusion restrictions,” leaving the AR parameter vector phi with r = 133 nonzero entries, so the identifiable VARMA parameter subset pi = (phi’, vech(Omega)’)’ is 133 + 7(8)/2 = 161-dimensional. (The wiki source flags that the stated exclusion-restriction description and the r=133 figure do not fully reconcile arithmetically from the excerpted passage alone; reproduced here as the paper states it – see review_flags.)
Q5. What does the concise VARMA representation imply for identifying the DSGE model’s structural parameters?
Morris shows that the largest subset of structural parameters locally identifiable from the model’s entire likelihood is also locally identifiable from the identifiable VARMA parameter subset pi alone – no information outside pi aids local identification of theta (Section 4, p. 32-33). For Smets-Wouters, under the Iskrev (2010) local-identification conditions at least 5 of the model’s 41 structural parameters must be fixed, and the Jacobian J(theta_0) = d(pi)/d(theta) evaluated at the remaining 36 is 161x36 with full column rank 36 “for a range of reasonable parameterizations” (Section 4, p. 32). This makes pi directly useful for GMM estimation of theta and produces 161-36=125 over-identifying restrictions that can be tested; the VARMA representation is described as useful independent of the realized rank of this Jacobian (Morris 2015, cited at Section 4, p. 33).
Q6. What are the main scope conditions and limitations on these results?
The results require Assumption 1 (n <= m, D invertible) and are explicitly local rather than global: the full-rank Jacobian claim holds “for a range of reasonable parameterizations,” not for all parameter values, so identification can still fail at measure-zero points in the parameter space (Section 4, p. 32; Section 3, p. 31). Proposition 1 also only establishes conditions under which a VARMA(kappa+2, kappa+1) representation exists – it does not prove this is the uniquely minimal representation in every case, though the corollaries identify further reductions when extra structure is present. Separately, the moving-average coefficient matrices {M_j} are not independently recoverable from the data; only phi and vech(Omega) – the identifiable subset pi – are directly useful for estimation or identification analysis (Section 2, p. 31).
Key terms in this paper
Definitions below follow the paper's own usage.
- ABCD representation
- the state-space form of a DSGE model used throughout the paper (following Fernandez-Villaverde et al. 2007), with an m x 1 state vector X_t, n x 1 observable vector Y_t, structural shocks eps_t entering through loading matrices B and D, and transition matrix A; the model is "stochastically singular" when D is not invertible (more shocks than observables), which the paper's main result explicitly rules out via Assumption 1.
- VARMA order (p, q)
- the number of autoregressive lags (p) and moving-average lags (q) needed to represent the observables Y_t as Y_t = sum of p AR terms in lagged Y plus a moving-average error with q lags of the innovation; the paper's central contribution is showing this order can be much smaller than the general VARMA(n+m, n+m-1) bound established by Ravenna (2007) for the same class of ABCD models.
- Observability index kappa and the matrix Psi(kappa)
- kappa is the minimum number of lags of the observable vector Y_t that must be stacked before the unobserved state vector X_t becomes indirectly recoverable ("observable") from that history, in the sense of classical systems theory (Kalman 1961); Psi(kappa) is the matrix (built from C, powers of A, and the state-transition blocks A_X, C_X) whose full column rank at a given kappa is the condition Proposition 1 checks, and the value of kappa that works directly sets the VARMA order to (kappa+2, kappa+1).
- Identifiable VARMA parameter subset pi(theta)
- the (r + n(n+1)/2)-dimensional vector pi = (phi', vech(Omega)')' stacking the r nonzero AR coefficients phi and the distinct elements of the moving-average-implied innovation covariance matrix Omega; this is the only part of a DSGE model's implied VARMA representation the paper treats as recoverable from data using lagged observables as instruments, since the individual moving-average coefficient matrices {M_j} are not separately identifiable.
- Local identifiability (of structural parameters via pi)
- the property, checked via the Jacobian J(theta_0) = d(pi)/d(theta) evaluated at a parameter point theta_0, that a subset of structural parameters can be distinguished from small perturbations around theta_0 when J(theta_0) has full column rank equal to that subset's dimension; the paper's identification result is that this local-identification property, when it holds for the entire likelihood, holds equally using pi alone -- but only "for a range of reasonable parameterizations," not at every point in the parameter space.