ABCs (and Ds) of Understanding VARs
📄 Summarized from the full manuscript · Human-reviewed for faithfulness before publication
In brief
Macroeconomists routinely fit a compact statistical model to a few time series and read off how the economy responds to a shock. When does that recover the true shocks of the theory that generated the data? This 2007 paper supplies a check that runs directly on a model's own equations. When it fails, the fitted model recovers only a filtered version of the economy, its shocks are noisier than the true ones, and the traced responses can differ even in sign, as a saving-and-income example shows. That matters because it tells researchers when their estimated responses cannot be trusted, and that watching more variables restores the link.
What this paper finds — and why it matters
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.
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 does the paper address, and why does it matter for interpreting structural VARs?
The paper asks when the economic shocks in a DSGE model’s state-space representation are recoverable from the one-step-ahead forecast errors (“innovations”) of a VAR fit to the model’s observables — the invertibility problem. If invertibility fails, a VAR’s identified “structural” shocks and impulse responses do not correspond to the underlying economic model’s shocks and responses, even if the VAR is estimated without error on population data. The paper revisits this known issue and proposes a simple diagnostic for detecting it (Introduction, p. 1021).
Q2. What is the “poor man’s invertibility condition,” and how is it derived?
In the square case — where the number of observables k equals the number of structural shocks m and the shock-loading matrix D is nonsingular — VAR innovations equal the model’s structural shocks if and only if the eigenvalues of the matrix A − BD^{-1}C are strictly less than one in modulus (Condition 1). Starting from the state equation x_{t+1} = Ax_t + Bw_{t+1} and observable equation y_{t+1} = Cx_t + Dw_{t+1}, nonsingular D lets one solve for w_{t+1} = D^{-1}(y_{t+1} − Cx_t) and substitute into the state equation, giving [I − (A − BD^{-1}C)L]x_{t+1} = BD^{-1}y_{t+1} (Eq. 4). If Condition 1 holds this lag-polynomial operator is invertible, so x_{t+1} is a square-summable linear combination of current and past y’s, var(x_t|y^t) = 0, and the complete state is effectively observed (Section I.C, p. 1022).
Q3. What happens when Condition 1 fails — what does a VAR estimated on the observables actually recover?
When at least one eigenvalue of A − BD^{-1}C is one or greater in modulus, the VAR instead uncovers the model’s “innovations representation” — a different state-space system in which the Kalman-filtered estimate x̂_t = E(x_t|y^t) replaces the true state x_t, and orthogonalized forecast errors ε_{t+1} replace the true shocks w_{t+1} (Eqs. 7-8). The steady-state Kalman filter recursions (Eqs. 9-12) imply D̂D̂’ = DD’ + CΣC’, where Σ = var(x_t|y^t) ≥ 0; when Condition 1 fails, Σ > 0, so D̂D̂’ > DD’ — the VAR’s innovations have strictly larger variance than the true structural shocks, because they combine the true shock’s contribution Dw_{t+1} with an uncorrelated state-estimation error term C(x_t − x̂_t) (Section I.D, pp. 1022-1023).
Q4. How does the permanent-income example set up the invertibility failure?
In the Sargent (1987) permanent-income consumption model calibrated with gross interest rate R = 1.2 and income-shock scale σ_w = 1, only the scalar surplus y_t − c_t is observed, and A − BD^{-1}C reduces to R > 1 — so Condition 1 fails and Σ = var(c_t | y^t − c^t) = σ²_w(1 − R^{-2}) > 0, meaning consumption cannot be perfectly inferred from the observed history of the surplus alone (Section II, p. 1025). In the true model, consumption c_{t+h} responds positively and permanently to an income shock (rising by σ_w(1 − R^{-1})), income y_{t+h} has a hump peaking on impact, and because consumers use all of their permanent income the surplus response has a present value of exactly zero (Figure 1, pp. 1023-1024).
Q5. How do the VAR’s impulse responses in this example differ from the model’s true responses?
The VAR recovers only an AR(1) for the surplus with innovations representation coefficient A − B̂D̂^{-1}C = R^{-1} < 1 (Eqs. 15-16), and its impulse responses are “markedly different” from the true model’s: consumption ĉ_{t+h} responds negatively to a VAR shock — the opposite sign from its true positive, permanent response — and the surplus’s present value is positive in the VAR rather than the true zero. The authors summarize this as the present value of the impulse response of consumption falling short of the present value of the impulse response of income in the VAR representation, unlike in the true model where the two exactly offset (Section II, p. 1025).
Q6. What does the paper conclude, and what are the scope limits of the analysis?
The authors conclude that observing additional variables — such as consumption, income, or the value of accumulated assets — can restore invertibility in the permanent-income example, and more generally argue that VARs remain “informative about the shapes of impulse-responses to some economic shocks that theories should attempt to match, while others are not,” useful for “coax[ing] interesting patterns from the data” even under model uncertainty. The analysis is explicitly restricted to the square case (k = m, D nonsingular); non-square systems require a more complex condition. It is also purely theoretical, offering the permanent-income economy as its only illustration with no empirical VAR estimation or Monte Carlo evidence on bias magnitudes in realistic DSGE systems, and it assumes Gaussian shocks and the time-invariant (steady-state) limits of the Kalman filter (Section III and Caveats, pp. 1021-1025).
Key terms in this paper
Definitions below follow the paper's own usage.
- invertibility (Condition 1)
- in this paper, the property that the state vector x_{t+1} can be recovered as a square-summable linear function of current and past observables y^{t+1}, which holds if and only if the eigenvalues of A − BD^{-1}C are strictly less than one in modulus; when it holds, VAR innovations equal the model's true structural shocks w_{t+1}.
- poor man's invertibility condition
- the authors' practical shorthand for Condition 1 — compute the matrix A − BD^{-1}C directly from the model's state-space matrices and check whether its eigenvalues lie strictly inside the unit circle, without needing to derive the model's full VARMA representation.
- innovations representation
- the alternative state-space system (Eqs. 7-8) in which the Kalman-filtered state estimate x̂_t = E(x_t|y^t) replaces the true state x_t and orthogonalized one-step-ahead forecast errors ε_{t+1} (from the steady-state Kalman filter) replace the true shocks w_{t+1}; this is what a VAR estimated on y_t always recovers, and it coincides with the true economic model only when Condition 1 holds.
- square case
- the paper's baseline setting, in which the number of observables k equals the number of economic shocks m and the shock-loading matrix D is nonsingular; this is what permits solving Eq. (2) for w_{t+1} in closed form and deriving Condition 1 directly, whereas for k ≠ m the invertibility condition is described as more complex.
- permanent-income surplus example
- the paper's single illustrative economy (Sargent 1987, chap. XII), in which only the scalar surplus y_t − c_t is observed; because A − BD^{-1}C = R > 1 there, Condition 1 fails, and the VAR's AR(1) representation of the surplus produces impulse responses (e.g., consumption falling rather than rising on impact, a nonzero present value of the surplus response) that differ sharply, including in sign, from the true model's budget-balanced responses.