Invertibility Problem

Definition

The invertibility problem arises when the history of a vector autoregression (VAR)'s one-step-ahead innovations $\{a_t\}$ does not span the same Hilbert space as the history of the underlying economic shocks $\{w_t\}$ . Formally, $H(a_t) \subsetneq H(w_t)$ . When non-invertibility holds, the VAR impulse responses cannot be made to match the economic model's impulse responses for any choice of identification matrix $G$ .

Key Ideas

By construction $H(a_t) \subseteq H(w_t)$ : VAR innovations are always linear combinations of current and past economic shocks. The question is whether the reverse holds.
Invertibility is a property of the pair (model, observable set) — the same model can be invertible for one choice of observables and non-invertible for another.
A unit eigenvalue of $A - BD^{-1}C$ is a benign borderline case: the model remains invertible but lacks an infinite-order VAR representation.
The practical consequence of non-invertibility: the VAR overestimates forecast error variance ( $GG' = C\Sigma C' + DD' > DD'$ when $\Sigma \neq 0$ ), and impulse response function (IRF) shapes diverge from the true economic model.
Omitted-variables interpretation (Stock-Watson 2018): invertibility holds iff knowledge of the past true shocks would not improve a VAR forecast of $Y_t$ — there is no forecasting gain from adding the shocks. An external instrument sidesteps invertibility: under lead–lag exogeneity, LP-IV identifies the structural IRF without invertibility, and comparing LP-IV to the invertibility-dependent SVAR-IV gives a Hausman-type test of invertibility. But a "no free lunch" result shows that once lagged controls are needed in LP-IV, its validity condition is again equivalent to invertibility.

How It Works

The Mapping from Economic to VAR Shocks

From the State-Space Representation $(A,B,C,D)$ , define:

$A^* = \begin{pmatrix} A & 0 \\ KC & A - KC \end{pmatrix}$

The mapping from economic shocks $w_t$ to VAR innovations $G\varepsilon_t$ is:

$G\varepsilon_t = \left[D + \begin{pmatrix}C \\ -C\end{pmatrix}(I - A^*L)^{-1}\begin{pmatrix}B \\ KD\end{pmatrix}L\right]w_t \tag{25}$

The model is invertible — $H(\varepsilon_t) = H(w_t)$ — if and only if the operator in brackets has a square-summable inverse in non-negative powers of $L$ .

The Invertibility Theorem (Square Case)

Assumptions: (1) all eigenvalues of $A$ are strictly less than 1 in modulus (stable system); (2) $D$ is square and invertible ( $k = m$ ).

Theorem (Fernández-Villaverde, Rubio-Ramírez, and Sargent [FVS] 2005, Theorem III.1): The zeros of $\det[I + C^*(zI - A^*)^{-1}B^*]$ equal the eigenvalues of $A - BD^{-1}C$ and the eigenvalues of $A$ .

Corollary: The mapping (25) is invertible if and only if all eigenvalues of $\mathbf{A - BD^{-1}C}$ are strictly less than 1 in modulus.

Check: Compute $A - BD^{-1}C$ from the $(A,B,C,D)$ tuple of the model and inspect its eigenvalues. No Kalman filter calculation is needed.

When Invertibility Holds: $K = BD^{-1}$ , $\Sigma = 0$

Theorem (FVS 2005, Theorem III.4): If $D^{-1}$ exists and $A - BD^{-1}C$ is stable, then the steady-state Kalman filter gives $K = BD^{-1}$ and $\Sigma = 0$ .

Consequences:

No hidden state variables: $x_t \in H(y_t)$ .
$G\varepsilon_t = Dw_t$ — VAR innovations are contemporaneously equal to the economic shocks scaled by $D$ .
The correct identification matrix is $G = D$ (up to an orthogonal rotation), and the VAR IRF exactly matches the economic model IRF.
Innovation covariance $GG' = DD'$ .

The Four-Level Hierarchy of Challenges

Even in the square case, mapping VAR shocks to economic shocks faces four nested challenges:

Level	Problem	Consequence
1	$k \neq m$ : wrong number of shocks vs. observables	VAR is stochastically singular or over-dimensioned
2	Invertibility fails ( $H(\varepsilon_t) \subsetneq H(w_t)$ )	IRFs cannot be matched for any $G$
3	Identifying $G$ from $GG' = \Omega$	Requires structural restrictions (see Structural Identification)
4	VAR is generically a vector autoregressive moving average (VARMA), not finite-order autoregressive (AR)	Truncated VARs are approximations; lag length matters

Scalar Intuition: MA(1) Example

The scalar process $y_t = w_t + \alpha w_{t-1}$ has state-space $(A,B,C,D) = (0, 1, \alpha, 1)$ so: $A - BD^{-1}C = -\alpha$

This is stable iff $|\alpha| < 1$ . When $|\alpha| > 1$ , the process has a non-fundamental Wold representation — the standard AR approximation assigns the wrong dynamics to $w_t$ .

Four Worked Examples

Model	Observables	Max $\lvert\operatorname{eig}(A-BD^{-1}C)\rvert$	Invertible?
Permanent income	$(c_t, d_t)$	$R = 1.05$	No — fatal
Fisher (2003) two-tech	$(\Delta\log p_t, \Delta(y_t-l_t), l_t)$	$< 1$	Yes
Erceg-Henderson-Levin (2000)	$(\Delta p_t, \Delta n_t, y_t, w_t - p_t)$	$= 1$	Yes (benign borderline)
Household production — Case I	$(y_m, l_m)$	$0.910$	Yes
Household production — Case II	$(c_m, l_m)$	$1.096$	No

The household production example confirms Watson's (1994) insight: non-invertibility is most likely when the observable set omits variables that load heavily on expectations of future values.

Posterior Probability of Non-Invertibility

For models where calibrated parameter values suggest invertibility, Markov Chain Monte Carlo (MCMC) can be used to compute the posterior probability that any draw of structural parameters violates the eigenvalue condition. For the Fisher and EHL models (Case I household production), this posterior probability is zero. For household production Case II, it is one.

Why It Matters

Non-invertibility means VAR-based inference about structural shocks is fundamentally compromised — no amount of cleverness in identification can recover the economic IRF. The eigenvalue check $\rho(A - BD^{-1}C) < 1$ is a necessary pre-condition for any Dynamic Stochastic General Equilibrium (DSGE)-to-VAR comparison to be interpretable.

Open Questions

The non-square case ( $k \neq m$ ) has no analogous clean criterion.
When $\rho(A - BD^{-1}C) > 1$ , changing the observable set can restore invertibility (as the household production example shows), but there is no systematic guidance on how to choose observables.
Non-linear models introduce additional invertibility challenges not covered by this framework.