Data-Driven Models
The impulse-response function is the primary output of any SVAR. Showing IRFs from two identification schemes (recursive vs. sign-restricted) on the same panel exposes how identification assumptions drive the shape, sign, and persistence of the estimated structural responses.
Peak recursive response
0.01%
Maximum absolute response of the first variable to structural shock 1 under recursive identification.
Sign-set width at impact
5.49%
Width of the identified set for the impact response under sign restrictions. Wider sets mean weaker identification.
Acceptance rate
1.76
Fraction of random rotations that satisfy the imposed sign restrictions. Low rates indicate tight constraints.
Half-life (recursive)
12.98
Number of periods for the recursive IRF to decay below half its impact value.
Diagonal elements of the VAR(1) coefficient matrix. Higher values mean more persistent dynamics and slower shock absorption.
Increasing persistence stretches the IRF horizontally, delays the peak response, and widens the sign-restricted identified set at long horizons.
Off-diagonal elements of A. Negative a12 means the interest rate depresses output; positive a21 means output feeds back into the interest rate.
Larger cross-feedback produces hump-shaped IRFs and amplifies the difference between recursive and sign-restricted responses.
Contemporaneous correlation between reduced-form residuals. This is what makes identification non-trivial -- high |rho| means the Cholesky ordering matters.
As |rho| rises, the recursive and sign-restricted IRFs diverge more, showing how identification assumptions bite harder when contemporaneous correlation is strong.
Number of Haar-uniform rotations sampled to characterize the sign-restricted identified set.
More draws produce a smoother identified-set envelope; fewer draws make the shaded region jagged and potentially miss corners of the set.
Maximum horizon for the impulse-response function.
Longer horizons reveal whether the response returns to zero (stationary VAR) or drifts (near-unit-root).
Kernel: time-series | Viz: Structural impulse-response functions with identification-scheme comparison
Transition logic
The response can jump on impact, then decay, reverse, or hump according to the transition matrix. The connecting line is a horizon guide.
Impact jump
Applies to: Horizon h=0
A nonzero point at h=0 means the shock changes the plotted variable on impact. It is not the end of a smooth pre-shock segment.
In a linear state-space representation, B maps the innovation into the contemporaneous state. A delayed variable has a zero row in B and moves only after A propagates the shock. A jump is therefore a timing restriction, not a charting artifact.
Decay or hump
Applies to: Horizons h>0
Post-impact movement follows the model recursion. A hump means delayed propagation, not interpolation by the chart.
Stable eigenvalues shrink repeated applications of A. If a variable is reached through another state first, the response rises before the stable decay dominates, producing a hump-shaped path.
Variable timing
Innovation
ε0The shock is dated at horizon zero. A response at h=0 is an impact effect, not the end of a hidden ramp.
An impulse-response experiment sets one innovation at the origin and traces conditional responses afterward. The timing is imposed by the experiment's information set.
State vector
xhA state can move on impact only through the relevant row of B; otherwise it moves after A propagates the shock.
Contemporaneous incidence is governed by B. Dynamic propagation is governed by powers of A, so different variables can jump, lag, reverse, or hump in the same model.
Reported response
rh=Cxh+DεhThe plotted variable can jump directly through D or indirectly through the state vector.
Measurement maps turn states into observables. Direct-feedthrough terms create impact movement even when a state stock itself is predetermined.
Stability checks
Stable propagation
ρ(A)<1Repeated applications of A shrink stable components, so responses decay after impact and any hump is temporary.
A unit root, explosive eigenvalue, or permanent component can keep the response from returning to zero.
For a finite-dimensional linear system, A^h approaches zero on stable modes if every eigenvalue lies inside the unit circle. That is the formal reason a stable IRF dies out.
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