Calculator inputs
Use the form below to model how a one-time shock moves through a lagged statistical process. The form uses a responsive three-column layout on large screens, two columns on smaller screens, and one column on mobile.
Example data table
The following example shows a compact interpretation of a typical response sequence from a positive shock in a moderately persistent process.
| Period | Shock | IR Weight | Estimated Response | Cumulative Effect |
|---|---|---|---|---|
| 0 | 1.00 | 1.0000 | 1.0000 | 1.0000 |
| 1 | 0.00 | 0.8075 | 0.8075 | 1.8075 |
| 2 | 0.00 | 0.4986 | 0.4986 | 2.3061 |
| 3 | 0.00 | 0.3079 | 0.3079 | 2.6140 |
| 4 | 0.00 | 0.1901 | 0.1901 | 2.8041 |
Formula used
ψ0 = 1
ψh = φh-1(φ + θ)δh, for h ≥ 1
IRFh = Shock × ψh
SEh = (σ / √n) × √(1 + h)
Lowerh = IRFh − z × SEh
Upperh = IRFh + z × SEh
CumIRFh = Σ IRFj, from j = 0 to h
LRM = 1 + [δ(φ + θ) / (1 − φδ)], when the denominator is not zero
This tool uses a practical ARMA-style response framework with an additional damping factor. It is suited for exploratory statistics, teaching, scenario testing, and fast comparisons across different parameter settings.
How to use this calculator
- Enter the one-time shock size you want to simulate.
- Select the number of future periods to trace.
- Provide AR and MA coefficients to control persistence and immediate spillover.
- Set a damping factor to reflect extra decay or amplification.
- Add baseline level, innovation standard deviation, and sample size.
- Choose a confidence level for uncertainty bands.
- Enable normalization when you want responses per shock unit.
- Submit the form to see results above the calculator, including the graph and downloadable table.
Frequently asked questions
1. What does an impulse response measure?
It measures how a one-time shock affects future periods in a dynamic process. The response can show persistence, reversal, damping, or amplification over time.
2. Why are AR and MA coefficients both included?
The AR coefficient controls persistence from prior states. The MA coefficient captures immediate propagation from the innovation. Together they give a richer short-run and medium-run response pattern.
3. What does the damping factor do?
The damping factor adds extra decay or growth across future periods. Values below one dampen the response. Values above one amplify it and may create unstable trajectories.
4. What does normalization change?
Normalization divides response values by the shock size, making outputs comparable across scenarios. It is useful when you want a response per unit shock instead of a raw effect.
5. How should I read the confidence bounds?
The bounds provide an approximate uncertainty range around each period response. Wider bands indicate greater uncertainty, while narrower bands suggest more precise period estimates.
6. What does the cumulative line mean?
The cumulative line adds all past period responses through the selected horizon. It helps you understand the total accumulated influence of the original shock.
7. When is a process considered stable here?
This tool flags the system as stable when |φ × damping| is below one. That suggests the effect should fade instead of exploding over time.
8. Is this suitable for formal model inference?
It is best for exploration, teaching, and quick scenario analysis. Formal inference should rely on a fully estimated model, validated assumptions, and robust diagnostic testing.