# Using Long-Run Restrictions

Long-run restrictions identify structural shocks by what they do *eventually*, not on impact: a shock is defined by having no permanent effect on the level of some variable. This is the Blanchard-Quah scheme.

## Define the restriction

Give the variables in order, most restricted first, and name the shocks:

```python
from impulso.identification import LongRunRestriction

scheme = LongRunRestriction(
    ordering=["output_growth", "unemployment"],
    shock_names=["supply", "demand"],
)
```

That single zero says the demand shock has no permanent effect on the level of output. Shock `j` has no long-run effect on any variable ordered before it, so the ordering is the restriction — with two variables there is exactly one.

If naming the zeros directly is clearer than reasoning about an ordering, use the alternative constructor. It recovers both orderings from the pattern — the variables' and the shocks', so neither list has to be given in order — and refuses patterns that no ordering can produce:

```python
scheme = LongRunRestriction.from_zero_restrictions(
    restrictions={"output_growth": ["demand"]},
    var_names=["output_growth", "unemployment"],
    shock_names=["supply", "demand"],
)
```

:::{admonition} Your variables must be differenced
:class: warning
The restriction is on the long-run level of the variables *as they enter
the model*. "Demand has no permanent effect on output" therefore requires
output to enter as a growth rate — the level then accumulates, and a
zero cumulative effect on the growth rate is a zero permanent effect on
the level. If you pass output in levels, the restriction says the demand
shock has no permanent effect on the *growth rate*, which is a much
weaker and rarely intended claim. Impulso cannot detect the difference.
:::

## Apply it

```python
identified = fitted.set_identification_strategy(scheme)
irf = identified.impulse_response(horizon=40)
```

The scheme needs the posterior lag coefficients to build the long-run multiplier; `set_identification_strategy` passes them through automatically.

Rows of the structural shock matrix stay in the data's variable order, whatever `ordering` says. Only the shock columns follow `shock_names`.

## Read the diagnostics

Identification succeeds or fails per posterior draw, so the diagnostics are posterior quantities. Summary statistics ride along on the shock matrix:

```python
P = identified.shock_matrix()
P.attrs["long_run_singular_draws"]      # draws where C(1) is undefined
P.attrs["long_run_explosive_draws"]     # draws with spectral radius > 1
P.attrs["long_run_condition_q95"]       # conditioning of I - sum_j A_j
P.attrs["long_run_spectral_radius_max"]
```

For the full per-draw picture:

```python
diagnostics = scheme.long_run_diagnostics(fitted.idata.posterior)
diagnostics["condition"]         # shape (chains, draws)
diagnostics["spectral_radius"]   # shape (chains, draws)
```

A spectral radius above one means that draw's moving-average sum diverges, so the long run does not exist for it. Impulso warns and reports those draws but keeps them: posteriors on persistent data routinely put mass above one, and dropping them would quietly condition the posterior on stability.

## When draws are undefined

The long-run multiplier is `C(1) = (I - sum_j A_j)^-1`. When that matrix is close to singular, the inverse is numerically meaningless. Those draws are blanked:

```python
scheme = LongRunRestriction(
    ordering=["output_growth", "unemployment"],
    shock_names=["supply", "demand"],
    on_undefined="nan",     # default; "raise" errors instead
    max_condition=1e8,      # condition number above which C(1) is refused
)
```

`NaN` draws propagate: impulse responses and forecast error variance decompositions (FEVD) report `NaN` for them rather than a misleading zero, and highest-density intervals over a mixed set of draws may come back `NaN` too. The scenario methods (`counterfactual`, `structural_scenario`) reject them outright. If any of that is in your plans, run with `on_undefined="raise"` so the problem surfaces at identification time instead of three steps later.

## A climate example

The scheme is not specific to macroeconomics. Take global-mean surface temperature and an El Niño-Southern Oscillation (ENSO) index, with temperature differenced:

```python
scheme = LongRunRestriction(
    ordering=["temperature_change", "enso_index"],
    shock_names=["forced", "internal"],
)
identified = fitted.set_identification_strategy(scheme)
```

The restriction: internal variability has no permanent effect on the level of global-mean temperature, while forced variability may. That is one assumption, stated plainly, and it is doing all the identifying work — so be clear about what it commits you to:

- Temperature must enter as a change, not a level, or the restriction means something else.
- With two variables this is exactly one restriction. Adding a third variable would assert three zeros at once, which is a far stronger joint claim.
- The zero is exact and permanent. If internal variability has a small but genuinely permanent effect, the scheme will attribute it to the forced shock.