VARMAResult.forecast

VARMAResult.forecast(steps=1)

Deterministic multi-step forecasts from the end of the sample.

The moving-average block contributes only while a forecast date is within q periods of the sample: the innovations after the sample end are set to their zero mean, and the last q estimated innovations decay out of the forecast one step at a time,

\[\hat y_{T+h} = D d_{T+h} + \sum_{i=1}^{p} A_i\, \hat y_{T+h-i} + \sum_{j=h+1}^{q} M_j\, \hat u_{T+h-j}.\]

Beyond q steps the iteration is exactly the autoregressive one.

Parameters:

steps (int) – Forecast horizon, at least one.

Return type:

NDArray[double]

Returns:

An array of shape (steps, k).

Raises:

SpecificationError – If steps is less than one.

Example

The first step carries the last innovation through M_1; the second is the pure autoregressive iteration of the first:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> u = rng.standard_normal((302, 2))
>>> y = np.zeros((302, 2))
>>> for t in range(1, 302):
...     y[t] = 0.5 * y[t - 1] + u[t] + 0.4 * u[t - 1]
>>> res = VARMA(y[2:], order=(1, 1)).fit()
>>> path = res.forecast(2)
>>> first = res.deterministic[0] + res.coefficients[0] @ y[-1]
>>> first = first + res.ma_coefficients[0] @ res.resid[-1]
>>> bool(np.allclose(path[0], first))
True
>>> bool(np.allclose(path[1], res.deterministic[0] + res.coefficients[0] @ path[0]))
True
>>> res.forecast(0)
Traceback (most recent call last):
cultivars.exceptions.SpecificationError: steps must be at least 1; got 0.