var.var_step()

Build the ssoe() mean and update of a VAR.

Usage

Source

var.var_step(
    phi,
    intercept=None,
)

The carry is the lag window (*batch, lags, obs). mean emits var_mean() of the window as the one-step-ahead mean and, given the row’s value, update drops the oldest row and appends the new one. Both ignore their exogenous input x_t; add regressors by wrapping mean (a VARX):

mean, update = var_step(phi, intercept)


def mean_x(carry, x_t):
    return mean(carry, x_t) + beta @ x_t

The pair knows nothing about priors: phi and intercept are whatever the model sampled (a weakly informative Normal, the moments of minnesota_prior(), a hierarchical prior, …), so changing the prior never touches the recursion.

Parameters

phi: Float[Array, " *#batch lags obs obs"]

Coefficient tensor (*batch, lags, obs, obs); see the module docstring.

intercept: Float[Array, " *#batch obs"] | None = None

Optional intercept of shape (*batch, obs).

Returns

tuple[
    SSOEMean[Float[Array, "*batch lags obs"]],
    SSOEUpdate[Float[Array, "*batch lags obs"]],
]

A (mean, update) pair for ssoe(): mean(carry, x_t) is var_mean() of the window and update(carry, y_t, eps_t, x_t) drops the oldest row and appends y_t.

Raises

ValueError

At step time, if the carry does not hold exactly phi.shape[-3] rows (the usual cause is an init_carry with the wrong number of lags).

Examples

An observed VAR(p) conditioned on its first p rows. y_init is the seed window, a constant closed over by the model; the likelihood rows travel through covariates (padded with pad_future() to fix the forecast horizon) and through data:

def var_model(covariates, data=None):
    h = Horizon.from_data(covariates, data)
    y = covariates[..., : h.t_obs, :]
    intercept = jnp.asarray(
        numpyro.sample("intercept", dist.Normal(0.0, 1.0).expand([k]).to_event(1))
    )
    sigma = jnp.asarray(numpyro.sample("sigma", dist.HalfNormal(1.0).expand([k]).to_event(1)))
    l_omega = jnp.asarray(numpyro.sample("l_omega", dist.LKJCholesky(k, concentration=1.0)))
    phi = jnp.asarray(
        numpyro.sample("phi", dist.Normal(0.0, 1.0).expand([p, k, k]).to_event(3))
    )
    scale_tril = sigma[..., :, None] * l_omega
    noise = dist.MultivariateNormal(jnp.zeros(k), scale_tril=scale_tril)
    mean, update = var_step(phi, intercept)
    r = ssoe(h, "eps", y, y_init, mean, update, noise)
    numpyro.sample("obs", dist.MultivariateNormal(r.mu, scale_tril=scale_tril), obs=h.data)
    if h.future > 0:
        numpyro.deterministic("forecast", r.y_future)

Closing over y_init is right for a single fit but wrong under backtest(), which slices covariates per window: for a backtest, ship the seed rows inside covariates and slice the carry from them in the model.