var.companion_matrix()

Stack the VAR coefficients into the companion form of a VAR(1).

Usage

Source

var.companion_matrix(phi)

F = \begin{bmatrix} \Phi_1 & \Phi_2 & \cdots & \Phi_{p-1} & \Phi_p \\ I & 0 & \cdots & 0 & 0 \\ 0 & I & \cdots & 0 & 0 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & I & 0 \end{bmatrix},

a square matrix of size lags * obs. The companion state stacks the lag window most recent first, the reverse of the lags layout: s = lags[..., ::-1, :].reshape(*batch, lags * obs), and then (F @ s)[..., :obs] equals var_mean() without intercept. The VAR is stable exactly when every eigenvalue of F has modulus below one; that is the condition for a finite unconditional mean (I - \sum_l \Phi_l)^{-1} c and for the impulse responses of impulse_response() to die out.

Parameters

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

Coefficient tensor (*batch, lags, obs, obs).

Returns

Float[Array, "*batch lags_obs lags_obs"]

The companion matrix per batch element.

Notes

JAX implements the eigenvalues of a nonsymmetric matrix on CPU only, so compute the spectral radius of posterior draws with NumPy: np.abs(np.linalg.eigvals(np.asarray(companion_matrix(phi)))).max(-1).