Initialization of a damped cycle

Decision. A damped cycle (\(\rho < 1\)) starts from its unconditional distribution, \(N(0, \sigma^2_\kappa / (1 - \rho^2) I)\).

Why. A damped cycle is stationary, and DK §3.2.4 give this distribution for it; DK §5.6 initialize stationary components from their unconditional distribution.

Alternatives. statsmodels’ UnobservedComponents starts every state, the damped cycle included, as diffuse. Both are valid models but give different log likelihoods: in one of our test fixtures, -65.72 with the stationary start against -63.60 with the diffuse one. To compare with statsmodels, the fixture re-initializes its cycle as stationary.

Where. cycle_init_blocks in statespace_structural.