Output coordinates of the univariate treatment

Decision. In periods processed element by element (DK §6.4), the filter reports \(v_t\), \(F_t\) and \(F_{\infty,t}\) in the original coordinates of \(y_t\), as in conventional periods. The per-element quantities of the univariate recursion, in the coordinates transformed by \(L^{-1}\) from \(H_{oo} = L D L'\), are kept separately (the uv_* arrays of filter_result_t) for the smoother. Smoothed observation disturbances are also in original coordinates, as full matrices: from the identity \(\varepsilon_{t,o} = y_{t,o} - d_{t,o} - Z_{t,o}\alpha_t\), \(\hat\varepsilon_{t,o} = y_{t,o} - d_{t,o} - Z_{t,o}\hat\alpha_t\) with variance \(Z_{t,o} V_t Z_{t,o}'\).

Why. A user reading \(v_t\) and \(F_t\) should not have to know which periods the filter treated element by element, which depends on the initialization and on the filter options. Original coordinates make the output of every period comparable, and make standardized residuals and diagnostics correct across the diffuse periods.

Alternatives. statsmodels reports these quantities in the transformed coordinates during diffuse and univariate periods, with diagonal disturbance variances only. For p = 1 the two coincide; the tests compare the transformed quantities (our uv_* arrays) with statsmodels’ in those periods and the original ones elsewhere.

Cost. Computing \(F_t\) in original coordinates adds a matrix product per univariate period; this is why the stored univariate filter output is slower than statsmodels’ (see Performance), while the log likelihood, which does not store it, is faster.

Where. observation_moments and store_elements in statespace_filter; measurement_disturbance in statespace_smoother.