property SmootherResults.standardized_forecasts_error

Standardized forecast errors


The forecast errors produced by the Kalman filter are

\[v_t \sim N(0, F_t)\]

Hypothesis tests are usually applied to the standardized residuals

\[v_t^s = B_t v_t \sim N(0, I)\]

where \(B_t = L_t^{-1}\) and \(F_t = L_t L_t'\); then \(F_t^{-1} = (L_t')^{-1} L_t^{-1} = B_t' B_t\); \(B_t\) and \(L_t\) are lower triangular. Finally, \(B_t v_t \sim N(0, B_t F_t B_t')\) and \(B_t F_t B_t' = L_t^{-1} L_t L_t' (L_t')^{-1} = I\).

Thus we can rewrite \(v_t^s = L_t^{-1} v_t\) or \(L_t v_t^s = v_t\); the latter equation is the form required to use a linear solver to recover \(v_t^s\). Since \(L_t\) is lower triangular, we can use a triangular solver (?TRTRS).

Last update: Jul 16, 2024