statsmodels.tsa.statespace.kalman_smoother.SmootherResults.standardized_forecasts_error

property SmootherResults.standardized_forecasts_error

Standardized forecast errors

Notes

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).