# statsmodels.tsa.statespace.kalman_filter.FilterResults.standardized_forecasts_error¶

property FilterResults.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).