phillips_drifts_volatilities: kalman_forward pairs each observation with the previous date's volatility #1108

kalman_forward builds r from the whole volatility path h, whose rows are dates 0 to T: run_sampler starts it with T + 1 rows, and draw_volatility_path updates log h_0 to log h_T. The filter's loop then uses r[t] for y[t], the observation at date t + 1, so each observation is filtered with R built from the previous date's volatility: R_1 comes from h_0, and h_T is never used. The rest of the sampler pairs h[t] with date t: draw_volatility_path matches h[t] with the residual at date t, draw_β standardises the residuals by h[1:], and the retained draws keep h[1:]. The model is R_t = B^{-1} H_t B^{-1}′ at the same date, which is also what the line's own comment (# R_1, ..., R_T) says.

This came in with the vectorised filter in #1105. The version before it paired h[t] with y[t - 1] in draw_coefficient_path.

The change is to build r from h[1:]:

r = np.einsum('ij,tj,kj->tik', b_inverse, h[1:], b_inverse)    # R_1, ..., R_T

The size of the effect on the posterior has not been measured: the figures and tables would need re-running with the change.

Found by Copilot's review of QuantEcon/lectures#36, which copies this lecture at 4d82fff: QuantEcon/lectures#36 (comment)