Predicting "Next-Day" Rate

We minimize the expected "next-day" squared deviation $\epsilon_{t+1}$ using the data available at the moment $t$

$$y_{t+1}^0=arg \min_{y_{t+1}} {\bf E}\ \epsilon_{t+1}^2.$$ (18)

Here

$${\bf E} \epsilon_{t+1}^2={\bf E}\ (B_{t+1}+\sum_{i=1}^p A_{t+1}(i) a_i(b^0))^2,$$ (19)

where the optimal parameter $b^0$ was obtained using the data available at the day $t$. Variance (1.20) is minimal, if

$$y_{t+1}^0= B_{t+1}+\sum_{i=1}^p A_{t+1}(i) a_i(b^0),$$ (20)

because the expectation of $y_{t+1}$ is $B_{t+1}+\sum_{i=1}^p A_{t+1}(i) a_i(b^0)$ under the assumptions.