Suppose I have multiple regressions, along with their r-squares, standard-errors, etc.:
$y(t) = \alpha_1 + \beta_1 x(t) + e_1$, where $t \in (\tau_0, \tau_1)$
$y(t) = \alpha_2 + \beta_2 x(t) + e_2$, where $t \in (\tau_0, \tau_2)$
$...$
$y(t) = \alpha_n + \beta_n x(t) + e_n$, where $t \in (\tau_0, \tau_n)$
The only difference between all these regressions is the interval used to carry out the estimation i.e. $t \in (\tau_0, \tau_i)$ for the $i$-th regression. These intervals can be overlapping.
Let's also assume that the true relationship is
$y(t) = \alpha^{\ast} + \beta^{\ast} x(t) + e^{\ast}$
We can assume $e_i$ and $e^{\ast}$ as Normal if needed.
How can I combine the individual $\beta_i$'s and $\alpha_i$'s to get $\alpha^{\ast}$ and $\beta^{\ast}$?
Do I weight them by r-square or (inverse of) standard-error or is there any name to techniques to combine such things?