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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?

uday
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  • If neither $\beta$ nor $\alpha$ depends on $t$ then, in theory at least, all results should be equal to the true approximation. The reason is because you've effectively drawn a random sample for each regression. In that case, all you need to do to recover $\beta^*$ is take the average and the only weighting would likely be a function of the sample sizes. – Greg Jul 27 '15 at 05:09

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