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There are 2 time series $X$ and $Y$ and 3 sets: the first set consists of $N_1$ observations, the second set contains $N_2$ observations right after the first set, and third set contains $N_1$ and $N_2$ observations, call it $N_{12}$.

Further suppose estimate $Y = \alpha + \beta X + e$ over these 3 sets, then we have equations $Y_1 = \hat{\alpha_1} + \hat{\beta_1} X_1$; $Y_2 = \hat{\alpha_2} + \hat{\beta_2} X_2$, $Y_{12} = \hat{\alpha_{12}} + \hat{\beta_{12}} X_{12}$

Is it possible to derive $\hat{\alpha_2}, \hat{\beta_2}$ somehow, provided we know $N_1, N_2, \hat{\alpha_1}, \hat{\beta_1}, \hat{\alpha_{12}}, \hat{\beta_{12}},Y_{12} = \begin{pmatrix} Y_1 \\ Y_2 \end{pmatrix}, X_{12} = \begin{pmatrix} X_1 \\ X_2 \end{pmatrix}$

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