# Time Series OLS

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 $$N_1, N_2, \hat{\alpha_1}, \hat{\beta_1}, \hat{\alpha_{12}}, \hat{\beta_{12}},Y_{12} = \begin{pmatrix} Y_1 \\ Y_2 \end{pmatrix}, X_{12} = \begin{pmatrix} X_1 \\ X_2 \end{pmatrix}$$