I have two data sets labeled $\alpha$ and $\beta$, with corresponding models that I want to fit to. $$ \textbf{Y}_{\alpha} = \textbf{X}_{\alpha}\cdot\vec{\theta}_{\alpha} + \epsilon \\ \textbf{Y}_{\beta} = \textbf{X}_{\beta}\cdot\vec{\theta}_{\beta} + \epsilon $$
Here the form of the design matrix $\textbf{X}$ is the same for both $\beta$ and $\alpha$. I expect $\theta_{\alpha}$ to be different from $\theta_{\beta}$. However, I know that data sets $\alpha$ and $\beta$ are highly correlated. I want to take advantage of this correlation to simultaneously determine $\theta_{\alpha}$ and $\theta_{\beta}$. Is there a way to do this?