Consider the following regression model:
\begin{equation} y = \alpha + \beta X + \gamma D + \epsilon, \end{equation}
where $y$ is the dependent variable, $X$ is a continuous variable, $D$ is a dummy variable and $\epsilon$ is the error term.
Now, suppose we decompose our dataset into two subsets where $D$ is either one or zero, and we denote the dependent variable as $y_{1}$ and $y_{0}$, respectively. Therefore, constructing the following models:
\begin{equation} y_{1} = \alpha^{'} + \beta^{'} X + \epsilon^{'}, \end{equation}
and,
\begin{equation} y_{0} = \alpha^{''} + \beta^{''} X + \epsilon^{''}, \end{equation}
and then using OLS to estimate those three models, separately, we come up with $\hat{\beta}, \hat{\beta^{'}}$ and $\hat{\beta^{''}}$. Now, my question is whether or not we can draw the following conclusion:
\begin{equation} \hat{\beta^{'}} < \hat{\beta} < \hat{\beta^{''}}, \end{equation}
Or,
\begin{equation} \hat{\beta^{'}} > \hat{\beta} > \hat{\beta^{''}}. \end{equation}
Thanks!