Linear regression model with a distribution over regression equations Suppose that the observations $(y_t, x_t, k_t)_{t=1}^N$ satisfy the linear regression equation:
\begin{equation}
\begin{split}
y_t = \begin{cases} 
x_t \beta + e_t & w.p. \; \theta \\
k_t \gamma + e_t & w.p. \; 1-\theta 
\end{cases}
\end{split}
\end{equation}
with $\mathbb{E}(e_t|x_t, k_t)=0$ and other usual assumptions.
Can I estimate $\beta*\theta$ and $\gamma*(1-\theta)$?

The conditional mean of $y_t$ is $\mathbb{E}(y_t|x_t, k_t) = x_t \beta \theta + k_t \gamma (1-\theta)$, so parameters can be identified with the regression $$y_t = a_1 x_t + a_2 k_t + e_t$$
