Classical least squares results in regression in statistics state that if $(Y, X)$ follow a model where $$\mathbb{E}[Y\mid X=x] = \alpha + \beta x,$$ we can estimate $\beta$ from a random sample using LS explicitly as the second component of $\hat{\beta} = (\textbf{X}^t\textbf{X})^{-1}\textbf{X}^t\textbf{Y}$, where $\textbf{X}$ is the matrix of ones and $X_i$.
My question is the following: Let $(Y, X, Z)$ follow a model $$\mathbb{E}[Y\mid Z=z, X=x] = (\alpha x) + (\beta x) z.$$
Consider a random sample from $(Y, X, Z)$. Can we estimate $\hat{\beta}$ and give an explicit expression for it?
