Identification of structural parameters in a linear model (treatment effect context) Suppose that we have $N$ observations indexed by $i=1,...,N$.
The observations are partitioned in three groups indexed by $g=1, 2,3$.
Here, we consider potential outcomes $Y_{ig}^0,Y_{ig}^1,Y_{ig}^2$.
That is, we are considering three potential treatment status,
and $Y_{ig}^0$: untreated potential outcome of an individual $i$ who is included in the group $g$, $Y_{ig}^1$: potential treated outcome of $i$ in $g$ who took the treatment of level one, and $Y_{ig}^2$: potential treated outcome of $i$ in $g$ who took the treatment of level two.
In this situation, suppose that the data generating process is as follows:
$$Y_{ig}^0=v_{ig}$$
$$Y_{ig}^1=\alpha_0D_{i0}+\alpha_1D_{i1}+\alpha_2D_{i2}+Y_{ig}^0$$
$$Y_{ig}^2=\beta_0D_{i0}+\beta_1D_{i1}+\beta_2D_{i2}+Y_{ig}^0,$$
where $v_{ig}$ is the usual error term and $D_{ig}$ is one if an individual $i$ is in a group $g$ and zero otherwise.
As we know, we can identify the structural parameters $\alpha_1$ and $\beta_2$ like below:
$\begin{align} Y_{ig}=&Y_{ig}^0+(Y_{ig}^1-Y_{ig}^0)D_{i1}+(Y_{ig}^2-Y_{ig}^0)D_{i2} \\
=&v_{ig}+(\alpha_0D_{i0}+\alpha_1D_{i1}+\alpha_2D_{i2})D_{i1}+(\beta_0D_{i0}+\beta_1D_{i1}+\beta_2D_{i2})D_{i2}
 \\
=&\alpha_1D_{i1}+\beta_2D_{i2}+v_{ig}\end{align},$
where $Y_{ig}$ is the observed (or realized) outcome.
Thus, we can estimate $\alpha_1$ and $\beta_2$ by using LSE (regression $Y_{ig}$ on $D_{i1}$ and $D_{i2}$).
However, I want to know other structural parameters, $\alpha_0, \alpha_2, \beta_0, \beta_1$, as well.
Although I tried to what all I know, I was not able to come up with how to identify the parameters.
So, how can I identify the parameters?
 A: You could add another column in your data matrix as a categorical variable which would be the treatment with 3 levels.
Then you could fit a model with the interaction between.
Let $x_{i,j}$ be the treatment for an individual $i$ with levels ${0, 1, 2}$.
We treat $x_{i,0}$ as a reference category and so it is simply covered by the intercept term $\alpha_0$
$x_{i,j}$ for $j=1,2$ are indicators which are one if individual $i$ received treatment $j$
For a model with just the categorical variable as a predictor we have:
$Y_i = \alpha_0 + \alpha_1*x_{i,1} + \alpha_2*x_{i,2} + \epsilon_i$
In this case $\alpha_1$ corresponds to the difference in the intercept between treatment 0 and treatment 1 and $\epsilon$ is the error term.
With another continuous predictor where we want to see the interaction between the treatment and the predictor say $z_i$ we fit a model of the form, where $\epsilon$ is our error term:
$Y_i = \alpha_0 + \alpha_1*x_{i,1} + \alpha_2*x_{i,2} + \beta_1*z_i + \beta_2*(z_i*x_{i,1}) + \beta_3*(z_i*x_{i,2}) + \epsilon_i$
This gives three regression lines which differ according to the treatment:
For treatment 0: $Y_i^0 = \alpha_0 + \beta_1*z_i + \epsilon_i$
For treatment 1: $Y_i^1 = (\alpha_0 + \alpha_1) + (\beta_1 + \beta_2)*z_i + \epsilon_i$
For treatment 2: $Y_i^2 = (\alpha_0 + \alpha_2) + (\beta_1 + \beta_3)*z_i + \epsilon_i$
We then think of $\beta_2$ and $\beta_3$ as the difference in the slope of the regression line of going from treatment 0 to treatment 1 and 2 respectively.
In R this would be implemented something like
lm1 <- lm(y ~ treatment + z + z:treatment)

