Synthetic control question on donors I'm using synthetic control methods.
The treatment group has covariates that are the extrema of the values across itself and all potential donor units.
So, say, on covariate $x_1$, the treatment unit has the lowest value of the group over time, and on $x_2$ the treatment unit has the highest and so on.
In such a case can you really find a donor group, or does the method break down? Or in using several covariates can we use those that provide a good synthetic control in some cases which compensates for poor balance for other (important and we believe important)covariates?
Is there a literature on this issue?
Welcome.
Do you mean: if $x^*_{k,it}$ are the panel variables, then $x_{1,i} = \min_t x^*_{1,it}$, $x_{2,i} = \max_t x^*_{2,it}$, etc.? Usually people use averages over time or values at certain points of time, while you mean the smallest (or largest) value during a period of time.
Computationally I do not see any differences. They are just numbers.
I am more curious (or doubtful) about interpretation, though. SCM balances on $\mathbf{x}_i$, where $\mathbf{x}_i = (x_{1,i}, x_{2,i}, ..., x_{k,i})$, i.e., SCM tries to find $w_2, \ldots, w_{J+1}$ such that $\mathbf{x}_1 - \sum_{j=2}^{J+1} w_j \mathbf{x}_j$ is close to zero (in terms of $v_j$-weighted sum of squares). But then what's the meaning of balancing on $x_{1,i}$? The treatment group may attain the minimum value in year 1960, while one donor in year 2000 and another in year 1990. Then what would it mean that $x^*_{1,1,1960}$ is close to $w_1 x^*_{1,2,2000} + w_2 x^*_{1,3,1990}$?
On the other hand, if it's the smallest during one business cycle, for example, that can make sense. If you have good reasons to use those min/max values, I think that's fine. I am very curious about what kind of min/max variables you have in mind.
Edit
If the treatment group's $x_1$ value is the smallest of $i=1,2,...,J+1$, then it is not possible to have $w_j$ such that $w_j\ge 0$, $\sum_j w_j=1$, and $x_{1,1} = \sum_{j=2}^{J+1} w_j x_{1,j}$. The convexity assumption (nonnegativity &amp; add-up) is violated and you cannot use AG (2003) or ADH's (2010 and after) SCM. If you still use their SCM, you will be harshly criticized, I am afraid.