Synthetic control question on donors
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.