# Estimating number of compliers

Be $X$ a dichotomous endogenous variable and $Z$ its dichotomous instrumental variable. Suppose that for compliers if $Z_i=0$ then $X_i=0$ and if $Z_i=1$ then $X_i=1$.

Assuming that defiers do not exist, we can easily estimate the number of compliers from the sample by the formula:
(number of individuals with $Z_i=0$ and $X_i=0$)-(number of individuals with $Z_i=1$ and $X_i=0$)+(number of individuals with $Z_i=1$ and $X_i=1$)-(number of individuals with $Z_i=0$ and $X_i=0$)
But how is it possible to estimate the number of compliers if the endogenous variable $X$ is not dichotomous? And what if both the endogenous variable $X$ and the instrumental variable $Z$ are not dichotomous?

For the binary treatment and instrument the size of the complier group is given by the Wald first-stage $$P[X_{i1} > X_{i0}] = P[X_{i}|Z_i=1] - P[X_{i}|Z_i=0]$$ If you take this first stage, multiply it by the probability that the instrument is switched on $P[Z_i=1]$ and divide by the number of all treated $P[D_i=1]$ you will get the proportion of compliers among the treated individuals.
With a binary instrument and a multivariate treatment you can learn about the size of the compliers using the average causal response (ACR) theorem. Suppose $X_i$ takes values $\{ 0,1,...,M \}$, then the relative size of compliers at each point $x$ in the support of $X$ is:$$P[x_{i1}\geq x>x_{i0}] = P[x_{i1}\geq x] - P[x_{i0} \geq x]$$ which you would need to calculate at every value of $X$. These are the people who are driven from a treatment intensity strictly below $x$ to a treatment intensity of at least $x$. However, this gives the actual size of the compliant subpopulation if and only if every individual moves up the treatment intensity by one unit and not more. If a complier goes from $x_{i0} = 2$ to $x_{i1} = 5$ then she will be counted in each of the subpopulations \begin{align} x_{i0} &< 3 \leq x_{i1} \newline x_{i0} &< 4 \leq x_{i1} \newline x_{i0} &< 5 \leq x_{i1} \newline \end{align} If you are interested in the size of the compliant subpopulation you would need to adjust for this double/multiple counting of individuals that switch treatment intensity by more than one unit. As a side note: for the LATE estimator this is a good thing because it will weight those observations more which are affected the most by the instrument.
Suppose your instrument $Z_i$ takes on values $\{ 0,1,...,J \}$. The compliant subpopulation for each value of the instrument is $$P[X_i=1|Z_i=j] - P[X_i=1|Z_i=j-1]$$ at every value that the instrument takes. The issue regarding the actual amount of compliers is the same as before. The count is correct if every treated subject was subjected to a one unit increase in the instrument only. An individual switching from instrument value $j-2$ to $j$ is counted twice.