IV: Average Treatment Effects I am estimating an instrumental variable regression (IV) where Y is my dependent, X my variable of interest, and Z the dicotomous instument.
I am interestend in verifying if the coefficient estimate of X is actually a Local Average Treatment Effect (LATE) or a Average Treatment Effect (ATE).
In order for the IV to produce a ATE the monotonicity assumption must old. That is no observation in the sample must be a defier(do the opposite of what they are supposed to do). Simply, Z has a causal effect on X.
For the case of LATE, the instrument Z has a heterogeneous effect on X.
Convincingly arguing monotonicity of the instrument on the instrumented is not a simple task.
However, I see that in R there exists the function 'ATE' from the homonimous package which computes ATE exploiting a matching technique.
Can someone explain the rationality behind this function and how to apply it for my case. Moreover, how do I check that my coefficient equals the LATE or ATE?
EDIT:
The ATE function acutally requires the dicotomous treatment vector, a numeric vector of the output of interest and a matrix object for the covariates to be used for the matching.
What actually the function does is to compute the mean effect for both the treated and not treated. It is able to do so because each treated is matched with a non-treated with the highest degree of similarity in the covariates.
The output of ATE mainly gives a difference in means, with a correspondent p-value where HP_null: mean of treated and mean non-treated are the same.
Going back to my specific example. I want to check if the effect of my instrument is monotonic for the treated on the instrumented variable.
In R i run: 
ate <- ATE(df$X, df$Z, socio_matrix, ATT = TRUE)

where socio_matrix is a matrix of socio-economic individual characteristics.
the output is the following: 
Estimate    StdErr 95%.Lower 95%.Upper Z.value   p.value    
E[Y(1)|T=1] 0.5820551 0.0071496 0.5680420 0.5960681  81.410 < 2.2e-16 ***
E[Y(0)|T=1] 0.2147127 0.0052483 0.2044262 0.2249992  40.911 < 2.2e-16 ***
ATT         0.3673423 0.0083993 0.3508799 0.3838047  43.735 < 2.2e-16 ***

This signals that on average the treated are likely to show higher values of the outcome (instrumented variable). The average treatment effect on
the treated is 0.36 (compared to their counterfactual, the non-treated)
What are the conclusions for my IV strategy?
 A: Estimating causal treatment effects is actually quite simple, or at least can be done several ways some of which are easy. Perhaps the easiest approach is using a regression model and making inference about the model coefficient beta. Other ways of doing this are propensity score match/weight/adjustment, structural equation models, and more sophisticated microsimulation models. It turns out they all estimate roughly the same thing. Thus the have the same basic, underlying assumption: proper identification and control of confounders. In regression models this is with multivariate adjustment.
The ATE package is from Gary Chan at the UW. Here is his paper. It's a nice paper and large sections if not the whole thing should be readable for you. Basically his approach is a modification of propensity weighting, IPW, where the instrumental variables can be chosen "empirically" by optimizing a set of weights so that the distribution of the covariates is approximately balanced in the "treated" and "untreated" samples. It's fairly straightforward and intuitive, but he goes one step further to establish some nice limit theorems for general distance functions, I would probably advocate using squared or Euclidean distance in almost all cases. But read on for some considerations about the IV matrix design.
I take your meaning to be that Y is the outcome of interest, X is a regressor of interest, and Z is (possibly many) confounding variables. To do a causal analysis, the approach advocated by Pearl is to draw a directed acyclic graph, to ensure that the causal relations of X, Y, and Z are such that Z is indeed a confounder and not a mediator or collider, or spurious variable. Indeed, any approach above will heavily favor such a variable, and for the wrong reasons. So fact check, fact check, fact check! Added to this, we must make the assumption of no interactions, or go about doing an analysis of effect modification. You have said the "the instrument has heterogeneous effect on X" without clarifying what you mean. You ought to describe your "X" and "Y" and "Z", are these measures of economic growth following policy implementation? anterior wall thickening measured by ultrasound? et cetera...
Thus I would believe the conclusions for your strategy are to think more deeply about the nature of confounding: sociodemographic constructs are just that: a construct and they go much deeper than can often be measured: maternal education does not suffice to predict, say, proximity to hospital when evaluating receipt of emergency medical services. Most non-statisticians would ask you to seek a better measure of proximity, like street address and GIS, and they'd be right! We have advanced new methods in causal modeling, but the rampant problem remiss in most basic regression models remains as problematic: you must have appropriate identification of confounders.
My general approach to data analysis is to fit the model I hope to be "as close to causal as possible", but interpret it strictly using basic inference and descriptive language and later connect my findings to the possibility of causal models proposed by others. I also make a point of discussing the possibility of omitted variable bias, even though there are reasons to I never make inference as to whether an effect is causal, nor do I call an observational measure a "treatment" except when it is actually a medical treatment. 
