When is a parameter considered an estimand? Context
Let us say we are interested in the Average Treatment Effect (ATE) as an estimand. Following the potential outcomes framework, we define it as:
$$\frac{1}{N} \sum_{1}^{N}(Y_i^0 - Y_i^1)$$
where $Y_i^a$ is the potential outcome for subject $i$ under condition $A = a$, $A$ is a treatment assignment variable that takes values $0$ or $1$ for control and treatment, respectively, and $N$ is the number of subjects in the population.
Let us define two working models for the causal structure:

*

*DAG 1, with a fork on $X1$ and on $X2$:

*

*$A \leftarrow X1 \rightarrow Y$,

*$A \leftarrow X2 \rightarrow Y$ and

*$A \rightarrow Y$



*DAG 2, with mediator $X1$ and a fork on $X2$:

*

*$A \rightarrow X1 \rightarrow Y$,

*$A \leftarrow X2 \rightarrow Y$ and

*$A \rightarrow Y$
Clearly, in DAG 1 both $X1$ and $X2$ are confounders, whereas in DAG 2 only $X2$ confounds the relationship between $A$ and $Y$. Using a regression model as estimation strategy (OLS), we arrive to two models that depend on our causal assumptions (adding a subscript to indicate each one):

*

*$Y_i = \beta_{10} + \beta_{11} A_i + \beta_{12} X1_i + \beta_{13} X2_i + \epsilon_i$, where we use our estimate of $\hat{\beta}_{11}$ as an estimate for the ATE.

*$Y_i = \beta_{20} + \beta_{21} A_i + \beta_{22} X2_i + \epsilon_i$, where we use our estimate of $\hat{\beta}_{21}$ as an estimate for the ATE.


Question
Considering that an estimand is a "quantity to be estimated in a statistical analysis", which of the following assertions is false and why?:

*

*$\beta_{11}$ and $\beta_{21}$ are different quantities being estimated and thus are different estimands

*ATE is the only estimand, with $\hat{\beta}_{11}$ and $\hat{\beta}_{21}$ being two estimates for it
(I know that the relationship between an estimand and its estimator can be fully arbitrary. For instance I could roll a die and have the result be the estimator for the average height of people in my country. It would be a useless estimator, but an estimator nonetheless. My question is about whether an estimand is uniquely defined in such context or if it is decidedly ambiguous. The motivation is thinking about its consequences in model averaging, for example)

I prefer to view it according to your second bullet point.  You have clearly identified the estimand of interest at the very top.  The two different models can be seen as sensitivity analyses, each one relying on an unverifiable assumption.  This is related to the ICH E9 addendum on estimands for clinical trials.  In that setting we would define a particular population-level treatment effect of interest as the estimand.  Whether we make a missing at random given treatment assumption, or missing at random given treatment and age assumption, or a missing not at random assumption for post-baseline confounding, these are all different missing data assumptions one could use in pursuit of the originally defined estimand.
