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You cannot express it in "do" notation! The conditioning event $X = 1$ would conflict with $do(X = 0)$. Indeed, the ATT is an example of a counterfactual/"rung 3" causal query that is strictly "deeper …

Judea Pearl makes this point very forcefully (e.g., in "Causality", 2009, Cambridge University Press). To me, multiple imputation of the potential outcome distribution does not make sense. …

This is a great question, because the important book by Judea Pearl ("Causality", 2nd ed.) actually calls $P(Y|do(X = x)) = P(Y(x))$ the "causal effect" of $X$ on $Y$ (Definition 3.2.1), so that $E[ …

First of all, there is no need for you to control for average hours played. If you run a simple regression just with the treatment indicator, this will give you an estimate of what the effect of your …

Two answers: 1) Included common causes often summarize the effects of more distant causes. In fact, this is behind the whole idea of instrumental variables: Some variables are causally far away from …

Let's say you are interested in the causal effect of $D$ on $Y$. The following statement are not quite precise but I think convey the intuition behind the two approaches: Back-door adjustment: Determ …

Potential outcomes are features of units $U$, so it helps in our case to index them: $$Y_u(1), Y_u(0)$$ The usual statement is that potential outcomes are fixed for a specific unit $u$, but may of c …

If you assume zero covariance between $\epsilon^1_t$ and $\epsilon^2_t$ as well as between $y_{t-1}$ and $\epsilon^2_t$, then $\beta$ is identified by a regression of $y_t$ on $x_t$ and $y_{t-1}$. Thi …

As noted in the comments, this were valid if $Y_x \perp X|Z$ would imply $Y \perp X|Z$. But it does not. The simplest counterexample is when $X$ affects $Y$. Then $Z$ blocks all back-door paths, but t …

In dagitty, when you indicate that a variable $A$ is "adjusted", you indicate that you will definitely adjust/control for it in the analysis. Dagitty will then tell you whether and how you can still e …

Lastly, you mention reverse causality. In your context, this would mean future performance influencing past gender of the CEO. This is physically impossible, and so not an issue. …

"Making adequate identification assumptions is sufficient for identifying causal relationships" is either tautologically true or obviously wrong. It is true if by "adequate identification assumptions" …

Your $H_{0}$ implies $E[Y_{i}(1) - Y_{i}(0)] = 0$, which is testable whenever we have identified this ATE, by whatever means. By elementary logic, rejecting an implication of a statement rejects this …

This "structural definition of counterfactuals" was proposed by Judea Pearl, see for example his book "Causality", or his book with Jewell and Glymour, "Causality: A Primer". …

The clearest approach to causality is the one using structural equations, potential outcomes, and causal graphs [1]. … Causality. Cambridge University Press. …