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So the first thing I would point out is that DAGs, as models, have no special status under the "all models are wrong" motto---if you don't write down the implied DAG of your model, your model will still …

Exact point identification is not possible here, but adjusting for Spend_in_Prev_Year does partially block the backdoor path, so that would be the rationale for it. As a general advice, you should ad …

Your model statement specifies a class of DAGs, not a single DAG. … with the error term with $x_2$, but note in this DAG the causal effect of $x_1$ is still identified): …

Whether "time" is an appropriate variable in a model depends on the phenomenon you are modeling. Thus, as you posed it, your question is about model misspecification, not a fundamental question about …

The basic IV dag is usually represented as: Where $U$ is unobserved and $Z$ is an instrument for the effect of $X$ on $Y$. … ; $S$ d-separates $Y$ from $Z$, in the DAG where the arrow $X\rightarrow Y$ is removed. …

Notice that even if the DAG were only $A \rightarrow Y \rightarrow C$ the post-interventional distribution $P(Y|do(A))$ is not non-parametrically identified, since $P(Y|do(A)) = P(Y|A) \neq P(Y|A, C)$ …

When we intervene on $X$, this means the parents of $X$ do not affect its value anymore, which corresponds to removing the arrows pointing to $X$.So let's represent this intervention on a new DAG. … It suffices to show that in your DAG: $$ \sum_{X'}P(Y|Z, X') P(X') = \sum_{U}P(Y|Z, U) P(U) $$ Notice the DAG implies $Y \perp X|U, Z$ and $U \perp Z|X$ then: $$ \begin{aligned} \sum_{X'}P(Y|Z, X') …

Yes, you can represent them in a DAG. In the structural framework, the DAG is nothing but a visual representation of the functional arguments that enter the structural equations. … You can thus write the DAG of the variables $T$, $X$ and $Y(1)$ as: In this DAG you can read that $Y(1) \perp\!\!\!\perp T \mid X$. …

So if that's the case, does statistical independence automatically mean lack of causation? No, and here's a simple counter example with a multivariate normal, set.seed(100) n <- 1e6 a <- 0.2 b …

Let the variables be stardadized to unit variance and let's use $\sigma_{ab}$ to denote the covariance of $A$ and $B$. You want the correlation of $A$ and $C$ when conditioning on the collider $B$. T …

I answered this once on twitter, I can reproduce the answer here. Derivation (graphs licensing each step are provided below). $$ \begin{align} P(y|do(x)) &= P(y|do(x), do(z)) \qquad &\text{Rule …