Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 1

In a standardized linear structural equation model, attach to every directed edge its path coefficient. Wright path tracing rule says that a covariance is the sum, over admissible unblocked paths between the two variables, of the product of the coefficients along each path; an admissible path does not pass through a collider and does not enter and later leave the same variable in a way that reverses direction twice. The total causal effect of $X$ on $Y$ is the sum of the products of edge coefficients over all directed paths from $X$ to $Y$.

The $X_1$ coefficient is the total causal effect of $X_1$ on $Y$. The node $X_1$ has no parents, so there is no backdoor path into it, and the unadjusted regression sums all directed paths from $X_1$ to $Y$.

The $X_2$ coefficient has no causal interpretation. The open backdoor path $X_2\leftarrow X_1\to Y$, together with paths through $X_3$, confounds the unadjusted association.

The $X_3$ coefficient has no causal interpretation. In particular, $X_3\leftarrow U\to Y$ is an open backdoor path, and $X_1$ and $X_2$ supply further confounding paths.

After adjusting for $X_1$, the $X_2$ coefficient is the total causal effect of $X_2$ on $Y$, consisting of $X_2\to Y$ and $X_2\to X_3\to Y$. The $X_1$ coefficient is the causal effect of $X_1$ with $X_2$ held fixed: it retains paths not passing through $X_2$, namely the direct path and $X_1\to X_3\to Y$.

Neither coefficient has a causal interpretation. Conditioning on $X_3$ opens the collider bias path $X_1\to X_3\leftarrow U\to Y$ for the $X_1$ coefficient, while the $X_3$ coefficient remains confounded by $U$ and by omitted $X_2$.

Neither coefficient has a causal interpretation. The $X_2$ coefficient is confounded by omitted $X_1$, and conditioning on the collider $X_3$ also opens $X_2\to X_3\leftarrow U\to Y$. The $X_3$ coefficient retains the open path through $U$.

None of the three coefficients has a causal interpretation. The $X_3$ coefficient is unavoidably confounded by unmeasured $U$. Conditioning on the collider $X_3$ opens paths through $U$ for both $X_1$ and $X_2$, so adding all measured regressors does not repair the bias.