Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 (Causal Inference)

In potential outcome notation, a valid instrumental variable $Z$ must satisfy:

In the graph, $Z_1$ has open noncausal paths to $Y$ that do not pass through $A$, includingEquivalently, these paths remain after deleting the causal edge $A\to Y$. Thus $Z_1$ is associated with potential outcomes through the latent variables $C$ and $S$, violating instrumental-variable independence; it is not a valid marginal instrument.

Among observed pretreatment variables, every sufficient set must contain $M_1$ to blockand $Z_3$ to blockConditioning on $Z_3$ opens the collider onso $M_3$ must also be included. The resulting minimal sufficient adjustment set isIt blocks every path from $Z_1$ to $Y$ that remains after removing $A\to Y$, while the open pathpreserves instrument relevance. Adding $M_2$ blocks no required relevance path, sois also sufficient.

There are no others. In particular, adding $Z_2$ blocks the displayed relevance path. Without $M_2$, conditioning on $Z_2$ also openswhich violates independence; adding $M_2$ closes that path but leaves no open path from $Z_1$ to $A$. Hence the complete list is $X_0$ and $X_1$.

For one unit write $M=(M_1,M_2,M_3)$ and $W=(M,Z_1,Z_3)$. Since the graph makes $S$ independent of $M$, the known distributions determine the conditional assignment lawUnder the sharp causal null hypothesis that $A$ has no effect on $Y$, delete $A\to Y$. The D-separation criterion then gives$M_2$ blocks the route through $C$, $Z_3$ blocks the route through $S$, $M_1$ and $M_3$ block the collider-opened detours through $Z_1$ and $Z_3$, and $A$ remains a collider on routes through $U$.

For independent units $i=1,\dots ,n$, choose a test statistic $T(Z_{2,1:n},Y_{1:n},W_{1:n})$ that measures residual association between $Z_2$ and $Y$. Hold $(Y_i,W_i)$ fixed and independently drawRecompute $T^{(b)}$ after each draw. A valid Monte Carlo conditional randomization test useswith a two-sided statistic or absolute value when appropriate.

Under the null, conditional on $(Y_{1:n},W_{1:n})$, the observed $Z_{2,1:n}$ and its $B$ resamples are exchangeable because they have the same product law ${\prod }_{i}q(\cdot \mid W_i)$. The rank of $T^{\mathrm{obs}}$ among the $B+1$ values is therefore uniform after randomized tie breaking and conservative without it. ConsequentlyTaking expectations proves unconditional Type I error control.

Take all components to be centered and let their marginal variances be $V_A,V_C,V_E$. A linear structural equation model iswhere $C_i,E_{i1},E_{i2}$ are mutually independent and the $E_{ij}$ are identically distributed. The causal directed acyclic graph has arrows

For monozygotic twins, set $A_{i1}=A_{i2}=G_i$ with $\mathrm{Var}(G_i)=V_A$; the genetic cause is completely shared. For dizygotic twins, one explicit construction iswhere $G_i,G_{i1},G_{i2}$ are independent with variance $V_A/2$. Then both additive genetic terms have variance $V_A$, while their covariance is $V_A/2$ and their correlation coefficient is $1/2$. In a fuller graph, the shared $G_i$ points to both genetic components and the unique $G_{ij}$ points only to its own component.

The common marginal trait variance isFor monozygotic pairs, shared genes and shared environment giveFor dizygotic pairs, the genetic covariance is halved while the common environment is unchanged:Subtracting cancels the common-environment variance and proves Falconer's formulathe heritability under the ACE model.

Suppose assortative mating raises the dizygotic genetic correlation from $1/2$ to $\rho >1/2$. Thenso the formula returnsThus using the nominal one-half genetic correlation produces downward bias in estimated heritability, assuming the other ACE assumptions remain valid.

The no unmeasured confounding assumption, or conditional exchangeability, isAlso assume consistency of potential outcomes, no interference in causal inference, positivity in causal inference, and finite expectations. For $a\in \{0,1\}$, the law of total expectation, exchangeability, and consistency givePositivity ensures that the observed conditional means exist on the covariate support being averaged. Subtracting the two cases identifies the average treatment effect as

Under the partially linear model,Inserting this constant conditional contrast into the identification formula from part i gives

For fixed $\beta$, conditional least squares is minimized byTherefore the remaining objective iswhose normal equation givesLetbe the conditional average treatment effect. Since $A$ is binary, conditional exchangeability impliesand $\mathrm{Var}(A\mid X)=e(X)\{1-e(X)\}$. HenceThus ${\beta }_3$ is the overlap-weighted average treatment effect. It weights covariate strata by the overlap weight and generally differs from the ordinary ATE ${\beta }_1=\mathbb{E}[\tau (X)]$ when treatment effects are heterogeneous and overlap varies with $X$.

Differentiating the residualized objective givesThis is the population Frisch–Waugh–Lovell theorem.

For a semiparametric estimator, estimate $e(x)=\mathbb{E}[A\mid X=x]$ and $\mu (x)=\mathbb{E}[Y\mid X=x]$ flexibly. With cross-fitting, obtain held-out predictions ${\hat{e}}_i,{\hat{\mu }}_i$ and regress the residualized outcome on the residualized treatment through the origin:Cross-fitting limits overfitting bias and permits flexible nuisance estimators under the usual convergence and overlap conditions.

The administrative data include only encounters with $M=1$. Conditioning on the stop indicator selects on the colliderwhich opens the noncausal path $D\leftrightarrow U\to Y$ and creates collider bias. The equal entries estimate only the selected risks $\mathbb{P}(Y=1\mid D=d,M=1)$. They ignore racial differences in the probability of being stopped and therefore do not identify the total causal effect of race on violence. This is also selection bias, because encounters with $M=0$ never enter the dataset.

Let $Y(d,m)$ be the potential violence outcome if race were set to $d$ and stop status to $m$. The substantive assumption is the structural zeroEquivalently, with the natural stop status $M(d)$, $M(d)=0$ implies $Y(d)=Y(d,M(d))=0$.

The graph has no cause of $D$, so $D$ is independent of its potential outcomes. By consistency of potential outcomes, the g-formula therefore givesCondition on $M$ and use the structural zero from part ii:Consequently the causal risk ratio isApplying Bayes theorem to the second factor giveswhich proves the stated formula. The unmeasured common cause $U$ of $M$ and $Y$ does not obstruct this total-effect argument because no mediator effect is being identified.

The stop records estimatebut they cannot estimate the population encounter probabilities $\mathbb{P}(D=1)$ and $\mathbb{P}(D=0)$ because encounters without a stop are absent. Equivalently, they do not determine the race-specific stop-probability ratio.

The scientist needs a representative denominator for all police-civilian encounters, including those with $M=0$. Suitable sources could include a carefully designed population or travel survey, systematic street and traffic observation, dispatch or body-camera sampling that records non-stop encounters, or an external administrative source measuring exposure to police by race. Combining its estimate of the population race odds with the estimable stop-data terms identifies the displayed causal risk ratio, provided the external sample targets the same city, period, and encounter population.