1 Introduction

Learning Objectives

By the end of this chapter, students should be able to:

Articulate the distinction between an interventional distribution $P(y \mid \doop(T{=}t))$ and an observational conditional distribution $P(y \mid T{=}t)$, and explain why they differ, in general, under confounding or endogeneity.
Describe the same causal question in all three languages — SEM, DAG graph surgery, and potential outcomes — and begin to translate between them.
Derive the two Gaussian laws explicitly in the linear confounded model, and identify $\alpha\gamma\sigma_U^2/\sigma_T^2$ as the difference between the observational and causal mean slopes.
State the two-step paradigm (identification then estimation) and locate each step in the causal inference pipeline.

This chapter is an orientation, not a complete treatment. It introduces four ideas — the interventional/observational distinction, the causal trinity, the Gaussian confounded model as a running example, and the identification-versus-estimation paradigm — that will be developed rigorously in the chapters that follow. After one reading, students should have a map of the territory; the details will become clear as the course proceeds. The remarks on NPSEM-IE and on scope, together with Section 1.2.6, are optional on a first reading.

1.1 Motivation: Why Causal Inference Is Hard

1.1.1 The Central Question

Causal inference is concerned with a deceptively simple question: what would happen to $Y$ if we were to set $T = t$? This is fundamentally different from the question that standard statistical procedures answer: what is the distribution of $Y$ among units observed to have $T = t$?

Definition: Interventional vs. Observational Distribution

The interventional distribution $P(y \mid \doop(T{=}t))$ is the distribution of $Y$ in a hypothetical world where $T$ has been externally set to $t$, regardless of the usual data-generating mechanism for $T$.
The observational conditional distribution $P(y \mid T{=}t)$ is the distribution of $Y$ among the subpopulation of units for whom $T = t$ was observed to hold.

These two distributions coincide when the treatment assignment is unconfounded with the outcome — in the simplest case, when $T$ shares no common causes with $Y$. Randomization is the canonical way to enforce unconfoundedness, but it is not the only way: any data-generating mechanism in which $T$ is unconfounded with $Y$ will produce the same coincidence. Under confounding, the two distributions generally differ. In the linear model of Section 1.3, this discrepancy appears as endogeneity bias in the OLS regression coefficient.

Example: Drug Trial with Unmeasured Severity

Does a new drug ($T=1$) reduce blood pressure ($Y$)? Suppose sicker patients are more likely to take the drug, so that unmeasured severity $U$ is a common cause of $T$ and $Y$. Then:

Observational study: $\E[Y \mid T{=}1] > \E[Y \mid T{=}0]$. Drug users have higher blood pressure on average. A naive analyst concludes the drug is harmful.
Randomized trial: $\E[Y \mid \doop(T{=}1)] < \E[Y \mid \doop(T{=}0)]$. Forcing everyone to take the drug reduces blood pressure. The drug is beneficial.

Same population. Different questions. Opposite answers. The entire discrepancy is caused by the back-door path $T \leftarrow U \to Y$, which the observational quantity $\E[Y \mid T{=}1]$ cannot remove.

The Identification Problem

The observed-data law contains $P(y \mid T{=}t)$, which can be estimated from data. The identification problem is whether that observational law also determines $P(y \mid \doop(T{=}t))$. The goal of identification theory is to find conditions under which it does.

1.2 The Causal Trinity: Three Languages for One Idea

The same causal question — what happens when we set $T = t$? — can be expressed in three closely related languages: the structural equation model (SEM), the directed acyclic graph (DAG) with graph surgery, and the potential outcomes framework. Collectively, we call these the causal trinity. The three frameworks are often intertranslatable, though exact equivalence requires additional assumptions about the causal model; we return to this point in the NPSEM-IE remark below and in Section 1.2.6. Each language illuminates a different facet of the same underlying idea, and fluency in all three is essential for modern causal reasoning.

The causal trinity: three languages that express the same causal question from different vantage points. Potential outcomes are especially convenient for defining estimands; DAGs make identification assumptions and intervention operations visible; SEMs provide generative and counterfactual constructions. These are native strengths rather than exclusive functions. Double-headed arrows show the translation assumptions connecting adjacent languages; all three converge on the same central question (ellipse).

1.2.1 Language 1: Structural Equation Models

A structural equation model (SEM) represents the causal data-generating mechanism as a system of equations, one for each modeled variable, together with a joint distribution for the background inputs.¹ The variables on the left-hand sides of the structural equations are generated within the modeled system. The background inputs have no structural equations in the model, and their joint distribution is taken as given; under the NPSEM-IE assumption adopted below, that joint distribution factorizes into independent marginals. The background inputs are the root sources of randomness; the modeled variables inherit their variability, directly or indirectly, from them.

The simplest interesting SEM has three observed variables: a covariate $X$, a treatment $T$, and an outcome $Y$, each generated by its own equation: \[X = f_X(\varepsilon_X), \qquad T = f_T(X, \delta), \qquad Y = f_Y(X, T, \varepsilon), \tag{1.1}\] where $\varepsilon_X$, $\delta$, and $\varepsilon$ are mutually independent background disturbances. In this system, $X$, $T$, and $Y$ are generated within the model, while the three disturbances are background inputs. As a concrete interpretation, consider a job-training program: background characteristics such as education and motivation ($X$) influence both enrollment in the program ($T$) and subsequent employment outcomes ($Y$), while $T$ may also affect $Y$ directly (Wang et al. 2026). On the surface, Equation 1.1 looks like a system of nonparametric regression models. What makes it structural is that each equation is read as an autonomous mechanism: it describes how its left-hand variable is generated, and it remains valid when other equations in the system are replaced by interventions.

Equation 1.1 poses no essential causal difficulty: the common cause $X$ of $T$ and $Y$ is observed, so it can be adjusted for (Chapters 3–5 develop this in detail). The hard case — and the running setup for the rest of this chapter — arises when the common cause is unobserved. Writing $U$ for such a variable, and suppressing its structural equation because we do not model how it arises, we obtain the three-variable confounded SEM over $(U, T, Y)$: \[T = f_T(U, \delta), \qquad Y = f_Y(T, U, \varepsilon), \tag{1.2}\] where $U$, $\delta$, and $\varepsilon$ are mutually independent background variables, with joint distribution $P_{U,\delta,\varepsilon} = P_U \otimes P_\delta \otimes P_\varepsilon$. The variable $U$ is an unobserved background cause of both $T$ and $Y$ (a confounder), while $\delta$ and $\varepsilon$ are equation-specific disturbances. Note the contrast with Equation 1.1: a variable whose generating equation we leave unspecified is treated as a background input. The marginal distributions of the background variables — including $p(u)$, to which we refer by this notation later — are suppressed in Equation 1.2.

Remark: Error Independence in the NPSEM-IE

The assumption that the background inputs in Equation 1.2 — $U$, $\delta$, and $\varepsilon$ — are jointly independent is a substantive restriction, not merely a technical convenience. Because $U$ is unobserved and enters both $f_T$ and $f_Y$, error independence does not remove the dependence between $T$ and $Y(t)$ induced by $U$; it restricts only the additional channel of dependence running through the equation-specific errors $(\delta, \varepsilon)$. Richardson and Robins (2014) call SEMs with this structure Nonparametric Structural Equation Models with Independent Errors (NPSEM-IEs).

A weaker single-world model is Robins’s FFRCISTG — the Finest Fully Randomized Causally Interpretable Structured Tree Graph (Robins 1986; Richardson and Robins 2014). For the version in which all modeled variables are subject to intervention, an FFRCISTG admits a structural-equation representation with possibly dependent equation-specific errors; the NPSEM-IE is a strict submodel obtained by imposing independence of those errors. The added independence is not innocuous: as shown in Chapter 4 and Appendix B, it introduces cross-world restrictions that are not empirically testable even under randomized treatment assignment. See Wang et al. (2026) for a careful side-by-side comparison.

These notes adopt the NPSEM-IE as a convenient sufficient model: error independence ensures the DAG’s Markov condition holds, making the Markov factorization of the joint density immediate, and the counterfactual construction integrates cleanly with the graphical identification theory developed in later chapters. The identification results we derive — the back-door criterion, the do-calculus, the IV assumptions — generally require less than the full collection of NPSEM-IE cross-world restrictions; the error-independence assumption is a modeling choice, not an identification necessity.

Observational regime. Throughout this chapter, $U$ is treated as unobserved. Because $U$ enters both $f_T$ and $f_Y$, treatment status carries information about $U$: conditioning on $T$ changes the distribution of $U$, so the conditional distribution $P(y \mid T{=}t)$ generally differs from the interventional distribution $P(y \mid \doop(T{=}t))$. This is confounding. If $U$ is omitted from an outcome regression, its contribution enters the regression disturbance, which is then correlated with $T$ — even though the structural disturbance $\varepsilon$ itself is independent of $T$ under the NPSEM-IE; this is endogeneity in the usual econometric sense. Common-cause confounding thus manifests as endogeneity in the specified outcome equation.

Remark: Equation-Specific Exogeneity

In these notes, exogeneity is defined relative to a particular equation. For an outcome equation with structural disturbance $\eta$, conditional mean exogeneity may be expressed as $\E(\eta \mid T, X) = 0$; for a particular linear estimator, a weaker moment restriction such as $\E[T\eta] = 0$ may suffice. Full statistical independence between $T$ and $\eta$ is stronger than either requirement. Exogeneity does not require the treatment to be independent of every background variable: in the confounded model of Section 1.3, either $\alpha = 0$ or $\gamma = 0$ makes $T$ exogenous for the outcome equation, although only the first makes $T$ independent of $U$.

Interventional regime. The do-operator replaces the equation for $T$ with a constant $t$, yielding the mutilated system: \[U \sim p(u), \qquad T = t \text{ (fixed)}, \qquad Y = f_Y(t, U, \varepsilon).\] Now $T$ and $U$ are independent by construction: the back-door path through $U$ has been severed. The interventional density $P(y \mid \doop(T{=}t))$ is therefore the distribution of $f_Y(t, U, \varepsilon)$ when $U$ is drawn from its marginal $p(u)$, not from the conditional $p(u \mid T{=}t)$. We define the potential outcome under treatment $t$ as the random variable $Y(t) \mathrel{:}= f_Y(t,\, U,\, \varepsilon)$; by construction, $Y(t)$ has distribution $P(y \mid \doop(T{=}t))$.

The surgery just performed relies on a substantive assumption that deserves a name. Only the equation for $T$ was replaced; the function $f_Y$, the distribution of $(U, \varepsilon)$, and every parameter they contain were carried over from the observational world unchanged. This invariance is autonomy (Haavelmo 1943): each structural equation describes a mechanism that continues to operate — with the same functional form and the same parameters — regardless of how its inputs come about, whether from the natural data-generating process or from an external intervention. It is precisely this across-world invariance that gives the mutilated system its predictive content — without autonomy, quantities estimated in the observational regime would say nothing about the interventional one. The three frameworks connect interventional quantities to observed outcomes in related but not identical ways: autonomy and modularity specify how mechanisms change under intervention, whereas consistency links observed outcomes to the potential outcomes corresponding to realized treatments. Section 1.2.4 compares these roles directly.

1.2.2 Language 2: Directed Acyclic Graphs

A DAG makes the same surgery graphically visible.

Definition: Graph Surgery

In the observational DAG, arrows into $T$ encode the mechanisms that determine $T$ (including back-door paths through unobserved variables). Under $\doop(T{=}t)$, all arrows into $T$ are deleted, producing the mutilated graph $\mathcal{G}_{\overline{T}}$. This removes spurious back-door associations into $T$ while preserving the directed causal pathways from $T$ to its descendants. The effect of $T$ on $Y$ then propagates along whatever directed $T \to \cdots \to Y$ paths remain.

Graph surgery has an exact distributional counterpart. Under the Markov factorization — immediate for an NPSEM-IE — the observational joint distribution factors into one conditional per node, $P(v_1, \dots, v_p) = \prod_j P(v_j \mid \mathrm{Pa}(v_j))$. Modularity states that the intervention $\doop(T{=}t)$ changes this factorization in exactly one place: the factor for $T$ is removed, and every remaining factor is kept as it is, now evaluated at the fixed value $T{=}t$. Arrow deletion at the level of the graph is factor deletion at the level of the distribution. For the confounded triangle of Equation 1.2, the observational joint is $p(u)\,P(t \mid u)\,P(y \mid t, u)$; removing the treatment factor $P(t \mid u)$ and setting $T{=}t$ in what remains gives \[P(u, y \mid \doop(T{=}t)) = p(u)\,P(y \mid t, u), \qquad\text{hence}\qquad P(y \mid \doop(T{=}t)) = \int P(y \mid t, u)\, p(u)\, du.\]

Two features of this calculation deserve emphasis. First, the right-hand side contains only pieces of the observational model — the marginal $p(u)$ and the conditional $P(y \mid t, u)$ — so the interventional distribution is determined by the observational joint of $(U, T, Y)$ together with the graph; no new distributions need to be postulated. Second, determined is not the same as estimable: the retained factors involve the unobserved $U$, so the display cannot be evaluated from data on $(T, Y)$ alone. Whether the same interventional quantity can be re-expressed using observed variables only is the identification problem of Chapter 3; when an observed back-door set is available, the display above — with that set in place of $U$ — becomes the adjustment formula. Modularity is thus the DAG’s counterpart of the autonomy property of Section 1.2.1; its general form, the truncated factorization, is developed in Chapter 3.

1.2.3 Language 3: Potential Outcomes

Definition: Potential Outcome (Rubin 1974)

The potential-outcomes framework takes the collection $\{Y(t) : t \in \mathcal{T}\}$ as primitive: $Y(t)$ is the outcome that would have been observed had $T$ been set to $t$, possibly contrary to fact. Within the SEM Equation 1.2, the same potential outcome is constructed as $Y(t) = f_Y(t, U, \varepsilon)$, the solution for $Y$ in the mutilated system.

Definition: SUTVA

The stable unit treatment value assumption (SUTVA) has two components: (i) no interference — unit $i$’s potential outcome depends only on unit $i$’s own treatment, not on the treatments of other units; and (ii) no hidden versions of treatment — each treatment level $t$ corresponds to a single, well-defined intervention. SUTVA makes the scalar notation $Y_i(t)$ adequate; the separate consistency assumption $Y_i = Y_i(T_i)$ then connects the observed outcome to the potential outcome at the treatment actually received. Both are developed more fully in Chapter 4.

The potential-outcomes framework has two practical advantages that explain its popularity in the statistics community. First, it is particularly useful for defining causal parameters precisely. The average treatment effect is $\E[Y(1)] - \E[Y(0)]$, and more refined estimands are equally direct; for example, the average treatment effect on the treated is $\E[Y(1) - Y(0) \mid T{=}1]$. The ATT conditions on the treatment actually received, so its second term, $\E[Y(0) \mid T{=}1]$, is a genuinely counterfactual quantity: it cannot be written as a do-expression, because the do-operator describes interventional distributions only. Addressing such estimands graphically requires augmenting the DAG with counterfactual variables, as in the single-world intervention graphs of Appendix B. Second, statistical inference is more convenient in the potential-outcomes framework: once an estimand is written as a functional of $(Y(1), Y(0), T, X)$, the machinery of estimation — estimators, standard errors, and the semiparametric efficiency theory of Part III — applies directly. These estimands are developed systematically in Chapter 4.

1.2.4 Two Questions Every Framework Must Answer

The three languages just introduced differ in notation and emphasis, but each must answer the same two questions. First, how is an intervention represented, and how is the interventional world connected to the observed data? Autonomy and modularity answer this by specifying how the system changes under intervention: the treatment mechanism is replaced while all remaining mechanisms are preserved. Consistency plays a different role: it links the observed outcome to the potential outcome corresponding to the treatment actually received. Second, what effect-specific condition licenses adjustment for observed covariates? The table below, adapted from Table 1 of Wang et al. (2026), summarizes both answers side by side. Here $T$ denotes the treatment, $Y$ the outcome, and $X$ observed pretreatment covariates.

Framework	Representation of intervention and link to observed outcomes	Effect-specific condition for adjustment given $X$
Potential outcomes	Define $Y(t)$; consistency links potential and observed outcomes through $Y = Y(T)$, assumed alongside SUTVA	Weak ignorability: $Y(t) \indep T \mid X$ for each treatment level $t$
SEM	Autonomy (Haavelmo 1943; Pearl 2009): $\doop(T{=}t)$ replaces the structural equation for $T$ with the constant $t$, while every other equation and the distribution of the background disturbances remain unchanged	Conditional on $X$, the treatment mechanism carries no information about the structural inputs determining $Y(t)$
Causal DAG	Modularity (graph surgery): $\doop(T{=}t)$ deletes all arrows into $T$, and the interventional distribution is given by the truncated factorization (Chapter 3)	$X$ blocks every back-door path from $T$ to $Y$ and contains no descendant of $T$ (Chapter 3)

In all three representations, an additional support condition — positivity — is required for nonparametric adjustment. For discrete treatment, $P(T{=}t \mid X{=}x) > 0$ for each relevant pair $(t, x)$; for continuous treatment, $t$ must lie in the conditional support of $T \mid X{=}x$, with positive conditional density in the relevant neighborhood (Chapter 4).

A note on terminology: the word modularity is used in the literature for both the SEM and the DAG rows of the table. In the SEM literature it is a synonym for autonomy — the mechanism generating $T$ can be replaced without altering the other equations — while in the graphical literature it refers to the arrow-deletion semantics of the do-operator. The two usages express the same idea in different languages, but readers should be prepared to encounter both.

Remark: Scope of the Assumptions

The assumptions used in the three languages differ in scope. Weak ignorability and the back-door condition are effect-specific: they concern a particular treatment–outcome pair and an adjustment set. The NPSEM-IE is system-wide: it jointly constructs counterfactual variables for interventions throughout the model and imposes restrictions across intervention worlds. Causal sufficiency is graph-wide: it rules out latent common causes among the variables represented in the graph. None of these system-wide properties by itself implies that every causal effect is identified from the observed-data distribution; identification remains a separate question, taken up in Section 1.4. See Wang et al. (2026) for an extended discussion.

1.2.5 Roadmap: How the Trinity Organizes These Notes

The three Parts of the course each draw on the three languages introduced above, with different languages moving into the foreground as the subject matter demands. Understanding this organization in advance will help you see why each language is introduced when it is.

Part I — Foundations: the graphical language (Chapters 1–3). Part I builds the graphical foundation. The DAG and the do-operator are the primary language here because they make the interventional/observational distinction syntactically explicit: confounding is visible as an open back-door path, and identifying assumptions are visible as graph criteria (back-door, front-door, do-calculus). A mistake that is easy to commit silently in potential-outcomes notation becomes a type error in do-notation, and is therefore caught before it propagates. The SEM appears as the generative backbone wherever closed-form calculations are needed, particularly in the Gaussian confounded running example.

Part II — Identification: designs and research strategies (Chapters 4–9). Part II introduces the potential outcomes framework (Chapter 4) and then studies the specific research designs through which observational and experimental data support identification: randomization and back-door adjustment (Chapter 5), propensity score methods (Chapter 6), instrumental variables (Chapter 7), mediation and front-door identification (Chapter 8), and sensitivity analysis when identifying assumptions are only partially credible (Chapter 9). This is where the potential-outcomes and graphical languages are most explicitly translated into each other: ignorability ($Y(t) \indep T \mid X$) is the potential-outcomes counterpart of the back-door criterion, and each design is stated both graphically and in potential-outcomes form before the resulting identification formula is derived.

Part III — Estimation: semiparametric and machine-learning methods (Chapters 10–13). Once identification has been established in the language of either DAGs or potential outcomes, Part III turns to estimation. The semiparametric efficiency framework — estimating equations, influence functions, doubly robust estimators, and double machine learning — is largely language-neutral: it works with whatever moment condition the identification step produces. Chapter 13 applies these tools to estimation under instrumental variables. Fluency in all three languages is expected by this point, as Part III draws freely on all of them.

1.2.6 Historical Roots of the Causal Trinity (Optional)

The three languages did not emerge from a single research program. They grew independently in different disciplines over more than a century, driven by different scientific questions and different communities. Understanding this history explains why the three schools sometimes use incompatible terminology for the same idea, and why unifying them is intellectually non-trivial.

Structural Equation Models: genetics and econometrics. The oldest strand begins with the biologist Sewall Wright (1921), who introduced path analysis to decompose correlations among hereditary traits into direct and indirect causal contributions — arguably the first use of a directed graph for causal reasoning. The framework was transplanted into economics by Haavelmo (1943), who argued that economic relationships should be modeled as autonomous structural equations robust to policy interventions — an early statement of what Pearl would later call “graph surgery”. The Cowles Commission (Koopmans, Klein, and others) formalized simultaneous equation systems throughout the 1940s–50s, and this SEM tradition became the standard language of causal inference in econometrics. Lucas’s (1976) critique — that reduced-form equations break down when policy changes the data-generating process — is essentially a statement of the interventional/observational distinction, decades before that phrase existed. SEMs were independently adopted by sociologists and psychologists (LISREL, factor analysis), making this the most interdisciplinary of the three schools.

Potential Outcomes: experimental statistics and epidemiology. Neyman (1923) introduced the notation $Y_i(t)$ in the context of randomized agricultural experiments, asking what a plot’s yield would have been under an unobserved treatment. Fisher (1935) placed randomization at the foundation of causal inference via the randomization distribution. The crucial extension to observational studies was made by Rubin (1974), who formalized the assignment mechanism; the stable unit treatment value assumption was subsequently labeled SUTVA in Rubin (1980). Holland’s (1986) JASA paper “Statistics and Causal Inference” brought the framework to mainstream statistics under the dictum “no causation without manipulation”. Robins (1986) extended potential outcomes to longitudinal treatments and time-varying confounding via $g$-computation. Imbens and Angrist (1994) connected the framework to instrumental variables and defined the LATE, bringing potential outcomes into empirical economics. Today the framework is dominant in statistics, epidemiology, and much of economics.

DAGs and do-calculus: computer science and philosophy. Pearl developed Bayesian networks through the 1980s as a representation for probabilistic reasoning in artificial intelligence. The pivotal step came in Pearl (1995), where the do-operator was introduced: a formal syntax for distinguishing interventional from observational distributions within a graph-based algebra. The 2000 monograph Causality synthesized DAGs, the do-calculus, identification theory, and the connections to potential outcomes into a unified framework. In parallel, Spirtes, Glymour, and Scheines (1993) — working from philosophy at Carnegie Mellon — developed constraint-based algorithms (PC, FCI) for learning causal structure from data. The DAG framework was rapidly adopted by epidemiology (Hernán and Robins) and is now the primary language of causal inference in computer science and AI.

Convergence. The three frameworks are closely related and often intertranslatable, but exact equivalence requires additional assumptions about the causal model; see Wang et al. (2026) for a recent side-by-side comparison. In these notes we work mainly in an NPSEM-IE setting, where translation between the languages is especially clean. Under weaker single-world models — for example, Robins’s FFRCISTG (Robins 1986) — some nested cross-world counterfactuals represented by an NPSEM-IE are not encoded without additional cross-world structure, so causal interpretations or identification results involving those quantities require assumptions beyond the single-world model. The historical differences between the three schools are primarily in notation, native strengths, and disciplinary culture, with substantive differences appearing mainly in how much cross-world structure each framework commits to.

Key milestones in the three schools of causal inference. SEM (top) originates in genetics and econometrics; PO (middle) in experimental statistics; DAG (bottom) in computer science and philosophy. The three developed largely in isolation until converging in the 2000s.

1.3 The Gaussian Linear Confounded Model

Section 1.2 introduced three complementary ways to represent an intervention. We now make the distinction between observation and intervention quantitatively explicit. In a confounded structural model, observing $T{=}t$ selects units according to a treatment-dependent distribution of the latent common cause $U$, whereas intervening to set $T{=}t$ leaves $U$ distributed according to its population law. In the linear model below, the coefficient of $t$ in the observational conditional mean decomposes into the causal slope $\beta$ plus an omitted-variable confounding term — association $=$ causation $+$ confounding, in the slogan of Wang et al. (2026) — while the full observational and interventional distributions additionally differ in variance. General conditions under which interventional distributions can be identified from observational data are deferred to Chapters 3–5.

1.3.1 Setup

The Gaussian linear confounded model is our primary running example throughout this course. It is simple enough to permit closed-form calculations yet rich enough to illustrate every key concept. We use it here not to develop any particular identification strategy, but to make the gap between observational and interventional reasoning mathematically explicit; various identification strategies for this model appear in later chapters.

Example: Gaussian Linear Confounded Model

The structural equations are \[Y = \beta T + \gamma U + \varepsilon, \qquad T = \alpha U + \delta, \tag{1.3}\] where $U$ is an unobserved confounder and the three primitive variables are mutually independent: \[U \sim \mathcal{N}(0,\sigma_U^2), \qquad \varepsilon \sim \mathcal{N}(0,\sigma_\varepsilon^2), \qquad \delta \sim \mathcal{N}(0,\sigma_\delta^2).\] Here $\beta$ is the causal effect of $T$ on $Y$, $\gamma$ is the direct effect of the confounder $U$ on $Y$, and $\alpha$ is the direct effect of $U$ on $T$. The joint independence of $(U, \varepsilon, \delta)$ is the NPSEM-IE condition.

The endogeneity of $T$ is immediate. From the analyst’s perspective, the composite error in the $Y$ equation is $\gamma U + \varepsilon$, so: \[\mathrm{Cov}(T,\; \gamma U + \varepsilon) = \gamma\,\mathrm{Cov}(\alpha U + \delta,\; U) = \gamma\alpha\sigma_U^2 \neq 0 \quad (\text{when } \alpha \neq 0 \text{ and } \gamma \neq 0).\] Consequently, OLS regressing $Y$ on $T$ does not estimate $\beta$.

Graph surgery on the Gaussian confounded model. The faded arrow in the mutilated DAG has been severed by the intervention $\doop(T{=}t)$. The back-door path $T \leftarrow U \to Y$ is eliminated; only the causal path $T \to Y$ remains active. The treatment node becomes a fixed constant (rectangle).

1.3.2 Two Explicit Densities

We now derive both densities in closed form. Write $\sigma_T^2 = \alpha^2\sigma_U^2 + \sigma_\delta^2$ for the marginal variance of $T$.

Interventional density. Set $T = t$ externally. The back-door path through $U$ is severed, so $U$ is independent of $T$ under the intervention and is drawn from its marginal $\mathcal{N}(0,\sigma_U^2)$. Then $Y \mid \doop(T{=}t) = \beta t + \gamma U + \varepsilon$ with $U \indep \varepsilon$. Hence \[Y \mid \doop(T{=}t) \sim \mathcal{N}\!\left(\beta t,\;\; \gamma^2\sigma_U^2 + \sigma_\varepsilon^2\right). \tag{1.4}\]

Observational density. Observe $T = t$. Since $Y$ and $T$ are both linear combinations of the independent normals $(U, \varepsilon, \delta)$, the pair $(Y, T)$ is jointly normal. Their covariance is \[\mathrm{Cov}(Y, T) = \mathrm{Cov}(\beta T + \gamma U + \varepsilon,\; T) = \beta\sigma_T^2 + \gamma\alpha\sigma_U^2.\] Applying the conditional normal formula: \[Y \mid T{=}t \sim \mathcal{N}\!\left(\left(\beta + \frac{\gamma\alpha\sigma_U^2}{\sigma_T^2}\right)t,\;\; \frac{\gamma^2\sigma_U^2\sigma_\delta^2}{\sigma_T^2} + \sigma_\varepsilon^2\right). \tag{1.5}\]

1.3.3 The Endogeneity Gap

Equations Equation 1.4 and Equation 1.5 differ in both mean and variance:

Equal when $\alpha = 0$ or $\gamma = 0$ (no confounding).
	Interventional	Observational
Mean	$\beta t$	$\left(\beta + \dfrac{\gamma\alpha\sigma_U^2}{\sigma_T^2}\right)t$
Variance	$\gamma^2\sigma_U^2 + \sigma_\varepsilon^2$	$\dfrac{\gamma^2\sigma_U^2\sigma_\delta^2}{\sigma_T^2} + \sigma_\varepsilon^2$
Distribution of $U$	$p(u)$	$p(u \mid T{=}t)$
Identified by	RCT; the slope $\beta$ also by IV (Chapter 7)	OLS

In this linear model, the coefficient of $t$ in the observational mean Equation 1.5 is also the probability limit of the OLS slope from regressing $Y$ on $T$, since both equal $\mathrm{Cov}(Y,T)/\mathrm{Var}(T)$. The comparison of the two means is therefore exactly the decomposition promised by the slogan in this section’s opening: \[\underbrace{\beta + \frac{\gamma\alpha\sigma_U^2}{\sigma_T^2}}_{\text{association: observational slope}} \;=\; \underbrace{\vphantom{\frac{\gamma\alpha\sigma_U^2}{\sigma_T^2}}\beta}_{\text{causation}} \;+\; \underbrace{\gamma \cdot \frac{\alpha\sigma_U^2}{\sigma_T^2}}_{\text{confounding: omitted-variable bias}}.\]

The confounding term is the classical omitted-variable bias (developed in Section 1.3.4 below) — the endogeneity bias of the OLS coefficient anticipated in the definition above — and the example shows exactly why: the omitted $U$ affects the outcome with coefficient $\gamma$, and its regression on the included regressor has slope $\E[U \mid T{=}t] = (\alpha\sigma_U^2/\sigma_T^2)\,t$ by the conditional normal formula. The bias is the product of the two — the effect of the omitted variable on $Y$ times the regression of the omitted variable on $T$ — and it vanishes when either factor does ($\gamma = 0$ or $\alpha = 0$). It takes the sign of $\gamma\alpha$ (positive here). Numerically, with $\alpha = \gamma = 1$ and $\sigma_U^2 = \sigma_\delta^2 = 1$: \[\text{bias} = \frac{1 \times 1 \times 1}{1^2 \times 1 + 1} = \frac{1}{2} = 0.5.\]

1.3.4 Omitted-Variable Bias

The decomposition just derived is an instance of a completely general regression identity that involves no causal assumptions — only second moments. Consider three population least-squares regressions of centered random variables, in notation mirroring the running model: the full model, regressing $Y$ on $(T, X)$ with coefficients $(\beta, \gamma)$; the reduced model, regressing $Y$ on $T$ alone with coefficient $\tilde{\beta}$; and the auxiliary model, regressing the omitted variable $X$ on $T$ with coefficient $\tilde{\alpha} = \mathrm{Cov}(X, T)/\mathrm{Var}(T)$. (Tilded coefficients come from regressions on $T$ alone.)

Theorem 1.1 (Omitted-Variable Bias) Let $(T, X, Y)$ be centered random variables with finite second moments, $\mathrm{Var}(T) > 0$, and $(T, X)$ not perfectly collinear. Then \[\tilde{\beta} = \beta + \gamma\,\tilde{\alpha}, \qquad\text{i.e.,}\qquad \underbrace{\tilde{\beta} - \beta}_{\text{omitted-variable bias}} = \gamma\,\tilde{\alpha}.\] The bias in the reduced-model coefficient is the product of the omitted variable’s coefficient in the full model and the coefficient from regressing the omitted variable on the included regressor.

Proof. By the normal equations of the full regression, $Y = \beta T + \gamma X + e$ with $\E[eT] = \E[eX] = 0$. Hence \[\tilde{\beta} = \frac{\mathrm{Cov}(Y, T)}{\mathrm{Var}(T)} = \beta + \gamma\,\frac{\mathrm{Cov}(X, T)}{\mathrm{Var}(T)} = \beta + \gamma\,\tilde{\alpha}. \qquad \square\]

The identity can equivalently be routed through the treatment model, the regression of $T$ on $X$ with coefficient $\alpha = \mathrm{Cov}(T, X)/\mathrm{Var}(X)$: since $\mathrm{Cov}(X, T) = \alpha\,\mathrm{Var}(X)$, the auxiliary coefficient is $\tilde{\alpha} = \alpha\,\mathrm{Var}(X)/\mathrm{Var}(T)$ and the bias is $\gamma\,\alpha\,\mathrm{Var}(X)/\mathrm{Var}(T)$.

Geometric interpretation. Population least squares is orthogonal projection in the Hilbert space of centered, square-integrable random variables with inner product $\langle A, B \rangle = \E[AB]$. The full model is the projection of $Y$ onto the plane $\mathrm{span}\{T, X\}$: the normal equations say exactly that $Y = \beta T + \gamma X + e$ with $e \perp \mathrm{span}\{T, X\}$. The reduced coefficient is the projection onto the line $\mathrm{span}\{T\}$, $\tilde{\beta}\,T = \Pi_T Y$. Because the line lies inside the plane, we may project in two stages: \[\Pi_T Y = \Pi_T\bigl(\beta T + \gamma X + e\bigr) = \beta T + \gamma\,\Pi_T X + 0 = (\beta + \gamma\tilde{\alpha})\,T,\] since $\Pi_T e = 0$ ($e$ is orthogonal to the whole plane, hence to the line) and $\Pi_T X = \tilde{\alpha}\,T$ — which is precisely the auxiliary regression. Theorem 1.1 is thus the tower property of nested projections, and the omitted-variable bias is the shadow that the omitted component $\gamma X$ casts on the line spanned by the included regressor. The bias vanishes precisely when $\gamma X \perp T$: either there is no omitted component ($\gamma = 0$) or it is orthogonal to the included regressor ($\tilde{\alpha} = 0$, equivalently $\alpha = 0$). A complementary fact, the Frisch–Waugh–Lovell theorem, reads off the full-model coefficient from the same picture: $\beta$ is the coefficient of $Y$ on the component of $T$ orthogonal to $X$. The same projection geometry, in infinite-dimensional form, underlies the semiparametric efficiency theory of Part III and Appendix C.

The omitted-variable bias theorem as projection geometry in the plane $\mathrm{span}\{T, X\}$; the residual $e$ is orthogonal to this plane and plays no role. Projecting $Y$ onto $\mathrm{span}\{T\}$ in two stages sends $\beta T$ to itself and the omitted component $\gamma X$ to its shadow $\gamma\tilde{\alpha}\,T$; the omitted-variable bias is the length of that shadow. It vanishes when $\gamma = 0$ (no omitted component) or $\tilde{\alpha} = 0$ (orthogonality).

Confirmation in the Gaussian model. Take $X = U$ in the running example Equation 1.3. Because $\varepsilon$ is independent of $(T, U)$, the structural coefficients $\beta$ and $\gamma$ are also the full-model regression coefficients. The auxiliary coefficient is $\tilde{\alpha} = \mathrm{Cov}(U, T)/\mathrm{Var}(T) = \alpha\sigma_U^2/\sigma_T^2$ — the slope of $\E[U \mid T{=}t]$ computed in the preceding subsection — where $\alpha$ enters because the treatment equation $T = \alpha U + \delta$ is itself the population regression of $T$ on $U$. Theorem 1.1 then gives \[\tilde{\beta} - \beta = \gamma\,\tilde{\alpha} = \gamma\,\alpha\,\frac{\sigma_U^2}{\sigma_T^2},\] exactly the confounding term of the decomposition display above. One caution: the theorem is assumption-free regression algebra, so it attaches no causal meaning to $\beta$ by itself. In this example the full-model coefficient is the causal slope because the structural model is linear and $U$ closes the only back-door path; in general, whether a full-model coefficient has a causal interpretation is precisely the identification question of Section 1.4.

1.3.5 Lab: Simulating the Two Densities

The analytical results of this section were illustrated numerically by simulation using $n = 10{,}000$ draws from the structural equations with parameters $\beta = 2$, $\alpha = 1$, $\gamma = 1$, $\sigma_U = \sigma_\varepsilon = \sigma_\delta = 1$. The R code is distributed separately as chapter1_lab.R.

Experiment 1: OLS bias. OLS was applied to the observational pairs $(Y_i, T_i)$. By the omitted-variable bias formula, the OLS probability limit is $\beta + \gamma\alpha\sigma_U^2/\sigma_T^2 = 2 + 0.5 = 2.5$. The results are consistent with this:

	Simulated	Theory	Formula
OLS slope	2.5147	2.5000	$\beta + \gamma\alpha\sigma_U^2/\sigma_T^2$
True $\beta$	—	2.0000	$\beta$
Bias	0.5147	0.5000	$\gamma\alpha\sigma_U^2/\sigma_T^2$

OLS overestimates the causal effect by 25%.

Experiment 2: Two-sample comparison at $t_0 = 1$. Since $(Y, T)$ is jointly normal, the exact observational conditional distribution $Y \mid T{=}1$ is available in closed form from Equation 1.5: it is $\mathcal{N}(2.5,\, 1.5)$. The observational sample was therefore drawn directly from this distribution; no bandwidth or neighborhood selection was required. The interventional sample was generated by setting $T = 1$ externally and drawing fresh $U^*_i \sim \mathcal{N}(0,1)$ and $\varepsilon^*_i \sim \mathcal{N}(0,1)$ independently, giving $Y^*_i = \beta + \gamma U^*_i + \varepsilon^*_i \sim \mathcal{N}(2,\,2)$.

	Mean (simulated)	Mean (theory)	SD (simulated)	SD (theory)
$Y \mid T{=}1$	2.497	2.500	1.229	$\sqrt{1.5} = 1.225$
$Y \mid \doop(T{=}1)$	2.001	2.000	1.428	$\sqrt{2} = 1.414$

The variance reduction in the observational distribution reflects the ratio $\sigma_{\mathrm{obs}}^2/\sigma_{\mathrm{int}}^2 = 1.5/2 = 0.75$: conditioning on $T{=}1$ pins down the value of $\alpha U + \delta$, which through the correlation with $\gamma U$ restricts the spread of $Y$.

Experiment 3: Density overlay for varying $\alpha$. The figure below shows the interventional density $\mathcal{N}(\beta t_0,\, \gamma^2\sigma_U^2 + \sigma_\varepsilon^2)$ (solid) and the observational density Equation 1.5 (broken line styles), both evaluated at $t_0 = 1$, for $\alpha \in \{0, 0.3, 0.7, 1.0\}$. The interventional density does not depend on $\alpha$ and remains fixed; as $\alpha$ increases over the plotted range, the observational curve shifts rightward (growing bias) and narrows (reduced variance). The bias is not globally monotone in $\alpha$, however: differentiating $b(\alpha) = \gamma\alpha\sigma_U^2/(\alpha^2\sigma_U^2 + \sigma_\delta^2)$ shows that it peaks at $\alpha = \sigma_\delta/\sigma_U$ — exactly $\alpha = 1$ under the figure’s parameters — and vanishes as $\alpha \to \infty$, because increasing $\alpha$ inflates $\mathrm{Var}(T)$ faster than $\mathrm{Cov}(T, U)$.

Figure 1.1: Interventional density $N(2,2)$ (solid black) versus observational density for $\alpha \in \{0, 0.3, 0.7, 1.0\}$ (broken line styles), evaluated at $t_0 = 1$. Parameters: $\beta=2$, $\gamma=1$, $\sigma_U=\sigma_\varepsilon=\sigma_\delta=1$. At $\alpha = 0$ there is no confounding and the two densities coincide.

1.4 The Two-Step Paradigm

Causal inference is a two-step discipline. The two steps are logically distinct and require different tools.

1.4.1 Step 1: Identification

Definition: Identification

A causal quantity $P(y \mid \doop(T{=}t))$ is identified from the observational distribution $P$ if there exists a functional $\Psi$, valid across the assumed model class, such that \[P(y \mid \doop(T{=}t)) = \Psi(P),\] and $\Psi(P)$ depends only on the observable joint distribution. Equivalently, relative to a class $\mathcal{M}$ of causal models, a causal quantity $Q$ is identified if any two models in $\mathcal{M}$ that induce the same observed-data distribution also agree on $Q$: \[P^{\mathrm{obs}}_{M_1} = P^{\mathrm{obs}}_{M_2} \;\Longrightarrow\; Q(M_1) = Q(M_2) \qquad \text{for all } M_1, M_2 \in \mathcal{M}.\]

The second formulation makes explicit what failure of identification means: two causal models that are indistinguishable from observational data yet disagree on the causal quantity. Chapter 3 exhibits exactly such a construction.

Example: Identification and Its Failure in the Gaussian Model

Return to the Gaussian confounded model Equation 1.3 and suppose only $(T, Y)$ is observed. Consider two parameter settings:

$M_1$ (all causal): $\beta = 1$, $\alpha = \gamma = 0$, $\sigma_\delta^2 = 1$, $\sigma_\varepsilon^2 = 2$;
$M_2$ (all confounding): $\beta = 0$, $\alpha = 1$, $\gamma = 2$, $\sigma_U^2 = \sigma_\delta^2 = \tfrac{1}{2}$, $\sigma_\varepsilon^2 = 1$.

Both settings induce the same observed law: $(T, Y)$ is bivariate normal with mean zero, $\mathrm{Var}(T) = 1$, $\mathrm{Var}(Y) = 3$, and $\mathrm{Cov}(T, Y) = 1$. In particular, the observed slope $\mathrm{Cov}(T,Y)/\mathrm{Var}(T) = 1$ in both — but it is all causation under $M_1$ and all confounding under $M_2$: the two terms of the slogan cannot be separated from the observed law alone. The interventional distributions differ accordingly: $Y \mid \doop(T{=}t) \sim \mathcal{N}(t, 2)$ under $M_1$, but $\mathcal{N}(0, 3)$ under $M_2$. No functional $\Psi$ of the law of $(T, Y)$ can return both answers, so $P(y \mid \doop(T{=}t))$ is not identified from the distribution of $(T, Y)$ alone. If instead $U$ is also observed, the two models induce different laws for $(U, T, Y)$, and the functional $\Psi(P) = \int P(y \mid t, u)\, p(u)\, du$ — the modularity calculation of Section 1.2.2 — identifies the interventional distribution. Observing the confounder restores identification.

Identification is a population-level question, posed relative to a causal model class, the observed-data law, and the support of that law: given the assumed graph and assumptions, can the interventional density be expressed as a functional of the observed-data distribution? It does not depend on sample size.

The main tools are the back-door criterion (introduced in Chapter 3, applied in Chapters 4–6), the front-door criterion (Chapters 3 and 8), the three rules of the do-calculus (Chapter 3), and IV assumptions (Chapter 7).

1.4.2 Step 2: Estimation

Once $\Psi(P)$ has been identified, the statistical problem is to estimate the functional $\Psi(P)$ from $n$ observations $O_1,\dots,O_n$ as efficiently as possible. The main tools are outcome regression and standardization (the g-formula, Chapter 5), efficient influence functions (EIF, Part III), inverse probability weighting (IPW, Chapter 6 and Part III), doubly robust estimators (Part III), and double machine learning (DML, Part III).

1.4.3 A Preview: Identification and Estimation in the Flu-Vaccination Example

The following example illustrates the two steps with a binary outcome and a fully observed confounder, so no advanced machinery is required.

Scenario. A public health agency observes 1,000 individuals over one flu season. Each person either received a flu vaccine ($T=1$) or did not ($T=0$), and either became infected ($Y=1$) or did not ($Y=0$). Age group $X$ (elderly: $X{=}1$; young: $X{=}0$) is recorded for all individuals. Elderly people are both more likely to be vaccinated (due to public health campaigns) and more susceptible to infection (due to weaker immune systems), so $X$ confounds the relationship between $T$ and $Y$.

Causal DAG for the vaccination example. The green arrow is the causal path $T \to Y$. The red dashed arrows form the back-door path $T \leftarrow X \to Y$: age $X$ causes both vaccination $T$ and infection $Y$, creating a spurious association. Because $X$ is observed, conditioning on it blocks the back-door path.

Observed data. The 1,000 individuals break down as follows (with within-cell infection rates in parentheses):

	Vaccinated ($T=1$): $n$	Infected	Unvaccinated ($T=0$): $n$	Infected	Total
Elderly ($X{=}1$)	360	108 (30%)	40	20 (50%)	400
Young ($X{=}0$)	60	3 (5%)	540	54 (10%)	600
Total	420	111 (26.4%)	580	74 (12.8%)	1,000

Simpson’s paradox at a glance. The infection-rate comparisons between vaccinated and unvaccinated — within each stratum and in aggregate — tell contradictory stories:

Stratum	Vaccinated	Unvaccinated	Difference	Conclusion
Elderly ($X{=}1$)	30%	50%	$-20$ pp	vaccine helps
Young ($X{=}0$)	5%	10%	$-5$ pp	vaccine helps
Aggregate (raw)	26.4%	12.8%	$\mathbf{+13.6}$ pp	vaccine harms??

Within every stratum the vaccine reduces infection. Yet in the aggregate the vaccinated group has a higher infection rate. The reversal occurs because the vaccinated group is disproportionately elderly (86% of vaccinated vs. 7% of unvaccinated): a high-risk group is systematically over-represented among the treated, making the raw comparison misleading. The back-door adjustment formula, developed in Chapter 3, corrects this by standardizing to the population age distribution rather than the treatment-conditional one. We use the back-door formula here as a preview: Chapters 3–5 derive its graphical and potential-outcomes justifications and develop the corresponding estimators systematically.

Step 1: Identification. The only back-door path is $T \leftarrow X \to Y$. The set $\{X\}$ satisfies the back-door criterion: it blocks this path (fork at $X$) and contains no descendant of $T$. We also assume positivity, $P(T{=}t \mid X{=}x) > 0$ for all four $(t, x)$ pairs — an assumption about the population with which the table is consistent, since every treatment-by-stratum cell is occupied. By the back-door formula: \[P\!\left(Y{=}1 \mid \doop(T{=}t)\right) = \sum_{x \in \{0,1\}} P(Y{=}1 \mid T{=}t, X{=}x)\,P(X{=}x). \tag{1.6}\] The causal quantity on the left has been expressed entirely as a functional of the observed-data distribution on the right.

Step 2: Estimation. Plugging sample proportions into Equation 1.6, with $\hat{P}(X{=}1) = 0.4$ and $\hat{P}(X{=}0) = 0.6$: \[\hat{P}(Y{=}1 \mid \doop(T{=}1)) = 0.30 \times 0.4 + 0.05 \times 0.6 = 0.15,\] \[\hat{P}(Y{=}1 \mid \doop(T{=}0)) = 0.50 \times 0.4 + 0.10 \times 0.6 = 0.26.\] The estimated causal risk difference is $\hat{\Delta}_{\mathrm{causal}} = 0.15 - 0.26 = -0.11$: under the assumed causal DAG, consistency, and positivity, the standardized estimate indicates an 11-percentage-point reduction in infection risk. This is a plug-in point estimate; quantifying its uncertainty is deferred to later chapters.

Why the naive comparison fails. The raw difference $0.264 - 0.128 = +0.136$ is precisely the aggregate row of the Simpson table above: it mixes strata in treatment-dependent proportions, importing the high elderly infection rate into the vaccinated group. Under the assumed causal model, the standardized estimate $-0.11$ is the correct target because it re-weights each stratum by $P(X{=}x)$, the population share, not by $P(X{=}x \mid T{=}t)$.

The Two Steps, Concretely

Step 1 (identification) established — from the assumed causal graph and its intervention semantics — that the back-door formula Equation 1.6 expresses the causal quantity as a functional of observable probabilities. Step 2 (estimation) replaced the population probabilities in that formula with sample proportions.

These two steps are logically separate: Step 1 is a mathematical argument about the causal model that does not depend on sample size; Step 2 is a statistical problem conditional on the identification formula obtained in Step 1. A researcher who skips Step 1 and reports the raw difference conflates the two and draws the wrong conclusion.

1.5 Summary

Interventional vs. observational distribution. Causal inference answers questions about interventions $P(y \mid \doop(T{=}t))$, not about observations $P(y \mid T{=}t)$. In the Gaussian confounded model, the two densities coincide when $\alpha\gamma = 0$. Only $\alpha = 0$ corresponds to exogeneity in the sense $T \indep U$; $\gamma = 0$ instead means that $U$ no longer affects $Y$. In either case, $T$ is exogenous for the outcome equation and OLS recovers $\beta$. When both channels are active ($\alpha \neq 0$ and $\gamma \neq 0$), the two distributions differ; the gap in mean slopes is the endogeneity bias of the OLS coefficient.
The causal trinity. The same causal question can be expressed in three languages — SEM (equation surgery), DAG (graph surgery), and potential outcomes ($Y(t)$). In the SEM framework, $Y(t)$ is constructed as the solution for $Y$ in the mutilated system, so $Y(t)$ has distribution $P(y \mid \doop(T{=}t))$ by construction. Fluency in all three languages is essential for modern causal reasoning.
Endogeneity bias in the Gaussian confounded model. In the Gaussian linear confounded model, the coefficient of $T$ in the observational mean $\E[Y \mid T{=}t]$ differs from the causal coefficient $\beta$ by $\gamma\alpha\sigma_U^2/\sigma_T^2$, where $\sigma_T^2 = \alpha^2\sigma_U^2 + \sigma_\delta^2$. This is the omitted-variable bias: $U$ drives both $T$ (via $\alpha$) and $Y$ (via $\gamma$), and its omission from the regression biases the OLS estimand. The observational variance $\gamma^2\sigma_U^2\sigma_\delta^2/\sigma_T^2 + \sigma_\varepsilon^2$ is smaller than the interventional variance $\gamma^2\sigma_U^2 + \sigma_\varepsilon^2$.
Two-step paradigm. Causal inference is a two-step discipline: identification (expressing $\Psi(P)$ as a functional of observable data, a mathematical question about the causal model) followed by estimation (constructing $\hat{\Psi}_n$ efficiently from $n$ observations, a statistical question about finite samples). The two steps are analytically separate, with Step 2 carried out conditional on the identification formula obtained in Step 1.

Causal inference is the study of when, and under what assumptions, the observational distribution contains enough information to determine an interventional distribution or causal estimand. Everything in this course — the back-door criterion, do-calculus, instrumental variables, sensitivity analysis, semiparametric estimation — is an answer to that question in a particular setting.

1.6 Problems

1. Do-notation fundamentals. For each statement below, decide whether it refers to an interventional or an observational distribution, and rewrite it unambiguously using do-notation where appropriate.

“The probability that a patient recovers given that they took the drug.”
“The probability that a patient would recover if we prescribed the drug to everyone.”
“Among students who attended tutoring, the average exam score was 85.”
“If we enrolled all students in tutoring, the average exam score would be 85.”

2. Deriving the two Gaussian laws. In the Gaussian confounded model Equation 1.3, verify Equation 1.5 and Equation 1.4 by carrying out the following steps. Let $\sigma_T^2 = \alpha^2\sigma_U^2 + \sigma_\delta^2$.

Show that $(Y, T)$ is jointly normal by writing both as linear combinations of the independent normals $(U, \varepsilon, \delta)$.
Compute $\mathrm{Cov}(Y, T)$ and $\mathrm{Var}(T)$.
Apply the conditional normal formula to derive $\E[Y \mid T{=}t]$ and $\mathrm{Var}[Y \mid T{=}t]$, and hence verify Equation 1.5.
Show that the OLS probability limit is $\beta + \gamma\alpha\sigma_U^2/\sigma_T^2$, and interpret this as an omitted-variable bias formula.
Derive the distribution of $Y \mid \doop(T{=}t)$ directly from the mutilated structural equation, and compare its mean and variance with those obtained in part (c).

3. The causal trinity. Consider the following verbal causal claim: “Aspirin ($T$) reduces the risk of heart attack ($Y$) because it inhibits platelet aggregation ($M$), but patients with pre-existing cardiovascular disease ($U$, unobserved) are both more likely to take aspirin and more likely to have a heart attack.” Assume that all effects of aspirin on heart-attack risk operate through platelet aggregation, so there is no direct $T \to Y$ edge.

Draw the causal DAG implied by this description.
Write down the recursive SEM with appropriate structural functions $f_T$, $f_M$, $f_Y$.
State the SUTVA assumption and discuss whether it is plausible in this setting.
Explain why $\E[Y \mid T{=}1] - \E[Y \mid T{=}0]$ does not identify the causal effect $\E[Y \mid \doop(T{=}1)] - \E[Y \mid \doop(T{=}0)]$ in this DAG.

4. Equation-specific exogeneity. In the Gaussian confounded model Equation 1.3, write the outcome equation as $Y = \beta T + \eta$ with structural disturbance $\eta = \gamma U + \varepsilon$. Verify that $\mathrm{Cov}(T, \eta) = \alpha\gamma\sigma_U^2$, and classify $T$ as exogenous or endogenous for the outcome equation under each of the following cases:

$\alpha \neq 0$ and $\gamma \neq 0$;
$\alpha = 0$ and $\gamma \neq 0$;
$\alpha \neq 0$ and $\gamma = 0$.

In which cases does OLS recover $\beta$? In which cases is $T$ independent of $U$?

5. Omitted-variable bias. In the setting of Theorem 1.1, with full model $Y = \beta T + \gamma X + e$, reduced-model coefficient $\tilde{\beta}$, and auxiliary coefficient $\tilde{\alpha} = \mathrm{Cov}(X,T)/\mathrm{Var}(T)$:

Show that the bias can equivalently be written in terms of the treatment model $T = \alpha X + v$, with $\alpha = \mathrm{Cov}(T,X)/\mathrm{Var}(X)$, as $\tilde{\beta} - \beta = \gamma\alpha\,\mathrm{Var}(X)/\mathrm{Var}(T)$, and explain why a variance ratio appears in this form but not in Theorem 1.1.
A researcher regresses log wage ($Y$) on years of schooling ($T$), omitting ability ($X$). Suppose ability raises both wages ($\gamma > 0$) and schooling ($\alpha > 0$). Determine the sign of the bias in the schooling coefficient and interpret the result.
Suppose instead that the omitted $X$ is uncorrelated with $T$ ($\alpha = 0$) but still affects $Y$ ($\gamma \neq 0$). Show that $\tilde{\beta} = \beta$, and show that the reduced model’s residual variance exceeds the full model’s by $\gamma^2\,\mathrm{Var}(X)$. What does omitting $X$ cost, if not bias?

Haavelmo, Trygve. 1943. “The Statistical Implications of a System of Simultaneous Equations.” Econometrica 11 (1): 1–12.

Pearl, Judea. 2009. Causality: Models, Reasoning, and Inference. 2nd ed. Cambridge University Press.

Richardson, Thomas S., and James M. Robins. 2014. ACE Bounds; Single World Intervention Graphs (SWIGs) and Identification of Causal Effects. University of Washington.

Robins, James M. 1986. “A New Approach to Causal Inference in Mortality Studies with a Sustained Exposure Period—Application to Control of the Healthy Worker Survivor Effect.” Mathematical Modelling 7 (9–12): 1393–512.

Rubin, Donald B. 1974. “Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies.” Journal of Educational Psychology 66 (5): 688–701.

Rubin, Donald B. 1980. “Randomization Analysis of Experimental Data: The Fisher Randomization Test.” Journal of the American Statistical Association 75 (371): 591–93.

Wang, Linbo, Thomas S. Richardson, and James M. Robins. 2026. “Causal Inference: A Tale of Three Frameworks.” Journal of Data Science 24 (1): 53–85. https://doi.org/10.6339/25-JDS1211.

SCM texts often call the background inputs exogenous variables and the variables generated by structural equations endogenous variables. To avoid conflict with econometric usage, these notes reserve the terms exogeneity and endogeneity primarily for equation-specific restrictions.↩︎

Framework	Representation of intervention and link to observed outcomes	Effect-specific condition for adjustment given \(X\)
Potential outcomes	Define \(Y(t)\); consistency links potential and observed outcomes through \(Y = Y(T)\), assumed alongside SUTVA	Weak ignorability: \(Y(t) \indep T \mid X\) for each treatment level \(t\)
SEM	Autonomy (Haavelmo 1943; Pearl 2009): \(\doop(T{=}t)\) replaces the structural equation for \(T\) with the constant \(t\), while every other equation and the distribution of the background disturbances remain unchanged	Conditional on \(X\), the treatment mechanism carries no information about the structural inputs determining \(Y(t)\)
Causal DAG	Modularity (graph surgery): \(\doop(T{=}t)\) deletes all arrows into \(T\), and the interventional distribution is given by the truncated factorization (Chapter 3)	\(X\) blocks every back-door path from \(T\) to \(Y\) and contains no descendant of \(T\) (Chapter 3)

	Interventional	Observational
Mean	\(\beta t\)	\(\left(\beta + \dfrac{\gamma\alpha\sigma_U^2}{\sigma_T^2}\right)t\)
Variance	\(\gamma^2\sigma_U^2 + \sigma_\varepsilon^2\)	\(\dfrac{\gamma^2\sigma_U^2\sigma_\delta^2}{\sigma_T^2} + \sigma_\varepsilon^2\)
Distribution of \(U\)	\(p(u)\)	\(p(u \mid T{=}t)\)
Identified by	RCT; the slope \(\beta\) also by IV (Chapter 7)	OLS