Linear Structural Equations for IV

Why IV Uses Several Equations

Instrumental variables (IV) analysis separates three relationships that are easy to conflate: how treatment relates to the outcome, how the instrument relates to treatment, and how the instrument relates to the outcome. Writing these relationships as equations makes their different roles explicit. In particular, the treatment is the exposure whose effect is of interest; the instrument is a variable used to generate or explain variation in that treatment.

The equations are a way to state a model, not a guarantee that its assumptions hold. An instrument’s presence in a first-stage equation does not by itself make it valid, and a treatment coefficient in an outcome equation is not automatically a causal effect. The equations help clarify what would need to be justified.

The Linear Outcome Equation

Let \(Y_i\) be an outcome for unit \(i\), \(D_i\) the treatment, \(Z_i\) an instrument, and \(X_i\) a vector of exogenous covariates. A common linear structural outcome equation is:

Here, \(\beta_0\) is an intercept, \(\beta\) is the coefficient on treatment, \(\gamma\) is a vector of covariate coefficients, and \(u_i\) collects influences on the outcome not represented by the included variables. The prime in \(X_i'\gamma\) denotes the sum of the covariate values multiplied by their corresponding coefficients.

The structural interpretation of \(\beta\) is clearest under a constant linear treatment-effect model. In that model, changing treatment from \(d\) to \(d+1\), holding the covariates and the unit’s other outcome determinants fixed, changes the outcome by \(\beta\) units. If \(D_i\) is binary, \(\beta\) represents the model’s treated-versus-untreated outcome contrast. If treatment is measured continuously, \(\beta\) is the modeled outcome change per one-unit increase in treatment.

The equation does not require treatment to be unrelated to the outcome error. In IV settings, \(D_i\) may be associated with \(u_i\), for example because unmeasured factors influence both treatment receipt and the outcome. That is one reason ordinary regression of \(Y_i\) on \(D_i\) and \(X_i\) may not recover the causal treatment coefficient.

The First-Stage Equation

The first stage describes how the instrument and covariates relate to treatment. For a single instrument, its linear form is:

The coefficient \(\pi_1\) describes the modeled change in treatment associated with a one-unit change in the instrument, conditional on the covariates. The error \(v_i\) collects treatment determinants not represented by the instrument and covariates. It is distinct from \(u_i\); the two errors may be related.

The first-stage equation can be used with either a binary or a continuous treatment. If treatment is binary, a linear equation is a linear-probability representation: when its conditional mean is correctly specified, its right-hand side gives \(P(D_i=1\mid Z_i,X_i)\). This is a model for the conditional probability, not a claim that each individual’s binary treatment value is itself continuous. If treatment is continuous, the equation instead describes its modeled conditional mean in the units of treatment.

An instrument is relevant when it predicts treatment after accounting for the included covariates; in this scalar equation, that means \(\pi_1\ne 0\). Relevance is a feature of the first stage, not a statement that the instrument is independent of unmeasured outcome causes or excluded from the outcome through every other pathway.

Worked Example: Binary Treatment and Binary Instrument

The Reduced-Form Equation

The reduced-form equation relates the outcome directly to the instrument and covariates, without including treatment as a regressor. Under the two equations above, it can be obtained algebraically by substituting the first-stage expression for \(D_i\) into the outcome equation. It is called “reduced form” because treatment has been removed from the right-hand side.

For the general equations, substitution gives:

Thus, the corresponding reduced-form equation is:

where \(\delta_0=\beta_0+\beta\pi_0\), \(\delta_1=\beta\pi_1\), \(\delta_2=\gamma+\beta\pi_2\), and \(\varepsilon_i=u_i+\beta v_i\). These equalities follow from substitution; they do not require the errors \(u_i\) and \(v_i\) to be independent.

If the equations represent conditional means, so that \(E(u_i\mid Z_i,X_i)=0\) and \(E(v_i\mid Z_i,X_i)=0\), then \(E(\varepsilon_i\mid Z_i,X_i)=0\) as well. In that case, the reduced-form equation also describes the conditional mean of the outcome given the instrument and covariates. Without the relevant mean conditions, the algebraic identity still holds, but the right-hand side need not equal that conditional mean.

Worked Example: Continuous Treatment and Instrument

Worked Example: Deriving the Reduced Form Numerically

What the Equations Do—and Do Not—Establish

The roles of the variables can be checked by asking where they appear. Treatment \(D_i\) appears in the outcome equation and is predicted by the instrument in the first stage. Instrument \(Z_i\) appears in the first stage and in the reduced form, but is excluded from the structural outcome equation. Exogenous covariates \(X_i\) are included in both structural equations to account for measured differences relevant to treatment and outcome. The errors \(u_i\) and \(v_i\) represent different omitted influences and need not be unrelated.

The word “exogenous” is an assumption about a variable’s relationship to unmeasured determinants in the relevant model; it is not a label made true merely by including the variable. Likewise, excluding \(Z_i\) from the outcome equation encodes the exclusion restriction, but writing that restriction does not verify it. Instrument independence and relevance also require justification. These distinctions matter because a well-formed set of equations can still describe an invalid IV design.

A further modeling choice is linearity. For a binary treatment, the linear first stage is a convenient representation of a treatment probability when its conditional mean is linear in \(Z_i\) and \(X_i\). For a continuous treatment, linearity means that the specified conditional mean changes at a constant rate over the modeled range. In either case, the equation is an approximation or assumption about the relationship, not a consequence of the variables’ definitions.

Common Mistake: Treating the First Stage as a Causal Result

A frequent error is to reason, “The instrument predicts treatment, so it is valid and the treatment coefficient is causal.” The first clause concerns relevance only. It says nothing by itself about whether the instrument is associated with unmeasured outcome causes or whether it affects the outcome through a route other than treatment. The correction is to keep the claims separate: the first stage describes instrument–treatment association; the structural outcome equation states the proposed treatment–outcome relationship; and validity requires additional assumptions about the instrument and the outcome process.