Reduced Forms And Structural Parameters

Why Reduced Forms Clarify an IV Model

An instrumental-variables model separates two relationships: how an instrument shifts an endogenous regressor, and how the instrument is associated with the outcome. The first is summarized by the first-stage reduced form; the second by the outcome reduced form. Their population slopes are connected to the structural slope, but only under specific conditions. Keeping the equations and assumptions explicit helps distinguish this population relationship from an estimate calculated in a sample.

This tutorial uses a scalar instrument $Z$, a scalar regressor $X$, and an outcome $Y$. The equations describe population variables, not necessarily conditional means: a reduced form here means a population linear projection on an intercept and $Z$. That distinction matters because a linear projection is defined by its means and covariances, even when a conditional expectation is not linear.

Structural Equation and First-Stage Reduced Form

Definition: Suppose $Y$, $X$, and $Z$ have finite second moments, and $\operatorname{Var}(Z)>0$. The scalar structural equation is $$ Y=\alpha+\beta X+u, $$ where $\alpha$ is the structural intercept, $\beta$ is the structural slope, and $u$ is the structural error. The first-stage reduced form is the population linear projection $$ X=\pi_0+\pi_1 Z+v, $$ where $\pi_0=E[X]-\pi_1 E[Z]$, $\pi_1=\operatorname{Cov}(Z,X)/\operatorname{Var}(Z)$, and $v=X-\pi_0-\pi_1Z$ is the projection residual.

By the defining properties of a population linear projection, $E[v]=0$ and $\operatorname{Cov}(Z,v)=0$. The first-stage slope $\pi_1$ describes the linear association between the instrument and the regressor. In this scalar setting, relevance means $\operatorname{Cov}(Z,X)\ne 0$, or equivalently $\pi_1\ne 0$, given $\operatorname{Var}(Z)>0$. Relevance says that the instrument is associated with variation in $X$; it does not by itself establish that the instrument is valid.

Deriving the Outcome Reduced Form

Define the outcome reduced form as the population linear projection of $Y$ on an intercept and $Z$:

Y=\gamma_0+\gamma_1 Z+e, \qquad \gamma_1=\frac{\operatorname{Cov}(Z,Y)}{\operatorname{Var}(Z)}, \qquad \gamma_0=E[Y]-\gamma_1 E[Z].

Substituting the structural equation into the covariance defining $\gamma_1$ gives

\gamma_1 =\frac{\operatorname{Cov}(Z,\alpha+\beta X+u)}{\operatorname{Var}(Z)} =\beta\frac{\operatorname{Cov}(Z,X)}{\operatorname{Var}(Z)} +\frac{\operatorname{Cov}(Z,u)}{\operatorname{Var}(Z)} =\beta\pi_1+\frac{\operatorname{Cov}(Z,u)}{\operatorname{Var}(Z)}.

The constant $\alpha$ has zero covariance with $Z$. The term involving $u$ makes explicit what the exclusion restriction contributes to this relationship: for the slope relation to reduce to $\gamma_1=\beta\pi_1$, it is sufficient that $\operatorname{Cov}(Z,u)=0$. Thus the instrument must not have a linear association with the structural error. Exclusion is not the same as relevance: relevance concerns $\operatorname{Cov}(Z,X)$, while exclusion concerns $\operatorname{Cov}(Z,u)$.

The intercept requires separate care. Write $m_u=E[u]$. Since $E[Y]=\alpha+\beta E[X]+m_u$, under exclusion the outcome reduced-form intercept is

\gamma_0 =E[Y]-\beta\pi_1E[Z] =\alpha+\beta\bigl(E[X]-\pi_1E[Z]\bigr)+m_u =\alpha+\beta\pi_0+m_u.

Consequently, $\gamma_0=\alpha+\beta\pi_0$ only if $E[u]=0$. Exclusion alone restricts the covariance of $Z$ and $u$; it does not require the error’s mean to be zero. If the stronger conditional-mean restrictions $E[v\mid Z]=0$ and $E[u\mid Z]=0$ hold, the linear reduced-form equations also describe the corresponding conditional means. The covariance restrictions used to derive the projection slopes do not, on their own, imply those stronger conditional statements.

Key result: For the structural equation $Y=\alpha+\beta X+u$ and the first-stage projection $X=\pi_0+\pi_1Z+v$, the outcome reduced-form slope satisfies $$ \gamma_1=\beta\pi_1+\frac{\operatorname{Cov}(Z,u)}{\operatorname{Var}(Z)}. $$ Under exclusion, $\operatorname{Cov}(Z,u)=0$, this becomes $\gamma_1=\beta\pi_1$. If relevance also holds, $\pi_1\ne 0$, the structural slope is determined by the two population reduced-form slopes: $\beta=\gamma_1/\pi_1$. The outcome intercept under exclusion is $\gamma_0=\alpha+\beta\pi_0+E[u]$.

Worked Applications of the Coefficient Relationship

Worked Example: Computing Both Reduced Forms

Suppose a hypothetical population has $E[Z]=10$, $\operatorname{Var}(Z)=4$, $E[X]=20$, and $\operatorname{Cov}(Z,X)=3$. Its structural equation is $Y=5+2X+u$, where $E[u]=1$ and $\operatorname{Cov}(Z,u)=0$. Find the first-stage and outcome reduced forms.

First-stage slope. The population projection slope is $$ \pi_1=\frac{\operatorname{Cov}(Z,X)}{\operatorname{Var}(Z)} =\frac{3}{4}=0.75. $$ Its intercept is $$ \pi_0=E[X]-\pi_1E[Z]=20-(0.75)(10)=12.5. $$ Thus the first-stage reduced form is $X=12.5+0.75Z+v$, where $v$ has mean zero and is uncorrelated with $Z$ by the projection definition.

Outcome slope. Exclusion holds because $\operatorname{Cov}(Z,u)=0$. Therefore $$ \gamma_1=\beta\pi_1=(2)(0.75)=1.5. $$ Outcome intercept. The error mean is one, not zero, so $$ \gamma_0=\alpha+\beta\pi_0+E[u] =5+(2)(12.5)+1=31. $$ The outcome reduced form is $Y=31+1.5Z+e$. As a check, $E[Y]=5+(2)(20)+1=46$, while $31+(1.5)(10)=46$. The intercept is 31 precisely because it includes $E[u]=1$.

Worked Example: Recovering the Structural Slope From Population Slopes

In a second hypothetical population, $E[Z]=0$, $\operatorname{Var}(Z)=9$, $E[X]=8$, and $\operatorname{Cov}(Z,X)=6$. Let $Y=10+4.5X+u$, with $E[u]=0$ and $\operatorname{Cov}(Z,u)=0$. Find the two reduced-form slopes and check what they imply about the structural slope.

The first-stage slope and intercept are $$ \pi_1=\frac{6}{9}=\frac{2}{3}, \qquad \pi_0=8-\left(\frac{2}{3}\right)(0)=8. $$ The outcome reduced-form slope is $$ \gamma_1=\beta\pi_1=(4.5)\left(\frac{2}{3}\right)=3. $$ Since $\pi_1\ne0$, relevance holds, and since $\operatorname{Cov}(Z,u)=0$, exclusion holds. The population slope relationship therefore gives $$ \frac{\gamma_1}{\pi_1}=\frac{3}{2/3}=4.5, $$ which matches the structural slope in the stated model. The outcome intercept is $10+(4.5)(8)+0=46$. Because $E[Z]=0$, the reduced-form intercept also equals $E[Y]=46$.

The units provide an additional check: if $X$ is measured in treatment units and $Y$ in outcome units, then $\pi_1$ has units of $X$ per unit of $Z$, $\gamma_1$ has units of $Y$ per unit of $Z$, and their quotient has units of $Y$ per unit of $X$, as required for $\beta$.

Worked Example: What Changes When Exclusion Fails?

Consider a hypothetical population with $E[Z]=4$, $\operatorname{Var}(Z)=5$, $E[X]=6$, and $\operatorname{Cov}(Z,X)=2.5$. Let $Y=3+2X+u$, with $E[u]=0$ but $\operatorname{Cov}(Z,u)=1$. Calculate the first-stage and outcome reduced-form coefficients, and identify the consequence of the nonzero covariance.

The first-stage slope and intercept are $$ \pi_1=\frac{2.5}{5}=0.5, \qquad \pi_0=6-(0.5)(4)=4. $$ The instrument is relevant because $\pi_1\ne0$. But exclusion fails. The additional term in the outcome slope is $$ \frac{\operatorname{Cov}(Z,u)}{\operatorname{Var}(Z)} =\frac{1}{5}=0.2. $$ Consequently, $$ \gamma_1=(2)(0.5)+0.2=1.2, \qquad \gamma_0=E[Y]-\gamma_1E[Z] =15-(1.2)(4)=10.2. $$ Here $E[Y]=3+(2)(6)+0=15$. The reduced-form slope of 1.2 is not just the product of the structural slope and the first-stage slope: that product is $2(0.5)=1$, and the remaining 0.2 comes from the association between $Z$ and $u$. Relevance alone therefore does not justify using the reduced-form slope relationship to identify $\beta$.

Justification of the Main Relationship

The central result follows directly from the projection definitions; it does not require assuming that either reduced form is a correct conditional-mean model. The argument uses only finite second moments, positive variance of $Z$, the structural equation, and the covariance conditions stated for the particular conclusion.

Write the projection slopes.
By definition, $\pi_1=\operatorname{Cov}(Z,X)/\operatorname{Var}(Z)$ and $\gamma_1=\operatorname{Cov}(Z,Y)/\operatorname{Var}(Z)$.

Use the structural equation.
Because $Y=\alpha+\beta X+u$, covariance linearity gives $\operatorname{Cov}(Z,Y)=\beta\operatorname{Cov}(Z,X)+\operatorname{Cov}(Z,u)$.

Divide by instrument variance.
Since $\operatorname{Var}(Z)>0$, $\gamma_1=\beta\pi_1+\operatorname{Cov}(Z,u)/\operatorname{Var}(Z)$.

Apply exclusion and relevance.
If $\operatorname{Cov}(Z,u)=0$, then $\gamma_1=\beta\pi_1$. If also $\pi_1\ne0$, algebra gives $\beta=\gamma_1/\pi_1$.

Each condition has a distinct role. Positive instrument variance makes the projection slopes well-defined. Relevance makes the first-stage slope nonzero. Exclusion removes the component of the outcome reduced-form slope that comes from the instrument’s association with the structural error. These are population statements; replacing population coefficients with sample coefficients introduces estimation and sampling questions that are separate from this derivation.

Common Mistakes and Interpretation

Dropping the structural error’s mean from the intercept. From exclusion alone, it does not follow that $E[u]=0$. The correct outcome intercept is $\alpha+\beta\pi_0+E[u]$; omit the last term only when a zero-mean assumption is stated.
Confusing exclusion with relevance. A nonzero first-stage slope does not show that $Z$ is uncorrelated with $u$. Both conditions are needed for the slope relationship to identify the structural slope.
Treating a reduced form as a conditional expectation automatically. The projection equations are linear summaries. A statement about $E[X\mid Z]$ or $E[Y\mid Z]$ requires appropriate conditional-mean assumptions, not merely the projection definitions.
Interpreting a coefficient relationship as proof of causation. The algebra gives the structural slope from population reduced-form slopes only under the model and instrument assumptions. It does not establish those assumptions from the coefficient values.

The reduced-form connection is useful because it isolates the two empirical links an instrument must provide: movement in $X$ and a corresponding movement in $Y$. Its strength is also its limitation. If exclusion fails, the outcome reduced-form slope combines the structural pathway through $X$ with the instrument’s direct association with the structural error.

Check Your Understanding

Use the definitions and assumptions in this tutorial to answer the following.

For the first-stage projection, write formulas for $\pi_0$ and $\pi_1$, and state the population properties of its residual $v$.
Starting from $Y=\alpha+\beta X+u$, derive the expression for the outcome reduced-form slope without imposing exclusion.
Why does exclusion alone not imply that the outcome reduced-form intercept is $\alpha+\beta\pi_0$?
If $\pi_1=0$, can the two reduced-form slopes determine $\beta$ through $\gamma_1=\beta\pi_1$? Explain.
Suppose $\pi_1=0.4$, $\beta=3$, and $\operatorname{Cov}(Z,u)/\operatorname{Var}(Z)=0.1$. What is $\gamma_1$, and which part of it reflects failure of exclusion?