Exclusion Restrictions As Identifying Variation

Why Exclusions Can Identify Structural Relationships

In a simultaneous-equations model, an observed outcome may be determined jointly by multiple structural relationships. A reduced form describes how the observed variables respond to exogenous variables, but those responses do not automatically reveal each structural coefficient. An exclusion restriction can make a difference: it says that a particular exogenous variable affects one structural equation but does not enter another. When the excluded variable actually shifts the system and there is enough independent variation, its effect on the observed outcomes can reveal a structural relationship that would otherwise be entangled with the others.

The key is not simply to count restrictions. The restrictions must create empirically distinct responses, and the relevant shifts must occur in the data. A variable excluded from an equation but with no effect on the system provides no identifying movement. Nor can two shifters that always move together necessarily reveal their separate effects.

A Two-Equation Model and Its Exclusions

Definition: An exclusion restriction specifies that an exogenous variable does not enter a particular structural equation, while it may enter another. In the linear model below, $y$ is excluded from the supply equation and $z$ is excluded from the demand equation: $$ \begin{aligned} q &= \alpha p + \beta y + u_d,\\ q &= \gamma p + \delta z + u_s. \end{aligned} $$ Here $q$ is quantity, $p$ is price, $y$ shifts demand, and $z$ shifts supply. The structural parameters are $\alpha,\beta,\gamma,\delta$; $u_d,u_s$ are structural disturbances.

The equations are structural descriptions: each gives a relationship that holds in the system, not a separate regression that treats every right-hand-side variable as exogenous. Because $p$ is jointly determined by the equations, its observed value can be related to the disturbances. For the identification argument, suppose the conditional means of both disturbances given $y,z$ are zero, the structural coefficients are stable, and $D=\alpha-\gamma$ is nonzero. Also suppose $y$ and $z$ have independent variation in the population. These conditions let us interpret the reduced-form coefficients as responses to the two shifters; they do not, by themselves, guarantee that the shifters are relevant.

The exclusions create different ways to move the system. Changing $y$, while holding $z$ fixed, shifts demand without directly shifting supply. The resulting movement in $p$ and $q$ follows the supply relationship. Changing $z$, while holding $y$ fixed, shifts supply without directly shifting demand, so the resulting movement follows the demand relationship. This is the identifying variation: a shift in one equation reveals the slope of the other, provided the shift changes the equilibrium.

From Shifts to Reduced-Form Responses

Equating the two expressions for $q$ and solving for $p$ gives the reduced form for price. Substituting it into the demand equation gives the reduced form for quantity:

\begin{aligned} p &= -\frac{\beta}{D}y+\frac{\delta}{D}z+\frac{u_s-u_d}{D},\\ q &= -\frac{\beta\gamma}{D}y+\frac{\alpha\delta}{D}z +\frac{\alpha u_s-\gamma u_d}{D}, \qquad D=\alpha-\gamma. \end{aligned}

Write $p_y,p_z,q_y,q_z$ for the population reduced-form coefficients: for example, $p_y$ is the change in the conditional mean of price for a one-unit increase in $y$, holding $z$ fixed. Under the stated conditional-mean assumption, they are $p_y=-\beta/D$, $p_z=\delta/D$, $q_y=-\beta\gamma/D$, and $q_z=\alpha\delta/D$.

If both shifters are relevant, so $p_y\ne 0$ and $p_z\ne 0$, the structural slopes follow from ratios of responses:

\gamma=\frac{q_y}{p_y}, \qquad \alpha=\frac{q_z}{p_z}.

For instance, a one-unit increase in $y$ changes quantity by $q_y$ and price by $p_y$. Since $y$ is excluded from supply, that movement traces the supply relationship, whose slope is $\gamma$. The corresponding movement from $z$ traces demand, whose slope is $\alpha$. Once these slopes are recovered, $D=\alpha-\gamma$, $\beta=q_y-\alpha p_y$, and $\delta=p_zD$. The observable responses thus contain enough information to recover all four structural coefficients under these conditions.

Identification result: In this linear model, the two exclusion restrictions support recovery of all four structural coefficients when $D\ne 0$, the conditional-mean assumptions hold, $y,z$ vary independently, and both excluded shifters generate nonzero price responses. Exclusions alone are not sufficient: relevance and independent variation matter too.

Worked Examples

Worked Example: Recovering Slopes From Excluded Shifters

Suppose hypothetical population reduced-form coefficients are $p_y=-1$, $p_z=4/3$, $q_y=1$, and $q_z=8/3$, with price measured in dollars and quantity in hundreds of units. Assume the model’s conditional-mean and stability conditions hold, $y,z$ vary independently, and both price responses are nonzero. Recover the structural coefficients.

The ratio for the supply slope is $$ \gamma=\frac{q_y}{p_y}=\frac{1}{-1}=-1. $$ The ratio for the demand slope is $$ \alpha=\frac{q_z}{p_z}=\frac{8/3}{4/3}=2. $$ Thus $D=\alpha-\gamma=2-(-1)=3$. The demand shifter coefficient is $$ \beta=q_y-\alpha p_y=1-(2)(-1)=3, $$ and the supply shifter coefficient is $$ \delta=p_zD=\left(\frac{4}{3}\right)(3)=4. $$ So the recovered equations have slopes $\alpha=2,\beta=3,\gamma=-1,\delta=4$ in the model’s stated units. As a check, the model implies $p_y=-3/3=-1$, $p_z=4/3$, $q_y=-3(-1)/3=1$, and $q_z=2(4)/3=8/3$, matching the given responses.

Worked Example: Removing One Exclusion Creates Multiple Solutions

Now allow $y$ to enter the supply equation as well: $$ \begin{aligned} q &= \alpha p+\beta y+u_d,\\ q &= \gamma p+\delta z+\kappa y+u_s. \end{aligned} $$ Use the same hypothetical reduced-form coefficients as above. Show why these responses no longer determine all structural coefficients.

Let $D=\alpha-\gamma\ne0$. Subtracting the equations gives $p_y=(\kappa-\beta)/D$ and $p_z=\delta/D$. The quantity responses still satisfy $q_y=\alpha p_y+\beta$ and $q_z=\alpha p_z$. The data therefore determine $\alpha=q_z/p_z=2$ and $\beta=q_y-\alpha p_y=1-2(-1)=3$, but leave $D$ free. For any nonzero $D$, set $$ \gamma=2-D,\qquad \kappa=3-D,\qquad \delta=\frac{4}{3}D. $$ These values reproduce the same four reduced-form coefficients.

For example, $D=3$ gives $(\alpha,\beta,\gamma,\delta,\kappa)=(2,3,-1,4,0)$. Choosing $D=2$ instead gives $(2,3,0,8/3,1)$. In the second model, $p_y=(1-3)/2=-1$, $p_z=(8/3)/2=4/3$, $q_y=2(-1)+3=1$, and $q_z=2(4/3)=8/3$, exactly as before. The distinct structural parameter values have identical reduced-form responses. Allowing $y$ into supply removed a restriction that had separated its role in the two equations; the parameters are not identified by these responses.

Worked Example: Exclusions Do Not Replace Independent Variation

Suppose the original exclusions hold, but the observed shifters always satisfy $y=z=t$. Compare two hypothetical structural models: $$ (\alpha,\beta,\gamma,\delta)=(1.5,2,-0.5,1.5) \quad\text{and}\quad (\alpha,\beta,\gamma,\delta)=(2,2.125,-1,1.375). $$

In the first model, $D=1.5-(-0.5)=2$, so the price responses to $y,z$ are $-2/2=-1$ and $1.5/2=0.75$. Along $y=z=t$, the observed price response is therefore $(-1+0.75)t=-0.25t$. Quantity responses are $-2(-0.5)/2=0.5$ and $1.5(1.5)/2=1.125$, giving a combined response of $1.625t$.

In the second model, $D=2-(-1)=3$. Its price responses are $-2.125/3$ and $1.375/3$, which sum to $-0.25$. Its quantity responses are $-2.125(-1)/3$ and $2(1.375)/3$, which sum to $0.7083+0.9167=1.625$. Thus both models imply the same observed responses when $y=z=t$, even though their structural coefficients differ. Because $y,z$ have no independent variation in this data pattern, the separate reduced-form coefficients cannot be distinguished. The exclusions are present, but the required variation is not.

A Short Identification Derivation

The ratio argument can be checked directly from the reduced form. Under the exclusions and $D\ne0$,

\frac{q_y}{p_y} = \frac{-\beta\gamma/D}{-\beta/D} =\gamma \quad\text{if }\beta\ne0, \qquad \frac{q_z}{p_z} = \frac{\alpha\delta/D}{\delta/D} =\alpha \quad\text{if }\delta\ne0.

The nonzero conditions are exactly what make the ratios available: $\beta\ne0$ implies $p_y\ne0$, and $\delta\ne0$ implies $p_z\ne0$, given finite nonzero $D$. After recovering $\alpha,\gamma$, the remaining coefficients follow from $q_y=\alpha p_y+\beta$ and $p_z=\delta/D$. If either excluded shifter has no effect on its own equation, its response cannot trace the other equation’s slope. The algebra therefore explains both the identifying role and the relevance requirement.

What the Exclusion Assumption Does—and Does Not—Say

An exclusion is a substantive restriction on the structural model, not a fact established merely by observing that a variable is absent from a written equation. Its credibility depends on whether the variable plausibly has no direct role in that equation, apart from the modeled system. If an omitted direct pathway exists, the restriction may fail and the recovery calculation may assign that pathway’s effect to another structural coefficient.

Exclusion and relevance answer different questions. Exclusion says where a shifter does not enter. Relevance says whether it moves the system enough to produce an observable response. Independent variation says whether separate shifters can be distinguished in the data. Identification requires these ingredients to work together; a mathematically valid exclusion with no relevant or independently varying shifter does not recover the structural parameters.

Common Mistake

Wrong reasoning: “There are two exclusions and two endogenous variables, so the structural parameters must be identified.” Counting restrictions does not establish that the shifters change outcomes or vary separately.
Correction: Derive the observable responses implied by the structural model. Check that the excluded shifters are relevant, that their variation can be separated, and that the resulting response map distinguishes the parameters. If removing an exclusion permits multiple structural parameter values to generate the same observed responses, identification fails.

Key takeaway: Exclusion restrictions identify by creating distinct shifts through the structural system. Their identifying force comes from observable response patterns, and it depends on relevance, independent variation, and the other stated model conditions—not on the restriction’s label alone.

Check Your Understanding

Answer using the model and conditions developed above.

In the two-equation model, why does a change in $y$ reveal the supply slope rather than the demand slope?
Which reduced-form response ratios recover $\gamma$ and $\alpha$, and what must be true for each ratio to be defined?
When $y$ is also allowed into the supply equation, what parameter remains free in the worked example’s reduced-form calculations?
How can both exclusion restrictions hold while structural parameters remain unidentified because of the observed shifters?
Why does counting restrictions alone not establish identification?