Why Sensitivity to Normalizations and Exclusions Matters
A structural estimate can change because the same equation was written with a different coefficient set equal to one, or because the model now imposes different restrictions on how variables affect one another. These changes have different meanings. A legitimate change of normalization should relabel parameters without changing the underlying structural relation. Changing an exclusion assumption can change which variation identifies a parameter, and may change the model’s predictions or even destroy identification.
Sensitivity analysis therefore asks more than whether two reported estimates differ. It asks whether the specifications represent the same restrictions in different coordinates, or whether they use different assumptions about the data-generating process. That distinction is important when structural parameters are interpreted or used to describe mechanisms.
What Should Remain Invariant Under a Normalization?
Consider a structural equation written as \(y_1=\beta y_2+\gamma x+u\). Its left-hand-side coefficient is one. If \(\beta\ne 0\), the same equation can instead be solved for \(y_2\). The coefficients and shock then change, but the set of values satisfying the equation does not.
This equivalence requires a nonzero coefficient for the variable used to solve the equation. It also requires transforming the shock consistently: the new shock is \(-u/\beta\), not the original \(u\). If a likelihood is used, its error distribution and density must likewise be transformed correctly. Treating the coefficients as changed while holding the shock definition fixed would not be a pure normalization change.
More generally, let a system of structural restrictions be \(F(\theta)=0\), where \(\theta\) collects its parameters. Multiplying one equation by a nonzero constant \(c\) leaves its zeros unchanged. The corresponding Jacobian row is multiplied by \(c\), which does not change the Jacobian’s rank. Thus a pure nonzero equation rescaling preserves local identification. Re-expressing parameters through a one-to-one differentiable mapping also preserves identification locally where that mapping has a nonsingular derivative. These invariance claims do not apply if the mapping is not invertible in the region of interest, or if restrictions are added or removed.
Worked Example: Rewriting a Structural Equation
Suppose a hypothetical structural equation is \(y_1=0.5y_2+2x+u\), with the coefficient on \(y_1\) normalized to one. Here \(\beta=0.5\ne 0\), so the equation can be solved for \(y_2\).
Starting with the equation and isolating \(y_2\) gives \(0.5y_2=y_1-2x-u\), and hence \(y_2=2y_1-4x-2u\). The second representation has a coefficient of 2 on \(y_1\), a coefficient of \(-4\) on \(x\), and a transformed shock of \(-2u\).
For example, if \(y_1=10\), \(x=1\), and \(u=0\), the original equation implies \(10=0.5y_2+2\), so \(y_2=16\). The rewritten equation gives \(y_2=2(10)-4(1)-2(0)=16\). The two representations agree for this value and, by the algebraic equivalence, for every value satisfying the equation.
If an estimator gives \(\hat\beta=0.5\) and \(\hat\gamma=2\) in the original representation, the corresponding transformed estimates are \(1/\hat\beta=2\) and \(-\hat\gamma/\hat\beta=-4\). Comparing \(0.5\) directly with 2 as if they estimated the same coefficient would be misleading: they refer to differently normalized equations. The underlying fitted relation should be compared after transforming estimates back to a common representation.
Exclusion Restrictions Change the Identifying Content
An exclusion restriction is different from an equation rescaling. For example, in \(y=\beta x+u\), if \(x\) is endogenous, an instrument \(z\) can identify \(\beta\) under assumptions that \(z\) is relevant for \(x\), is uncorrelated with \(u\), and has no direct effect on \(y\). With centered variables, the population moment condition is \(E[zy]=\beta E[zx]\). If \(E[zx]\ne 0\), this condition identifies \(\beta\) as the ratio of those moments.
The no-direct-effect condition is an exclusion restriction. If it is relaxed, the equation may instead be \(y=\beta x+\theta z+u\). Then the same moment becomes \(E[zy]=\beta E[zx]+\theta E[z^2]\), and it no longer isolates \(\beta\). The restriction \(\theta=0\) is doing identifying work; it is not simply choosing a convenient way to write the equation.
In practice, alternative exclusions may be defensible but lead to different estimates. The estimates’ difference is evidence of sensitivity to identifying assumptions, not evidence that one specification is automatically correct. Sample moments can show whether estimates agree with the restrictions used; they generally cannot prove that a proposed exclusion is substantively valid.
Worked Example: Comparing Two Candidate Exclusions
Consider a hypothetical centered sample for the structural equation \(y=\beta x+u\), where \(x\) is endogenous. Suppose each candidate instrument is assumed to be uncorrelated with \(u\) and to have no direct effect on \(y\). The relevant hypothetical sample covariances are: \(s_{z_1y}=2.00\), \(s_{z_1x}=1.00\), \(s_{z_2y}=6.00\), and \(s_{z_2x}=2.00\). Both instruments are relevant in this sample because each covariance with \(x\) is nonzero.
Using \(z_1\) alone gives the just-identified instrumental-variables estimate \(\hat\beta_1=s_{z_1y}/s_{z_1x}=2.00/1.00=2.00\). Using \(z_2\) alone gives \(\hat\beta_2=s_{z_2y}/s_{z_2x}=6.00/2.00=3.00\). Thus the point estimates differ by \(3.00-2.00=1.00\) unit of \(y\) per unit of \(x\).
These calculations check the sample relevance condition, but not the assumed exogeneity or exclusion conditions. If both instruments were instead imposed as valid moments for the same \(\beta\), the sample moments would not both hold exactly at either estimate: at \(\hat\beta=2.00\), the \(z_2\) moment discrepancy is \(6.00-2.00(2.00)=2.00\); at \(\hat\beta=3.00\), the \(z_1\) discrepancy is \(2.00-3.00(1.00)=-1.00\). A formal assessment of multiple overidentifying moments requires an appropriate estimation and testing procedure, but the arithmetic already shows why the two candidate exclusions cannot be treated as mere relabelings.
The conclusion is limited but useful: estimates are sensitive to which candidate exclusion supplies identifying variation. The covariance calculations do not establish which instrument is valid, and they do not establish that either estimate is a causal effect.
How Relaxing an Exclusion Can Remove Identification
A particularly important sensitivity comparison asks what happens when an excluded variable is allowed a direct effect in the equation it was previously excluded from. The effect of that change can be seen in a simple reduced-form relationship. Suppose a first-stage equation has a coefficient \(\pi_2\) on \(z_2\), while the reduced-form coefficient of \(z_2\) in \(y\) is \(\rho_2\).
If the exclusion restriction sets \(\theta=0\) and \(\pi_2\ne 0\), then \(\beta=\rho_2/\pi_2\). If \(\theta\) is unrestricted, this one reduced-form relationship contains two unknowns, \(\beta\) and \(\theta\). A nonzero first-stage coefficient alone no longer identifies \(\beta\).
Worked Example: Sensitivity to a Direct-Effect Exclusion
Suppose a hypothetical first stage gives \(\pi_2=0.400\), and the reduced-form coefficient of \(z_2\) in \(y\) is \(\rho_2=0.600\). Assume \(z_2\) is uncorrelated with the structural error \(u\), and that the reported coefficients are defined on compatible scales. Relevance holds because \(\pi_2\ne 0\).
Under the exclusion restriction that \(z_2\) has no direct effect on \(y\), set \(\theta=0\). The structural coefficient is then \(\hat\beta=0.600/0.400=1.50\), in units of \(y\) per unit of \(x\).
If the direct effect is allowed, the reduced-form relationship is \(0.600=0.400\beta+\theta\). For \(\beta=1.00\), it requires \(\theta=0.200\); for \(\beta=2.00\), it requires \(\theta=-0.200\). Both pairs satisfy the same observed reduced-form coefficient. Without further identifying restrictions or information, the relationship does not choose between them.
This is not a coefficient transformation: changing \(\theta=0\) to unrestricted \(\theta\) changes the model’s restrictions and removes the information that identified \(\beta\) in this example. The value 1.50 is therefore conditional on the exclusion assumption, not a conclusion that survives its relaxation automatically.
A Practical Sensitivity Analysis
A useful comparison makes the role of each change explicit. First write down the structural equation, its normalization, and the restrictions that determine which variables may enter it. Then separate algebraically equivalent representations from specifications with different exclusions. For equivalent normalizations, transform parameters and shocks into a common representation before comparing estimates. For different exclusions, state which moment or source of variation identifies the parameter in each case.
Write the structural equations, parameter normalization, included variables, exclusions, and the assumptions that support the identifying moments.
Check whether it only rescales an equation or reparameterizes it invertibly. If a variable is added to or removed from a structural equation, treat that as a change in substantive restrictions.
Transform estimates from equivalent normalizations to a common parameterization. For alternative exclusions, identify the moments used and report the resulting estimates separately.
Verify that the relevant variation remains nonzero and that each parameter is identified under the stated restrictions. Describe conclusions as conditional on those restrictions.
This process does not make an exclusion credible merely because the resulting estimate is precise or stable. Credibility depends on the substantive setting and the assumptions about direct effects and error dependence. Nor does disagreement alone identify which specification is preferable. It shows that conclusions depend on choices that should be made visible.
Common Mistake: Treating Every Coefficient Change as Sensitivity
A common error is to see a coefficient change after rewriting an equation and conclude that the estimate is fragile. If the rewrite is an invertible normalization change, the raw coefficients may differ by design; compare the transformed parameters or the implied structural relation instead. The reverse mistake is equally serious: calling an added direct effect a “new normalization” conceals a changed restriction. If that change alters the identifying moments, compare it as a distinct model and explain which assumption supplies identification.
A clear report should therefore specify (1) the normalization, (2) the exclusion assumptions, (3) the identifying variation or moments, and (4) whether the comparison preserves the model’s restrictions. A structural estimate is interpretable only relative to those choices.
Check Your Understanding
Use the definitions and calculations in this tutorial to answer the following questions.
- Why must the shock be transformed when the equation \(y_1=\beta y_2+\gamma x+u\) is solved for \(y_2\)?
- For what condition on \(\beta\) is that rearrangement valid as a representation solved for \(y_2\)?
- In the candidate-instrument example, what assumption beyond relevance is needed to justify using \(z_1\) to identify \(\beta\)?
- If \(\rho_2=0.600\) and \(\pi_2=0.400\), what value of \(\beta\) follows under the exclusion \(\theta=0\)?
- Why does allowing \(\theta\) to be unrestricted mean that the single relationship \(\rho_2=\beta\pi_2+\theta\) does not identify \(\beta\)?