Self-Normalizing Denominators
in Rational Causal Estimation

Graphical identification in linear structural equation models often yields a causal coefficient of the form $N(\Sigma)/D(\Sigma)$. Here $\Sigma\in\mathbb{S}^{d}_{++}$ is the observational covariance matrix of $d$ observed coordinates, and $N$ and $D$ are polynomial functions on the space $\operatorname{Sym}_{d}$ of symmetric matrices of order $d$. Instrumental variables, conditional instruments, front-door adjustment, half-trek identification and proximal formulas all produce such expressions (31; 6; 14; 23; 39; 29; 38; 21). At regular distributions, substituting the sample covariance matrix into such a formula is routine. However, when $D$ is small, weak-instrument theory suggests a ratio-of-normals limit, distorted Wald inference and confidence sets that must sometimes be unbounded (16; 11; 36; 37; 2; 3). The standard safeguard compares the estimated denominator with its estimated standard error, as in first-stage relevance screening.

Two of these strategies display the contrast that motivates this paper. Throughout this preview, the observed vector is centred Gaussian with covariance matrix $\Sigma\in\mathbb{S}^{d}_{++}$. The scalar $\sigma_{UV}=\operatorname{cov}(U,V)$ denotes the covariance of two observed variables $U$ and $V$, and $\hat{\Sigma}$ denotes the sample covariance matrix from $n$ independent observations. Section 2 gives the remaining conventions. For the front-door strategy, the observed vector is $(X,M,Y)^{\top}$ and

At every such $\Sigma$, the asymptotic variance of $n^{1/2}\{D_{\mathrm{fd}}(\hat{\Sigma})-D_{\mathrm{fd}}(\Sigma)\}$ is $10D_{\mathrm{fd}}(\Sigma)^{2}$. Consequently, the statistic that divides $nD_{\mathrm{fd}}(\hat{\Sigma})^{2}$ by the plug-in value of this variance equals $n/10$ for every positive-definite sample covariance. Screening this denominator reveals nothing about the underlying distribution, although the precision of the plug-in estimator may deteriorate without bound. By contrast, consider a linear proximal strategy with observed coordinates $(A,Z,W)^{\top}$, comprising a treatment, a treatment proxy and an outcome proxy. Its denominator is

which equals $\sigma_{AA}$ times the partial covariance of $Z$ and $W$ given $A$. This denominator vanishes at interior covariance matrices at which its asymptotic variance does not. There are therefore sequences of covariance matrices along which the plug-in ratio can have the ratio-of-normals limit of weak-instrument theory. The relevant screening direction is the partial covariance, not the marginal treatment–proxy association.

These examples motivate the question of which denominators make the first-order ratio regime impossible along every sequence of underlying covariance matrices. Under centred Gaussian sampling, the first-order variance of a covariance polynomial is a classical quadratic form in its symmetric gradient. We show that, for a special class of polynomials, this quadratic form is exactly proportional to the square of the polynomial. Denominator noise then contracts at the same rate as the denominator itself, and the standardized denominator is a samplewise constant rather than a relevance statistic. We call such denominators exactly self-normalizing.

Two neighbouring phenomena are excluded from this notion. A polynomial whose first-order variance merely vanishes on its zero set can still generate a nondegenerate second-order local limit when that variance is not proportional to the square of the polynomial. Section 5 gives an example. Numerator singularities are also distinct. The mediation null remains nonregular even though the front-door denominator is exactly self-normalizing (10). Therefore, the classification below concerns denominators, not universal regularity of the associated estimators.

The first contribution is an all-dimensional sufficient class. Products of powers of covariance determinants along a nested flag of subspaces are exactly self-normalizing. Their relative gradients have a fixed spectrum, and their sample-to-population ratios factor into independent chi-squared variables. In recursive coordinates, these polynomials are monomials in successive conditional variances, which links the algebra directly to variance information.

The second contribution is a low-dimensional converse. The converse is complete in dimension two. In dimension three, determinant stripping reduces every solution to a rank-one branch, a pure plane-minor branch or a mixed branch with constant relative spectrum and a polynomial kernel line. The first two branches are flag powers, while rigidity of the mixed kernel line remains open. For the practically important class whose determinant-free irreducible factors have degree at most two, only one rank-one variance factor and one nested plane-minor factor can occur. This gives a coefficient-level factor-and-rank diagnostic and classifies every reducible self-normalizing cubic.

The statistical results connect this algebra to weak-denominator inference. A local result recovers the classical ratio-of-normals limit under a drift condition that no exactly self-normalizing denominator can satisfy. The standardized denominator is the noncentrality diagnostic of this limit and also determines whether Fieller inversion is bounded. For marginal and partial covariance denominators, it is an explicit increasing function of the corresponding first-stage statistic. For linear front-door adjustment, the Gaussian delta Wald statistic remains asymptotically standard normal when the mediator residual variance tends to zero at any rate, provided the treatment–mediator coefficient is nonzero.

Together, the results give a diagnostic pathway for graph-derived rational formulas. One factors the symbolic denominator, checks whether its factors form a nested variance flag, and then chooses between conventional studentization and weak-identification-robust inversion. Simulations and a diagnostic audit of right-heart-catheterization data from the SUPPORT study illustrate the two branches of this pathway.

Graphical criteria for instrumental sets, front-door adjustment and proximal identification determine whether a causal effect can be expressed from the observational law. The half-trek criterion and computer-algebra methods provide rational certificates for linear structural equation models (31; 6; 15; 14; 23; 39; 29; 21; 38). Generic and rational identifiability are distinct algebraic notions, and a parameter may be generically unique without being represented by the particular rational formula under study (9). Recent semiparametric work derives regular influence functions for half-trek estimators at fixed interior distributions (28). Our analysis starts after a rational certificate has been selected. It classifies the denominator as a polynomial on the ambient covariance space and determines which first-order boundary regime that certificate can generate.

Fieller and Anderson–Rubin inversion provide the classical ratio-based confidence constructions (1; 13). The impossibility of uniformly bounded confidence sets and the local theory of weak instruments and weak moments explain why a noisy denominator near zero produces non-Gaussian ratio limits (16; 11; 36; 37; 2). Modern work develops first-stage-dependent corrections and robust procedures, including settings with many weak moments and weakly identified nuisance functions (3; 24; 40; 4). The local result in Section 5 is a covariance-polynomial specialization of this literature. The new point is structural and precedes the limit calculation. The self-normalization identity decides when the first-order ratio regime is algebraically impossible.

Condition-number analyses quantify sensitivity of an identified functional or structural parameter to perturbations of the observational law (34; 19; 33). Exact self-normalization instead compares the gradient noise of one denominator with the denominator itself through a global polynomial identity. The two notions need not agree. In the front-door boundary sequence, the standard error diverges and the formula becomes badly conditioned, yet the denominator cannot enter the first-order Fieller regime. Conversely, a well-scaled slope denominator may have an interior zero with nondegenerate first-order noise.

Products of nested covariance minors are generalized power functions of a symmetric cone (12), and their exact sampling factorization follows from the Bartlett decomposition of a Wishart matrix (30). The affine-invariant metric on the positive-definite cone, which agrees up to scale with the Fisher–Rao metric of the centred Gaussian family, gives the geometric form of the logarithmic gradient flow used in the converse proofs (5). The separation between Gaussian and elliptical covariance operators is familiar in covariance-structure analysis (35; 22). Here it identifies which part of the classification is distributional and which part is algebraic.

The dimension-two converse uses the Gordan–Noether theorem for forms with vanishing Hessian (18; 27). That route fails for the six variables of a symmetric matrix of order three because noncone forms with vanishing Hessian exist in this dimension (32; 7; 17). Supplementary Section I.1 explains why Hessian degeneracy does not force a cone on the six-dimensional space $\operatorname{Sym}_{3}$, while Supplementary Section I.2 identifies the information lost when the relevant tensor relation is passed through polynomial multiplication. These two obstructions explain why neither route presently closes the mixed branch (25). This distinction explains the complete dimension-two result and the mixed-kernel problem left in dimension three.

We use $\mathbb{R}$ and $\mathbb{C}$ for the real and complex fields. Vectors are columns, $A^{\top}$ denotes matrix transpose, $I_{m}$ is the identity matrix of order $m$, and $e_{j}$ is the $j$th standard basis vector. The subscript on $I_{m}$ is omitted when the order is clear. The symbols $\operatorname{tr}(A)$, $\det(A)$, $\operatorname{rank}(A)$ and $\operatorname{spec}(A)$ denote trace, determinant, rank and the multiset of eigenvalues of $A$. Expectations, variances, covariances and probabilities under a probability measure $Q$ are denoted by $\mathbb{E}_{Q}$, $\operatorname{var}_{Q}$, $\operatorname{cov}_{Q}$ and $\mathrm{pr}_{Q}$. The subscript is suppressed once the probability measure is fixed. Convergence in probability and in distribution are denoted by $\longrightarrow_{\rm p}$ and $\rightsquigarrow$, and limits are as $n\to\infty$ unless stated otherwise. The indicator of a statement $A$ is $\mathrm{1}(A)$, and $\operatorname{sgn}(x)$ is the sign of $x\in\mathbb{R}$.

For a positive integer $d$, let $[d]=\{1,\ldots,d\}$, $p_{d}=d(d+1)/2$ and $\mathcal{I}_{d}=\{(i,j):1\leq i\leq j\leq d\}$. We write $\operatorname{Sym}_{d}=\{A\in\mathbb{R}^{d\times d}:A=A^{\top}\}$ and denote its positive-definite and positive-semidefinite cones by $\mathbb{S}^{d}_{++}$ and $\mathbb{S}^{d}_{+}$. The notation $A\succ 0$ and $A\succeq 0$ means $A\in\mathbb{S}^{d}_{++}$ and $A\in\mathbb{S}^{d}_{+}$, respectively. For $\Sigma\in\operatorname{Sym}_{d}$, its entries are $\sigma_{ij}=\sigma_{ji}$. For $I,J\subseteq[d]$, $A_{I,J}$ is the submatrix with rows in $I$ and columns in $J$, $A_{I}=A_{I,I}$, and $A_{1:r}=A_{\{1,\ldots,r\},\{1,\ldots,r\}}$.

We order $\mathcal{I}_{d}$ lexicographically, first by $i$ and then by $j$, and set

Vectors and matrices indexed by $\mathcal{I}_{d}$ follow this ordering. We identify $\mathbb{R}[\operatorname{Sym}_{d}]=\mathbb{R}[\sigma_{ij}:(i,j)\in\mathcal{I}_{d}]$ with the ring of polynomial maps from $\operatorname{Sym}_{d}$ to $\mathbb{R}$. The notation $P\equiv 0$ means that $P$ is the zero polynomial in this ring. A polynomial $P$ is homogeneous of degree $K$ when $P(t\Sigma)=t^{K}P(\Sigma)$ for every $t\in\mathbb{R}$. Its total degree is $\deg(P)$, and two polynomials are coprime when their only common divisors are nonzero constants.

For $P\in\mathbb{R}[\operatorname{Sym}_{d}]$, the coordinate gradient $\nabla P(\Sigma)\in\mathbb{R}^{p_{d}}$ consists of the partial derivatives in the preceding $\operatorname{vech}$ order. The Fréchet differential of $P$ at $\Sigma$ in the direction $H$ is

The symmetric gradient $G_{P}(\Sigma)\in\operatorname{Sym}_{d}$ is the unique matrix satisfying $\mathrm{d}P_{\Sigma}(H)=\operatorname{tr}\{G_{P}(\Sigma)H\}$ for every $H\in\operatorname{Sym}_{d}$. Thus, $(G_{P})_{ii}=\partial P/\partial\sigma_{ii}$ and $(G_{P})_{ij}=2^{-1}\partial P/\partial\sigma_{ij}$ for $i<j$, with all quantities evaluated at $\Sigma$. For $P(\Sigma)\neq 0$, the relative gradient is $R_{P}(\Sigma)=G_{P}(\Sigma)\Sigma/P(\Sigma)$.

We formulate the statistical problem in the ambient covariance space. Fix $d\geq 1$ and $\Sigma\in\mathbb{S}^{d}_{++}$. The notation $N_{m}(\mu,\Omega)$ denotes the Gaussian law on $\mathbb{R}^{m}$ with mean $\mu\in\mathbb{R}^{m}$ and covariance matrix $\Omega\in\mathbb{S}^{m}_{+}$. Write $\mathcal{P}_{\Sigma}=N_{d}(0,\Sigma)$ and $\mathcal{P}_{\Sigma}^{(n)}=\mathcal{P}_{\Sigma}^{\otimes n}$. Let $X_{1},\ldots,X_{n}$ be independent observations with joint law $\mathcal{P}_{\Sigma}^{(n)}$. We use the known-mean sample covariance matrix

For an integer $\nu\geq 1$ and $\Omega\in\mathbb{S}^{m}_{+}$, $\mathcal{W}_{m}(\nu,\Omega)$ denotes the law of $\sum_{\ell=1}^{\nu}Z_{\ell}Z_{\ell}^{\top}$ for independent $Z_{\ell}\sim N_{m}(0,\Omega)$. Its scalar specialization with $\Omega=1$ is $\chi^{2}_{\nu}$. Consequently, $n\hat{\Sigma}\sim\mathcal{W}_{d}(n,\Sigma)$ and $\hat{\Sigma}\succ 0$ almost surely.

The identification equality is required only on a covariance model, whereas the polynomial pair representing it governs off-model sampling behaviour. The following definition separates these two objects.

This distinction is necessary for two reasons. First, $\hat{\Sigma}$ need not satisfy the model equations, so the sampling behaviour of $\hat{\tau}$ depends on $N$ and $D$ as polynomials on the whole of $\operatorname{Sym}_{d}$. Second, multiplying both $N$ and $D$ by a nonconstant common factor leaves the identified ratio unchanged wherever both representations are defined, yet alters the off-model sampling behaviour of the denominator. Coprimality removes this ambiguity. Two coprime pairs representing the same rational function coincide up to a common nonzero scalar, while distinct rational certificates for the same target remain distinct strategies.

We now introduce the central property of the paper. Define the polynomial matrix map $\Gamma:\operatorname{Sym}_{d}\to\operatorname{Sym}_{p_{d}}$ by

Under $\mathcal{P}_{\Sigma}^{(n)}$, the matrix $\Gamma(\Sigma)$ is the covariance matrix of $n^{1/2}\operatorname{vech}(\hat{\Sigma}-\Sigma)$. For $D\in\mathbb{R}[\operatorname{Sym}_{d}]$, define its Gaussian first-order variance functional by

The second equality is the Gaussian quadratic-form variance identity, verified in Supplementary Lemma B.1. Both sides are polynomial in $\Sigma$, so the equality extends from $\mathbb{S}^{d}_{++}$ to all of $\operatorname{Sym}_{d}$. For $\Sigma\in\mathbb{S}^{d}_{+}$ the matrix $\Gamma(\Sigma)$ is positive semidefinite, and we write $s_{D}(\Sigma)=\{s_{D}^{2}(\Sigma)\}^{1/2}$ there. The standardized denominator is $\widehat{F}_{D}=nD(\hat{\Sigma})^{2}/s_{D}^{2}(\hat{\Sigma})$, defined on the event $\{s_{D}^{2}(\hat{\Sigma})>0\}$.

The constant $c$ is unique because $D$ is not the zero polynomial. Below, self-normalizing always means exactly self-normalizing. Equation (4) states that the Gaussian first-order noise of $D$ contracts in exact proportion to $|D|$ throughout the ambient covariance space. In particular, evaluating (4) at $\hat{\Sigma}$ gives $\widehat{F}_{D}=n/c$ whenever $s_{D}^{2}(\hat{\Sigma})>0$. Thus, the standardized denominator of a self-normalizing polynomial $D$ is constant over samples rather than a relevance statistic. Exact self-normalization is invariant under congruence transformations. Every rational strategy for a scale-invariant coefficient in a linear structural equation model has a homogeneous representative, and a nonzero self-normalizing polynomial has no zero in $\mathbb{S}^{d}_{++}$. Supplementary Lemmas B.1–B.3 prove these facts. If $D$ depends on $\Sigma$ only through its compression to an $r$-dimensional subspace, its self-normalization status and constant are unchanged in every ambient dimension $d\geq r$ by Supplementary Lemma B.4. The dimension-three classification below therefore applies to any denominator supported on at most three directions, whatever the number of observed coordinates.

We construct self-normalizing denominators in every dimension. The building blocks are covariance volumes of subspaces. For an $r$-dimensional subspace $V\subseteq\mathbb{R}^{d}$, let $M_{V}$ be any full-row-rank matrix whose row space is $V$, and write $\Delta_{V}(\Sigma)=\det(M_{V}\Sigma M_{V}^{\top})$. If a random vector $X$ has covariance matrix $\Sigma$, then $\Delta_{V}(\Sigma)$ is the generalized variance of the reduced vector $M_{V}X$. A different choice of $M_{V}$ multiplies $\Delta_{V}$ by a positive constant and affects no statement below.

A flag is a strictly increasing chain $V_{1}\subset\cdots\subset V_{k}$ of nonzero subspaces of $\mathbb{R}^{d}$. Let $a_{j}=\dim(V_{j})$, so that $1\leq a_{1}<\cdots<a_{k}\leq d$, and let $e_{1},\ldots,e_{k}$ be positive integers. The associated flag power and its multiplicities are

Thus, $m_{i}$ is the total exponent carried by factors whose subspaces have dimension at least $i$, and $m_{1}\geq\cdots\geq m_{a_{k}}\geq 1$. The following theorem shows that every flag power is exactly self-normalizing and gives its exact sampling law.

The nesting hypothesis cannot be weakened. If neither of two subspaces $V$ and $W$ contains the other, then no product $\Delta_{V}^{p}\Delta_{W}^{q}$ with positive integer exponents $p$ and $q$ is exactly self-normalizing. Supplementary Section C proves both this two-subspace criterion and Theorem 1.

Flag powers aggregate conditional-variance information. Let $X=(X^{(1)},\ldots,X^{(d)})^{\top}$ have the sampling law $\mathcal{P}_{\Sigma}$, and write $v_{i}(\Sigma)=\operatorname{var}_{\Sigma}\{X^{(i)}\mid X^{(1)},\ldots,X^{(i-1)}\}$ for the successive conditional variances, with $v_{1}(\Sigma)=\sigma_{11}$. For every $r\in[d]$,

Consider the coordinate flag, whose $j$th subspace is spanned by the first $a_{j}$ coordinate directions. By (7), the flag power (5) is then the monomial $\prod_{i=1}^{a_{k}}v_{i}(\Sigma)^{m_{i}}$, so the multiplicity $m_{i}$ is the exponent attached to the $i$th conditional variance. A general flag reduces to this case, up to a positive constant, after one fixed nonsingular linear transformation of the observation vector. For the front-door strategy of Section 1, the first two conditional variances are the treatment variance and the mediator residual variance. Its denominator is the monomial $v_{1}(\Sigma)^{2}v_{2}(\Sigma)$, and part (ii) of Theorem 1 returns the constant $c=2(2^{2}+1^{2})=10$ behind the value $n/10$.

The same representation explains the exact sampling law. Under $\mathcal{P}_{\Sigma}^{(n)}$, the sample conditional variances computed from $\hat{\Sigma}$ along the coordinate flag are mutually independent. The $i$th is distributed as $v_{i}(\Sigma)$ times a $\chi^{2}_{n-i+1}/n$ variable. This is the Bartlett decomposition of the Wishart matrix $n\hat{\Sigma}$ (30, Ch. 3), and evaluating the monomial at these independent factors gives the law in (6). To first order, the relative variance of every sample conditional variance is $2/n$, whatever the value of $\Sigma$. By independence, the relative first-order variance of the sample flag power is $2\sum_{i=1}^{a_{k}}m_{i}^{2}/n$, which equals $c/n$. This constancy in $\Sigma$ is the sampling mechanism behind exact self-normalization. The sample flag power depends on $\hat{\Sigma}$ only through these conditional variances. The regression coefficients of each coordinate on its predecessors, which complete the recursive decomposition of $\hat{\Sigma}$, do not enter. This contrast between conditional variances and regression coefficients is the sample form of the distinction between variance information and slope information that organizes the converse results of Section 4.

We first obtain a complete converse for covariance matrices of order two. Through face restriction, this result also drives the three-variable analysis.

Thus, a two-variable denominator can self-normalize only by combining one variance direction with the full covariance volume.

We summarize the proof mechanism, which explains the rigidity. Euler’s identity and (4) fix the first two power sums of the eigenvalues of $R_{D}(\Sigma)$. The elementary symmetric identity converts them into a determinant identity with an explicit constant coefficient. When that coefficient is nonzero, the determinant divides $D$, the factor can be stripped, and induction on the degree applies. When it vanishes, the gradient map takes values in the rank-one quadric, and the Hessian determinant of $D$ vanishes identically. The Gordan–Noether theorem reduces $D$ to a power of one linear form, which symmetry and reality identify with a variance direction (18; 27). Supplementary Section D.2 gives the full proof.

We now reduce the three-variable converse to a single mixed branch. Put $\delta=\det\Sigma$ and call $D\in\mathbb{R}[\operatorname{Sym}_{3}]$ determinant-free when $\delta\nmid D$. For homogeneous $D$ of degree $K$, define the rank-one restriction $p_{D}:\mathbb{R}^{3}\to\mathbb{R}$ by $p_{D}(u)=D(uu^{\top})$. For a plane $V\subseteq\mathbb{R}^{3}$ with basis matrix $P\in\mathbb{R}^{3\times 2}$, define the face restriction $D_{V}:\operatorname{Sym}_{2}\to\mathbb{R}$ by $D_{V}(S)=D(PSP^{\top})$. The plane $V$ is called $D$-good when $D_{V}\not\equiv 0$, and simply good when the polynomial is clear from context. A polynomial vector is primitive when its entries have no nonconstant common divisor.

A nonzero plane restriction of a self-normalizing polynomial is again self-normalizing with the same constant. Supplementary Lemma D.3 provides this restriction principle, and Theorem 2 then determines the form of every such restriction.

A determinant-free solution has at least one good plane, and its degree and constant admit at most one pair satisfying (9), so the face type exists and is common to all good planes. Removing a factor of $\delta$ preserves exact self-normalization with a shifted constant, so the residue in the next theorem is itself self-normalizing. Supplementary Lemmas D.4 and D.6 provide both facts.

The polynomials in cases (i) and (ii) are flag powers built on a single subspace. A mixed type has degree $a+2b\geq 3$, so every solution of degree at most two falls under the first two cases and is a flag power. Therefore, the unresolved solutions lie in case (iii) and carry one possibly moving null direction together with the fixed spectrum in (10).

Supplementary Section D also gives an independent closure of the mixed branch that does not presuppose a constant kernel line. If the restrictions of the determinant-free mixed residue to rank-two faces agree with those of one fixed flag power built from a nested line and plane, the residue is that flag power whenever its mixed type has degree at most eight. For higher-degree types, the same conclusion holds under this boundary agreement condition outside an explicit arithmetic resonance set. Therefore, the unresolved difficulty is the geometric gluing of the face directions rather than an uncontrolled interior perturbation.

The constancy of the relative spectrum that drives the three-variable reduction is special to dimension three. Euler’s identity fixes the trace of $R_{D}(\Sigma)$ at the degree of $D$, and (4) fixes the trace of its square, so in dimension three the eigenvalues are confined to a circle. The polynomial character of $D$ forces at least one further linear relation with integer coefficients among the eigenvalues. A line meets a circle in at most two points, so only finitely many spectra are possible, and continuity on the connected cone makes the spectrum constant. This finiteness underlies the fixed spectrum displayed in (10). In dimension $d\geq 4$ the same two traces leave a sphere of dimension $d-2$, a linear relation cuts out a set of dimension $d-3$, and no finiteness follows.

A conditional converse nevertheless holds in every dimension. If the relative gradients $R_{D}(\Sigma)$ preserve one fixed complete flag for every $\Sigma\succ 0$ and induce constant weights on the successive one-dimensional quotients, then $D$ is the corresponding flag power. This is an intrinsic condition on the gradient rather than an assumed factorization, and the precise statement is Supplementary Proposition D.5. Exact self-normalization alone is not claimed to create such an invariant flag when $d\geq 4$. The obstruction in higher dimensions is thus the emergence of one common invariant recursive ordering, not the integration step once that ordering is present, and dimension three is a sharp low-dimensional target rather than an arbitrary truncation.

Although the mixed branch obstructs an unconditional converse, the denominators produced by graphical identification typically have a simple factor structure. The two denominators of Section 1 factor into linear and quadratic pieces after determinant stripping, and the same holds for the formulas revisited in Section 6. We now show that this class is classified completely and that membership can be decided by exact symbolic computation.

Under this condition the mixed branch disappears and the classification closes.

Therefore, linear and quadratic factors cannot assemble into any self-normalizing configuration other than a nested flag.

For a square matrix $A$, $\operatorname{adj}(A)$ denotes its classical adjugate, characterized by $A\operatorname{adj}(A)=\operatorname{adj}(A)A=\det(A)I$. Writing the plane minor through the adjugate turns Theorem 4 into a normal form whose ingredients can be tested one by one.

This corollary reduces the audit of a candidate denominator to a finite procedure. One factors the polynomial symbolically, tests the ranks of the coefficient matrices of the linear and adjugate-linear factors, and checks orthogonality and nesting of the resulting directions. The globalization of the generic face factors and the resulting coefficient-level characterization are proved in Supplementary Section D.8.

The two formulas of Section 1 illustrate the procedure.

Thus, the audit separates the two formulas before any data are collected.

We formulate the first-order regime of a weak denominator whose sampling noise does not degenerate with it. Throughout this section, $(N,D)$ is a fixed identification strategy in the sense of Definition 1, and $\hat{\tau}=N(\hat{\Sigma})/D(\hat{\Sigma})$ is its plug-in estimator. For a deterministic sequence $\Sigma_{n}\in\mathbb{S}^{d}_{++}$ with $D(\Sigma_{n})\neq 0$, define

and put $s_{D,n}=s_{D}(\Sigma_{n})$ and $s_{g,n}=\{g_{n}^{\top}\Gamma(\Sigma_{n})g_{n}\}^{1/2}$. Whenever $s_{D,n}s_{g,n}>0$, define

These three quantities are the standardized drift of the denominator, the correlation between the moment direction and the denominator gradient, and their scale ratio.

By continuity of the polynomial gradients and of $\Gamma$, Assumption 1 implies $s_{D,n}\to s_{D,\ast}$, $s_{g,n}\to s_{g,\ast}$ and

Assumption 1 describes a denominator that drifts to zero at exactly the rate of its nondegenerate first-order noise. Continuity gives $D(\Sigma_{n})\to D(\Sigma_{\ast})$, and the finite limit of $n^{1/2}D(\Sigma_{n})$ then forces $D(\Sigma_{\ast})=0$, so the sequence approaches a zero of the denominator, which may lie in the interior of the cone or on its boundary. The condition $s_{D,\ast}>0$ makes this zero first-order nondegenerate, and the condition $s_{g,\ast}>0$ excludes a simultaneous first-order degeneracy of the moment function itself. The limit $\mu_{\ast}$ records how many first-order standard errors separate the drifting denominator from zero, and it acts as the noncentrality parameter of the limit experiment.

The associated Wald procedure studentizes the plug-in estimator along the estimated moment direction. On the event where $D(\hat{\Sigma})\neq 0$ and the quadratic form below is positive, define

The next proposition identifies the joint limits.

Thus, a first-order nondegenerate small denominator produces the classical Fieller ratio limit. The limiting standardized denominator $(\mu_{\ast}+Z_{2})^{2}$ is a noncentral chi-squared variable with one degree of freedom and noncentrality $\mu_{\ast}^{2}$, so $\widehat{F}_{D}$ measures the local strength of identification. The Wald statistic converges to a non-Gaussian law governed by $\mu_{\ast}$ and $r_{\ast}$. Supplementary Section E proves Proposition 1.

We connect the ratio experiment to robust inference. For $\alpha\in(0,1)$, inverting the single moment $N-tD$ gives the confidence set

where

is the $(1-\alpha)$ quantile of the $\chi_{1}^{2}$ law.

Supplementary Section E proves Proposition 2. This is the classical Fieller and Anderson–Rubin analysis specialized to covariance polynomials (1; 13). It gives the standardized denominator a second role. Proposition 1 identifies $\widehat{F}_{D}$ as the noncentrality diagnostic of the local experiment, and Proposition 2 makes the same statistic decide whether the inverted set is bounded. Combining the two results, the probability that $\mathcal{C}_{1-\alpha}$ is bounded converges to $\mathrm{pr}\{(\mu_{\ast}+Z_{2})^{2}>q_{1-\alpha}\}$, so unbounded sets retain a positive limiting frequency under weak drift, in accordance with the impossibility results for uniformly bounded confidence sets (16; 11).

We now delimit what exact self-normalization removes. As noted after Assumption 1, the drift forces $D(\Sigma_{\ast})=0$ while requiring $s_{D}(\Sigma_{\ast})=s_{D,\ast}>0$. For a self-normalizing denominator, evaluating (4) at $\Sigma_{\ast}$ gives $s_{D}(\Sigma_{\ast})=c^{1/2}|D(\Sigma_{\ast})|=0$, so the two requirements are incompatible and no drifting sequence satisfies Assumption 1. Therefore, exact self-normalization excludes the first-order ratio experiment, not every nonregular limit.

The exclusion concerns the first order only. On $\operatorname{Sym}_{2}$, the polynomial $D=\sigma_{12}^{2}$ has $s_{D}^{2}=4\sigma_{12}^{2}(\sigma_{11}\sigma_{22}+\sigma_{12}^{2})$, so its first-order variance vanishes on the entire zero set and Assumption 1 cannot hold, although the ratio $s_{D}^{2}/D^{2}$ is not constant. Under the drift $\sigma_{12,n}=\zeta n^{-1/2}$ with $\sigma_{11,n}=\sigma_{22,n}=1$, the standardized denominator converges in distribution to $(\zeta+Z)^{2}/4$ for a standard normal variable $Z$, a nondegenerate limit generated at second order. Supplementary Section E verifies this example. Hence, the self-normalizing and the first-order nondegenerate regimes are two extremes of a broader classification rather than an exhaustive dichotomy.

We translate the algebra into familiar covariance diagnostics. For distinct indices $a,b\in[d]$, put $\rho_{ab}=\sigma_{ab}/(\sigma_{aa}\sigma_{bb})^{1/2}$ and let $\hat{\rho}_{ab}$ be the same function evaluated at $\hat{\Sigma}$. For the marginal covariance $D=\sigma_{ab}$, the Gaussian first-order variance is $s_{D}^{2}=\sigma_{aa}\sigma_{bb}+D^{2}$, so every zero of $D$ in the open cone is first-order nondegenerate and the denominator conditions of Assumption 1 can hold along suitable drifting sequences. The standardized denominator is $\widehat{F}_{D}=n\hat{\rho}_{ab}^{2}/(1+\hat{\rho}_{ab}^{2})$, an increasing function of the familiar first-stage statistic $n\hat{\rho}_{ab}^{2}/(1-\hat{\rho}_{ab}^{2})$.

For indices $e,a,b\in[d]$ with $e\neq a$ and $e\neq b$, allowing $a=b$, define $\rho_{ab\cdot e}$ as the correlation of the residuals from the population linear projections of coordinates $a$ and $b$ on coordinate $e$, and let $\hat{\rho}_{ab\cdot e}$ be its sample-covariance analogue. For the partial minor $D=\sigma_{ee}\sigma_{ab}-\sigma_{ea}\sigma_{eb}$,

When $a=b$ the minor is a principal $2\times 2$ minor, the identity reduces to $s_{D}^{2}=4D^{2}$, and $\widehat{F}_{D}=n/4$ in agreement with Theorem 1. When $a\neq b$, define

so the standardized denominator is again an increasing transform of the familiar partial first-stage index $F_{\mathrm{par}}$. In both cases the standardized denominator reproduces the relevant marginal or partial screening direction without any modelling input. Supplementary Proposition F.1 gives the calculation and the exact partial-correlation formula.

We compare four representative strategies in Table 1. The table separates variance-type flag powers from slope-type marginal or nonprincipal minors. The structural parameterizations and substitutions yielding the third column are given in Supplementary Section F.1, so that each displayed pullback can be checked directly from the stated linear reduced form. The three slope-type rows can realize the denominator conditions of Assumption 1, whereas the front-door row is a flag power. The next subsection develops the front-door strategy in detail.

We now show that the front-door studentized statistic survives arbitrary collinearity drift when the treatment–mediator coefficient is nonzero. The exact regression decomposition and the proof of the boundary result are given in Supplementary Section F.3. Consider the Gaussian reduced form

in which $X$, $\varepsilon_{M}$ and $\varepsilon$ are independent centred Gaussian variables with $\operatorname{var}(X)=\nu_{X}>0$ and $\operatorname{var}(\varepsilon)>0$, and the observed vector is $(X,M,Y)^{\top}$. The coefficients $a$, $b$ and $\gamma$ are fixed, while the mediator residual variance $v_{M,n}=\operatorname{var}(\varepsilon_{M})$ may change with $n$ and takes values in a fixed interval $(0,\bar{v}]$. Observations are $n$ independent copies of the observed vector, and $\hat{\Sigma}$ is the known-mean sample covariance (1). Substituting the reduced form gives $\sigma_{XX}=\nu_{X}$ and $\det\Sigma_{\{X,M\}}=\nu_{X}v_{M,n}$, so the population denominator equals $\nu_{X}^{2}v_{M,n}$, which is the pullback recorded in Table 1 and which tends to zero whenever $v_{M,n}$ does.