The Rank Condition For Multiple Regressors

Why Identification Becomes a Matrix Question

With one endogenous regressor, a valid instrument must provide relevant variation in that regressor for the IV moment to determine its coefficient. With several endogenous regressors, the question is more demanding: the instruments must provide enough distinct directions of variation to distinguish all the coefficients. Counting instruments is a useful first check, but it does not show whether their information is independent.

This tutorial formulates that issue as a population rank condition. The focus is identification from the IV population moments: whether those moments determine one coefficient vector, rather than several. The condition concerns the relationship between the instruments and the endogenous regressors; it does not by itself establish that the instruments satisfy the required validity assumptions.

Population Moments and Their Dimensions

Suppose the structural equation, with variables centered for simplicity, is

y=x'\beta+u.

Here $y$ is a scalar outcome, $x$ is a $k\times 1$ vector of endogenous regressors, $\beta$ is the $k\times 1$ vector of structural coefficients, and $u$ is the structural error. Let $z$ be a $q\times 1$ vector of instruments. Assume the relevant expectations exist and that the instruments satisfy the population orthogonality condition $E[zu]=0$. In words, each instrument has zero population covariance with the structural error.

Multiplying the structural equation by $z$ and taking expectations gives the population IV moments. Define $c=E[zy]$, a $q\times 1$ vector, and $A=E[zx']$, a $q\times k$ matrix. The moments can then be written as

E[z(y-x'\beta)]=0 \qquad\Longleftrightarrow\qquad c=A\beta.

Each row of $A$ corresponds to an instrument, and each column corresponds to an endogenous regressor. Its entries are the population cross-moments between those variables. The matrix $A$ summarizes which combinations of the regressors move with the instruments. The question is whether this information distinguishes every possible change in $\beta$.

Definition: The population rank condition for identifying all $k$ coefficients from these IV moments is $$ \operatorname{rank}(E[zx'])=k. $$ Equivalently, $A$ must have full column rank: its $k$ columns must be linearly independent. Because $A$ has $q$ rows, this requires $q\geq k$, but that count alone is not sufficient.

Why Full Column Rank Gives Uniqueness

A coefficient vector is identified by the moments when no other coefficient vector satisfies the same population moment equations. The rank condition is exactly the condition that ensures this uniqueness. The argument uses only linear algebra, but it is useful to see each step.

Start with two candidate vectors.
Suppose $\beta_1$ and $\beta_2$ both satisfy the same moments, so $c=A\beta_1=A\beta_2$.

Subtract the moment equations.
The difference gives $A(\beta_1-\beta_2)=0$. Thus any ambiguity between candidates must lie in the null space of $A$.

Apply full column rank.
If $\operatorname{rank}(A)=k$, the null space of this $q\times k$ matrix contains only the zero vector. Therefore $\beta_1-\beta_2=0$, and the two candidates are identical.

Consider failure of the condition.
If $\operatorname{rank}(A)<k$, there is a nonzero vector $d$ with $Ad=0$. Whenever $\beta$ solves $c=A\beta$, every vector $\beta+td$, for scalar $t$, gives $A(\beta+td)=c$. The moments cannot distinguish those candidates.

This proves that full column rank is necessary and sufficient for uniqueness from these linear population moment equations, when a solution exists. It does not say that a solution exists for an arbitrary vector $c$. In the structural model, the validity condition and the equation $c=A\beta$ ensure that the true coefficient vector supplies a solution. The rank condition then makes that solution unique.

Worked Examples: Checking Identification

Worked Example: Two Instruments and Two Endogenous Regressors

Suppose a hypothetical population has two endogenous regressors and two instruments. Let the population cross-moment matrix and outcome moment vector be

A=E[zx']= \begin{pmatrix} 2&1\\ 1&3 \end{pmatrix}, \qquad c=E[zy]= \begin{pmatrix} 7\\ 8 \end{pmatrix}.

Assume these moments are finite and the instruments satisfy $E[zu]=0$. The dimensions are $A:2\times2$, $\beta:2\times1$, and $c:2\times1$, so $A\beta=c$ is dimensionally consistent. To test the rank condition, calculate the determinant:

\det(A)=(2)(3)-(1)(1)=5\ne 0.

Thus $A$ has rank 2, equal to the number of coefficients. The moment equations are $2\beta_1+\beta_2=7$ and $\beta_1+3\beta_2=8$. Multiplying the second equation by 2 gives $2\beta_1+6\beta_2=16$; subtracting the first equation yields $5\beta_2=9$, so $\beta_2=1.8$. Substitution into the first equation gives $2\beta_1+1.8=7$, hence $\beta_1=2.6$. Therefore the moments uniquely identify $\beta=(2.6,1.8)'$. This conclusion is about the stated hypothetical population moments, not a claim about any real study.

Worked Example: Enough Instruments by Count, but Not by Rank

Now consider two endogenous regressors and three instruments, with hypothetical moments

A= \begin{pmatrix} 1&2\\ 2&4\\ -1&-2 \end{pmatrix}, \qquad c= \begin{pmatrix} 5\\ 10\\ -5 \end{pmatrix}.

The dimensions are $A:3\times2$, so the instrument count $q=3$ is at least the coefficient count $k=2$. But the second column is twice the first column. Therefore the columns are dependent and $\operatorname{rank}(A)=1$, not 2. The moment equations all reduce to the same restriction: $\beta_1+2\beta_2=5$.

For example, $\beta=(5,0)'$ satisfies the moments. So does $\beta=(3,1)'$, because $3+2(1)=5$. More generally, $d=(-2,1)'$ satisfies $Ad=0$, and adding any multiple of $d$ to a solution leaves the moments unchanged. The instruments are sufficient in number but fail to supply two independent directions of information. The coefficients are not uniquely identified by these moments.

Worked Example: Three Instruments with Two Independent Directions

Consider two endogenous regressors and three instruments with the hypothetical matrix and outcome moments

A= \begin{pmatrix} 1&0\\ 0&1\\ 1&1 \end{pmatrix}, \qquad c= \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix}.

Here $A$ is $3\times2$. Its first two rows form the $2\times2$ identity matrix, whose determinant is $1$, so $\operatorname{rank}(A)=2$. The rank condition holds for $k=2$ coefficients. The first two moment equations give $\beta_1=2$ and $\beta_2=-1$. Substitution into the third gives $\beta_1+\beta_2=2+(-1)=1$, which matches the third entry of $c$. Thus the moments are consistent and uniquely determine $\beta=(2,-1)'$. The third equation does not add another coefficient to identify; the key fact is that the matrix has two independent columns.

Interpreting the Condition

The rank condition is about independent instrument–regressor relationships, not simply the number of instruments. If two instruments have relationships with all regressors that are proportional, they do not create two distinct directions in the cross-moment matrix. Likewise, a single instrument cannot identify two unrestricted coefficients from these moments: a $1\times2$ matrix has rank at most 1. A necessary implication is $q\geq k$, but a matrix can have many rows and still have dependent columns.

A useful geometric interpretation is that each column of $A$ records how one endogenous regressor relates to the full instrument vector. Full column rank means that no regressor’s instrument relationship can be reproduced as a linear combination of the others. If one column can be so reproduced, a change in the corresponding coefficient can be offset by changes in other coefficients without changing the instrument moments. That is the source of non-uniqueness.

For another way to express the same condition, suppose $E[zz']$ exists and is nonsingular. The linear projection coefficient matrix for $x$ on $z$, under the convention $x=\Pi'z+v$, is $\Pi=(E[zz'])^{-1}E[zx']$, with dimensions $q\times k$. Multiplication on the left by a nonsingular matrix does not change rank, so $\Pi$ has full column rank exactly when $E[zx']$ does. This connects the condition to whether the instruments’ linear predictions span independent directions across the endogenous regressors. The condition can be stated directly using $E[zx']$, without requiring this projection representation.

Common Mistake: Confusing Instrument Count with Identification

A common mistake is to reason, “There are at least as many instruments as endogenous regressors, so all coefficients are identified.” That conclusion skips the rank check. Instrument count only establishes the dimensional possibility of full column rank. If the columns of $E[zx']$ are linearly dependent, the moment equations still leave some coefficient direction unresolved, even when $q>k$.

The correction is to write down the $q\times k$ matrix $E[zx']$ and check whether its $k$ columns are linearly independent. Also keep the scope of the conclusion precise: full rank addresses whether the population IV moments uniquely determine the coefficient vector, assuming valid moments. It does not, by itself, establish instrument validity or show that the rank is strong enough for reliable finite-sample estimation.

Key takeaway: For $k$ endogenous regressors, the population IV moments identify all $k$ coefficients uniquely when $E[zx']$ has full column rank $k$. Having at least $k$ instruments is necessary, but independent instrument–regressor relationships are what make identification possible.

Check Your Understanding

Use the dimensions and rank of the population cross-moment matrix to answer each question.

If there are four endogenous regressors and three instruments, what is the largest possible rank of $E[zx']$, and can the full-column-rank condition hold?
A $3\times2$ cross-moment matrix has two proportional columns. What does this imply about identification from its population IV moments?
Why is having $q\geq k$ necessary but not sufficient for identifying $k$ coefficients?
If $Ad=0$ for some nonzero vector $d$, explain why $\beta$ and $\beta+td$ give the same moments for every scalar $t$.
Does full column rank alone establish that the instruments are valid? Explain the separate role of the orthogonality condition.