Why the Moment Jacobian Matters
Population moment restrictions identify parameters only when different parameter values can be distinguished by the restrictions. The moment Jacobian gives a local way to assess that distinction: it describes how the vector of population moments changes, to first order, when the parameter vector changes. Its rank therefore provides a useful condition for local identification, but interpreting that condition correctly requires care.
Let \(\theta\) be a \(k\)-dimensional parameter vector, and let \(g(\theta)\) be a \(q\)-dimensional vector of population moments. The maintained moment restrictions are \(g(\theta_0)=0\), where \(\theta_0\) is the parameter value of interest. The moments may be overidentified, so \(q\) can exceed \(k\). This tutorial concerns identification by these population moments, not the numerical behavior of a particular estimator.
For a small parameter change \(h\), differentiability gives the local approximation \(g(\theta_0+h)\approx g(\theta_0)+D(\theta_0)h\). Since \(g(\theta_0)=0\), the Jacobian describes the first-order response \(D(\theta_0)h\). The key rank condition asks whether any nonzero parameter change \(h\) produces zero first-order movement in every moment.
Local and Global Identification
Local identification rules out other solutions sufficiently close to \(\theta_0\). It does not rule out a distant solution elsewhere in the parameter space. Global identification is the stronger property. Both definitions concern solutions to the population moment equations; a small or large sample estimate is not itself a definition of identification.
A standard local rank condition is that the Jacobian have full column rank, meaning \(\operatorname{rank}D(\theta_0)=k\). Since \(D(\theta_0)\) has \(k\) columns, this condition requires that the columns be linearly independent. It implies that the only vector \(h\) satisfying \(D(\theta_0)h=0\) is \(h=0\). Under regularity conditions, this is sufficient for local identification.
This is a sufficient condition, not a necessary one. If the Jacobian has deficient column rank, the first-order test fails, but the parameter may still be identified by higher-order changes in the moments. Conversely, full column rank gives local identification, not necessarily global identification.
Worked Examples
Worked Example: Full Column Rank in an Overidentified System
Suppose \(\theta=(\beta,\gamma)'\) and two hypothetical population moments are \(g_1(\beta,\gamma)=4-2\beta-\gamma\) and \(g_2(\beta,\gamma)=1-\beta-2\gamma\). For example, these could be moments of the form \(E[Z_j(Y-\beta X-\gamma Q)]\), with stipulated population cross-moments. The parameter space is \(\mathbb{R}^2\), and the restrictions require both moments to equal zero. Find the solution and check the rank condition.
Find the moment solution. Setting the moments to zero gives \(2\beta+\gamma=4\) and \(\beta+2\gamma=1\). Subtracting the second equation from the first gives \(\beta-\gamma=3\). Substituting \(\beta=\gamma+3\) into the second equation yields \(3\gamma+3=1\), so \(\gamma=-2/3\) and \(\beta=7/3\).
Compute the Jacobian. Differentiating each moment with respect to \((\beta,\gamma)\) gives
The determinant is nonzero, so the \(2\times2\) Jacobian has rank 2, its full column rank. The theorem implies local identification. In this example the equations are linear, and the nonzero determinant also shows directly that the displayed solution is the unique solution on \(\mathbb{R}^2\), so identification is global as well.
Worked Example: Redundant Moments Do Not Necessarily Cause Rank Failure
Consider a hypothetical Jacobian with three moments and two parameters:
The third row repeats the first, so the third moment adds no independent first-order restriction beyond the first. But the columns remain independent: if \(Dh=0\) for \(h=(h_1,h_2)'\), the first two rows give \(h_1=0\) and \(h_2=0\). Therefore the only null direction is \(h=0\), and \(\operatorname{rank}D=2\), the full column rank.
The number of independent rows can be smaller than the number of rows without undermining local identification. What matters for the rank condition is whether the columns are independent. Redundant rows mean some moments add no independent information; they do not, by themselves, mean that distinct parameter changes produce the same first-order movement.
Worked Example: A Null Direction and Actual Nonidentification
Let \(\theta=(\beta,\gamma)'\), and suppose the hypothetical population moments are \(g_1(\beta,\gamma)=\beta+\gamma-3\) and \(g_2(\beta,\gamma)=2\beta+2\gamma-6\). Take \(\theta_0=(1,2)'\). Both moments are zero there. Compute the Jacobian and check whether the restrictions identify this value locally.
The Jacobian is
Its columns are dependent and its rank is 1, less than \(k=2\). In particular, the nonzero direction \(v=(1,-1)'\) satisfies
Here the rank failure corresponds to actual nonidentification, not just a failed first-order test. For every real \(t\), \(\theta_0+tv=(1+t,2-t)'\) still has \(\beta+\gamma=3\), so both moments remain exactly zero. Values with arbitrarily small nonzero \(t\) are distinct from \(\theta_0\), which means \(\theta_0\) is not locally identified.
Worked Example: Rank Failure Can Coexist With Identification
Consider one parameter \(\theta\in\mathbb{R}\) and the hypothetical moment restriction \(g(\theta)=(\theta-2)^3=0\). Does the zero derivative at the solution imply that \(\theta=2\) is unidentified?
The moment is zero if and only if \((\theta-2)^3=0\), which holds if and only if \(\theta=2\). Thus the restriction identifies \(\theta=2\) globally. But the Jacobian at that value is the scalar
The Jacobian has deficient column rank, so the full-rank condition does not establish identification here. It does not establish nonidentification either. Moment changes occur at higher than first order: for any nonzero \(h\), \(g(2+h)=h^3\ne0\), even though the derivative at \(2\) is zero.
Worked Example: Local Identification Without Global Identification
Let \(\theta\in\mathbb{R}\) and \(g(\theta)=\theta^2-1\). At \(\theta_0=1\), the moment equals zero and the Jacobian is \(g'(1)=2\), which has full column rank for one parameter. The rank condition therefore establishes local identification.
Indeed, in the neighborhood \((0,2)\), the only solution to \(\theta^2-1=0\) is \(1\). But on the full parameter space \(\mathbb{R}\), both \(1\) and \(-1\) solve the moment restriction. Thus the parameter is locally identified at \(1\), but not globally identified on \(\mathbb{R}\).
Why the Rank Condition Works
The sufficient condition follows by reducing the overidentified system to a square system of moments. If \(D(\theta_0)\) has rank \(k\), some \(k\) rows form a nonsingular \(k\times k\) submatrix. Select those \(k\) moments and call their vector \(g_S(\theta)\). Its derivative at \(\theta_0\) is nonsingular. By the inverse function theorem, \(g_S\) is one-to-one in a neighborhood of \(\theta_0\). Therefore no other nearby \(\theta\) can have \(g_S(\theta)=g_S(\theta_0)=0\). Since \(g(\theta)=0\) implies \(g_S(\theta)=0\), no other nearby parameter can solve the full set of moments either.
This argument also explains why a redundant row need not be a problem. The proof needs a set of \(k\) moments whose derivatives are independent with respect to the \(k\) parameters; it does not require every moment to add a new direction. In contrast, if the columns are dependent, there is a nonzero \(v\) with \(D(\theta_0)v=0\). Such a direction produces no first-order change in any moment. That fact motivates concern about identification, but alone it does not prove that the exact moments stay unchanged along that direction.
Common Mistakes and Practical Interpretation
- Confusing redundant rows with dependent columns. Repeated or linearly dependent moment rows mean some restrictions are redundant. The Jacobian may still have full column rank, as the three-row example shows.
- Treating rank failure as proof of nonidentification. It means the first-order rank condition cannot establish local identification. Higher-order changes can still make the moment solution unique, as in the cubic example.
- Treating a null direction as proof of an exact solution path. A nonzero \(v\) with \(Dv=0\) gives no first-order movement, not necessarily zero movement at every order. Check the moment equations themselves before claiming nonidentification.
- Calling local identification global identification. Full column rank is a local condition. Other solutions may exist far from \(\theta_0\), as the quadratic example demonstrates.
- Confusing identification with moment fit. The question is whether the population restrictions distinguish parameter values. The Jacobian rank is not a measure of whether the moment equations are substantively plausible.
In applications, full column rank is a useful benchmark for local distinguishability. A Jacobian can have full rank while being close to rank deficient; then some parameter changes produce only small first-order moment changes. This makes the rank condition a qualitative identification check, not a guarantee that the parameters will be precisely estimated in finite samples.
Check Your Understanding
Use the definitions and rank condition to answer these questions.
- For a Jacobian with \(q=5\) rows and \(k=3\) columns, what rank is required by the full-column-rank condition for local identification?
- Can a Jacobian with a repeated row still have full column rank? Explain what the rows and columns represent.
- If \(D(\theta_0)v=0\) for a nonzero vector \(v\), what does this establish about first-order moment changes, and what does it not establish by itself?
- Why does \(g(\theta)=(\theta-2)^3\) show that the full-rank condition is not necessary for identification?
- How can a parameter be locally identified but not globally identified? Give the solutions in the quadratic example.