Population Moment Restrictions And The GMM Criterion

From Population Restrictions to a Sample Criterion

Many econometric models are expressed through restrictions on expectations. For example, an instrument may be required to have zero expected product with a structural error at the true parameter value. The population restriction describes the model; observed data can only provide an estimate of its expectation. Generalized method of moments (GMM) makes this distinction explicit: it forms a vector of population moments, estimates that vector with sample averages, and measures how far the sample moments are from zero.

The GMM criterion is a way to combine multiple moment discrepancies. Its weighting matrix determines how the components are scaled and how their joint discrepancy is measured. The key construction is simple, but it is important to keep the population restrictions, their sample counterparts, and the criterion itself distinct.

Definition: Let \(W\) denote an observation and let \(\theta\) be a \(k\)-dimensional parameter vector. A \(q\)-dimensional moment function \(g(W,\theta)\) has population moment vector \[ m(\theta)=E[g(W,\theta)]. \] The population moment restrictions are \(m(\theta_0)=0\), where \(\theta_0\) is the parameter value of interest. Given observations \(W_1,\ldots,W_n\), the sample moment vector is \[ \bar g_n(\theta)=\frac{1}{n}\sum_{i=1}^n g(W_i,\theta). \]

The expectation defining \(m(\theta)\) is taken under the population distribution of \(W\). The sample vector \(\bar g_n(\theta)\) replaces that expectation with an average over the observed data. For these definitions, assume the relevant expectations exist; in particular, each component of \(g(W,\theta)\) must have a finite expectation at the parameter values being considered. In the examples below, observations are treated as independent draws from the stated population.

The vector has \(q\) components, one for each moment restriction. The parameter has \(k\) components. Counting them gives the standard distinction between exactly identified and overidentified systems. This count describes the number of restrictions relative to the number of unknown parameters; it does not, by itself, establish that the restrictions determine one unique parameter value.

Exactly Identified and Overidentified Systems

Definition: A moment system is exactly identified by count when it has the same number of moment conditions as parameters, \(q=k\). It is overidentified by count when it has more moment conditions than parameters, \(q>k\). When \(q<k\), it has fewer moment conditions than parameters. These counts alone do not establish whether a unique parameter value satisfies the population restrictions.

In an exactly identified system, the number of sample moment equations equals the number of parameter components. In an overidentified system, there are more sample moment equations than parameter components. A particular parameter value might make every sample moment zero, but in an overidentified system the equations will generally not all be satisfied exactly by one value. GMM therefore needs a way to summarize their joint discrepancy rather than requiring every sample moment to vanish.

Worked Example: Counting Moments in an Instrumental-Variables Model

Suppose a model has one unknown scalar coefficient \(\beta\), and two proposed instruments give the moment vector \[ g(W_i,\beta)= \begin{pmatrix} Z_{i1}(Y_i-\beta X_i)\\ Z_{i2}(Y_i-\beta X_i) \end{pmatrix}. \] Assume the population restrictions are \(E[Z_{i1}(Y_i-\beta_0 X_i)]=0\) and \(E[Z_{i2}(Y_i-\beta_0 X_i)]=0\), with the expectations finite. Here \(k=1\) and \(q=2\), so the system is overidentified by count. The fact that there are two restrictions for one coefficient does not guarantee that both sample averages equal zero at the same value of \(\beta\). The criterion below gives a scalar measure of their joint discrepancy.

The Quadratic GMM Criterion

To combine the \(q\) sample moments, choose a \(q\)-by-\(q\) weighting matrix \(A\). For the basic criterion, take \(A\) to be fixed, symmetric, and positive definite: \(A=A'\), and \(v'Av>0\) for every nonzero \(q\)-vector \(v\). The criterion is a scalar, even though the sample moment vector has \(q\) components.

Definition: Given a symmetric positive-definite weighting matrix \(A\), the (normalized) GMM criterion at \(\theta\) is \[ Q_n(\theta)=\bar g_n(\theta)'A\bar g_n(\theta). \] Another common scaling is \(nQ_n(\theta)\). For a fixed \(n\), this positive scaling changes the criterion's numerical value but not which parameter values minimize it.

The matrix allows the criterion to account for the relative scales and joint weighting of the moment components. With \(A\) equal to the identity matrix, the criterion is the sum of squared sample moments. A different positive-definite matrix can give different relative weights and include cross-products between moment components. The criterion is not itself a probability or a test statistic; it is an objective function measuring the discrepancy from the zero-moment restrictions.

Evaluate the moment contributions.
For a candidate parameter value, calculate \(g(W_i,\theta)\) for each observation.

Average across observations.
Compute \(\bar g_n(\theta)=n^{-1}\sum_{i=1}^{n}g(W_i,\theta)\), retaining all \(q\) components.

Apply the weight and form the quadratic.
Calculate \(\bar g_n(\theta)'A\bar g_n(\theta)\). The result is one nonnegative number for a positive-definite \(A\).

Why the Criterion Is Nonnegative

The nonnegativity and zero condition follow directly from positive definiteness. One way to see this is to use the spectral decomposition of a symmetric positive-definite matrix: its eigenvalues are all positive, so it has a symmetric square root \(A^{1/2}\). For any sample moment vector, write \(v=\bar g_n(\theta)\). Then

Q_n(\theta) =v'Av =v'A^{1/2}A^{1/2}v =\left\|A^{1/2}v\right\|^2 \geq 0.

The squared norm is zero exactly when \(A^{1/2}v=0\). Since \(A^{1/2}\) is invertible, this happens exactly when \(v=0\). Thus \(Q_n(\theta)=0\) if and only if every component of \(\bar g_n(\theta)\) is zero. In an overidentified sample, the criterion can be positive because the available parameter values do not make every sample moment vanish simultaneously.

Worked Example: An Exactly Identified Sample Moment

Consider the scalar moment \(g(W_i,\beta)=Z_i(Y_i-\beta X_i)\) with one parameter and one moment, so \(k=q=1\). For this hypothetical sample, let the observed values be:

Observation	\(Z_i\)	\(X_i\)	\(Y_i\)
1	1	1	2
2	2	1	3
3	1	1	2.4

Use the scalar weighting matrix \(A=1\), which is positive definite. At the candidate \(\beta=2.6\), the three moment contributions are \[ 1(2-2.6\cdot1)=-0.6,\qquad 2(3-2.6\cdot1)=0.8,\qquad 1(2.4-2.6\cdot1)=-0.2. \] Their average is \[ \bar g_n(2.6)=\frac{-0.6+0.8-0.2}{3}=0. \] Therefore \(Q_n(2.6)=0\cdot1\cdot0=0\). This candidate makes the sample moment zero. That arithmetic alone does not establish that the population restriction is true or that \(\beta=2.6\) is uniquely determined by the population model.

Worked Example: A Nonzero Criterion at a Candidate Value

For a separate hypothetical sample, again use \(g(W_i,\beta)=Z_i(Y_i-\beta X_i)\) and \(A=1\). Let the three observations have \((Z_i,X_i,Y_i)=(1,1,2),(1,2,4),(1,1,4)\). There is one parameter and one moment, so this system is exactly identified by count. At the candidate \(\beta=2\), the residuals \(Y_i-\beta X_i\) are \[ 2-2\cdot1=0,\qquad 4-2\cdot2=0,\qquad 4-2\cdot1=2. \] Because each \(Z_i=1\), the moment contributions are \(0,0,2\), and the sample moment is \(\bar g_n(2)=(0+0+2)/3=2/3\). Thus \[ Q_n(2)=\left(\frac{2}{3}\right)^2=\frac{4}{9}\approx0.4444. \] The positive criterion indicates a nonzero sample moment at this candidate; it does not, by itself, say whether the population restriction holds.

Worked Example: Weighting Two Overidentifying Moments

Consider a hypothetical model with one coefficient \(\beta\) and two instruments. At the candidate \(\beta=1\), suppose three observations have \(X_i=1\) and \((Y_i,Z_{i1},Z_{i2})\) equal to \((2,1,0)\), \((3,0,1)\), and \((1,-1,0.5)\), respectively. The moment function is \[ g(W_i,\beta)= \begin{pmatrix} Z_{i1}(Y_i-\beta X_i)\\ Z_{i2}(Y_i-\beta X_i) \end{pmatrix}. \] The residuals at \(\beta=1\) are \(1,2,0\), so the three moment vectors are \((1,0)'\), \((0,2)'\), and \((0,0)'\). Therefore \[ \bar g_n(1)= \begin{pmatrix} 1/3\\ 2/3 \end{pmatrix}. \] There are \(q=2\) moments and \(k=1\) parameter, making this overidentified by count.

Take \[ A= \begin{pmatrix} 2&0.5\\ 0.5&1 \end{pmatrix}. \] This matrix is symmetric and positive definite: its leading diagonal entry is \(2>0\), and its determinant is \(2(1)-(0.5)^2=1.75>0\). Multiplying gives \[ A\bar g_n(1)= \begin{pmatrix} 2(1/3)+0.5(2/3)\\ 0.5(1/3)+1(2/3) \end{pmatrix} = \begin{pmatrix} 1\\ 5/6 \end{pmatrix}. \] Hence \[ Q_n(1)= \begin{pmatrix}1/3&2/3\end{pmatrix} \begin{pmatrix}1\\5/6\end{pmatrix} =\frac{1}{3}+\frac{5}{9} =\frac{8}{9}\approx0.8889. \] The criterion is positive because the sample moment vector is not zero. The off-diagonal entries of \(A\) contribute to the quadratic calculation; they do not mean that either sample moment has been omitted.

What the Criterion Does—and Does Not—Say

For a chosen weighting matrix, GMM uses the criterion to favor parameter values with smaller weighted sample moment discrepancies. The matrix matters: changing it can change the relative importance assigned to the moment components and can therefore change the criterion's minimizer. The definition of the criterion does not, on its own, supply a rule for choosing an optimal weight matrix or establish the statistical properties of a resulting estimator.

It is also essential to distinguish population restrictions from sample calculations. A population model asserts \(m(\theta_0)=0\); a finite sample will generally have nonzero sample moments even at the true parameter. Conversely, a zero sample moment at a candidate value does not prove that the corresponding population moment is zero. Those are different statements about different quantities.

Common Mistake

Wrong reasoning: “There is one parameter and two instruments, so both sample moments must equal zero at the fitted coefficient.”
Correction: The count \(q>k\) means the system is overidentified by count, not that all sample equations can necessarily be solved simultaneously. The GMM criterion combines their discrepancies; it equals zero only when the entire sample moment vector is zero.
Wrong reasoning: “A small positive criterion proves the population restrictions are correct.”
Correction: The criterion is a sample objective at a specified parameter and weighting matrix. A positive value records a nonzero weighted sample discrepancy, while a zero value records zero sample moments. Neither observation alone proves a population claim.

Key takeaway: Write the population moment vector as an expectation and the sample moment vector as an average. Count moment conditions and parameters to describe the system, then use a symmetric positive-definite matrix to turn the sample moment vector into a nonnegative quadratic criterion.

Check Your Understanding

Answer each question by distinguishing population restrictions from sample quantities.

A model has three parameters and three moment conditions. How is the system classified by count, and what does that classification not guarantee?
For a two-component sample moment vector \(\bar g_n(\theta)=(0,1)'\) and \(A\) equal to the identity matrix, calculate \(Q_n(\theta)\).
State the condition under which a positive-definite GMM criterion equals zero.
A scalar parameter is paired with four moment conditions. Is the system exactly identified or overidentified by count?
Why does observing \(\bar g_n(\theta)=0\) at one candidate value not by itself establish the corresponding population restriction?