Deriving The Efficient GMM Weight

Why the GMM Weight Matters

GMM chooses parameter values that make a vector of sample moments close to zero. When there are more moment conditions than parameters, there are different ways to measure how far that vector is from zero. The weighting matrix determines which combinations of moments matter most to the criterion, and therefore affects the estimator’s asymptotic precision.

The central result is that, under correct specification and regularity conditions, the most precise fixed-weight GMM estimator uses the inverse long-run covariance matrix of the moments. This choice accounts both for differences in moment variability and for dependence among moments. The result is a covariance comparison: the efficient weight yields an asymptotic covariance matrix no larger than that from any other admissible fixed weight, in the positive-semidefinite sense.

Definition: Let \(g_i(\theta)\) be a \(q\)-dimensional vector of moment contributions for observation \(i\), and let \(\bar g_n(\theta)=n^{-1}\sum_{i=1}^n g_i(\theta)\). A fixed-weight GMM estimator minimizes \(\bar g_n(\theta)'W\bar g_n(\theta)\) over \(\theta\), where \(W\) is a fixed, symmetric, positive-definite \(q\times q\) matrix. “Fixed” means that the weight does not vary with the candidate parameter during this minimization or with the sample size in the asymptotic comparison.

Conditions and the General Covariance Formula

Write the parameter as a \(p\)-dimensional vector, with \(q\geq p\) moment conditions. Correct specification means that there is a true parameter value \(\theta_0\) satisfying \(E[g_i(\theta_0)]=0\). Define the moment Jacobian and the long-run covariance matrix at that value by

G=E\!\left[\frac{\partial g_i(\theta_0)}{\partial\theta'}\right], \qquad \Omega=\sum_{k=-\infty}^{\infty} \operatorname{Cov}\!\left(g_i(\theta_0),g_{i-k}(\theta_0)\right).

For independent observations, the off-diagonal-lag terms in \(\Omega\) are zero, so it is the covariance matrix of one observation’s moments. For dependent observations, it incorporates serial dependence. The earlier material on long-run covariance estimation gives ways to estimate this matrix; here it is treated as the population covariance governing the comparison.

The standard fixed-weight GMM asymptotic result applies when the estimator is consistent for \(\theta_0\), the sample moments satisfy a central limit theorem, the moments are sufficiently smooth near \(\theta_0\), and \(G\) has full column rank. Assume also that \(\Omega\) is finite and positive definite and that \(W\) is symmetric and positive definite. These conditions ensure that the moment variation is nondegenerate and that the local quadratic criterion identifies the parameter.

Theorem: Under these conditions, the fixed-weight GMM estimator has asymptotic covariance matrix

V(W) = (G'WG)^{-1}G'W\Omega WG(G'WG)^{-1}, $$ meaning that \(\sqrt n(\hat\theta_W-\theta_0)\) converges in distribution to a mean-zero normal vector with covariance \(V(W)\). The efficient fixed weight is \(W=\Omega^{-1}\), or any positive scalar multiple of it. Its asymptotic covariance is $$ V_{\mathrm{eff}}=(G'\Omega^{-1}G)^{-1}.

This formula separates the role of the weight from the role of the data. The matrix \(G\) describes how the expected moments change with the parameter. The matrix \(\Omega\) describes the variability and dependence of the moments. A weight that ignores \(\Omega\) can give excessive influence to noisy moment combinations, or fail to account for correlated combinations.

Why the Inverse Covariance Is Optimal

The covariance comparison can be proved by viewing each fixed-weight estimator’s first-order behavior as a linear combination of the sample moments. For any fixed \(W\), define

B_W=(G'WG)^{-1}G'W.

Then \(B_WG=I_p\), and the covariance formula is \(V(W)=B_W\Omega B_W'\). The inverse-covariance choice corresponds to

B_0=(G'\Omega^{-1}G)^{-1}G'\Omega^{-1}.

It also satisfies \(B_0G=I_p\). Any other matrix \(B_W\) satisfying that restriction can be written as \(B_W=B_0+D\), where \(DG=0\). Expanding its covariance gives

\begin{aligned} B_W\Omega B_W' &=B_0\Omega B_0'+B_0\Omega D'+D\Omega B_0'+D\Omega D'\\ &=B_0\Omega B_0'+D\Omega D'. \end{aligned}

To see why the cross terms vanish, \(B_0\Omega=(G'\Omega^{-1}G)^{-1}G'\). Thus \(B_0\Omega D'=(G'\Omega^{-1}G)^{-1}(DG)'=0\), and the other cross term is its transpose. Since \(\Omega\) is positive definite, \(D\Omega D'\) is positive semidefinite: for every vector \(a\), \(a'D\Omega D'a=(D'a)'\Omega(D'a)\geq0\). Therefore

V(W)-V_{\mathrm{eff}}=D\Omega D'\ \text{ is positive semidefinite}.

This establishes the efficiency result: no fixed-weight GMM estimator satisfying the stated conditions has a smaller asymptotic covariance matrix in any direction than the inverse-covariance estimator. The result concerns asymptotic covariance, not a claim that the efficient estimator must have lower error in every finite sample.

Worked Comparisons

Worked Example: Comparing Identity and Efficient Weights

Suppose a hypothetical model has one parameter and two correctly specified moments. Let the moment Jacobian and long-run covariance be

G=\begin{pmatrix}1\\1\end{pmatrix}, \qquad \Omega=\begin{pmatrix}4&1\\1&2\end{pmatrix}.

The covariance matrix is positive definite: its leading diagonal entry is \(4>0\), and its determinant is \(4(2)-1(1)=7>0\). Compare the identity weight \(W=I_2\) with the efficient weight.

For \(W=I_2\), \(G'G=2\) and \(\Omega G=(5,3)'\), so \(G'\Omega G=8\). The scalar asymptotic variance is

V(I_2)=\frac{G'\Omega G}{(G'G)^2} =\frac{8}{2^2}=2.

The efficient weight is

\Omega^{-1}=\frac{1}{7} \begin{pmatrix}2&-1\\-1&4\end{pmatrix}.

Multiplying gives \(\Omega^{-1}G=(1/7)(1,3)'\), and hence \(G'\Omega^{-1}G=4/7\). Therefore

V_{\mathrm{eff}}=(G'\Omega^{-1}G)^{-1}=\frac{7}{4}=1.750.

The efficient weight reduces asymptotic variance from \(2\) to \(1.750\), a reduction of \(0.250\) in the parameter’s squared units. The off-diagonal entries of \(\Omega^{-1}\) matter here: the efficient weighting accounts for the positive correlation between the two moment contributions rather than treating their information as independent.

Worked Example: Unequal Moment Variances

Consider a hypothetical scalar parameter with two moment conditions, with

G=\begin{pmatrix}1\\2\end{pmatrix}, \qquad \Omega=\begin{pmatrix}9&0\\0&1\end{pmatrix}.

The moments are uncorrelated, but the first has variance \(9\), while the second has variance \(1\). With identity weighting, \(G'G=5\) and \(G'\Omega G=9+4=13\). Thus

V(I_2)=\frac{13}{5^2}=\frac{13}{25}=0.5200.

The inverse covariance is \(\Omega^{-1}=\operatorname{diag}(1/9,1)\). It gives

G'\Omega^{-1}G=\frac{1}{9}+4=\frac{37}{9}, \qquad V_{\mathrm{eff}}=\left(\frac{37}{9}\right)^{-1} =\frac{9}{37}\approx0.2432.

The inverse-covariance weight scales down the contribution from the highly variable first moment and retains more of the comparatively precise second moment. The resulting asymptotic variance, approximately \(0.2432\) in squared parameter units, is smaller than \(0.5200\).

Worked Example: Why a Scalar Multiple Is Equally Efficient

Return to the first example, where \(G'\Omega^{-1}G=4/7\), and multiply the efficient weight by \(2\): \(W=2\Omega^{-1}\). Then \(G'WG=8/7\). Also,

G'W\Omega WG =4G'\Omega^{-1}\Omega\Omega^{-1}G =4G'\Omega^{-1}G =\frac{16}{7}.

For a scalar parameter, the sandwich formula becomes

V(2\Omega^{-1}) =\frac{G'W\Omega WG}{(G'WG)^2} =\frac{16/7}{(8/7)^2} =\frac{7}{4}=1.750.

Thus multiplying the efficient weight by \(2\) leaves its asymptotic covariance unchanged. In general, replacing \(W\) by \(cW\), for \(c>0\), multiplies \(G'WG\) by \(c\) and \(G'W\Omega WG\) by \(c^2\); those factors cancel in the sandwich covariance. A positive scalar multiple also leaves the minimizer of the sample criterion unchanged.

What the Result Does—and Does Not—Say

The optimality claim is conditional on correct specification: at \(\theta_0\), all population moments equal zero. It also compares fixed positive-definite weighting matrices under the same moment conditions. It does not say that adding arbitrary moment conditions always improves precision, nor does it establish optimality when the moments are misspecified. Those are different questions.

There is also an important special case. If \(q=p\) and \(G\) is nonsingular, the model is exactly identified. Locally, the estimator solves the \(p\) moment equations, and every nonsingular fixed weight gives the same first-order covariance. Indeed, \(B_WG=I_p\) then forces \(B_W=G^{-1}\). The choice of weight has a distinctive efficiency role when the model is overidentified, with more moments than parameters.

Common Pitfall

Wrong reasoning: “The efficient weight is the covariance matrix \(\Omega\), because high-variance moments should count more.”
Correction: The efficient weight is \(\Omega^{-1}\), not \(\Omega\). Inverse weighting reduces the influence of noisy moment directions and accounts for covariance among moments. The claim is about the covariance of the resulting parameter estimator, not about making the sample moments themselves less variable.
Another error: Comparing only diagonal variances and concluding one matrix is smaller without checking all parameter combinations. Matrix efficiency means the difference in covariance matrices is positive semidefinite; equivalently, every linear combination of parameter estimates has no larger asymptotic variance under the efficient weight.

Key takeaway: Under correct specification and the stated regularity conditions, the fixed-weight GMM covariance is the sandwich \(V(W)\). Choosing \(W=\Omega^{-1}\), up to positive scale, minimizes that covariance in the positive-semidefinite ordering, with minimum \((G'\Omega^{-1}G)^{-1}\).

Check Your Understanding

Use the assumptions and covariance comparison in this tutorial to answer the following.

What does \(\Omega\) represent when the moment contributions are serially dependent?
For a fixed positive-definite weight \(W\), write the asymptotic covariance matrix of the GMM estimator.
Why does the term \(D\Omega D'\) establish that the efficient covariance is no larger than the covariance from another fixed weight?
In the unequal-variance example, which moment is downweighted by \(\Omega^{-1}\), and why?
Why do nonsingular fixed weights give the same first-order covariance in an exactly identified model?