Why Use More Than One G-Estimating Moment?
In g-estimation, a candidate blip parameter transforms the observed outcome into a blipped-down outcome. At the true parameter, treatment residuals multiplied by that transformed outcome have mean zero under the identifying assumptions. A single valid moment can identify a scalar parameter, but a history may supply several valid functions of treatment and covariate history. Each gives another moment condition. The resulting system is overidentified when it has more moment equations than unknown parameters.
Having more equations does not mean that all should be treated as equally informative. Moments may be correlated, have different variability, or carry very different amounts of information about the blip parameter. Generalized method of moments (GMM) combines them using a weighting matrix. With suitable assumptions, a covariance-aware choice improves precision compared with an arbitrary combination.
These moments are valid at the true parameter \(\psi_0\) when the structural nested model is correctly specified and the identifying assumptions imply that \(U_t(\psi_0)\) is conditionally mean-independent of \(A_t\) given \(H_t\). Sequential exchangeability is one standard condition used to obtain that implication. Consistency and positivity are also required for the relevant treatment histories. In addition, the moments must have finite second moments for covariance-based weighting, and the moments must distinguish the parameter values being estimated.
From Valid Moments to an Overidentified Estimator
For independent observations \(O_1,\ldots,O_n\), let \(m(O_i,\psi)\) be an \(r\)-vector of moment functions and define its sample average by \(\bar m_n(\psi)=n^{-1}\sum_{i=1}^n m(O_i,\psi)\). If \(\psi\) has \(p\) components and \(r>p\), the sample equations \(\bar m_n(\psi)=0\) will generally have no exact solution. GMM instead selects the parameter that makes the moment vector small under a chosen positive-definite weighting matrix \(W\):
The identity matrix weights moment coordinates equally, but equal coordinate weights are not generally equal information weights: the coordinates can have different scales and covariance. The population moment covariance is \(\Omega=E[m(O,\psi_0)m(O,\psi_0)^\mathsf{T}]\), since \(E[m(O,\psi_0)]=0\). The key optimal choice is \(W=\Omega^{-1}\), when \(\Omega\) is finite and nonsingular. The next derivation explains both what that choice means and why it is optimal.
To derive the variance factor, expand the sample equation around \(\psi_0\). Differentiability gives \(\bar m_n(\hat\psi)=\bar m_n(\psi_0)+D(\hat\psi-\psi_0)+o_p(n^{-1/2})\) for a consistent regular solution. Multiplication by \(a^\mathsf{T}\), followed by solving for the parameter difference, yields
The central limit theorem gives covariance \(\Omega\) for \(\sqrt n\,\bar m_n(\psi_0)\). Thus the leading variance of \(\sqrt n(\hat\psi-\psi_0)\) is the stated factor. For the minimization, use the inner product \(\langle x,y\rangle_\Omega=x^\mathsf{T}\Omega y\). Cauchy–Schwarz applied to \(\Omega^{1/2}a\) and \(\Omega^{-1/2}D\) gives
Rearranging proves the lower bound \(a^\mathsf{T}\Omega a/(a^\mathsf{T}D)^2\geq1/(D^\mathsf{T}\Omega^{-1}D)\). Equality holds when \(\Omega^{1/2}a\) is proportional to \(\Omega^{-1/2}D\), or \(a\) is proportional to \(\Omega^{-1}D\). This proves the optimal-combination claim. \(\square\)
For a vector parameter, write \(D=E[\partial m(O,\psi_0)/\partial\psi^\mathsf{T}]\), an \(r\)-by-\(p\) matrix. Under the corresponding differentiability, rank, and regularity conditions, the GMM criterion with \(W=\Omega^{-1}\) has first-order covariance matrix \((D^\mathsf{T}\Omega^{-1}D)^{-1}\) for \(\sqrt n(\hat\psi-\psi_0)\). This follows by linearizing the criterion’s first-order condition: its leading parameter change is \(-(D^\mathsf{T}WD)^{-1}D^\mathsf{T}W\sqrt n\,\bar m_n(\psi_0)\). Substituting \(W=\Omega^{-1}\) gives the stated covariance. These formulas describe large-sample precision, not a guarantee of better finite-sample behavior.
Worked Examples
Worked Example: Correcting for Correlated, Unequally Noisy Moments
Suppose two valid moments estimate a scalar parameter, with derivative vector \(D=(1,1)^\mathsf{T}\) and covariance matrix
The determinant is \(1(4)-0.5^2=3.75>0\), so this covariance matrix is positive definite. If we give the two moments equal normalized weight, \(a=(0.5,0.5)^\mathsf{T}\), then \(a^\mathsf{T}D=1\). Its variance factor is
For optimal weighting, calculate
The entries of \(\Omega^{-1}D\) sum to \(4/3.75\), so the normalized optimal combination is \(a=(0.875,0.125)^\mathsf{T}\). Its derivative is still one. Directly, \(\Omega a=(0.9375,0.9375)^\mathsf{T}\), and therefore \(a^\mathsf{T}\Omega a=0.9375\). Equivalently, \(D^\mathsf{T}\Omega^{-1}D=4/3.75=16/15\), whose reciprocal is \(15/16=0.9375\). The identity-weighted factor, \(1.5\), is larger. The less variable first moment receives most of the weight, while the covariance is accounted for rather than ignored.
Worked Example: Checking Two Treatment-History Moments
Let \(H\) be binary with \(P(H=0)=P(H=1)=1/2\). Given \(H\), treatment \(A\) is Bernoulli with \(p(H)=1/4\) when \(H=0\) and \(p(H)=3/4\) when \(H=1\). Suppose the true blipped-down outcome is \(U=H+\varepsilon\), where \(\varepsilon\) is independent of \(A,H\), has mean zero and variance one. Set \(R=A-p(H)\) and use \(q_1(H)=1\), \(q_2(H)=H\).
Conditional on either history, \(E[R\mid H]=0\), and \(U\) is independent of \(A\) given \(H\). Hence \(E[q_j(H)RU\mid H]=q_j(H)E[R\mid H]E[U\mid H]=0\) for both moments; averaging over \(H\) gives \(E[q_j(H)RU]=0\). Thus both instruments yield valid population moments in this specified example.
Their covariance matrix can also be checked. Since \(E[R^2\mid H]=p(H)\{1-p(H)\}=3/16\) in both groups, and \(E[U^2\mid H=0]=1\), \(E[U^2\mid H=1]=2\), the entries for \(m_1=RU\) and \(m_2=HRU\) are
The determinant is \((9/32)(3/16)-(3/16)^2=9/512>0\), so the two moments are not exact linear duplicates in this population. That fact alone does not guarantee that both contribute much information about a particular blip parameter: their derivatives with respect to that parameter also matter.
Worked Example: A Valid but Weak Additional Moment
Suppose two uncorrelated valid moments for a scalar parameter have covariance matrix \(\Omega=\operatorname{diag}(1,4)\) and derivative vector \(D=(1,0.2)^\mathsf{T}\). The second moment is less informative in this setup: its variance is 4 and its derivative is only 0.2. The information quantity is
The optimal variance factor is \(1/1.01=100/101\), approximately \(0.9901\), compared with factor \(1\) from the first moment alone. The normalized optimal weights are proportional to \(\Omega^{-1}D=(1,0.05)^\mathsf{T}\), so they are \((1/1.05,0.05/1.05)^\mathsf{T}\). The extra moment gives a small theoretical gain, not a dramatic one. If its derivative were zero and it were uncorrelated with the informative moments, it would add no information for this parameter; otherwise, it can affect precision through its covariance with them.
Choosing Weights and Recognizing Failure Modes
The optimality result is a statement within a correctly specified moment model. It does not make an invalid instrument valid. It also assumes that the covariance matrix and parameter sensitivity are appropriately represented. In practice, \(\Omega\) is usually estimated, and the weight matrix may be formed from preliminary estimates. If a propensity or other nuisance model is estimated, the relevant large-sample covariance must account for that estimation; using a covariance formula as if the nuisance quantities were known can give incorrect weights and uncertainty calculations. See the earlier material on g-estimation with known and estimated propensities for that distinction.
Several checks help identify problems before interpreting a precision gain:
- Check the moment definitions. Each instrument must be a function of the appropriate observed history, and the treatment residual must use the conditional treatment probability for that history.
- Check scale and redundancy. Very different scales can make identity weighting arbitrary. Nearly redundant moments make \(\Omega\) ill-conditioned, so its estimated inverse can be unstable.
- Check sensitivity. Compare estimates under a few sensible weighting choices and inspect whether one moment dominates the result. A large change can signal weak information, unstable weights, or moment incompatibility.
- Separate validity from precision. A moment can have a small estimated variance yet be invalid under the assumed structural model or exchangeability conditions. Optimal weighting cannot repair that failure.
Exactly identified g-estimation uses just enough independent moments to identify the parameter and does not require choosing among excess equations. Overidentified GMM can gain precision by combining additional valid moments. Identity weighting is simple but generally not efficient; inverse-covariance weighting is asymptotically optimal under the stated conditions. In finite samples, estimating and inverting a noisy covariance matrix can erase theoretical gains, especially when many moments are nearly dependent. The later material on asymptotic variance develops uncertainty calculations further; the present comparison concerns how the choice of moments and their weights affects information.
Check Your Understanding
Answer each question, then check the short solutions below.
- What makes a g-estimating system overidentified, and why may its sample equations have no exact common root?
- For scalar \(\psi\), explain what \(D\) and \(\Omega\) represent in the optimal-combination result.
- Suppose two uncorrelated moments have derivatives \(D=(1,1)^\mathsf{T}\) and variances 1 and 9. What are the normalized optimal weights and the optimal variance factor?
- Why does adding a valid moment with derivative zero fail to improve precision for a scalar parameter when it is uncorrelated with the informative moments?
- Name one reason estimated optimal weights may perform poorly in a finite sample.
Solutions. (1) There are more moment equations than unknown parameters; sample averages fluctuate, so the equations need not share a root. GMM minimizes a weighted discrepancy instead. (2) \(D\) is the derivative of the population moment vector with respect to the parameter at the truth; \(\Omega\) is the covariance matrix of the valid moment vector. (3) \(\Omega=\operatorname{diag}(1,9)\), so \(\Omega^{-1}D=(1,1/9)^\mathsf{T}\). Normalizing gives weights \((9/10,1/10)^\mathsf{T}\). The information is \(1+1/9=10/9\), so the variance factor is \(9/10\). (4) The moment adds no derivative information, so it cannot increase \(D^\mathsf{T}\Omega^{-1}D\) when uncorrelated with the informative moments. (5) For example, a nearly singular estimated covariance matrix makes its inverse unstable, or nuisance estimation may be ignored when estimating the covariance.