Why Moment Selection Is a Modeling Decision
GMM can use several restrictions at once, but having more candidate moment conditions does not automatically make an analysis better. A moment can add identifying information and improve precision if it is valid. If it is invalid, including it can instead change the estimand or undermine consistency. Even among valid moments, a weak or noisy condition may contribute little useful information. The choice is therefore not simply “use all available moments” or “use the largest set.”
Moment selection combines economic or scientific reasoning about which restrictions are credible with statistical reasoning about identification and precision. It also requires care about how the choice was made. Standard GMM inference generally treats the moment set as fixed; choosing that set after looking at the same data can make reported tests and intervals unreliable.
Validity, Identification, and Precision
Write the \(q\)-vector of selected moment functions as \(g_{i,S}(\theta)\). Its sample average is \(\bar g_S(\theta)=n^{-1}\sum_{i=1}^n g_{i,S}(\theta)\). GMM chooses parameter values that make this vector small according to a weight matrix. The population derivative matrix is \(D_S=E[\partial g_{i,S}(\theta_0)/\partial\theta']\), and the long-run covariance matrix of the moments is \(\Omega_S\). For independent observations, \(\Omega_S\) is the covariance matrix of \(g_{i,S}(\theta_0)\); for dependent data, it incorporates serial covariances as well.
Under the usual regularity conditions for GMM, a fixed valid set identifies a \(k\)-dimensional parameter locally when \(D_S\) has column rank \(k\). With efficient weighting, the asymptotic covariance matrix of the estimator, expressed as the covariance of \(\sqrt n(\hat\theta_S-\theta_0)\), is
This expression shows why simply counting moments is not enough. A moment is useful when it adds information after accounting for its variability and its covariance with the other moments. A condition that is nearly redundant may add little precision; one with a small derivative relative to its noise may add very little identifying information. The formula presumes the included moments are valid, the relevant matrices are well behaved, and the selected set is treated as fixed. If validity fails, the formula around \(\theta_0\) does not justify the usual interpretation.
A Short Derivation: The Value of an Additional Valid Moment
For one scalar parameter, consider combining the moments linearly. Let \(a\) be a vector of weights. The combination \(a'\bar g_S(\theta)\) has derivative \(a'D_S\). Normalize it so that \(a'D_S=1\); then its asymptotic variance is \(a'\Omega_Sa\). The most precise normalized combination solves a constrained minimization problem:
Using a Lagrange multiplier \(\lambda\), the first-order condition is \(2\Omega_Sa-\lambda D_S=0\), so \(a=(\lambda/2)\Omega_S^{-1}D_S\). Applying the constraint gives \((\lambda/2)D_S'\Omega_S^{-1}D_S=1\). Therefore
Thus, with valid moments and correctly accounted covariance, efficient GMM uses the informative combination rather than giving every condition equal influence. The result does not make an invalid moment safe: optimization of precision is meaningful only over moment conditions that are defensible as restrictions of the model.
Worked Example: Screening Candidate Instrument Moments
Worked Example: Screening Candidate Instrument Moments
Consider a hypothetical structural equation \(y_i=\beta_0x_i+u_i\), with three candidate instrument moments \(E[z_ju]=0\), \(j=1,2,3\). Suppose the research design gives a substantive basis for treating \(z_1\) and \(z_2\) as uncorrelated with the structural error, but not \(z_3\). In a hypothetical population, the derivative magnitudes \(|E[z_jx]|\) are 1.00, 0.15, and 0.80, respectively. The corresponding moment variances are 1.00, 1.00, and 1.00. The sample size is \(n=400\).
For a single scalar moment with derivative \(d\) and variance \(\Omega\), the asymptotic variance of the estimator, on the \(\sqrt n\) scale, is \(\Omega/d^2\). Using \(z_1\) alone gives
Using \(z_2\) alone gives
Both moments are assumed valid here, but \(z_2\) has a much smaller derivative relative to its variability. It is weakly informative in this comparison. The stated standard errors are asymptotic approximations, not guarantees about finite-sample performance. The third moment has a reasonably large derivative, but the design does not justify its zero-expectation restriction. Its apparent information cannot compensate for its lack of a validity argument. On these assumptions, \(z_1\) is the defensible choice; whether to add \(z_2\) depends on its incremental information and the reliability of the approximations, not on the raw number of moments.
Comparing Precision With More Than One Valid Moment
When candidate moments are all defensible, their joint covariance matters. The next calculation illustrates the gain from combining an additional valid condition. It assumes a fixed moment set, independent observations, and the stated population derivatives and covariance; it is an asymptotic comparison, not an assertion that sample covariance estimates are known exactly.
Worked Example: Combining Two Valid Moments
Suppose a scalar parameter has two valid moments with derivative vector \(D=(1,0.8)'\) and long-run covariance matrix \(\Omega=\begin{pmatrix}4&0\\0&9\end{pmatrix}\). The moments are uncorrelated in this hypothetical population. With the first moment alone, the asymptotic variance is \(4/1^2=4\). With the second alone, it is \(9/0.8^2=14.0625\).
For both moments, efficient GMM gives
At \(n=100\), the asymptotic variance of the estimator is approximately \(3.11418685/100=0.0311419\), and its standard error is \(\sqrt{0.0311419}\approx0.1765\). For comparison, the first moment alone gives standard error \(\sqrt{4/100}=0.200\); the second alone gives \(\sqrt{14.0625/100}=0.375\). The combined set is more precise than either single moment because the second condition contributes some independent information. The increase in precision is modest relative to what the first moment already supplies. This calculation would not justify adding the second moment if its validity were doubtful.
What Diagnostic Tests Can—and Cannot—Decide
An overidentification test, such as Hansen’s \(J\) test under its standard assumptions, can assess whether the sample evidence is broadly compatible with the restrictions in an overidentified model. A rejection is evidence against the joint maintained restrictions, but it does not identify which condition is wrong. A non-rejection is not proof that every moment is valid: tests can have limited power, and different failures can offset one another in the test statistic.
It is tempting to try candidate sets, retain the one with a non-significant \(J\) statistic, and then report its ordinary GMM inference as though the set had been fixed in advance. This procedure uses the same data both to select the restrictions and to estimate and test the model. The selected set is consequently related to sample noise. The usual reference distribution for a test conducted as if that set were fixed need not describe the full selection procedure. Repeatedly trying alternatives also creates a multiple-selection problem.
Worked Example: Why Selection on the Estimate Changes Inference
Worked Example: Why Selection on the Estimate Changes Inference
Suppose two candidate moment sets each produce an unbiased estimator of a scalar parameter \(\theta\), and, for illustration, their estimates are independent with distributions \(X_1,X_2\sim N(\theta,1)\). A researcher inspects both estimates and reports the smaller one, perhaps because a smaller estimated effect is more convenient for the argument. If the observed estimates are 10.4 and 9.6 units, the reported estimate is 9.6 units.
Even though each estimate is unbiased before selection, the selected minimum is not. Use the identity \(\min(X_1,X_2)=(X_1+X_2-|X_1-X_2|)/2\). Since \(X_1-X_2\sim N(0,2)\), its expected absolute value is \(2/\sqrt{\pi}\). Therefore
The selected estimator is biased downward by about 0.5642 units in this example, despite both original estimators being unbiased with standard error 1. This simplified calculation is not a model for every moment-selection procedure. It demonstrates the general issue: conditional on being selected, an estimate may have a distribution different from its unconditional distribution. Likewise, choosing a moment set because it produces a favorable coefficient, standard error, or test result can make conventional post-selection inference misleading.
Principled Strategies for Choosing Moments
A defensible strategy separates the scientific question of validity from the statistical question of efficiency. Practical steps include:
- Start with the model and design. State why each proposed restriction should have expectation zero, using the substantive assumptions that support it. Do not treat a convenient instrument or covariate as valid merely because it is available.
- Check local identification. Consider whether the derivative matrix for the proposed set has the required rank and whether the moments provide meaningful variation in the directions relevant to the parameter.
- Compare incremental information. For plausible valid sets, consider the derivative and covariance structure together. A large moment count is not itself a precision criterion; noisy or redundant moments may contribute little.
- Specify the main set before examining results where feasible. Use secondary candidate sets as sensitivity analyses, and report how the estimates and conclusions change rather than concealing the alternatives tried.
- If selection must use the data, account for that selection. Independent data for choosing and estimating moments can help separate the decisions, provided the split and analysis are designed appropriately. Alternatively, use inferential methods designed for the actual selection procedure. In either case, describe the selection rule and its limitations.
These strategies do not turn a statistical diagnostic into proof of validity. They make the assumptions and tradeoffs visible, and they reduce the risk of treating a data-dependent choice as if it were fixed in advance.
Common Mistakes
- “More moments must mean more precision.” This ignores validity, redundancy, and covariance. Add a condition only when it is credible and contributes useful information.
- “A non-rejected \(J\) test proves the moments are valid.” Non-rejection is limited evidence, not confirmation of every restriction.
- “The strongest-looking instrument is automatically best.” Relevance alone does not establish the zero-correlation restriction needed for validity.
- “I can select the best-looking set and use fixed-set standard errors.” Selection based on the same data can change the sampling distribution. Report and account for the selection process.
Check Your Understanding
Try these questions before reviewing the explanations in the tutorial.
- What is the difference between a valid moment condition and an identifying moment condition?
- For one scalar moment with derivative \(d=0.2\) and variance \(\Omega=1\), what is its asymptotic variance on the \(\sqrt n\) scale? What does the result suggest about weak information?
- Why does adding a valid but highly redundant moment condition not necessarily produce a substantial precision gain?
- What can a non-rejection of an overidentification test tell you, and what can it not establish?
- Give one reason that choosing a moment set after inspecting the same data can invalidate conventional fixed-set inference.