Evaluating Difference-In-Hansen Tests For System GMM

Why Test the Level Moments Separately?

System GMM combines moment conditions from a differenced equation with additional moment conditions for the equation in levels. Those added moments can improve estimation when the lagged dependent variable is persistent, but they rely on restrictions beyond those needed for the differenced equation. A test of the added moments is therefore useful: it asks whether the extra restrictions are consistent with the data, rather than asking whether every instrument in the system is valid.

A difference-in-Hansen test answers this subset question by comparing the overidentifying-restriction statistic for the full system with the corresponding statistic when the level-moment subset is excluded. Its interpretation depends on the subset being tested, the moments maintained for comparison, and a valid nested-GMM calculation.

System-GMM Moments and the Test Hypothesis

For orientation, consider a dynamic panel model such as \(y_{it}=\rho y_{i,t-1}+\eta_i+v_{it}\), where \(i\) indexes units, \(t\) indexes time, \(\eta_i\) is a time-invariant unit effect, and \(v_{it}\) is an idiosyncratic error. Differencing removes \(\eta_i\). Under appropriate restrictions on the errors, lagged levels can then provide instruments for the differenced equation. System GMM also uses an equation in levels, with additional instruments such as lagged differences.

Those level-equation moments require additional assumptions about the initial conditions or changes in the series. In the simple illustration, a moment might assert that a lagged difference is uncorrelated with the unit effect and the relevant equation error. The exact set depends on which regressors are endogenous, predetermined, or exogenous and on the instrument design. The key point for the test is that these are additional restrictions: their validity does not follow merely from the validity of the differenced-equation moments.

Let \(g_D(\theta)\) denote the sample moments maintained from the differenced equation and \(g_L(\theta)\) the additional sample moments from the levels equation. The full system uses both sets. The subset test asks whether the population level moments are zero, while continuing to rely on the validity of the maintained differenced moments.

The test requires a correctly specified nested comparison. The restricted specification omits the level-moment subset being tested, while retaining the intended comparison moments. The sample, outcome equation, regressor specification, and instrument construction for the maintained moments must be aligned across the two calculations. Inference also requires the usual GMM regularity conditions, including valid maintained moments under the null, an appropriate covariance estimate for the moments, and sufficient rank. The number of moments should be controlled: instrument proliferation can weaken Hansen-based diagnostics and undermine their practical informativeness.

Statistic, Degrees of Freedom, and Calibration

Write \(J_{\mathrm{full}}\) for the Hansen overidentification statistic from the full system and \(J_{\mathrm{restricted}}\) for the corresponding statistic from the nested specification that excludes the tested level moments. With a compatible difference-in-Hansen construction, the subset statistic is

If the tested subset contributes \(q\) independent moment restrictions, then under \(H_0\), \(C\) has an asymptotic chi-square distribution with \(q\) degrees of freedom. Equivalently, if the full and restricted Hansen statistics have degrees of freedom \(df_{\mathrm{full}}\) and \(df_{\mathrm{restricted}}\), then \(q=df_{\mathrm{full}}-df_{\mathrm{restricted}}\), when the two specifications are nested and the relevant ranks are as expected. The p-value is the upper-tail probability \(P(\chi^2_q\ge C_{\mathrm{obs}})\).

This calibration is asymptotic, not an exact finite-sample guarantee. The simple subtraction is meaningful only when the two statistics are computed using a valid, compatible GMM procedure. In robust two-step GMM, the implementation must account appropriately for the covariance of the moment conditions and the restricted estimation. Software may report a dedicated difference-in-Hansen statistic rather than asking the user to subtract arbitrary Hansen statistics from unrelated runs. Follow the software’s definition and documentation, and verify that the tested subset and maintained set match the intended question.

Why the Difference Has a Chi-Square Reference Distribution

The reasoning is easiest to see after expressing the sample moments in units of their covariance. Near the true parameter, the efficient GMM criterion is a quadratic measure of the discrepancy between sample moments and their population restrictions. Estimating the model parameters uses up directions in that moment space. The restricted test checks the remaining overidentifying directions among the maintained moments; the full test checks those directions plus the additional directions contributed by the level moments.

Under the null and regularity conditions, the standardized sample-moment discrepancy is asymptotically normal. In a nested, efficiently weighted comparison, the additional directions are separated from the maintained overidentifying directions. The quadratic contribution from the additional \(q\) independent directions is therefore asymptotically a sum of \(q\) squared independent standard normal variables, which is a chi-square variable with \(q\) degrees of freedom.

The degrees of freedom count independent restrictions added by the tested subset, not the number of time periods and not necessarily the raw number of instrument columns. If some moments are redundant or linearly dependent, the effective rank can be smaller than the nominal count. The chi-square result also explains why a standard subset statistic should be nonnegative in the ideal nested construction: it is a quadratic contribution. In finite samples, however, a difference formed from separately estimated or incompatible statistics can be negative. A negative result is a warning to check the calculation, not evidence that the level moments are especially valid.

Reading the Result in Context

A difference-in-Hansen p-value is most useful when reported with the subset tested and its degrees of freedom. “The Hansen test passed” is not a sufficient description of what a subset test evaluated. For system GMM, state whether the tested subset consists of all additional level-equation moments or a narrower block, such as moments associated with a particular group of regressors. Then state the conditional nature of the conclusion.

The overall Hansen test and the difference-in-Hansen test answer distinct questions. The overall test concerns the joint set of overidentifying restrictions in the full instrument set. The subset test focuses on the incremental restrictions in the selected block, conditional on the maintained block. A rejection by one need not imply rejection by the other. Conversely, a non-rejection of the subset does not certify the system as a whole.

These tests are diagnostics, not automatic instrument-selection rules. A rejection may reflect invalid level moments, but it can also arise when other aspects of the model or maintained moments are misspecified. A non-rejection may have limited power, particularly when the instrument count is large relative to the information in the panel. Interpret the result alongside the moment assumptions, instrument design, and other relevant specification checks.

Common Mistakes

• Overstating a rejection. “The difference-in-Hansen test rejects the instruments” is too broad when only level moments were tested. Name the level-moment subset and say the rejection is conditional on the maintained moments.
• Treating non-rejection as validation. A large p-value is not proof of validity. The test may have limited power, and its conclusion depends on the maintained assumptions.
• Using the wrong degrees of freedom. Use the number of independent restrictions added by the tested subset, or the difference in degrees of freedom for the valid nested statistics. Do not use the full-model degrees of freedom alone.
• Subtracting unrelated Hansen statistics. The statistic is not automatically valid just because two reported J statistics can be subtracted. The nested specifications and weighting or covariance treatment must support the difference-in-Hansen construction.
• Ignoring instrument proliferation. A large instrument set can weaken the usefulness of Hansen diagnostics. Keep the instrument design interpretable and consider sensitivity to reasonable instrument restrictions.