Level-Equation Moments In System GMM

Why Add Moments for the Level Equation?

In a dynamic panel, differencing removes a time-invariant unit effect, but it can leave lagged outcome levels weakly informative about changes when the outcome is highly persistent. System GMM addresses this by combining the differenced equation with the equation in levels. The extra level-equation instruments are lagged first differences of the outcome. Their validity, however, does not follow just from the assumptions that justify the difference-equation instruments. It requires additional restrictions on how the panel began.

This tutorial focuses on those restrictions: what the level-equation moments say, how mean stationarity and initial conditions can support them, and what they do—and do not—guarantee. They are substantive assumptions about the data-generating process, not an automatic consequence of using GMM.

Dynamic Model and the Additional Moment

Consider the first-order dynamic panel model

Here \(y_{it}\) is the outcome for unit \(i\) at time \(t\), \(\alpha\) is the common persistence parameter, \(\eta_i\) is a time-invariant unit effect, and \(u_{it}\) is a time-varying innovation. Assume \(|\alpha|<1\), the innovations have conditional mean zero, and they are serially uncorrelated and orthogonal to the unit effects. The initial outcome \(y_{i1}\) may be related to \(\eta_i\); that relationship is central to the question of whether the extra moments hold.

The level equation has composite error \(\eta_i+u_{it}\). The proposed instrument for that equation is the previous change in the outcome, \(\Delta y_{i,t-1}=y_{i,t-1}-y_{i,t-2}\). For panels observed from \(t=1\), this instrument is available for level equations from \(t=3\) onward.

The second orthogonality condition is supported by the usual timing assumptions: the current innovation is uncorrelated with past outcomes and past innovations. The first is different. A change in the outcome can be related to the unit effect if the outcome is adjusting from an initial value that depends on that effect. Thus, the extra level-equation moment imposes a restriction about the initial conditions and their evolution.

Difference GMM uses moments for the differenced equation, such as moments that make suitable lagged levels orthogonal to differenced innovations. Those restrictions do not, by themselves, imply that changes are orthogonal to \(\eta_i\). System GMM adds the level equation and its moments only when the additional orthogonality is defensible.

Connection to Initial Conditions and Mean Stationarity

For a fixed unit effect \(\eta_i\), the long-run mean of the outcome under the model is \(\eta_i/(1-\alpha)\). One way to motivate the additional restriction is to suppose that the initial outcome is already at this conditional mean, in expectation:

With mean-zero innovations conditional on \(\eta_i\), the expected outcome remains at that value. In particular, the expected change has conditional mean zero. This is a mean-stationarity condition: the process does not begin, on average, away from its long-run mean in a way that makes its changes systematically related to the unit effect. It does not require every realized outcome or change to be constant.

A useful formulation for the moment restrictions is directly in terms of changes:

Under the model’s innovation assumptions, this initial orthogonality propagates forward. To see why, subtract the equation at \(t-1\) from the equation at \(t\):

Multiplying by \(\eta_i\) and taking expectations gives

If each innovation is orthogonal to \(\eta_i\), the last two terms are zero. Therefore, if \(E[\Delta y_{i2}\eta_i]=0\), then \(E[\Delta y_{it}\eta_i]=0\) for later \(t\) by repeated application of this equality. Together with the timing assumption that \(E[\Delta y_{i,t-1}u_{it}]=0\), this yields the level-equation moment.

This argument is a sufficient justification, not a claim that mean stationarity is the only possible data-generating process consistent with the moments. The precise requirement for the instrument is its orthogonality to the level error. Researchers must decide whether the implied initial-condition restriction is plausible in their application.

Why Mean Stationarity Is Substantive

Suppose instead that the initial outcome is systematically below its conditional long-run mean for units with large positive \(\eta_i\). Those units will tend to rise toward their long-run means. Their changes will then be positively related to \(\eta_i\), making \(E[\Delta y_{i,t-1}\eta_i]\) nonzero. The lagged change would carry information about the unit effect in the level equation’s error, so treating it as an instrument would violate the required moment condition.

Mean stationarity is not the same as strict stationarity of every part of the process, nor does it say that outcomes do not change over time. It restricts the relation between initial conditions, subsequent changes, and unit effects in a way that supports the orthogonality condition. The model can have substantial individual variation and nonzero innovations while satisfying a zero expected change conditional on the unit effect.

Reading the Moment as an Instrument Restriction

For any valid instrument \(z_{it}\), the defining population requirement is that its expected product with the equation error is zero. Here \(z_{it}=\Delta y_{i,t-1}\) and the level error is \(\eta_i+u_{it}\). The moment can be written as

The equality on the right separates two sources of validity. The first is the additional restriction involving the unit effect; the second comes from the timing of the innovation. A zero sum could, as a mathematical matter, occur through offsetting nonzero terms, but standard justification establishes each orthogonality separately rather than relying on accidental cancellation.

A sample analogue for a given time period is the average of the instrument-residual products across units. For a candidate parameter value \(\alpha\), the level residual is \(y_{it}-\alpha y_{i,t-1}\). It equals \(\eta_i+u_{it}\) at the true parameter, although the components of that composite error need not be separately observed. In estimation, GMM uses the sample moments across the available instruments and periods; the construction of the stacked estimator is a separate step.

Common Mistakes and Practical Checks

• Assuming differencing validates the level instrument. Differencing eliminates \(\eta_i\) from the transformed equation, but that does not make changes orthogonal to \(\eta_i\) in the level equation. Check the additional restriction directly as a substantive assumption.
• Equating mean stationarity with no dynamics. Mean stationarity concerns the conditional mean and its relationship to unit effects. The outcome may still vary substantially because of innovations and unit-specific realizations.
• Using a plausible lag without checking its timing. The instrument must be dated before the level-equation innovation and must be available from the observed panel. With observations beginning at \(t=1\), \(\Delta y_{i,t-1}\) requires \(t\geq3\).
• Treating a small sample moment as validation. A sample average near zero is not evidence that the population moment is guaranteed. Moment-based diagnostics can be informative, but their interpretation depends on the assumptions and the available information.

The practical question is therefore not merely whether lagged differences can be entered as instruments. It is whether the process plausibly began in a condition that makes changes unrelated to persistent unit effects, and whether the innovation timing assumptions are credible. When that justification is weak, the additional moments may be invalid even if they would improve numerical instrument relevance.