Collapsing And Restricting Dynamic-GMM Instruments

Why Reshape the Instrument Set?

In difference-GMM, lagged levels of an outcome can instrument its first difference when the model’s moment conditions make those levels orthogonal to the differenced error. As the number of periods grows, using every eligible lag at every equation date can create a large set of instruments. Two direct ways to control that set are to collapse the instrument matrix and to restrict the eligible lag distances.

These operations are related but not interchangeable. Collapsing combines time-specific instruments that share a lag distance into a common column. Restricting a lag set removes instruments at selected distances, and therefore removes their associated moments. Neither operation, by itself, proves that the retained instruments are valid or that the estimator is well behaved. The goal is to state clearly which moment conditions the estimator uses, rather than to maximize the instrument count.

Setup and Instrument-Matrix Definitions

Consider the dynamic-panel model \(y_{it}=\rho y_{i,t-1}+\eta_i+\varepsilon_{it}\), where \(i\) indexes units and \(t\) indexes time. First differencing removes the time-invariant component \(\eta_i\), giving \(\Delta y_{it}=\rho\Delta y_{i,t-1}+\Delta\varepsilon_{it}\). For this tutorial, focus on the lagged dependent variable as the endogenous regressor and on its lagged levels as instruments.

For an equation dated \(t\), define the lag distance \(d=t-j\) for an instrument \(y_{ij}\). Under the standard difference-GMM timing conditions, \(y_{ij}\) is eligible for the differenced equation at \(t\) when \(j\leq t-2\), or equivalently \(d\geq2\). With data beginning at period 1, the available distances also satisfy \(d\leq t-1\). These are validity conditions, not just a convenient way to index columns: the chosen levels must be orthogonal to \(\Delta\varepsilon_{it}\).

For example, a lag-distance set \(2\) through \(L\) means that an equation at \(t\) can use \(y_{i,t-2},\ldots,y_{i,t-L}\), subject to the observed panel boundary. For unbalanced panels, availability is determined separately for each unit and date. Instrument counts below refer to the balanced, single-endogenous-regressor case; adding other instrumented regressors or included exogenous instruments changes the total.

Worked Example: Uncollapsed and Collapsed Matrices

Counting Instruments and Restricting Lag Distances

With periods 1 through \(T\), equations dated \(3,\ldots,T\), and all eligible lagged levels, the uncollapsed count is the sum of the number of available lags at each date. At date \(t\), this number is \(t-2\). The collapsed matrix instead has one column for each available distance, \(2,\ldots,T-1\).

If the retained lag distances are \(a,\ldots,b\), where \(2\leq a\leq b\), then the collapsed count is \(b-a+1\), provided each distance is available in at least one equation. The uncollapsed count is date-specific: at equation date \(t\), retain the distances in \(a,\ldots,b\) that also satisfy \(d\leq t-1\). Thus early equations may contribute fewer columns than later equations. Restriction and collapse can be combined: first select the lag distances, then use one collapsed column for each selected distance.

What Happens to the Moment Conditions?

Let \(g_{t,d}(\rho)\) denote the sample moment contribution for an instrument at distance \(d\) in the equation dated \(t\). In an uncollapsed specification, the date-specific moment conditions are kept as separate components. Collapsing distance \(d\) replaces those components with their pooled sum, over dates where that distance is available:

This expression describes the moment aggregation induced by a shared column. It is not generally equivalent to retaining every date-specific moment: several separate restrictions have been replaced by one pooled restriction. By contrast, restricting the lag window removes the moment components associated with omitted distances altogether. The retained moments still need their original orthogonality assumptions.

Worked Example: Comparing Illustrative GMM Estimates

Construction and Reporting

A transparent implementation begins by defining the equation dates and the eligible lag distances, then mapping each instrument to its column. A useful workflow is:

The starting assumptions matter. For the lagged outcome, the standard moment conditions require that the eligible lagged levels be orthogonal to the differenced disturbance. If serial dependence or the assumed timing structure undermines that orthogonality, fewer columns do not repair the invalid moments that remain. Conversely, a lag restriction can remove some questionable moments only when the retained distances are justified; it is not an automatic validity test.

Common Mistake and Why the Choice Matters

A common mistake is to say that collapsing “keeps all the same moment conditions but writes them more compactly.” That is generally false. Separate uncollapsed columns encode separate date-specific moments; a collapsed column pools their contributions and imposes a different, lower-dimensional set of restrictions. Another mistake is to assume that reducing the instrument count guarantees more reliable estimates. Reduction addresses the size and structure of the instrument set, not every source of misspecification or weak identification.

For interpretation, report enough detail to make the instrument construction reproducible: the lag range, whether the matrix is collapsed, the total number of instruments, and which regressor blocks contribute instruments. When estimates change across constructions, that difference is informative about dependence on the chosen moments, but it is not itself proof that one result is correct. The choice should be substantively defensible and made with the underlying orthogonality conditions in view.