Why Lag Depth Is a Modeling Choice
Difference GMM uses lagged levels of panel variables as internal instruments for equations in first differences. Choosing which lags to include is consequential: very recent lags may violate the required moment conditions, while very distant lags may carry little information about the differenced regressors. Adding many lag columns can also make the instrument set large relative to the number of panel units. A single preferred lag window can therefore hide important dependence on modeling choices.
A sensitivity analysis compares estimates and diagnostics across a small set of substantively defensible starting and ending lags. It is not a search for the window that produces the most attractive coefficient or p-value. The aim is to learn whether the substantive conclusion is stable and whether the instruments used in each candidate specification appear relevant and compatible with the maintained assumptions.
The Moment Conditions and Candidate Windows
Consider the dynamic panel model
where \(i\) indexes units, \(t\) indexes time, \(\eta_i\) is a time-invariant unit effect, and \(\varepsilon_{it}\) is an idiosyncratic error. First differencing removes \(\eta_i\), giving a regression that includes \(\Delta y_{i,t-1}\) and \(\Delta x_{it}\). Because \(\Delta y_{i,t-1}\) contains \(\varepsilon_{i,t-1}\), it is generally correlated with the differenced error. Lagged levels provide instruments only under suitable timing and error assumptions.
For illustration, suppose the idiosyncratic errors are serially uncorrelated and the initial conditions and timing of \(y\) imply that \(y_{i,t-2}\) and earlier levels are uncorrelated with both \(\varepsilon_{it}\) and \(\varepsilon_{i,t-1}\). Then the moment condition for the lagged dependent variable is
The condition follows because \(y_{i,t-s}\) is uncorrelated with each error in the difference when \(s\geq2\); linearity of expectation then gives a zero covariance with their difference. The same timing argument does not generally justify \(y_{i,t-1}\), which can be related to \(\varepsilon_{i,t-1}\). For an endogenous regressor \(x_{it}\), its own timing assumptions must be used to determine which lag distances are admissible; do not automatically copy the dependent-variable window.
A window is defensible only when its lags satisfy the relevant moment conditions. The diagnostics can reveal incompatibility or weak information, but they cannot make an invalid lag valid. In particular, serial correlation in \(\varepsilon_{it}\) can change which lag distances are orthogonal to \(\Delta\varepsilon_{it}\). Candidate windows should therefore be chosen from the model’s timing assumptions and design, before comparing results.
Worked Example: Counting Instruments Under Different Windows
Suppose a hypothetical balanced panel has observations at \(t=1,\ldots,6\), and one lagged dependent variable is instrumented in the differenced equations for \(t=3,4,5,6\). Consider uncollapsed instruments, so each available lag in each equation is a separate instrument. Under the basic validity assumption above, a candidate window must start at lag 2 or later.
With no upper restriction beyond the available history, the lag distances in the four equations are \(2\) at \(t=3\); \(2,3\) at \(t=4\); \(2,3,4\) at \(t=5\); and \(2,3,4,5\) at \(t=6\). The total is \(1+2+3+4=10\) instrument columns.
Restricting the window to \([2,3]\) gives counts \(1,2,2,2\), for \(1+2+2+2=7\) columns. If instead instruments are collapsed by lag distance, that same window has two columns, one for lag 2 and one for lag 3. These counts describe this single-variable illustration only; additional endogenous variables and included instruments add columns. The calculation shows why both lag depth and instrument construction matter when assessing instrument-set size.
What to Compare Across Specifications
A useful comparison changes the lag window while holding other choices fixed as far as possible: the estimation sample, included regressors, time effects, instrument-collapsing decision, and GMM estimator. If changing the window also changes the sample, a coefficient change may reflect either the instruments or the observations. State any unavoidable differences explicitly. It is often informative to compare a few nested windows and a narrower window that excludes potentially weak distant lags.
Consider four kinds of evidence, not a single pass/fail rule:
- Relevance: assess whether the instruments contain meaningful information about the differenced endogenous regressors. Sample correlations or first-stage summaries can be useful descriptive checks, but there is no universal cutoff that certifies strong internal instruments. Persistent outcomes can make distant levels weak predictors of changes.
- Serial correlation: the Arellano–Bond tests assess serial correlation in differenced residuals. First-order correlation is commonly expected after differencing even when the level errors are serially uncorrelated. Evidence of second-order correlation is a warning for the usual lag-2 moment conditions.
- Overidentifying restrictions: Sargan or Hansen tests assess joint compatibility of overidentifying moment restrictions with the data and maintained assumptions. A small p-value is evidence against those restrictions; a large p-value is not proof that they are true. Instrument proliferation can also weaken the informativeness of the Hansen test.
- Estimate stability: compare coefficient estimates, standard errors, and the substantive conclusion. Material changes across defensible windows mean the result depends on instrument choice, even if no diagnostic rejects at a chosen significance level.
Relevance is not established merely because an instrument is valid, and validity is not established merely because an instrument predicts a regressor. The moment assumptions and the amount of identifying information are distinct concerns. Likewise, a coefficient moving within its individual confidence intervals is not, by itself, a formal test that the estimates from two specifications are equal: estimates from the same data are generally dependent.
Worked Example: Comparing Estimates Across Lag Windows
Suppose three collapsed difference-GMM specifications are fit to the same hypothetical panel sample, with the same regressors and estimator. The dependent-variable instrument windows are \([2,3]\), \([2,5]\), and \([3,4]\). Suppose the reported estimates of \(\rho\) and standard errors are respectively \(0.62\) and \(0.10\), \(0.48\) and \(0.08\), and \(0.39\) and \(0.14\). For this single instrumented variable, the collapsed lag windows contribute two, four, and two lag columns.
Using \([2,3]\) as a prespecified reference, the absolute relative change for \([2,5]\) is
Thus the point estimate is about 22.6% lower than the reference estimate. For \([3,4]\), the corresponding change is
The coefficient estimates range from \(0.39\) to \(0.62\), so the estimated persistence is not stable across these candidates. This calculation is a descriptive sensitivity measure, not a hypothesis test. Since \(y\) and its lag are measured in the same units, \(\rho\) is dimensionless; if the substantive claim is that persistence is positive, all three point estimates support that direction, but their magnitudes differ substantially. The standard errors should be reported alongside the estimates, not used as a shortcut for testing equality across specifications.
A Short Decision Process
Use the model’s timing assumptions to rule out lags that are not valid instruments. For example, lag 2 is the conventional earliest candidate for the lagged dependent variable under serially uncorrelated level errors.
Compare a baseline with a narrower and/or wider defensible window. State the rationale for the endpoints, and consider collapsing or restricting instruments to control their number.
Hold the sample and other specification choices fixed when possible. Record any differences that cannot be held fixed.
Examine relevance, serial-correlation tests, overidentification tests, and coefficient stability. Report concerns even when a test does not reject.
Worked Example: Interpreting Diagnostics Together
Suppose the same hypothetical panel yields the following output for two collapsed windows. The Arellano–Bond \(p\)-values are for the first- and second-order serial-correlation tests, respectively. The partial \(R^2\) is a descriptive first-stage summary for \(\Delta y_{i,t-1}\), not a universal strength test.
| Window | Partial \(R^2\) | AR(1) \(p\) | AR(2) \(p\) | Hansen \(p\) | \(\hat{\rho}\) |
|---|---|---|---|---|---|
| [2,3] | 0.21 | 0.01 | 0.31 | 0.24 | 0.62 |
| [2,5] | 0.27 | 0.02 | 0.27 | 0.08 | 0.48 |
At \(\alpha=0.05\), both specifications reject no first-order serial correlation, which is commonly expected for differenced errors, and neither rejects no second-order serial correlation. Neither Hansen test rejects its null of joint validity of the overidentifying restrictions: the p-values are \(0.24\) and \(0.08\). The latter is closer to 0.05 and warrants attention, but it does not cross that threshold. These outcomes do not prove the moment conditions.
The wider window has a somewhat larger partial \(R^2\), but that alone does not establish strong instruments. Its Hansen p-value is less reassuring than the narrower window’s, and its \(\hat{\rho}\) is \(0.14\) lower. A careful report would describe this evidence and investigate whether the wider window is substantively justified; it would not declare \([2,5]\) best simply because its partial \(R^2\) is larger. Nor would it treat the Hansen p-value of \(0.24\) as proof that \([2,3]\) is valid.
Common Pitfall and Reporting
A common mistake is to select the window with the largest Hansen p-value, or to keep adding lags until that test no longer rejects. The reasoning is wrong because the Hansen test is not a score for instrument quality: failure to reject does not certify validity, and a large instrument set can make the test weak. The correction is to justify candidate lags from timing assumptions, keep the instrument set manageable, and report all relevant diagnostics alongside the coefficient sensitivity.
A concise report should name the lag window for each instrumented variable, whether instruments were collapsed, the estimator and sample, instrument count, coefficient estimates with standard errors, and the relevant serial-correlation and overidentification results. Explain why each window is admissible and describe how the substantive conclusion changes, or does not change, across them. If estimates vary materially, present that dependence as a limitation rather than hiding it by selecting one preferred result.
Check Your Understanding
Use the assumptions and comparisons in this tutorial to answer the following.
- Why is lag 2 commonly the earliest instrument for the differenced lagged dependent variable when level errors are serially uncorrelated?
- In the six-period example, how many uncollapsed instruments are contributed by the \([2,3]\) window, and how many lag columns does the collapsed version have?
- What does a rejection of the Arellano–Bond AR(2) null suggest about the usual lag-2 moment conditions?
- Why does a larger descriptive first-stage partial \(R^2\) not, by itself, establish that a wider window is preferable?
- What can and cannot be concluded when a Hansen test has a p-value above 0.05?