A Worked Clustered Panel 2SLS Analysis

Why the Panel Structure Matters

In panel data, repeated observations on the same unit may share unobserved characteristics and errors. Fixed effects remove time-constant unit characteristics; cluster-robust uncertainty allows the remaining errors to be dependent within a unit. When a regressor is endogenous, the coefficient is estimated using instruments after removing the fixed effects. These are distinct parts of the analysis: fixed effects address time-invariant confounding, an instrument addresses endogeneity under its validity assumptions, and clustering changes the estimated uncertainty.

This worked analysis uses a deliberately small, hypothetical, two-period panel with one endogenous regressor and one excluded instrument. Its small size makes the arithmetic transparent, not suitable for drawing a real-world conclusion. The model includes unit fixed effects and no time-varying exogenous controls or time fixed effects.

Model and Conditions

For this scalar, just-identified model, the estimate is the ratio of the within-transformed instrument-outcome cross-product to the instrument-regressor cross-product. The fixed effect disappears because \(\tilde\alpha_i=0\). This calculation requires a nonzero within first-stage cross-product, \(\sum_i\sum_t\tilde z_{it}\tilde x_{it}\ne 0\), as well as a valid instrument.

Instrument validity must concern the transformed variables, not merely a contemporaneous level moment. A sufficient moment condition for this model is \(E[\sum_t\tilde z_{it}\tilde u_{it}]=0\), along with relevance and suitable regularity conditions. Strict exogeneity of the instrument across periods, \(E[z_{is}u_{it}]=0\) for every pair of periods \(s,t\), is one stronger condition that can imply the needed within moment. Contemporaneous orthogonality alone need not do so: within transformation combines observations across time. These assumptions are substantive and cannot be verified from the calculation below.

For cluster-robust inference, take units as the clusters. The usual large-cluster justification requires independent clusters, allows dependence of errors within a cluster, and requires appropriate finite-moment and identification conditions. The number of clusters should be sufficiently large for the chosen reference approximation to be credible. In the numerical example, we use the uncorrected cluster sandwich variance and a \(t\) reference distribution with \(G-1\) degrees of freedom as an explicitly stated small-sample approximation.

Worked Example: Transforming a Two-Period Panel

Cluster-Robust Uncertainty

The cluster sandwich uses a score contribution from each unit rather than treating all unit-period observations as independent. In this just-identified scalar case, define the within residual \(\hat{\tilde u}_{it}=\tilde y_{it}-\hat\beta\tilde x_{it}\). Unit \(i\)'s instrument score is \(q_i=\sum_t\tilde z_{it}\hat{\tilde u}_{it}\). The uncorrected cluster-robust variance is the sum of squared cluster scores divided by the squared instrument-regressor cross-product.

This formula retains within-unit dependence by forming one score for each entire unit. In a model with additional included exogenous regressors, the corresponding matrix sandwich uses the full instrument and regressor matrices; one must not apply this scalar formula unchanged.

Testing a Coefficient Restriction

A linear restriction on a scalar coefficient can be tested with a Wald statistic. For \(H_0:\beta=\beta_0\), divide the estimate’s distance from the null value by its standard error. Under the stated cluster-based approximation, compare the statistic with a \(t\) distribution using \(G-1\) degrees of freedom. The two-sided p-value is the probability, under \(H_0\) and this reference approximation, of a statistic at least as extreme in absolute value as the observed one.

Justification, Diagnostics, and Alternatives

The within calculation follows directly from demeaning the model. Because \(\alpha_i\) is constant over time, subtracting the unit mean gives \(\tilde y_{it}=\beta\tilde x_{it}+\tilde u_{it}\). In the just-identified case, the instrument moment sets the sample cross-product \(\sum_{i,t}\tilde z_{it}(\tilde y_{it}-\hat\beta\tilde x_{it})\) to zero. Solving this equation for \(\hat\beta\) gives the ratio used above. For two periods, the within values are \((-\Delta v_i/2,+\Delta v_i/2)\), so summing their products gives \(\Delta z_i\Delta v_i/2\). This explains both the ratio calculation and why differencing gives the same point estimate in this special two-period setting.

Several checks are essential before interpreting a result. First, verify that \(z\) varies within units and predicts the endogenous regressor after removing fixed effects. An instrument that varies only between units is eliminated by this transformation. Second, examine whether the instrument’s exclusion and exogeneity arguments are credible for the transformed model. In particular, contemporaneous level orthogonality is not enough if cross-period relationships between instruments and errors violate the transformed moment. Third, ensure the cluster definition matches the dependence structure: clustering by unit does not protect against correlation across units, such as a common shock, unless the model or inference strategy addresses it. Finally, assess whether the number of independent clusters supports the approximation used. Eight clusters are few, so the example’s \(t_7\) inference should be treated cautiously, even though the arithmetic is fully specified.

Fixed-effects OLS would use within variation but would generally remain inconsistent if \(x_{it}\) is correlated with \(u_{it}\). Pooled 2SLS does not remove \(\alpha_i\) and can be confounded if the unit effect is related to the regressors or instrument. Fixed-effects 2SLS instead instruments the within-transformed endogenous regressor, relying on valid transformed moments. With exactly two periods, first-difference IV and within 2SLS yield the same coefficient in this simple model, though the presentation and small-sample variance conventions must still be specified.

A common mistake is to argue, “The instrument is uncorrelated with the error in each period, so it is valid after demeaning.” That conclusion does not follow: demeaning combines values across periods, and the transformed error is related to every period’s error. State and defend the moment condition after transformation, or give a stronger cross-period condition that implies it. Another mistake is to use a conventional second-stage OLS standard error. The fitted regressor is generated from the instruments, and that naive formula does not provide the required 2SLS uncertainty. Use a 2SLS sandwich estimator with the appropriate cluster-level scores.