Confidence Intervals From 2SLS Estimates

From a 2SLS Estimate to an Interval

A 2SLS estimate is not reported in isolation: its uncertainty is part of the result. A confidence interval gives a range of parameter values compatible with the estimate and its estimated sampling variability under a specified model. The interval’s width depends on both the standard error and the critical value, so changing the covariance estimator can change the interval even when the 2SLS coefficient estimate stays the same.

Let the linear structural model be \(y=X\beta+u\), where \(y\) is an \(n\)-vector, \(X\) is an \(n\times k\) matrix of regressors, \(\beta\) is the \(k\)-vector of coefficients, and \(u\) is the error vector. Some regressors in \(X\) may be endogenous. Let \(Z\) be the \(n\times L\) instrument matrix, including the exogenous regressors, and let \(P_Z=Z(Z'Z)^{-1}Z'\) when \(Z\) has full column rank. The 2SLS estimate is

The inverse requires the relevant rank condition: the instruments must provide enough independent variation to identify the coefficients in \(\beta\). For the usual large-sample confidence intervals, also require valid instruments (orthogonal to the structural error), a correctly specified linear model for the target coefficients, and regularity conditions supporting the asymptotic approximation. These conditions do not follow merely from computing 2SLS.

For an individual coefficient, the target is \(\beta_j\), and \(\hat\theta=\hat\beta_j\). More generally, a linear contrast is \(\theta=c'\beta\), where \(c\) is a specified \(k\)-vector of constants. Its estimate is \(c'\hat\beta_{2SLS}\). If \(\widehat V\) estimates the covariance matrix of \(\hat\beta_{2SLS}\), then

For a contrast involving more than one coefficient, the off-diagonal covariances in \(\widehat V\) matter. Adding coefficient variances alone is generally incorrect.

Why the Wald Formula Works

The 2SLS estimator can be written as \(\hat\beta_{2SLS}=\beta+(X'P_ZX)^{-1}X'P_Zu\). Under identification, valid instruments, and suitable large-sample conditions, its sampling distribution is approximately normal around \(\beta\), with a covariance matrix estimated using a variance estimator appropriate to the error structure. A linear contrast is also approximately normal because it is a linear transformation of the coefficient vector.

To see how this yields the interval, suppose \(\widehat V\) consistently estimates the sampling covariance matrix \(V\). Then the standardized contrast is approximately standard normal:

A standard normal variable lies between \(-z_{1-\alpha/2}\) and \(z_{1-\alpha/2}\) with probability \(1-\alpha\). Applying those bounds to the standardized contrast and rearranging gives

This is an approximate large-sample justification, not a guarantee of exact coverage in every sample. In particular, ordinary 2SLS Wald intervals can be unreliable when identification is weak. The usual standard error and critical value do not remove that limitation.

Choose the Covariance Estimator Before Calculating the Interval

The coefficient estimate is the same across the common covariance choices, but the estimated covariance matrix reflects different assumptions about the errors. Write \(\hat u=y-X\hat\beta_{2SLS}\) and \(\hat X=P_ZX\). The following expressions omit optional finite-sample corrections; software may apply such corrections, so the convention should be reported.

Here \(\hat\sigma^2\) is a residual-based estimate of the common error variance. The subscript \(g\) denotes the observations in cluster \(g\), and \(\hat X_g\) and \(\hat u_g\) are the fitted-regressor matrix and residual vector for that cluster. In practice, cluster covariance estimators often include a finite-sample multiplier. Such a multiplier changes the estimated standard error and therefore the interval endpoints.

The formula for a contrast uses the selected covariance matrix, not a separate standard error for each coefficient. For an individual coefficient, set \(c\) to select that coefficient. For example, \(c=(1,0,\ldots,0)'\) selects the first coefficient; a contrast such as \(\beta_1-\beta_2\) uses \(c=(1,-1,0,\ldots,0)'\).

Intervals for Linear Contrasts

A contrast can express a difference, sum, or other linear combination relevant to the scientific question. Its variance must account for covariance among the estimated coefficients. For two coefficients, the general calculation is

For a difference, the covariance term is subtracted twice: \(\widehat{\operatorname{Var}}(\hat\beta_1-\hat\beta_2)=\widehat{\operatorname{Var}}(\hat\beta_1)+\widehat{\operatorname{Var}}(\hat\beta_2)-2\widehat{\operatorname{Cov}}(\hat\beta_1,\hat\beta_2)\). A covariance can be positive or negative, so omitting it may either overstate or understate uncertainty.

Clustered Intervals and Their Conditions

Use cluster-robust uncertainty when observations within a cluster may share shocks—for example, people in the same clinic or observations from the same school. The cluster-robust formula aggregates the fitted-regressor-weighted residual contributions within each cluster before forming the covariance estimate. That aggregation preserves within-cluster dependence rather than treating every observation as independent.

The key asymptotic requirement is generally a sufficiently large number of independent clusters, not merely a large total number of observations. With few clusters, the cluster-robust covariance estimate and normal critical value can give poor coverage. Small-sample corrections or a suitable finite-cluster method may help, but do not make a very small number of clusters automatically adequate.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting an interval, check whether its covariance estimator matches the sampling structure. If residual variation differs across observations, the homoskedastic formula may be unsuitable; if observations share within-group shocks, an observation-level robust estimator does not account for that clustering. Also verify that the contrast vector matches the intended question, that the reported standard error is for 2SLS rather than a naive second-stage regression, and that the chosen critical value and any finite-sample correction are stated consistently.

A confidence interval and a Wald test are two presentations of the same large-sample calculation: for a two-sided test at level \(\alpha\), the null value is outside the corresponding \(1-\alpha\) Wald interval exactly when the test rejects, subject to using the same covariance estimator and critical value. An interval is often more informative because it shows both the range of estimates compatible with the data and the precision. Neither presentation repairs invalid instruments or weak identification. When identification is weak, ordinary Wald intervals can be misleading; other inference procedures may be needed.

• Pitfall: choosing a covariance estimator for convenience. The reasoning “the robust option is always safer” ignores dependence between observations. The correction is to choose an estimator that reflects the plausible error structure and sampling design.
• Pitfall: treating an interval as a probability statement about a fixed parameter. A 95% interval does not assign a 95% probability to the parameter being inside this realized interval. Its coverage statement describes the long-run performance of the interval procedure under the assumptions.
• Pitfall: ignoring covariance in a contrast. Adding component standard errors or variances does not generally yield the standard error of a sum or difference. Use \(c'\widehat Vc\), including all covariance terms.
• Pitfall: assuming many observations solve few-cluster problems. If the number of independent clusters is small, the total observation count alone does not ensure reliable cluster-robust coverage.

Finally, statistical coverage is not a substitute for substantive interpretation. If the model and instrument assumptions support a causal interpretation, state the interval in terms of that causal target. Otherwise, describe it as uncertainty about the specified 2SLS coefficient, without implying that the interval by itself establishes causality.