Overidentified IV Inference From Moments to Wald Tests

From Instrumental-Variables Moments to Inference

With more instruments than endogenous regressors, the sample moment conditions generally cannot all equal zero at the same parameter value. The estimator must combine those moments, and inference must account both for the resulting estimation uncertainty and for the possibility that the instruments’ restrictions are incompatible with the data. This tutorial follows those steps in one small numerical example.

Consider the linear model \(y_i=x_i'\beta+u_i\), with \(k\) regressors and \(q\) instruments, where \(q>k\). Write \(X\) for the \(n\times k\) regressor matrix and \(Z\) for the \(n\times q\) instrument matrix. The population moment restriction is \(E[z_i u_i]=0\). Identification requires that \(Q_{zx}=E[z_i x_i']\) have full column rank \(k\); in particular, the instruments must be sufficiently informative about the regressors. Assume independent, identically distributed observations, finite moments adequate for the central limit theorem and covariance estimation, and a positive-definite \(Q_{zz}=E[z_i z_i']\). For the usual asymptotic inference below, also assume the model and moment restrictions are correctly specified and identification is strong.

Two-stage least squares (2SLS) combines these moments using the weight matrix corresponding to the sample instrument cross-products. Let \(P_Z=Z(Z'Z)^{-1}Z'\), assuming \(Z'Z\) is nonsingular. The estimator is

This is also a generalized method of moments (GMM) estimator using the weight \((Z'Z/n)^{-1}\). In an overidentified model, \(\hat\beta_{\mathrm{2SLS}}\) ordinarily makes the weighted combination of sample moments as small as possible, not every moment zero.

Worked Through: Sample Moments and 2SLS

Robust Covariance and a Coefficient Wald Test

Under heteroskedasticity, the variance of the moment vector depends on the observation-specific error variance. A heteroskedasticity-robust covariance estimator uses the squared 2SLS residuals rather than imposing a common error variance. Define \(\hat u_i=y_i-x_i'\hat\beta_{\mathrm{2SLS}}\), and let \(\widehat\Omega=n^{-1}\sum_{i=1}^n z_i z_i'\hat u_i^2\). Under the regularity and identification conditions stated above, the asymptotic covariance of \(\sqrt n(\hat\beta_{\mathrm{2SLS}}-\beta)\) is

where “sandwiched” means multiplying the displayed middle matrix on both sides by \((Q_{xz}Q_{zz}^{-1}Q_{zx})^{-1}\). In sample-matrix form, the estimated covariance of \(\hat\beta_{\mathrm{2SLS}}\) is

This is the heteroskedasticity-robust sandwich covariance. It estimates uncertainty around 2SLS; it does not change the 2SLS point estimate.

To see the source of this result, the sample moment equation can be linearized around the true \(\beta\). With \(Q_{zx}=E[z_i x_i']\), its first-order form is

The central limit theorem gives the sum a limiting covariance \(\Omega=E[z_i z_i'u_i^2]\). Multiplying that covariance on the left and right by the displayed linear transformation gives the stated sandwich. The sample version substitutes \(n^{-1}Z'Z\), \(n^{-1}Z'X\), and \(n^{-1}\sum_i z_i z_i'\hat u_i^2\); under the stated conditions, these converge to their population counterparts. This derivation explains why a homoskedastic covariance formula is not generally valid when error variances differ across observations. \(\square\)

Checking the Overidentifying Restrictions

A coefficient Wald test asks whether a specified value of \(\beta\) is plausible. An overidentification test asks a different question: are the extra instrument restrictions jointly compatible with the model? A robust Hansen \(J\) test uses efficient GMM weighting. Under the null, with valid moments, strong identification, and consistent estimation of the moment covariance, its statistic has an asymptotic \(\chi^2_{q-k}\) distribution.

The distinction in weighting matters. 2SLS uses the weight based on \(Q_{zz}^{-1}\). Under heteroskedasticity, that is not generally the efficient GMM weight, which is \(\Omega^{-1}\). A robust \(J\) test is obtained by minimizing the sample moments using a consistent estimate of \(\Omega^{-1}\), then evaluating the minimized criterion. The degrees of freedom are \(q-k\), the number of moment restrictions left after estimating \(k\) parameters.

More explicitly, the central limit theorem makes the scaled moment vector converge to a \(q\)-dimensional normal vector with covariance \(\Omega\). Premultiplying by \(\Omega^{-1/2}\) gives a standard normal vector. Efficient estimation removes its component in the \(k\)-dimensional space that can be changed by varying \(\beta\). The residual component lies in a space of dimension \(q-k\); its squared Euclidean length has the \(\chi^2_{q-k}\) distribution. Consistent estimation of \(\Omega\) preserves this limit.

Common Pitfall and Interpretation

A common mistake is to calculate \(n\bar g_n'\widehat\Omega^{-1}\bar g_n\) using the 2SLS residual moments and then automatically compare it with \(\chi^2_{q-k}\). That statistic does not generally have the robust Hansen \(J\) reference distribution: under heteroskedasticity, 2SLS does not minimize the criterion using \(\widehat\Omega^{-1}\). The correction is to use a robust efficient-GMM estimator for the \(J\) check, or to use a test procedure whose stated assumptions match the estimator and covariance actually used. The robust 2SLS covariance remains appropriate for the 2SLS coefficient Wald test; it does not turn 2SLS into efficient GMM.

Nor does failure to reject the overidentifying restrictions prove that every instrument is valid. The test assesses their joint compatibility with the model and can have limited power. A rejection signals that at least one maintained restriction or part of the model is questionable; it does not identify which instrument is responsible. These checks complement substantive justification for instrument validity rather than replace it.