Wald Tests And Confidence Regions In GMM

Why Wald Inference Uses the GMM Sandwich Covariance

A GMM estimate is often used to answer questions about combinations of parameters, not just about individual coefficients. For example, an analyst may want to test whether two coefficients sum to a specified value, whether several coefficients jointly equal a benchmark vector, or whether a nonlinear function of the coefficients satisfies a restriction. Wald inference turns these questions into comparisons between an estimated restriction and its hypothesized value, scaled by its estimated uncertainty.

The uncertainty calculation must account for the covariance among the GMM estimates. Two estimates may be correlated, so the uncertainty of a contrast or a joint hypothesis cannot generally be found by treating each coefficient separately. The GMM sandwich covariance provides the required joint covariance matrix.

Setup and Wald Statistic

Let \(\theta_0\) be a \(k\)-dimensional parameter vector and let \(\hat{\theta}\) be a GMM estimator based on \(n\) observations. Suppose the estimator is consistent and asymptotically normal:

Here \(V\) is the asymptotic covariance matrix of the scaled estimator. In standard GMM notation, if \(G\) is the Jacobian of the population moments, \(S\) is their covariance matrix, and \(W\) is the limiting GMM weight matrix, the sandwich covariance is

The estimate \(\hat V\) estimates the covariance of \(\sqrt n(\hat{\theta}-\theta_0)\); the estimated covariance of \(\hat{\theta}\) itself is \(\hat V/n\). This distinction determines where the factor \(n\) belongs in a Wald statistic. We assume the GMM estimator satisfies the regularity and identification conditions for the stated asymptotic normality, \(\hat V\) consistently estimates \(V\), and the covariance matrices used below are nonsingular.

The rank condition means the \(q\) restrictions provide \(q\) locally independent pieces of information. The matrix \(R\hat V R'\) estimates the covariance of the scaled restriction error \(\sqrt n\{r(\hat{\theta})-r(\theta_0)\}\). A large Wald statistic indicates that the estimated restrictions are far from their hypothesized values relative to their uncertainty.

Linear Contrasts

For a linear restriction \(r(\theta)=C\theta\), where \(C\) has \(q\) independent rows, the Jacobian is simply \(C\). The test of \(H_0:C\theta_0=r_0\) becomes

For one restriction, this is the square of a standard-normal Wald statistic. For several restrictions, it is a quadratic form that uses the full covariance among the estimated contrasts.

Joint Hypotheses and Confidence Regions

A joint hypothesis tests several restrictions together. For example, \(H_0:\theta_1=a_1,\theta_2=a_2\) has two restrictions and uses a chi-square reference distribution with two degrees of freedom. The off-diagonal entries of the covariance matrix matter: they describe how the estimates move together and affect the distance from the null vector.

The corresponding \(100(1-\alpha)\%\) Wald confidence region for the full parameter vector is the set of values \(\theta\) satisfying

where \(\chi^2_{k,1-\alpha}\) is the \(1-\alpha\) quantile of a chi-square distribution with \(k\) degrees of freedom. The region is typically an ellipsoid centered at \(\hat{\theta}\). Its shape and orientation reflect the estimated variances and covariances. It is a joint region, not a collection of separate intervals with the same nominal coverage.

Smooth Nonlinear Restrictions

Restrictions need not be linear. If \(r(\theta)\) is differentiable, a first-order expansion around the true value gives

Consequently, asymptotic normality of \(\hat{\theta}\) implies that \(\sqrt n\{r(\hat{\theta})-r(\theta_0)\}\) is asymptotically normal with covariance \(R(\theta_0)VR(\theta_0)'\). Replacing the unknown quantities by consistent estimates and forming the corresponding squared, covariance-scaled distance yields the chi-square Wald statistic. This is the delta-method justification for testing smooth restrictions.

Intervals and Regions for Smooth Functions

A scalar smooth function \(\psi(\theta)\) can also be reported with a delta-method Wald interval. Let \(a(\theta)=\partial\psi(\theta)/\partial\theta\), written as a column vector. Its estimated standard error is

An approximate 95% interval is \(\psi(\hat{\theta})\pm1.96\,\widehat{SE}\{\psi(\hat{\theta})\}\). For a vector-valued smooth function with a full-row-rank Jacobian, the analogous Wald confidence region is obtained by replacing the scalar standard error with its estimated covariance matrix and using a chi-square critical value with degrees of freedom equal to the number of function components.

These confidence regions are asymptotic: their coverage approaches the stated level as the sample size grows under the model and regularity conditions. They are not guarantees that a particular realized region contains the fixed parameter, nor do they measure the probability that a null hypothesis is true.

Common Pitfalls and Practical Checks

• Using the wrong covariance scale. If \(\hat V\) estimates the covariance of the scaled estimator, the covariance of \(\hat{\theta}\) is \(\hat V/n\). Equivalently, use \(n\) in the quadratic Wald formula. Do not divide by \(n\) and then multiply by \(n\) a second time.
• Ignoring covariance. For a contrast or joint test, using only the diagonal variances discards dependence between estimates. Use the full matrix in the quadratic form.
• Replacing a joint test with separate tests. Testing each coefficient separately does not generally test the joint null at the same error rate or account for their dependence. Use the joint Wald statistic with the appropriate number of degrees of freedom.
• Using a singular restriction covariance. The inverse in the statistic requires \(R\hat V R'\) to be nonsingular. Redundant restrictions or a rank-deficient Jacobian violate the stated setup; the restriction system must be expressed in terms of independent restrictions.
• Reading a p-value as the probability the null is true. A p-value is calculated under the null model and describes how extreme the statistic is relative to its reference distribution. It is not the probability that the null holds.

Wald inference is useful because the same principle applies to linear contrasts, nonlinear functions, and joint restrictions. Its reliability depends on a suitable asymptotic approximation, a consistently estimated sandwich covariance, and restrictions that are locally independent. In finite samples, the resulting tests and regions can be only approximate, so the assumptions and the scale of the covariance estimate should be checked carefully.