Wald Tests for 2SLS Coefficients

Why Wald Tests Are Useful in 2SLS

A 2SLS analysis often asks whether one estimated coefficient differs from a specified value, or whether a group of coefficients is jointly consistent with a set of specified values. A Wald test answers these questions using the 2SLS estimate and an estimated covariance matrix. The central calculation is straightforward, but the conclusion depends on using a covariance matrix appropriate to the data and on the large-sample approximation being adequate.

Write the structural coefficient vector as \(\beta\), its 2SLS estimate as \(\hat{\beta}\), and the estimated covariance matrix of that estimate as \(\widehat{\operatorname{Var}}(\hat{\beta})\). This tutorial treats inference conditional on a correctly specified instrumental-variables model, valid instruments, identification of the coefficients being tested, and regularity conditions supporting the relevant asymptotic approximation. A Wald test does not establish those conditions.

A Test of One Coefficient

For coefficient \(\beta_j\), consider \(H_0:\beta_j=\beta_{j0}\) against a two-sided alternative \(H_a:\beta_j\ne\beta_{j0}\). Let \(SE(\hat{\beta}_j)\) be the square root of the \(j\)th diagonal element of the covariance matrix used for inference. The single-coefficient Wald statistic is

Under the null and the stated large-sample conditions, \(z\) is approximately standard normal and \(W\) is approximately chi-square with one degree of freedom. The two-sided p-value can be calculated as \(2P(Z\geq |z|)\) for \(Z\sim N(0,1)\), or equivalently as the upper-tail probability for a chi-square random variable with one degree of freedom. At significance level \(\alpha\), reject \(H_0\) when the p-value is less than \(\alpha\).

The coefficient’s estimated variance must come from the covariance estimator chosen for the error structure: for example, homoskedastic, heteroskedasticity-robust, cluster-robust, or HAC. The point estimate may be the same across these choices while its standard error, test statistic, and p-value differ. Use the 2SLS covariance estimate, not the ordinary least-squares second-stage standard error from a regression of the outcome on fitted endogenous regressors.

Joint Wald Tests

A joint test asks whether several restrictions hold simultaneously. Suppose the restrictions select \(q\) coefficients from a \(k\)-coefficient vector and set them to specified values. Let \(R\) be the \(q\)-by-\(k\) matrix that selects those coefficients, and let \(r\) be the \(q\)-vector of null values. The hypotheses are \(H_0:R\beta=r\) and \(H_a:R\beta\ne r\), where the alternative means that at least one restriction fails. The rows of \(R\) must be linearly independent so that the test contains \(q\) distinct restrictions.

The estimated restrictions are \(R\hat{\beta}\). Their estimated covariance matrix is \(R\widehat{\operatorname{Var}}(\hat{\beta})R'\). The joint Wald statistic is

Under \(H_0\), if the covariance estimate is consistent, the restrictions are identified, and the relevant asymptotic approximation holds, \(W\) has an approximate chi-square distribution with \(q\) degrees of freedom. Reject at level \(\alpha\) if \(W\) exceeds the \(1-\alpha\) quantile of that distribution, or equivalently if its upper-tail p-value is less than \(\alpha\). The full covariance matrix matters: when tested coefficient estimates are correlated, separate variances alone do not determine the joint statistic.

Why the Joint Reference Distribution Applies

The justification follows from the asymptotic distribution of the 2SLS estimate. Under the null, the restriction error \(R\hat{\beta}-r\) is centered around zero, and its asymptotic covariance is estimated by \(R\widehat{\operatorname{Var}}(\hat{\beta})R'\). Standardizing the restriction error therefore gives an approximately multivariate standard normal vector. Its squared length is the Wald statistic.

For justification, if \(A\) is the covariance matrix of a mean-zero multivariate normal vector \(U\), then \(A^{-1/2}U\) has covariance equal to the identity matrix. Its \(q\) components are standard normal, so the sum of their squares has a chi-square distribution with \(q\) degrees of freedom. Substituting the estimated covariance for \(A\) yields the large-sample Wald approximation. This argument explains why covariance estimation and adequate approximation matter; it does not make the test exact in a finite sample.

Choosing the Covariance Estimate and Checking the Test

The Wald formula does not determine which covariance estimate is appropriate. That choice follows the sampling and error structure, as in the earlier material on 2SLS inference. Homoskedastic standard errors require a homoskedastic error model; heteroskedasticity-robust standard errors allow nonconstant error variance; cluster-robust standard errors allow dependence within the specified clusters; and HAC standard errors address suitable forms of serial dependence. In every case, use the covariance estimator for the 2SLS estimator and apply the same choice throughout the test.

Before interpreting a result, check that the tested restrictions are genuinely distinct, the restricted covariance matrix is invertible, and the chosen estimator matches the data structure. If the matrix is singular or nearly singular, the restrictions may be redundant or the estimated covariance may not support the calculation reliably. Check that the software is testing the intended null and reports the intended covariance type, degrees of freedom, and reference distribution.

• Weak identification: Conventional Wald tests can have poor size and misleading p-values when instruments provide little information about the endogenous regressors. A robust covariance estimator does not, by itself, repair weak-identification problems.
• Few independent clusters: A cluster-robust covariance estimate and a large-sample chi-square reference may be unreliable when the number of independent clusters is small. Finite-sample corrections and an appropriate reference distribution may be needed, but no correction guarantees accurate inference in every design.
• Wrong dependence assumptions: Treating correlated observations as independent can understate uncertainty. Conversely, choosing a covariance estimator that does not match the design does not make a Wald test valid.
• Multiple separate tests: Testing each coefficient separately is not a substitute for a joint test. The separate tests do not account for the covariance among estimates and answer different questions.

Large-Sample Tests and Alternatives

For one coefficient, a two-sided Wald test is equivalent to comparing the absolute \(z\) statistic with a standard-normal critical value. For \(q\) restrictions, the large-sample reference is \(\chi^2_q\). In finite samples, some software reports an \(F\)-type statistic or uses a finite-sample reference approximation. Such a convention should be identified explicitly: it is not a universal exact \(F\) result for 2SLS, and a variance correction alone does not make the Wald distribution exact.

Wald tests are convenient because they use an unrestricted estimate and its covariance matrix. Score-type tests instead evaluate departures from the null using estimation under the restrictions. Likelihood-ratio tests compare likelihoods and require an appropriate likelihood framework; they are not a generic replacement for Wald tests in every IV analysis. The choice among available procedures depends on the model and assumptions, while weak identification requires particular caution with conventional Wald inference.

A common mistake is to say, “The first coefficient is individually insignificant, so the coefficients are jointly insignificant.” This does not follow: a joint test uses the entire vector of estimates and their covariance, and can reach a different conclusion. Another mistake is to use the ordinary second-stage OLS standard error because the second-stage regression output displays one. That standard error generally fails to account for estimation of the fitted endogenous regressor; the appropriate 2SLS covariance estimate is required.