Waldtest

The Wald test is a statistical method used to assess the significance of one or more parameters in a model. It tests a null hypothesis that specifies linear restrictions on the parameter vector, typically written as H0: Rβ = r, where R is a known matrix and r is a known vector.

The test statistic is based on the estimated parameters and their covariance. Let β̂ be the estimator of β, and V̂ be its estimated covariance matrix. The Wald statistic for testing Rβ = r is W = (Rβ̂ − r)ᵀ [R V̂ Rᵀ]⁻¹ (Rβ̂ − r). Under H0 and in large samples, W follows a chi-square distribution with q degrees of freedom, where q is the number of restrictions (the rank of R). For a single parameter, this reduces to W = (β̂ − β0)² / Var(β̂). In regression contexts, V̂ is the estimated covariance matrix of the estimator (e.g., MLE or GMM).

Wald tests are common in many models, including generalized linear models, where they can test multiple coefficients jointly or test linear hypotheses about parameters. They can also be performed with robust (sandwich) covariance estimators to accommodate heteroskedasticity or model misspecification, yielding a robust Wald test.

Advantages of the Wald test include computational convenience, since it requires only estimates and their covariance and does not require refitting the model under the alternative hypothesis. Limitations include reliance on large-sample approximations and sensitivity to model misspecification or parameters on the boundary; in small samples or non-regular settings, the test may be less reliable than alternatives such as the likelihood ratio or score tests.