Wald test

In statistics, the Wald test (named after Abraham Wald) assesses constraints on statistical parameters based on the weighted distance between the unrestricted estimate and its hypothesized value under the null hypothesis, where the weight is the precision of the estimate.[1][2] Intuitively, the larger this weighted distance, the less likely it is that the constraint is true. While the finite sample distributions of Wald tests are generally unknown,[3]: 138  it has an asymptotic χ2-distribution under the null hypothesis, a fact that can be used to determine statistical significance.[4]

Together with the Lagrange multiplier test and the likelihood-ratio test, the Wald test is one of three classical approaches to hypothesis testing. An advantage of the Wald test over the other two is that it only requires the estimation of the unrestricted model, which lowers the computational burden as compared to the likelihood-ratio test. However, a major disadvantage is that (in finite samples) it is not invariant to changes in the representation of the null hypothesis; in other words, algebraically equivalent expressions of non-linear parameter restriction can lead to different values of the test statistic.[5][6] That is because the Wald statistic is derived from a Taylor expansion,[7] and different ways of writing equivalent nonlinear expressions lead to nontrivial differences in the corresponding Taylor coefficients.[8] Another aberration, known as the Hauck–Donner effect,[9] can occur in binomial models when the estimated (unconstrained) parameter is close to the boundary of the parameter space—for instance a fitted probability being extremely close to zero or one—which results in the Wald test no longer monotonically increasing in the distance between the unconstrained and constrained parameter.[10][11]

  1. ^ Fahrmeir, Ludwig; Kneib, Thomas; Lang, Stefan; Marx, Brian (2013). Regression : Models, Methods and Applications. Berlin: Springer. p. 663. ISBN 978-3-642-34332-2.
  2. ^ Ward, Michael D.; Ahlquist, John S. (2018). Maximum Likelihood for Social Science : Strategies for Analysis. Cambridge University Press. p. 36. ISBN 978-1-316-63682-4.
  3. ^ Martin, Vance; Hurn, Stan; Harris, David (2013). Econometric Modelling with Time Series: Specification, Estimation and Testing. Cambridge University Press. ISBN 978-0-521-13981-6.
  4. ^ Davidson, Russell; MacKinnon, James G. (1993). "The Method of Maximum Likelihood : Fundamental Concepts and Notation". Estimation and Inference in Econometrics. New York: Oxford University Press. p. 89. ISBN 0-19-506011-3.
  5. ^ Gregory, Allan W.; Veall, Michael R. (1985). "Formulating Wald Tests of Nonlinear Restrictions". Econometrica. 53 (6): 1465–1468. doi:10.2307/1913221. JSTOR 1913221.
  6. ^ Phillips, P. C. B.; Park, Joon Y. (1988). "On the Formulation of Wald Tests of Nonlinear Restrictions" (PDF). Econometrica. 56 (5): 1065–1083. doi:10.2307/1911359. JSTOR 1911359.
  7. ^ Hayashi, Fumio (2000). Econometrics. Princeton: Princeton University Press. pp. 489–491. ISBN 1-4008-2383-8.,
  8. ^ Lafontaine, Francine; White, Kenneth J. (1986). "Obtaining Any Wald Statistic You Want". Economics Letters. 21 (1): 35–40. doi:10.1016/0165-1765(86)90117-5.
  9. ^ Hauck, Walter W. Jr.; Donner, Allan (1977). "Wald's Test as Applied to Hypotheses in Logit Analysis". Journal of the American Statistical Association. 72 (360a): 851–853. doi:10.1080/01621459.1977.10479969.
  10. ^ King, Maxwell L.; Goh, Kim-Leng (2002). "Improvements to the Wald Test". Handbook of Applied Econometrics and Statistical Inference. New York: Marcel Dekker. pp. 251–276. ISBN 0-8247-0652-8.
  11. ^ Yee, Thomas William (2022). "On the Hauck–Donner Effect in Wald Tests: Detection, Tipping Points, and Parameter Space Characterization". Journal of the American Statistical Association. 117 (540): 1763–1774. arXiv:2001.08431. doi:10.1080/01621459.2021.1886936.