In statistics, the Wald test is one of three classical approaches to hypothesis testing, together with the Lagrange multiplier and the likelihood-ratio test. It is based on the asymptotic normality of the estimator, specifically in that it tests whether the difference between the unrestricted parameter estimate and the hypothesized value (scaled by the unrestricted precision matrix) is statistically significant; under the null hypothesis, the test statistic has an asymptotic χ2-distribution with degrees of freedom equal to the number of restrictions. If the hypothesis involves only a single restriction, then reduces to a squared (pseudo) t-ratio that is, however, not actually t-distributed. The finite sample distributions of Wald tests are generally unknown.
An advantage of the Wald test 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. That is because the Wald statistic is derived from a Taylor expansion, and different ways of writing equivalent nonlinear expressions lead to nontrivial differences in the corresponding Taylor coefficients. Another aberration, known as the Hauck–Donner effect, can occur in binomial models when 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 and .
Under the Wald statistical test, the maximum likelihood estimate of the parameter(s) of interest is compared with the proposed value , with the assumption that the difference between the two will be approximately normally distributed. Typically the square of the difference is compared to a chi-squared distribution.
Test on a single parameter
In the univariate case, the Wald statistic is
which is compared against a chi-squared distribution.
Alternatively, the difference can be compared to a normal distribution. In this case, the test statistic is
Test(s) on multiple parameters
The Wald test can be used to test a single hypothesis on multiple parameters, as well as to test jointly multiple hypotheses on single/multiple parameters. Let be our sample estimator of P parameters (i.e., is a P 1 vector), which is supposed to follow asymptotically a normal distribution with covariance matrix V, . The test of Q hypotheses on the P parameters is expressed with a Q P matrix R:
The test statistic is:
where is an estimator of the covariance matrix.
In the standard form, the Wald test is used to test linear hypotheses, that can be represented by a single matrix R. If one wishes to test a non-linear hypothesis of the form:
The test statistic becomes:
Non-invariance to re-parameterisations
The fact that one uses an approximation of the variance has the drawback that the Wald statistic is not-invariant to a non-linear transformation/reparametrisation of the hypothesis: it can give different answers to the same question, depending on how the question is phrased. For example, asking whether R = 1 is the same as asking whether log R = 0; but the Wald statistic for R = 1 is not the same as the Wald statistic for log R = 0 (because there is in general no neat relationship between the standard errors of R and log R, so it needs to be approximated).
Alternatives to the Wald test
There exist several alternatives to the Wald test, namely the likelihood-ratio test and the Lagrange multiplier test (also known as the score test). Robert F. Engle showed that these three tests, the Wald test, the likelihood-ratio test and the Lagrange multiplier test are asymptotically equivalent. Although they are asymptotically equivalent, in finite samples, they could disagree enough to lead to different conclusions.
- Non-invariance: As argued above, the Wald test is not invariant to a reparametrization, while the Likelihood ratio tests will give exactly the same answer whether we work with R, log R or any other monotonic transformation of R.
- The other reason is that the Wald test uses two approximations (that we know the standard error, and that the distribution is χ2), whereas the likelihood ratio test uses one approximation (that the distribution is χ2).
- The Wald test requires an estimate under the alternative hypothesis, corresponding to the "full" model. In some cases, the model is simpler under the zero hypothesis, so that one might prefer to use the score test (also called Lagrange Multiplier test), which has the advantage that it can be formulated in situations where the variability is difficult to estimate; e.g. the Cochran–Mantel–Haenzel test is a score test.
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