Instatistics, theWald test (named afterAbraham Wald) assessesconstraints onstatistical parameters based on the weighted distance between theunrestricted estimate and its hypothesized value under thenull hypothesis, where the weight is theprecision of the estimate.^[1][2] Intuitively, the larger this weighted distance, the less likely it is that the constraint is true. While thefinite sample distributions of Wald tests are generally unknown,^[3]: 138 it has an asymptoticχ²-distribution under the null hypothesis, a fact that can be used to determinestatistical significance.^[4]

Together with theLagrange multiplier test and thelikelihood-ratio test, the Wald test is one of three classical approaches tohypothesis testing. An advantage of the Wald test over the other two is that it only requires the estimation of the unrestricted model, which lowers thecomputational 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 equivalentexpressions 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 aTaylor 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 inbinomial models when the estimated (unconstrained) parameter is close to theboundary of theparameter space—for instance a fitted probability being extremely close to zero or one—which results in the Wald test no longermonotonically increasing in the distance between the unconstrained and constrained parameter.^[10][11]

Under the Wald test, the estimated${\displaystyle \hat{\theta }}$ that was found as themaximizing argument of the unconstrainedlikelihood function is compared with a hypothesized value${\displaystyle {\theta }_0}$. In particular, the squared difference${\displaystyle \hat{\theta }-{\theta }_0}$ is weighted by the curvature of the log-likelihood function.

If the hypothesis involves only a single parameter restriction, then the Wald statistic takes the following form:

${\displaystyle W=\frac{{(\hat{\theta }-{\theta }_0)}^2}{\mathrm{var}(\hat{\theta })}}$

which under the null hypothesis follows an asymptotic χ²-distribution with one degree of freedom. The square root of the single-restriction Wald statistic can be understood as a (pseudo)t-ratio that is, however, not actuallyt-distributed except for the special case of linear regression withnormally distributed errors.^[12] In general, it follows an asymptoticz distribution.^[13]

${\displaystyle \sqrt{W}=\frac{\hat{\theta }-{\theta }_0}{\mathrm{se}(\hat{\theta })}}$

where${\displaystyle \mathrm{se}(\hat{\theta })}$ is thestandard error (SE) of themaximum likelihood estimate (MLE), the square root of the variance. There are several ways toconsistently estimate thevariance matrix which in finite samples leads to alternative estimates of standard errors and associated test statistics andp-values.^[3]: 129 The validity of still getting an asymptotically normal distribution after plugin-in theMLE estimator of${\displaystyle \hat{\theta }}$ into theSE relies onSlutsky's theorem.

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${\displaystyle {\hat{\theta }}_n}$ be our sample estimator ofP parameters (i.e.,${\displaystyle {\hat{\theta }}_n}$ is a${\displaystyle P\times 1}$ vector), which is supposed to follow asymptotically a normal distribution withcovariance matrix V,${\displaystyle \sqrt{n}({\hat{\theta }}_n-\theta )\stackrel{\mathcal{D}}{\to }N(0,V)}$.The test ofQ hypotheses on theP parameters is expressed with a${\displaystyle Q\times P}$ matrix R:

${\displaystyle H_0:R\theta =r}$

${\displaystyle H_1:R\theta \ne r}$

The distribution of the test statistic under the null hypothesis is

${\displaystyle (R{\hat{\theta }}_n-r{)}^{\prime }[R({\hat{V}}_n/n)R^{\prime }{]}^{-1}(R{\hat{\theta }}_n-r)/Q\stackrel{\mathcal{D}}{\to }F(Q,n-P)\underset{n\to \mathrm{\infty }}{\overset{\mathcal{D}}{\to }}{\chi }_Q^2/Q,}$

which in turn implies

${\displaystyle (R{\hat{\theta }}_n-r{)}^{\prime }[R({\hat{V}}_n/n)R^{\prime }{]}^{-1}(R{\hat{\theta }}_n-r)\underset{n\to \mathrm{\infty }}{\overset{\mathcal{D}}{\to }}{\chi }_Q^2,}$

where${\displaystyle {\hat{V}}_n}$ is an estimator of the covariance matrix.^[14]

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:

${\displaystyle H_0:c(\theta )=0}$

${\displaystyle H_1:c(\theta )\ne 0}$

The test statistic becomes:

${\displaystyle c{\left({\hat{\theta }}_n\right)}^{\prime }{\left[c^{\prime }\left({\hat{\theta }}_n\right)\left({\hat{V}}_n/n\right)c^{\prime }{\left({\hat{\theta }}_n\right)}^{\prime }\right]}^{-1}c\left({\hat{\theta }}_n\right)\stackrel{\mathcal{D}}{\to }{\chi }_Q^2}$

where${\displaystyle c^{\prime }({\hat{\theta }}_n)}$ is thederivative of c evaluated at the sample estimator. This result is obtained using thedelta method, which uses a first order approximation of the variance.

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.^[15][5] For example, asking whetherR = 1 is the same as asking whether log R = 0; but the Wald statistic forR = 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 ofR and log R, so it needs to be approximated).^[16]

There exist several alternatives to the Wald test, namely thelikelihood-ratio test and theLagrange multiplier test (also known as the score test).Robert F. Engle showed that these three tests, the Wald test, thelikelihood-ratio test and theLagrange multiplier test areasymptotically equivalent.^[17] Although they are asymptotically equivalent, in finite samples, they could disagree enough to lead to different conclusions.

There are several reasons to prefer the likelihood ratio test or the Lagrange multiplier to the Wald test:^[18][19][20]

• Non-invariance: As argued above, the Wald test is not invariant under reparametrization, while the likelihood ratio tests will give exactly the same answer whether we work withR, log R or any othermonotonic transformation of R.^[5]
• The other reason is that the Wald test uses two approximations (that we know the standard error orFisher information and the maximum likelihood estimate), whereas the likelihood ratio test depends only on the ratio of likelihood functions under the null hypothesis and alternative hypothesis.
• The Wald test requires an estimate using the maximizing argument, corresponding to the "full" model. In some cases, the model is simpler under the null hypothesis, so that one might prefer to use thescore test (also called Lagrange multiplier test), which has the advantage that it can be formulated in situations where the variability of the maximizing element is difficult to estimate or computing the estimate according to the maximum likelihood estimator is difficult; e.g. theCochran–Mantel–Haenzel test is a score test.^[21]

• Chow test
• Sequential probability ratio test
• Sup-Wald test
• Student'st-test
• Welch'st-test

• Greene, William H. (2012).Econometric Analysis (Seventh international ed.). Boston: Pearson. pp. 155–161.ISBN 978-0-273-75356-8.
• Kmenta, Jan (1986).Elements of Econometrics (Second ed.). New York: Macmillan. pp. 492–493.ISBN 0-02-365070-2.
• Thomas, R. L. (1993).Introductory Econometrics: Theory and Application (Second ed.). London: Longman. pp. 73–77.ISBN 0-582-07378-2.

• Wald test on theEarliest known uses of some of the words of mathematics

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