Wald test

In

Under the Wald test, the estimated ${\displaystyle {\hat {\theta }}}$ that was found as the maximizing argument of the unconstrained likelihood 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.

Test on a single parameter

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

${\displaystyle W={\frac {{({\widehat {\theta }}-\theta _{0})}^{2}}{\operatorname {var} ({\hat {\theta }})}}}$

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)

${\displaystyle {\sqrt {W}}={\frac {{\widehat {\theta }}-\theta _{0}}{\operatorname {se} ({\hat {\theta }})}}}$

where ${\displaystyle \operatorname {se} ({\widehat {\theta }})}$ is the

MLE

estimator of ${\displaystyle {\hat {\theta }}}$ into the

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

${\displaystyle H_{0}:R\theta =r}$

${\displaystyle H_{1}:R\theta \neq r}$

The distribution of the test statistic under the null hypothesis is

${\displaystyle (R{\hat {\theta }}_{n}-r)'[R({\hat {V}}_{n}/n)R']^{-1}(R{\hat {\theta }}_{n}-r)/Q\quad \xrightarrow {\mathcal {D}} \quad F(Q,n-P)\quad {\xrightarrow[{n\rightarrow \infty }]{\mathcal {D}}}\quad \chi _{Q}^{2}/Q,}$

which in turn implies

${\displaystyle (R{\hat {\theta }}_{n}-r)'[R({\hat {V}}_{n}/n)R']^{-1}(R{\hat {\theta }}_{n}-r)\quad {\xrightarrow[{n\rightarrow \infty }]{\mathcal {D}}}\quad \chi _{Q}^{2},}$

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

Nonlinear hypothesis

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:

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

${\displaystyle H_{1}:c(\theta )\neq 0}$

The test statistic becomes:

${\displaystyle c\left({\hat {\theta }}_{n}\right)'\left[c'\left({\hat {\theta }}_{n}\right)\left({\hat {V}}_{n}/n\right)c'\left({\hat {\theta }}_{n}\right)'\right]^{-1}c\left({\hat {\theta }}_{n}\right)\quad {\xrightarrow {\mathcal {D}}}\quad \chi _{Q}^{2}}$

where ${\displaystyle c'({\hat {\theta }}_{n})}$ is the derivative of c evaluated at the sample estimator. This result is obtained using the delta method, which uses a first order approximation of the variance.

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.^[15][5] 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).^[16]

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.^[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 with R, log R or any other
  monotonic transformation of R.^[5]
• The other reason is that the Wald test uses two approximations (that we know the standard error or Fisher 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 the
  Cochran–Mantel–Haenzel test is a score test.^[21]

• Chow test
• Sequential probability ratio test
• Sup-Wald test
• Student's t-test
• Welch's t-test

References