I am reading possibly conflicting statements in different sources.
Sometimes I think I understand that the word "Wald test" refers to
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any kind of test of whether the value of a parameter (whether in a linear model or not) is significantly removed from a null hypothesis. i.e. the Wald test outputs one of $\{\text {Reject},\text{Not-Reject} \}$ given a rejection threshold $\alpha$ (usually 0.05 or 0.01), of the likelihood that a test statistic $S$ farther away from $S_0$ would be obtained if the null hypothesis is true,
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The above, but specifically to coefficent parameters in a linear regression model.
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The above, but specifically taking the population variance $\sigma^2$ as given. (i.e. a $t$ test is not a Wald test).
Which of these is it, or is it something else?
Best Answer
The last one seems pretty close.
It may help to look at the relationship between the Wald, likelihood ratio, and score test, e.g. https://stats.idre.ucla.edu/other/mult-pkg/faq/general/faqhow-are-the-likelihood-ratio-wald-and-lagrange-multiplier-score-tests-different-andor-similar/