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I've heard that one can't do a one-sided Wald test for multiple linear hypotheses, and I found that a Stata FAQ page wrote:

... The "test" command can perform Wald tests for simple and composite linear hypotheses on the parameters, but these Wald tests are also limited to tests of equality...

Why would it be so hard to do a one-sided Wald test?

Does this have anything to do with the definition of $\geq $ and $ \leq $ for vectors?

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There are two problems here, and they both have to do with defining orderings for vectors.

The first is deciding what it would even mean to do a one-sided test. Suppose you have just two parameters $\beta_x$ and $\beta_y$ and you want to test $\leq 0$ vs $>0$. Do you want to reject when one is positive and you're not sure about the other or do they both have to be positive? The possibilities grow exponentially with the number of tests.

The second problem is technical. The theory for the Wald multi-parameter test doesn't apply to null hypotheses with corners. Suppose your alternative is $\beta_x$ and $\beta_y$ are both positive and your null is that at least one of them is zero or negative. The boundary of the null hypothesis is made up of two line segments that meet at right angles at the origin. The distribution of the Wald test statistic will not be the usual $\chi^2$ distribution; it will be a mixture of $\chi^2$ distributions.

Both these problems can be overcome, but they mean there isn't a simple generalisation of the one-sided tests to multiple parameters.

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  • $\begingroup$ Would you happen to have references handy for these two points? $\endgroup$
    – dimitriy
    Aug 20, 2021 at 23:04
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    $\begingroup$ I don't know what a reference for the first one would look like. I learned about the second one from the book Order Restricted Statistical Inference by Robertson, but it would be easier to find jstor.org/stable/2289471 by Self and Liang -- they write about likelihood ratio tests, but the issues are basically the same. $\endgroup$ Aug 20, 2021 at 23:31
  • $\begingroup$ I had a textbook treatment of one-sided tests in mind (ideally, on the applied side of things) that covers these topics. Thank you for these references! $\endgroup$
    – dimitriy
    Aug 20, 2021 at 23:38

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