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I am referring to the question here: Determining the Number of Restrictions in an F-test.

But I want to derive a more general rule than counting the equality signs. If I denote the null hypothesis in matrices and vectors as:

$Rβ = q$

Is J now always equal to the number of rows of the matrix $R$?

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    $\begingroup$ No. What matters is the rank of $R,$ as you can see (for instance) by considering an $R$ with two identical rows. $\endgroup$
    – whuber ♦
    Commented Oct 15, 2023 at 15:58
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    $\begingroup$ That is technically correct of course, but it may be worth noting that such cases are, sensibly, often ruled out when stating $R$ as that would amount to testing redundant or incompatible restrictions and lead to test statistics with reduced rank matrices that we however aim to invert in the construction of the test statistic. $\endgroup$
    – Christoph Hanck
    Commented Oct 16, 2023 at 9:54
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    $\begingroup$ @ChristophHanck This is not a mere technical quibble, because in complex settings there often is a genuine question of the rank of a collection of constraints. It would be impractical merely to "rule out" such applications by fiat. $\endgroup$
    – whuber ♦
    Commented Oct 16, 2023 at 13:01

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