In a mixed model, should all random effects be also fixed?

Question

In my textbook (in french), the formula of mixed model is written as:

$y = X \beta + Z u + \epsilon$

Here, the author says that $Z$ should be a sub-vector of $X$, without explaining further why.

His example is a mixed model for repeated measures, where fixed effect are intercept + X + Y + time + X:time and random effects are individual_ID + time. As he presents the example, he considers individual_ID as an intercept.

However, I could not find this information anywhere else. Indeed, in numerous courses and tutorials (this one for instance), there can be random effects (group) which are not included in fixed effects.

Is this rule commonly accepted?

If it is, what is the problem of indluding a random effect without its fixed effect? Are above-mentioned tutorials wrong?

If it is not, why would the author of my textbook write something like this?

Answer 1

The reason why commonly the columns of $Z$ are a subset of the columns of $X$ is because you typically assume that the random effects $u$ have mean zero. The fixed effects that correspond to the random effects (e.g., in your example the fixed-effects intercept and time) quantify the average $y$ for these variables.

If you would put a variable, say $w$, into $Z$ but not in $X$ you would assume that the average change in $y$ for a unit increase in $w$ is zero, which is a strong assumption unless you have a-priori reasons that support that.