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We have a three-level model with time-of-day (binary: morning versus afternoon, Level 1) nested in days (Level 2) nested in persons (Level 3). We want to test a cross-level interaction with the Level-1 variable moderating the relationship between a Level-2 predictor variable and the Level-1 outcome variable. Note: this cross-level interaction is different from the "usual" cross-level interaction where the Level-2 variable is the moderator.

In MLwiN we can build an interaction term between the Level-2 predictor and the Level-1 moderator, this is straightforward. But we are wondering how we should treat fixed versus random slopes: in our model the effect from the Level-2 predictor variable on the Level-1 outcome is contingent on the score of the Level-1 moderator. Thus, would it make sense to specify a random slope of the Level-2 predictor (i.e., the slope varying at Level 1)? It is possible in technical terms, but appears strange within the multilevel logic.

It typically doesn't make sense to predict a level 1 outcome using something defined at level 2. It can be specified but what happens is that such a model effectively induces dependencies among level 1 residuals. Also not easy to interpret.
Harvey Goldstein

Ignore previous post - should have read that it is inappropriate to predict a level 2 using a level 1 variable - this seems implicitly what is happening in your model.

Hi Harvey,
thank you very much for your quick reply. How would you then specify the slopes? As fixed? Because content-wise it makes sense to propose that time of day (Level 1) influences the relationship between a day-spefic variable (e.g., sleep quality during the previous night; Level 2) and momentary affect (Level 1) - i.e., sleep quality would matter more for for affect in the morning than for affect in the afternoon.
Also Aguinis et al. (2013) briefly mention a situation of a Level-1 variable moderating the relationship between a Level-2 variable and a Level-1 outcome. But is seems that there are not many studies capturing this.
Best
Sabine

If I understand your model it seems to me that what you call the level 1 classification (am/pm) is actually a predictor variable and if you treat it as such then your analysis is straightforward, isn't it?