Cross-Level-Interaction without varying slope in Multilevel Analysis?

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Is it useful to include cross-level-interaction in multilevel (hierarchical) models without varying slopes (only varying intercept)? A short example in R:

m1 <- lme(y~x_1+x_2+z_1+z_2+x_1:z_1, random = ~1|county, data = data)

Where:

• y = Response variable
• x = covariate from individual-Level
• z = covariate from group-Level
  

Technically these models can be fitted in R without warnings. But is it useful from a theoretical point of view? I once read that cross-level-interactions are only possible with individual-Level variables that possess a varying slope. Is that correct?

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关键词：interaction Multilevel interact Analysis without without example include useful

While the model can be estimated, it doesn't make sense to have a cross-level interaction without a random slope, as the slope variability is what cross-level interaction is trying to predict.