I suppose the language of lags unnecessarily muddied the water. Let’s retract that part.
Say a and b are anxiety and depression ratings. They’re collected once daily for 100 days across a good number of participants. I’m interested in person-specific correlations between the two.
Eventually, the model would have predictors for the two variables (e.g., time, day of the week, workday…). But that’s down the road.
I imagine the correlations between the two as having a population mean and person-specific deviations. So I suppose I’m looking for something like \rho_{[a, b]_\text{id}}.