Suppose I want to fit the following model to data:
Beta1 ~ Zeta1 X variable2
y[i, t] ~ Intercept + Beta1[i] X [t] + Beta2 X variable1
Where [i] indexes person, [t] indexes time, and variable2 is a time invariant characteristic of a person that inform the growth rate of y[i, t] via Beta1.
Once I fit the model, I want to use it on new data. However, I will observe y[i,t], [t],[i] and variable1. What I really want to do is use this observed information to infer what variable2 is for a given person [i] at a given time [t].
I can set up the model and understand how to predict y[i,t]. How would I use the same model to predict variable2?