Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 218 / 5 / b / Solution

Repeated measurements from one person can share persistent unobserved spending tendencies, violating model1's independent-error assumption and shifting that person's baseline. Model2 represents this clustered data through a common random intercept $b_i$, which induces within-person covariance $\mathrm{Cov}(Y_{ij},Y_{ik}\mid X)={\sigma }^2$ for $j\ne k$.