It’s because I want the mean of each observation in group i to be u[i]. The model is
Y_{i,t} = \mu_i + \delta (Y_{i,t-1} - \mu_i) + \epsilon_{i,t}\\
\epsilon_{i,t} \sim \text{Normal}(0,1)
Taking expectations of each side:
\mathbb{E}[Y_{i,t}] = \mu_i + \delta \mathbb{E}[(Y_{i,t-1} - \mu_i)] \\
\mathbb{E}[Y_{i,t}] = \mu_i + \delta (\mathbb{E}[Y_{i,t-1}] - \mu_i) \\
\mathbb{E}[Y_{i,t}] = \mu_i