Consider the following model: $$ y_{it}=\nu_{it}+\epsilon_{it}$$ $$\nu_{it}=\rho \nu_{it-1}+\zeta_{it}$$ Where $y_{it}$ is the income for $i$ at time $t$. $\epsilon_{it}$ is the idiosyncratic income shock. $\nu_{it}$ denotes the permanent component that follows an AR(1) process, like unobserved productivity. The only observed data is $\{y_{it}\}$. We assume that $\epsilon_{it}\sim N(0,\sigma_\epsilon^2)$ and $\zeta_{it}\sim N(0,\sigma_{\zeta}^2)$, and $0<|\rho|<1$ so that the process is stationary.
Can we identify $\rho, \sigma_{\epsilon}^2$ and $\sigma_{\zeta}^2$ in this case? And how do we estimate the parameters exactly?
