How do we identify parameters in this simple model? 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.
I think $\rho$ can be identified as follows:
We have $v_{it}=y_{it}-\epsilon_{it}$ from the first equation, and $v_{it}=\rho v_{it-1}+\zeta_{it}$ from the second. Therefore, we have
\begin{equation*}
    y_{it}-\epsilon_{it}=\rho v_{it-1}+\zeta_{it}
\end{equation*}
We also have $v_{it-1}=y_{it-1}-\epsilon_{it-1}$. Thus,
\begin{equation*}
    y_{it}-\epsilon_{it}=\rho(y_{it-1}-\epsilon_{it-1})+\zeta_{it}
\end{equation*}
Then we have
\begin{equation*}
    \rho=\frac{ y_{it}-\epsilon_{it}-\zeta_{it}}{y_{it-1}-\epsilon_{it-1}}
\end{equation*}
Thus, $\rho$ is identified.
But how can we identify $\sigma_{\epsilon}^2$ and $\sigma_{\zeta}^2$, the two variances, in this case?


I think $\rho$ can be identified as follows:
We have $v_{it}=y_{it}-\epsilon_{it}$ from the first equation [...]

No, you can't. For that, you would need to know $\epsilon_{it}$, while you don't as it is not observed. The rest of the equations also doesn't work as they are based on variables that are not known. To identify $\nu_{it}$ you need to know $\nu_{it-1}$ and $\rho$, both unknown. All those variables are unknown, so there is an infinite number of solutions that "match" them. For it to be identifiable you would need additional sources of information to identify at least some of the parameters.
