# Identification Problem in Minimum Distance Estimation

I have the following problem with a system of minimum distance equations I want to solve. The objective is to estimate the parameters of the random variables in the following DGP:

$$x_t= \phi_t(\zeta^{u}_t+\zeta^{\nu}_t)+\zeta^{\nu}_t+\epsilon^{\nu}_t+(\theta^{\nu}-1)\epsilon^{\nu}_{t-1}+\theta^{\nu}\epsilon^{\nu}_{t-2}$$ $$z_t= \zeta^{u}_t+\epsilon^{u}_t+(\theta^{u}-1)\epsilon^{u}_{t-1}+\theta^{u}\epsilon^{u}_{t-2}$$ $\zeta^{\nu},\zeta^{u},\epsilon^{u}$ and $\epsilon^{\nu}$ are mean-zero normal random variables; so I need to estimate their variances. Their variances are the same over time. $\theta^{\nu}$ and $\theta^{u}$ are parameters smaller than $1$ in absolute value that I need to estimate. $\phi_t$ is a normal random variable with a non-zero mean, so that I need to estimate its mean and variance.

I have panel-data on 2000 units running for 20 periods.

The parameters of the variables in the process $z_t$ are identified by the variance moment ($Var(z_t)$) and the first and second autocovariance ($Cov(z_t,z_{t-1})$ and $Cov(z_t,z_{t-2})$). Accordingly, I have an estimate of $\sigma^2_{\zeta_u}$.

Now I can find the mean of $\phi_t$, $\mu_{\phi}$, by calculating the $Cov(x_t,z_t)=\mu_{\phi}\sigma^2_{\zeta_u}$. So $\mu_{\phi}$ is identified.

However, the $Var(x_t)$ still contains two parameters not separately identified: $\sigma^2_{\zeta^{\nu}}$ and $\sigma^2_{\phi}$. $$Var(x_t)=2(1-\theta^{\nu}+(\theta^{\nu})^2)\sigma^2_{\epsilon}+\sigma^2_{\phi}\sigma^2_{\zeta^{u}}+\sigma^2_{\zeta^{\nu}}+2\mu_{\phi}\sigma^2_{\zeta^{\nu}}+\mu^2_{\phi}(\sigma^2_{\zeta^{u}}+\sigma^2_{\zeta^{\nu}})$$

I have thought about the following strategies to identify $\sigma^2_{\phi}$:

1. Assume that $\phi_t$ is constant for a given unit for a couple of periods so that I may estimate the cross-sectional variance of $\phi$ from the difference between the estimate $\mu_{\phi}$ and the unit specific estimate $\mu^i_{\phi}$ that I get, when I generate $Cov(x_t,z_t)$ within a certain unit. I simulated this, but it didn't work out, since I couldn't just divide by $\sigma^2_{\zeta^{u}}$ to get the two differing estimates.
2. I wanted to get $\sigma^2_{\phi}$ from the standard errors of $\mu_{\phi}$. I calculate these via the Delta-Method. However, the standard errors depend on other variances than just $\sigma^2_{\phi}$ and I don't know the precise relationship.

Please help me to find and estimate of $\sigma^2_{\phi}$.

• Please ask me any question if I left something unclear, since I am very interested in resolving this. – option_select Feb 1 '17 at 11:51