# Unit roots and GMM estimation

I want to estimate panel models of the following structure:

$y_{it} = \rho y_{i,t-1} + \beta_1 x_1 + \dots + \beta_k x_k + c_i + \gamma_t + \epsilon_{it}$,

where $c_i$ are time constant country specific effects, $\gamma_t$ are year dummies and $\epsilon_{it}$ is the usual, well behaved error term.

If I understand the concept correctly, a unit root would mean $\rho = 1$. I wonder if it is actually necessary to run separate unit root test if my estimation results will tell whether or not $\rho$ is close to one anyway. The only difficulty arising will be that standard $t$ and $p$-Tests report the results of testing $\text{H}_0$: $\rho = 0$, not $\text{H}_0$: $\rho = 1$.

Where is the error in my thinking?

$$y_{it}=\rho y_{i,t-1} + c_i +\gamma_t+\epsilon_{it}$$
In this case you're moment conditions might be $E[\Delta\epsilon_{it} y_{it-1}]=0$. This instrument relies on some correlation between $y_{it}$ and $\Delta y_{it}$ to avoid the weak instruments problem. However, $\Delta y_{it}$ can be written as: $$\Delta y_{it} = (\rho - 1) y_{i,t-1}+c_i+\epsilon_t$$
If your series is near stationary, then $\rho$ will be near 1 and your instruments will be very weak. Moreover, if the correlation between $y_{i,t-1}$ and $c_i$ is positive, then your estimate of $\rho$ will be biased towards one. This may suggest to you that if you have $\rho$ far away from one you're probably fine. However you may not know the correlation of $y_{i,t}$ and $c_i$, and this problem may apply to other persistent regressors - so it may be difficult to figure out exactly how the bias would affect your estimates. Blundell and Bond addressed the problem for near-stationary processes in two articles, Blundell and Bond (1998) and Blundell and Bond (1999). Their simple way to solve the weak instrument problem is to use first differences as instruments for levels instead, $E[\Delta y_{i,t-1} \epsilon_{it}]=0$. There are some additional restrictions on the initial conditions of the process you need to impose, which you can find in either of those references, so this does not come free. The logic was extended to the unit root case for the same instruments by Binder, Hsiao and Pesaran (2000), but the restrictions required look strange to me.
If $\rho$ is less than 1, you can use either or both of first differences and levels as instruments. The two together are commonly called the system GMM estimator and it should be available to you. However, all these moment conditions are assumptions of your model, so if you are not certain which ($\rho<1$ or $\rho = 1$) is true you should indeed use a unit root test to determine which moment conditions are appropriate. The $t$-test should be inappropriate, because your model likely imposes the restriction that $\rho<1$, and may impose $\rho\leq 1$, which would require a one-sided test. You might want to also check out this paper or this one.