# difference GMM with trending variables

I want to estimate a panel equation of the form $$y_{it}=ρ y_{i,t−1}+β_1 x_{1,t}+⋯+β_k x_{k,t}+ c_i+γ_t+ϵ_{it}$$ where $c_i$ are country specific effects, $γ_t$ period effects and $ϵ_{it}$ error term.

the problem is that some of individual time series have certain trend, but the first difference of the variables are stationary

Can I perform difference GMM estimation proposed by Arellano-Bond(1991) on this equation given the fact that it performs first difference transformation or I need stationarity in levels in order to proceed to estimation?

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, $E[\Delta y_{i,t-1} \epsilon_{it}]=0$.