I am attempting to produce out-of-sample forecasts based on a dynamic panel data model with fixed effects.

I have a short panel consisting of T=12 and N=150, and have settled on using Mr. Roodman's xtabond2 suite of estimators, namely the first differenced GMM estimator proposed by Arellano and Bond (1991), as well as the expanded system GMM estimator by Blundell and Bond (1998). I have succesfully estimated the parameters of an AR(1) panel data model for the first nine periods - using both estimators - and am now looking to apply them in out-of-sample forecasts for the remaining three periods.

In the case of a simple AR(1) panel data model, stated as

yit = ayi,t-1 + ui + vit ,

the optimal predictor at time t = T + K should be the expectation of yi,T+k conditional on the information set IT = (y,a,u,v), stated as

yi,T+k | T = E[ yi,T+k | IT ] = E[ ayi,T+k-1 + ui + vi,T+k | IT ] = ayi,T+k-1 + ui .

Because the expectation of future shocks of the idiosyncratic error term vit is assumed to zero, it is expunged through the conditional expectation - however, the fixed effect term ui remains. I am confused as to how I can implement the fixed effect in the forecast. Contrary to xtreg, Mr. Roodman's xtabond2 does not return a fixed effect estimate for either of the aforementioned estimators in the postestimation. Currently, I can produce forecasts with the command

Code:
forecast est y
forecast solve, pre(f_) begin(tm(2018m10))
which solves the forecasts for each panel for the remaining three periods. However, the root mean square forecast errors are greater than those obtained by the inconsistent fixed effect estimator (using xtreg, fe). Following the logic of above, I attribute the systematically poor performance of the forecasts to the exclusion of the fixed effects. Ideally, I would adjust the forecasts as follows

Code:
forecast est y
forecast adjust y = y + u
forecast solve, pre(f_) begin(tm(2018m10))
where u is the predicted fixed effects from the xtabond2 estimations. But how would I obtain these predictions? From preliminary research on this topic, I found a paper on a proposed estimator for the fixed effects by Liu, Moon and Schorfheide (2018). This, however, does not seem trivial to implement and time constrictions won't allow me to dive deeper into this issue. At this point, I am simply curious to know if there is another way to obtain the fixed effects from the xtabond2 suite of estimators? Or are forecasts simply not feasible in case of fixed effects in the dynamic panel data model, without the implementation of more sophisticated estimators for the fixed effects?

Thank you for your time.
Citations:

Arellano, M. and Bond, S. (1991). Some tests of specification for panel data: Monte carlo evidence and an application to employment equations. The Review of Economic Studies, 58(194).
Blundell, R. and Bond, S. (1998). Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics, 87(1):115–143.
Roodman, D. (2009). How to do xtabond2: An introduction to difference and system GMM in Stata. Stata Journal, 9(1):86–136.
Liu, L., Moon, H., and Schorfheide, F. (2018). Forecasting with dynamic panel data models NBER Working Paper Series.