Forecast applications of dynamic panel data models with fixed effects

[Val Eggers]

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

y_it = ay_i,t-1 + u_i + v_it ,

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

y_{i,T+k | T} = E[ y_i,T+k | I_T ] = E[ ay_i,T+k-1 + u_i + v_i,T+k | I_T ] = ay_i,T+k-1 + u_i .

Because the expectation of future shocks of the idiosyncratic error term v_it is assumed to zero, it is expunged through the conditional expectation - however, the fixed effect term u_i remains, as it is included in the information set. 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

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

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.

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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.

[Sebastian Kripfganz]

You can replicate the xtabond2 estimates with the xtdpdgmm command:
XTDPDGMM: new Stata command for efficient GMM estimation of linear (dynamic) panel models with nonlinear moment conditions
This command allows to use the predict command as you know it from xtreg.

However, there needs to be a warning made: In such short-T, large-N panel data models, the fixed effects u_i are not consistently estimated and any forecasts based on them will be unreliable to some extent.

[Val Eggers]

Dear Mr. Kripfganz.

Thank you for your hasty reply! I've read several of your other posts in the past, and it is truly comforting to have your competent input on this matter.

I am still curious to know how exactly the fixed effects u_i are estimated. I understand that since they are time-invariant and assumed independent across groups, they require a number of periods to estimate them reliably. I wonder if there is a "breaking point" where a panel data set becomes too short to reliably estimate the fixed effects?

I appreciate your time and effort, and thank you again for your input.

[Sebastian Kripfganz]

Those estimated fixed effects are essentially just the group-specific averages of the overall residual.

Regarding the "breaking point", that is a tricky question. It obviously depends on your specific needs but also on the context. In macroeconomic analysis, these fixed effects might be much more important than in microeconomic studies. If you want to analyse differences in the levels of the forecasts across groups, then clearly you want those effects to be precise as possible. If you just care about the forecasts on average, then the precision of the individual effects matters much less. (The average effect is consistently estimated by the intercept.) If you just care about changes over time, then they do not matter at all. But generally, would you trust any effect that is measured based on only 12 observations? Even in a simple linear regression, you would typically want to have at least 30 observations.