Fixed Effects and Dynamic Panel Data I have a theoretical model that suggests I should estimate the following regression using longitudinal data:
$s_{it} = \eta_{i} + \beta_0 x_{it} + \beta_1 x_{it}^2 + \epsilon_{it}$
where
$x_{it} \equiv \displaystyle\sum_{\tau=0}^{t-1} s_{i\tau}$.
I want allow for fixed effects, but since the panel data is "large $N$, small $T$", I think the above estimation will suffer from Nickell bias. 
My questions:
1) Can I simply instrument for $x_{it}$ with $x_{it-1}$ or $x_{it-2}$? Will this solve the problem?
2) Are there other, better ways to solve the problem?
Any other thoughts would be helpful. Thanks in advance!
 A: You are right, fixed effect and first differencing are inconsistent with substantial downwards bias in small $T$.
The standard approach for a dynamic model and an unobserved fixed effect is to remove the fixed effect by first differencing and then finding instruments for the transformed regressors. All this assumes no serial correlation of the errors. If this is not the case, the parameters in your model are not identified and cannot be consistently estimated.
For your model, we get:
$s_{it} - s_{it-1} = \beta_0 s_{it-1} + \beta_1 ( s_{it-1}^2 + \text{  cross terms } ) + \epsilon_{it}-\epsilon_{it-1}$
As it is, both regressors must correlate with the error term. The only valid instruments will be $s_{it-2}$ or further back in time. Of course, instruments have to be good predictors of the regressors as well, otherwise you can have large biases ("weak instruments").
In theory you could use many of the valid lags as instruments (or in GMM terminology, moment conditions) as you want and there are ways of cleverly doing that using GMM estimation that do not make your $T$ smaller than it already is (1 observation is lost by first differencing alone)$.
References for these approaches would be the Arellano-Bond estimator and the Blundell-Bond estimator.
