# 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?

You are right, fixed effect and first differencing are inconsistent with substantial downwards bias in small $T$.
$s_{it} - s_{it-1} = \beta_0 s_{it-1} + \beta_1 ( s_{it-1}^2 + \text{ cross terms } ) + \epsilon_{it}-\epsilon_{it-1}$
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)$. References for these approaches would be the Arellano-Bond estimator and the Blundell-Bond estimator. • Thank you, this is helpful. As a follow-up: If I have an alternative (equally plausible from a theoretical point of view) measure of$s$at my disposal, call it$s^{'}$, can I simply instrument for$s_{i,t-1}$with$s_{i,t-1}^{'}$? – John Jan 27, 2015 at 20:10 • If$E[s'\Delta \epsilon_{it}]=0$, then yes. This is a difficult condition to satisfy, though. Think of everything that determines$s$other than your regressors. That's what goes into$\epsilon$in a structural model. Are you sure that stuff doesn't correlate with your different measure and that your measure is not also something that affects$s$? Then it's a valid instrument. If it is also strongly correlated with$s$is a strong instrument and everything should work. Jan 27, 2015 at 20:22 • The data I'm using are from a survey. The$s^{'}$of which I speak is the "time diary" measure of time spent on leisure (ie. "How much time did you spend on leisure activities yesterday?"), whereas$s$is the "recall" measure of time spent on leisure (ie. "How much time did you spent on leisure since the last survey?"). Does this change your answer? – John Jan 28, 2015 at 19:05 • Assuming you model actual time spent on leisure (as opposed to the response to a survey), it seems both$s$and$s'$are proxies for the real variable of interest. I don't see how one of them could be a valid instrument but not the other, but if you can tell a convincing story to that effect, it would be feasible. Also keep in mind that the independent variable being measured with error ($s= \text{ actual time spent on leisure} + \text{noise}\$ is yet another source of potential endogeneity! Jan 31, 2015 at 16:51