# Fixed effects versus first difference in panel data

In panel data model, both fixed effects model and first difference remove unobserved heterogeneity. If this is the case, when which technique is more appropriate and under what circumstances?

Consider a panel model of the form $$Y_{it} = \alpha_i + X_{it}\beta + \varepsilon_{it}$$ A typical concern when writing down a model such as this one is that one worries that $$\alpha_i$$ is correlated in some way with the $$X_{it}$$'s, so ignoring the panel structure of the data and estimating the OLS model with $$Y$$ as the outcome and the $$X$$'s as the predictor may be biased due to omitted variables bias (in particular, in this case, due to omitting the indicator for which individual $$i$$ the observation comes from). Fixed effects and first differences are two ways to deal with this issue, and to understand their relative appropriateness, it helps to understand how each is derived.
The fixed effects estimator has a fairly straightforward motivation. Given the panel structure of the data, one can think of the "ommitted variable" as the indicator for which individual the observation comes from, so the solution is to just include these indicators as coefficients to be estimated. A straightforward argument using the Frisch-Waugh-Lovell theorem shows that this idea is numerically equivalent to fitting the regression $$\tilde Y_{it} = \tilde X_{it}\beta + \tilde\varepsilon_{it}$$ where here, for a generic variable $$A_{it}$$, we define $$\tilde A_{it} = A_{it} - \frac1T\sum_{\tau=1}^T A_{i\tau}$$.
The first difference approach is motivated by the slightly different observation that under this model, if we define for a generic variable $$A_{it}$$, $$\Delta A_{it} = A_{it} - A_{i(t-1)}$$, then the following model is also true: $$\Delta Y_{it} = \Delta X_{it}\beta + \Delta\varepsilon_{it}$$
Let us now consider the bias of the OLS estimators corresonding to the above two ideas, and study its asymptotic properties. Because panels are two-dimensional data structures (individuals and time), there are actually some non-trivial choices one can make in terms of what one means by "asymptotics". Historically, due to the way the typical dataset panel datas were used on looked, a lot of emphasis was placed on asymptotics when $$n\to\infty$$ while $$T$$ remained fixed. With the fixed effects estimator, we have that as $$n\to\infty$$, we have that $$E[\hat\beta_{fe}] \to \beta + E\left[\sum_{t=1}^T \tilde X_{it}\tilde X_{it}'\right]^{-1}E\left[\sum_{t=1}^T\tilde X_{it}\tilde e_{it}\right]$$ Similarly, with the first differences estimator, we have $$E[\hat\beta_{fe}] \to \beta + E\left[\sum_{t=2}^T \Delta X_{it}\Delta X_{it}'\right]^{-1}E\left[\sum_{t=1}^T\Delta X_{it}\Delta e_{it}\right]$$ Consider now, the conditions under which the respective terms in the two expressions above are zero. In either case, we will at least maintain the basic exogeneity condition that $$E[\varepsilon_{it} | X_{it}] = 0$$ For the fixed effects estimator, we will typically need a stronger condition, sometimes known as strict exogeneity: $$E[\varepsilon_{it} | X_{i1},\ldots, X_{iT}] = 0$$ To see why, note that $$\tilde \varepsilon_{it}$$ is defined by subtracting off the mean of all the $$\varepsilon_{i\tau}$$'s, so in a small way, the error term in every period will have some effect on $$\tilde\varepsilon_{it}$$ for each $$t$$. The same observation also holds, of course, for $$\tilde X_{it}$$. On the other hand, the first difference estimator only requires $$E[\varepsilon_{it} | X_{i(t-1)}, X_{it}] = 0$$ This conditions on far fewer of the $$X$$'s and thus is generally thought to be a weaker condition than strong exogeneity. In particular, this assumption allows for the possibility that $$X_{it}$$ was chosen in anticipation of future shocks, i.e. in anticipation of $$\varepsilon_{i\tau}$$ for $$\tau > t$$. In some cases, this may be a very valid concern, and the first differences estimator is robust to it.
Given the seeming advantage of the first difference estimator, one might wonder why the fixed effects estimator is used. Two possible explanations are as follows. First, the fixed effects estimator tends to use the data more efficiently, since each observation is used for the differencing. To demonstrate a second consideration, suppose (as a simplest case) that one assumes that $$\varepsilon_{it}$$ follows an $$AR(1)$$ process. Then neither the assumption justifying fixed effects nor the one justifying finite differences holds. But note that when $$T$$ is very large, one would expect that the correlation between $$X_{it}$$ and $$\varepsilon_{i\tau}$$ is very small as long as $$t$$ and $$\tau$$ are far apart. As a result, one might suspect that the bias from the fixed effects estimator goes to 0 as $$T\to\infty$$, and this suspicion would be correct (see, e.g. chapter 10 of Wooldridge, which also contains a much richer discussion of some of the issues at play). By contrast, this does not happen with the first difference estimator, since we never compare far-away neighbors who are unlikely to be too correalted with one another. As a result, if one has enough time periods that $$T\to\infty$$ asymptotics are expected to perform reasonably, then one might still prefer the fixed effects estimator, even if strong exogeneity is not assumed.