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The first-difference estimator eliminates time-invariant unobserved heterogeneity by taking successive period differences, then running OLS on differenced variables. Simple and intuitive, it loses information and performs poorly with persistent outcomes, motivating alternative estimators.
You already know from your study of panel data that observing the same unit over multiple time periods gives you leverage that cross-sectional data cannot: you can control for stable, unobserved unit-level characteristics by exploiting within-unit variation over time. The first-difference (FD) estimator is one specific technique for doing this, and its logic is beautifully transparent: subtract yesterday from today.
Start with a two-period panel model: Yᵢₜ = αᵢ + βXᵢₜ + εᵢₜ. The term αᵢ is the individual fixed effect — every stable characteristic of unit i that affects Y but that you cannot observe (innate ability, firm culture, neighborhood quality). The problem you learned about in fixed-effects models is that if Xᵢₜ is correlated with αᵢ, OLS on the pooled data gives biased estimates. The FD estimator's solution: write the equation for period 2 and subtract the equation for period 1. The αᵢ terms cancel exactly — they are the same number in both periods, so the difference is zero. What remains is: ΔYᵢ = βΔXᵢ + Δεᵢ, where Δ denotes change from period 1 to period 2. Now run OLS on this differenced equation. Any time-invariant confounder is gone.
The intuition is concrete. Suppose you want to estimate the effect of job training on wages and you worry that more motivated workers both seek training and earn higher wages regardless. With two periods of data, take each worker's wage change and ask whether it is larger for workers whose training status changed. A motivated worker's motivation is the same in both periods — it differences out. What you are left with is: among workers who are otherwise comparable in their stable traits, do wage changes track changes in training participation? This is within-unit identification, the same logic as fixed effects, but implemented by subtracting rather than demeaning.
The key practical difference between FD and the within (demeaning) estimator is what happens with more than two periods. With T periods, FD uses T−1 differences per unit; the within estimator uses all T observations demeaned around the unit average. When the error term εᵢₜ is serially uncorrelated, the within estimator is more efficient — it uses more information. But when errors follow a random walk (each period's shock persists), FD differencing produces white-noise errors while the within estimator's errors become correlated, making FD more appropriate. The choice between them is not mechanical: it requires thinking about whether shocks are transitory or persistent.
The FD estimator also has a structural weakness: it discards level information entirely. If your outcome variable is very persistent — meaning its level changes little from period to period — there is very little signal in the differences, and the FD estimator becomes imprecise. A firm whose productivity barely changes year to year reveals little about the effect of policy in differenced form. This is why the FD estimator motivates alternatives like the Arellano-Bond GMM estimator, which uses lagged levels as instruments for the differenced equation, recovering more information from the panel structure. Understanding FD deeply is the prerequisite for understanding why those more sophisticated approaches are necessary.