First-difference estimator

In statistics and econometrics, the first-difference (FD) estimator is an estimator used to address the problem of omitted variables with panel data. It is consistent under the assumptions of the fixed effects model. In certain situations it can be more efficient than the standard fixed effects (or "within") estimator.

The estimator requires data on a dependent variable, $$y_{it}$$, and independent variables, $$x_{it}$$, for a set of individual units $$i = 1, \dots, N$$ and time periods $$t = 1, \dots, T$$. The estimator is obtained by running a pooled ordinary least squares (OLS) estimation for a regression of $$\Delta y_{it}$$ on $$\Delta x_{it}$$.

Derivation
The FD estimator avoids bias due to some unobserved, time-invariant variable $$c_{i}$$, using the repeated observations over time:


 * $$y_{it}=x_{it}\beta + c_{i}+ u_{it}, t=1,...T ,$$
 * $$y_{it-1}=x_{it-1}\beta + c_{i}+u_{it-1}, t=2,...T .$$

Differencing the equations, gives:
 * $$\Delta y_{it}=y_{it}-y_{it-1}=\Delta x_{it}\beta + \Delta u_{it}, t=2,...T ,$$

which removes the unobserved $$c_{i}$$.

The FD estimator $$\hat{\beta}_{FD}$$ is then obtained by using the differenced terms for x and u in OLS:
 * $$\hat{\beta}_{FD} = (\Delta X'\Delta X)^{-1}\Delta X' \Delta y=\beta + (\Delta X'\Delta X)^{-1}\Delta X' \Delta u$$
 * Where $$X,y,$$ and $$u$$, are notation for matrices of relevant variables. Note that the rank condition must be met for $$\Delta X'\Delta X$$ to be invertible ($$rank[\Delta X'\Delta X]=k$$) where $$k$$ is the number of regressors.
 * Let $$\Delta X_i =[\Delta X_{i2}, \Delta X_{i3}, ..., \Delta X_{iT}] $$ and define $$\Delta u_i $$ analogously. If $$E[u_{it}|x_{i1}, x_{i2}, .., x_{iT}]=0  $$, by the Central limit theorem, Law of large numbers, and Slutsky's theorem, the estimator is distributed normally with asymptotic variance of $$E[\Delta X_i' \Delta X_i]^{-1}E[\Delta X_i' \Delta u_i \Delta u_i' \Delta X_i]E[\Delta X_i'\Delta X_i]^{-1}  $$.

Under the assumption of homoskedasticity and no serial correlation, mathematically that, $$Var(\Delta u | X)=\sigma^2_{\Delta u}$$, the asymptotic variance can be estimated with
 * $$\widehat{Avar}(\hat{\beta}_{FD})=\hat{\sigma}^{2}_{\Delta u}(\Delta X'\Delta X)^{-1} ,$$

where $$\hat{\sigma}^{2}_{u}$$ is given by
 * $$\hat{\sigma}^{2}_{\Delta u} = [n(T-1)-K]^{-1}\sum_{i=1}^n\sum_{t=2}^T \widehat{\Delta u_{it}}^2 $$
 * and $$\widehat{\Delta u_{it}}=\Delta y_{it}-\hat{\beta}_{FD}\Delta x_{it}$$.

Properties
To be unbiased, the fixed effects estimator (FE) requires strict exogeneity, $$E[u_{it}|x_{i1}, x_{i2}, .., x_{iT}]=0 $$. The first difference estimator is also unbiased under this assumption. Under the weaker assumption that $$E[(u_{it}-u_{it-1})(x_{it}-x_{it-1})]=0$$, the FD estimator is consistent. Note that this assumption is less restrictive than the assumption of strict exogeneity which is required for consistency using the FE estimator when T is fixed. If T goes to infinity, then both FE and FD are consistent with the weaker assumption of contemporaneous exogeneity.

Relation to fixed effects estimator
For $$T=2$$, the FD and fixed effects estimators are numerically equivalent.

Under the assumption of homoscedasticity and no serial correlation in $$u_{it}$$, the FE estimator is more efficient than the FD estimator. This is because the FD estimator induces no serial correlation when differencing the errors. If $$u_{it}$$ follows a random walk, however, the FD estimator is more efficient as $$\Delta u_{it}$$ are serially uncorrelated.