Solved – How to prove FD and FE will give the same estimates when T = 2

The fixed effects model is defined as:

$$ y_{it} = alpha_i + X_{it}^prime beta + epsilon_{it} $$

where the $ alpha_i $ defines the unobserved indvidual-specific effects.

The Fixed Effects (FE) estimator is obtained by eliminating $alpha_i$ and time-invariant regressors via subtraction of the time average:

$$ y_{it} – bar{y}_i = (X_{it} – bar{X}_i)^prime beta + (epsilon_{it} – bar{epsilon}_i) $$

Alternatively, one can subtract the one-period lag of the model to obtain the First-Differences (FD) estimator:

$$ y_{it} – y_{it-1} = (X_{it} – X_{it-1})^prime beta + (epsilon_{it} – epsilon_{it-1}) $$

Now we have to show that in the two-period case (T=2) both estimators are equivalent. In this case we can write the equation for the FD estimator as follows (omitting subscript i for simplicity):

$$ y_2 – y_1 = (X_2 – X_1)^prime beta + epsilon_2 – epsilon_1 $$

The equation for FE estimator is defined as:

$$ y_{1} – bar{y} = (X_{1} – bar{X})^prime beta + (epsilon_{1} – bar{epsilon}) $$

where we can substitute the time averages:

$$ y_{1} – (y_{1}+y_{2})frac{1}{2} = (X_{1} – (X_{1}+X_{2})frac{1}{2})^prime beta + (epsilon_{1} – (epsilon_{1}+epsilon_{2})frac{1}{2}) $$

which yields the following relationship:

$$ y_{1} – y_{2} = (X_{1} – X_{2})^prime beta + (epsilon_{1} – epsilon_{2}) $$

Finally, multiplying both sides by -1:

$$ y_{2} – y_{1} = (X_{2} – X_{1})^prime beta + (epsilon_{2} – epsilon_{1}) $$

Note that the last equation is equivalent to the model for the FD estimator.