Deriving the Fixed-Effects Within Estimator

Why Remove Unit-Specific Intercepts?

In a panel, the same unit is observed in multiple periods. A unit may have persistent characteristics—such as baseline productivity or long-run health—that affect its outcome but are not recorded as regressors. If those characteristics are related to the observed regressors, pooled OLS can confound their effects with the slope of interest. The fixed-effects within estimator addresses this particular problem by removing each unit’s time-constant component before estimating the slope.

The operation is simple: subtract a unit’s time average from each of its observations. The resulting equation contains changes relative to that unit’s own average, not comparisons of levels across different units. This removes the unit-specific intercept, but it does not remove every source of endogeneity or establish a causal interpretation by itself.

The unit effect \(\alpha_i\) may be associated with the entire regressor history for that unit. The within transformation does not require it to be independent of \(x_{it}\). Its key algebraic feature is that \(\alpha_i\) is constant over time for unit \(i\), so subtracting the time average removes it exactly.

Time Demeaning and the Within Equation

For each unit, define its time averages as \(\bar y_i=T^{-1}\sum_{t=1}^T y_{it}\), \(\bar x_i=T^{-1}\sum_{t=1}^T x_{it}\), and \(\bar u_i=T^{-1}\sum_{t=1}^T u_{it}\). Subtract the unit’s average model equation from each period’s equation.

The intercept disappears because \(\alpha_i-\alpha_i=0\). Write \(\tilde y_{it}=y_{it}-\bar y_i\), \(\tilde x_{it}=x_{it}-\bar x_i\), and \(\tilde u_{it}=u_{it}-\bar u_i\). The transformed model is \(\tilde y_{it}=\tilde x_{it}'\beta+\tilde u_{it}\). In particular, \(\sum_{t=1}^T\tilde x_{it}=0\) and \(\sum_{t=1}^T\tilde y_{it}=0\) for every unit.

In sample-moment form, the estimator is

The matrix in parentheses is \(K\times K\); its invertibility requires enough independent within-unit regressor variation to distinguish the \(K\) slope coefficients. With one regressor, this reduces to a ratio: the sum of demeaned regressor–outcome products divided by the sum of squared demeaned regressors.

Why This Formula Follows From Least Squares

The within estimator can also be derived by first considering least squares with a separate intercept for each unit. For any proposed \(\beta\), choose the intercepts to minimize the residual sum of squares. The part of that sum for unit \(i\) is

For fixed \(\beta\), differentiate this expression with respect to \(\alpha_i\), set the derivative to zero, and solve:

Substituting this intercept into each residual gives

Thus, after minimizing over each unit’s intercept, the remaining least-squares objective as a function of \(\beta\) is \(\sum_i\sum_t(\tilde y_{it}-\tilde x_{it}'\beta)^2\). Differentiating this objective with respect to \(\beta\) and setting the resulting \(K\)-vector equal to zero gives

When the cross-product matrix is invertible, solving these normal equations yields the stated within-estimator formula. This derivation explains why demeaning is not an arbitrary adjustment: it is the least-squares way to remove the unit intercepts while estimating a common slope.

Several Regressors: The Matrix Calculation

With multiple regressors, each component is demeaned separately, but the regressors must be considered jointly. Their within-unit cross-products form a matrix, so the slope estimates account for overlap in the regressors’ within-unit movements.

Conditions for Consistency

The transformation removes \(\alpha_i\), but consistency still requires assumptions about the remaining error and the available within variation. A standard setting lets the number of independent units \(N\) grow while the number of periods \(T\) is fixed. The slope vector is common across units, observations are generated from the stated model, and units are independent of one another (or satisfy suitable weak-dependence conditions). Dependence among a unit’s errors over time need not be ruled out for consistency.

To see the central argument, substitute the demeaned model into the estimator and subtract \(\beta\). The result is

Because \(\sum_t\tilde x_{it}=0\), the unit’s demeaned-error contribution satisfies \(\sum_t\tilde x_{it}\tilde u_{it}=\sum_t\tilde x_{it}u_{it}\): the term involving \(\bar u_i\) cancels. Strict exogeneity implies that each \(u_{it}\) has conditional mean zero given the unit’s full regressor history and \(\alpha_i\). Consequently, the average regressor–error cross-product converges to zero under the sampling and moment conditions. Meanwhile, the normalized cross-product matrix converges to the positive-definite matrix \(Q\), whose inverse exists. The estimation error therefore converges to zero.

Strict exogeneity is stronger than requiring only that the current regressor be uncorrelated with the current error. It rules out, for example, an error in one period predicting a regressor in another period. The fixed-effects transformation does not make this requirement unnecessary. Nor does it require \(\alpha_i\) to be independent of the regressors: that persistent unit component is precisely what the transformation eliminates.

Common Pitfall: Demeaning Does Not Remove Everything

• Wrong reasoning: “Fixed effects removes unobserved confounding, so the slope is automatically causal.”
• Correction: Demeaning removes additive factors that are constant within a unit over time. Time-varying omitted factors, reverse feedback, measurement error, and violations of strict exogeneity can still invalidate the estimate or its causal interpretation.
• Wrong reasoning: “A regressor that differs across units provides information for the within slope.”
• Correction: A time-invariant regressor has \(\tilde x_{it}=0\) for every period and contributes no within variation. A slope is identified only when the demeaned regressors have adequate variation and their aggregate cross-product matrix has full rank.