Choosing Between Fixed Effects And First Differences

Why Compare the Two Transformations?

Fixed effects and first differences both remove a time-constant unit effect, but they retain and weight the remaining within-unit information differently. The choice can therefore affect precision even when both estimators target the same slope. The key question is not simply how many observations each method uses: it is how the transformation interacts with the covariance of the errors over time.

The comparison below concerns a static panel with a common slope. It does not establish that either estimator is always preferable. Under independent, equal-variance errors, fixed effects have a standard efficiency advantage. With sufficiently persistent errors, first differences can instead be more precise for a particular regressor pattern.

Model and Estimators

Suppose there are \(N\) units observed in each of \(T\) periods. For unit \(i\) in period \(t\), consider

Here \(y_{it}\) is the outcome, \(x_{it}\) is a vector of \(K\) regressors, \(\beta\) is a common slope vector, \(\alpha_i\) is an unrestricted time-constant unit effect, and \(u_{it}\) is the time-varying error. Assume a balanced panel, a common \(\beta\), and strict exogeneity: conditional on all regressors for unit \(i\) and its unit effect, the errors have mean zero. Assume units are independent for the variance calculations below. The within-unit covariance of \(u_{it}\), however, need not be the same for every comparison.

Write \(y_i\) for the \(T\)-vector of outcomes and \(X_i\) for the \(T\)-by-\(K\) matrix of regressors. Let \(M_T=I_T-\mathbf{1}\mathbf{1}'/T\) be the within-demeaning matrix, and let \(D\) be the first-difference matrix, so that \(Dy_i\) contains \(y_{i,t}-y_{i,t-1}\) for \(t=2,\ldots,T\). Both transformations remove \(\alpha_i\): \(M_T\mathbf{1}=0\) and \(D\mathbf{1}=0\).

These expressions require the relevant transformed regressors to have full column rank. Both estimators eliminate the unit effect, but their transformed errors are \(M_Tu_i\) and \(Du_i\), respectively. That difference is central to precision.

What Determines Relative Precision?

For a transparent comparison, first consider a single regressor and condition on the observed regressor values. Let \(q_{FE}=\sum_i x_i'M_Tx_i\) and \(q_{FD}=\sum_i (Dx_i)'Dx_i\), both positive. If the within-unit error covariance matrix is \(\Omega\), and it is common across independent units, then

Thus, the comparison depends on both the regressor paths and the error covariance. With several regressors, the corresponding conditional variance is a matrix; precision then concerns the variance of particular coefficient contrasts or an explicitly chosen comparison criterion.

Independent Errors: Why Fixed Effects Have the Advantage

Suppose \(u_i\) has covariance \(\sigma^2 I_T\): errors have a common variance and are uncorrelated across periods, conditional on the regressors. On the within-demeaned subspace, the covariance of \(M_Tu_i\) is \(\sigma^2 M_T\). The fixed-effects regression is therefore ordinary least squares with spherical errors on the space that remains after removing the unit effects.

The Gauss–Markov result implies that, under these conditions, the within estimator has minimum variance among linear unbiased estimators of the common slope in the model with unit effects. First differencing still gives a linear unbiased estimator under strict exogeneity, but its transformed errors are not spherical when \(T>2\). In fact, their covariance is

which has \(2\sigma^2\) on the diagonal and \(-\sigma^2\) on the immediately adjacent off-diagonals. Ordinary least squares on these changes does not use that covariance structure to weight the observations. It is generally less efficient than fixed effects in this setting, though the two estimators coincide when \(T=2\).

A short calculation illustrates the difference. With one regressor and independent errors, fixed effects has conditional variance \(\sigma^2/q_{FE}\). In a panel where every unit has the same regressor path \((0,0,1)'\), the FE slope weights that unit’s three errors by \((-1/2,-1/2,1)\). The FD slope uses only the final change, weighting its errors by \((0,-1,1)\). The squared weights sum to \(1.5\) for FE and \(2\) for FD. With independent errors, that gives FE the lower variance in this example.

Persistent Errors: When First Differences Can Win

Positive serial correlation changes the comparison. Differencing replaces \(u_{it}\) with \(u_{it}-u_{i,t-1}\). If adjacent errors move together, much of their shared variation cancels. Demeaning also removes a unit’s common component, but it does not generally produce the same cancellation or error weights as differencing.

There is no universal rule that positive serial correlation favors FD for every regressor path. The relevant question is how the covariance affects the actual estimator weights. The next example keeps the regressor pattern and sample size from the independent-error example and changes only the error covariance.

A Practical Comparison

The calculations lead to a useful, qualified summary. Under independent, equal-variance errors and the stated linear model, FE is at least as precise as FD among linear unbiased estimators. With persistent errors, differencing can reduce the contribution of common adjacent error movements and may deliver lower variance. Other covariance patterns and regressor paths can change the result again.

Precision is not the only consideration. Both transformations require within-unit regressor variation; neither can identify a slope on a time-invariant regressor after removing \(\alpha_i\). If the conditional mean-zero assumption fails, the fact that one estimator has a smaller variance does not make it unbiased. Inference must also account for the panel’s error dependence: an estimator’s point estimate and its standard error are separate issues.

Common Mistake

• Wrong reasoning: “First differences use fewer transformed observations, so they must be less precise,” or, conversely, “differencing removes persistent errors, so it must always be more precise.”
• Correction: Compare the conditional variances implied by the regressor paths and the within-unit error covariance. Under independent errors, FE has the efficiency advantage described above. Under sufficiently persistent errors, FD can have lower variance in some designs. Neither slogan is a general ranking.